Computational Physics

9.2 The molecular dynamics method 271

of the Fock matrix (seeSection 4.5.2and above): forψ ̈k=0, Eq. (9.21) reduces to
an eigenvalue equation after an appropriate unitary transformation of the set{ψk}

and of the Lagrange parameters (^) kl.
The values of the Lagrange parameters (^) kldepend on time: they must be cal-
culated at each MD step such that they guarantee the orthonormality constraint
(9.6). This calculational procedure is related to the particular integration scheme
used (the Verlet algorithm in our case). In Section 8.6.2 we have encountered this
problem already. Car and Parrinello have used the iterative SHAKE algorithm of
Ryckaertet al.[5] (see Section 8.6.3) to solve for the (^) kl. We return to the problem
of calculating the (^) klin more detail below.
If the nuclear equilibrium configuration is searched for, starting from an initial
configuration which might be far off the equilibrium, we are likely to end up in
a local energy minimum instead of the global minimum. In this case, we might
use the simulated annealing method [4] which allows the system to hop over local
energy barriers to arrive at the global minimum.
It is interesting to compare the equations obtained here with the time-dependent
Hartree–Fock (TDHF) equations. These are obtained from a variational treatment
of the time-dependent Schrödinger equation using Slater determinants constructed
from time-dependent spin-orbitals. The time-dependent Schrödinger equation can
be derived as the stationarity condition of the functional

S=

∫

dt

∫

dX∗(X,t)

(

i

∂

∂t

−H

)

(X,t) (9.24)

withX = (x 1 ,...,xN). By taking for(R,t)a Slater determinant with time-
dependent orbitals ψk(x,t), the stationarity condition leads to the following
equation for the spin-orbitals [6]:

i

∂

∂t

ψk(x,t)=Fψk(x,t). (9.25)

The TDHF equations lead to a conservation law for the overlap matrixSkl(t)=
〈ψk(t)|ψl(t)〉. Hence, if we choose an orthonormal set to start off with att=0, the
set will remain orthonormal in the course of time.
In comparison with the MD equation of motion for the electrons, Eq. (9.21),
we see that the second derivative with respect to time is replaced by a first order
one, and that there is no Lagrange parameter as a result of the overlap matrix being
conserved.
Time-dependent Hartree–Fock is used for studying the quantum dynamics of
scattering processes, for example in nuclear physics and in studies of scattering of
electrons from atoms.