Computational Physics

6.7 The pseudopotential method 151

wherefj is theoccupancy, which is usually the Fermi–Dirac function with an
additional factor of 2 in the closed-shell case. To check that the normalisation
is indeed correct, we calculate the total charge for the closed-shell system atT=0:
∫
n(r)d^3 r= 2

∑

jocc

|φ(j)(r)|^2 d^3 r= 2

N^3

∑

jocc

∑

r

|φ(j)(r)|^2 =Nel (6.67)

whereNelis the number of electrons (the factor 2 in front of the second and third
expressions is due to the spin degeneracy). Note that the prefactor
/N^3 results
from the transition of the integral to a sum.
We can formulate Fourier transformsin continuous, real spaceby writing oper-
ators and vectors with respect to the basis functions exp(iK·r)/

√

. In that case,
the discrete representation converges to the continuum one for fine grids, and there
is no ambiguity concerning the prefactors in the Fourier transforms (powers of ).
The only exception is the density, which is not a vector or operator in Hilbert space,
and we have therefore some freedom in the definition of its Fourier transform. We
adopt the convention usually taken in this field, writing

n(r)=

∑

K

n(K)eiK·r. (6.68)

It then follows fromEqs. (6.63)and(6.66)that

n(K)=

1

∫

n(r)e−iK·rd^3 r=

1

∑

j

fj

∑

K′

c(Kj)+K′cK∗(′j). (6.69)

An important issue concerns the truncation of the sums overK. The point is that
the potential is expressed in terms of thedensity, and not of the wave function. Now
suppose that we can safely assume that the wave function vanishes forK-vectors
beyond some maximum valueKmax. In that case, from working out the density in
real space,

n(r)=

∑

j

fj|φj(r)|^2 =

∑

j

fj

∑

K,K′

c(Kj)eiK·rc∗K(′j)e−iK

′·r

=

1

∑

j

fj

∑

K,K′

c(Kj)+K′c∗K(′j)eiK·r=

∑

K

n(K)eiK·r, (6.70)

we see thatn(K)contains contributionsK−K′running up to 2Kmax! Therefore,
the potential also contains nonzero components forKup to 2Kmax. To see that these
terms occur in the Hamiltonian matrix, we consider now alocalpotential: this is a
potential which depends only onr. Fourier transforming leads to a potentialVK,K′
in reciprocal space which is translationally invariant:

VK,K′=V(K−K′). (6.71)