The quantity ϕ is an azimuthal angle around the molecular axis (i.e., the straight
line connecting the two nuclei). Then, we have
H 11 ¼ AjHjA
h
i¼ A À
ħ
2
2m
∇
2 À
e
2
4πε 0 r A
A
(
)
þ A À
e
2
4πε 0 r B
A
(
)
þ A
e
2
4πε 0 R
A
(
)
¼ E 1s À
e
2
4πε 0
A
1
r B
A
(
)
þ
e
2
4πε 0 R
,
ð19:218Þ
where E 1s is the same as that given in (3.258) with Z ¼ n ¼ 1 and μ replaced with
m (i.e., E 1s ¼ À
ħ
2
2ma 2 ). Using a coordinate representation of (3.301), we have
j Ai ¼
ffiffiffi
1
π
r
a
À3=2 e
Àr A =a
:
ð19:219Þ
Moreover considering (19.217), we have
A
1
r B
A
(
)
¼
1
πa 3
ð
dτe
À2r A =a 1
r B
:
Converting Cartesian coordinates to ellipsoidal coordinates [3] such that
ð
dτ ¼
R
2
3 ð 2π
0
dϕ
ð 1
1
dμ
ð 1
À1
dν μ
2
À ν
2
À
Á
,
we have
A
B
e
Fig. 19.15 Configuration of electron (e) and hydrogen nuclei (A and B). r A and r B denote a
separation between the electron and A and that between the electron and B, respectively. R denotes
a separation between A and B
19.5 MO Calculations of Methane
791
line connecting the two nuclei). Then, we have
H 11 ¼ AjHjA
h
i¼ A À
ħ
2
2m
∇
2 À
e
2
4πε 0 r A
A
(
)
þ A À
e
2
4πε 0 r B
A
(
)
þ A
e
2
4πε 0 R
A
(
)
¼ E 1s À
e
2
4πε 0
A
1
r B
A
(
)
þ
e
2
4πε 0 R
,
ð19:218Þ
where E 1s is the same as that given in (3.258) with Z ¼ n ¼ 1 and μ replaced with
m (i.e., E 1s ¼ À
ħ
2
2ma 2 ). Using a coordinate representation of (3.301), we have
j Ai ¼
ffiffiffi
1
π
r
a
À3=2 e
Àr A =a
:
ð19:219Þ
Moreover considering (19.217), we have
A
1
r B
A
(
)
¼
1
πa 3
ð
dτe
À2r A =a 1
r B
:
Converting Cartesian coordinates to ellipsoidal coordinates [3] such that
ð
dτ ¼
R
2
3 ð 2π
0
dϕ
ð 1
1
dμ
ð 1
À1
dν μ
2
À ν
2
À
Á
,
we have
A
B
e
Fig. 19.15 Configuration of electron (e) and hydrogen nuclei (A and B). r A and r B denote a
separation between the electron and A and that between the electron and B, respectively. R denotes
a separation between A and B
19.5 MO Calculations of Methane
791