Page 264 - PRINCIPLES OF QUANTUM MECHANICS as Applied to Chemistry and Chemical Physics

9.5 Degenerate perturbation theory 255
ø (0) ˆj2si, ø (0) ˆj2p 0 i
1 2
(9:73)
ø (0) ˆj2p 1 i, ø (0) ˆj2p ÿ1 i
3 4
The `correct'set of unperturbed eigenfunction ö (0) are, then
á
4
X
ö (0) ˆ c áâ ø (0) ˆ c á1 j2si‡ c á2 j2p 0 i‡ c á3 j2p 1 i‡ c á4 j2p ÿ1 i,
á
â
âˆ1
á ˆ 1, 2, 3, 4 (9:74)
^
The matrix elements H (1) in this example are
áâ
(0)
(0)
(0)
^
(0)
H (1) ˆ eE hø jzjø iˆ eE hø jr cos èjø i
á
á
â
áâ
â

2ð ð 1
(0)
2
ˆ eE ø (0) ø r cos èr sin è dr dè dj (9:75)
á
â
0 0 0
These matrix elements vanish unless Äm ˆ 0 and Äl ˆ 1. Thus, only the
(1)
(1)
^
^
matrix element H , which equals H , is non-zero.
21
12
(1)
^
To evaluate the matrix element H , we substitute the 2s wave function from
12
equation (6.59) and the 2p 0 wave function from equation (6.60a) into (9.75)
ð 2ð
1 r
^
4 ÿ1
2
^
H (1) ˆ H (1) ˆ eE [ð(2a 0 ) ] r 4 1 ÿ e ÿr=a 0 dr cos è sin è dè dj
12
21
0 2a 0 0 0
ˆÿ3eE a 0
where equations (A.26) and (A.28) are used.
The secular determinant (9.65) is

ÿE (1) ÿ3eE a 0 0 0
2
(1) 0 0
ÿ3eE a 0 ÿE 2 ˆ 0
(1)
0 0 ÿE 0
2
(1)
0 0 0 ÿE 2

which expands to
(1) 2 2 (1) 2
[(E ) ÿ (3eE a 0 ) ](E ) ˆ 0
2
2
(1)
The four roots are E 2 ˆÿ3eE a 0 ,3eE a 0 , 0, 0, so that to ®rst order the
perturbed energy levels are
ÿe9 2 ÿe9 2
E 21 ˆ ÿ 3eE a 0 , E 23 ˆ
8a 0 8a 0
(9:76)
ÿe9 2 ÿe9 2
E 22 ˆ ‡ 3eE a 0 , E 24 ˆ
8a 0 8a 0
The four linear homogeneous simultaneous equations (9.64) are