expresses a secular equation with respect to two SALCs that belong to the same
irreducible representation.
The first equation of (19.199) reads as
1 À S 12
2
À
Á λ
2
À H 11 þ H 22 À 2H 12 S 12
ð
Þ λ þ H 11 H 22 À H 12
2
¼ 0:
ð19:200Þ
In (19.200) we define quantities as follows:
S 12
ð
H
A 1
ð Þ C2sdτ hH
A 1
ð Þ C2s
j
i¼
hH 1 þ H 2 þ H 3 þ H 4 C2s
j
i
2
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi
1 þ 3S
HH
p
¼
2hH 1 C2s
j
i
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi
1 þ 3S
HH
p
¼
2S
CH
A 1
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi
1 þ 3S
HH
p
:
ð19:201Þ
In (19.201), an overlap integral between hydrogen atomic orbitals and C2s is
identical from the symmetry requirement and it is defined as
S
CH
A 1
H 1 jC2s
h
i :
ð19:202Þ
Also in (19.200), other quantities are defined as follows:
H 11 H
A 1
ð Þ
jHH
A 1
ð Þ
D
E
¼
α
H
þ 3β
HH
1 þ 3S
HH ,
ð19:203Þ
H 22 C2sjHC2s
h
i ,
ð19:204Þ
H 12 H
A 1
ð Þ
jHC2s
D
E
¼
H 1 þ H 2 þ H 3 þ H 4 jHC2s
h
i
2
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi
1 þ 3S
HH
p
¼
2 H 1 jHC2s
h
i
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi
1 þ 3S
HH
p
¼
2β
CH
A 1
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi
1 þ 3S
HH
p
,
ð19:205Þ
where H is a Hamiltonian of a methane molecule. In (19.203) and (19.205),
moreover, we define the quantities as
α
H
H 1 HH 1 i, β
HH
hH 1
HH 2
, β
CH
A 1
H 1 jHC2s
h
i :
ð19:206Þ
The quantity of H 11 is a “Coulomb” integral of the hydrogen-based SALC that
involves four hydrogen atoms. Solving the first equation of (19.199), we get
λ¼
H 11 þH 22 À2H 12 S 12 Æ
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
H 11 ÀH 22
ð
Þ
2 þ4 H 12
2
þH 11 H 22 S 12
2
ÀH 12 S 12 H 11 þH 22
ð
Þ
Â
Ã
q
2 1ÀS 12
2
À
Á
:
ð19:207Þ
788
19 Applications of Group Theory to Physical Chemistry
irreducible representation.
The first equation of (19.199) reads as
1 À S 12
2
À
Á λ
2
À H 11 þ H 22 À 2H 12 S 12
ð
Þ λ þ H 11 H 22 À H 12
2
¼ 0:
ð19:200Þ
In (19.200) we define quantities as follows:
S 12
ð
H
A 1
ð Þ C2sdτ hH
A 1
ð Þ C2s
j
i¼
hH 1 þ H 2 þ H 3 þ H 4 C2s
j
i
2
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi
1 þ 3S
HH
p
¼
2hH 1 C2s
j
i
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi
1 þ 3S
HH
p
¼
2S
CH
A 1
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi
1 þ 3S
HH
p
:
ð19:201Þ
In (19.201), an overlap integral between hydrogen atomic orbitals and C2s is
identical from the symmetry requirement and it is defined as
S
CH
A 1
H 1 jC2s
h
i :
ð19:202Þ
Also in (19.200), other quantities are defined as follows:
H 11 H
A 1
ð Þ
jHH
A 1
ð Þ
D
E
¼
α
H
þ 3β
HH
1 þ 3S
HH ,
ð19:203Þ
H 22 C2sjHC2s
h
i ,
ð19:204Þ
H 12 H
A 1
ð Þ
jHC2s
D
E
¼
H 1 þ H 2 þ H 3 þ H 4 jHC2s
h
i
2
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi
1 þ 3S
HH
p
¼
2 H 1 jHC2s
h
i
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi
1 þ 3S
HH
p
¼
2β
CH
A 1
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi
1 þ 3S
HH
p
,
ð19:205Þ
where H is a Hamiltonian of a methane molecule. In (19.203) and (19.205),
moreover, we define the quantities as
α
H
H 1 HH 1 i, β
HH
hH 1
HH 2
, β
CH
A 1
H 1 jHC2s
h
i :
ð19:206Þ
The quantity of H 11 is a “Coulomb” integral of the hydrogen-based SALC that
involves four hydrogen atoms. Solving the first equation of (19.199), we get
λ¼
H 11 þH 22 À2H 12 S 12 Æ
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
H 11 ÀH 22
ð
Þ
2 þ4 H 12
2
þH 11 H 22 S 12
2
ÀH 12 S 12 H 11 þH 22
ð
Þ
Â
Ã
q
2 1ÀS 12
2
À
Á
:
ð19:207Þ
788
19 Applications of Group Theory to Physical Chemistry