Computational Physics

292 Quantum molecular dynamics

must be minimised subject to the normalisation constraint

∑

rs

CrCsSrs=1.

The Lagrangian function for this problem is

L[Cr]=

∑

rs

CrCsHrs−λ

∑

rs

CrCsSrs

whereλshould be such that the normalisation remains guaranteed when moving in

the steepest descent direction. The steepest descent directionζis given by

ζ=

∑

r

Drχr(x);

Dr=− 2

∑

s

CsHrs+ 2 λ

∑

s

CsSrs.

(a) Show that

λ=〈ψ|H|ψ〉.

(b) Use this in applying the conjugate gradients method in order to find the ground

state. Note that convergence is slow as no preconditioning is applied.

Now we consider the problem of finding more energy eigenstatesψkwhich are

expanded in the basis set states as

ψk(x)=

∑

r

Crkχp(x).

ForNeigenstates we haveN(N+ 1 )/2 constraints.

The Lagrange function now reads

L[Cr]=

∑

k

∑

rs

CrkCskHrs−

∑

kl

(^) kl
∑
rs
CrkCslSrs.
(c) Show that
(^) kl=〈ψk|H|ψl〉.
(d) Use this form to find the four lowest eigenstates.
9.4 [C] Use the conjugate gradients technique for finding the minimum of the electronic
energy of the hydrogen molecule with a fixed configuration of nuclei. This is a
straightforward extension of the first program of the previous problem: there is only
one normalisation constraint. Note however that the energy functional contains a term
which is quartic in theCr– seeEq. (9.26). Show that the steepest descent direction
subject to the constraint is given by
Dr=− 2
∑
s
FrsCs+ 2 λ
∑
s
SrsCs
with
λ=
∑
rs
CrFrsCs.