Computational Physics

286 Quantum molecular dynamics

system and that the negative charge cloud would therefore oscillate more and more
violently around the nuclei. To avoid this, the total simulation time should be kept
short, or the electrons and the nuclei should both be coupled to a Nosé–Hoover
thermostat (see Section 8.5.1), perhaps at different temperatures [21].
In the conjugate gradients method, the errors are more erratic than oscillatory in
nature. Therefore this cancellation effect will not occur in that case, and it is neces-
sary to keep the electronic charge distribution very close to the energy minimum at
any time in order to avoid an unstable propagation of the errors spoiling the results.
This means that we must perform many conjugate gradient steps, which slows the
calculation down considerably. The only way to achieve good performance using
the conjugate gradient method is by extrapolating the orbitals at the next time step
from the previous ones, so that the conjugate gradients iterations start off from a
configuration which is close to the exact energy minimum [22].
DetailsconcerningtheconjugategradientsmethodcanbefoundinRefs.[ 8 , 14 ,
16 , 17 ].

9.4.3 The RM-DIIS technique

Another widely used technique for finding the optimal orbitals is the RM-DIIS
technique. The abbreviation stands for residual minimisation by direct inversion
of the iterative subspace. This can be applied to any problem in which a set of
orthonormal eigenstates of a large Hamiltonian must be found. We describe the
methodbriefly–fordetailsseeRefs.[ 9 , 23, 24].
The eigenfunctions that we seek satisfy
(H−εn)|ψn〉=0; n=1,...,N. (9.78)

We now want to quantify the deviation of some approximate set of states from this.
To this end we reformulate(9.78)as

|ψn〉=|ψn〉+

1

Hnn

(H−Hnn)|ψn〉, (9.79)

with

Hnn=

〈ψn|H|ψn〉

〈ψn|ψn〉

. (9.80)

For the eigenstates we obviously haveεn=Hnn. This form may seem a bit arbitrary
but it naturally leads to an iterative scheme:

|ψn(j+^1 )〉=|ψn(j)〉+

1

Hnn

(H−Hnn)|ψn(j)〉. (9.81)

The reason the correction term has a prefactor 1/Hnnis that in this way all eigen-
vectors have a similar correction, independent of the energy eigenvalue. The main