4.3 Accurate Computational Models
125
On the contrary, the diagonal elements do have a contribution from the 2h-2p
excitations. Continuing with the external determinants Φ R and Φ S , it is easily shown
that the former only contributes to Φ I | ˆ
H eff |Φ I and the latter to Φ J | ˆ
H eff |Φ J
Φ R :
||hhab| ˆ
V |ppab| 2
E I − E R
= 0
||hhba| ˆ
V |ppab| 2
E J − E R
= 0
(4.33a)
Φ S :
||hhab| ˆ
V |ppba| 2
E I − E S
= 0
||hhba| ˆ
V |ppba| 2
E J − E S
= 0
(4.33b)
Both non-zero integrals are identical and through the Slater–Condon rules we arrive
at the following second-order contribution of Φ R and Φ S to the diagonal elements
Φ I | ˆ
H eff |Φ I and Φ J | ˆ
H eff |Φ J
||pp|
1− ˆ
P 12
r 12
|hh| 2
2ε h − 2ε p
(4.34)
where the denominator is obtained by assuming the Møller–Plesset division for ˆ
H =
ˆ
H (0) + ˆ
V . In the general case the contribution of all the 2h-2p determinants to the
diagonal elements is given by
h,h ′
p,p ′
||pp ′ |
1− ˆ
P 12
r 12
|hh ′ | 2
ε h + ε h ′ − ε p − ε p ′
(4.35)
The summation only involves integrals that depend on the inactive (h, h ′ ) and virtual
(p, p ′ ) orbitals, and hence, is exactly the same for all the diagonal elements in the
model space. This uniform shift of the diagonal elements does not affect the energy
differences of the eigenstates of the model space and in combination with the zero
contribution to the off-diagonal elements, this shows that the 2h-2p determinants can
be skipped in the CI expansion of the wave function. Note that this argument is based
on second-order perturbation theory, the inclusion of higher-order interactions gives
rise to small contributions and strictly speaking the mutual interaction between the
2h-2p determinants could affect the energy differences.
4.9 Consider a (non-degenerate) model space with neutral and ionic determinants. (a) Show that the 2h-1p determinant Φ R =|aapb| introduces non-zero
off-diagonal elements between the ionic and neutral determinants of the model
space. (b) Are the diagonal elements of the model space shifted uniformly by
Φ R ?
The number of external determinants can even be more reduced when the model
space is reduced to the neutral determinants Φ I ={ | hhab|, |hhba|}. Under these
125
On the contrary, the diagonal elements do have a contribution from the 2h-2p
excitations. Continuing with the external determinants Φ R and Φ S , it is easily shown
that the former only contributes to Φ I | ˆ
H eff |Φ I and the latter to Φ J | ˆ
H eff |Φ J
Φ R :
||hhab| ˆ
V |ppab| 2
E I − E R
= 0
||hhba| ˆ
V |ppab| 2
E J − E R
= 0
(4.33a)
Φ S :
||hhab| ˆ
V |ppba| 2
E I − E S
= 0
||hhba| ˆ
V |ppba| 2
E J − E S
= 0
(4.33b)
Both non-zero integrals are identical and through the Slater–Condon rules we arrive
at the following second-order contribution of Φ R and Φ S to the diagonal elements
Φ I | ˆ
H eff |Φ I and Φ J | ˆ
H eff |Φ J
||pp|
1− ˆ
P 12
r 12
|hh| 2
2ε h − 2ε p
(4.34)
where the denominator is obtained by assuming the Møller–Plesset division for ˆ
H =
ˆ
H (0) + ˆ
V . In the general case the contribution of all the 2h-2p determinants to the
diagonal elements is given by
h,h ′
p,p ′
||pp ′ |
1− ˆ
P 12
r 12
|hh ′ | 2
ε h + ε h ′ − ε p − ε p ′
(4.35)
The summation only involves integrals that depend on the inactive (h, h ′ ) and virtual
(p, p ′ ) orbitals, and hence, is exactly the same for all the diagonal elements in the
model space. This uniform shift of the diagonal elements does not affect the energy
differences of the eigenstates of the model space and in combination with the zero
contribution to the off-diagonal elements, this shows that the 2h-2p determinants can
be skipped in the CI expansion of the wave function. Note that this argument is based
on second-order perturbation theory, the inclusion of higher-order interactions gives
rise to small contributions and strictly speaking the mutual interaction between the
2h-2p determinants could affect the energy differences.
4.9 Consider a (non-degenerate) model space with neutral and ionic determinants. (a) Show that the 2h-1p determinant Φ R =|aapb| introduces non-zero
off-diagonal elements between the ionic and neutral determinants of the model
space. (b) Are the diagonal elements of the model space shifted uniformly by
Φ R ?
The number of external determinants can even be more reduced when the model
space is reduced to the neutral determinants Φ I ={ | hhab|, |hhba|}. Under these