Computational Physics - Department of Physics

15.4 Derivation of the Hartree-Fock equations 505

15.4.2 Varying the single-particle wave functions

If we generalize the Euler-Lagrange equations to more variables and introduceN^2 Lagrange
multipliers which we denote byεμ ν, we can write the variational equation for the functional
of Eq. (15.23) as

δE−

N

∑

μ= 1

N

∑

ν= 1

εμ νδ

∫

ψμ∗ψν= 0. (15.23)

For the orthogonal wave functionsψμthis reduces to

δE−

N

∑

μ= 1

εμδ

∫

ψμ∗ψμ= 0. (15.24)

Variation with respect to the single-particle wave functionsψμyields then

N

∑

μ= 1

∫

δ ψμ∗hˆiψμdxi+^1

2

N

∑

μ= 1

N

∑

ν= 1

[∫

δ ψ∗μψν∗^1

ri j

ψμψνd(xixj)−

∫

δ ψμ∗ψν∗^1

ri j

ψνψμdridrj

]

+

N

∑

μ= 1

∫

ψμ∗hˆiδ ψμdri+

1

2

N

∑

μ= 1

N

∑

ν= 1

[∫

ψ∗μψν∗

1

ri j

δ ψμψνdridrj−

∫

ψμ∗ψν∗

1

ri j

ψνδ ψμdridrj

]

−

N

∑

μ= 1

Eμ

∫

δ ψ∗μψμdxi−

N

∑

μ= 1

Eμ

∫

ψμ∗δ ψμdri= 0.

(15.25)

Although the variationsδ ψandδ ψ∗are not independent, they may in fact be treated as
such, so that the terms dependent on eitherδ ψandδ ψ∗individually may be set equal to
zero. To see this, simply replace the arbitrary variationδ ψbyiδ ψ, so thatδ ψ∗is replaced by
−iδ ψ∗, and combine the two equations. We thus arrive at the Hartree-Fock equations
[
−

1

2

∇^2 i−

Z

ri

+

N

∑

ν= 1

∫

ψν∗(rj)

1

ri j

ψν(rj)drj

]

ψμ(xi)

−

[N

∑

ν= 1

∫

ψν∗(rj)

1

ri j

ψμ(rj)drj

]

ψν(ri) =εμψμ(ri).

(15.26)

Notice that the integration

∫
drjimplies an integration over the spatial coordinatesrjand
a summation over the spin-coordinate of electronj.
The two first terms are the one-body kinetic energy and the electron-nucleus potential.
The third ordirectterm is the averaged electronic repulsion of the other electrons. This
term includes the ’self-interaction’ of electrons wheni=j. The self-interaction is cancelled in
the fourth term, or theexchangeterm. The exchange term results from our inclusion of the
Pauli principle and the assumed determinantal form of the wave-function. The effect of the
exchange is for electrons of like-spin to avoid each other. Atheoretically convenient form of
the Hartree-Fock equation is to regard the direct and exchange operator defined through the
following operators

Vμd(ri) =

∫

ψμ∗(rj)^1

ri j

ψμ(rj)drj (15.27)

and

Vμex(ri)g(ri) =

(∫

ψ∗μ(rj)

1

ri jg(rj)drj

)

ψμ(ri), (15.28)