镧系离子间无辐射能量传递速率的教程综述 增强出版

A Tutorial Review on the Nonradiative Energy Transfer Rates between Lanthanide Ions 镧系离子间无辐射能量传递速率的教程综述 1 first-author Albano N. Carneiro Neto （1985-）, received his Ph.D. in Chemistry from Federal University of Pernambuco （Brazil） in 2018, started as a pos⁃doc from 2019 to 2021 at University of Aveiro （Portugal） and nowadays he is a Researcher at the Physics Department and CICECO⁃Aveiro Institute of Materials, University of Aveiro. He acted as guest editor of Journal of Luminescence in 2018 and he is a member of the topical advisory panel of Applied Sciences. His scientific interests are lanthanides chemistry; quantum chemistry calculations; covalency and chemical bonding; thermometry; f⁃f intensities; Judd⁃Ofelt theory; ligand⁃ and lanthanide⁃to⁃lanthanide energy transfer. He authored 47 papers in international journals and 04 book chapters. According to Google Scholar （November 2022）, he has more than 920 citations with an h⁃index of 18. E-mail： albanoneto@ua.pt http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39818830&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39818838&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39818833&type= 24.89200020 29.88732910 Albano N. Carneiro Neto （1985-）, received his Ph.D. in Chemistry from Federal University of Pernambuco （Brazil） in 2018, started as a pos⁃doc from 2019 to 2021 at University of Aveiro （Portugal） and nowadays he is a Researcher at the Physics Department and CICECO⁃Aveiro Institute of Materials, University of Aveiro. He acted as guest editor of Journal of Luminescence in 2018 and he is a member of the topical advisory panel of Applied Sciences . His scientific interests are lanthanides chemistry; quantum chemistry calculations; covalency and chemical bonding; thermometry; f⁃f intensities; Judd⁃Ofelt theory; ligand⁃ and lanthanide⁃to⁃lanthanide energy transfer. He authored 47 papers in international journals and 04 book chapters. According to Google Scholar （November 2022）, he has more than 920 citations with an h⁃index of 18. E-mail： albanoneto@ua.pt albanoneto@ua.pt 2 3 4 5 4 5 4 5 4 corresp E-mail： oscar.malta@ufpe.br E-mail： oscar.malta@ufpe.br Oscar L. Malta （1954-）, Ph.D. at Université Pierre et Marie Curie-Paris 6 （France, 1981）. He is a Full Professor in the Department of Fundamental Chemistry of the UFPE （Brazil）, researcher 1A of the National Council of Scientific and Technological Development, and member of the Brazilian Academy of Science. He was awarded the Prof. Ricardo Ferreira Award for Scientific Merit （granted by Pernambuco State Foundation for Science and Technology） and the Prof. Paulo Duarte medal （granted by The Brazilian Association of Chemistry）. Received the medal of the National Order of the Scientific Merit （Brazilian Government, 2018）. He had a Special Issue in the Journal of Luminescence dedicated to his 60s. He was the Chairman of the 18th International Conference on Luminescence, ICL-2017. He is Doctor Honoris Causa by the University of Wrocław. His research interests are in Chemistry and Physics, with emphasis on atomic and molecular spectroscopy, compounds with lanthanide ions, and nanostructured materials.oscar. malta@ufpe. br http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39818840&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39818847&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39818844&type= 24.89202881 29.88732147 Oscar L. Malta （1954-）, Ph.D. at Université Pierre et Marie Curie-Paris 6 （France, 1981）. He is a Full Professor in the Department of Fundamental Chemistry of the UFPE （Brazil）, researcher 1A of the National Council of Scientific and Technological Development, and member of the Brazilian Academy of Science. He was awarded the Prof. Ricardo Ferreira Award for Scientific Merit （granted by Pernambuco State Foundation for Science and Technology） and the Prof. Paulo Duarte medal （granted by The Brazilian Association of Chemistry）. Received the medal of the National Order of the Scientific Merit （Brazilian Government, 2018）. He had a Special Issue in the Journal of Luminescence dedicated to his 60s. He was the Chairman of the 18th International Conference on Luminescence, ICL-2017. He is Doctor Honoris Causa by the University of Wrocław. His research interests are in Chemistry and Physics, with emphasis on atomic and molecular spectroscopy, compounds with lanthanide ions, and nanostructured materials. oscar. malta@ufpe. br oscar. malta@ufpe. br 4 corresp ricardo.longo@ufpe.br ricardo.longo@ufpe.br Ricardo L. Longo （1964-）, Ph. D. from University of Florida（USA, 1993）and pos-doc at the QuantumTheory Project（U. of Florida）. Since 1994 he is a professor at the Department of Fundamental Chemistry, Federal University at Pernambuco, Recife, Brazil. He was a visiting at McGill University, Montrèal,Canada（2016）and visiting consultant at University of Aveiro, Portugal（2019）. Served as associate editor of Journal of the Brazilian Chemical Society（2006-2012）and guest editor of Journal of Luminescence （2016 and 2018）. He is an advisor/reviewer in Brazilian funding agencies（CAPES, FACEPE）. His scientific interests cover the development of theoretical and computational methods to physical chemistry, particularly to molecular spectroscopy, thermometry, chemical reactions, hydrogen bonded systems and complex networks. Supervised 10 post-docs, 22 Ph. D. , 21 MSc, and 26 undergraduate students in the Chemistry and Materials Science Programs. Currently, he is supervising 08 Ph. D. and 04 MSc students. He au‑thored 04 book chapters and about 110 papers.ricardo. longo@ufpe. br http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39818854&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39818854&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39818854&type= 24.89202881 29.88732147 Ricardo L. Longo （1964-）, Ph. D. from University of Florida（USA, 1993）and pos-doc at the QuantumTheory Project（U. of Florida）. Since 1994 he is a professor at the Department of Fundamental Chemistry, Federal University at Pernambuco, Recife, Brazil. He was a visiting at McGill University, Montrèal,Canada（2016）and visiting consultant at University of Aveiro, Portugal（2019）. Served as associate editor of Journal of the Brazilian Chemical Society（2006-2012）and guest editor of Journal of Luminescence （2016 and 2018）. He is an advisor/reviewer in Brazilian funding agencies（CAPES, FACEPE）. His scientific interests cover the development of theoretical and computational methods to physical chemistry, particularly to molecular spectroscopy, thermometry, chemical reactions, hydrogen bonded systems and complex networks. Supervised 10 post-docs, 22 Ph. D. , 21 MSc, and 26 undergraduate students in the Chemistry and Materials Science Programs. Currently, he is supervising 08 Ph. D. and 04 MSc students. He au‑thored 04 book chapters and about 110 papers. ricardo. longo@ufpe. br ricardo. longo@ufpe. br 阿威罗大学 物理系和 CICECO⁃阿威罗材料研究所， 阿威罗， 葡萄牙， 3810-193 阿威罗大学 物理系和 CICECO⁃阿威罗材料研究所， 阿威罗， 葡萄牙， 3810-193 Physics Department and CICECO-Aveiro Institute of Materials， University of Aveiro， Aveiro， Portugal， 3810-193 Physics Department and CICECO-Aveiro Institute of Materials， University of Aveiro， Aveiro， Portugal， 3810-193 南卫理公会大学 化学系， 达拉斯， 美国， 75275⁃0314 南卫理公会大学 化学系， 达拉斯， 美国， 75275⁃0314 Department of Chemistry， Southern Methodist University， Dallas， United States， 75275-0314 Department of Chemistry， Southern Methodist University， Dallas， United States， 75275-0314 帕拉伊巴联邦大学 化学与物理系， 阿雷亚， 巴西， 58397⁃000 帕拉伊巴联邦大学 化学与物理系， 阿雷亚， 巴西， 58397⁃000 Department of Chemistry and Physics， Federal University of Paraíba， Areia， Brazil， 58397-000 （permanent address） Department of Chemistry and Physics， Federal University of Paraíba， Areia， Brazil， 58397-000 （permanent address） 伯南布哥联邦大学 基础化学系， 累西腓， 巴西， 50740⁃560 伯南布哥联邦大学 基础化学系， 累西腓， 巴西， 50740⁃560 Departamento de Química Fundamental， Universidade Federal de Pernambuco， Recife， Brazil， 50740-560 Departamento de Química Fundamental， Universidade Federal de Pernambuco， Recife， Brazil， 50740-560 伯南布哥联邦大学 材料科学研究生院， 累西腓， 巴西， 50670⁃901 伯南布哥联邦大学 材料科学研究生院， 累西腓， 巴西， 50670⁃901 Programa de Pós-Graduação em Ciência de Materiais， Universidade Federal de Pernambuco， Recife， Brazil， 50670-901 Programa de Pós-Graduação em Ciência de Materiais， Universidade Federal de Pernambuco， Recife， Brazil， 50670-901 In this tutorial review， we present nonradiative energy transfer（ET） rates between lanthanides in a rearranged form. We emphasize the nature of the contributions which are different from those developed by Förster and Dexter theories because of the unique properties of the lanthanide ions. The expressions discussed here were based on Kushida's approach （electric multipolar mechanisms： dipole‑dipole（ <math id="M1"><msub><mrow><mi>W</mi></mrow><mrow><mi mathvariant="normal">d</mi><mi mathvariant="normal">‑</mi><mi mathvariant="normal">d</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39818858&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39818856&type= 4.99533367 3.21733332 ）， dipole‑quadrupole（ <math id="M2"><msub><mrow><mi>W</mi></mrow><mrow><mi mathvariant="normal">d</mi><mi mathvariant="normal">‑</mi><mi mathvariant="normal">q</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39818861&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39818859&type= 4.99533367 3.21733332 ）， and quadrupole‑quadrupole（ <math id="M3"><msub><mrow><mi>W</mi></mrow><mrow><mi mathvariant="normal">q</mi><mi mathvariant="normal">‑</mi><mi mathvariant="normal">q</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39818864&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39818863&type= 4.99533367 3.21733332 ）） within the Judd-Ofelt framework for 4f-4f transitions. Notice that these mechanisms were extended to include the exchange（ <math id="M4"><msub><mrow><mi>W</mi></mrow><mrow><mi mathvariant="normal">e</mi><mi mathvariant="normal">x</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39818867&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39818866&type= 4.23333359 3.21733332 ） and magnetic dipole‑magnetic dipole mechanisms（ <math id="M5"><msub><mrow><mi>W</mi></mrow><mrow><mi mathvariant="normal">m</mi><mi mathvariant="normal">d</mi><mi mathvariant="normal">‑</mi><mi mathvariant="normal">m</mi><mi mathvariant="normal">d</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39818870&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39818869&type= 7.45066643 3.21733332 ）， and were improved to include shielding effects as well as an analytical expression for the F -factor （density of states in Fermi’s golden rule）. Similar to the original approach of Kushida， only the Forced Electric Dipole（FED） contributions to the Judd-Ofelt intensity parameters should be considered. A detailed discussion of selection rules and matrix elements calculations for the magnetic dipole‑magnetic dipole interaction is presented. In addition， step-by-step examples of Tb（Ⅲ）-Eu（Ⅲ） and Yb（Ⅲ）-Er（Ⅲ） energy transfer rates calculations are provided， with extensive Supporting Information， including scripts for calculations. 在本教程综述中，我们重新阐述和表达了镧系离子间无辐射能量传递（ET） 速率的理论形式，并且强调了考虑镧系离子本身特异性所引发的与 Förster 和 Dexter 理论不同之处。所给出的表达式遵循了Judd‑Ofelt的4f‑4f跃迁理论框架之下的 Kushida 方法，并计入了如下电多极机制：偶极‑偶极（ <math id="M6"><msub><mrow><mi>W</mi></mrow><mrow><mi mathvariant="normal">d</mi><mi mathvariant="normal">‑</mi><mi mathvariant="normal">d</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39818873&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39818872&type= 4.99533367 3.21733332 ）、偶极‑四极（ <math id="M7"><msub><mrow><mi>W</mi></mrow><mrow><mi mathvariant="normal">d</mi><mi mathvariant="normal">‑</mi><mi mathvariant="normal">q</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39818877&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39818876&type= 4.99533367 3.21733332 ）和四极-四极（ <math id="M8"><msub><mrow><mi>W</mi></mrow><mrow><mi mathvariant="normal">q</mi><mi mathvariant="normal">‑</mi><mi mathvariant="normal">q</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39818882&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39818880&type= 4.99533367 3.21733332 ）。更为重要的是，当前的机制也扩展包括了交换（ <math id="M9"><msub><mrow><mi>W</mi></mrow><mrow><mi mathvariant="normal">e</mi><mi mathvariant="normal">x</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39818887&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39818884&type= 4.23333359 3.21733332 ）和磁偶极子⁃磁偶极子（ <math id="M10"><msub><mrow><mi>W</mi></mrow><mrow><mi mathvariant="normal">m</mi><mi mathvariant="normal">d</mi><mi mathvariant="normal">⁃</mi><mi mathvariant="normal">m</mi><mi mathvariant="normal">d</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39818892&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39818888&type= 7.45066643 3.21733332 ）作用，并经过改进进一步包含了屏蔽效应以及给出了 F 因子（费米黄金规则中的态密度）的解析表达式。与 Kushida 的原始方法类似，我们只考虑了强制电偶极子 （FED） 对 Judd⁃Ofelt 强度参数的贡献，并细节性地讨论了磁偶极⁃磁偶极相互作用的选择定则以及相关矩阵元的计算。此外，我们还以Tb（Ⅲ）‑Eu（Ⅲ） 和 Yb（Ⅲ）⁃Er（Ⅲ） 的能量传递速率计算为例，逐步展示了计算过程、主要的计算支撑信息以及所使用的计算脚本。 无辐射能量传递 镧系 理论计算 Ln - Ln 能量传递速率 选择定则 nonradiative energy transfer lanthanides theoretical calculations Ln - Ln energy transfer rates selection rules 1　Introduction 1 Energy transfer（ET） processes are ubiquitous in nature and are essential to life [ 1 1 - 2 2 1-2 ] . Despite being difficult to visualize, these ET processes are always present in our everyday life and obey the laws of thermodynamics（for macroscopic systems）. Many modern technologies are based on ET processes, for instance, those involving luminescence, where an excited sensitizer transfers energy to an activator that emits light（or electromagnetic radiation）. Most results of these ET processes are visible and quantifiable, but their detailed understanding and description, at the microscopic level, requires quantum mechanics. As a result, at least two concepts are important, namely, （quantum） states and transitions between these states. Photoluminescence involving lanthanide ions has some unique and remarkable features, which have been explored in many applications and new technologies, from biomedicine to engineering, and public health to public safety [ 3 3 - 7 7 3-7 ] . Energy transfer between lanthanide ions is particularly relevant to several processes such as long-range energy migration and energy upconversion. Resonances present in systems containing lanthanide ions of the same type, e . g ., Yb（Ⅲ）, allow the energy to be efficiently transferred between these ions, which can migrate for long distances depending on their concentration and the host matrix [ 8 8 - 11 11 8-11 ] . On the other hand, a lanthanide ion with relatively large oscillator strength, e . g ., Yb（Ⅲ） and Nd（Ⅲ）, can be employed as a sensitizer of another lanthanide ion, e . g ., Er（Ⅲ） and Tm（Ⅲ）, called an activator, which can be excited by sequential energy transfer processes, followed by emission of photons with higher energy than those absorbed photons; hence energy upconversion [ 10 10 , 12 12 - 13 13 12-13 ] . In this context, energy transfer between lanthanide ions （ Ln - Ln ET） plays a fundamental and essential role in several processes that are explored and exploited in applications and new technologies [ 14 14 - 16 16 14-16 ] . Therefore, the focus of this contribution will be the quantitative description of the Ln - Ln ET processes. A common picture used for a general ET process employs two subsystems denoted as donor（D） and acceptor（A）. Initially, the donor is prepared in an excited state, D * , by absorbing energy（light, heat, and mechanical）, which may be represented as <math id="M11"><mfenced open="" close=">" separators="|"><mrow><mfenced open="|" close="" separators="|"><mrow><mi>i</mi></mrow></mfenced></mrow></mfenced><mo>=</mo><mfenced open="" close=">" separators="|"><mrow><mfenced open="|" close="" separators="|"><mrow><msup><mrow><mi>D</mi></mrow><mrow><mi>*</mi></mrow></msup><mi>A</mi></mrow></mfenced></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819200&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819199&type= 17.10266685 5.07999992 . In other words, in the initial state, the energy absorbed by the system is localized at the donor subunit. Then, the system undergoes a transition to another state with the energy localized at the acceptor subsystem, represented as <math id="M12"><mfenced open="" close=">" separators="|"><mrow><mfenced open="|" close="" separators="|"><mrow><mi>f</mi></mrow></mfenced></mrow></mfenced><mo>=</mo><mfenced open="" close=">" separators="|"><mrow><mfenced open="|" close="" separators="|"><mrow><mi>D</mi><msup><mrow><mi>A</mi></mrow><mrow><mi>*</mi></mrow></msup></mrow></mfenced></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39818900&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39818899&type= 18.28800011 5.33400011 . This general description requires some clarification and specification of different categories of ET processes as well as additional phenomena accompanying these processes. For instance, energy transfer can induce or be caused by electron（or charge） transfer between the subsystems. These coupled processes are important and relevant for many biological systems and applications, but they will not be considered here. The ET processes of interest are uncoupled energy transfer processes in the sense that are not caused nor can induce electron transfer. This is an important distinction that is not always clear in the literature and has caused some confusion. The transition mentioned above describes an ET process, represented by <math id="M13"><mfenced open="" close=">" separators="|"><mrow><mfenced open="|" close="" separators="|"><mrow><msup><mrow><mi>D</mi></mrow><mrow><mi>*</mi></mrow></msup><mi>A</mi></mrow></mfenced></mrow></mfenced><mo>→</mo><mfenced open="" close=">" separators="|"><mrow><mfenced open="|" close="" separators="|"><mrow><mi>D</mi><msup><mrow><mi>A</mi></mrow><mrow><mi>*</mi></mrow></msup></mrow></mfenced></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39818904&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39818902&type= 23.53733444 5.07999992 , can be pictured as a deexcitation of the donor and an excitation of the acceptor, which may or may not involve photon（s）. When the deexcitation of <math id="M14"><msup><mrow><mi>D</mi></mrow><mrow><mi>*</mi></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39818908&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39818906&type= 3.55599999 2.96333337 occurs by photon emission（with energy ћω=hν ）, <math id="M15"><mfenced open="" close=">" separators="|"><mrow><mfenced open="|" close="" separators="|"><mrow><msup><mrow><mi>D</mi></mrow><mrow><mi>*</mi></mrow></msup></mrow></mfenced></mrow></mfenced><mo>→</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39818911&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39818909&type= 11.51466751 5.07999992 <math id="M16"><mfenced open="" close=">" separators="|"><mrow><mfenced open="|" close="" separators="|"><mrow><mi>D</mi></mrow></mfenced></mrow></mfenced><mo>+</mo><mi>h</mi><mi>ν</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39818915&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39818914&type= 13.37733269 4.48733330 , and the emitted photon is absorbed by <math id="M17"><mi>A</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39818927&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39818926&type= 2.28600001 2.62466669 , <math id="M18"><mfenced open="" close=">" separators="|"><mrow><mfenced open="|" close="" separators="|"><mrow><mi>A</mi></mrow></mfenced></mrow></mfenced><mo>+</mo><mi>h</mi><mi>ν</mi><mo>→</mo><mfenced open="" close=">" separators="|"><mrow><mfenced open="|" close="" separators="|"><mrow><msup><mrow><mi>A</mi></mrow><mrow><mi>*</mi></mrow></msup></mrow></mfenced></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39818921&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39818919&type= 25.39999962 5.07999992 , then the ET from <math id="M19"><msup><mrow><mi>D</mi></mrow><mrow><mi>*</mi></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39818924&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39818923&type= 3.55599999 2.96333337 to <math id="M20"><mi>A</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39818927&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39818926&type= 2.28600001 2.62466669 is known as radiative energy transfer. This is also an important category of ET processes; however, they are usually not operative when the acceptor（or activator） is a lanthanide ion. This is because the oscillator strengths associated with 4f-4f transitions are very small（~10 -6 ） [ 17 17 - 19 19 17-19 ] , except for some transitions in Nd（Ⅲ） and Yb（Ⅲ）, which preclude radiative absorption. Thus, for those interested in lanthanide ions, the dominant ET processes are nonradiative（or radiationless）. Recently, the intramolecular ET processes in lanthanide complexes, namely, ligand- Ln （Ⅲ） transfer rates, were revised and reviewed [ 20 20 ] . Thus, the present review will focus on radiationless or nonradiative ET processes between lanthanide ions, Ln - Ln . The main goal is to present a review for non-specialists, providing sufficient concepts to critically ascertain the approximations involved in the equations of the energy transfer rate mechanisms. Enough details will be provided for a qualitative assessment of the main factors affecting the ET processes in Ln - Ln systems as well as to perform estimations and calculations of these ET rates. The original equations were reorganized to ease their applications as well as their interpretations. In the Supporting Information, a derivation of Fermi's golden rule is provided together with the physical constants and unit conversions, details matrix elements calculations, and dimensional analysis of the main quantities. In the following sections, some concepts and equations involved in the quantum treatment of transitions will be present culminating in Fermi's golden rule for transition rates. This expression will then be particularized for nonradiative transitions between two lanthanide ions, namely, the Ln - Ln energy transfer rates. Detailed applications of these equations will be presented as tutorial examples for ET processes involved in the Yb（Ⅲ）-Er（Ⅲ） and Tb（Ⅲ）-Eu（Ⅲ） pairs. 2　Quantum Transitions 1 An energy transfer process can be regarded as a transition from a state representing the energy localized initially at the donor subsystem to another state characterized by the energy localized in the acceptor unit. This process can be described quantitatively by an energy transfer rate, that is, the probability of transition induced by a perturbation per unit of time, in units of <math id="M21"><msup><mrow><mi mathvariant="normal">s</mi></mrow><mrow><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39818931&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39818929&type= 3.64066648 2.96333337 , because the probability is dimensionless. Thus, to comprehend the equations for the ET rates, representations of the initial and final states are required, which will be addressed in detail later, as well as the transition probability between these states per unit of time. For most transitions occurring at the molecular level, their rates can be quantitatively described by Fermi's golden rule. A detailed derivation of the expression for the transition rate, in particular Fermi's golden rule, can be found in many textbooks [ 21 21 - 22 22 21-22 ] and is provided in the Supporting Information. Here we will present the main equations and the approximations and assumptions employed. Transitions at the molecular level are described by quantum mechanics. A system is described by its state represented by a （vector） state or wavefunction, which depends on the spatial and spin coordinates of its particles and on the time <math id="M22"><mi>t</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819028&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819027&type= 1.10066664 2.62466669 . This state can be expressed as a linear combination of a complete set of eigenfunctions <math id="M23"><mo stretchy="false">{</mo><msub><mrow><mi>Ψ</mi></mrow><mrow><mi>n</mi></mrow></msub><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo stretchy="false">}</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39818937&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39818935&type= 12.02266598 4.57200003 with corresponding energy eigenvalues <math id="M24"><mo stretchy="false">{</mo><msub><mrow><mi>E</mi></mrow><mrow><mi>n</mi></mrow></msub><mo stretchy="false">}</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39818939&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39818938&type= 6.77333355 4.40266657 . If the system is described by a time-independent Hamiltonian, e . g ., a molecule with contributions from the kinetic energy of the nuclei and electrons and Coulomb interactions between the charged particles （nuclei-nuclei, nuclei-electrons, and electrons-electrons）, the time-dependent Schrödinger equation can be separated and have solutions <math id="M25"><msub><mrow><mi>Ψ</mi></mrow><mrow><mi>n</mi></mrow></msub><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo>=</mo><msub><mrow><mi>ψ</mi></mrow><mrow><mi>n</mi></mrow></msub><msup><mrow><mi>e</mi></mrow><mrow><mo>-</mo><mrow><mrow><mi mathvariant="normal">i</mi><msub><mrow><mi>E</mi></mrow><mrow><mi>n</mi></mrow></msub><mi>t</mi></mrow><mo>/</mo><mrow><mi>ћ</mi></mrow></mrow></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39818941&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39818940&type= 24.55333328 5.07999992 , where <math id="M26"><msub><mrow><mi>ψ</mi></mrow><mrow><mi>n</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819042&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819041&type= 3.55599999 4.48733330 are the stationary （time-independent） eigenfunctions of this Hamiltonian. So, if the system is （prepared） at an eigenstate <math id="M27"><msub><mrow><mi>Ψ</mi></mrow><mrow><mi>m</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39818946&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39818945&type= 4.48733330 3.80999994 , with energy <math id="M28"><msub><mrow><mi>E</mi></mrow><mrow><mi>m</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39818948&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39818947&type= 3.89466691 3.72533321 , then it is stationary, <math id="M29"><msubsup><mrow><mi>Ψ</mi></mrow><mrow><mi>m</mi></mrow><mrow><mi>*</mi></mrow></msubsup><mfenced separators="|"><mrow><mi>t</mi></mrow></mfenced><msub><mrow><mi>Ψ</mi></mrow><mrow><mi>m</mi></mrow></msub><mfenced separators="|"><mrow><mi>t</mi></mrow></mfenced><mo>=</mo><msubsup><mrow><mi>ψ</mi></mrow><mrow><mi>m</mi></mrow><mrow><mi>*</mi></mrow></msubsup><msup><mrow><mi>e</mi></mrow><mrow><mrow><mrow><mi mathvariant="normal">i</mi><msub><mrow><mi>E</mi></mrow><mrow><mi>m</mi></mrow></msub><mi>t</mi></mrow><mo>/</mo><mrow><mi>ћ</mi></mrow></mrow></mrow></msup><msub><mrow><mi>ψ</mi></mrow><mrow><mi>m</mi></mrow></msub><msup><mrow><mi>e</mi></mrow><mrow><mo>-</mo><mrow><mrow><mi mathvariant="normal">i</mi><msub><mrow><mi>E</mi></mrow><mrow><mi>m</mi></mrow></msub><mi>t</mi></mrow><mo>/</mo><mrow><mi>ћ</mi></mrow></mrow></mrow></msup><mo>=</mo><msubsup><mrow><mi>ψ</mi></mrow><mrow><mi>m</mi></mrow><mrow><mi>*</mi></mrow></msubsup><msub><mrow><mi>ψ</mi></mrow><mrow><mi>m</mi></mrow></msub><mo>=</mo><msup><mrow><mfenced open="|" close="|" separators="|"><mrow><msub><mrow><mi>ψ</mi></mrow><mrow><mi>m</mi></mrow></msub></mrow></mfenced></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mo>=</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39818951&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39818950&type= 74.42199707 5.07999992 constant, and it will remain in this state indefinitely. Thus, a perturbation <math id="M30"><mover accent="true"><mi>V</mi><mo>^</mo></mover><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39818977&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39818976&type= 6.77333355 4.23333359 is needed to promote a transition from this state to another state. This perturbation causes the temporal evolution of a non-stationary state <math id="M31"><mi>Ψ</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39818981&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39818980&type= 7.61999989 3.47133350 , which can still be represented as a linear combination of eigenfunctions <math id="M32"><mo stretchy="false">{</mo><msub><mrow><mi>Ψ</mi></mrow><mrow><mi>n</mi></mrow></msub><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo stretchy="false">}</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39818985&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39818984&type= 12.02266598 4.57200003 with coefficients <math id="M33"><mo stretchy="false">{</mo><msub><mrow><mi>c</mi></mrow><mrow><mi>n</mi></mrow></msub><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo stretchy="false">}</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39818987&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39818986&type= 10.83733368 4.57200003 . As a result, the probability of finding the system at time <math id="M34"><mi>t</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819028&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819027&type= 1.10066664 2.62466669 in state <math id="M35"><mi>k</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819725&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819724&type= 1.69333339 2.62466669 is <math id="M36"><msup><mrow><mfenced open="|" close="|" separators="|"><mrow><msub><mrow><mi>c</mi></mrow><mrow><mi>k</mi></mrow></msub><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo></mrow></mfenced></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mo>=</mo><msubsup><mrow><mi>c</mi></mrow><mrow><mi>k</mi></mrow><mrow><mi>*</mi></mrow></msubsup><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><msub><mrow><mi>c</mi></mrow><mrow><mi>k</mi></mrow></msub><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39818971&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39818970&type= 29.97200012 4.99533367 . As a result of this perturbation, the system can be described by a Hamiltonian, <math id="M37"><mover accent="true"><mi>H</mi><mo>^</mo></mover></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39818990&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39818989&type= 2.70933342 3.64066648 , consisting of the time-independent Hamiltonian, <math id="M38"><msub><mrow><mover accent="true"><mi>H</mi><mo>^</mo></mover></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39818975&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39818974&type= 3.72533321 4.74133301 , and the perturbation <math id="M39"><mover accent="true"><mi>V</mi><mo>^</mo></mover><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39818977&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39818976&type= 6.77333355 4.23333359 , i . e ., <math id="M40"><mover accent="true"><mi>H</mi><mo>^</mo></mover><mo>=</mo><msub><mrow><mover accent="true"><mi>H</mi><mo>^</mo></mover></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><mo>+</mo><mover accent="true"><mi>V</mi><mo>^</mo></mover><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39818979&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39818978&type= 21.92866707 5.33400011 . A procedure developed by Dirac starts by replacing the wavefunction <math id="M41"><mi>Ψ</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39818981&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39818980&type= 7.61999989 3.47133350 , in the time-dependent Schrödinger equation employing <math id="M42"><mover accent="true"><mi>H</mi><mo>^</mo></mover></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39818990&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39818989&type= 2.70933342 3.64066648 , by an expansion in terms of <math id="M43"><mo stretchy="false">{</mo><msub><mrow><mi>Ψ</mi></mrow><mrow><mi>n</mi></mrow></msub><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo stretchy="false">}</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39818985&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39818984&type= 12.02266598 4.57200003 and <math id="M44"><mo stretchy="false">{</mo><msub><mrow><mi>c</mi></mrow><mrow><mi>n</mi></mrow></msub><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo stretchy="false">}</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39818987&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39818986&type= 10.83733368 4.57200003 . After applying the time derivative and the full Hamiltonian <math id="M45"><mover accent="true"><mi>H</mi><mo>^</mo></mover></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39818990&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39818989&type= 2.70933342 3.64066648 on this expansion and projecting the resulting equation onto state <math id="M46"><mi>k</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819725&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819724&type= 1.69333339 2.62466669 （‘multiplication’ or scalar product with the bra <math id="M47"><mfenced open="<" close="" separators="|"><mrow><mfenced open="" close="|" separators="|"><mrow><mi>k</mi></mrow></mfenced></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39818994&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39818993&type= 4.40266657 4.48733330 ）, an exact equation for the time evolution <math id="M48"><msub><mrow><mover accent="true"><mi>c</mi><mo>˙</mo></mover></mrow><mrow><mi>n</mi></mrow></msub><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39818996&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39818995&type= 7.28133297 4.57200003 is obtained. By integrating the resulting equation from <math id="M49"><mi>t</mi><mo>=</mo><mn mathvariant="normal">0</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819062&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819061&type= 7.45066643 2.70933342 , when the coefficients are <math id="M50"><msub><mrow><mi>c</mi></mrow><mrow><mi>n</mi></mrow></msub><mo stretchy="false">(</mo><mn mathvariant="normal">0</mn><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819006&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819005&type= 8.04333401 4.57200003 , to the time <math id="M51"><mi>t</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819028&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819027&type= 1.10066664 2.62466669 of interest, an exact expression for the coefficients <math id="M52"><msub><mrow><mi>c</mi></mrow><mrow><mi>n</mi></mrow></msub><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819004&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819003&type= 7.28133297 4.57200003 is derived, so the probability could, in principle, be calculated. Thus, a uniquely define exact solution is obtained when the boundary conditions, namely, the initial coefficients <math id="M53"><msub><mrow><mi>c</mi></mrow><mrow><mi>n</mi></mrow></msub><mo stretchy="false">(</mo><mn mathvariant="normal">0</mn><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819006&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819005&type= 8.04333401 4.57200003 are known. These conditions are usually expressed as <math id="M54"><msub><mrow><mi>c</mi></mrow><mrow><mi>n</mi></mrow></msub><mo stretchy="false">(</mo><mi>t</mi><mo>=</mo><mn mathvariant="normal">0</mn><mo stretchy="false">)</mo><mo>=</mo><msub><mrow><mi>δ</mi></mrow><mrow><mi>n</mi><mi>i</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819008&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819007&type= 21.25133324 4.57200003 , where <math id="M55"><mi>i</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819727&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819726&type= 1.10066664 2.62466669 is the initial state, or <math id="M56"><msub><mrow><mi>c</mi></mrow><mrow><mi>i</mi></mrow></msub><mo stretchy="false">(</mo><mi>t</mi><mo>=</mo><mn mathvariant="normal">0</mn><mo stretchy="false">)</mo><mo>=</mo><mn mathvariant="normal">1</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819012&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819011&type= 19.38866615 4.57200003 and <math id="M57"><msub><mrow><mi>c</mi></mrow><mrow><mi>n</mi><mo>≠</mo><mi>i</mi></mrow></msub><mo stretchy="false">(</mo><mi>t</mi><mo>=</mo><mn mathvariant="normal">0</mn><mo stretchy="false">)</mo><mo>=</mo><mn mathvariant="normal">0</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819014&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819013&type= 22.77533340 4.57200003 . However, the exact determination of the coefficient <math id="M58"><msub><mrow><mi>c</mi></mrow><mrow><mi>k</mi></mrow></msub><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819018&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819017&type= 7.19666624 4.57200003 requires all the coefficients, including <math id="M59"><msub><mrow><mi>c</mi></mrow><mrow><mi>k</mi></mrow></msub><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819018&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819017&type= 7.19666624 4.57200003 , i . e ., the exact solution is a set of coupled integral equations. So, approximations and assumptions are needed to obtain a computable expression. Suppose that the perturbation is so weak and applied during a time interval so short that all the coefficients remain close to their initial values and are approximated as <math id="M60"><msub><mrow><mi>c</mi></mrow><mrow><mi>n</mi></mrow></msub><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo>≅</mo><mn mathvariant="normal">0</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819020&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819019&type= 13.46199989 4.57200003 for <math id="M61"><mi>n</mi><mo>≠</mo><mi>i</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819022&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819021&type= 7.45066643 2.62466669 and <math id="M62"><msub><mrow><mi>c</mi></mrow><mrow><mi>i</mi></mrow></msub><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo>≅</mo><mn mathvariant="normal">1</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819024&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819023&type= 13.03866673 4.57200003 . This is the so-called first-order perturbation or first-order approximation and yields the following expression for the coefficients （ <math id="M63"><mi>k</mi><mo>≠</mo><mi>i</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819026&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819025&type= 7.28133297 2.62466669 ） at time <math id="M64"><mi>t</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819028&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819027&type= 1.10066664 2.62466669 [ 21 21 - 22 22 21-22 ] : <math display="block" id="M65"><msub><mrow><mi>c</mi></mrow><mrow><mi>k</mi></mrow></msub><mfenced separators="|"><mrow><mi>t</mi></mrow></mfenced><mo>≅</mo><msubsup><mrow><mi>c</mi></mrow><mrow><mi>k</mi></mrow><mrow><mfenced separators="|"><mrow><mn mathvariant="normal">1</mn></mrow></mfenced></mrow></msubsup><mo>=</mo><mfrac><mrow><mn mathvariant="normal">1</mn></mrow><mrow><mi mathvariant="normal">i</mi><mi>ћ</mi></mrow></mfrac><msubsup><mo>∫</mo><mrow><mn mathvariant="normal">0</mn></mrow><mrow><mi>t</mi></mrow></msubsup><mrow><msub><mrow><mi>V</mi></mrow><mrow><mi>k</mi><mi>i</mi></mrow></msub></mrow><mfenced separators="|"><mrow><msup><mrow><mi>t</mi></mrow><mrow><mi mathvariant="normal">'</mi></mrow></msup></mrow></mfenced><msup><mrow><mi>e</mi></mrow><mrow><mi mathvariant="normal">i</mi><msub><mrow><mi>ω</mi></mrow><mrow><mi>k</mi><mi>i</mi></mrow></msub><msup><mrow><mi>t</mi></mrow><mrow><mi mathvariant="normal">'</mi></mrow></msup></mrow></msup><mi mathvariant="normal">d</mi><msup><mrow><mi>t</mi></mrow><mrow><mi mathvariant="normal">'</mi></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819030&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819029&type= 46.65133286 7.53533268 ， where <math id="M66"><msub><mrow><mi>ω</mi></mrow><mrow><mi>k</mi><mi>i</mi></mrow></msub><mo>=</mo><mrow><mrow><mo stretchy="false">(</mo><msub><mrow><mi>E</mi></mrow><mrow><mi>k</mi></mrow></msub><mo>-</mo><msub><mrow><mi>E</mi></mrow><mrow><mi>i</mi></mrow></msub><mo stretchy="false">)</mo></mrow><mo>/</mo><mrow><mi>ћ</mi></mrow></mrow></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819032&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819031&type= 26.07733345 3.97933316 and the generic matrix element, <math id="M67"><msub><mrow><mi>V</mi></mrow><mrow><mi>k</mi><mi>n</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819034&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819033&type= 4.14866638 3.72533321 , of the perturbation operator <math id="M68"><mover accent="true"><mi>V</mi><mo>^</mo></mover></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819302&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819301&type= 2.37066650 3.64066648 is expressed as <math display="block" id="M69"><msub><mrow><mi>V</mi></mrow><mrow><mi>k</mi><mi>n</mi></mrow></msub><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo>=</mo><mo>∫</mo><msubsup><mrow><mi>ψ</mi></mrow><mrow><mi>k</mi></mrow><mrow><mi mathvariant="normal">*</mi></mrow></msubsup><mover accent="true"><mi>V</mi><mo>^</mo></mover><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><msub><mrow><mi>ψ</mi></mrow><mrow><mi>n</mi></mrow></msub><mi mathvariant="normal">d</mi><mi>τ</mi><mo>≡</mo><mfenced open="<" close=">" separators="|"><mrow><mi>k</mi><mo stretchy="false">|</mo><mover accent="true"><mi>V</mi><mo>^</mo></mover><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo stretchy="false">|</mo><mi>n</mi></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819038&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819037&type= 51.90066528 4.99533367 ， with the respect to the stationary eigenfunctions <math id="M70"><msub><mrow><mi>ψ</mi></mrow><mrow><mi>k</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819040&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819039&type= 3.47133350 4.48733330 and <math id="M71"><msub><mrow><mi>ψ</mi></mrow><mrow><mi>n</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819042&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819041&type= 3.55599999 4.48733330 . Notice that the first equality employs the integral representation of the matrix element, which is equivalent to that in the ‘bra-ket’（or Dirac） notation. 3　Fermi’s Golden Rule for Radiationless（or Nonradiative） Transitions 1 Several processes are induced by a coupling between subsystems or between states and radiation is not required, representing the so-called radiationless or nonradiative transitions. For example, in a time <math id="M72"><mi>t</mi><mo>=</mo><mn mathvariant="normal">0</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819062&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819061&type= 7.45066643 2.70933342 the system （instantaneously） absorbs energy （ e . g ., light absorption, mechanical or thermal excitations） and is promoted to an excited state, thus initiating a coupling or interaction. This is the initial condition of the system, and this excited state is its initial state <math id="M73"><mi>i</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819727&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819726&type= 1.10066664 2.62466669 . The nonadiabatic electronic coupling or the spin-orbit coupling between the initial state <math id="M74"><mi>i</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819727&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819726&type= 1.10066664 2.62466669 and another state <math id="M75"><mi>k</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819725&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819724&type= 1.69333339 2.62466669 causes a radiationless transition from <math id="M76"><mi>i</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819727&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819726&type= 1.10066664 2.62466669 to <math id="M77"><mi>k</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819725&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819724&type= 1.69333339 2.62466669 , <math id="M78"><mi>i</mi><mo>→</mo><mi>k</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819056&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819055&type= 8.89000034 2.62466669 . Another example is a system of two uncoupled subunits, e . g ., donor and acceptor D - A . Upon excitation, at t =0, localized on the donor subsystem, the system can be represented as D * - A and a coupling or interaction between D and A appears and promotes a nonradiative transition to another state that can lead to energy transfer or electron transfer, depending on the nature of the donor and acceptor, and on the initial and final states. In these radiationless transitions, the coupling or interaction initiated at t =0 does not depend explicitly on time, so it is constant in time: <math id="M79"><mover accent="true"><mi>V</mi><mo>^</mo></mover><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo>≡</mo><mover accent="true"><mi>V</mi><mo>^</mo></mover><mo>=</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819058&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819057&type= 16.34066582 4.23333359 constant. Thus, for nonradiative energy transfer processes, a constant perturbation <math id="M80"><mover accent="true"><mi>V</mi><mo>^</mo></mover></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819302&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819301&type= 2.37066650 3.64066648 （time-independent） is considered, which is turned on at <math id="M81"><mi>t</mi><mo>=</mo><mn mathvariant="normal">0</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819062&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819061&type= 7.45066643 2.70933342 and off at <math id="M82"><mi>t</mi><mo>=</mo><mi>τ</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819064&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819063&type= 7.36600018 2.62466669 . For this perturbation, the coefficient describing state <math id="M83"><mi>k</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819725&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819724&type= 1.69333339 2.62466669 at time <math id="M84"><mi>t</mi><mo>≥</mo><mi>τ</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819068&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819067&type= 7.36600018 2.62466669 becomes [ 21 21 ] : <math display="block" id="M85"><msub><mrow><mi>c</mi></mrow><mrow><mi>k</mi></mrow></msub><mo>=</mo><mo>-</mo><mfrac><mrow><mi mathvariant="normal">i</mi></mrow><mrow><mi>ћ</mi></mrow></mfrac><msub><mrow><mi>V</mi></mrow><mrow><mi>k</mi><mi>i</mi></mrow></msub><msubsup><mo>∫</mo><mrow><mn mathvariant="normal">0</mn></mrow><mrow><mi>τ</mi></mrow></msubsup><mrow><msup><mrow><mi>e</mi></mrow><mrow><mi mathvariant="normal">i</mi><msub><mrow><mi>ω</mi></mrow><mrow><mi>k</mi><mi>i</mi></mrow></msub><mi>t</mi></mrow></msup></mrow><mi mathvariant="normal">d</mi><mi>t</mi><mo>,</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819070&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819069&type= 30.05666542 7.53533268 <math display="block" id="M86"><mtext>         </mtext><msub><mrow><mi>V</mi></mrow><mrow><mi>k</mi><mi>i</mi></mrow></msub><mo>=</mo><mo>∫</mo><msubsup><mrow><mi>ψ</mi></mrow><mrow><mi>k</mi></mrow><mrow><mi>*</mi></mrow></msubsup><mover accent="true"><mi>V</mi><mo>^</mo></mover><msub><mrow><mi>ψ</mi></mrow><mrow><mi>i</mi></mrow></msub><mi mathvariant="normal">d</mi><mi>τ</mi><mo>≡</mo><mfenced open="<" close=">" separators="|"><mrow><mi>k</mi><mo stretchy="false">|</mo><mover accent="true"><mi>V</mi><mo>^</mo></mover><mo stretchy="false">|</mo><mi>i</mi></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819072&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819071&type= 44.61933517 6.43466663 ， where it was considered that the matrix element <math id="M87"><msub><mrow><mi>V</mi></mrow><mrow><mi>k</mi><mi>i</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819132&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819131&type= 3.72533321 3.72533321 has a weak dependence on the quasi-continuum states around <math id="M88"><mi>k</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819725&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819724&type= 1.69333339 2.62466669 . The duration of the perturbation, denoted as <math id="M89"><mi>τ</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819170&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819169&type= 1.77800000 2.62466669 , should not be confused with the lifetime of an excited state, which is usually also denoted as <math id="M90"><mi>τ</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819170&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819169&type= 1.77800000 2.62466669 . The integral can be performed analytically, so the transition probability to state <math id="M91"><mi>k</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819725&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819724&type= 1.69333339 2.62466669 is <math display="block" id="M92"><msubsup><mrow><mi>P</mi></mrow><mrow><mi>k</mi></mrow><mrow><mi>i</mi></mrow></msubsup><mo>=</mo><msup><mrow><mfenced open="|" close="|" separators="|"><mrow><msub><mrow><mi>V</mi></mrow><mrow><mi>k</mi><mi>i</mi></mrow></msub></mrow></mfenced></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><msubsup><mrow><mi>F</mi></mrow><mrow><mi>k</mi></mrow><mrow><mi>i</mi></mrow></msubsup><mfenced separators="|"><mrow><msub><mrow><mi>ω</mi></mrow><mrow><mi>k</mi><mi>i</mi></mrow></msub><mo>,</mo><mi>τ</mi></mrow></mfenced><mtext> </mtext><mo>,</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819084&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819083&type= 32.34266663 4.99533367 <math display="block" id="M93"><msubsup><mrow><mi>F</mi></mrow><mrow><mi>k</mi></mrow><mrow><mi>i</mi></mrow></msubsup><mfenced separators="|"><mrow><msub><mrow><mi>ω</mi></mrow><mrow><mi>k</mi><mi>i</mi></mrow></msub><mo>,</mo><mi>τ</mi></mrow></mfenced><mo>=</mo><mfrac><mrow><mn mathvariant="normal">1</mn></mrow><mrow><msup><mrow><mi>ћ</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup></mrow></mfrac><mfrac><mrow><mi mathvariant="normal">s</mi><mi mathvariant="normal">i</mi><msup><mrow><mi mathvariant="normal">n</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mtext> </mtext><mfenced separators="|"><mrow><mrow><mrow><msub><mrow><mi>ω</mi></mrow><mrow><mi>k</mi><mi>i</mi></mrow></msub><mi>τ</mi></mrow><mo>/</mo><mrow><mn mathvariant="normal">2</mn></mrow></mrow></mrow></mfenced></mrow><mrow><msup><mrow><mfenced separators="|"><mrow><mrow><mrow><msub><mrow><mi>ω</mi></mrow><mrow><mi>k</mi><mi>i</mi><mtext>  </mtext></mrow></msub></mrow><mo>/</mo><mrow><mn mathvariant="normal">2</mn></mrow></mrow></mrow></mfenced></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup></mrow></mfrac><mtext> </mtext><mo>.</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819086&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819085&type= 44.70399857 12.44600010 This expression yields the same probability for the reverse transition: <math id="M94"><msubsup><mrow><mi>P</mi></mrow><mrow><mi>k</mi></mrow><mrow><mi>i</mi></mrow></msubsup><mo>=</mo><msubsup><mrow><mi>P</mi></mrow><mrow><mi>i</mi></mrow><mrow><mi>k</mi></mrow></msubsup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819088&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819087&type= 11.17599964 4.06400013 , because <math id="M95"><msup><mrow><mfenced open="|" close="|" separators="|"><mrow><msub><mrow><mi>V</mi></mrow><mrow><mi>k</mi><mi>i</mi></mrow></msub></mrow></mfenced></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mo>=</mo><msup><mrow><mfenced open="|" close="|" separators="|"><mrow><msub><mrow><mi>V</mi></mrow><mrow><mi>i</mi><mi>k</mi></mrow></msub></mrow></mfenced></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819090&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819089&type= 18.79599953 4.99533367 and <math id="M96"><msubsup><mrow><mi>F</mi></mrow><mrow><mi>k</mi></mrow><mrow><mi>i</mi></mrow></msubsup><mo>=</mo><msubsup><mrow><mi>F</mi></mrow><mrow><mi>i</mi></mrow><mrow><mi>k</mi></mrow></msubsup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819092&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819091&type= 11.43000031 4.06400013 . For <math id="M97"><mi>τ</mi><mo>≳</mo><mrow><mrow><mn mathvariant="normal">2</mn><mi>π</mi><mi mathvariant="normal">ℏ</mi></mrow><mo>/</mo><mrow><msub><mrow><mi>E</mi></mrow><mrow><mi>k</mi></mrow></msub></mrow></mrow></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819094&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819093&type= 17.44133186 3.80999994 , the function <math id="M98"><msubsup><mrow><mi>F</mi></mrow><mrow><mi>k</mi></mrow><mrow><mi>i</mi></mrow></msubsup><mo stretchy="false">(</mo><msub><mrow><mi>ω</mi></mrow><mrow><mi>k</mi><mi>i</mi></mrow></msub><mo>,</mo><mtext> </mtext><mi>τ</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819162&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819161&type= 14.73200035 4.82600021 becomes sharply peaked, with the height of the peak around <math id="M99"><msub><mrow><mi>E</mi></mrow><mrow><mi>k</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819138&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819137&type= 3.38666677 3.72533321 increasing as <math id="M100"><mrow><mrow><msup><mrow><mi>τ</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup></mrow><mo>/</mo><mrow><msup><mrow><mi>ћ</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup></mrow></mrow></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819100&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819099&type= 7.45066643 3.38666677 and width decreasing as <math id="M101"><mrow><mrow><mn mathvariant="normal">2</mn><mi mathvariant="normal">π</mi><mi>ћ</mi></mrow><mo>/</mo><mrow><mi>τ</mi></mrow></mrow></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819102&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819101&type= 9.48266602 3.21733332 , so the area of this peak is approximately <math id="M102"><mrow><mrow><mn mathvariant="normal">2</mn><mi mathvariant="normal">π</mi><mi>τ</mi></mrow><mo>/</mo><mrow><mi>ћ</mi></mrow></mrow></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819104&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819103&type= 9.48266602 3.21733332 , which grows linearly with <math id="M103"><mi>τ</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819170&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819169&type= 1.77800000 2.62466669 . This indicates that a time <math id="M104"><mi>τ</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819170&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819169&type= 1.77800000 2.62466669 after the initial coupling （or interaction） between the states or the subsystems, their energy eigenvalues will present dispersions of <math id="M105"><mi>δ</mi><mi>E</mi><mo>=</mo><mrow><mrow><mn mathvariant="normal">2</mn><mi mathvariant="normal">π</mi><mi mathvariant="normal">ℏ</mi></mrow><mo>/</mo><mrow><mi>τ</mi></mrow></mrow></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819110&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819109&type= 17.94933319 3.13266683 , which is consistent with the energy-time uncertainty principle because <math id="M106"><mi>δ</mi><mi>E</mi><mi>τ</mi><mo>=</mo><mn mathvariant="normal">2</mn><mi mathvariant="normal">π</mi><mi>ћ</mi><mo>></mo><mrow><mrow><mi>ћ</mi></mrow><mo>/</mo><mrow><mn mathvariant="normal">2</mn></mrow></mrow></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819112&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819111&type= 26.24666595 3.21733332 [ 21 21 ] . So, the final states after the perturbation are better described as nearly a continuum of energy values with a narrow distribution represented by a density of states, <math id="M107"><mi>ρ</mi><mo stretchy="false">(</mo><mi>E</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819124&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819123&type= 7.78933382 3.55599999 , which is the number of states per energy unit around energy <math id="M108"><mi>E</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819120&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819119&type= 2.28600001 2.62466669 and <math id="M109"><mi>ρ</mi><mo stretchy="false">(</mo><mi>E</mi><mo stretchy="false">)</mo><mi mathvariant="normal">d</mi><mi>E</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819118&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819117&type= 12.02266598 3.55599999 is the number of （continuum） states in the range <math id="M110"><mi>E</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819120&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819119&type= 2.28600001 2.62466669 to <math id="M111"><mi>E</mi><mo>+</mo><mi mathvariant="normal">d</mi><mi>E</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819122&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819121&type= 11.43000031 2.70933342 [ 21 21 - 22 22 21-22 ] . In other words, the perturbation, represented by the interaction （or coupling）, has generated many final states closely spaced in energy. For a molecular system, other mechanisms of energy dispersion will also be present, such as a distribution of closely spaced rotational‑vibrational energy levels. Thus, if the final states have a density of states <math id="M112"><mi>ρ</mi><mo stretchy="false">(</mo><mi>E</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819124&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819123&type= 7.78933382 3.55599999 , then the total probability of transition to state <math id="M113"><mi>k</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819126&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819125&type= 1.69333339 2.62466669 becomes an integral of <math id="M114"><msubsup><mrow><mi>P</mi></mrow><mrow><mi>k</mi></mrow><mrow><mi>i</mi></mrow></msubsup><mi>ρ</mi><mo stretchy="false">(</mo><mi>E</mi><mo stretchy="false">)</mo><mi mathvariant="normal">d</mi><mi>E</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819128&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819127&type= 15.66333294 4.99533367 over all final states accessible under the influence of the perturbation: <math display="block" id="M115"><mtext>                                          </mtext><msubsup><mrow><mi>P</mi></mrow><mrow><mover accent="true"><mi>k</mi><mo>¯</mo></mover></mrow><mrow><mi>i</mi></mrow></msubsup><mo>=</mo><mo>∫</mo><msup><mrow><mfenced open="|" close="|" separators="|"><mrow><msub><mrow><mi>V</mi></mrow><mrow><mi>k</mi><mi>i</mi></mrow></msub></mrow></mfenced></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><msubsup><mrow><mi>F</mi></mrow><mrow><mi>k</mi></mrow><mrow><mi>i</mi></mrow></msubsup><mo stretchy="false">(</mo><msub><mrow><mi>ω</mi></mrow><mrow><mi>k</mi><mi>i</mi></mrow></msub><mo>,</mo><mi>τ</mi><mo stretchy="false">)</mo><mi>ρ</mi><mo stretchy="false">(</mo><msub><mrow><mi>E</mi></mrow><mrow><mi>k</mi></mrow></msub><mo stretchy="false">)</mo><mi mathvariant="normal">d</mi><msub><mrow><mi>E</mi></mrow><mrow><mi>k</mi></mrow></msub><mo>≅</mo><msup><mrow><mfenced open="|" close="|" separators="|"><mrow><msub><mrow><mi>V</mi></mrow><mrow><mi>k</mi><mi>i</mi></mrow></msub></mrow></mfenced></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mi>ρ</mi><mo stretchy="false">(</mo><msub><mrow><mi>E</mi></mrow><mrow><mi>k</mi></mrow></msub><mo stretchy="false">)</mo><mo>∫</mo><msubsup><mrow><mi>F</mi></mrow><mrow><mi>k</mi></mrow><mrow><mi>i</mi></mrow></msubsup><mo stretchy="false">(</mo><msub><mrow><mi>ω</mi></mrow><mrow><mi>k</mi><mi>i</mi></mrow></msub><mo>,</mo><mi>τ</mi><mo stretchy="false">)</mo><mi mathvariant="normal">d</mi><msub><mrow><mi>E</mi></mrow><mrow><mi>k</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819130&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819129&type= 124.20600128 6.43466663 ， where the last approximation considers that the matrix element <math id="M116"><msub><mrow><mi>V</mi></mrow><mrow><mi>k</mi><mi>i</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819132&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819131&type= 3.72533321 3.72533321 and the density of state <math id="M117"><mi>ρ</mi><mo stretchy="false">(</mo><msub><mrow><mi>E</mi></mrow><mrow><mi>k</mi></mrow></msub><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819166&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819165&type= 8.80533314 4.65666676 are constant in the integration range. Because the function <math id="M118"><msubsup><mrow><mi>F</mi></mrow><mrow><mi>k</mi></mrow><mrow><mi>i</mi></mrow></msubsup><mo stretchy="false">(</mo><msub><mrow><mi>ω</mi></mrow><mrow><mi>k</mi><mi>i</mi></mrow></msub><mo>,</mo><mtext> </mtext><mi>τ</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819162&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819161&type= 14.73200035 4.82600021 is sharply peaked, the integrand has significant values only around <math id="M119"><msub><mrow><mi>E</mi></mrow><mrow><mi>k</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819138&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819137&type= 3.38666677 3.72533321 , which justifies the last approximation. This property of the function <math id="M120"><msubsup><mrow><mi>F</mi></mrow><mrow><mi>k</mi></mrow><mrow><mi>i</mi></mrow></msubsup><mo stretchy="false">(</mo><msub><mrow><mi>ω</mi></mrow><mrow><mi>k</mi><mi>i</mi></mrow></msub><mo>,</mo><mtext> </mtext><mi>τ</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819162&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819161&type= 14.73200035 4.82600021 also allows for extending the integration from <math id="M121"><mo>-</mo><mi mathvariant="normal">∞</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819142&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819141&type= 4.74133301 2.62466669 to <math id="M122"><mo>+</mo><mi mathvariant="normal">∞</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819144&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819143&type= 4.74133301 2.62466669 , so the last integral is just <math id="M123"><mrow><mrow><mn mathvariant="normal">2</mn><mi mathvariant="normal">π</mi><mi>τ</mi></mrow><mo>/</mo><mrow><mi mathvariant="normal">ℏ</mi></mrow></mrow></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819146&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819145&type= 9.48266602 3.13266683 and <math id="M124"><msubsup><mrow><mi>P</mi></mrow><mrow><mover accent="true"><mi>k</mi><mo>¯</mo></mover></mrow><mrow><mi>i</mi></mrow></msubsup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819148&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819147&type= 3.38666677 4.06400013 has a linear dependence on <math id="M125"><mi>τ</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819170&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819169&type= 1.77800000 2.62466669 , where <math id="M126"><mover accent="true"><mi>k</mi><mo>¯</mo></mover></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819152&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819151&type= 1.69333339 3.13266683 represents the summation over over the quasi-continuum （ e.g ., rovibrational） contributions.. As a result, the transition rate <math id="M127"><msub><mrow><mi>W</mi></mrow><mrow><mi>i</mi><mo>→</mo><mi>k</mi></mrow></msub><mo>≡</mo><msub><mrow><mi>W</mi></mrow><mrow><mi>k</mi><mi>i</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819154&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819153&type= 16.76399994 3.80999994 from an initial state <math id="M128"><mi>i</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819727&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819726&type= 1.10066664 2.62466669 to a final state <math id="M129"><mi>k</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819725&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819724&type= 1.69333339 2.62466669 , which is the change of the probability per unit of time, can be written as [ 21 21 - 22 22 21-22 ] ： <math display="block" id="M130"><msub><mrow><mi>W</mi></mrow><mrow><mi>i</mi><mo>→</mo><mi>k</mi></mrow></msub><mo>≡</mo><msub><mrow><mi>W</mi></mrow><mrow><mi>k</mi><mi>i</mi></mrow></msub><mo>=</mo><mfrac><mrow><mi mathvariant="normal">d</mi><msubsup><mrow><mi>P</mi></mrow><mrow><mi>k</mi></mrow><mrow><mi>i</mi></mrow></msubsup></mrow><mrow><mi mathvariant="normal">d</mi><mi>t</mi></mrow></mfrac><mo>≅</mo><mfrac><mrow><msubsup><mrow><mi>P</mi></mrow><mrow><mover accent="true"><mi>k</mi><mo>¯</mo></mover></mrow><mrow><mi>i</mi></mrow></msubsup></mrow><mrow><mi>τ</mi></mrow></mfrac><mo>=</mo><mfrac><mrow><mn mathvariant="normal">2</mn><mi>π</mi></mrow><mrow><mi>ћ</mi></mrow></mfrac><msup><mrow><mfenced open="|" close="|" separators="|"><mrow><msub><mrow><mi>V</mi></mrow><mrow><mi>k</mi><mi>i</mi></mrow></msub></mrow></mfenced></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mi>ρ</mi><mfenced separators="|"><mrow><msub><mrow><mi>E</mi></mrow><mrow><mi>k</mi></mrow></msub></mrow></mfenced><mo>.</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819160&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819159&type= 64.17733002 7.78933382 This is called Fermi’s golden rule and provides the rate of transition. Despite this derivation being performed for constant perturbation, the same expression is obtained for periodic potentials that describe, for instance, the electromagnetic radiation. For a proper application of this expression for the transition rate, it is important to present the approximations and assumptions as well as their consequences （see Supporting Information）. The task of applying Eq.（6） Eq.（6） to a given process becomes （ⅰ） the proper representation of the initial and final states of the system, （ⅱ） defining and establishing the perturbation, and （ⅲ） determining the density of states. The sharply peaked function <math id="M131"><msubsup><mrow><mi>F</mi></mrow><mrow><mi>k</mi></mrow><mrow><mi>i</mi></mrow></msubsup><mo stretchy="false">(</mo><msub><mrow><mi>ω</mi></mrow><mrow><mi>k</mi><mi>i</mi></mrow></msub><mo>,</mo><mtext> </mtext><mi>τ</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819162&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819161&type= 14.73200035 4.82600021 appearing in Eq.（5） Eq.（5） becomes a Dirac delta function at the <math id="M132"><mi>τ</mi><mo>→</mo><mi mathvariant="normal">∞</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819164&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819163&type= 10.58333302 2.62466669 limit. As a result, the density of states <math id="M133"><mi>ρ</mi><mo stretchy="false">(</mo><msub><mrow><mi>E</mi></mrow><mrow><mi>k</mi></mrow></msub><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819166&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819165&type= 8.80533314 4.65666676 in Eq.（6） Eq.（6） can be replaced by the delta function <math id="M134"><mi>δ</mi><mo stretchy="false">(</mo><msub><mrow><mi>E</mi></mrow><mrow><mi>k</mi></mrow></msub><mo>-</mo><msub><mrow><mi>E</mi></mrow><mrow><mi>i</mi></mrow></msub><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819180&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819179&type= 16.00200081 4.57200003 for long <math id="M135"><mi>τ</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819170&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819169&type= 1.77800000 2.62466669 [ 21 21 - 22 22 21-22 ] : <math display="block" id="M136"><msub><mrow><mi>w</mi></mrow><mrow><mi>k</mi><mi>i</mi></mrow></msub><mo>≅</mo><mfrac><mrow><mn mathvariant="normal">2</mn><mi mathvariant="normal">π</mi></mrow><mrow><mi>ћ</mi></mrow></mfrac><msup><mrow><mfenced open="|" close="|" separators="|"><mrow><msub><mrow><mi>V</mi></mrow><mrow><mi>k</mi><mi>i</mi></mrow></msub></mrow></mfenced></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mi>δ</mi><mfenced separators="|"><mrow><msub><mrow><mi>E</mi></mrow><mrow><mi>k</mi></mrow></msub><mo>-</mo><msub><mrow><mi>E</mi></mrow><mrow><mi>i</mi></mrow></msub></mrow></mfenced><mo>,</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819172&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819171&type= 40.38600159 7.36600018 <math display="block" id="M137"><mtext> </mtext><msub><mrow><mi>W</mi></mrow><mrow><mi>k</mi><mi>i</mi></mrow></msub><mo>=</mo><mo>∫</mo><msub><mrow><mi>w</mi></mrow><mrow><mi>k</mi><mi>i</mi></mrow></msub><mi>ρ</mi><mfenced separators="|"><mrow><msub><mrow><mi>E</mi></mrow><mrow><mi>k</mi></mrow></msub></mrow></mfenced><mi mathvariant="normal">d</mi><msub><mrow><mi>E</mi></mrow><mrow><mi>k</mi></mrow></msub><mo>.</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819174&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819173&type= 32.25799942 6.43466663 This expression is known as state-to-state Fermi's golden rule. The delta function <math id="M138"><mi>δ</mi><mo stretchy="false">(</mo><msub><mrow><mi>E</mi></mrow><mrow><mi>k</mi></mrow></msub><mo>-</mo><msub><mrow><mi>E</mi></mrow><mrow><mi>i</mi></mrow></msub><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819176&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819175&type= 15.40933418 4.57200003 expresses the condition that at the <math id="M139"><mi>τ</mi><mo>→</mo><mi mathvariant="normal">∞</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819178&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819177&type= 9.90600014 2.62466669 limit, only transitions that satisfy energy conservation can be caused by a secular（time-independent） interaction. The presence of <math id="M140"><mi>δ</mi><mo stretchy="false">(</mo><msub><mrow><mi>E</mi></mrow><mrow><mi>k</mi></mrow></msub><mo>-</mo><msub><mrow><mi>E</mi></mrow><mrow><mi>i</mi></mrow></msub><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819180&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819179&type= 16.00200081 4.57200003 in Eq.（7） Eq.（7） does not violate the energy-time uncertainty principle, because as τ→∞ , the energy dispersion <math id="M141"><mi>δ</mi><mi>E</mi><mo>=</mo><mrow><mrow><mn mathvariant="normal">2</mn><mi mathvariant="normal">π</mi><mi>ћ</mi></mrow><mo>/</mo><mrow><mi>τ</mi></mrow></mrow></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819182&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819181&type= 17.94933319 3.21733332 vanishes. 4　Rates of Energy Transfer Processes 1 As mentioned, the application of Eq.（6） Eq.（6） for an energy transfer process between a donor and acceptor D-A requires the proper representation of the initial and final states. Usually, the subsystems D and A are considered uncoupled or weakly coupled, so that the state of the system can be represented by a （tensor） product of wavefunctions describing the subunits, namely, <math id="M142"><mi>Ψ</mi><mo>≡</mo><msub><mrow><mi>Ψ</mi></mrow><mrow><mi mathvariant="normal">D</mi><mi mathvariant="normal">A</mi></mrow></msub><mo>≅</mo><msub><mrow><mi>Ψ</mi></mrow><mrow><mi mathvariant="normal">D</mi></mrow></msub><msub><mrow><mi>Ψ</mi></mrow><mrow><mi mathvariant="normal">A</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819184&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819183&type= 25.23066711 3.80999994 or <math id="M143"><mfenced open="" close=">" separators="|"><mrow><mfenced open="|" close="" separators="|"><mrow><mi>Ψ</mi></mrow></mfenced></mrow></mfenced><mo>≡</mo><mfenced open="" close=">" separators="|"><mrow><mfenced open="|" close="" separators="|"><mrow><mi>D</mi><mi>A</mi></mrow></mfenced></mrow></mfenced><mo>≅</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819186&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819185&type= 19.81200027 4.65666676 <math id="M144"><mfenced open="" close=">" separators="|"><mrow><mfenced open="|" close="" separators="|"><mrow><mi>D</mi></mrow></mfenced></mrow></mfenced><mfenced open="" close=">" separators="|"><mrow><mfenced open="|" close="" separators="|"><mrow><mi>A</mi></mrow></mfenced></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819188&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819187&type= 10.92199993 4.48733330 . In this representation, the wavefunction <math id="M145"><msub><mrow><mi>Ψ</mi></mrow><mrow><mi mathvariant="normal">A</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819190&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819189&type= 4.40266657 3.80999994 depends on the spatial and spin coordinates of the nuclei and electrons constituting the acceptor and similarly <math id="M146"><msub><mrow><mi>Ψ</mi></mrow><mrow><mi mathvariant="normal">D</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819192&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819191&type= 4.40266657 3.80999994 the donor. Despite this simple notation, <math id="M147"><msub><mrow><mi>Ψ</mi></mrow><mrow><mi mathvariant="normal">D</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819194&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819193&type= 4.40266657 3.80999994 and <math id="M148"><msub><mrow><mi>Ψ</mi></mrow><mrow><mi mathvariant="normal">A</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819196&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819195&type= 4.40266657 3.80999994 represent multielectronic states that may require more than one determinant （multideterminantal） for an adequate description. Employing a single particle or monoelectronic representation may lead to improper descriptions and an unreliable picture of the energy transfer process. The initial state <math id="M149"><mfenced open="" close=">" separators="|"><mrow><mfenced open="|" close="" separators="|"><mrow><mi>i</mi></mrow></mfenced></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819198&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819197&type= 3.72533321 4.48733330 of an energy transfer process can be described by the donor at a higher or excited energy level and the acceptor at a lower or ground level denoted as <math id="M150"><mfenced open="" close=">" separators="|"><mrow><mfenced open="|" close="" separators="|"><mrow><mi>i</mi></mrow></mfenced></mrow></mfenced><mo>=</mo><mfenced open="" close=">" separators="|"><mrow><mfenced open="|" close="" separators="|"><mrow><msup><mrow><mi>D</mi></mrow><mrow><mi>*</mi></mrow></msup><mi>A</mi></mrow></mfenced></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819200&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819199&type= 17.10266685 5.07999992 . This represents the additional energy to be localized at the donor subsystem. Whereas the final state <math id="M151"><mfenced open="" close=">" separators="|"><mrow><mfenced open="|" close="" separators="|"><mrow><mi>f</mi></mrow></mfenced></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819202&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819201&type= 5.33400011 5.33400011 would have this additional energy localized at the acceptor subunit and is represented as <math id="M152"><mfenced open="" close=">" separators="|"><mrow><mfenced open="|" close="" separators="|"><mrow><mi>f</mi></mrow></mfenced></mrow></mfenced><mo>=</mo><mfenced open="" close=">" separators="|"><mrow><mfenced open="|" close="" separators="|"><mrow><mi>D</mi><msup><mrow><mi>A</mi></mrow><mrow><mi>*</mi></mrow></msup></mrow></mfenced></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819204&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819203&type= 18.28800011 5.33400011 . As a result, the energy transfer process <math id="M153"><mfenced open="" close=">" separators="|"><mrow><mfenced open="|" close="" separators="|"><mrow><mi>i</mi></mrow></mfenced></mrow></mfenced><mo>→</mo><mfenced open="" close=">" separators="|"><mrow><mfenced open="|" close="" separators="|"><mrow><mi>f</mi></mrow></mfenced></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819206&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819205&type= 15.15533447 5.33400011 can be pictured as a simultaneous deexcitation of the donor and excitation of the acceptor <math id="M154"><mfenced open="" close=">" separators="|"><mrow><mfenced open="|" close="" separators="|"><mrow><msup><mrow><mi>D</mi></mrow><mrow><mi>*</mi></mrow></msup><mi>A</mi></mrow></mfenced></mrow></mfenced><mo>→</mo><mfenced open="" close=">" separators="|"><mrow><mfenced open="|" close="" separators="|"><mrow><mi>D</mi><msup><mrow><mi>A</mi></mrow><mrow><mi>*</mi></mrow></msup></mrow></mfenced></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819208&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819207&type= 23.53733444 5.07999992 , without the presence of any photon or charge transferred. The ET process can then be represented as <math id="M155"><mfenced open="" close=">" separators="|"><mrow><mfenced open="|" close="" separators="|"><mrow><msup><mrow><mi>D</mi></mrow><mrow><mi>*</mi></mrow></msup></mrow></mfenced></mrow></mfenced><mfenced open="" close=">" separators="|"><mrow><mfenced open="|" close="" separators="|"><mrow><mi>A</mi></mrow></mfenced></mrow></mfenced><mo>→</mo><mfenced open="" close=">" separators="|"><mrow><mfenced open="|" close="" separators="|"><mrow><mi>D</mi></mrow></mfenced></mrow></mfenced><mfenced open="" close=">" separators="|"><mrow><mfenced open="|" close="" separators="|"><mrow><msup><mrow><mi>A</mi></mrow><mrow><mi>*</mi></mrow></msup></mrow></mfenced></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819210&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819209&type= 30.56466866 5.07999992 to emphasize the independence of the subsystems. In addition, this representation allows for selection rules of the ET process. In other words, depending on the perturbation <math id="M156"><mover accent="true"><mi>V</mi><mo>^</mo></mover></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819302&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819301&type= 2.37066650 3.64066648 , a given deexcitation <math id="M157"><mfenced open="" close=">" separators="|"><mrow><mfenced open="|" close="" separators="|"><mrow><msup><mrow><mi>D</mi></mrow><mrow><mi mathvariant="normal">*</mi></mrow></msup></mrow></mfenced></mrow></mfenced><mo>→</mo><mfenced open="" close=">" separators="|"><mrow><mfenced open="|" close="" separators="|"><mrow><mi>D</mi></mrow></mfenced></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819214&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819213&type= 18.03400040 5.07999992 or excitation <math id="M158"><mfenced open="" close=">" separators="|"><mrow><mfenced open="|" close="" separators="|"><mrow><mi>A</mi></mrow></mfenced></mrow></mfenced><mo>→</mo><mfenced open="" close=">" separators="|"><mrow><mfenced open="|" close="" separators="|"><mrow><msup><mrow><mi>A</mi></mrow><mrow><mi>*</mi></mrow></msup></mrow></mfenced></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819216&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819215&type= 17.27199936 5.07999992 may not be allowed, so this energy transfer pathway is forbidden. However, it is important to remember that the selection rules will also depend on the nature of the donor and acceptor subsystems. Because the D-A subunits are considered independent, the perturbation can be promptly and easily pictured as the interaction between these subsystems. So, the perturbation <math id="M159"><mover accent="true"><mi>V</mi><mo>^</mo></mover></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819302&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819301&type= 2.37066650 3.64066648 can be represented by the Coulomb （or electrostatic） interactions between electron densities and nuclear charges of the donor and the acceptor. This perturbation leads to a mechanism known as multipolar [ 23 23 ] or, more restricted, dipolar or Förster [ 24 24 ] . On the other hand, approximating the wavefunction of the system by a （tensor） product causes the electrons of the donor to be distinguishable from the electrons of the acceptor, which violates the indistinguishability and antisymmetry principles of quantum mechanics. So, the exchanges between the electrons of the donor and the acceptor can be regarded as a perturbation of the D-A subunits. This perturbation can be described by an exchange operator and originates a mechanism known as exchange or Dexter [ 25 25 ] . A step further in the development of a workable expression for the energy transfer rate can be taken by realizing that, usually, the time scale of the nuclear motion is much longer than of the electronic motion. As a result, the molecular wavefunction can be approximately factored into a product of electronic and nuclear wavefunctions. This is the well-known Born-Oppenheimer approximation, which has been used since the dawn of quantum mechanics and its successes and limitations are well-established [ 26 26 - 27 27 26-27 ] . So, the wavefunctions describing the initial state with the energy localized at the donor, <math id="M160"><mfenced open="" close=">" separators="|"><mrow><mfenced open="|" close="" separators="|"><mrow><mi>i</mi></mrow></mfenced></mrow></mfenced><mo>=</mo><mfenced open="" close=">" separators="|"><mrow><mfenced open="|" close="" separators="|"><mrow><msup><mrow><mi>D</mi></mrow><mrow><mi>*</mi></mrow></msup><mi>A</mi></mrow></mfenced></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819220&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819219&type= 17.44133186 5.07999992 , and the final state related to the energy localized at the acceptor, <math id="M161"><mfenced open="" close=">" separators="|"><mrow><mfenced open="|" close="" separators="|"><mrow><mi>f</mi></mrow></mfenced></mrow></mfenced><mo>=</mo><mfenced open="" close=">" separators="|"><mrow><mfenced open="|" close="" separators="|"><mrow><mi>D</mi><msup><mrow><mi>A</mi></mrow><mrow><mi>*</mi></mrow></msup></mrow></mfenced></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819222&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819221&type= 18.54199982 5.33400011 , can be approximated as <math display="block" id="M162"><mfenced open="" close=">" separators="|"><mrow><mfenced open="|" close="" separators="|"><mrow><mi>i</mi></mrow></mfenced></mrow></mfenced><mo>≅</mo><msub><mrow><mfenced open="" close=">" separators="|"><mrow><mfenced open="|" close="" separators="|"><mrow><msup><mrow><mi>D</mi></mrow><mrow><mi>*</mi></mrow></msup><mi>A</mi></mrow></mfenced></mrow></mfenced></mrow><mrow><mi mathvariant="normal">e</mi></mrow></msub><msub><mrow><mfenced open="" close=">" separators="|"><mrow><mfenced open="|" close="" separators="|"><mrow><msup><mrow><mi>D</mi></mrow><mrow><mi>*</mi></mrow></msup><mi>A</mi></mrow></mfenced></mrow></mfenced></mrow><mrow><mi mathvariant="normal">n</mi><mi mathvariant="normal">u</mi><mi mathvariant="normal">c</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819224&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819223&type= 31.24200058 5.67266655 ， <math display="block" id="M163"><mfenced open="" close=">" separators="|"><mrow><mfenced open="|" close="" separators="|"><mrow><mi>f</mi></mrow></mfenced></mrow></mfenced><mo>≅</mo><msub><mrow><mfenced open="" close=">" separators="|"><mrow><mfenced open="|" close="" separators="|"><mrow><mi>D</mi><msup><mrow><mi>A</mi></mrow><mrow><mi>*</mi></mrow></msup></mrow></mfenced></mrow></mfenced></mrow><mrow><mi mathvariant="normal">e</mi></mrow></msub><msub><mrow><mfenced open="" close=">" separators="|"><mrow><mfenced open="|" close="" separators="|"><mrow><mi>D</mi><msup><mrow><mi>A</mi></mrow><mrow><mi>*</mi></mrow></msup></mrow></mfenced></mrow></mfenced></mrow><mrow><mi mathvariant="normal">n</mi><mi mathvariant="normal">u</mi><mi mathvariant="normal">c</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819226&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819225&type= 31.91933441 5.75733376 ， where <math id="M164"><msub><mrow><mfenced open="" close=">" separators="|"><mrow><mfenced open="|" close="" separators="|"><mrow><msup><mrow><mi>D</mi></mrow><mrow><mi>*</mi></mrow></msup><mi>A</mi></mrow></mfenced></mrow></mfenced></mrow><mrow><mi mathvariant="normal">e</mi></mrow></msub><mo>≡</mo><mtext> </mtext><msub><mrow><mi>ψ</mi></mrow><mrow><mi mathvariant="normal">e</mi></mrow></msub><mo stretchy="false">(</mo><msup><mrow><mi>D</mi></mrow><mrow><mi>*</mi></mrow></msup><mi>A</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819228&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819227&type= 25.90800095 5.67266655 is the electronic wavefunction of the donor in an excited state and the acceptor in the ground state, while <math id="M165"><msub><mrow><mfenced open="" close=">" separators="|"><mrow><mfenced open="|" close="" separators="|"><mrow><mi>D</mi><msup><mrow><mi>A</mi></mrow><mrow><mi>*</mi></mrow></msup></mrow></mfenced></mrow></mfenced></mrow><mrow><mi mathvariant="normal">e</mi></mrow></msub><mo>≡</mo><msub><mrow><mi>ψ</mi></mrow><mrow><mi mathvariant="normal">e</mi></mrow></msub><mo stretchy="false">(</mo><mi>D</mi><msup><mrow><mi>A</mi></mrow><mrow><mi>*</mi></mrow></msup><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819230&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819229&type= 27.00866890 5.67266655 is the electronic wavefunction of the donor in the electronic ground state and the acceptor in an excited state. These electronic states have corresponding nuclear wavefunctions represented as <math id="M166"><msub><mrow><mfenced open="" close=">" separators="|"><mrow><mfenced open="|" close="" separators="|"><mrow><msup><mrow><mi>D</mi></mrow><mrow><mi>*</mi></mrow></msup><mi>A</mi></mrow></mfenced></mrow></mfenced></mrow><mrow><mi mathvariant="normal">n</mi><mi mathvariant="normal">u</mi><mi mathvariant="normal">c</mi></mrow></msub><mo>≡</mo><msub><mrow><mi>ψ</mi></mrow><mrow><mi mathvariant="normal">n</mi><mi mathvariant="normal">u</mi><mi mathvariant="normal">c</mi></mrow></msub><mo stretchy="false">(</mo><msup><mrow><mi>D</mi></mrow><mrow><mi>*</mi></mrow></msup><mi>A</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819232&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819231&type= 31.91933441 5.67266655 and <math id="M167"><msub><mrow><mfenced open="" close=">" separators="|"><mrow><mfenced open="|" close="" separators="|"><mrow><mi>D</mi><msup><mrow><mi>A</mi></mrow><mrow><mi>*</mi></mrow></msup></mrow></mfenced></mrow></mfenced></mrow><mrow><mi mathvariant="normal">n</mi><mi mathvariant="normal">u</mi><mi mathvariant="normal">c</mi></mrow></msub><mo>≡</mo><msub><mrow><mi>ψ</mi></mrow><mrow><mi mathvariant="normal">n</mi><mi mathvariant="normal">u</mi><mi mathvariant="normal">c</mi></mrow></msub><mo stretchy="false">(</mo><mi>D</mi><msup><mrow><mi>A</mi></mrow><mrow><mi>*</mi></mrow></msup><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819234&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819233&type= 30.90333366 5.67266655 , which are usually described as vibrational wavefunctions. It is important to emphasize that the electronic wavefunction <math id="M168"><msub><mrow><mfenced open="" close=">" separators="|"><mrow><mfenced open="|" close="" separators="|"><mrow><mi>X</mi></mrow></mfenced></mrow></mfenced></mrow><mrow><mi mathvariant="normal">e</mi></mrow></msub><mo>≡</mo><msub><mrow><mi>ψ</mi></mrow><mrow><mi mathvariant="normal">e</mi></mrow></msub><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819236&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819235&type= 20.91266632 5.07999992 is a parametric function of the nuclear positions <math id="M169"><msub><mrow><mi mathvariant="bold-italic">R</mi></mrow><mrow><mi>α</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819274&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819273&type= 3.72533321 3.72533321 , with <math id="M170"><mi>α</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819240&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819239&type= 2.20133328 2.62466669 being the label of each nucleus in the D-A system. As a result, any expectation value concerning the electronic wavefunction will be a function of the set <math id="M171"><mo stretchy="false">{</mo><msub><mrow><mi mathvariant="bold-italic">R</mi></mrow><mrow><mi>α</mi></mrow></msub><mo stretchy="false">}</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819242&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819241&type= 7.02733374 4.40266657 , namely, <math id="M172"><mi>A</mi><mfenced separators="|"><mrow><mo stretchy="false">{</mo><msub><mrow><mi mathvariant="bold-italic">R</mi></mrow><mrow><mi>α</mi></mrow></msub><mo stretchy="false">}</mo></mrow></mfenced><mo>=</mo><mfenced open="<" close=">" separators="|"><mrow><mover accent="true"><mi>A</mi><mo>^</mo></mover></mrow></mfenced><mo>=</mo><mo>∫</mo><msubsup><mrow><mi>ψ</mi></mrow><mrow><mi>e</mi></mrow><mrow><mi>*</mi></mrow></msubsup><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mover accent="true"><mi>A</mi><mo>^</mo></mover><msub><mrow><mi>ψ</mi></mrow><mrow><mi mathvariant="normal">e</mi></mrow></msub><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mi mathvariant="normal">d</mi><mi>τ</mi><mo>≡</mo><msub><mrow><mfenced open="<" close=">" separators="|"><mrow><mi>X</mi><mo stretchy="false">|</mo><mover accent="true"><mi>A</mi><mo>^</mo></mover><mo stretchy="false">|</mo><mi>X</mi></mrow></mfenced></mrow><mrow><mi mathvariant="normal">e</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819244&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819243&type= 71.62799835 6.43466663 .Substitution of these factored representations of the initial and final states in Eq.（6） Eq.（6） yields the following expression for the energy transfer rate <math display="block" id="M173"><msub><mrow><mi>W</mi></mrow><mrow><mi mathvariant="normal">E</mi><mi mathvariant="normal">T</mi></mrow></msub><mo>≅</mo><mfrac><mrow><mn mathvariant="normal">2</mn><mi mathvariant="normal">π</mi></mrow><mrow><mi>ћ</mi></mrow></mfrac><msup><mrow><mfenced open="|" close="|" separators="|"><mrow><msub><mrow><mi>V</mi></mrow><mrow><mi>f</mi><mi>i</mi></mrow></msub></mrow></mfenced></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mi>F</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819246&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819245&type= 25.14599991 7.36600018 ， <math display="block" id="M174"><msub><mrow><mi>V</mi></mrow><mrow><mi>f</mi><mi>i</mi></mrow></msub><mo>=</mo><mo>∫</mo><msubsup><mrow><mi>ψ</mi></mrow><mrow><mi>e</mi></mrow><mrow><mi>*</mi></mrow></msubsup><mo stretchy="false">(</mo><mi>D</mi><msup><mrow><mi>A</mi></mrow><mrow><mi mathvariant="normal">*</mi></mrow></msup><mo stretchy="false">)</mo><mover accent="true"><mi>V</mi><mo>^</mo></mover><msub><mrow><mi>ψ</mi></mrow><mrow><mi mathvariant="normal">e</mi></mrow></msub><mo stretchy="false">(</mo><msup><mrow><mi>D</mi></mrow><mrow><mi>*</mi></mrow></msup><mi>A</mi><mo stretchy="false">)</mo><mi mathvariant="normal">d</mi><mi>τ</mi><mo>≡</mo><msub><mrow><mfenced open="<" close=">" separators="|"><mrow><mi>D</mi><msup><mrow><mi>A</mi></mrow><mrow><mi mathvariant="normal">*</mi></mrow></msup><mo stretchy="false">|</mo><mover accent="true"><mi>V</mi><mo>^</mo></mover><mo stretchy="false">|</mo><msup><mrow><mi>D</mi></mrow><mrow><mi mathvariant="normal">*</mi></mrow></msup><mi>A</mi></mrow></mfenced></mrow><mrow><mi mathvariant="normal">e</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819248&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819247&type= 66.71733856 6.43466663 ， where <math id="M175"><msub><mrow><mi>V</mi></mrow><mrow><mi>f</mi><mi>i</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819272&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819271&type= 3.38666677 3.72533321 is the electronic matrix element due to the time-independent part <math id="M176"><mover accent="true"><mi>V</mi><mo>^</mo></mover></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819302&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819301&type= 2.37066650 3.64066648 of the perturbation operator, while the <math id="M177"><mi>F</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819986&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819985&type= 2.37066650 2.62466669 -factor includes contributions from the overlap of the nuclear wavefunctions, <math id="M178"><mo>∫</mo><msubsup><mrow><mi>ψ</mi></mrow><mrow><mi mathvariant="normal">n</mi><mi mathvariant="normal">u</mi><mi mathvariant="normal">c</mi></mrow><mrow><mi>*</mi></mrow></msubsup><mtext> </mtext><mo stretchy="false">(</mo><mi>D</mi><msup><mrow><mi>A</mi></mrow><mrow><mi>*</mi></mrow></msup><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819256&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819255&type= 16.84866714 6.43466663 · <math id="M179"><msub><mrow><mi>ψ</mi></mrow><mrow><mi mathvariant="normal">n</mi><mi mathvariant="normal">u</mi><mi mathvariant="normal">c</mi></mrow></msub><msup><mrow><mi>D</mi></mrow><mrow><mi>*</mi></mrow></msup><mi>A</mi><mo stretchy="false">)</mo><mi mathvariant="normal">d</mi><mi>τ</mi><mo>≡</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819258&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819257&type= 21.50533485 4.99533367 <math id="M180"><mtext> </mtext></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819260&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819259&type= 0.84666669 0.93133330 <math id="M181"><mo stretchy="false">(</mo><mfenced open="<" close=">" separators="|"><mrow><msup><mrow><mi>D</mi></mrow><mrow><mi mathvariant="normal">*</mi></mrow></msup><mi>A</mi></mrow><mrow><mi mathvariant="normal">D</mi><msup><mrow><mi mathvariant="normal">A</mi></mrow><mrow><mi mathvariant="normal">*</mi></mrow></msup></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819262&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819261&type= 17.35666656 3.89466691 , and from the density of states <math id="M182"><mi>ρ</mi><mo stretchy="false">(</mo><mi>E</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819264&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819263&type= 7.78933382 3.55599999 . This separation into an electronic contribution <math id="M183"><msub><mrow><mi>V</mi></mrow><mrow><mi>f</mi><mi>i</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819272&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819271&type= 3.38666677 3.72533321 and a <math id="M184"><mi>F</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819986&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819985&type= 2.37066650 2.62466669 -factor is possible only if the time-independent perturbation <math id="M185"><mover accent="true"><mi>V</mi><mo>^</mo></mover></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819302&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819301&type= 2.37066650 3.64066648 acts in the electronic （spatial and spin） coordinates, so <math id="M186"><msub><mrow><mi>V</mi></mrow><mrow><mi>f</mi><mi>i</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819272&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819271&type= 3.38666677 3.72533321 is a function of the nuclear positions <math id="M187"><msub><mrow><mi mathvariant="bold-italic">R</mi></mrow><mrow><mi>α</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819274&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819273&type= 3.72533321 3.72533321 , <math id="M188"><msub><mrow><mi>V</mi></mrow><mrow><mi>f</mi><mi>i</mi></mrow></msub><mo>≡</mo><msub><mrow><mi>V</mi></mrow><mrow><mi>f</mi><mi>i</mi></mrow></msub><mo stretchy="false">(</mo><mfenced open="{" close="}" separators="|"><mrow><msub><mrow><mi mathvariant="bold-italic">R</mi></mrow><mrow><mi>α</mi></mrow></msub></mrow></mfenced><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819276&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819275&type= 22.43666649 4.40266657 . Notice that in the wavefunction representation of the matrix element, the notation <math id="M189"><msubsup><mrow><mi>ψ</mi></mrow><mrow><mi mathvariant="normal">e</mi></mrow><mrow><mi>*</mi></mrow></msubsup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819278&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819277&type= 3.55599999 4.74133301 represents the complex conjugate of the function <math id="M190"><msub><mrow><mi>ψ</mi></mrow><mrow><mi mathvariant="normal">e</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819280&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819279&type= 3.47133350 4.48733330 , whereas <math id="M191"><mi>D</mi><msup><mrow><mi>A</mi></mrow><mrow><mi>*</mi></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819282&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819281&type= 5.50333309 2.96333337 or <math id="M192"><msup><mrow><mi>D</mi></mrow><mrow><mi>*</mi></mrow></msup><mi>A</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819284&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819283&type= 6.18066692 2.96333337 represents the acceptor or the donor in an excited electronic state. This confusion does not arise in the ‘bra-ket’ notation. So far, the proposed model, expressed in Eq.（9） Eq.（9） , to describe energy transfer processes is general and can be applied to intermolecular D…A organic and/or inorganic pairs [ 28 28 - 29 29 28-29 ] as well as energy transfer in intramolecular D-A pair such as lanthanide complexes L- Ln （ligand-lanthanide ion） [ 20 20 , 30 30 - 33 33 30-33 ] and lanthanide-lanthanide ion pairs Ln D -Ln A [ 34 34 - 37 37 34-37 ] . For the latter case, it is important to distinguish two situations: Ln D -L- Ln A , when there are bridging ligands between the lanthanide ions; and Ln D - Ln A , when no ligands bridge the ions. The case of Ln D -L- Ln A will not be considered here because the direct energy transfer pathways between the lanthanide ions are very inefficient compared to the ligand-lanthanide energy transfer pathways. So, most likely this case involves nonradiative energy transfer processes between the ligands and the lanthanide ions. Recently, the theoretical modeling of intramolecular ligand-lanthanide energy transfer was reviewed [ 20 20 ] . So, here the focus will be on the direct energy transfer in Ln D - Ln A pairs with no bridging ligands will be described. 5　Energy Transfer Rates Between Lanthanide Ions 1 A quantitative description of the nonradiative energy transfer between lanthanide ions （ Ln - Ln ET） was developed in the early 1970s [ 38 38 ] , after the success of the Judd-Ofelt theory for the 4f-4f transition intensities [ 39 39 - 40 40 39-40 ] and way before the development of the ligand-lanthanide ET model [ 20 20 , 32 32 , 41 41 ] . This sequence of events may be explained by the very similar nature of the donor and acceptor states in the Ln - Ln ET, in contrast to the very dissimilar nature of the states involved in ligand-lanthanide ET. Lanthanide ions have a unique feature related to the shielding of the N electrons in the 4f subshell, 4f N . Because the filled subshells 5s 2 5p 6 are more radially external than the 4f N subshell, the electronic density of 5s 2 5p 6 shields the 4f electrons from the interactions with the surroundings （ e . g ., ligating atoms, ligands, ions, lattice, solvent, etc .） of the lanthanide. As a result, the 4f states are practically those of the free-ion, where the effects of the surroundings can be treated as a small perturbation, if necessary. This feature was explored in the development of the Judd-Ofelt theory, and the successes of this formulation validate this picture of the 4f states. Therefore, donor and acceptor states in the energy transfer processes between 4f levels of lanthanide ions are well-defined and several theoretical tools developed previously could be employed. These Ln - Ln ET processes can be pictured as an excited state within the <math id="M193"><mn mathvariant="normal">4</mn><msup><mrow><mi mathvariant="normal">f</mi></mrow><mrow><msub><mrow><mi>N</mi></mrow><mrow><mi mathvariant="normal">D</mi></mrow></msub></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819286&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819285&type= 5.50333309 2.87866688 configuration of the donor lanthanide being deexcited to a lower level （usually the ground state） and its energy is transferred to the acceptor lanthanide causing an excitation within its <math id="M194"><mn mathvariant="normal">4</mn><msup><mrow><mi mathvariant="normal">f</mi></mrow><mrow><mi>N</mi></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819288&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819287&type= 4.57200003 2.87866688 configuration. Because the 4f donor and acceptor states are well localized on each lanthanide, the donor-acceptor distance is taken as the distance between the lanthanide ions. This feature is not present in a ligand- Ln system, because the states of the ligand are usually delocalized, the donor-acceptor distance is much more difficult to define and determine. Because of the localized nature of the 4f states, the donor and acceptor electronic wavefunctions can be approximately treated as independent or uncoupled, that is, <math display="block" id="M195"><msub><mrow><mfenced open="" close=">" separators="|"><mrow><mfenced open="|" close="" separators="|"><mrow><msup><mrow><mi>D</mi></mrow><mrow><mi>*</mi></mrow></msup><mi>A</mi></mrow></mfenced></mrow></mfenced></mrow><mrow><mi mathvariant="normal">e</mi></mrow></msub><mo>≡</mo><msubsup><mrow><mi>ψ</mi></mrow><mrow><mi mathvariant="normal">e</mi></mrow><mrow><mi>*</mi></mrow></msubsup><mfenced separators="|"><mrow><mi>D</mi><msup><mrow><mi>A</mi></mrow><mrow><mi>*</mi></mrow></msup></mrow></mfenced><mo>≅</mo><mfenced open="" close=">" separators="|"><mrow><mfenced open="|" close="" separators="|"><mrow><msup><mrow><mi>D</mi></mrow><mrow><mi>*</mi></mrow></msup></mrow></mfenced></mrow></mfenced><mfenced open="" close=">" separators="|"><mrow><mfenced open="|" close="" separators="|"><mrow><mi>A</mi></mrow></mfenced></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819290&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819289&type= 44.02666473 5.67266655 ， <math display="block" id="M196"><msub><mrow><mfenced open="" close=">" separators="|"><mrow><mfenced open="|" close="" separators="|"><mrow><mi mathvariant="normal">D</mi><msup><mrow><mi mathvariant="normal">A</mi></mrow><mrow><mi>*</mi></mrow></msup></mrow></mfenced></mrow></mfenced></mrow><mrow><mi mathvariant="normal">e</mi></mrow></msub><mo>≡</mo><msubsup><mrow><mi>ψ</mi></mrow><mrow><mi mathvariant="normal">e</mi></mrow><mrow><mi>*</mi></mrow></msubsup><mo stretchy="false">(</mo><mi mathvariant="normal">D</mi><msup><mrow><mi mathvariant="normal">A</mi></mrow><mrow><mi>*</mi></mrow></msup><mo stretchy="false">)</mo><mo>≅</mo><mfenced open="" close=">" separators="|"><mrow><mfenced open="|" close="" separators="|"><mrow><mi mathvariant="normal">D</mi></mrow></mfenced></mrow></mfenced><mfenced open="" close=">" separators="|"><mrow><mfenced open="|" close="" separators="|"><mrow><msup><mrow><mi mathvariant="normal">A</mi></mrow><mrow><mi>*</mi></mrow></msup></mrow></mfenced></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819292&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819291&type= 44.28066635 5.67266655 ， where <math id="M197"><mfenced open="" close=">" separators="|"><mrow><mfenced open="|" close="" separators="|"><mrow><msup><mrow><mi>D</mi></mrow><mrow><mi>*</mi></mrow></msup></mrow></mfenced></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819294&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819293&type= 6.51933336 5.07999992 and <math id="M198"><mfenced open="" close=">" separators="|"><mrow><mfenced open="|" close="" separators="|"><mrow><mi>D</mi></mrow></mfenced></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819296&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819295&type= 5.33400011 4.48733330 represent the electronic wavefunction of the isolated donor in the excited and ground states, respectively, whereas <math id="M199"><mfenced open="" close=">" separators="|"><mrow><mfenced open="|" close="" separators="|"><mrow><mi>A</mi></mrow></mfenced></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819298&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819297&type= 4.91066647 4.48733330 and <math id="M200"><mfenced open="" close=">" separators="|"><mrow><mfenced open="|" close="" separators="|"><mrow><msup><mrow><mi>A</mi></mrow><mrow><mi mathvariant="normal">*</mi></mrow></msup></mrow></mfenced></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819300&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819299&type= 6.18066692 5.07999992 of the isolated acceptor in the ground and excited states. As a result, the perturbation operator <math id="M201"><mover accent="true"><mi>V</mi><mo>^</mo></mover></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819302&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819301&type= 2.37066650 3.64066648 in Eq.（9） Eq.（9） becomes well defined, namely, it represents the interactions between these two independent D-A subunits. The immediate interaction is the electron repulsions between the electronic densities of <math id="M202"><mfenced open="" close=">" separators="|"><mrow><mfenced open="|" close="" separators="|"><mrow><msup><mrow><mi>D</mi></mrow><mrow><mi>*</mi></mrow></msup></mrow></mfenced></mrow></mfenced><mfenced open="" close=">" separators="|"><mrow><mfenced open="|" close="" separators="|"><mrow><mi>A</mi></mrow></mfenced></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819336&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819335&type= 12.19200039 5.07999992 and <math id="M203"><mfenced open="" close=">" separators="|"><mrow><mfenced open="|" close="" separators="|"><mrow><mi>D</mi></mrow></mfenced></mrow></mfenced><mfenced open="" close=">" separators="|"><mrow><mfenced open="|" close="" separators="|"><mrow><msup><mrow><mi>A</mi></mrow><mrow><mi>*</mi></mrow></msup></mrow></mfenced></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819338&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819337&type= 12.19200039 5.07999992 . This has a classical analogue related to Coulomb interactions between the charge densities and can be represented by the following operator <math display="block" id="M204"><msub><mrow><mover accent="true"><mi>V</mi><mo>^</mo></mover></mrow><mrow><mi mathvariant="normal">C</mi></mrow></msub><mo>=</mo><munder><mo movablelimits="false">∑</mo><mrow><mi>i</mi><mo>∈</mo><mi mathvariant="normal">D</mi></mrow></munder><mrow><munder><mo movablelimits="false">∑</mo><mrow><mi>j</mi><mo>∈</mo><mi mathvariant="normal">A</mi></mrow></munder><mrow><mfrac><mrow><msup><mrow><mi>e</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup></mrow><mrow><mfenced open="|" close="|" separators="|"><mrow><msub><mrow><mi mathvariant="bold-italic">r</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>-</mo><msub><mrow><mi mathvariant="bold-italic">r</mi></mrow><mrow><mi>j</mi></mrow></msub></mrow></mfenced></mrow></mfrac></mrow></mrow><mo>≡</mo><munder><mo movablelimits="false">∑</mo><mrow><mi>i</mi><mo>∈</mo><mi mathvariant="normal">D</mi></mrow></munder><mrow><munder><mo movablelimits="false">∑</mo><mrow><mi>j</mi><mo>∈</mo><mi mathvariant="normal">A</mi></mrow></munder><mrow><mfrac><mrow><msup><mrow><mi>e</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup></mrow><mrow><msub><mrow><mi>r</mi></mrow><mrow><mi>i</mi><mi>j</mi></mrow></msub></mrow></mfrac></mrow></mrow></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819308&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819307&type= 45.21200180 10.15999985 ， where <math id="M205"><msub><mrow><mi mathvariant="bold-italic">r</mi></mrow><mrow><mi>i</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819310&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819309&type= 2.11666679 3.72533321 and <math id="M206"><msub><mrow><mi mathvariant="bold-italic">r</mi></mrow><mrow><mi>j</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819312&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819311&type= 1.94733346 3.72533321 are the position vectors of electrons <math id="M207"><mi>i</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819727&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819726&type= 1.10066664 2.62466669 （belonging to the donor） and <math id="M208"><mi>j</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819330&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819329&type= 1.52400005 3.47133350 （belonging to the acceptor）, respectively. Another more subtle interaction comes from the principles of quantum mechanics, which has no classical analogue. In the approximation expressed by Eq.（10） Eq.（10） , it is implicitly assumed that the donor has a fixed number of electrons, e . g ., <math id="M209"><msub><mrow><mi>N</mi></mrow><mrow><mi mathvariant="normal">D</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819318&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819317&type= 3.97933316 3.72533321 , as well as the acceptor has <math id="M210"><msub><mrow><mi>N</mi></mrow><mrow><mi mathvariant="normal">A</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819320&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819319&type= 3.97933316 3.72533321 -electrons. However, when these subsystems interact with each other, the electrons of the donor must become indistinguishable from the electrons of the acceptor, as a principle of quantum mechanics. This can be accomplished by an exchange operator that permutes the indices （or labels） of each electron of the donor with each electron of the acceptor, which can be represented by the following operator <math display="block" id="M211"><msub><mrow><mover accent="true"><mi>V</mi><mo>^</mo></mover></mrow><mrow><mi mathvariant="normal">e</mi><mi mathvariant="normal">x</mi></mrow></msub><mo>=</mo><munder><mo movablelimits="false">∑</mo><mrow><mi>i</mi><mo>∈</mo><mi mathvariant="normal">D</mi></mrow></munder><mrow><munder><mo movablelimits="false">∑</mo><mrow><mi>j</mi><mo>∈</mo><mi mathvariant="normal">A</mi></mrow></munder><mrow><mfrac><mrow><msup><mrow><mi>e</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup></mrow><mrow><msub><mrow><mi>r</mi></mrow><mrow><mi>i</mi><mi>j</mi></mrow></msub></mrow></mfrac><msub><mrow><mi>P</mi></mrow><mrow><mi>i</mi><mi>j</mi></mrow></msub></mrow></mrow><mo>,</mo><mtext> </mtext><msub><mrow><mi>P</mi></mrow><mrow><mi>i</mi><mi>j</mi></mrow></msub><mo>=</mo><mfrac><mrow><mn mathvariant="normal">1</mn></mrow><mrow><mn mathvariant="normal">2</mn></mrow></mfrac><mo>+</mo><msub><mrow><mover accent="true"><mi mathvariant="bold-italic">s</mi><mo>^</mo></mover></mrow><mrow><mi>i</mi></mrow></msub><mo>·</mo><msub><mrow><mover accent="true"><mi mathvariant="bold-italic">s</mi><mo>^</mo></mover></mrow><mrow><mi>j</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819322&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819321&type= 48.68333054 8.80533314 ， where <math id="M212"><msub><mrow><mover accent="true"><mi mathvariant="bold-italic">s</mi><mo>^</mo></mover></mrow><mrow><mi>i</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819324&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819323&type= 2.11666679 3.89466691 and <math id="M213"><msub><mrow><mover accent="true"><mi mathvariant="bold-italic">s</mi><mo>^</mo></mover></mrow><mrow><mi>j</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819326&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819325&type= 2.03200006 3.89466691 are the spin operators of electrons <math id="M214"><mi>i</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819727&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819726&type= 1.10066664 2.62466669 and <math id="M215"><mi>j</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819330&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819329&type= 1.52400005 3.47133350 . It is important to realize that the wavefunctions of the isolated donor and acceptor are normalized and antisymmetrized, so they individually satisfy the principles of quantum mechanics. The electronic wavefunctions of 4f states of an isolated lanthanide ion are accurately described by the total electronic angular momentum quantum number <math id="M216"><mi>J</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819735&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819734&type= 1.77800000 2.96333337 and additional quantum numbers <math id="M217"><mi>ψ</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819334&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819333&type= 2.37066650 3.38666677 . Thus, the initial <math id="M218"><mfenced open="" close=">" separators="|"><mrow><mfenced open="|" close="" separators="|"><mrow><msup><mrow><mi>D</mi></mrow><mrow><mi>*</mi></mrow></msup></mrow></mfenced></mrow></mfenced><mfenced open="" close=">" separators="|"><mrow><mfenced open="|" close="" separators="|"><mrow><mi>A</mi></mrow></mfenced></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819336&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819335&type= 12.19200039 5.07999992 and final <math id="M219"><mfenced open="" close=">" separators="|"><mrow><mfenced open="|" close="" separators="|"><mrow><mi>D</mi></mrow></mfenced></mrow></mfenced><mfenced open="" close=">" separators="|"><mrow><mfenced open="|" close="" separators="|"><mrow><msup><mrow><mi>A</mi></mrow><mrow><mi>*</mi></mrow></msup></mrow></mfenced></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819338&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819337&type= 12.19200039 5.07999992 states, Eq. （10） Eq. （10） , in the ET process are represented as <math id="M220"><mfenced open="" close=">" separators="|"><mrow><mfenced open="|" close="" separators="|"><mrow><msubsup><mrow><mi>ψ</mi></mrow><mrow><mi mathvariant="normal">D</mi></mrow><mrow><mi>*</mi></mrow></msubsup><msubsup><mrow><mi>J</mi></mrow><mrow><mi mathvariant="normal">D</mi></mrow><mrow><mi>*</mi></mrow></msubsup></mrow></mfenced></mrow></mfenced><mfenced open="" close=">" separators="|"><mrow><mfenced open="|" close="" separators="|"><mrow><msub><mrow><mi>ψ</mi></mrow><mrow><mi mathvariant="normal">A</mi></mrow></msub><msub><mrow><mi>J</mi></mrow><mrow><mi mathvariant="normal">A</mi></mrow></msub></mrow></mfenced></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819340&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819339&type= 21.42066574 5.24933338 and <math id="M221"><mfenced open="" close=">" separators="|"><mrow><mfenced open="|" close="" separators="|"><mrow><msub><mrow><mi>ψ</mi></mrow><mrow><mi mathvariant="normal">D</mi></mrow></msub><msub><mrow><mi>J</mi></mrow><mrow><mi mathvariant="normal">D</mi></mrow></msub></mrow></mfenced></mrow></mfenced><mfenced open="" close=">" separators="|"><mrow><mfenced open="|" close="" separators="|"><mrow><msubsup><mrow><mi>ψ</mi></mrow><mrow><mi mathvariant="normal">A</mi></mrow><mrow><mi>*</mi></mrow></msubsup><msubsup><mrow><mi>J</mi></mrow><mrow><mi mathvariant="normal">A</mi></mrow><mrow><mi>*</mi></mrow></msubsup></mrow></mfenced></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819342&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819341&type= 21.42066574 5.24933338 . Notice that the projection <math id="M222"><msub><mrow><mi>M</mi></mrow><mrow><mi>J</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819344&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819343&type= 3.97933316 3.72533321 of the angular momentum <math id="M223"><mi>J</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819735&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819734&type= 1.77800000 2.96333337 is also an appropriate quantum number. In addition, it is possible to represent these wavefunctions as a linear combination of the wavefunctions represented in the total spin <math id="M224"><mi>S</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819348&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819347&type= 1.86266661 2.62466669 and orbital <math id="M225"><mi>L</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819350&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819349&type= 1.94733346 2.62466669 angular momenta （and their projections, <math id="M226"><msub><mrow><mi>M</mi></mrow><mrow><mi>S</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819352&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819351&type= 4.14866638 3.72533321 and <math id="M227"><msub><mrow><mi>M</mi></mrow><mrow><mi>L</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819354&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819353&type= 4.14866638 3.72533321 ） basis as the so-called intermediate coupling scheme. This transformation is quite convenient because it allows the choice of a diagonal matrix representation of the operator of interest. Once the wavefunctions describing the initial and final states have been properly represented, the matrix element involving the perturbation operator <math id="M228"><mover accent="true"><mi>V</mi><mo>^</mo></mover><mo>=</mo><msub><mrow><mover accent="true"><mi>V</mi><mo>^</mo></mover></mrow><mrow><mi mathvariant="normal">C</mi></mrow></msub><mo>+</mo><msub><mrow><mover accent="true"><mi>V</mi><mo>^</mo></mover></mrow><mrow><mi mathvariant="normal">e</mi><mi mathvariant="normal">x</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819356&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819355&type= 18.54199982 4.74133301 can be determined. Notice that this operator involves the two-electron interaction <math id="M229"><mrow><mrow><msup><mrow><mi>e</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup></mrow><mo>/</mo><mrow><msub><mrow><mi>r</mi></mrow><mrow><mn mathvariant="normal">12</mn></mrow></msub></mrow></mrow></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819416&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819415&type= 7.28133297 4.06400013 , which can be expanded in terms of the spherical harmonics or equivalently in terms of Racah's irreducible spherical tensor operators [ 23 23 , 42 42 ] . This allows the factorization of the matrix element into radial and angular parts, where this latter part can be calculated analytically. As a result, the following equations for the energy transfer rates between lanthanide ions can be obtained [ 16 16 , 34 34 - 37 37 34-37 , 43 43 ] and classified the dipole⁃dipole, <math id="M230"><msub><mrow><mi>W</mi></mrow><mrow><mi mathvariant="normal">d</mi><mi mathvariant="normal">⁃</mi><mi mathvariant="normal">d</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819360&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819359&type= 5.67266655 3.72533321 <math display="block" id="M231"><msub><mrow><mi>W</mi></mrow><mrow><mi mathvariant="normal">d</mi><mi mathvariant="normal">⁃</mi><mi mathvariant="normal">d</mi></mrow></msub><mo>=</mo><mfrac><mrow><mn mathvariant="normal">4</mn><mi mathvariant="normal">π</mi></mrow><mrow><mn mathvariant="normal">3</mn><mi>ћ</mi></mrow></mfrac><mfrac><mrow><msubsup><mrow><mi>S</mi></mrow><mrow><mi mathvariant="normal">d</mi></mrow><mrow><mi mathvariant="normal">D</mi></mrow></msubsup><msubsup><mrow><mi>S</mi></mrow><mrow><mi mathvariant="normal">d</mi></mrow><mrow><mi mathvariant="normal">A</mi></mrow></msubsup></mrow><mrow><msubsup><mrow><mi>R</mi></mrow><mrow><mi mathvariant="normal">D</mi><mi mathvariant="normal">A</mi></mrow><mrow><mn mathvariant="normal">6</mn></mrow></msubsup></mrow></mfrac><mi>F</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819362&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819361&type= 26.75466537 8.55133343 ， the dipole⁃quadrupole and quadrupole⁃dipole, <math display="block" id="M232"><msub><mrow><mi>W</mi></mrow><mrow><mi mathvariant="normal">d</mi><mi mathvariant="normal">q</mi><mi mathvariant="normal">⁃</mi><mi mathvariant="normal">q</mi><mi mathvariant="normal">d</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819364&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819363&type= 7.61999989 3.72533321 , <math display="block" id="M233"><msub><mrow><mi>W</mi></mrow><mrow><mi mathvariant="normal">d</mi><mi mathvariant="normal">q</mi><mi mathvariant="normal">⁃</mi><mi mathvariant="normal">q</mi><mi mathvariant="normal">d</mi></mrow></msub><mo>=</mo><mfrac><mrow><mi mathvariant="normal">π</mi></mrow><mrow><mi>ћ</mi></mrow></mfrac><mfrac><mrow><mo stretchy="false">(</mo><msubsup><mrow><mi>S</mi></mrow><mrow><mi mathvariant="normal">d</mi></mrow><mrow><mi mathvariant="normal">D</mi></mrow></msubsup><msubsup><mrow><mi>S</mi></mrow><mrow><mi mathvariant="normal">q</mi></mrow><mrow><mi mathvariant="normal">A</mi></mrow></msubsup><mo>+</mo><msubsup><mrow><mi>S</mi></mrow><mrow><mi mathvariant="normal">q</mi></mrow><mrow><mi mathvariant="normal">D</mi></mrow></msubsup><msubsup><mrow><mi>S</mi></mrow><mrow><mi mathvariant="normal">d</mi></mrow><mrow><mi mathvariant="normal">A</mi></mrow></msubsup><mo stretchy="false">)</mo></mrow><mrow><msubsup><mrow><mi>R</mi></mrow><mrow><mi mathvariant="normal">D</mi><mi mathvariant="normal">A</mi></mrow><mrow><mn mathvariant="normal">8</mn></mrow></msubsup></mrow></mfrac><mi>F</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819366&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819365&type= 41.31733322 8.80533314 ， the quadrupole⁃quadrupole, <math id="M234"><msub><mrow><mi>W</mi></mrow><mrow><mi mathvariant="normal">q</mi><mi mathvariant="normal">⁃</mi><mi mathvariant="normal">q</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819802&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819801&type= 5.67266655 3.72533321 , <math display="block" id="M235"><msub><mrow><mi>W</mi></mrow><mrow><mi mathvariant="normal">q</mi><mi mathvariant="normal">⁃</mi><mi mathvariant="normal">q</mi></mrow></msub><mo>=</mo><mfrac><mrow><mn mathvariant="normal">28</mn><mi mathvariant="normal">π</mi></mrow><mrow><mn mathvariant="normal">5</mn><mi>ћ</mi></mrow></mfrac><mfrac><mrow><msubsup><mrow><mi>S</mi></mrow><mrow><mi mathvariant="normal">q</mi></mrow><mrow><mi mathvariant="normal">D</mi></mrow></msubsup><msubsup><mrow><mi>S</mi></mrow><mrow><mi mathvariant="normal">q</mi></mrow><mrow><mi mathvariant="normal">A</mi></mrow></msubsup></mrow><mrow><msubsup><mrow><mi>R</mi></mrow><mrow><mi mathvariant="normal">D</mi><mi mathvariant="normal">A</mi></mrow><mrow><mn mathvariant="normal">10</mn></mrow></msubsup></mrow></mfrac><mi>F</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819370&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819369&type= 28.61733437 8.80533314 ， the magnetic dipole⁃magnetic dipole <math id="M236"><msub><mrow><mi>W</mi></mrow><mrow><mi mathvariant="normal">m</mi><mi mathvariant="normal">d</mi><mi mathvariant="normal">⁃</mi><mi mathvariant="normal">m</mi><mi mathvariant="normal">d</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819372&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819371&type= 8.55133343 3.72533321 , <math display="block" id="M237"><msub><mrow><mi>W</mi></mrow><mrow><mi mathvariant="normal">m</mi><mi mathvariant="normal">d</mi><mi mathvariant="normal">⁃</mi><mi mathvariant="normal">m</mi><mi mathvariant="normal">d</mi></mrow></msub><mo>=</mo><mfrac><mrow><mn mathvariant="normal">4</mn><mi mathvariant="normal">π</mi></mrow><mrow><mn mathvariant="normal">3</mn><mi>ћ</mi></mrow></mfrac><mfrac><mrow><msubsup><mrow><mi>S</mi></mrow><mrow><mi mathvariant="normal">m</mi><mi mathvariant="normal">d</mi></mrow><mrow><mi mathvariant="normal">D</mi></mrow></msubsup><msubsup><mrow><mi>S</mi></mrow><mrow><mi mathvariant="normal">m</mi><mi mathvariant="normal">d</mi></mrow><mrow><mi mathvariant="normal">A</mi></mrow></msubsup></mrow><mrow><msubsup><mrow><mi>R</mi></mrow><mrow><mi mathvariant="normal">D</mi><mi mathvariant="normal">A</mi></mrow><mrow><mn mathvariant="normal">6</mn></mrow></msubsup></mrow></mfrac><mi>F</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819374&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819373&type= 31.75000000 8.55133343 ， and the exchange, <math id="M238"><msub><mrow><mi>W</mi></mrow><mrow><mi mathvariant="normal">e</mi><mi mathvariant="normal">x</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819749&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819748&type= 4.82600021 3.72533321 , mechanisms <math display="block" id="M239"><msub><mrow><mi>W</mi></mrow><mrow><mi mathvariant="normal">e</mi><mi mathvariant="normal">x</mi></mrow></msub><mo>=</mo><mfrac><mrow><mn mathvariant="normal">2</mn><mi mathvariant="normal">π</mi></mrow><mrow><mi>ћ</mi></mrow></mfrac><msup><mrow><mfenced open="[" close="]" separators="|"><mrow><mfenced separators="|"><mrow><mfrac><mrow><msup><mrow><mi>e</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup></mrow><mrow><msub><mrow><mi>R</mi></mrow><mrow><mi mathvariant="normal">D</mi><mi mathvariant="normal">A</mi></mrow></msub></mrow></mfrac></mrow></mfenced><msubsup><mrow><mi>ρ</mi></mrow><mrow><mi mathvariant="normal">f</mi><mi mathvariant="normal">f</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msubsup></mrow></mfenced></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mi>F</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819378&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819377&type= 35.22133255 10.41399956 ， where <math id="M240"><msub><mrow><mi>S</mi></mrow><mrow><mi mathvariant="normal">d</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819848&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819847&type= 3.04800010 3.72533321 （in <math id="M241"><mi mathvariant="normal">e</mi><mi mathvariant="normal">r</mi><mi mathvariant="normal">g</mi><mo>·</mo><mi mathvariant="normal">c</mi><msup><mrow><mi mathvariant="normal">m</mi></mrow><mrow><mn mathvariant="normal">3</mn></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819382&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819381&type= 11.76866722 3.72533321 ） is the modified forced electric dipole strength <math display="block" id="M242"><msubsup><mrow><mi>S</mi></mrow><mrow><mi mathvariant="normal">d</mi></mrow><mrow><mi>x</mi></mrow></msubsup><mo>=</mo><mfrac><mrow><msup><mrow><mi>e</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><msup><mrow><mfenced separators="|"><mrow><mn mathvariant="normal">1</mn><mo>-</mo><msubsup><mrow><mi>σ</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow><mrow><mi>x</mi></mrow></msubsup></mrow></mfenced></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup></mrow><mrow><mn mathvariant="normal">2</mn><msub><mrow><mi>J</mi></mrow><mrow><mi>x</mi></mrow></msub><mo>+</mo><mn mathvariant="normal">1</mn></mrow></mfrac><munder><mo movablelimits="false">∑</mo><mrow><mi>K</mi><mo>=</mo><mn mathvariant="normal">2,4</mn><mo>,</mo><mn mathvariant="normal">6</mn></mrow></munder><mrow><msubsup><mrow><mi>Ω</mi></mrow><mrow><mi>K</mi></mrow><mrow><mi>x</mi></mrow></msubsup><mfenced open="<" close="" separators="|"><mrow><msubsup><mrow><mi>ψ</mi></mrow><mrow><mi>x</mi></mrow><mrow><mi mathvariant="normal">*</mi></mrow></msubsup><msubsup><mrow><mi>J</mi></mrow><mrow><mi>x</mi></mrow><mrow><mi mathvariant="normal">*</mi></mrow></msubsup></mrow></mfenced><mfenced open="‖" close="‖" separators="|"><mrow><msup><mrow><mi>U</mi></mrow><mrow><mfenced separators="|"><mrow><mi>K</mi></mrow></mfenced></mrow></msup></mrow></mfenced><msup><mrow><mfenced open="" close=">" separators="|"><mrow><msub><mrow><mi>ψ</mi></mrow><mrow><mi>x</mi></mrow></msub><msub><mrow><mi>J</mi></mrow><mrow><mi>x</mi></mrow></msub></mrow></mfenced></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup></mrow><mtext> </mtext></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819384&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819383&type= 68.83399963 9.90600014 ， while <math id="M243"><msub><mrow><mi>S</mi></mrow><mrow><mi mathvariant="normal">q</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819386&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819385&type= 3.04800010 3.72533321 （in <math id="M244"><mi mathvariant="normal">e</mi><mi mathvariant="normal">r</mi><mi mathvariant="normal">g</mi><mo>·</mo><mi mathvariant="normal">c</mi><msup><mrow><mi mathvariant="normal">m</mi></mrow><mrow><mn mathvariant="normal">5</mn></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819388&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819387&type= 11.76866722 3.72533321 ） is the modified quadrupole strength <math display="block" id="M245"><mtext>                 </mtext><msubsup><mrow><mi>S</mi></mrow><mrow><mi mathvariant="normal">q</mi></mrow><mrow><mi>x</mi></mrow></msubsup><mo>=</mo><mfrac><mrow><msup><mrow><mi>e</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mo stretchy="false">(</mo><mn mathvariant="normal">1</mn><mo>-</mo><msubsup><mrow><mi>σ</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow><mrow><mi>x</mi></mrow></msubsup><msup><mrow><mo stretchy="false">)</mo></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup></mrow><mrow><mn mathvariant="normal">2</mn><msub><mrow><mi>J</mi></mrow><mrow><mi mathvariant="normal">x</mi></mrow></msub><mo>+</mo><mn mathvariant="normal">1</mn></mrow></mfrac><mfenced open="<" close="" separators="|"><mrow><mi>f</mi></mrow></mfenced><mfenced open="‖" close="‖" separators="|"><mrow><msup><mrow><mi>C</mi></mrow><mrow><mfenced separators="|"><mrow><mn mathvariant="normal">2</mn></mrow></mfenced></mrow></msup></mrow></mfenced><msup><mrow><mfenced open="" close=">" separators="|"><mrow><mi>f</mi></mrow></mfenced></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mo>×</mo><mtext>               </mtext><msubsup><mrow><mfenced open="<" close=">" separators="|"><mrow><msup><mrow><mi>r</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup></mrow></mfenced></mrow><mrow><mi>x</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msubsup><mfenced open="<" close="" separators="|"><mrow><msubsup><mrow><mi>ψ</mi></mrow><mrow><mi>x</mi></mrow><mrow><mi mathvariant="normal">*</mi></mrow></msubsup><msubsup><mrow><mi>J</mi></mrow><mrow><mi>x</mi></mrow><mrow><mi mathvariant="normal">*</mi></mrow></msubsup></mrow></mfenced><mfenced open="‖" close="‖" separators="|"><mrow><msup><mrow><mi>U</mi></mrow><mrow><mfenced separators="|"><mrow><mn mathvariant="normal">2</mn></mrow></mfenced></mrow></msup></mrow></mfenced><msup><mrow><mfenced open="" close=">" separators="|"><mrow><msub><mrow><mi>ψ</mi></mrow><mrow><mi>x</mi></mrow></msub><msub><mrow><mi>J</mi></mrow><mrow><mi>x</mi></mrow></msub></mrow></mfenced></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819390&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819389&type= 109.89733887 8.29733276 ， whereas <math id="M246"><msub><mrow><mi>S</mi></mrow><mrow><mi mathvariant="normal">m</mi><mi mathvariant="normal">d</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819392&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819391&type= 4.48733330 3.72533321 （in <math id="M247"><mi mathvariant="normal">e</mi><mi mathvariant="normal">r</mi><mi mathvariant="normal">g</mi><mo>·</mo><mi mathvariant="normal">c</mi><msup><mrow><mi mathvariant="normal">m</mi></mrow><mrow><mn mathvariant="normal">3</mn></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819394&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819393&type= 11.76866722 3.72533321 ） is the magnetic dipole strength analogue <math display="block" id="M248"><mtable columnalign="left"><mtr><mtd><mtext>          </mtext><msubsup><mrow><mi>S</mi></mrow><mrow><mi mathvariant="normal">m</mi><mi mathvariant="normal">d</mi></mrow><mrow><mi mathvariant="normal">x</mi></mrow></msubsup><mo>=</mo><mfrac><mrow><msubsup><mrow><mi>μ</mi></mrow><mrow><mi mathvariant="normal">B</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msubsup><mo stretchy="false">(</mo><mn mathvariant="normal">1</mn><mo>-</mo><msubsup><mrow><mi>σ</mi></mrow><mrow><mi mathvariant="normal">m</mi><mi mathvariant="normal">d</mi></mrow><mrow><mi>x</mi></mrow></msubsup><msup><mrow><mo stretchy="false">)</mo></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup></mrow><mrow><mo stretchy="false">(</mo><mn mathvariant="normal">2</mn><msub><mrow><mi>J</mi></mrow><mrow><mi mathvariant="normal">x</mi></mrow></msub><mo>+</mo><mn mathvariant="normal">1</mn><mo stretchy="false">)</mo></mrow></mfrac><mfenced open="<" close="" separators="|"><mrow><msubsup><mrow><mi>ψ</mi></mrow><mrow><mi>x</mi></mrow><mrow><mi mathvariant="normal">*</mi></mrow></msubsup><msubsup><mrow><mi>J</mi></mrow><mrow><mi>x</mi></mrow><mrow><mi mathvariant="normal">*</mi></mrow></msubsup></mrow></mfenced><mfenced open="‖" close="‖" separators="|"><mrow><mover accent="true"><mi>M</mi><mo>^</mo></mover></mrow></mfenced><msup><mrow><mfenced open="" close=">" separators="|"><mrow><msub><mrow><mi>ψ</mi></mrow><mrow><mi>x</mi></mrow></msub><msub><mrow><mi>J</mi></mrow><mrow><mi>x</mi></mrow></msub></mrow></mfenced></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mo>,</mo><mtext> </mtext></mtd></mtr><mtr><mtd><mtext>                        </mtext><mover accent="true"><mi>M</mi><mo>^</mo></mover><mo>≡</mo><mover accent="true"><mi>L</mi><mo>^</mo></mover><mo>+</mo><msub><mrow><mi>g</mi></mrow><mrow><mi>S</mi></mrow></msub><mover accent="true"><mi>S</mi><mo>^</mo></mover></mtd></mtr></mtable></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819396&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819395&type= 66.71733856 15.83266735 ， with <math id="M249"><mi>x</mi><mo>=</mo><mi mathvariant="normal">A</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819398&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819397&type= 8.72066593 2.62466669 for the acceptor and <math id="M250"><mi>x</mi><mo>=</mo><mi mathvariant="normal">D</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819400&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819399&type= 8.72066593 2.62466669 for the donor, <math id="M251"><msub><mrow><mi>J</mi></mrow><mrow><mi>x</mi></mrow></msub><mo>=</mo><msubsup><mrow><mi>J</mi></mrow><mrow><mi mathvariant="normal">D</mi></mrow><mrow><mi>*</mi></mrow></msubsup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819402&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819401&type= 10.41399956 4.40266657 being the total angular momentum quantum number of the excited state of the donor or <math id="M252"><msub><mrow><mi>J</mi></mrow><mrow><mi>x</mi></mrow></msub><mo>=</mo><msub><mrow><mi>J</mi></mrow><mrow><mi mathvariant="normal">A</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819404&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819403&type= 10.41399956 4.14866638 of the ground state of the acceptor, and <math id="M253"><msub><mrow><mi>g</mi></mrow><mrow><mi>S</mi></mrow></msub><mo>=</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819406&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819405&type= 5.92666674 4.57200003 2.002 319 3 being the electron spin g -factor [ 44 44 ] . See the Supporting Information for dimensional analysis and unit conversion. The total energy transfer rate will be a summation of each contribution in Eqs.（ 13 13 ）-（ 17 17 ）. Usually, there is one contribution that dominates this sum, however, there are exceptions, and the relative contributions change strongly with the donor-acceptor distance, <math id="M254"><msub><mrow><mi>R</mi></mrow><mrow><mi mathvariant="normal">D</mi><mi mathvariant="normal">A</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819940&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819939&type= 5.16466665 3.72533321 . It is important to emphasize that <math id="M255"><msub><mrow><mi>S</mi></mrow><mrow><mi mathvariant="normal">d</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819848&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819847&type= 3.04800010 3.72533321 is denoted as forced electric dipole strength analogue because only the forced electric dipole （FED） mechanism contributes to the calculation of <math id="M256"><msub><mrow><mi>Ω</mi></mrow><mrow><mi>K</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819852&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819851&type= 4.06400013 3.89466691 . This is due to the contribution of the opposite parity configuration mixing by the odd components of the ligand field to the intensity parameters <math id="M257"><msub><mrow><mi>Ω</mi></mrow><mrow><mi>K</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819414&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819413&type= 4.06400013 3.89466691 , like in the Judd-Ofelt theory [ 38 38 ] . These equations are the same as those published in the revisited versions of the energy transfer rates between lanthanide ions [ 34 34 , 36 36 - 37 37 36-37 ] . However, those involving multipole and magnetic dipole mechanisms were rearranged and separated into more symmetrical and easily interpretable forms. In addition, these separations allow a stepwise calculation of the contributing terms, which can assist the analysis of the transfer rates and determine if there are dominant contributions from either donor or acceptor. The energy transfer rates are expressed in terms of well-defined and calculable quantities. It must be emphasized that the choice of the two-electron interaction as <math id="M258"><mrow><mrow><msup><mrow><mi>e</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup></mrow><mo>/</mo><mrow><msub><mrow><mi>r</mi></mrow><mrow><mn mathvariant="normal">12</mn></mrow></msub></mrow></mrow></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819416&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819415&type= 7.28133297 4.06400013 , implies that the CGS system of units is employed. The quantities in these expressions depend on physical constants and other properties independent of the nature of the lanthanide ion: ⋅ <math id="M259"><mi mathvariant="normal">ℏ</mi><mo>=</mo><mrow><mrow><mi>h</mi></mrow><mo>/</mo><mrow><mo stretchy="false">(</mo><mn mathvariant="normal">2</mn><mi mathvariant="normal">π</mi><mo stretchy="false">)</mo></mrow></mrow></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819418&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819417&type= 16.93333435 3.89466691 is the reduced Planck constant （see the Supporting Information for the values of the physical constants and unit conversion）, ⋅ <math id="M260"><mi>e</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819420&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819419&type= 1.43933344 2.62466669 is the elementary charge in <math id="M261"><mi mathvariant="normal">e</mi><mi mathvariant="normal">s</mi><mi mathvariant="normal">u</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819422&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819421&type= 4.99533367 2.70933342 , ⋅ <math id="M262"><msub><mrow><mi>μ</mi></mrow><mrow><mi mathvariant="normal">B</mi></mrow></msub><mo>=</mo><mrow><mrow><mi>e</mi><mi>ћ</mi></mrow><mo>/</mo><mrow><mo stretchy="false">(</mo><mn mathvariant="normal">2</mn><msub><mrow><mi>m</mi></mrow><mrow><mi mathvariant="normal">e</mi></mrow></msub><mi>c</mi></mrow></mrow><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819424&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819423&type= 23.79133224 3.97933316 is the Bohr magneton, ⋅ <math id="M263"><mfenced open="<" close="" separators="|"><mrow><mi>f</mi></mrow></mfenced><mfenced open="‖" close="‖" separators="|"><mrow><msup><mrow><mi>C</mi></mrow><mrow><mfenced separators="|"><mrow><mn mathvariant="normal">2</mn></mrow></mfenced></mrow></msup></mrow></mfenced><mfenced open="" close=">" separators="|"><mrow><mi>f</mi></mrow></mfenced><mo>=</mo><mo>-</mo><mn mathvariant="normal">1.366</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819426&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819425&type= 33.61266708 4.65666676 is the one-electron doubly reduced spherical tensor matrix element （dimensionless） [ 23 23 ] , on intrinsic quantities of the lanthanide ion : ⋅ <math id="M264"><msubsup><mrow><mi>J</mi></mrow><mrow><mi mathvariant="normal">D</mi></mrow><mrow><mi>*</mi></mrow></msubsup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819428&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819427&type= 3.13266683 4.40266657 is the total angular momentum quantum number of the excited state of the donor and <math id="M265"><msub><mrow><mi>J</mi></mrow><mrow><mi mathvariant="normal">A</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819430&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819429&type= 3.13266683 4.06400013 of the ground state of the acceptor, ⋅ <math id="M266"><mfenced open="<" close="" separators="|"><mrow><msup><mrow><mi>ψ</mi></mrow><mrow><mi>*</mi></mrow></msup><msup><mrow><mi>J</mi></mrow><mrow><mi>*</mi></mrow></msup></mrow></mfenced><mfenced open="‖" close="‖" separators="|"><mrow><msup><mrow><mi>U</mi></mrow><mrow><mfenced separators="|"><mrow><mi>K</mi></mrow></mfenced></mrow></msup></mrow></mfenced><mfenced open="" close=">" separators="|"><mrow><mi>ψ</mi><mi>J</mi></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819432&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819431&type= 24.72266769 4.65666676 （dimensionless） are the doubly reduced matrix elements of the unit irreducible tensor operator <math id="M267"><msup><mrow><mi>U</mi></mrow><mrow><mfenced separators="|"><mrow><mi>K</mi></mrow></mfenced></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819442&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819441&type= 5.75733376 3.04800010 of rank <math id="M268"><mi>K</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819782&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819781&type= 2.53999996 2.62466669 evaluated in the intermediate coupling scheme. Their square, <math id="M269"><mfenced open="<" close="" separators="|"><mrow><msup><mrow><mi>ψ</mi></mrow><mrow><mi>*</mi></mrow></msup><msup><mrow><mi>J</mi></mrow><mrow><mi>*</mi></mrow></msup></mrow></mfenced><mtext> </mtext></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819438&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819437&type= 9.31333351 4.48733330 <math id="M270"><mfenced open="‖" close="‖" separators="|"><mrow><msup><mrow><mi>U</mi></mrow><mrow><mfenced separators="|"><mrow><mi>K</mi></mrow></mfenced></mrow></msup></mrow></mfenced><msup><mrow><mfenced open="" close=">" separators="|"><mrow><mi>ψ</mi><mi>J</mi></mrow></mfenced></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819440&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819439&type= 17.01799965 5.16466665 , are often abbreviated as <math id="M271"><msup><mrow><mi>U</mi></mrow><mrow><mfenced separators="|"><mrow><mi>K</mi></mrow></mfenced></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819442&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819441&type= 5.75733376 3.04800010 [ 45 45 ] or <math id="M272"><mi>U</mi><mo stretchy="false">(</mo><mi>K</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819444&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819443&type= 8.63599968 3.47133350 [ 46 46 ] and tabulated values can be found in reference [ 47 47 ] , e . g ., Eu（Ⅲ） has <math id="M273"><mfenced open="<" close="" separators="|"><mrow><mmultiscripts><mrow/><mprescripts/><mrow><mtext> </mtext></mrow><mrow><mn mathvariant="normal">7</mn></mrow></mmultiscripts><msub><mrow><mi mathvariant="normal">F</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub></mrow></mfenced><mfenced open="‖" close="‖" separators="|"><mrow><msup><mrow><mi>U</mi></mrow><mrow><mfenced separators="|"><mrow><mn mathvariant="normal">2</mn></mrow></mfenced></mrow></msup></mrow></mfenced><msup><mrow><mfenced open="" close=">" separators="|"><mrow><mmultiscripts><mrow/><mprescripts/><mrow><mtext> </mtext></mrow><mrow><mn mathvariant="normal">5</mn></mrow></mmultiscripts><msub><mrow><mi mathvariant="normal">D</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub></mrow></mfenced></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mo>=</mo><mtext> </mtext></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819446&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819445&type= 28.70199966 5.16466665 0.003 2 and <math id="M274"><mfenced open="<" close="" separators="|"><mrow><mmultiscripts><mrow/><mprescripts/><mrow><mtext> </mtext></mrow><mrow><mn mathvariant="normal">7</mn></mrow></mmultiscripts><msub><mrow><mi mathvariant="normal">F</mi></mrow><mrow><mn mathvariant="normal">4</mn></mrow></msub></mrow></mfenced><mfenced open="‖" close="‖" separators="|"><mrow><msup><mrow><mi>U</mi></mrow><mrow><mfenced separators="|"><mrow><mn mathvariant="normal">4</mn></mrow></mfenced></mrow></msup></mrow></mfenced><msup><mrow><mfenced open="" close=">" separators="|"><mrow><mmultiscripts><mrow/><mprescripts/><mrow><mtext> </mtext></mrow><mrow><mn mathvariant="normal">5</mn></mrow></mmultiscripts><msub><mrow><mi mathvariant="normal">D</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub></mrow></mfenced></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819448&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819447&type= 24.63800049 5.16466665 =0.002 3, ⋅ <math id="M275"><mfenced open="<" close="" separators="|"><mrow><msup><mrow><mi>ψ</mi></mrow><mrow><mi>*</mi></mrow></msup><msup><mrow><mi>J</mi></mrow><mrow><mi>*</mi></mrow></msup></mrow></mfenced><mfenced open="‖" close="‖" separators="|"><mrow><mover accent="true"><mi>M</mi><mo>^</mo></mover></mrow></mfenced><mfenced open="" close=">" separators="|"><mrow><mi>ψ</mi><mi>J</mi></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819450&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819449&type= 22.35199928 5.75733376 （dimensionless）, <math id="M276"><mover accent="true"><mi>M</mi><mo>^</mo></mover><mo>≡</mo><mover accent="true"><mi>L</mi><mo>^</mo></mover><mo>+</mo><msub><mrow><mi>g</mi></mrow><mrow><mi>S</mi></mrow></msub><mover accent="true"><mi>S</mi><mo>^</mo></mover><mo>≅</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819452&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819451&type= 20.65866661 5.58799982 <math id="M277"><mover accent="true"><mi>L</mi><mo>^</mo></mover><mo>+</mo><mn mathvariant="normal">2</mn><mover accent="true"><mi>S</mi><mo>^</mo></mover></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819454&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819453&type= 10.15999985 3.64066648 is the doubly reduced matrix element of the angular <math id="M278"><mover accent="true"><mi>L</mi><mo>^</mo></mover></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819456&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819455&type= 1.94733346 3.64066648 and spin <math id="M279"><mover accent="true"><mi>S</mi><mo>^</mo></mover></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819458&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819457&type= 1.86266661 3.64066648 operators calculated in the intermediate coupling scheme [ 48 48 - 50 50 48-50 ] , have typical values of 0.1-0.6 for states with different L and S terms [ 37 37 ] and assume high values into the same L and S terms （ e . g ., Tb（Ⅲ） has <math id="M280"><mfenced open="|" close="|" separators="|"><mrow><mfenced open="<" close="" separators="|"><mrow><mmultiscripts><mrow/><mprescripts/><mrow><mtext> </mtext></mrow><mrow><mn mathvariant="normal">7</mn></mrow></mmultiscripts><msub><mrow><mi mathvariant="normal">F</mi></mrow><mrow><mn mathvariant="normal">5</mn></mrow></msub></mrow></mfenced><mfenced open="‖" close="‖" separators="|"><mrow><mover accent="true"><mi>M</mi><mo>^</mo></mover></mrow></mfenced><mfenced open="" close=">" separators="|"><mrow><mmultiscripts><mrow/><mprescripts/><mrow><mtext> </mtext></mrow><mrow><mn mathvariant="normal">7</mn></mrow></mmultiscripts><msub><mrow><mi mathvariant="normal">F</mi></mrow><mrow><mn mathvariant="normal">6</mn></mrow></msub></mrow></mfenced></mrow></mfenced><mo>=</mo><mn mathvariant="normal">4.38</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819460&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819459&type= 34.96733475 6.51933336 ） [ 51 51 ] . It is important to emphasize that this matrix element is also expressed in units of ћ , in which case, the Bohr magneton <math id="M281"><msub><mrow><mi>μ</mi></mrow><mrow><mi mathvariant="normal">B</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819462&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819461&type= 3.80999994 4.48733330 cannot include <math id="M282"><mi>ћ</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819464&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819463&type= 1.94733346 2.70933342 , ⋅ <math id="M283"><mfenced open="<" close=">" separators="|"><mrow><msup><mrow><mi>r</mi></mrow><mrow><mi>t</mi></mrow></msup></mrow></mfenced><mo>≡</mo><mfenced open="<" close=">" separators="|"><mrow><mn mathvariant="normal">4</mn><mi mathvariant="normal">f</mi><mo stretchy="false">|</mo><msup><mrow><mi>r</mi></mrow><mrow><mi>t</mi></mrow></msup><mo stretchy="false">|</mo><mn mathvariant="normal">4</mn><mi mathvariant="normal">f</mi></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819466&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819465&type= 22.69066620 3.64066648 is the expectation value （integral） of <math id="M284"><msup><mrow><mi>r</mi></mrow><mrow><mi>t</mi></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819468&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819467&type= 2.03200006 2.96333337 with respect to the 4f radial function and the values for <math id="M285"><mi>t</mi><mo>=</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819470&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819469&type= 4.48733330 2.62466669 2, 4, and 6 can be found in Ref.[ 52 52 ], e . g ., <math id="M286"><mfenced open="<" close=">" separators="|"><mrow><msup><mrow><mi>r</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819472&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819471&type= 5.75733376 3.64066648 are 0.997, 0.893, 0.710, 0.773 a.u. for Eu（Ⅲ）, Tb（Ⅲ）, Yb（Ⅲ）, and Er（Ⅲ）, respectively [ 52 52 ] , ⋅ <math id="M287"><msub><mrow><mi>Ω</mi></mrow><mrow><mi>K</mi></mrow></msub><mo>≡</mo><msub><mrow><mi>Ω</mi></mrow><mrow><mi>K</mi></mrow></msub><mo stretchy="false">(</mo><mi mathvariant="normal">F</mi><mi mathvariant="normal">E</mi><mi mathvariant="normal">D</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819474&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819473&type= 22.69066620 4.57200003 is the forced electric dipole contribution to the intensity parameter of rank <math id="M288"><mi>K</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819782&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819781&type= 2.53999996 2.62466669 , in units of <math id="M289"><msup><mrow><mn mathvariant="normal">10</mn></mrow><mrow><mo>-</mo><mn mathvariant="normal">20</mn></mrow></msup><mtext> </mtext><mi mathvariant="normal">c</mi><msup><mrow><mi mathvariant="normal">m</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819478&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819477&type= 13.03866673 3.04800010 [ 20 20 , 34 34 , 36 36 ] , on the surroundings of the lanthanide ion : ⋅ <math id="M290"><mfenced separators="|"><mrow><mn mathvariant="normal">1</mn><mo>-</mo><msub><mrow><mi>σ</mi></mrow><mrow><mi>k</mi></mrow></msub></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819480&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819479&type= 12.53066730 4.31799984 , with <math id="M291"><mi>k</mi><mo>=</mo><mn mathvariant="normal">1</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819482&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819481&type= 8.04333401 2.62466669 and <math id="M292"><mn mathvariant="normal">2</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819484&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819483&type= 1.86266661 2.62466669 , is the shielding factor for donor and acceptor approximated as [ 34 34 , 53 53 - 54 54 53-54 ] <math display="block" id="M293"><mtext>    </mtext><mo stretchy="false">(</mo><mn mathvariant="normal">1</mn><mo>-</mo><msub><mrow><mi>σ</mi></mrow><mrow><mi>k</mi></mrow></msub><mo stretchy="false">)</mo><mo>=</mo><mi>ρ</mi><msup><mrow><mfenced separators="|"><mrow><mn mathvariant="normal">2</mn><mi>β</mi></mrow></mfenced></mrow><mrow><mi>k</mi><mo>+</mo><mn mathvariant="normal">1</mn></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819486&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819485&type= 33.69733429 5.07999992 ， where <math id="M294"><mi>ρ</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819488&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819487&type= 2.03200006 3.38666677 is the overlap integral between the valence subshells of the ligating atom and the 4f subshell of the lanthanide ion and <math id="M295"><mi>β</mi><mo>=</mo><mrow><mrow><mn mathvariant="normal">1</mn><mtext> </mtext></mrow><mo>/</mo><mrow><mfenced separators="|"><mrow><mn mathvariant="normal">1</mn><mo>±</mo><mi>ρ</mi></mrow></mfenced></mrow></mrow></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819490&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819489&type= 21.84399986 4.91066647 [ 53 53 , 55 55 ] , ⋅ <math id="M296"><msub><mrow><mi>σ</mi></mrow><mrow><mi mathvariant="normal">m</mi><mi mathvariant="normal">d</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819492&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819491&type= 4.99533367 3.72533321 is the shielding factor, for donor and acceptor, which shields the magnetic dipole interactions, on the lanthanide pair : ⋅ <math id="M297"><msub><mrow><mi>R</mi></mrow><mrow><mi mathvariant="normal">D</mi><mi mathvariant="normal">A</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819940&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819939&type= 5.16466665 3.72533321 is the distance between the lanthanides, <math display="block" id="M298"><msub><mrow><mi>R</mi></mrow><mrow><mi mathvariant="normal">D</mi><mi mathvariant="normal">A</mi></mrow></msub><mo>=</mo><mroot><mrow><mo stretchy="false">(</mo><msub><mrow><mi>x</mi></mrow><mrow><mi mathvariant="normal">D</mi></mrow></msub><mo>-</mo><msub><mrow><mi>x</mi></mrow><mrow><mi mathvariant="normal">A</mi></mrow></msub><msup><mrow><mo stretchy="false">)</mo></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mo>+</mo><mo stretchy="false">(</mo><msub><mrow><mi>y</mi></mrow><mrow><mi mathvariant="normal">D</mi></mrow></msub><mo>-</mo><msub><mrow><mi>y</mi></mrow><mrow><mi mathvariant="normal">A</mi></mrow></msub><msup><mrow><mo stretchy="false">)</mo></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mo>+</mo><mo stretchy="false">(</mo><msub><mrow><mi>z</mi></mrow><mrow><mi mathvariant="normal">D</mi></mrow></msub><mo>-</mo><msub><mrow><mi>z</mi></mrow><mrow><mi mathvariant="normal">A</mi></mrow></msub><msup><mrow><mo stretchy="false">)</mo></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup></mrow><mrow/></mroot></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819496&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819495&type= 65.78600311 5.41866684 ， with <math id="M299"><mo stretchy="false">(</mo><msub><mrow><mi>x</mi></mrow><mrow><mi mathvariant="normal">D</mi></mrow></msub><mo>,</mo><mtext> </mtext><msub><mrow><mi>y</mi></mrow><mrow><mi mathvariant="normal">D</mi></mrow></msub><mo>,</mo><mtext> </mtext><msub><mrow><mi>z</mi></mrow><mrow><mi mathvariant="normal">D</mi></mrow></msub><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819498&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819497&type= 15.74800014 4.74133301 and <math id="M300"><mo stretchy="false">(</mo><msub><mrow><mi>x</mi></mrow><mrow><mi mathvariant="normal">A</mi></mrow></msub><mo>,</mo><mtext> </mtext><msub><mrow><mi>y</mi></mrow><mrow><mi mathvariant="normal">A</mi></mrow></msub><mo>,</mo><mtext> </mtext><msub><mrow><mi>z</mi></mrow><mrow><mi mathvariant="normal">A</mi></mrow></msub><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819500&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819499&type= 15.74800014 4.74133301 being the Cartesian coordinates of the donor and acceptor centers, respectively, ⋅ <math id="M301"><msub><mrow><mi>ρ</mi></mrow><mrow><mi mathvariant="normal">f</mi><mi mathvariant="normal">f</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819745&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819744&type= 3.30200005 4.48733330 is the overlap integral between the 4f subshells of the donor and acceptor lanthanide ions [ 53 53 ] . The last quantity to determine the ET transfer rates is the <math id="M302"><mi>F</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819986&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819985&type= 2.37066650 2.62466669 -factor in Eq. （9） Eq. （9） , which includes contributions from the nuclear wavefunctions and the density of states. It is important to recall that when the donor subunit is excited to generate the initial state in the energy transfer process, the energy eigenvalues will be broadened and will have a quasi-continuum behavior that is described by the density of states of the donor <math id="M303"><msub><mrow><mi>ρ</mi></mrow><mrow><mi mathvariant="normal">D</mi></mrow></msub><mo stretchy="false">(</mo><mi>E</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819506&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819505&type= 9.56733322 4.65666676 and of the acceptor <math id="M304"><msub><mrow><mi>ρ</mi></mrow><mrow><mi mathvariant="normal">A</mi></mrow></msub><mo stretchy="false">(</mo><mi>E</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819508&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819507&type= 9.56733322 4.65666676 . In the original model for the nonradiative energy transfer between organic species via the dipole-dipole mechanism [ 24 24 ] , this factor was assigned to and determined as the spectral overlap between the absorption band of the acceptor and the emission band of the donor. As a result, the emission spectrum of the donor and the absorption spectrum of the acceptor are recorded separately and properly rescaled to determine the area under which these spectra overlap. However, this requires the spectra of the isolated donor and acceptor, which might not be viable, as well as a delicate procedure for determining the area of the spectral overlap. This would also preclude the development of a quantitative and predictive model. Considering that the density of states <math id="M305"><msub><mrow><mi>ρ</mi></mrow><mrow><mi mathvariant="normal">D</mi></mrow></msub><mo stretchy="false">(</mo><mi>E</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819548&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819547&type= 9.56733322 4.65666676 and <math id="M306"><msub><mrow><mi>ρ</mi></mrow><mrow><mi mathvariant="normal">A</mi></mrow></msub><mo stretchy="false">(</mo><mi>E</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819542&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819541&type= 9.56733322 4.65666676 are Gaussian functions, then the area of their overlap can be expressed as a sum of two error functions that depend on the widths of the Gaussian functions, their energy maxima, and the point of their intersection. Despite this area being calculable, there is not an analytical expression for it, which requires numerical modeling. It should be noticed that a typo occurred in the expression of <math id="M307"><msub><mrow><mi>W</mi></mrow><mrow><mi mathvariant="normal">m</mi><mi mathvariant="normal">d</mi><mi mathvariant="normal">⁃</mi><mi mathvariant="normal">m</mi><mi mathvariant="normal">d</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819828&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819827&type= 8.55133343 3.72533321 in previous publications [ 16 16 , 37 37 , 56 56 ] ,where <math id="M308"><msubsup><mrow><mi>σ</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow><mrow><mi>x</mi></mrow></msubsup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819516&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819515&type= 3.64066648 4.06400013 was used instead of <math id="M309"><msubsup><mrow><mi>σ</mi></mrow><mrow><mi mathvariant="normal">m</mi><mi mathvariant="normal">d</mi></mrow><mrow><mi>x</mi></mrow></msubsup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819530&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819529&type= 4.99533367 4.06400013 as shown in Eq.（20） Eq.（20） . Indeed, because the electric dipole interactions, which are shielded by electron densities represented as <math id="M310"><msubsup><mrow><mi>σ</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow><mrow><mi>x</mi></mrow></msubsup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819520&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819519&type= 3.64066648 4.06400013 , have different nature than the magnetic dipole interactions, the shielding of these interactions, represented as <math id="M311"><msubsup><mrow><mi>σ</mi></mrow><mrow><mi mathvariant="normal">m</mi><mi mathvariant="normal">d</mi></mrow><mrow><mi>x</mi></mrow></msubsup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819522&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819521&type= 4.99533367 4.06400013 , should not be the same. Hence, the introduction of the new notation <math id="M312"><msubsup><mrow><mi>σ</mi></mrow><mrow><mi mathvariant="normal">m</mi><mi mathvariant="normal">d</mi></mrow><mrow><mi>x</mi></mrow></msubsup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819524&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819523&type= 4.99533367 4.06400013 . However, because the shielding of the 4f electrons by the remaining electrons of the lanthanide ion and of the ligands involves closed-shells （ e . g ., 5s 2 5p 6 ）, then it is expected that the magnetic shielding <math id="M313"><msubsup><mrow><mi>σ</mi></mrow><mrow><mi mathvariant="normal">m</mi><mi mathvariant="normal">d</mi></mrow><mrow><mi>x</mi></mrow></msubsup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819530&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819529&type= 4.99533367 4.06400013 should be very small. This reasoning, associated with the difficulties in modeling <math id="M314"><msubsup><mrow><mi>σ</mi></mrow><mrow><mi mathvariant="normal">m</mi><mi mathvariant="normal">d</mi></mrow><mrow><mi>x</mi></mrow></msubsup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819530&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819529&type= 4.99533367 4.06400013 , has prompted the complete neglect of <math id="M315"><msubsup><mrow><mi>σ</mi></mrow><mrow><mi mathvariant="normal">m</mi><mi mathvariant="normal">d</mi></mrow><mrow><mi>x</mi></mrow></msubsup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819530&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819529&type= 4.99533367 4.06400013 in the equation for <math id="M316"><msub><mrow><mi>W</mi></mrow><mrow><mi mathvariant="normal">m</mi><mi mathvariant="normal">d</mi><mi mathvariant="normal">⁃</mi><mi mathvariant="normal">m</mi><mi mathvariant="normal">d</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819532&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819531&type= 8.55133343 3.72533321 [ 57 57 - 58 58 57-58 ] or to consider <math id="M317"><msubsup><mrow><mi>σ</mi></mrow><mrow><mi mathvariant="normal">m</mi><mi mathvariant="normal">d</mi></mrow><mrow><mi>x</mi></mrow></msubsup><mo>≈</mo><mn mathvariant="normal">0</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819534&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819533&type= 11.43000031 4.06400013 in the calculations of <math id="M318"><msub><mrow><mi>W</mi></mrow><mrow><mi mathvariant="normal">m</mi><mi mathvariant="normal">d</mi><mi mathvariant="normal">⁃</mi><mi mathvariant="normal">m</mi><mi mathvariant="normal">d</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819828&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819827&type= 8.55133343 3.72533321 [ 16 16 , 37 37 , 56 56 ] , which will cause a slight overestimation of the <math id="M319"><msub><mrow><mi>W</mi></mrow><mrow><mi mathvariant="normal">m</mi><mi mathvariant="normal">d</mi><mi mathvariant="normal">⁃</mi><mi mathvariant="normal">m</mi><mi mathvariant="normal">d</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819828&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819827&type= 8.55133343 3.72533321 rate. During the study of the plasmon effects on the luminescence of lanthanide ions, an alternative model was developed to determine the contributions of the density of states <math id="M320"><msub><mrow><mi>ρ</mi></mrow><mrow><mi mathvariant="normal">D</mi></mrow></msub><mo stretchy="false">(</mo><mi>E</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819548&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819547&type= 9.56733322 4.65666676 and <math id="M321"><msub><mrow><mi>ρ</mi></mrow><mrow><mi mathvariant="normal">A</mi></mrow></msub><mo stretchy="false">(</mo><mi>E</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819542&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819541&type= 9.56733322 4.65666676 to the <math id="M322"><mi>F</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819986&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819985&type= 2.37066650 2.62466669 -factor [ 59 59 ] .Later, the overlaps of the vibrational states were taken into consideration to develop a more complete formulation of the <math id="M323"><mi>F</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819986&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819985&type= 2.37066650 2.62466669 -factor, which becomes a function of the temperature by assuming a Boltzmann distribution for the vibrational states of the initial and final electronic states [ 20 20 , 34 34 ] . This formulation is relevant because it explicitly accounts for the creation of phonons when the donor state has energy higher than the acceptor. It also accounts for the annihilation of phonons when the donor state is below in energy relative to the acceptor state because the energy mismatch must be compensated by phonons [ 20 20 ] . By considering the bands of the donor and acceptor, <math id="M324"><msub><mrow><mi>ρ</mi></mrow><mrow><mi mathvariant="normal">D</mi></mrow></msub><mo stretchy="false">(</mo><mi>E</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819548&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819547&type= 9.56733322 4.65666676 and <math id="M325"><msub><mrow><mi>ρ</mi></mrow><mrow><mi mathvariant="normal">A</mi></mrow></msub><mo stretchy="false">(</mo><msup><mrow><mi>E</mi></mrow><mrow><mi>'</mi></mrow></msup><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819550&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819549&type= 10.32933331 4.99533367 , have Gaussian profiles, an analytical expression for the integral of a product of two Gaussian functions can be obtained [ 60 60 ] . This comes from the fact that a product of two Gaussian functions is a well-defined Gaussian function, whose define integral has analytical expression and the <math id="M326"><mi>F</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819986&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819985&type= 2.37066650 2.62466669 -factor can be expressed approximately as （in <math id="M327"><mi mathvariant="normal">e</mi><mi mathvariant="normal">r</mi><msup><mrow><mi mathvariant="normal">g</mi></mrow><mrow><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819554&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819553&type= 6.77333355 3.72533321 ） [ 34 34 ] <math display="block" id="M328"><mi>F</mi><mo>=</mo><mfrac><mrow><mi>Υ</mi></mrow><mrow><msup><mrow><mfenced separators="|"><mrow><msubsup><mrow><mi>γ</mi></mrow><mrow><mi mathvariant="normal">D</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msubsup><mo>+</mo><msubsup><mrow><mi>γ</mi></mrow><mrow><mi mathvariant="normal">A</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msubsup></mrow></mfenced></mrow><mrow><mrow><mrow><mn mathvariant="normal">1</mn></mrow><mo>/</mo><mrow><mn mathvariant="normal">2</mn></mrow></mrow></mrow></msup></mrow></mfrac><mo>×</mo><msup><mrow><mi>e</mi></mrow><mrow><mo>-</mo><msup><mrow><mo stretchy="false">(</mo><mrow><mrow><mi>Δ</mi><mtext> </mtext></mrow><mo>/</mo><mrow><msub><mrow><mi>γ</mi></mrow><mrow><mi mathvariant="normal">D</mi></mrow></msub></mrow></mrow><mo stretchy="false">)</mo></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mi>Γ</mi></mrow></msup><mi>G</mi><mo stretchy="false">(</mo><mi>Δ</mi><mo>,</mo><mi>T</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819556&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819555&type= 53.42466354 9.82133293 ， where <math display="block" id="M329"><mi>Γ</mi><mo>=</mo><mfenced open="[" close="]" separators="|"><mrow><mn mathvariant="normal">1</mn><mo>-</mo><mfrac><mrow><mn mathvariant="normal">1</mn></mrow><mrow><mn mathvariant="normal">1</mn><mo>+</mo><msup><mrow><mfenced separators="|"><mrow><mrow><mrow><msub><mrow><mi>γ</mi></mrow><mrow><mi mathvariant="normal">D</mi></mrow></msub></mrow><mo>/</mo><mrow><msub><mrow><mi>γ</mi></mrow><mrow><mi mathvariant="normal">A</mi></mrow></msub></mrow></mrow></mrow></mfenced></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup></mrow></mfrac></mrow></mfenced><mi mathvariant="normal">l</mi><mi mathvariant="normal">n</mi><mtext> </mtext><mn mathvariant="normal">2</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819558&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819557&type= 42.84133148 14.39333248 ， <math display="block" id="M330"><mi>Υ</mi><mo>=</mo><msup><mrow><mfenced separators="|"><mrow><mfrac><mrow><mi mathvariant="normal">l</mi><mi mathvariant="normal">n</mi><mtext> </mtext><mn mathvariant="normal">2</mn></mrow><mrow><mi mathvariant="normal">π</mi></mrow></mfrac></mrow></mfenced></mrow><mrow><mrow><mrow><mn mathvariant="normal">1</mn></mrow><mo>/</mo><mrow><mn mathvariant="normal">2</mn></mrow></mrow></mrow></msup><mfrac><mrow><mn mathvariant="normal">1</mn></mrow><mrow><mi>c</mi><mi>h</mi></mrow></mfrac></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819560&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819559&type= 23.96066666 9.05933285 ， with <math id="M331"><msub><mrow><mi>γ</mi></mrow><mrow><mi mathvariant="normal">D</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819719&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819718&type= 3.55599999 4.48733330 and <math id="M332"><msub><mrow><mi>γ</mi></mrow><mrow><mi mathvariant="normal">A</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819721&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819720&type= 3.55599999 4.48733330 being the bandwidths at half-height （in <math id="M333"><mi mathvariant="normal">c</mi><msup><mrow><mi mathvariant="normal">m</mi></mrow><mrow><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819574&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819573&type= 6.51933336 2.96333337 ） of the donor and acceptor, respectively, so <math id="M334"><mn mathvariant="normal">2</mn><mi mathvariant="normal">π</mi><mi>c</mi><mi>ћ</mi><msub><mrow><mi>γ</mi></mrow><mrow><mi>x</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819568&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819567&type= 10.75266743 4.48733330 has units of energy （in <math id="M335"><mi mathvariant="normal">e</mi><mi mathvariant="normal">r</mi><mi mathvariant="normal">g</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819570&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819569&type= 4.57200003 3.47133350 ）, <math id="M336"><mi>Δ</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819733&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819732&type= 2.45533323 2.79399991 （in <math id="M337"><mi mathvariant="normal">c</mi><msup><mrow><mi mathvariant="normal">m</mi></mrow><mrow><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819574&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819573&type= 6.51933336 2.96333337 ） being the energy difference （mismatch） between the donor and acceptor states, <math id="M338"><mi>Δ</mi><mo>=</mo><msub><mrow><mi>E</mi></mrow><mrow><mi mathvariant="normal">D</mi></mrow></msub><mo>-</mo><msub><mrow><mi>E</mi></mrow><mrow><mi mathvariant="normal">A</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819576&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819575&type= 18.96533203 3.89466691 , and <math id="M339"><mi>Υ</mi><mo>=</mo><mn mathvariant="normal">2.3646184</mn><mo>×</mo><msup><mrow><mn mathvariant="normal">10</mn></mrow><mrow><mn mathvariant="normal">15</mn></mrow></msup><mtext> </mtext><mi mathvariant="normal">c</mi><msup><mrow><mi mathvariant="normal">m</mi></mrow><mrow><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msup><mo>·</mo><mi mathvariant="normal">e</mi><mi mathvariant="normal">r</mi><msup><mrow><mi mathvariant="normal">g</mi></mrow><mrow><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819578&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819577&type= 49.52999878 3.80999994 . The function <math id="M340"><mi>G</mi><mo stretchy="false">(</mo><mi>Δ</mi><mo>,</mo><mi>T</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819677&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819676&type= 11.43000031 3.47133350 in Eq.（23） Eq.（23） collects contributions from the overlaps of the vibrational wavefunctions, which can be expressed in terms of the Huang-Rhys parameters with dependence on the average vibrational quantum number and on the number of phonons required for energy conservation. An approximated expression for this function is <math display="block" id="M341"><mi>G</mi><mfenced separators="|"><mrow><mi>Δ</mi><mo>,</mo><mi>T</mi></mrow></mfenced><mo>=</mo><mfenced open="{" close="" separators="|"><mrow><mtable><mtr><mtd><mn mathvariant="normal">1</mn><mo>,</mo><mtext>          </mtext><mi>Δ</mi><mo>></mo><mn mathvariant="normal">0</mn></mtd></mtr><mtr><mtd><msup><mrow><mi>e</mi></mrow><mrow><mrow><mrow><mi>Δ</mi><mtext> </mtext></mrow><mo>/</mo><mrow><mo stretchy="false">(</mo><msub><mrow><mi>k</mi></mrow><mrow><mi mathvariant="normal">B</mi></mrow></msub><mi>T</mi><mo stretchy="false">)</mo></mrow></mrow></mrow></msup><mo>,</mo><mtext>    </mtext><mi>Δ</mi><mo><</mo><mn mathvariant="normal">0</mn></mtd></mtr></mtable></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819582&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819581&type= 41.14799881 9.90600014 ， which represents an approximation to the multiphonon mechanism used to compensate for the energy conservation, with <math id="M342"><msub><mrow><mi>k</mi></mrow><mrow><mi mathvariant="normal">B</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819584&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819583&type= 3.04800010 3.72533321 being the Boltzmann constant. This reorganized form of the <math id="M343"><mi>F</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819986&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819985&type= 2.37066650 2.62466669 -factor shows explicitly its dependence on the energy mismatch <math id="M344"><mi>Δ</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819588&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819587&type= 2.45533323 2.79399991 and the bandwidths ћ <math id="M345"><msub><mrow><mi>γ</mi></mrow><mrow><mi mathvariant="normal">D</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819590&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819589&type= 3.55599999 4.48733330 and <math id="M346"><mi>ћ</mi><msub><mrow><mi>γ</mi></mrow><mrow><mi mathvariant="normal">A</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819592&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819591&type= 5.41866684 4.48733330 . For instance, in the case of Ln - Ln energy transfer, the bandwidths are narrow and practically the same, <math id="M347"><msub><mrow><mi>γ</mi></mrow><mrow><mi mathvariant="normal">D</mi></mrow></msub><mo>≅</mo><msub><mrow><mi>γ</mi></mrow><mrow><mi mathvariant="normal">A</mi></mrow></msub><mo>≡</mo><mi>γ</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819594&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819593&type= 18.11866570 4.48733330 , so <math id="M348"><mi>F</mi><mo>=</mo><mrow><mrow><mo stretchy="false">(</mo><mi>Υ</mi></mrow><mo>/</mo><mrow><mi>γ</mi><mroot><mrow><mn mathvariant="normal">2</mn></mrow><mrow/></mroot></mrow></mrow><mo stretchy="false">)</mo><msup><mrow><mi>e</mi></mrow><mrow><msup><mrow><mo>-</mo><mrow><mrow><mo stretchy="false">(</mo><mi>Δ</mi><mtext> </mtext></mrow><mo>/</mo><mrow><mi>γ</mi></mrow></mrow><mo stretchy="false">)</mo></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mrow><mrow><mfenced separators="|"><mrow><mi mathvariant="normal">l</mi><mi mathvariant="normal">n</mi><mtext> </mtext><mn mathvariant="normal">2</mn></mrow></mfenced></mrow><mo>/</mo><mrow><mn mathvariant="normal">2</mn></mrow></mrow></mrow></msup><mi>G</mi><mo stretchy="false">(</mo><mi>Δ</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819596&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819595&type= 46.39733505 5.16466665 . For narrow bands, the restriction on the energy mismatch <math id="M349"><mi>Δ</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819598&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819597&type= 2.45533323 2.79399991 is quite severe. For instance, <math id="M350"><mi>γ</mi><mo>≅</mo><mn mathvariant="normal">200</mn><mtext> </mtext><mi mathvariant="normal">c</mi><msup><mrow><mi mathvariant="normal">m</mi></mrow><mrow><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819600&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819599&type= 19.47333336 3.64066648 gives <math id="M351"><mi>F</mi><mo>≅</mo><mn mathvariant="normal">8</mn><mo>×</mo><msup><mrow><mn mathvariant="normal">10</mn></mrow><mrow><mn mathvariant="normal">12</mn></mrow></msup><msup><mrow><mi>e</mi></mrow><mrow><msup><mrow><mo>-</mo><mn mathvariant="normal">0.35</mn><mrow><mrow><mo stretchy="false">(</mo><mi>Δ</mi><mtext> </mtext></mrow><mo>/</mo><mrow><mi>γ</mi></mrow></mrow><mo stretchy="false">)</mo></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup></mrow></msup><mtext> </mtext><mi mathvariant="normal">e</mi><mi mathvariant="normal">r</mi><msup><mrow><mi mathvariant="normal">g</mi></mrow><mrow><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819602&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819601&type= 39.11600113 4.48733330 , for <math id="M352"><mi>Δ</mi><mo>></mo><mn mathvariant="normal">0</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819604&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819603&type= 8.55133343 2.79399991 . If <math id="M353"><mi>Δ</mi><mo>≅</mo><mn mathvariant="normal">1</mn><mtext> </mtext><mn mathvariant="normal">000</mn><mtext> </mtext><mi mathvariant="normal">c</mi><msup><mrow><mi mathvariant="normal">m</mi></mrow><mrow><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msup><mo>=</mo><mn mathvariant="normal">5</mn><mi>γ</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819606&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819605&type= 29.46400070 3.80999994 ,then <math id="M354"><msup><mrow><mi>e</mi></mrow><mrow><mo>-</mo><mn mathvariant="normal">0.3466</mn><mo>·</mo><msup><mrow><mn mathvariant="normal">5</mn></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup></mrow></msup><mo>~</mo><msup><mrow><mn mathvariant="normal">10</mn></mrow><mrow><mo>-</mo><mn mathvariant="normal">4</mn></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819608&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819607&type= 19.13466644 3.55599999 and <math id="M355"><mi>F</mi><mo>~</mo><msup><mrow><mn mathvariant="normal">10</mn></mrow><mrow><mn mathvariant="normal">9</mn></mrow></msup><mtext> </mtext><mi mathvariant="normal">e</mi><mi mathvariant="normal">r</mi><msup><mrow><mi mathvariant="normal">g</mi></mrow><mrow><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819610&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819609&type= 16.59466553 3.72533321 ; however, if <math id="M356"><mi>Δ</mi><mo>≅</mo><mn mathvariant="normal">2</mn><mtext> </mtext><mn mathvariant="normal">000</mn><mtext> </mtext><mi mathvariant="normal">c</mi><msup><mrow><mi mathvariant="normal">m</mi></mrow><mrow><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msup><mo>=</mo><mn mathvariant="normal">10</mn><mi>γ</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819612&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819611&type= 31.41133499 3.80999994 , then <math id="M357"><msup><mrow><mi>e</mi></mrow><mrow><mo>-</mo><mn mathvariant="normal">0.3466</mn><mo>·</mo><msup><mrow><mn mathvariant="normal">10</mn></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup></mrow></msup><mo>~</mo><msup><mrow><mn mathvariant="normal">10</mn></mrow><mrow><mo>-</mo><mn mathvariant="normal">15</mn></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819614&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819613&type= 21.16666603 3.55599999 and <math id="M358"><mi>F</mi><mo>~</mo><msup><mrow><mn mathvariant="normal">10</mn></mrow><mrow><mo>-</mo><mn mathvariant="normal">2</mn></mrow></msup><mtext> </mtext></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819616&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819615&type= 11.00666618 3.04800010 <math id="M359"><mi mathvariant="normal">e</mi><mi mathvariant="normal">r</mi><msup><mrow><mi mathvariant="normal">g</mi></mrow><mrow><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819618&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819617&type= 6.77333355 3.72533321 . So, a twofold increase in <math id="M360"><mi>Δ</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819642&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819641&type= 2.45533323 2.79399991 leads to a <math id="M361"><msup><mrow><mn mathvariant="normal">10</mn></mrow><mrow><mn mathvariant="normal">11</mn></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819622&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819621&type= 5.75733376 2.96333337 decrease of <math id="M362"><mi>F</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819986&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819985&type= 2.37066650 2.62466669 , because of the quadratic dependence on <math id="M363"><mi>Δ</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819642&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819641&type= 2.45533323 2.79399991 in the argument of the exponential. This situation is quite distinct from that involving a sensitizer with a broad band, where <math id="M364"><msub><mrow><mi>γ</mi></mrow><mrow><mi mathvariant="normal">D</mi></mrow></msub><mo>≫</mo><msub><mrow><mi>γ</mi></mrow><mrow><mi mathvariant="normal">A</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819628&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819627&type= 12.19200039 4.48733330 . In this case, the <math id="M365"><mi>F</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819630&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819629&type= 2.37066650 2.62466669 -factor becomes <math id="M366"><mi>F</mi><mo>=</mo><mrow><mrow><mo stretchy="false">(</mo><mi>Υ</mi></mrow><mo>/</mo><mrow><msub><mrow><mi>γ</mi></mrow><mrow><mi mathvariant="normal">D</mi></mrow></msub></mrow></mrow><mo stretchy="false">)</mo><msup><mrow><mi>e</mi></mrow><mrow><msup><mrow><mo>-</mo><mrow><mrow><mo stretchy="false">(</mo><mi>Δ</mi><mtext> </mtext></mrow><mo>/</mo><mrow><msub><mrow><mi>γ</mi></mrow><mrow><mi mathvariant="normal">D</mi></mrow></msub></mrow></mrow><mo stretchy="false">)</mo></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mi mathvariant="normal">l</mi><mi mathvariant="normal">n</mi><mn mathvariant="normal">2</mn></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819632&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819631&type= 28.78666496 4.65666676 <math id="M367"><mi>G</mi><mo stretchy="false">(</mo><mi>Δ</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819634&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819633&type= 8.04333401 3.47133350 .So, for a donor bandwidth of <math id="M368"><msub><mrow><mi>γ</mi></mrow><mrow><mi mathvariant="normal">D</mi></mrow></msub><mo>≅</mo><mn mathvariant="normal">2</mn><mtext> </mtext><mn mathvariant="normal">000</mn><mtext> </mtext><mi mathvariant="normal">c</mi><msup><mrow><mi mathvariant="normal">m</mi></mrow><mrow><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819636&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819635&type= 23.53733444 4.74133301 and <math id="M369"><mi>Δ</mi><mo>≅</mo><mn mathvariant="normal">6</mn><mtext> </mtext><mn mathvariant="normal">000</mn><mtext> </mtext><mi mathvariant="normal">c</mi><msup><mrow><mi mathvariant="normal">m</mi></mrow><mrow><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msup><mo>=</mo><mn mathvariant="normal">3</mn><msub><mrow><mi>γ</mi></mrow><mrow><mi mathvariant="normal">D</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819638&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819637&type= 31.07266808 4.91066647 , then <math id="M370"><mi>F</mi><mo>~</mo><mn mathvariant="normal">2</mn><mo>×</mo><msup><mrow><mn mathvariant="normal">10</mn></mrow><mrow><mn mathvariant="normal">9</mn></mrow></msup><mtext> </mtext><mi mathvariant="normal">e</mi><mi mathvariant="normal">r</mi><msup><mrow><mi mathvariant="normal">g</mi></mrow><mrow><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819640&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819639&type= 22.35199928 3.72533321 . Despite this energy mismatch <math id="M371"><mi>Δ</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819642&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819641&type= 2.45533323 2.79399991 being six times larger than that of narrow bands, the <math id="M372"><mi>F</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819986&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819985&type= 2.37066650 2.62466669 -factor is the same order of magnitude because it corresponds to three times the broader bandwidth, whereas in the latter case it corresponds to five times the narrower bandwidth. The dependence of the <math id="M373"><mi>F</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819986&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819985&type= 2.37066650 2.62466669 -factor on the energy mismatch, bandwidths, and temperature is depicted in the following Figs. Fig.1 Fig.1 shows that the <math id="M374"><mi>F</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819986&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819985&type= 2.37066650 2.62466669 -factor is slightly asymmetric because the function <math id="M375"><mi>G</mi><mo stretchy="false">(</mo><mi>Δ</mi><mo>,</mo><mi>T</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819677&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819676&type= 11.43000031 3.47133350 differs when <math id="M376"><mi>Δ</mi><mo><</mo><mn mathvariant="normal">0</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819652&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819651&type= 8.80533314 2.79399991 or <math id="M377"><mi>Δ</mi><mo>></mo><mn mathvariant="normal">0</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819654&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819653&type= 8.80533314 2.79399991 （ Eq.25 Eq.25 ）. For very narrow bandwidths values （ e . g ., <math id="M378"><msub><mrow><mi>γ</mi></mrow><mrow><mi mathvariant="normal">D</mi></mrow></msub><mo>=</mo><msub><mrow><mi>γ</mi></mrow><mrow><mi mathvariant="normal">A</mi></mrow></msub><mo>≈</mo><mn mathvariant="normal">100</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819656&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819655&type= 21.59000015 4.48733330 cm -1 ）, the <math id="M379"><mi>F</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819986&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819985&type= 2.37066650 2.62466669 -factor assumes high values only if <math id="M380"><mi>Δ</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819733&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819732&type= 2.45533323 2.79399991 is in the same order as bandwidths. Temperature effects on the energy transfer rates can be rationalized from the changes of the <math id="M386"><mi>F</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819986&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819985&type= 2.37066650 2.62466669 -factor. Thus, two outcomes （direct and indirect） are expected: the energy mismatch barrier in the <math id="M387"><mi>G</mi><mo stretchy="false">(</mo><mi>Δ</mi><mo>,</mo><mi>T</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819677&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819676&type= 11.43000031 3.47133350 when Δ < 0（direct） and the widening of the bandwidths due to the temperature rising （indirect）. Fig.2 Fig.2 shows the <math id="M388"><mi>F</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819986&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819985&type= 2.37066650 2.62466669 -factor behaves as a function of temperature （direct） and of <math id="M389"><mi>Δ</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819681&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819680&type= 2.45533323 2.79399991 , for fixed values of <math id="M390"><msub><mrow><mi>γ</mi></mrow><mrow><mi mathvariant="normal">D</mi></mrow></msub><mo>=</mo><msub><mrow><mi>γ</mi></mrow><mrow><mi mathvariant="normal">A</mi></mrow></msub><mo>≈</mo><mn mathvariant="normal">400</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819683&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819682&type= 21.59000015 4.48733330 cm -1 . For the temperature effects on the bandwidth broadening, consider the following hypothetical case: a co-doped Nd（Ⅲ）,Yb（Ⅲ）-containing material at an initial temperature <math id="M391"><msub><mrow><mi>T</mi></mrow><mrow><mi>i</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819685&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819684&type= 2.79399991 3.72533321 presents <math id="M392"><msub><mrow><mi>γ</mi></mrow><mrow><mi mathvariant="normal">D</mi></mrow></msub><mo stretchy="false">(</mo><msub><mrow><mi>T</mi></mrow><mrow><mi>i</mi></mrow></msub><mo stretchy="false">)</mo><mo>=</mo><msub><mrow><mi>γ</mi></mrow><mrow><mi mathvariant="normal">A</mi></mrow></msub><mo stretchy="false">(</mo><msub><mrow><mi>T</mi></mrow><mrow><mi>i</mi></mrow></msub><mo stretchy="false">)</mo><mo>≈</mo><mn mathvariant="normal">300</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819687&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819686&type= 33.69733429 4.65666676 cm -1 for a pair of donor and acceptor transitions （ e . g ., Nd（Ⅲ） 4 G 5/2 → 4 I 15/2 as donor and Yb（Ⅲ） 2 F 7/2 → 2 F 5/2 as acceptor, with <math id="M393"><mi>Δ</mi><mo>≅</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819689&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819688&type= 5.58799982 2.79399991 1 100 cm -1[ 47 47 ] ）, and the sample is uniformly heated to a final temperature <math id="M394"><msub><mrow><mi>T</mi></mrow><mrow><mi>f</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819691&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819690&type= 2.79399991 3.72533321 and both bandwidths increase by 50 cm -1 , so （ <math id="M395"><msub><mrow><mi>γ</mi></mrow><mrow><mi mathvariant="normal">D</mi></mrow></msub><mo stretchy="false">(</mo><msub><mrow><mi>T</mi></mrow><mrow><mi>f</mi></mrow></msub><mo stretchy="false">)</mo><mo>=</mo><msub><mrow><mi>γ</mi></mrow><mrow><mi mathvariant="normal">A</mi></mrow></msub><mo stretchy="false">(</mo><msub><mrow><mi>T</mi></mrow><mrow><mi>f</mi></mrow></msub><mo stretchy="false">)</mo><mo>≈</mo><mn mathvariant="normal">350</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819693&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819692&type= 33.69733429 4.65666676 cm -1 ）. According to Eq.（23） Eq.（23） and approximating <math id="M396"><mi>Δ</mi><mo>≅</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819695&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819694&type= 5.58799982 2.79399991 1 000 cm -1 （as illustrated in Fig.3 Fig.3 ）, the <math id="M397"><mi>F</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819986&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819985&type= 2.37066650 2.62466669 -factor increases from 1.18×10 11 to 2.82×10 11 erg -1 . Notice that the graph in Fig.3 Fig.3 presents symmetry with respect to the interchange of <math id="M405"><msub><mrow><mi>γ</mi></mrow><mrow><mi mathvariant="normal">D</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819719&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819718&type= 3.55599999 4.48733330 and <math id="M406"><msub><mrow><mi>γ</mi></mrow><mrow><mi mathvariant="normal">A</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819721&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819720&type= 3.55599999 4.48733330 values, which is not evident in Eqs.（ 23 23 ） and （ 24 24 ） because of the <math id="M407"><msup><mrow><mi>e</mi></mrow><mrow><mo>-</mo><msup><mrow><mo stretchy="false">(</mo><mrow><mrow><mi>Δ</mi><mtext> </mtext></mrow><mo>/</mo><mrow><msub><mrow><mi>γ</mi></mrow><mrow><mi mathvariant="normal">D</mi></mrow></msub></mrow></mrow><mo stretchy="false">)</mo></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mi>Γ</mi></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819723&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819722&type= 11.17599964 3.47133350 term. However, it makes sense once the Dirac delta function acting in <math id="M408"><mi>k</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819725&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819724&type= 1.69333339 2.62466669 and <math id="M409"><mi>i</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819727&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819726&type= 1.10066664 2.62466669 states should be the same when the order of states is reversed, i . e ., <math id="M410"><mi>δ</mi><mfenced separators="|"><mrow><msub><mrow><mi>E</mi></mrow><mrow><mi>k</mi></mrow></msub><mo>-</mo><msub><mrow><mi>E</mi></mrow><mrow><mi>i</mi></mrow></msub></mrow></mfenced><mo>=</mo><mi>δ</mi><mfenced separators="|"><mrow><msub><mrow><mi>E</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>-</mo><msub><mrow><mi>E</mi></mrow><mrow><mi>k</mi></mrow></msub></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819729&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819728&type= 37.08399963 4.31799984 in Eq.（7） Eq.（7） . 6　Selection Rules for Energy Transfer Rates 1 Lanthanide ions have a wealthy number of levels in the optical region that can be active in energy transfer processes. So, the number of ET pathways between two lanthanide ions can be substantially large and the selection rules are important to exclude those pathways that are null. In fact, the selection rules of the <math id="M411"><mi>J</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819735&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819734&type= 1.77800000 2.96333337 quantum number, together with the energy mismatch <math id="M412"><mi>Δ</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819733&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819732&type= 2.45533323 2.79399991 , in Eq.（23） Eq.（23） , are very useful to select the donor and acceptor states for adequate ET pathways. The exchange contribution, Eq.（17） Eq.（17） , to the ET rate does not have selection rules for the <math id="M413"><mi>J</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819735&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819734&type= 1.77800000 2.96333337 quantum number, that is, it always present, regardless of the donor or acceptor states. This is particular to lanthanide-lanthanide ET processes because in this case, the isotropic contribution does not vanish, so no dependence on the angular momentum appears. For the case of ligand-lanthanide ET processes, because the ligand states are usually restricted to singlets and triplets, with a singlet ground state, the contribution from the spin operator vanishes and only transitions with <math id="M414"><mfenced open="|" close="|" separators="|"><mrow><mo>∆</mo><mi>J</mi></mrow></mfenced><mo>=</mo><mn mathvariant="normal">0</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819737&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819736&type= 11.76866722 3.80999994 or 1 are allowed （nonvanishing） [ 20 20 ] .Despite this lack of selection rules, the exchange contribution is strongly dependent on the lanthanide-lanthanide distance, <math id="M415"><msub><mrow><mi>R</mi></mrow><mrow><mi mathvariant="normal">D</mi><mi mathvariant="normal">A</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819940&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819939&type= 5.16466665 3.72533321 , because of <math id="M416"><msubsup><mrow><mi>R</mi></mrow><mrow><mi mathvariant="normal">D</mi><mi mathvariant="normal">A</mi></mrow><mrow><mo>-</mo><mn mathvariant="normal">2</mn></mrow></msubsup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819741&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819740&type= 5.16466665 4.06400013 in Eq. （17） Eq. （17） , and, mainly of <math id="M417"><msubsup><mrow><mi>ρ</mi></mrow><mrow><mi mathvariant="normal">f</mi><mi mathvariant="normal">f</mi></mrow><mrow><mn mathvariant="normal">4</mn></mrow></msubsup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819743&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819742&type= 3.30200005 4.74133301 dependence. Indeed, the overlap integral between 4f-4f radial functions, <math id="M418"><msub><mrow><mi>ρ</mi></mrow><mrow><mi mathvariant="normal">f</mi><mi mathvariant="normal">f</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819745&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819744&type= 3.30200005 4.48733330 , decreases rapidly with the lanthanide-lanthanide distance （ Fig.4 Fig.4 ） [ 53 53 ] . As a result, depending on the value of the <math id="M419"><mi>F</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819986&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819985&type= 2.37066650 2.62466669 -factor, the exchange contribution, <math id="M420"><msub><mrow><mi>W</mi></mrow><mrow><mi mathvariant="normal">e</mi><mi mathvariant="normal">x</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819749&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819748&type= 4.82600021 3.72533321 , can be promptly ruled out and would be considered mostly when the other contributions are null. In the case of the multipolar mechanisms, <math id="M423"><msub><mrow><mi>W</mi></mrow><mrow><mi mathvariant="normal">d</mi><mi mathvariant="normal">⁃</mi><mi mathvariant="normal">d</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819758&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819757&type= 5.67266655 3.72533321 , <math id="M424"><msub><mrow><mi>W</mi></mrow><mrow><mi mathvariant="normal">d</mi><mi mathvariant="normal">q</mi><mi mathvariant="normal">⁃</mi><mi mathvariant="normal">q</mi><mi mathvariant="normal">d</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819800&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819799&type= 7.61999989 3.72533321 , and <math id="M425"><msub><mrow><mi>W</mi></mrow><mrow><mi mathvariant="normal">q</mi><mi mathvariant="normal">⁃</mi><mi mathvariant="normal">q</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819802&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819801&type= 5.67266655 3.72533321 , the selection rules are: <math id="M426"><mfenced open="|" close="|" separators="|"><mrow><msup><mrow><mi>J</mi></mrow><mrow><mi>*</mi></mrow></msup><mo>-</mo><mi>J</mi></mrow></mfenced><mo>≤</mo><mi>K</mi><mo>≤</mo><msup><mrow><mi>J</mi></mrow><mrow><mi>*</mi></mrow></msup><mo>+</mo><mi>J</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819764&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819763&type= 30.98800087 4.31799984 .These come from the fact that these rates will vanish when the dipole <math id="M427"><msubsup><mrow><mi>S</mi></mrow><mrow><mi mathvariant="normal">d</mi></mrow><mrow><mi>x</mi></mrow></msubsup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819766&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819765&type= 3.04800010 4.06400013 or quadruple <math id="M428"><msubsup><mrow><mi>S</mi></mrow><mrow><mi mathvariant="normal">q</mi></mrow><mrow><mi>x</mi></mrow></msubsup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819768&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819767&type= 3.04800010 4.06400013 strengths are null. On the other hand, these quantities will be null when all reduced matrix elements <math id="M429"><mfenced open="<" close="" separators="|"><mrow><msubsup><mrow><mi>ψ</mi></mrow><mrow><mi>x</mi></mrow><mrow><mi>*</mi></mrow></msubsup><msubsup><mrow><mi>J</mi></mrow><mrow><mi>x</mi></mrow><mrow><mi>*</mi></mrow></msubsup></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819770&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819769&type= 8.80533314 4.48733330 <math id="M430"><mfenced open="‖" close="‖" separators="|"><mrow><msup><mrow><mi>U</mi></mrow><mrow><mfenced separators="|"><mrow><mi>K</mi></mrow></mfenced></mrow></msup></mrow></mfenced><mfenced open="" close=">" separators="|"><mrow><msub><mrow><mi>ψ</mi></mrow><mrow><mi>x</mi></mrow></msub><msub><mrow><mi>J</mi></mrow><mrow><mi>x</mi></mrow></msub></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819772&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819771&type= 17.86466599 4.65666676 are zero, hence these selection rules.For instance, consider the <math id="M431"><mmultiscripts><mrow/><mprescripts/><mrow><mtext> </mtext></mrow><mrow><mn mathvariant="normal">5</mn></mrow></mmultiscripts><msub><mrow><mi mathvariant="normal">D</mi></mrow><mrow><mn mathvariant="normal">4</mn></mrow></msub><mo>→</mo><mmultiscripts><mrow><mi mathvariant="normal">F</mi></mrow><mprescripts/><mrow><mtext> </mtext></mrow><mrow><mn mathvariant="normal">7</mn></mrow></mmultiscripts><msub><mrow><mi mathvariant="normal">F</mi></mrow><mrow><mn mathvariant="normal">5</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819774&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819773&type= 14.73200035 4.65666676 transition of Tb（Ⅲ） as a donor state, then <math id="M432"><mfenced open="|" close="|" separators="|"><mrow><msubsup><mrow><mi>J</mi></mrow><mrow><mi mathvariant="normal">D</mi></mrow><mrow><mi>*</mi></mrow></msubsup><mo>-</mo><msub><mrow><mi>J</mi></mrow><mrow><mi mathvariant="normal">D</mi></mrow></msub></mrow></mfenced><mo>=</mo><mn mathvariant="normal">1</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819776&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819775&type= 17.01799965 4.31799984 and <math id="M433"><msubsup><mrow><mi>J</mi></mrow><mrow><mi mathvariant="normal">D</mi></mrow><mrow><mi>*</mi></mrow></msubsup><mo>+</mo><msub><mrow><mi>J</mi></mrow><mrow><mi mathvariant="normal">D</mi></mrow></msub><mo>=</mo><mn mathvariant="normal">9</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819778&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819777&type= 17.18733406 4.40266657 , so <math id="M434"><mn mathvariant="normal">1</mn><mo>≤</mo><mi>K</mi><mo>≤</mo><mn mathvariant="normal">9</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819780&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819779&type= 15.07066536 2.70933342 which covers all possible values of <math id="M435"><mi>K</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819782&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819781&type= 2.53999996 2.62466669 and increases the chance of having nonzero matrix elements. On the other hand, if the acceptor state is the <math id="M436"><mmultiscripts><mrow/><mprescripts/><mrow><mtext> </mtext></mrow><mrow><mn mathvariant="normal">7</mn></mrow></mmultiscripts><msub><mrow><mi mathvariant="normal">F</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><mo>→</mo><mmultiscripts><mrow><mi mathvariant="normal">D</mi></mrow><mprescripts/><mrow><mtext> </mtext></mrow><mrow><mn mathvariant="normal">5</mn></mrow></mmultiscripts><msub><mrow><mi mathvariant="normal">D</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819784&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819783&type= 14.73200035 4.65666676 transition of Eu（Ⅲ）, then <math id="M437"><mfenced open="|" close="|" separators="|"><mrow><msubsup><mrow><mi>J</mi></mrow><mrow><mi mathvariant="normal">A</mi></mrow><mrow><mi>*</mi></mrow></msubsup><mo>-</mo><msub><mrow><mi>J</mi></mrow><mrow><mi mathvariant="normal">A</mi></mrow></msub></mrow></mfenced><mo>=</mo><mn mathvariant="normal">1</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819786&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819785&type= 19.38866615 4.31799984 and <math id="M438"><msubsup><mrow><mi>J</mi></mrow><mrow><mi mathvariant="normal">A</mi></mrow><mrow><mi>*</mi></mrow></msubsup><mo>+</mo><msub><mrow><mi>J</mi></mrow><mrow><mi mathvariant="normal">A</mi></mrow></msub><mo>=</mo><mn mathvariant="normal">1</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819788&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819787&type= 17.18733406 4.40266657 , so <math id="M439"><mi>K</mi><mo>=</mo><mn mathvariant="normal">1</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819790&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819789&type= 8.72066593 2.62466669 and the transition is forbidden because matrix elements <math id="M440"><mfenced open="<" close="" separators="|"><mrow><msubsup><mrow><mi>ψ</mi></mrow><mrow><mi>x</mi></mrow><mrow><mi>*</mi></mrow></msubsup><msubsup><mrow><mi>J</mi></mrow><mrow><mi>x</mi></mrow><mrow><mi>*</mi></mrow></msubsup></mrow></mfenced><mfenced open="‖" close="‖" separators="|"><mrow><msup><mrow><mi>U</mi></mrow><mrow><mfenced separators="|"><mrow><mn mathvariant="normal">1</mn></mrow></mfenced></mrow></msup></mrow></mfenced><mfenced open="" close=">" separators="|"><mrow><msub><mrow><mi>ψ</mi></mrow><mrow><mi>x</mi></mrow></msub><msub><mrow><mi>J</mi></mrow><mrow><mi>x</mi></mrow></msub></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819792&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819791&type= 26.58533096 4.65666676 are zero. However, these selection rules may be slightly relaxed by the J -mixing effect, providing very small values of <math id="M441"><mfenced open="<" close="" separators="|"><mrow><msubsup><mrow><mi>ψ</mi></mrow><mrow><mi>x</mi></mrow><mrow><mi>*</mi></mrow></msubsup><msubsup><mrow><mi>J</mi></mrow><mrow><mi>x</mi></mrow><mrow><mi>*</mi></mrow></msubsup></mrow></mfenced><mfenced open="‖" close="‖" separators="|"><mrow><msup><mrow><mi>U</mi></mrow><mrow><mfenced separators="|"><mrow><mn mathvariant="normal">2</mn></mrow></mfenced></mrow></msup></mrow></mfenced><msup><mrow><mfenced open="" close=">" separators="|"><mrow><msub><mrow><mi>ψ</mi></mrow><mrow><mi>x</mi></mrow></msub><msub><mrow><mi>J</mi></mrow><mrow><mi>x</mi></mrow></msub></mrow></mfenced></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mo>≈</mo><mn mathvariant="normal">0.0032</mn><mo>×</mo><mn mathvariant="normal">0</mn><mo>.</mo><msup><mrow><mn mathvariant="normal">05</mn></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819794&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819793&type= 56.98066711 5.16466665 for the specific cases involving the Eu（Ⅲ） <math id="M442"><mmultiscripts><mrow/><mprescripts/><mrow><mtext> </mtext></mrow><mrow><mn mathvariant="normal">7</mn></mrow></mmultiscripts><msub><mrow><mi mathvariant="normal">F</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819978&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819977&type= 4.65666676 4.65666676 level. Therefore, this effect is negligible in the energy transfer between lanthanide ions. So, the contributions from multipolar mechanisms, <math id="M443"><msub><mrow><mi>W</mi></mrow><mrow><mi mathvariant="normal">d</mi><mi mathvariant="normal">⁃</mi><mi mathvariant="normal">d</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819854&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819853&type= 5.67266655 3.72533321 , <math id="M444"><msub><mrow><mi>W</mi></mrow><mrow><mi mathvariant="normal">d</mi><mi mathvariant="normal">q</mi><mi mathvariant="normal">⁃</mi><mi mathvariant="normal">q</mi><mi mathvariant="normal">d</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819800&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819799&type= 7.61999989 3.72533321 , and <math id="M445"><msub><mrow><mi>W</mi></mrow><mrow><mi mathvariant="normal">q</mi><mi mathvariant="normal">⁃</mi><mi mathvariant="normal">q</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819802&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819801&type= 5.67266655 3.72533321 , to the ET rate via the pathway Tb（Ⅲ） <math id="M446"><mmultiscripts><mrow/><mprescripts/><mrow><mtext> </mtext></mrow><mrow><mn mathvariant="normal">5</mn></mrow></mmultiscripts><msub><mrow><mi mathvariant="normal">D</mi></mrow><mrow><mn mathvariant="normal">4</mn></mrow></msub><mo>→</mo><mmultiscripts><mrow><mi mathvariant="normal">F</mi></mrow><mprescripts/><mrow><mtext> </mtext></mrow><mrow><mn mathvariant="normal">7</mn></mrow></mmultiscripts><msub><mrow><mi mathvariant="normal">F</mi></mrow><mrow><mn mathvariant="normal">5</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819880&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819879&type= 14.39333248 4.65666676 to Eu（Ⅲ） <math id="M447"><mmultiscripts><mrow/><mprescripts/><mrow><mtext> </mtext></mrow><mrow><mn mathvariant="normal">7</mn></mrow></mmultiscripts><msub><mrow><mi mathvariant="normal">F</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><mo>→</mo><mmultiscripts><mrow><mi mathvariant="normal">D</mi></mrow><mprescripts/><mrow><mtext> </mtext></mrow><mrow><mn mathvariant="normal">5</mn></mrow></mmultiscripts><msub><mrow><mi mathvariant="normal">D</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819806&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819805&type= 14.73200035 4.65666676 are null. However, this pathway is allowed by the exchange mechanism and might be allowed by the magnetic dipole mechanism. Hence, for the multipolar mechanisms, another acceptor state of Eu（Ⅲ） needs to be considered, e . g ., <math id="M448"><mmultiscripts><mrow/><mprescripts/><mrow><mtext> </mtext></mrow><mrow><mn mathvariant="normal">7</mn></mrow></mmultiscripts><msub><mrow><mi mathvariant="normal">F</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>→</mo><mmultiscripts><mrow><mi mathvariant="normal">D</mi></mrow><mprescripts/><mrow><mtext> </mtext></mrow><mrow><mn mathvariant="normal">5</mn></mrow></mmultiscripts><msub><mrow><mi mathvariant="normal">D</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819808&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819807&type= 15.83266735 4.65666676 . In this case, <math id="M449"><mfenced open="|" close="|" separators="|"><mrow><msubsup><mrow><mi>J</mi></mrow><mrow><mi mathvariant="normal">A</mi></mrow><mrow><mi>*</mi></mrow></msubsup><mo>-</mo><msub><mrow><mi>J</mi></mrow><mrow><mi mathvariant="normal">A</mi></mrow></msub></mrow></mfenced><mo>=</mo><mn mathvariant="normal">0</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819810&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819809&type= 19.38866615 4.31799984 and <math id="M450"><msubsup><mrow><mi>J</mi></mrow><mrow><mi mathvariant="normal">A</mi></mrow><mrow><mi>*</mi></mrow></msubsup><mo>+</mo><msub><mrow><mi>J</mi></mrow><mrow><mi mathvariant="normal">A</mi></mrow></msub><mo>=</mo><mn mathvariant="normal">2</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819812&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819811&type= 17.18733406 4.40266657 , so <math id="M451"><mi>K</mi><mo>=</mo><mn mathvariant="normal">2</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819814&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819813&type= 8.72066593 2.62466669 and the <math id="M452"><mmultiscripts><mrow/><mprescripts/><mrow><mtext> </mtext></mrow><mrow><mn mathvariant="normal">7</mn></mrow></mmultiscripts><msub><mrow><mi mathvariant="normal">F</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>→</mo><mmultiscripts><mrow><mi mathvariant="normal">D</mi></mrow><mprescripts/><mrow><mtext> </mtext></mrow><mrow><mn mathvariant="normal">5</mn></mrow></mmultiscripts><msub><mrow><mi mathvariant="normal">D</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819816&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819815&type= 14.73200035 4.65666676 transition is allowed because matrix element <math id="M453"><msup><mrow><mi>U</mi></mrow><mrow><mfenced separators="|"><mrow><mn mathvariant="normal">2</mn></mrow></mfenced></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819818&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819817&type= 5.33400011 3.13266683 of Eu（Ⅲ） is nonzero. The remaining mechanism for ET between lanthanide ions is the magnetic dipole⁃magnetic dipole, <math id="M454"><msub><mrow><mi>W</mi></mrow><mrow><mi mathvariant="normal">m</mi><mi mathvariant="normal">d</mi><mi mathvariant="normal">⁃</mi><mi mathvariant="normal">m</mi><mi mathvariant="normal">d</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819828&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819827&type= 8.55133343 3.72533321 , in Eq.（ 16 16 ） and （ 20 20 ）. Thus, the selection rules are given by the <math id="M455"><mfenced open="<" close="" separators="|"><mrow><msup><mrow><mi>ψ</mi></mrow><mrow><mi>*</mi></mrow></msup><msup><mrow><mi>J</mi></mrow><mrow><mi>*</mi></mrow></msup></mrow></mfenced><mfenced open="‖" close="‖" separators="|"><mrow><mover accent="true"><mi>L</mi><mo>^</mo></mover><mo>+</mo><msub><mrow><mi>g</mi></mrow><mrow><mi>S</mi></mrow></msub><mover accent="true"><mi>S</mi><mo>^</mo></mover></mrow></mfenced><mfenced open="" close=">" separators="|"><mrow><mi>ψ</mi><mi>J</mi></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819822&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819821&type= 30.90333366 5.75733376 matrix elements, which will be zero unless <math id="M456"><mo>∆</mo><mi>J</mi><mo>=</mo><mn mathvariant="normal">0</mn><mo>,</mo><mo>±</mo><mn mathvariant="normal">1</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819824&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819823&type= 17.44133186 3.21733332 , except <math id="M457"><msup><mrow><mi>J</mi></mrow><mrow><mi>*</mi></mrow></msup><mo>=</mo><mi>J</mi><mo>=</mo><mn mathvariant="normal">0</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819826&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819825&type= 15.49400043 3.30200005 . As a result, the <math id="M458"><msub><mrow><mi>W</mi></mrow><mrow><mi mathvariant="normal">m</mi><mi mathvariant="normal">d</mi><mi mathvariant="normal">⁃</mi><mi mathvariant="normal">m</mi><mi mathvariant="normal">d</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819828&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819827&type= 8.55133343 3.72533321 contribution to the ET rate via the Tb（Ⅲ） <math id="M459"><mmultiscripts><mrow/><mprescripts/><mrow><mtext> </mtext></mrow><mrow><mn mathvariant="normal">5</mn></mrow></mmultiscripts><msub><mrow><mi mathvariant="normal">D</mi></mrow><mrow><mn mathvariant="normal">4</mn></mrow></msub><mo>→</mo><mmultiscripts><mrow><mi mathvariant="normal">F</mi></mrow><mprescripts/><mrow><mtext> </mtext></mrow><mrow><mn mathvariant="normal">7</mn></mrow></mmultiscripts><msub><mrow><mi mathvariant="normal">F</mi></mrow><mrow><mn mathvariant="normal">5</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819830&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819829&type= 14.39333248 4.65666676 to Eu（Ⅲ） <math id="M460"><mmultiscripts><mrow/><mprescripts/><mrow><mtext> </mtext></mrow><mrow><mn mathvariant="normal">7</mn></mrow></mmultiscripts><msub><mrow><mi mathvariant="normal">F</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><mo>→</mo><mmultiscripts><mrow><mi mathvariant="normal">D</mi></mrow><mprescripts/><mrow><mtext> </mtext></mrow><mrow><mn mathvariant="normal">5</mn></mrow></mmultiscripts><msub><mrow><mi mathvariant="normal">D</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819832&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819831&type= 14.13933277 4.65666676 pathway will be nonzero because <math id="M461"><mo>∆</mo><mi>J</mi><mo>=</mo><mo>±</mo><mn mathvariant="normal">1</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819834&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819833&type= 11.51466751 3.04800010 for both transitions, which satisfy the selection rules. Notice that for Eu（Ⅲ）, the first excited state <math id="M462"><mmultiscripts><mrow/><mprescripts/><mrow><mtext> </mtext></mrow><mrow><mn mathvariant="normal">7</mn></mrow></mmultiscripts><msub><mrow><mi mathvariant="normal">F</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819936&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819935&type= 4.65666676 4.65666676 is close to ground level <math id="M463"><mmultiscripts><mrow/><mprescripts/><mrow><mtext> </mtext></mrow><mrow><mn mathvariant="normal">7</mn></mrow></mmultiscripts><msub><mrow><mi mathvariant="normal">F</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819978&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819977&type= 4.65666676 4.65666676 , so the populations of these states will depend on the temperature via the Boltzmann factor. When comparing the relative contributions of each mechanism to the total ET rate, it is common to weigh those rates involving the <math id="M464"><mmultiscripts><mrow/><mprescripts/><mrow><mtext> </mtext></mrow><mrow><mn mathvariant="normal">7</mn></mrow></mmultiscripts><msub><mrow><mi mathvariant="normal">F</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819978&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819977&type= 4.65666676 4.65666676 and <math id="M465"><mmultiscripts><mrow/><mprescripts/><mrow><mtext> </mtext></mrow><mrow><mn mathvariant="normal">7</mn></mrow></mmultiscripts><msub><mrow><mi mathvariant="normal">F</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819936&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819935&type= 4.65666676 4.65666676 levels by ca. 0.65 and 0.33 （at 300 K）, respectively. However, care must be exercised when the ET rates are going to be employed in the rate equations. In this case, the thermal effects are already included in the initial populations when solving the rate equations, so the weighting procedure should not be overcounted. 7　Forced Electric Dipole Intensity Parameter， Ω K （FED） 1 The forced electric dipole （FED） contribution to the intensity parameter <math id="M466"><msub><mrow><mi>Ω</mi></mrow><mrow><mi>K</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819844&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819843&type= 4.06400013 3.89466691 （with <math id="M467"><mi>K</mi><mo>=</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819846&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819845&type= 5.41866684 2.62466669 2, 4, and 6）, that appears in the （forced） electric dipole strength <math id="M468"><msub><mrow><mi>S</mi></mrow><mrow><mi mathvariant="normal">d</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819848&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819847&type= 3.04800010 3.72533321 , Eq.（18） Eq.（18） ,is perhaps one of the most difficult quantities to determine. It cannot be determined from fitted values of experimental absorption or emission spectra,because the fitted intensity parameters <math id="M469"><msub><mrow><mi>Ω</mi></mrow><mrow><mi>K</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819850&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819849&type= 4.06400013 3.89466691 contains contributions from both FED and dynamic coupling （DC） mechanisms. Usually, the FED contribution to <math id="M470"><msub><mrow><mi>Ω</mi></mrow><mrow><mi>K</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819852&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819851&type= 4.06400013 3.89466691 is very small; however, it depends on the lanthanide ion and its environment, so no trends have been observed [ 37 37 , 67 67 - 69 69 67-69 ] . It cannot be neglected, because despite it being small, the contributions of the <math id="M471"><msub><mrow><mi>W</mi></mrow><mrow><mi mathvariant="normal">d</mi><mi mathvariant="normal">⁃</mi><mi mathvariant="normal">d</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819854&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819853&type= 5.67266655 3.72533321 and <math id="M472"><msub><mrow><mi>W</mi></mrow><mrow><mi mathvariant="normal">d</mi><mi mathvariant="normal">q</mi><mi mathvariant="normal">⁃</mi><mi mathvariant="normal">q</mi><mi mathvariant="normal">d</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819856&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819855&type= 7.61999989 3.72533321 rates can be relevant or even dominant for several ET pathways. In addition, <math id="M473"><msub><mrow><mi>Ω</mi></mrow><mrow><mi>K</mi></mrow></msub><mo stretchy="false">(</mo><mi mathvariant="normal">F</mi><mi mathvariant="normal">E</mi><mi mathvariant="normal">D</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819864&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819863&type= 14.22399998 4.57200003 contains effects from the environment of the lanthanide ion, which can be used to design systems with more efficient ET pathways. Therefore, values of <math id="M474"><msub><mrow><mi>Ω</mi></mrow><mrow><mi>K</mi></mrow></msub><mo stretchy="false">(</mo><mi mathvariant="normal">F</mi><mi mathvariant="normal">E</mi><mi mathvariant="normal">D</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819860&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819859&type= 14.22399998 4.57200003 have to be obtained from quantitative models. For instance, the simple overlap model （SOM） [ 55 55 , 70 70 ] yields quantitative expressions for <math id="M475"><msub><mrow><mi>Ω</mi></mrow><mrow><mi>K</mi></mrow></msub><mo stretchy="false">(</mo><mi mathvariant="normal">F</mi><mi mathvariant="normal">E</mi><mi mathvariant="normal">D</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819924&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819923&type= 14.22399998 4.57200003 , which can be evaluated using the JOYSpectra web platform [ 71 71 ] . For Eu（Ⅲ） and Tb（Ⅲ） in a unique nine-coordination site of the doped material Sr 3 Tb 0.90 Eu 0.10 ⁃ （PO 4 ） 3 [ 36 36 , 71 71 ] , two sets of <math id="M476"><msub><mrow><mi>Ω</mi></mrow><mrow><mi>K</mi></mrow></msub><mo stretchy="false">(</mo><mi mathvariant="normal">F</mi><mi mathvariant="normal">E</mi><mi mathvariant="normal">D</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819864&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819863&type= 14.22399998 4.57200003 <math id="M477"><mfenced open="{" close="}" separators="|"><mrow><msub><mrow><mi>Ω</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo>;</mo><mtext> </mtext><msub><mrow><mi>Ω</mi></mrow><mrow><mn mathvariant="normal">4</mn></mrow></msub><mo>;</mo><mtext> </mtext><msub><mrow><mi>Ω</mi></mrow><mrow><mn mathvariant="normal">6</mn></mrow></msub></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819866&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819865&type= 20.15066719 4.40266657 （in units of 10 -20 cm 2 ） are obtained: {0.60; 0.21; 0.26} for Tb（Ⅲ） and {0.82; 0.38; 0.52} for Eu（Ⅲ）. These calculations were performed using the values of the force constant, <math id="M478"><msub><mrow><mi>k</mi></mrow><mrow><mi>L</mi><mi>n</mi><mi mathvariant="normal">⁃</mi><mi mathvariant="normal">L</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819868&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819867&type= 5.58799982 3.72533321 , of the Ln -Ligating atom bond obtained from density functional theory （DFT） calculations and, therefore, the values of the <math id="M479"><mi>g</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819874&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819873&type= 1.94733346 3.47133350 -charge factors were determined for all Ln ⁃O pairs with Eq.S80 [ 36 36 ] .In the specific case of Sr 3 Tb 0.90 Eu 0.10 ⁃ （PO 4 ） 3 materials, the calculated <math id="M480"><msub><mrow><mi>k</mi></mrow><mrow><mi mathvariant="normal">L</mi><mi mathvariant="normal">n</mi><mi mathvariant="normal">⁃</mi><mi mathvariant="normal">L</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819872&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819871&type= 5.67266655 3.72533321 is in the range of 0.38-0.42 mdyn/Å and it provides <math id="M481"><mi>g</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819874&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819873&type= 1.94733346 3.47133350 -charge factors around 1.29. For further details on applying the SOM model, see Supporting Information and references [20, 35, 55, 67, 70-76]. 8　Illustrative Examples of How to Calculate Energy Transfer Rates 1 Here we will illustrate step-by-step examples of some energy transfer pathways between Ln （Ⅲ） ions. The first example （ E1 ） involves the Tb（Ⅲ） [ <math id="M482"><mmultiscripts><mrow/><mprescripts/><mrow><mtext> </mtext></mrow><mrow><mn mathvariant="normal">5</mn></mrow></mmultiscripts><msub><mrow><mi mathvariant="normal">D</mi></mrow><mrow><mn mathvariant="normal">4</mn></mrow></msub><mo>→</mo><mmultiscripts><mrow><mi mathvariant="normal">F</mi></mrow><mprescripts/><mrow><mtext> </mtext></mrow><mrow><mn mathvariant="normal">7</mn></mrow></mmultiscripts><msub><mrow><mi mathvariant="normal">F</mi></mrow><mrow><mn mathvariant="normal">5</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819946&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819945&type= 14.73200035 4.65666676 ] → Eu（Ⅲ） [ <math id="M483"><mmultiscripts><mrow/><mprescripts/><mrow><mtext> </mtext></mrow><mrow><mn mathvariant="normal">7</mn></mrow></mmultiscripts><msub><mrow><mi mathvariant="normal">F</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>→</mo><mmultiscripts><mrow><mi mathvariant="normal">D</mi></mrow><mprescripts/><mrow><mtext> </mtext></mrow><mrow><mn mathvariant="normal">5</mn></mrow></mmultiscripts><msub><mrow><mi mathvariant="normal">D</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819878&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819877&type= 13.97000027 4.65666676 ] energy transfer pathway which is the dominant one for Tb-Eu processes [ 36 36 - 37 37 36-37 ] . As an exercise of a direct application of selection rules on J , the Tb（Ⅲ）[ <math id="M484"><mmultiscripts><mrow/><mprescripts/><mrow><mtext> </mtext></mrow><mrow><mn mathvariant="normal">5</mn></mrow></mmultiscripts><msub><mrow><mi mathvariant="normal">D</mi></mrow><mrow><mn mathvariant="normal">4</mn></mrow></msub><mo>→</mo><mmultiscripts><mrow><mi mathvariant="normal">F</mi></mrow><mprescripts/><mrow><mtext> </mtext></mrow><mrow><mn mathvariant="normal">7</mn></mrow></mmultiscripts><msub><mrow><mi mathvariant="normal">F</mi></mrow><mrow><mn mathvariant="normal">5</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819880&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819879&type= 14.39333248 4.65666676 ]→Eu（Ⅲ）[ <math id="M485"><mmultiscripts><mrow/><mprescripts/><mrow><mtext> </mtext></mrow><mrow><mn mathvariant="normal">7</mn></mrow></mmultiscripts><msub><mrow><mi mathvariant="normal">F</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><mo>→</mo><mmultiscripts><mrow><mi mathvariant="normal">D</mi></mrow><mprescripts/><mrow><mtext> </mtext></mrow><mrow><mn mathvariant="normal">5</mn></mrow></mmultiscripts><msub><mrow><mi mathvariant="normal">D</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819882&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819881&type= 14.39333248 4.65666676 ] （ E2 ） will also be addressed. These two pathways are illustrated in Fig.5 Fig.5 . Since the discovery of the upconversion process by F. Auzel [ 77 77 - 78 78 77-78 ] , ET between lanthanide ions has become an important issue regarding luminescent materials based on Ln （Ⅲ） ions [ 79 79 - 84 84 79-84 ] . Thus, we will also show how to calculate two energy transfer steps that are in good energy resonance conditions: Yb（Ⅲ） [ <math id="M486"><mmultiscripts><mrow/><mprescripts/><mrow><mtext> </mtext></mrow><mrow><mn mathvariant="normal">2</mn></mrow></mmultiscripts><msub><mrow><mi mathvariant="normal">F</mi></mrow><mrow><mrow><mrow><mn mathvariant="normal">5</mn></mrow><mo>/</mo><mrow><mn mathvariant="normal">2</mn></mrow></mrow></mrow></msub><mo>→</mo><mmultiscripts><mrow><mi mathvariant="normal">F</mi></mrow><mprescripts/><mrow><mtext> </mtext></mrow><mrow><mn mathvariant="normal">2</mn></mrow></mmultiscripts><msub><mrow><mi mathvariant="normal">F</mi></mrow><mrow><mrow><mrow><mn mathvariant="normal">7</mn></mrow><mo>/</mo><mrow><mn mathvariant="normal">2</mn></mrow></mrow></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819887&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819886&type= 18.20333290 5.58799982 ] → Er（Ⅲ） [ <math id="M487"><mmultiscripts><mrow/><mprescripts/><mrow><mtext> </mtext></mrow><mrow><mn mathvariant="normal">4</mn></mrow></mmultiscripts><msub><mrow><mi mathvariant="normal">I</mi></mrow><mrow><mrow><mrow><mn mathvariant="normal">15</mn></mrow><mo>/</mo><mrow><mn mathvariant="normal">2</mn></mrow></mrow></mrow></msub><mo>→</mo><mmultiscripts><mrow><mi mathvariant="normal">I</mi></mrow><mprescripts/><mrow><mtext> </mtext></mrow><mrow><mn mathvariant="normal">4</mn></mrow></mmultiscripts><msub><mrow><mi mathvariant="normal">I</mi></mrow><mrow><mrow><mrow><mn mathvariant="normal">11</mn></mrow><mo>/</mo><mrow><mn mathvariant="normal">2</mn></mrow></mrow></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819889&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819888&type= 17.78000069 5.58799982 ] （ E3 ） and Yb（Ⅲ）[ <math id="M488"><mmultiscripts><mrow/><mprescripts/><mrow><mtext> </mtext></mrow><mrow><mn mathvariant="normal">2</mn></mrow></mmultiscripts><msub><mrow><mi mathvariant="normal">F</mi></mrow><mrow><mrow><mrow><mn mathvariant="normal">5</mn></mrow><mo>/</mo><mrow><mn mathvariant="normal">2</mn></mrow></mrow></mrow></msub><mo>→</mo><mmultiscripts><mrow><mi mathvariant="normal">F</mi></mrow><mprescripts/><mrow><mtext> </mtext></mrow><mrow><mn mathvariant="normal">2</mn></mrow></mmultiscripts><msub><mrow><mi mathvariant="normal">F</mi></mrow><mrow><mrow><mrow><mn mathvariant="normal">7</mn></mrow><mo>/</mo><mrow><mn mathvariant="normal">2</mn></mrow></mrow></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819891&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819890&type= 17.78000069 5.58799982 ] → Er（Ⅲ） [ <math id="M489"><mmultiscripts><mrow/><mprescripts/><mrow><mtext> </mtext></mrow><mrow><mn mathvariant="normal">4</mn></mrow></mmultiscripts><msub><mrow><mi mathvariant="normal">I</mi></mrow><mrow><mrow><mrow><mn mathvariant="normal">11</mn></mrow><mo>/</mo><mrow><mn mathvariant="normal">2</mn></mrow></mrow></mrow></msub><mo>→</mo><mmultiscripts><mrow><mi mathvariant="normal">I</mi></mrow><mprescripts/><mrow><mtext> </mtext></mrow><mrow><mn mathvariant="normal">4</mn></mrow></mmultiscripts><msub><mrow><mi mathvariant="normal">I</mi></mrow><mrow><mrow><mrow><mn mathvariant="normal">7</mn></mrow><mo>/</mo><mrow><mn mathvariant="normal">2</mn></mrow></mrow></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39820058&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39820057&type= 16.76399994 5.58799982 ] （ E4 ） pathways [ 16 16 , 35 35 ] . Fig.6 Fig.6 illustrates these pathways in E3 and E4 . ⋅ E1 Tab.1 Tab.1 contains the input data considering Tb（Ⅲ） [ <math id="M490"><mmultiscripts><mrow/><mprescripts/><mrow><mtext> </mtext></mrow><mrow><mn mathvariant="normal">5</mn></mrow></mmultiscripts><msub><mrow><mi mathvariant="normal">D</mi></mrow><mrow><mn mathvariant="normal">4</mn></mrow></msub><mo>→</mo><mmultiscripts><mrow><mi mathvariant="normal">F</mi></mrow><mprescripts/><mrow><mtext> </mtext></mrow><mrow><mn mathvariant="normal">7</mn></mrow></mmultiscripts><msub><mrow><mi mathvariant="normal">F</mi></mrow><mrow><mn mathvariant="normal">5</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819898&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819897&type= 14.81666565 4.65666676 ] and Eu（Ⅲ） [ <math id="M491"><mmultiscripts><mrow/><mprescripts/><mrow><mtext> </mtext></mrow><mrow><mn mathvariant="normal">7</mn></mrow></mmultiscripts><msub><mrow><mi mathvariant="normal">F</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>→</mo><mmultiscripts><mrow><mi mathvariant="normal">D</mi></mrow><mprescripts/><mrow><mtext> </mtext></mrow><mrow><mn mathvariant="normal">5</mn></mrow></mmultiscripts><msub><mrow><mi mathvariant="normal">D</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819900&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819899&type= 14.81666565 4.65666676 ] transitions as donor and acceptor, respectively. Eq. Donor Acceptor <math id="M492"><mi>J</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39820042&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39820041&type= 1.35466671 2.28600001 18⁃20 4 1 ρ 21 0.057 0.057 <math id="M493"><mfenced open="<" close="" separators="|"><mrow><mtext> </mtext></mrow></mfenced><mfenced open="‖" close="‖" separators="|"><mrow><msup><mrow><mi>U</mi></mrow><mrow><mfenced separators="|"><mrow><mn>2</mn></mrow></mfenced></mrow></msup></mrow></mfenced><msup><mrow><mfenced open="" close=">" separators="|"><mrow><mtext> </mtext></mrow></mfenced></mrow><mrow><mn>2</mn></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39820044&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39820043&type= 12.95400047 4.06400013 18⁃19 0.014 2 0.002 5 <math id="M494"><mfenced open="<" close="" separators="|"><mrow><mtext> </mtext></mrow></mfenced><mfenced open="‖" close="‖" separators="|"><mrow><msup><mrow><mi>U</mi></mrow><mrow><mfenced separators="|"><mrow><mn>4</mn></mrow></mfenced></mrow></msup></mrow></mfenced><msup><mrow><mfenced open="" close=">" separators="|"><mrow><mtext> </mtext></mrow></mfenced></mrow><mrow><mn>2</mn></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39820046&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39820045&type= 12.95400047 4.06400013 18 0.001 3 0 <math id="M495"><mfenced open="<" close="" separators="|"><mrow><mtext> </mtext></mrow></mfenced><mfenced open="‖" close="‖" separators="|"><mrow><msup><mrow><mi>U</mi></mrow><mrow><mfenced separators="|"><mrow><mn>6</mn></mrow></mfenced></mrow></msup></mrow></mfenced><msup><mrow><mfenced open="" close=">" separators="|"><mrow><mtext> </mtext></mrow></mfenced></mrow><mrow><mn>2</mn></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39820048&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39820047&type= 12.95400047 4.06400013 18 0.002 2 0 <math id="M496"><mfenced open="<" close=">" separators="|"><mrow><msup><mrow><mi>r</mi></mrow><mrow><mn>2</mn></mrow></msup></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39820008&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39820007&type= 4.57200003 3.04800010 （10 -17 cm 2 ） 19 2.30 2.57 <math id="M497"><mfenced open="<" close="" separators="|"><mrow><mtext> </mtext></mrow></mfenced><mfenced open="‖" close="‖" separators="|"><mrow><mi>L</mi><mo>+</mo><mn>2</mn><mi>S</mi></mrow></mfenced><msup><mrow><mfenced open="" close=">" separators="|"><mrow><mtext> </mtext></mrow></mfenced></mrow><mrow><mn>2</mn></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39820050&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39820049&type= 16.51000023 3.30200005 20 0.696 0.000 9 <math id="M498"><mi>γ</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39820012&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39820011&type= 1.60866666 2.53999996 （cm -1 ） 23⁃24 350 350 Eq. 2 Donor⁃Acceptor <math id="M499"><msub><mrow><mi>R</mi></mrow><mrow><mi mathvariant="normal">D</mi><mi mathvariant="normal">A</mi></mrow></msub><mo>/</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819916&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819915&type= 5.33400011 2.87866688 cm 22 2 3.97×10 -8 <math id="M500"><mi>Δ</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39820016&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39820015&type= 1.86266661 2.11666679 / cm -1 23， 25 2 259 <math id="M501"><msub><mrow><mi>ρ</mi></mrow><mrow><mi mathvariant="normal">f</mi><mi mathvariant="normal">f</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819920&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819919&type= 2.70933342 3.38666677 17 2 6.70×10 -4 at 3.97 Å Thus, when the data in Tab.1 Tab.1 is applied to the indicated equations with the sets of <math id="M502"><msub><mrow><mi>Ω</mi></mrow><mrow><mi>K</mi></mrow></msub><mo stretchy="false">(</mo><mi mathvariant="normal">F</mi><mi mathvariant="normal">E</mi><mi mathvariant="normal">D</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819924&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819923&type= 14.22399998 4.57200003 , mentioned previously for the case of Sr 3 Tb 0.90 Eu 0.10 （PO 4 ） 3 [ 36 36 ] , we obtain the following rates for each mechanism: <math display="block" id="M503"><msub><mrow><mi>W</mi></mrow><mrow><mi mathvariant="normal">d</mi><mi mathvariant="normal">⁃</mi><mi mathvariant="normal">d</mi></mrow></msub><mo>=</mo><mn mathvariant="normal">6.6</mn><mo>×</mo><msup><mrow><mn mathvariant="normal">10</mn></mrow><mrow><mo>-</mo><mn mathvariant="normal">2</mn></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819926&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819925&type= 25.23066711 4.06400013 s -1 ， <math display="block" id="M504"><msub><mrow><mi>W</mi></mrow><mrow><mi mathvariant="normal">d</mi><mi mathvariant="normal">q</mi><mi mathvariant="normal">⁃</mi><mi mathvariant="normal">q</mi><mi mathvariant="normal">d</mi></mrow></msub><mo>=</mo><mn mathvariant="normal">4.2</mn><mo>×</mo><msup><mrow><mn mathvariant="normal">10</mn></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819928&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819927&type= 26.07733345 4.06400013 s -1 ， <math display="block" id="M505"><msub><mrow><mi>W</mi></mrow><mrow><mi mathvariant="normal">q</mi><mi mathvariant="normal">⁃</mi><mi mathvariant="normal">q</mi></mrow></msub><mo>=</mo><mn mathvariant="normal">5.1</mn><mo>×</mo><msup><mrow><mn mathvariant="normal">10</mn></mrow><mrow><mn mathvariant="normal">4</mn></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819930&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819929&type= 24.12999916 4.06400013 s -1 ， <math display="block" id="M506"><msub><mrow><mi>W</mi></mrow><mrow><mi mathvariant="normal">m</mi><mi mathvariant="normal">d</mi><mi mathvariant="normal">⁃</mi><mi mathvariant="normal">m</mi><mi mathvariant="normal">d</mi></mrow></msub><mo>=</mo><mn mathvariant="normal">6.8</mn><mo>×</mo><msup><mrow><mn mathvariant="normal">10</mn></mrow><mrow><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819932&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819931&type= 28.02466965 4.06400013 s -1 ， <math display="block" id="M507"><msub><mrow><mi>W</mi></mrow><mrow><mi mathvariant="normal">e</mi><mi mathvariant="normal">x</mi></mrow></msub><mo>=</mo><mn mathvariant="normal">1.6</mn><mo>×</mo><msup><mrow><mn mathvariant="normal">10</mn></mrow><mrow><mn mathvariant="normal">5</mn></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819934&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819933&type= 23.28333473 4.06400013 s -1 . The total energy transfer rate is given by the sum of all these contributions and multiplied, in this case, by the <math id="M508"><mmultiscripts><mrow/><mprescripts/><mrow><mtext> </mtext></mrow><mrow><mn mathvariant="normal">7</mn></mrow></mmultiscripts><msub><mrow><mi mathvariant="normal">F</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819936&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819935&type= 4.65666676 4.65666676 population fraction （ca. 0.33 at room temperature）: <math display="block" id="M509"><mtable columnalign="left"><mtr><mtd><mtext>   </mtext><msub><mrow><mi>W</mi></mrow><mrow><mi mathvariant="normal">T</mi><mi mathvariant="normal">O</mi><mi mathvariant="normal">T</mi><mi mathvariant="normal">A</mi><mi mathvariant="normal">L</mi></mrow></msub><mo>=</mo><mn mathvariant="normal">0</mn><mo>.</mo><mtext> </mtext><mn mathvariant="normal">33</mn><mo>×</mo><mfenced separators="|"><mrow><msub><mrow><mi>W</mi></mrow><mrow><mi mathvariant="normal">d</mi><mi mathvariant="normal">⁃</mi><mi mathvariant="normal">d</mi></mrow></msub><mo>+</mo><msub><mrow><mi>W</mi></mrow><mrow><mi mathvariant="normal">d</mi><mi mathvariant="normal">q</mi><mi mathvariant="normal">⁃</mi><mi mathvariant="normal">q</mi><mi mathvariant="normal">d</mi></mrow></msub><mo>+</mo><msub><mrow><mi>W</mi></mrow><mrow><mi mathvariant="normal">q</mi><mi mathvariant="normal">⁃</mi><mi mathvariant="normal">q</mi></mrow></msub><mo>+</mo><msub><mrow><mi>W</mi></mrow><mrow><mi mathvariant="normal">m</mi><mi mathvariant="normal">d</mi><mi mathvariant="normal">⁃</mi><mi mathvariant="normal">m</mi><mi mathvariant="normal">d</mi></mrow></msub><mo>+</mo><msub><mrow><mi>W</mi></mrow><mrow><mi mathvariant="normal">e</mi><mi mathvariant="normal">x</mi></mrow></msub></mrow></mfenced></mtd></mtr><mtr><mtd><mtext>     </mtext><mo>=</mo><mn mathvariant="normal">7.0</mn><mo>×</mo><msup><mrow><mn mathvariant="normal">10</mn></mrow><mrow><mn mathvariant="normal">4</mn></mrow></msup></mtd></mtr></mtable></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819938&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819937&type= 75.01467133 9.82133293 s -1 . These rates are in agreement with those presented in reference [ 36 36 ] for the same pathway at the same <math id="M510"><msub><mrow><mi>R</mi></mrow><mrow><mi mathvariant="normal">D</mi><mi mathvariant="normal">A</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819940&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819939&type= 5.16466665 3.72533321 distance. A slight deviation in the quadrupole-quadrupole mechanism from reference [ 36 36 ] （ <math id="M511"><msub><mrow><mi>W</mi></mrow><mrow><mi mathvariant="normal">q</mi><mi mathvariant="normal">⁃</mi><mi mathvariant="normal">q</mi></mrow></msub><mo>=</mo><mn mathvariant="normal">5.67</mn><mo>×</mo><msup><mrow><mn mathvariant="normal">10</mn></mrow><mrow><mn mathvariant="normal">4</mn></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819942&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819941&type= 26.92399979 4.06400013 s -1 ） was observed. This is due to the use of <math id="M512"><mfenced open="<" close=">" separators="|"><mrow><msup><mrow><mi>r</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup></mrow></mfenced><mo>=</mo><mn mathvariant="normal">2.57</mn><mo>×</mo><msup><mrow><mn mathvariant="normal">10</mn></mrow><mrow><mo>-</mo><mn mathvariant="normal">17</mn></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819944&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819943&type= 29.04066467 3.64066648 for the Tb（Ⅲ）. ⋅ E2 To estimate each contribution to the rate of the Tb（Ⅲ） [ <math id="M513"><mmultiscripts><mrow/><mprescripts/><mrow><mtext> </mtext></mrow><mrow><mn mathvariant="normal">5</mn></mrow></mmultiscripts><msub><mrow><mi mathvariant="normal">D</mi></mrow><mrow><mn mathvariant="normal">4</mn></mrow></msub><mo>→</mo><mmultiscripts><mrow><mi mathvariant="normal">F</mi></mrow><mprescripts/><mrow><mtext> </mtext></mrow><mrow><mn mathvariant="normal">7</mn></mrow></mmultiscripts><msub><mrow><mi mathvariant="normal">F</mi></mrow><mrow><mn mathvariant="normal">5</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819946&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819945&type= 14.73200035 4.65666676 ] → Eu（Ⅲ） [ <math id="M514"><mmultiscripts><mrow/><mprescripts/><mrow><mtext> </mtext></mrow><mrow><mn mathvariant="normal">7</mn></mrow></mmultiscripts><msub><mrow><mi mathvariant="normal">F</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><mo>→</mo><mmultiscripts><mrow><mi mathvariant="normal">D</mi></mrow><mprescripts/><mrow><mtext> </mtext></mrow><mrow><mn mathvariant="normal">5</mn></mrow></mmultiscripts><msub><mrow><mi mathvariant="normal">D</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819950&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819949&type= 13.97000027 4.65666676 ] pathway, some values in Tab.1 Tab.1 need to be updated to match the <math id="M515"><mmultiscripts><mrow/><mprescripts/><mrow><mtext> </mtext></mrow><mrow><mn mathvariant="normal">7</mn></mrow></mmultiscripts><msub><mrow><mi mathvariant="normal">F</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><mo>→</mo><mmultiscripts><mrow><mi mathvariant="normal">D</mi></mrow><mprescripts/><mrow><mtext> </mtext></mrow><mrow><mn mathvariant="normal">5</mn></mrow></mmultiscripts><msub><mrow><mi mathvariant="normal">D</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819950&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819949&type= 13.97000027 4.65666676 transition （ Tab.2 Tab.2 ）. Eq. Donor Acceptor <math id="M516"><mi>J</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39820042&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39820041&type= 1.35466671 2.28600001 18⁃20 4 0 <math id="M517"><mfenced open="<" close="" separators="|"><mrow><mtext> </mtext></mrow></mfenced><mfenced open="‖" close="‖" separators="|"><mrow><msup><mrow><mi>U</mi></mrow><mrow><mfenced separators="|"><mrow><mn>2</mn></mrow></mfenced></mrow></msup></mrow></mfenced><msup><mrow><mfenced open="" close=">" separators="|"><mrow><mtext> </mtext></mrow></mfenced></mrow><mrow><mn>2</mn></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39820044&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39820043&type= 12.95400047 4.06400013 18⁃19 0.014 2 0 <math id="M518"><mfenced open="<" close="" separators="|"><mrow><mtext> </mtext></mrow></mfenced><mfenced open="‖" close="‖" separators="|"><mrow><msup><mrow><mi>U</mi></mrow><mrow><mfenced separators="|"><mrow><mn>4</mn></mrow></mfenced></mrow></msup></mrow></mfenced><msup><mrow><mfenced open="" close=">" separators="|"><mrow><mtext> </mtext></mrow></mfenced></mrow><mrow><mn>2</mn></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39820046&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39820045&type= 12.95400047 4.06400013 18 0.001 3 0 <math id="M519"><mfenced open="<" close="" separators="|"><mrow><mtext> </mtext></mrow></mfenced><mfenced open="‖" close="‖" separators="|"><mrow><msup><mrow><mi>U</mi></mrow><mrow><mfenced separators="|"><mrow><mn>6</mn></mrow></mfenced></mrow></msup></mrow></mfenced><msup><mrow><mfenced open="" close=">" separators="|"><mrow><mtext> </mtext></mrow></mfenced></mrow><mrow><mn>2</mn></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39820048&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39820047&type= 12.95400047 4.06400013 18 0.002 2 0 <math id="M520"><mfenced open="<" close="" separators="|"><mrow><mtext> </mtext></mrow></mfenced><mfenced open="‖" close="‖" separators="|"><mrow><mi>L</mi><mo>+</mo><mn>2</mn><mi>S</mi></mrow></mfenced><msup><mrow><mfenced open="" close=">" separators="|"><mrow><mtext> </mtext></mrow></mfenced></mrow><mrow><mn>2</mn></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39820050&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39820049&type= 16.51000023 3.30200005 20 0.696 0.027 3 <math id="M521"><mi>γ</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39820012&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39820011&type= 1.60866666 2.53999996 （cm -1 ） 23⁃24 350 106 Eq. 2 Donor⁃Acceptor <math id="M522"><mi>Δ</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39820016&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39820015&type= 1.86266661 2.11666679 /cm -1 23， 25 2 -631 It can be noticed that for this pathway, according to the selection rules, the <math id="M523"><mmultiscripts><mrow/><mprescripts/><mrow><mtext> </mtext></mrow><mrow><mn mathvariant="normal">7</mn></mrow></mmultiscripts><msub><mrow><mi mathvariant="normal">F</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><mo>→</mo><mmultiscripts><mrow><mi mathvariant="normal">D</mi></mrow><mprescripts/><mrow><mtext> </mtext></mrow><mrow><mn mathvariant="normal">5</mn></mrow></mmultiscripts><msub><mrow><mi mathvariant="normal">D</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819968&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819967&type= 14.73200035 4.65666676 transition has all <math id="M524"><mfenced open="<" close="" separators="|"><mrow><mtext> </mtext></mrow></mfenced><mfenced open="‖" close="‖" separators="|"><mrow><msup><mrow><mi>U</mi></mrow><mrow><mfenced separators="|"><mrow><mi>K</mi></mrow></mfenced></mrow></msup></mrow></mfenced><msup><mrow><mfenced open="" close=">" separators="|"><mrow><mtext> </mtext></mrow></mfenced></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mo>=</mo><mn mathvariant="normal">0</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819970&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819969&type= 23.19866753 4.65666676 . Therefore, <math id="M525"><msub><mrow><mi>W</mi></mrow><mrow><mi mathvariant="normal">d</mi><mi mathvariant="normal">⁃</mi><mi mathvariant="normal">d</mi></mrow></msub><mo>=</mo><msub><mrow><mi>W</mi></mrow><mrow><mi mathvariant="normal">d</mi><mi mathvariant="normal">q</mi><mi mathvariant="normal">⁃</mi><mi mathvariant="normal">q</mi><mi mathvariant="normal">d</mi></mrow></msub><mo>=</mo><msub><mrow><mi>W</mi></mrow><mrow><mi mathvariant="normal">q</mi><mi mathvariant="normal">⁃</mi><mi mathvariant="normal">q</mi></mrow></msub><mo>=</mo><mn mathvariant="normal">0</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819972&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819971&type= 34.37466812 3.80999994 and this pathway has only the contributions of the <math id="M526"><msub><mrow><mi>W</mi></mrow><mrow><mi mathvariant="normal">e</mi><mi mathvariant="normal">x</mi></mrow></msub><mo>=</mo><mn mathvariant="normal">3.3</mn><mo>×</mo><msup><mrow><mn mathvariant="normal">10</mn></mrow><mrow><mn mathvariant="normal">3</mn></mrow></msup><mtext> </mtext><msup><mrow><mi mathvariant="normal">s</mi></mrow><mrow><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819974&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819973&type= 27.85533524 4.14866638 and <math id="M527"><msub><mrow><mi>W</mi></mrow><mrow><mi mathvariant="normal">m</mi><mi mathvariant="normal">d</mi><mi mathvariant="normal">⁃</mi><mi mathvariant="normal">m</mi><mi mathvariant="normal">d</mi></mrow></msub><mo>=</mo><mn mathvariant="normal">1.3</mn><mo>×</mo><msup><mrow><mn mathvariant="normal">10</mn></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msup><mtext> </mtext><msup><mrow><mi mathvariant="normal">s</mi></mrow><mrow><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819976&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819975&type= 31.58066559 4.14866638 , in accordance with those in reference [ 36 36 ] . Adding these rates, considering the <math id="M528"><mmultiscripts><mrow/><mprescripts/><mrow><mtext> </mtext></mrow><mrow><mn mathvariant="normal">7</mn></mrow></mmultiscripts><msub><mrow><mi mathvariant="normal">F</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819978&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819977&type= 4.65666676 4.65666676 population fraction （ca. 0.64 at room temperature）, and the energy barrier factor <math id="M529"><mi>G</mi><mfenced separators="|"><mrow><mi>Δ</mi><mo>,</mo><mi>T</mi></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819980&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819979&type= 11.76866722 4.14866638 , yields the total rate: <math display="block" id="M530"><mtable columnalign="left"><mtr><mtd><mtext>             </mtext><msub><mrow><mi>W</mi></mrow><mrow><mi mathvariant="normal">T</mi><mi mathvariant="normal">O</mi><mi mathvariant="normal">T</mi><mi mathvariant="normal">A</mi><mi mathvariant="normal">L</mi></mrow></msub><mo>=</mo><mn mathvariant="normal">0.64</mn><mo>×</mo><mi>G</mi><mfenced separators="|"><mrow><mi>Δ</mi><mo>,</mo><mi>T</mi></mrow></mfenced><mo>×</mo><mfenced separators="|"><mrow><msub><mrow><mi>W</mi></mrow><mrow><mi mathvariant="normal">m</mi><mi mathvariant="normal">d</mi><mi mathvariant="normal">⁃</mi><mi mathvariant="normal">m</mi><mi mathvariant="normal">d</mi></mrow></msub><mo>+</mo><msub><mrow><mi>W</mi></mrow><mrow><mi mathvariant="normal">e</mi><mi mathvariant="normal">x</mi></mrow></msub></mrow></mfenced></mtd></mtr><mtr><mtd><mtext>                </mtext><mo>=</mo><mn mathvariant="normal">1.0</mn><mo>×</mo><msup><mrow><mn mathvariant="normal">10</mn></mrow><mrow><mn mathvariant="normal">3</mn></mrow></msup></mtd></mtr></mtable></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819982&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819981&type= 72.22066498 8.97466660 s -1 . It is important to pay attention that, in this case, <math id="M531"><mi>G</mi><mfenced separators="|"><mrow><mi>Δ</mi><mo>,</mo><mtext> </mtext><mi>T</mi></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819984&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819983&type= 12.53066730 4.14866638 was considered outside of the <math id="M532"><mi>F</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819986&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819985&type= 2.37066650 2.62466669 -factor just to match with the data in the literature. But it must be considered for each case appropriately. ⋅ E3 The first step in the Yb-Er upconversion process is given by the Yb（Ⅲ）[ 2 F 5/2 → 2 F 7/2 ]→Er（Ⅲ） [ <math id="M533"><mmultiscripts><mrow/><mprescripts/><mrow><mtext> </mtext></mrow><mrow><mn mathvariant="normal">4</mn></mrow></mmultiscripts><msub><mrow><mi mathvariant="normal">I</mi></mrow><mrow><mrow><mrow><mn mathvariant="normal">15</mn></mrow><mo>/</mo><mrow><mn mathvariant="normal">2</mn></mrow></mrow></mrow></msub><mo>→</mo><mmultiscripts><mrow><mi mathvariant="normal">I</mi></mrow><mprescripts/><mrow><mtext> </mtext></mrow><mrow><mn mathvariant="normal">4</mn></mrow></mmultiscripts><msub><mrow><mi mathvariant="normal">I</mi></mrow><mrow><mrow><mrow><mn mathvariant="normal">11</mn></mrow><mo>/</mo><mrow><mn mathvariant="normal">2</mn></mrow></mrow></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819988&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819987&type= 18.54199982 5.58799982 ] pathway. Tab.3 Tab.3 contains the data for calculating the pairwise energy transfer for the shorter Yb⁃Er distance （ <math id="M534"><msub><mrow><mi>R</mi></mrow><mrow><mi mathvariant="normal">D</mi><mi mathvariant="normal">A</mi></mrow></msub><mo>=</mo><mn mathvariant="normal">3.5</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819990&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819989&type= 15.15533447 3.80999994 Å） in NaYF 4 ∶Yb,Er material. Eq. Donor Acceptor <math id="M536"><mi>J</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39820042&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39820041&type= 1.35466671 2.28600001 18⁃20 5/2 15/2 ρ 21 0.065 0.065 <math id="M537"><mfenced open="<" close="" separators="|"><mrow><mtext> </mtext></mrow></mfenced><mfenced open="‖" close="‖" separators="|"><mrow><msup><mrow><mi>U</mi></mrow><mrow><mfenced separators="|"><mrow><mn>2</mn></mrow></mfenced></mrow></msup></mrow></mfenced><msup><mrow><mfenced open="" close=">" separators="|"><mrow><mtext> </mtext></mrow></mfenced></mrow><mrow><mn>2</mn></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39820044&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39820043&type= 12.95400047 4.06400013 18⁃19 0.122 5 0.027 6 <math id="M538"><mfenced open="<" close="" separators="|"><mrow><mtext> </mtext></mrow></mfenced><mfenced open="‖" close="‖" separators="|"><mrow><msup><mrow><mi>U</mi></mrow><mrow><mfenced separators="|"><mrow><mn>4</mn></mrow></mfenced></mrow></msup></mrow></mfenced><msup><mrow><mfenced open="" close=">" separators="|"><mrow><mtext> </mtext></mrow></mfenced></mrow><mrow><mn>2</mn></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39820046&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39820045&type= 12.95400047 4.06400013 18 0.409 6 0.000 2 <math id="M539"><mfenced open="<" close="" separators="|"><mrow><mtext> </mtext></mrow></mfenced><mfenced open="‖" close="‖" separators="|"><mrow><msup><mrow><mi>U</mi></mrow><mrow><mfenced separators="|"><mrow><mn>6</mn></mrow></mfenced></mrow></msup></mrow></mfenced><msup><mrow><mfenced open="" close=">" separators="|"><mrow><mtext> </mtext></mrow></mfenced></mrow><mrow><mn>2</mn></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39820048&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39820047&type= 12.95400047 4.06400013 18 0.857 5 0.394 2 <math id="M540"><msub><mrow><mi>Ω</mi></mrow><mrow><mn>2</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39820002&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39820001&type= 3.04800010 2.96333337 （10 -20 cm 2 ） 18 0.067 0.065 <math id="M541"><msub><mrow><mi>Ω</mi></mrow><mrow><mn>4</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39820004&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39820003&type= 3.04800010 2.96333337 （10 -20 cm 2 ） 18 0.229 0.259 <math id="M542"><msub><mrow><mi>Ω</mi></mrow><mrow><mn>6</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39820006&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39820005&type= 3.04800010 2.96333337 （10 -20 cm 2 ） 18 0.551 0.637 <math id="M543"><mfenced open="<" close=">" separators="|"><mrow><msup><mrow><mi>r</mi></mrow><mrow><mn>2</mn></mrow></msup></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39820008&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39820007&type= 4.57200003 3.04800010 （10 -17 cm 2 ） 19 1.82 1.99 <math id="M544"><mfenced open="<" close="" separators="|"><mrow><mtext> </mtext></mrow></mfenced><mfenced open="‖" close="‖" separators="|"><mrow><mi>L</mi><mo>+</mo><mn>2</mn><mi>S</mi></mrow></mfenced><msup><mrow><mfenced open="" close=">" separators="|"><mrow><mtext> </mtext></mrow></mfenced></mrow><mrow><mn>2</mn></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39820050&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39820049&type= 16.51000023 3.30200005 20 30.86 0 <math id="M545"><mi>γ</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39820012&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39820011&type= 1.60866666 2.53999996 （cm -1 ） 23⁃24 400 450 Eq. 2 Donor⁃Acceptor <math id="M546"><msub><mrow><mi>R</mi></mrow><mrow><mi mathvariant="normal">D</mi><mi mathvariant="normal">A</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39820014&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39820013&type= 4.23333359 2.87866688 /cm 22 2 3.50×10 -8 <math id="M547"><mi>Δ</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39820016&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39820015&type= 1.86266661 2.11666679 /cm -1 23， 25 2 129 <math id="M548"><msub><mrow><mi>ρ</mi></mrow><mrow><mi mathvariant="normal">f</mi><mi mathvariant="normal">f</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39820018&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39820017&type= 2.70933342 3.38666677 17 2 1.38×10 -3 at 3.50 Å Applying the data in Tab.3 Tab.3 leads to: <math display="block" id="M549"><msub><mrow><mi>W</mi></mrow><mrow><mi mathvariant="normal">d</mi><mi mathvariant="normal">⁃</mi><mi mathvariant="normal">d</mi></mrow></msub><mo>=</mo><mn mathvariant="normal">5.2</mn><mo>×</mo><msup><mrow><mn mathvariant="normal">10</mn></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39820022&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39820021&type= 24.12999916 4.06400013 s -1 ， <math display="block" id="M550"><msub><mrow><mi>W</mi></mrow><mrow><mi mathvariant="normal">d</mi><mi mathvariant="normal">q</mi><mi mathvariant="normal">⁃</mi><mi mathvariant="normal">q</mi><mi mathvariant="normal">d</mi></mrow></msub><mo>=</mo><mn mathvariant="normal">3.1</mn><mo>×</mo><msup><mrow><mn mathvariant="normal">10</mn></mrow><mrow><mn mathvariant="normal">4</mn></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39820024&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39820023&type= 26.07733345 4.06400013 s -1 ， <math display="block" id="M551"><msub><mrow><mi>W</mi></mrow><mrow><mi mathvariant="normal">q</mi><mi mathvariant="normal">⁃</mi><mi mathvariant="normal">q</mi></mrow></msub><mo>=</mo><mn mathvariant="normal">3.2</mn><mo>×</mo><msup><mrow><mn mathvariant="normal">10</mn></mrow><mrow><mn mathvariant="normal">6</mn></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39820026&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39820025&type= 24.12999916 4.06400013 s -1 ， <math display="block" id="M552"><msub><mrow><mi>W</mi></mrow><mrow><mi mathvariant="normal">m</mi><mi mathvariant="normal">d</mi><mi mathvariant="normal">⁃</mi><mi mathvariant="normal">m</mi><mi mathvariant="normal">d</mi></mrow></msub><mo>=</mo><mn mathvariant="normal">0</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39820028&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39820027&type= 14.90133286 3.80999994 ， <math display="block" id="M553"><msub><mrow><mi>W</mi></mrow><mrow><mi mathvariant="normal">e</mi><mi mathvariant="normal">x</mi></mrow></msub><mo>=</mo><mn mathvariant="normal">3.6</mn><mo>×</mo><msup><mrow><mn mathvariant="normal">10</mn></mrow><mrow><mn mathvariant="normal">6</mn></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39820030&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39820029&type= 23.28333473 4.06400013 s -1 . The total pairwise energy transfer Yb（Ⅲ） [ <math id="M554"><mmultiscripts><mrow/><mprescripts/><mrow><mtext> </mtext></mrow><mrow><mn mathvariant="normal">2</mn></mrow></mmultiscripts><msub><mrow><mi mathvariant="normal">F</mi></mrow><mrow><mrow><mrow><mn mathvariant="normal">5</mn></mrow><mo>/</mo><mrow><mn mathvariant="normal">2</mn></mrow></mrow></mrow></msub><mo>→</mo><mmultiscripts><mrow><mi mathvariant="normal">F</mi></mrow><mprescripts/><mrow><mtext> </mtext></mrow><mrow><mn mathvariant="normal">2</mn></mrow></mmultiscripts><msub><mrow><mi mathvariant="normal">F</mi></mrow><mrow><mrow><mrow><mn mathvariant="normal">7</mn></mrow><mo>/</mo><mrow><mn mathvariant="normal">2</mn></mrow></mrow></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39820032&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39820031&type= 18.20333290 5.58799982 ] → Er（Ⅲ） [ <math id="M555"><mmultiscripts><mrow/><mprescripts/><mrow><mtext> </mtext></mrow><mrow><mn mathvariant="normal">4</mn></mrow></mmultiscripts><msub><mrow><mi mathvariant="normal">I</mi></mrow><mrow><mrow><mrow><mn mathvariant="normal">15</mn></mrow><mo>/</mo><mrow><mn mathvariant="normal">2</mn></mrow></mrow></mrow></msub><mo>→</mo><mmultiscripts><mrow><mi mathvariant="normal">I</mi></mrow><mprescripts/><mrow><mtext> </mtext></mrow><mrow><mn mathvariant="normal">4</mn></mrow></mmultiscripts><msub><mrow><mi mathvariant="normal">I</mi></mrow><mrow><mrow><mrow><mn mathvariant="normal">11</mn></mrow><mo>/</mo><mrow><mn mathvariant="normal">2</mn></mrow></mrow></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39820034&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39820033&type= 18.11866570 5.58799982 ] pathway is <math id="M556"><msub><mrow><mi>W</mi></mrow><mrow><mi mathvariant="normal">T</mi><mi mathvariant="normal">O</mi><mi mathvariant="normal">T</mi><mi mathvariant="normal">A</mi><mi mathvariant="normal">L</mi></mrow></msub><mo>=</mo><mn mathvariant="normal">6.9</mn><mo>×</mo><msup><mrow><mn mathvariant="normal">10</mn></mrow><mrow><mn mathvariant="normal">6</mn></mrow></msup><mtext> </mtext><msup><mrow><mi mathvariant="normal">s</mi></mrow><mrow><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39820036&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39820035&type= 32.59666443 4.14866638 , which also reproduces the value for the shortest Yb⁃Er distance for the first upconverting step from reference [ 16 16 ] . ⋅ E4 Repeating the same procedure as E3 but updating the acceptor transition from <math id="M557"><mmultiscripts><mrow/><mprescripts/><mrow><mtext> </mtext></mrow><mrow><mn mathvariant="normal">4</mn></mrow></mmultiscripts><msub><mrow><mi mathvariant="normal">I</mi></mrow><mrow><mrow><mrow><mn mathvariant="normal">15</mn></mrow><mo>/</mo><mrow><mn mathvariant="normal">2</mn></mrow></mrow></mrow></msub><mo>→</mo><mmultiscripts><mrow><mi mathvariant="normal">I</mi></mrow><mprescripts/><mrow><mtext> </mtext></mrow><mrow><mn mathvariant="normal">4</mn></mrow></mmultiscripts><msub><mrow><mi mathvariant="normal">I</mi></mrow><mrow><mrow><mrow><mn mathvariant="normal">11</mn></mrow><mo>/</mo><mrow><mn mathvariant="normal">2</mn></mrow></mrow></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39820038&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39820037&type= 17.78000069 5.58799982 to <math id="M558"><mmultiscripts><mrow/><mprescripts/><mrow><mtext> </mtext></mrow><mrow><mn mathvariant="normal">4</mn></mrow></mmultiscripts><msub><mrow><mi mathvariant="normal">I</mi></mrow><mrow><mrow><mrow><mn mathvariant="normal">11</mn></mrow><mo>/</mo><mrow><mn mathvariant="normal">2</mn></mrow></mrow></mrow></msub><mo>→</mo><mmultiscripts><mrow><mi mathvariant="normal">I</mi></mrow><mprescripts/><mrow><mtext> </mtext></mrow><mrow><mn mathvariant="normal">4</mn></mrow></mmultiscripts><msub><mrow><mi mathvariant="normal">I</mi></mrow><mrow><mrow><mrow><mn mathvariant="normal">7</mn></mrow><mo>/</mo><mrow><mn mathvariant="normal">2</mn></mrow></mrow></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39820040&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39820039&type= 17.18733406 5.58799982 , we obtain the input data in Tab.4 Tab.4 . Eq. Donor Acceptor <math id="M559"><mi>J</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39820042&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39820041&type= 1.35466671 2.28600001 18⁃20 5/2 11/2 <math id="M560"><mfenced open="<" close="" separators="|"><mrow><mtext> </mtext></mrow></mfenced><mfenced open="‖" close="‖" separators="|"><mrow><msup><mrow><mi>U</mi></mrow><mrow><mfenced separators="|"><mrow><mn>2</mn></mrow></mfenced></mrow></msup></mrow></mfenced><msup><mrow><mfenced open="" close=">" separators="|"><mrow><mtext> </mtext></mrow></mfenced></mrow><mrow><mn>2</mn></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39820044&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39820043&type= 12.95400047 4.06400013 18⁃19 0.122 5 0.032 0 <math id="M561"><mfenced open="<" close="" separators="|"><mrow><mtext> </mtext></mrow></mfenced><mfenced open="‖" close="‖" separators="|"><mrow><msup><mrow><mi>U</mi></mrow><mrow><mfenced separators="|"><mrow><mn>4</mn></mrow></mfenced></mrow></msup></mrow></mfenced><msup><mrow><mfenced open="" close=">" separators="|"><mrow><mtext> </mtext></mrow></mfenced></mrow><mrow><mn>2</mn></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39820046&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39820045&type= 12.95400047 4.06400013 18 0.409 6 0.265 3 <math id="M562"><mfenced open="<" close="" separators="|"><mrow><mtext> </mtext></mrow></mfenced><mfenced open="‖" close="‖" separators="|"><mrow><msup><mrow><mi>U</mi></mrow><mrow><mfenced separators="|"><mrow><mn>6</mn></mrow></mfenced></mrow></msup></mrow></mfenced><msup><mrow><mfenced open="" close=">" separators="|"><mrow><mtext> </mtext></mrow></mfenced></mrow><mrow><mn>2</mn></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39820048&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39820047&type= 12.95400047 4.06400013 18 0.857 5 0.154 5 <math id="M563"><mfenced open="<" close="" separators="|"><mrow><mtext> </mtext></mrow></mfenced><mfenced open="‖" close="‖" separators="|"><mrow><mi>L</mi><mo>+</mo><mn>2</mn><mi>S</mi></mrow></mfenced><msup><mrow><mfenced open="" close=">" separators="|"><mrow><mtext> </mtext></mrow></mfenced></mrow><mrow><mn>2</mn></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39820050&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39820049&type= 16.51000023 3.30200005 20 30.86 0 Eq. 2 Donor⁃Acceptor <math id="M564"><mi>Δ</mi><mo>/</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39820052&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39820051&type= 2.70933342 2.11666679 cm -1 23， 25 2 -111 In this case, the total pairwise energy transfer of the Yb（Ⅲ） [ <math id="M565"><mmultiscripts><mrow/><mprescripts/><mrow><mtext> </mtext></mrow><mrow><mn mathvariant="normal">2</mn></mrow></mmultiscripts><msub><mrow><mi mathvariant="normal">F</mi></mrow><mrow><mrow><mrow><mn mathvariant="normal">5</mn></mrow><mo>/</mo><mrow><mn mathvariant="normal">2</mn></mrow></mrow></mrow></msub><mo>→</mo><mmultiscripts><mrow><mi mathvariant="normal">F</mi></mrow><mprescripts/><mrow><mtext> </mtext></mrow><mrow><mn mathvariant="normal">2</mn></mrow></mmultiscripts><msub><mrow><mi mathvariant="normal">F</mi></mrow><mrow><mrow><mrow><mn mathvariant="normal">7</mn></mrow><mo>/</mo><mrow><mn mathvariant="normal">2</mn></mrow></mrow></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39820056&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39820055&type= 17.44133186 5.58799982 ] → Er（Ⅲ） [ <math id="M566"><mmultiscripts><mrow/><mprescripts/><mrow><mtext> </mtext></mrow><mrow><mn mathvariant="normal">4</mn></mrow></mmultiscripts><msub><mrow><mi mathvariant="normal">I</mi></mrow><mrow><mrow><mrow><mn mathvariant="normal">11</mn></mrow><mo>/</mo><mrow><mn mathvariant="normal">2</mn></mrow></mrow></mrow></msub><mo>→</mo><mmultiscripts><mrow><mi mathvariant="normal">I</mi></mrow><mprescripts/><mrow><mtext> </mtext></mrow><mrow><mn mathvariant="normal">4</mn></mrow></mmultiscripts><msub><mrow><mi mathvariant="normal">I</mi></mrow><mrow><mrow><mrow><mn mathvariant="normal">7</mn></mrow><mo>/</mo><mrow><mn mathvariant="normal">2</mn></mrow></mrow></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39820058&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39820057&type= 16.76399994 5.58799982 ] pathway is <math id="M567"><msub><mrow><mi>W</mi></mrow><mrow><mi mathvariant="normal">T</mi><mi mathvariant="normal">O</mi><mi mathvariant="normal">T</mi><mi mathvariant="normal">A</mi><mi mathvariant="normal">L</mi></mrow></msub><mo>=</mo><mn mathvariant="normal">5.1</mn><mo>×</mo><msup><mrow><mn mathvariant="normal">10</mn></mrow><mrow><mn mathvariant="normal">6</mn></mrow></msup><mtext> </mtext><msup><mrow><mi mathvariant="normal">s</mi></mrow><mrow><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39820060&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39820059&type= 32.59666443 4.14866638 , in agreement with the value for the shortest Yb⁃Er distance for the second upconverting step from reference [ 16 16 ] . All these examples are worked out as separated Mathcad® script files available in the Supporting Information （Examples_Mathcad.zip）. 9　Conclusions 1 In this tutorial review, we presented the equations for the energy transfer rates between lanthanide ions in an alternative form, which aimed at an easier interpretation and a simpler step-by-step calculation. Four examples were provided to illustrate this procedure for relevant lanthanide pairs. We expect that the present work will guide students and stimulate the research fn the field of energy transfer between lanthanide ions. References JAHNKE T ， SANN H ， HAVERMEIER T ， et al . Ultrafast energy transfer between water molecules ［J］. Nat. Phys. ， 2010 ， 6 （ 2 ）： 139 - 142 . Ultrafast energy transfer between water molecules XUAN M J ， ZHAO J ， SHAO J X ， et al . Perspective of energy transfer from light energy into biological energy ［J］. Green Energy Environ. ， 2017 ， 2 （ 1 ）： 18 - 22 . 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Effect of Yb3+ concentration on upconversion luminescence in Yb3+， Tm3+ co-doped Lu2O3 nanophosphors LIANG L L ， TEH D B L ， DINH N D ， et al . Upconversion amplification through dielectric superlensing modulation ［J］. Nat. Commun. ， 2019 ， 10 （ 1 ）： 1391 . Upconversion amplification through dielectric superlensing modulation TU L P ， LIU X M ， WU F ， et al . Excitation energy migration dynamics in upconversion nanomaterials ［J］. Chem. Soc. Rev. ， 2015 ， 44 （ 6 ）： 1331 - 1345 . Excitation energy migration dynamics in upconversion nanomaterials The authors are grateful for the financial support from CNPq, CAPES, FACEPE, and FINEP agencies. This work was funded by the Public Call n. 03 Produtividade em Pesquisa PROPESQ/PRPG/UFPB project number PVN13305⁃2020, and PROPESQ/CNPq/UFPB PIN11132⁃2019. This work was developed within the scope of the project CICECO⁃Aveiro Institute of Materials, UIDB/50011/2020 and UIDP/50011/2020, financed by Portuguese funds through the FCT/MEC and when appropriate co⁃financed by FEDER under the PT2020 Partnership Agreement. RLL is grateful for the partial financial support under grants: Pronex APQ⁃0675⁃1.06/14, INCT⁃NANOMARCS APQ⁃0549⁃1.06/17, APQ⁃1007⁃1.06/15, and CNPq⁃PQ fellowship (Proc. 309177/2018⁃9). Author Statement Albano N. Carneiro Neto: Conceptualization, Methodology, Software, Validation, Formal analysis, Investigation, Data Curation, Writing⁃Original Draft, Writing⁃Review & Editing. Renaldo T. Moura Jr.: Conceptualization, Writing⁃Review & Editing. Jorge A. A. Coelho: Conceptualization, Methodology, Writing⁃Review & Editing. Mauro E. Silva⁃Junior: Conceptualization, Writing⁃Review & Editing. Janderson L. Costa: Conceptualization, Writing⁃Review & Editing. Oscar L. Malta: Conceptualization, Methodology, Formal analysis, Writing⁃Original Draft, Writing⁃Review & Editing, Supervision. Ricardo L. Longo: Conceptualization, Methodology, Investigation, Formal analysis, Writing⁃Original Draft, Writing⁃Review & Editing, Validation, Supervision, Project administration. 10.37188/CJL.EN20220007 O482.31 A 1000-7032（2022）12-1871-21 Albano N Carneiro Neto,Renaldo T Moura Jr,Jorge A A Coelho等.镧系离子间无辐射能量传递速率的教程综述[J].发光学报,2022,43(12):1871-1891. Albano N Carneiro Neto,Renaldo T Moura Jr,Jorge A A Coelho等.镧系离子间无辐射能量传递速率的教程综述[J].发光学报,2022,43(12):1871-1891. Neto Albano N Carneiro,Moura Renaldo T,Coelho Jorge A A,et al.A Tutorial Review on the Nonradiative Energy Transfer Rates between Lanthanide Ions[J].Chinese Journal of Luminescence,2022,43(12):1871-1891. Neto Albano N Carneiro,Moura Renaldo T,Coelho Jorge A A,et al.A Tutorial Review on the Nonradiative Energy Transfer Rates between Lanthanide Ions[J].Chinese Journal of Luminescence,2022,43(12):1871-1891. 2022-05-20 2022-06-19 2022-12-05 2023-02-13 发光学报 12 43