Computational Physics

9.4 Orthonormalisation; conjugate gradient and RM-DIIS techniques 279

these values depends on the particular integration algorithm used, which is usu-
ally the Verlet algorithm. In the previous section you have seen how this is done
in the case of one orbital, where only the normalisation matters – for more orbitals,
understanding the different procedures is quite subtle.
In the following we shall use the notation
ε=

∑

k

〈ψk|H|ψk〉 (9.49)

for the total energy for a set of orthonormal orbitalsψk.Hstands for the Fock
matrix in HF, and in DFT it is the Kohn–Sham Hamiltonian. Let us write down the
Verlet equations of motion for the electronic orbitals:

ψk(t+h)= 2 ψk(t)−ψk(t−h)−

2 h^2

μ

(Hψk−

∑

l

(^) klψl). (9.50)
The – as yet unknown – multipliers (^) klare symmetric, (^) kl= (^) lk, and therefore
representN(N+ 1 )/2 independent values, which are determined by theN(N+ 1 )/ 2
orthonormality conditions. Hence the Lagrange multipliers are uniquely defined.
It might therefore be surprising that several different orthogonalisation algorithms
exist [ 8 , 10 ]. The reason is that a unitary transformation of the set of orbitals leaves
the set orthonormal: the set{ψk′}defined by
ψk′=

∑

l

Uklψl (9.51)

is orthonormal. Moreover, a unitary transformation leaves the charge density
unchanged – remember the DFT energy depends on the density and not on the
individual orbitals. Also, the Slater determinants forming the basis functions in the
Hartree–Fock theory are invariant under unitary transformations (see Problem 4.7).
It should be noted that such a transformation of the setψkis accompanied by a
similarity transform of the Lagrange parameters:

′kl=

∑

mn

Ukm† (^) mnUnl (9.52)
as can be verified directly from the equation of motion(9.21). Different orthonorm-
alisation algorithms result in sets of orbitals which span the same space of functions
but which are slightly rotated with respect to each other.
Such a rotation may have a tremendous effect on the performance of the Verlet
algorithm. To see this, consider a permutation of the orbitals (which is a special
case of a unitary transformation), carried out between two time steps. This per-
mutation does not affect the density but it may have a disastrous effect on the
integration of the equations of motion: the (fictitious) velocities of the permuted
orbitals increase suddenly to values ofO(h−^1 ), because the permutation disrupts