50415 Many-body approaches to studies of electronic systems: Hartree-Fock theory and Density Functional Theory
∂f
∂xi
+∑
kλk
∂ φk
∂xi= 0.
Let us specialize to the expectation value of the energy for one particle in three-dimensions.
This expectation value reads
E=∫
dxdydzψ∗(x,y,z)Hˆψ(x,y,z),with the constraint ∫
dxdydzψ∗(x,y,z)ψ(x,y,z) = 1 ,
and a Hamiltonian
Hˆ=−^1
2 ∇
(^2) +V(x,y,z).
The integral involving the kinetic energy can be written as,if we assume periodic boundary
conditions or that the functionψvanishes strongly for large values ofx,y,z,
∫
dxdydzψ∗
(
−
1
2
∇^2
)
ψdxdydz=ψ∗∇ψ|+∫
dxdydz1
2
∇ψ∗∇ψ.Inserting this expression into the expectation value for the energy and taking the variational
minimum (usingV(x,y,z) =V) we obtain
δE=δ{∫
dxdydz(
1
2
∇ψ∗∇ψ+Vψ∗ψ)}
= 0.
The requirement that the wave functions should be orthogonal gives
∫
dxdydzψ∗ψ=constant,and multiplying it with a Lagrangian multiplierλand taking the variational minimum we
obtain the final variational equation
δ{∫
dxdydz(
1
2
∇ψ∗∇ψ+Vψ∗ψ−λ ψ∗ψ)}
= 0.
We introduce the functionf
f=1
2 ∇ψ∗∇ψ+Vψ∗ψ−λ ψ∗ψ=^1
2 (ψ∗
xψx+ψ
∗
yψy+ψ
∗
zψz)+Vψ
∗ψ−λ ψ∗ψ.In our notation here we have dropped the dependence onx,y,zand introduced the shorthand
ψx,ψyandψzfor the various first derivatives.
Forψ∗the Euler equation results in
∂f
∂ ψ∗
−∂
∂x∂f
∂ ψx∗−∂
∂y∂f
∂ ψy∗−∂
∂z∂f
∂ ψz∗= 0 ,
which yields
−^1
2
(ψxx+ψyy+ψzz)+Vψ=λ ψ.We can then identify the Lagrangian multiplier as the energyof the system. The last equation
is nothing but the standard Schrödinger equation and the variational approach discussed
here provides a powerful method for obtaining approximate solutions of the wave function.