Computational Physics

9.5 Implementation of the Car–Parrinello technique for pseudopotential DFT 289

nuclear positions remain fixed, the method scales asNB, but with a large pre-
factor. The method converges much faster for insulators and semiconductors than
for metals [27]. Finally, discretising the degrees of freedom on a cubic lattice can
lead to methods with a favourable time scaling [ 28 , 29 ].

*9.5 Implementation of the Car–Parrinello technique

for pseudopotential DFT

In this section we describe how the self-consistent pseudopotential code described
in Section 6.7.3 can be extended to include Car–Parrinello dynamics. In the code of
Section 6.7.3, we constructed the DFT Hamiltonian, diagonalised it and calculated
the density which was then used to construct a new Hamiltonian, and so on until
convergence was achieved. We then calculated the total energy from the orbitals,
the density and the nuclear positions (which were taken to be fixed).
Now we want to use the derivative of the total energy with respect to the orbital
degrees of freedomandwith respect to the nuclear positions in order to formulate
equations of motions which can then be solved using the (velocity) Verlet algorithm.
The rather complicated expression for the energy leads to even more complicated
expressions for these forces. We start with the orbital forces. They follow directly
from the expression for the total energy given in Section 6.7.7. If we calculate the
orbital forces from this energy, we obtain by direct differentiation: [9]

∂Etotal

∂c∗j(K)

=

K^2

2

cj(K)+

∑

K′

Vloc∗ (K−K′)cj(K′)

+

∑

n

∑

lm

Fjlmn e−iK·RnYlm(Kˆ)hnlmplm(K), (9.88)

whereVlocallis the total local potential:

Vloc(K)=

∑

n

Vloc(K)+Vxc(K)+ 4 π

ntot(K)

K^2

. (9.89)

The potentials occurring in this expressions are given in Section 6.7.3.
If we implement this force in a Verlet algorithm in which we reduce the velocities
at each time step such as to mimic frictional forces which bring us to the stationary
energy minimum, the result should be equal to that obtained in the Kohn–Sham
program. For two silicon atoms placed at a distance of 1.05 a.u. in a cell of size
8 a.u., the converged energy is−21.235 347 29 atomic units.
For the dynamics of the nuclei, we need the gradients of the energy with respect
to the nuclear coordinates. These can be obtained directly from the expression for
the total energy. The nuclear gradients are generated by the local and nonloncal