Computational Physics

270 Quantum molecular dynamics

have the opportunity to adapt themselves to the changing nuclear configuration at
any time), there is no reason to believe that the retardation effect implied by the
Car–Parrinello Lagrangian is related to real physical behaviour.
The details of the kinetic energy of the electrons do not matter: what matters
is the fact that the massμused in this kinetic energy should be small enough
to enable the electronic wave function to adapt reasonably well to the changing
nuclear configurations. This mass should therefore be much smaller than the nuclear
masses. The choice of the massμis determined by a trade-off between accuracy and
efficiency. If we include friction in the equations of motion, the particular values
of neither electronic nor nuclear masses matter, as we shall always end up with
zero kinetic energy, at the minimum of the total energy of the system (which is
the potential of the Car–Parrinello Lagrangian), although different choices of these
masses lead to different rates of convergence towards the energy minimum.
Let us write down the equations of motion for the Car–Parrinello Lagrangian. We
must take the orthogonality constraint(9.6)into account using Lagrange parameters

(^) kl(t). The Euler–Lagrange equations now read
μψ ̈k=−
∂Etot
∂ψk

+ 2

∑

l

(^) klψl(r) (9.21)
and
MnR ̈n=−
∂Etot
∂Rn

+

∑

kl

(^) kl
∂〈ψk|ψl〉
∂Rn

. (9.22)

The last term on the right hand side of the last equation vanishes if the basis functions
do not depend on the nuclear positionsS. As we know the total energy in both DFT
and HF in terms of the orbitalsψkandRn, the energy derivatives occurring in these
equations can be evaluated – see the next section.
Instead of assigning a kinetic energy to the orbitalsψk, leading toEq. (9.21), we
can assign a kinetic energy to the expansion coefficientsCrk. In that case,Eq. (9.21)
becomes

μC ̈rk=−

∂Etot

∂Crk

+ 2

∑

l

(^) kl

∑

s

SrsCsl. (9.23)

Ifμis allowed to depend onrandk, this equation can be made equivalent to(9.21)
but, as argued above, the details of the kinetic energy do not matter that much
as long as the electronic degrees of freedom can adapt themselves to the nuclear
positions.
If a frictional term is added to the equations of motion, the solution will become
stationary after some time, and the left hand side vanishes. Equation(9.21)then
becomes an equation similar to the Fock and the Kohn–Sham equations ((9.11)and

(9.16)), except for the eigenvaluesεkbeing replaced by the matrix elements (^) kl.
This is precisely the same difference as we have encountered in the diagonalisation