somewhat complicated. To simplify the situation, (i) in, e.g., (19.207) let us consider
a case where jH 11 j ) j H 22 j or jH 11 j ( j H 22 j. In that case, (H 11 À H 22 )
2
dominates inside a square root, and so ignoring À2H 12 S 12 and ÀS 12
2 we have
λ % H 11 or λ % H 22 . Inserting these values into (19.199), we have either Ψ %
H
A 1
ð Þ or Ψ % C2s, where Ψ is a resulting MO. This implies that no interaction would
arise between H
A 1
ð Þ and C2s. (ii) If, on the other hand, H 11 ¼ H 22 , we would get
λ ¼
H 11 À H 12 S 12 Æ j H 12 À H 11 S 12 j
1 À S 12
2
:
ð19:215Þ
As H 12 À H 11 S 12 is positive or negative, we have a following alternative:
Case I: H 12 À H 11 S 12 > 0. We have λ H ¼
H 11 þH 12
1þS 12
and λ L ¼
H 11 ÀH 12
1ÀS 12
, where
λ H > λ L .
Case II: H 12 À H 11 S 12 < 0. We have λ H ¼
H 11 ÀH 12
1ÀS 12
and λ L ¼
H 11 þH 12
1þS 12
, where
λ H > λ L .
In this case, we would anticipate maximum orbital mixing between H
A 1
ð Þ and
C2s. (iii) If H 11 and H 22 are moderately different in between the above (i) and (ii), the
orbital mixing is likely to be moderate. This is an actual situation. With (19.209) we
have eigenvalues expressed similarly to those of (19.207). Therefore, classifications
related to the above Cases I and II hold with G 12 À G 11 T 12 , F 12 À F 11 V 12 , and
K 12 À K 11 W 12 as well.
In spite of simplicity of (19.215), the quantities H 11 , H 12 , and S 12 are hard to
calculate. In general cases including the present example (i.e., methane), the difficulty in calculating these quantities results essentially from the fact that we are
dealing with a many-particle interactions that include electron repulsion. Nonetheless, for a simplest case of a hydrogen molecular ion H
þ
2 the estimation is feasible
[3]. Let us estimate H 12 À H 11 S 12 quantitatively according to Atkins and
Friedman [3].
The Hamiltonian of H
þ
2 is described as
H ¼ À
ħ
2
2m
∇
2 À
e
2
4πε 0 r A
À
e
2
4πε 0 r B
þ
e
2
4πε 0 R
,
ð19:216Þ
where m is an electron rest mass and other symbols are defined in Fig. 19.15; the last
term represents the repulsive interaction between the two hydrogen nuclei. To
estimate the quantities in (19.215), it is convenient to use dimensionless ellipsoidal
coordinates
μ
ν
ϕ
0
B
@
1
C
A such that [3]
μ ¼
r A þ r B
R
and ν ¼
r A À r B
R
:
ð19:217Þ
790
19 Applications of Group Theory to Physical Chemistry
a case where jH 11 j ) j H 22 j or jH 11 j ( j H 22 j. In that case, (H 11 À H 22 )
2
dominates inside a square root, and so ignoring À2H 12 S 12 and ÀS 12
2 we have
λ % H 11 or λ % H 22 . Inserting these values into (19.199), we have either Ψ %
H
A 1
ð Þ or Ψ % C2s, where Ψ is a resulting MO. This implies that no interaction would
arise between H
A 1
ð Þ and C2s. (ii) If, on the other hand, H 11 ¼ H 22 , we would get
λ ¼
H 11 À H 12 S 12 Æ j H 12 À H 11 S 12 j
1 À S 12
2
:
ð19:215Þ
As H 12 À H 11 S 12 is positive or negative, we have a following alternative:
Case I: H 12 À H 11 S 12 > 0. We have λ H ¼
H 11 þH 12
1þS 12
and λ L ¼
H 11 ÀH 12
1ÀS 12
, where
λ H > λ L .
Case II: H 12 À H 11 S 12 < 0. We have λ H ¼
H 11 ÀH 12
1ÀS 12
and λ L ¼
H 11 þH 12
1þS 12
, where
λ H > λ L .
In this case, we would anticipate maximum orbital mixing between H
A 1
ð Þ and
C2s. (iii) If H 11 and H 22 are moderately different in between the above (i) and (ii), the
orbital mixing is likely to be moderate. This is an actual situation. With (19.209) we
have eigenvalues expressed similarly to those of (19.207). Therefore, classifications
related to the above Cases I and II hold with G 12 À G 11 T 12 , F 12 À F 11 V 12 , and
K 12 À K 11 W 12 as well.
In spite of simplicity of (19.215), the quantities H 11 , H 12 , and S 12 are hard to
calculate. In general cases including the present example (i.e., methane), the difficulty in calculating these quantities results essentially from the fact that we are
dealing with a many-particle interactions that include electron repulsion. Nonetheless, for a simplest case of a hydrogen molecular ion H
þ
2 the estimation is feasible
[3]. Let us estimate H 12 À H 11 S 12 quantitatively according to Atkins and
Friedman [3].
The Hamiltonian of H
þ
2 is described as
H ¼ À
ħ
2
2m
∇
2 À
e
2
4πε 0 r A
À
e
2
4πε 0 r B
þ
e
2
4πε 0 R
,
ð19:216Þ
where m is an electron rest mass and other symbols are defined in Fig. 19.15; the last
term represents the repulsive interaction between the two hydrogen nuclei. To
estimate the quantities in (19.215), it is convenient to use dimensionless ellipsoidal
coordinates
μ
ν
ϕ
0
B
@
1
C
A such that [3]
μ ¼
r A þ r B
R
and ν ¼
r A À r B
R
:
ð19:217Þ
790
19 Applications of Group Theory to Physical Chemistry