Computational Physics - Department of Physics

15.4 Derivation of the Hartree-Fock equations 509

N

∑

μ= 1

Vμd(ri)ψ 200 ↑(ri) = 2

∫∞

0

drj

(

φ 100 ∗ (rj)^1

ri j

φ 100 (rj)+φ 200 ∗ (rj)^1

ri j

φ 200 (rj)

)

ψ 200 ↑(ri) =

( 2 V 10 d(ri)+ 2 V 20 d(ri))ψ 200 ↑(ri)

For the Fock term we get

N

∑

μ= 1

Vμex(ri)ψ 200 ↑(ri) =

∫∞

0

drjφ 100 ∗ (rj)^1

ri j

φ 200 (rj)ψ 100 ↑(ri)+

∫∞

0

drjφ 200 ∗ (rj)

1

ri jφ^200 (rj)ψ^200 ↑(ri) =V

10 ex(ri)+V 20 ex(ri)

The second term is the same as we have for the Hartree term with 2 s. The final differential
equation is (

−

1

2

d^2

dr^2

−

4

r

+ 2 V 10 d(r)+V 20 d(r)

)

u 20 (r)−V 10 ex(r) =e 20 u 20 (r).

Note again thatV 10 ex(r)contains the 2 sfunction in the integral, that is

V 10 ex(r) =

∫∞

0

drjφ 100 ∗ (rj)^1

r−rj

φ 200 (rj)ψ 100 ↑(r).

We have two coupled differential equations
(
−

1

2

d^2

dr^2

−

4

r

+V 10 d(r)+ 2 V 20 d(r)

)

u 10 (r)−V 20 ex(r) =e 10 u 10 (r),

and (

−^1

2

d^2

dr^2

−^4

r

+ 2 V 10 d(r)+V 20 d(r)

)

u 20 (r)−V 10 ex(r) =e 20 u 20 (r).

Recall again that the interaction does not depend on spin. This means that the single-particle
energies and single-particle functionudo not depend on spin.

15.4.4Hartree-Fock by variation of basis function coefficients

Another possibility is to expand the single-particle functions in a known basis and vary the
coefficients, that is, the new single-particle wave function is written as a linear expansion in
terms of a fixed chosen orthogonal basis (for example harmonic oscillator, Laguerre polyno-
mials etc)
ψa=∑
λ

Caλψλ. (15.32)

In this case we vary the coefficientsCaλ.
The single-particle wave functionsψλ(r), defined by the quantum numbersλandrare
defined as the overlap
ψα(r) =〈r|α〉.

We will omit the radial dependence of the wave functions and introduce first the following
shorthands for the Hartree and Fock integrals

〈μ ν|V|μ ν〉=

∫

ψ∗μ(ri)ψ∗ν(rj)V(ri j)ψμ(ri)ψν(rj)drirj,

and