Computational Physics

4.5 Self-consistency and exchange: Hartree–Fock theory 57

therefore calculate the expectation value of the energy for an arbitrary Slater determ-
inant using the Born–Oppenheimer Hamiltonian and then minimise the result with
respect to the spin-orbitals in the determinant.
We write the Hamiltonian as follows:

H=

∑

i

h(i)+

1

2

∑

i,j;i
=j

g(i,j) with

g(i,j)=

1

|ri−rj|

and

h(i)=−

1

2

∇i^2 −

∑

n

Zn

|ri−Rn|

.

(4.34)

h(i)depends onrionly andg(i,j)onriandrj. Writing the Slater determinantψas
a sum of products of spin-orbitals and using the orthonormality of the latter, it can
easily be verified that this determinant is normalised, and for the matrix element of
the one-electron part of the Hamiltonian, we find (see Problem 4.3)
〈
AS

∣∣

∣∣

∣

∑

i

h(i)

∣∣

∣∣

∣

AS

〉

=N·

(N− 1 )!

N!

∑

k

〈ψk|h|ψk〉

=

∑

k

〈ψk|h|ψk〉=

∑

k

∫

dxψk∗(x)h(r)ψk(x). (4.35)

By

∫

dxwe denote an integral over the spatial coordinates and a sum over the
spin-degrees of freedom as usual.
The matrix element of the two-electron termg(i,j)for a Slater determinant not
only gives a nonzero contribution when the spin-orbitals in the left and right hand
sides of the inner product occur in the same order, but also forkandlinterchanged
on one side (the derivation is treated in Problem 4.3):
〈
AS

∣∣

∣∣

∣∣

∑

i,j

g(i,j)

∣∣

∣∣

∣∣AS

〉

=

∑

k,l

〈ψkψl|g|ψkψl〉−

∑

k,l

〈ψkψl|g|ψlψk〉. (4.36)

In this equation, the following notation is used:

〈ψkψl|g|ψmψn〉=

∫

dx 1 dx 2 ψk∗(x 1 )ψl∗(x 2 )

1

|r 1 −r 2 |

ψm(x 1 )ψn(x 2 ). (4.37)

In summary, we obtain for the expectation value of the energy:

E=

∑

k

〈ψk|h|ψk〉+

1

2

∑

kl

[〈ψkψl|g|ψkψl〉−〈ψkψl|g|ψlψk〉]. (4.38)