98 Density functional theory
where we have used the definition
Fλ[n]=min
ψ|n〈|H0,λ|〉 (5.31)The minimisation is carried out on the set of wave functions compatible with the
given densitiesn(r).
We now need a theorem that plays an important role in the quantum molecular
dynamics method (seeChapter 9): theHellmann–Feynman theorem. Here we shall
prove this theorem for the simple case in which we have a Hamiltonian depending
on a single parameterλ. The theorem tells us how the energy eigenvalues of a
HamiltonianHλ, depending on a parameterλ, vary withλ. Differentiating the
Schrödinger equation
Hλ|ψλ〉=Eλ|ψλ〉 (5.32)
with respect toλwe obtain (the prime indicates derivative with respect toλ):
Hλ′‖ψλ〉+Hλ‖ψλ′〉=Eλ′|ψλ〉+Eλ|ψλ′〉. (5.33)Taking the inner product from the left with〈ψλ|and using the Hermitian conjugate
of (5.32), we see that
dEλ
dλ
=
〈ψλ|dHλ/dλ|ψλ〉
〈ψλ|ψλ〉. (5.34)
We can write the derivative ofFλfrom the Hellmann–Feynman theorem, by
realising that, since|ψλ〉is the variational ground state ofFλ, it must be the lowest
eigenstate ofH0,λ. We then obtain
dFλ
dλ=〈λ|Vc|λ〉. (5.35)From this, and from the fact that forλ=0 we have trivial, noninteracting electron
gas, we have
Fλ= 1 [n]=TKS[n]+∫ 1
0〈ψλ|Vc|λ〉dλ. (5.36)We now find the exchange correlation potential as the difference between the
interacting and noninteracting electron gas including the Hartree energyEH:
Exc=Fλ= 1 [n]−TKS[n]−∑ 1
2
∫
n(r)1
|r−r′|
n(r′)d^3 rd^3 r′=
∫ 1
0〈ψλ|Vc|λ〉dλ−EH. (5.37)The main point of the derivation is that in(5.36), which holds for the interacting gas,
the kinetic energy is that of the noninteracting gas; therefore, we find the exchange
correlation correction only in terms of the Coulomb interaction. Forλ=0, the XC
correction term is nonzero as the Hartree energy does not take the antisymmetry of