The Variation Method - Methods of Molecular Quantum Mechanics - page 72

<

5

c s
2

3c 0 þ c s
2

c s
c 0 þ c s

ð s 2 j 1s 2 Þ¼

2
4

3
5

! 3

c 0 þ c s
3

6c 0 þ c s
3c 0 þ c s

c 0 þ c s
3c 0 þ c s

ð 4 : 69 Þ

ð s1s j 1s 2 Þ¼

S s1s 1

:

c 0 c s
ð c 0 þ c s Þ

ð s1s j s1s Þ¼ 44

7

<
:

5

c p
2

3c 0 þ c p
2

c p
c 0 þ c p

ð 2p z j 1s 2 Þ¼

ð 4 : 70 Þ

c 0 c p
ð c 0 þ c p Þ

28
3

ð 1s 2p z j 1s2p z Þ¼

7

where the nonorthogonality integral S s1s ¼h 1s j s i¼ S is given by
Equation 3.68.
Then:

(

Þ 1

J ¼ð 2s 2 j 1s 2 Þ¼ð 1 S 2 ½ð s 2 j 1s 2 Þþ S 2 ð 1s 2 j 1s 2 Þ 2S ð s1s j 1s 2 Þ

Þ 1

K ¼ð 1s 2s j 1s 2s Þ¼ð 1 S 2 ½ð s1s j s1s Þþ S 2 ð 1s 2 j 1s 2 Þ 2S ð s1s j 1s 2 Þ

ð 4 : 71 Þ

J 0 ¼ð 2p z j 1s 2 K 0 ¼ð 1s 2p z j 1s 2p z Þ

Þ; ð 4 :

72 Þ

As a final example, we give below the calculation of J 0 and K 0 .

(i) Evaluation of J 0

3 c p ¥
0

4

dr exp ð 2c p r Þ½ r 3 exp ð 2c 0 r Þð r 3 þ c 0 r 4

J 0 ¼ð 2p z j J 1s 2 Þ¼

Þ

< =

2
4

3
5

4
3 c p

3 2
ð 2c p Þ

3 2
ð 2c 0 þ 2c p Þ

4 3 2
ð 2c 0 þ 2c p Þ

¼

4 4 þ c 0

:

5

;

5

c p
2

3c 0 þ c p
2

c p
c 0 þ c p

ð 4 : 73 Þ

¼