Computational Physics

282 Quantum molecular dynamics

which is repeated until the orbitals do not change any more. By inspection of this
algorithm it is seen that if the orbitals are orthonormal up to orderh^4 , they will
change to an extent within that order over a time step. The same holds for the
Gram–Schmidt orthonormalisation, which is given by the algorithm

ψk′=ψk−

∑

l<k

〈ψl|ψk〉ψl. (9.63)

In contrast to the previous algorithm, the Gram–Schmidt algorithm depends on
the way in which the orbitals are ordered. In particular, the orbital which we take
as the first in the Gram–Schmidt process remains unchanged. It is clear from the
foregoing analysis that this does not really matter if the states are already orthogonal
to orderh^4.
Another possible way to orthogonalise the orbitals is to calculate the Lagrange
parameters such that the final orbitals will be orthonormal. For example, we first
calculate the new orbitals without taking the Lagrange multipliers into account:

|ψ ̃k(t+h)〉= 2 |ψk(t)〉−|ψk(t−h)〉−

2 h^2

μ

H|ψk(t)〉, (9.64)

and then we calculate the Lagrange parametersXijsuch that the orbitals

|ψk(t+h)〉=|ψ ̃k(t+h)〉+

∑

l

Xkl|ψl(t)〉 (9.65)

form an orthonormal set. Obviously, theXklare related to the (^) klby
Xkl=
2 h^2
μ
(^) kl. (9.66)
The parametersXijshould satisfy a matrix equation which can conveniently
be formulated after introducing the matricesAkl =〈ψ ̃k(t+h)|ψ ̃l(t+h)〉and
Bkl=〈ψk(t)|ψ ̃l(t+h)〉as:
XX†+XB+B†X†=I−A. (9.67)
This can be solved iteratively by the straightforward reformulation
X(n+^1 )=^12 [I−A+X(n)(I−B)+(I−B†)X(n)−X(n)X(n)†]. (9.68)
As an initial guess, we takeX=^12 (I−A), which is close to the first guess for (^) kl
found above.
Summarising the integration algorithm so far:
We h ave|ψk(t)〉and|ψk(t−h)〉.
Find|ψ ̃ ̇k(t)〉=(|ψk(t)〉−|ψk(t−h)〉)/h.