Computational Physics

268 Quantum molecular dynamics

In density functional theory the energy can be written as a function of the ground
state density, which in turn is written in terms of the basis functions as

n(r)=

∑N

k= 1

|ψk(r)|^2 , (9.12)

assuming that the states are ordered according to increasing energy. We have seen
already in Chapter 5 that there is no direct expression of the total energy as a
function of the density, as the form of the kinetic energy functional of the density
is unknown. The energy can however be obtained indirectly via the solutionsψkof
the Kohn–Sham equations:

−

1

2

∇^2 ψk(r)+Veff(r)ψk(r)=εkψk(r) (9.13)

where

Veff(r)=Vion(r)+

∫

d^3 r′

n(r′)

|r−r′|

+Vxc[n](r). (9.14)

The exchange correlation potentialVxcon the right hand side is the derivative of
the exchange correlation energyExcwith respect ton(r).
In terms of theψk, the total DFT energy is written as

EDFT=−

∑

k

1

2

〈ψk|∇^2 |ψk〉+

∑

k

〈ψk|Vion|ψk〉

+

1

2

∫

d^3 rd^3 r′

n(r)n(r′)

|r−r′|

+Exc[n](r). (9.15)

The Kohn–Sham equations can be written as

δEDFT

δψk(r)

=εkψk(r), (9.16)

i.e. the same form as(9.11).
Summarising, the total energy, which is the electronic energy (eitherEDFTor
EHF) plus the electrostatic energy of the nuclei, can be written as a functional
depending on the orbitalsψkand of the nuclear coordinates, collected together in
the variableS:
Etot=Etot({ψk},S), (9.17)

where the orbitalsψkform an orthonormal set. In both the Hartree–Fock and the
density functional theory approach we minimise this energy with respect to the
orbitalsψk, according to the variational principle. Usually, a finite basis set{χr}is
used, in terms of which the orbitals are given as

ψk(r)=

∑

r

Crkχr(r), (9.18)