1.2 Matrix Elements for Many-Electron States
15
corresponds to a singly excited state (with respect to | 0 ), in which the kth electron
is excited to the virtual orbital a. The matrix element
ak | ˆ
V | 0 =
N
i=1
V ia[ik]
(1.60)
may serve as an example for the SC rule c2.
To evaluate matrix elements for differing Slater determinants, one has to assure
that the orbitals are ordered in the form supposed in the derivation of the SC rules.
At the end of the next section, we shall demonstrate how this can be achieved quite
conveniently using second quantization.
Spin-Free Expressions:
Finally, we take a look at the derivation of spin-free expressions for many-electron
matrix elements, which applies to operators acting on the spatial variables only. Let
us again expand the spin-orbital quantum numbers used so far into the pair of spatial
and spin quantum numbers:
q → qγ
and recall that the spin-orbitals are products of spatial and spin-orbitals,
ψ qγ (ξ) = ϕ q (x)χ γ (σ)
(1.61)
For a spatial operator, the one-particle integrals (1.46) simplify according to
pγ| ˆ
w|qγ
= =γ|γ
ϕ
∗
p (x) ˆ
wϕ q (x)dx
= δ γγ w pq
(1.62)
where w pq is a spatial one-particle integral, and the spin-integral becomes a trivial
Kronecker delta. In a similar way, the general two-particle integrals can be evaluated
to become
pγqσ| ˆ
v|r ρsτ = =γ|ρσ|τ
dx 1 dx 2 ϕ
∗
p (x 1 )ϕ
∗
q (x 2 )
e
2
|x 1 − x 2 |
ϕ r (x 1 )ϕ s (x 2 )
= δ γρ δ στ V pqrs
(1.63)
The simplification of the spin–orbit integrals can readily be exploited in the manyelectron matrix elements. As an example, let us consider the ground-state expectation
value (1.58) of the hamiltonian. Supposing that there are n =
1
2
N spatial orbitals,
each occupied by a spin-α and spin-β electron,
| 0 = |ϕ 1α ϕ 1β . . . ϕ nα ϕ nβ
15
corresponds to a singly excited state (with respect to | 0 ), in which the kth electron
is excited to the virtual orbital a. The matrix element
ak | ˆ
V | 0 =
N
i=1
V ia[ik]
(1.60)
may serve as an example for the SC rule c2.
To evaluate matrix elements for differing Slater determinants, one has to assure
that the orbitals are ordered in the form supposed in the derivation of the SC rules.
At the end of the next section, we shall demonstrate how this can be achieved quite
conveniently using second quantization.
Spin-Free Expressions:
Finally, we take a look at the derivation of spin-free expressions for many-electron
matrix elements, which applies to operators acting on the spatial variables only. Let
us again expand the spin-orbital quantum numbers used so far into the pair of spatial
and spin quantum numbers:
q → qγ
and recall that the spin-orbitals are products of spatial and spin-orbitals,
ψ qγ (ξ) = ϕ q (x)χ γ (σ)
(1.61)
For a spatial operator, the one-particle integrals (1.46) simplify according to
pγ| ˆ
w|qγ
= =γ|γ
ϕ
∗
p (x) ˆ
wϕ q (x)dx
= δ γγ w pq
(1.62)
where w pq is a spatial one-particle integral, and the spin-integral becomes a trivial
Kronecker delta. In a similar way, the general two-particle integrals can be evaluated
to become
pγqσ| ˆ
v|r ρsτ = =γ|ρσ|τ
dx 1 dx 2 ϕ
∗
p (x 1 )ϕ
∗
q (x 2 )
e
2
|x 1 − x 2 |
ϕ r (x 1 )ϕ s (x 2 )
= δ γρ δ στ V pqrs
(1.63)
The simplification of the spin–orbit integrals can readily be exploited in the manyelectron matrix elements. As an example, let us consider the ground-state expectation
value (1.58) of the hamiltonian. Supposing that there are n =
1
2
N spatial orbitals,
each occupied by a spin-α and spin-β electron,
| 0 = |ϕ 1α ϕ 1β . . . ϕ nα ϕ nβ