Computational Physics

9.2 The molecular dynamics method 267

method by recalling the energy functionals of the Hartree–Fock and the density
functional theory (see Chapters 4 and 5 ).
The ground state Hartree–Fock wave function forNelectrons can be written as
the Slater determinant

G(R)=det[ψk(xi)]=

1

√

N!

∣

∣

∣∣

∣∣

∣∣

∣

ψ 1 (x 1 )ψ 2 (x 1 ) ··· ψN(x 1 )

ψ 1 (x 2 )ψ 2 (x 2 ) ··· ψN(x 2 )

..

.

..

.

..

.

ψ 1 (xN)ψ 2 (xN) ··· ψN(xN)

∣

∣

∣∣

∣∣

∣∣

∣

, (9.5)

where theψkare one-electron spin-orbitals;xiis the combined spin and orbital
coordinate of particle i. The spin-orbitals should satisfy the orthonormality
requirements

〈ψk|ψl〉=δkl. (9.6)

The energy is given in terms of theψkas

EHF=

∑

k

〈ψk|h|ψk〉+

1

2

∑

kl

[〈ψkψl|g|ψkψl〉−〈ψkψl|g|ψlψk〉]. (9.7)

his the one-electron Hamiltonian andgis the electron–electron Coulomb repulsion
(seeChapter 4). Minimisation of this expression with respect to theψksubject to
the constraint (9.6) requires the Fock equation to be satisfied:

Fψk=

∑

l

(^) klψl (9.8)
with
Fψk=

[

−

1

2

∇^2 −

∑

n

Zn

|r−Rn|

]

ψk(x)+

∑N

l= 1

∫

dx′|ψl(x′)|^2

1

|r−r′|

ψk(x)

−

∑N

l= 1

∫

dx′ψl∗(x′)

1

|r−r′|

ψk(x′)ψl(x). (9.9)

After a unitary transformation of the set{ψk}(seeSection 4.5.2and Problem 4.7),
Eq. (9.8)transforms into

Fψk=εkψk. (9.10)

UsingFψk=δEHF/δψk, we can rewrite this as

δEHF

δψk(x)

=εkψk(x). (9.11)

The eigenvaluesεkare the Fock levels; the energy can be calculated from these.