The Hamiltonian describing a rare-earth impurity ion is assumed to be composed of the free ion Hamiltonian

H

F

and the one-electron crystal-field Hamiltonian

H

CF

·We write

H

CF

=

V

0

+

H

’

CF

with

<

img style="vertical-align:middle" alt="" order="0" src="http://www.rhhz.net/qikantupian/fgxb-19870101-1.gif" /

>

here

V

0

is the part of

H

CF

that is independent of angular variables.Thus the Hamiltionian of the system can be written as

H

=

H

F

+

V

0

+

H

’

CF

(3)The eigenvalues of

H

F

+

V

0

are usually called free ion levels. Because of the existence of

H

’

CF

each free ion level will split into Stark levels

and the center of gravity of the Stark levels will be shifted from correspondent free ion level. It is not concluded whether these shifts are comparable with the contributions of configuration interaction

spin-spin interaction and other interactions. The systematical calculation on these shifts has not been reported so far as we know. In this paper we present the results for Er

3+

in Y

3

Al

5

O

12

.