Computational Physics

158 Solving the Schrödinger equation in periodic solids

6.7.7 Evaluating the energy

If you have obtained the correct spectrum, the density should necessarily be correct
too. One major task remains, however: evaluating the total energy. The energy can
be evaluated either by adding all the Kohn–Sham eigenvalues and subtracting the
appropriate corrections as in Eq. (5.3), or by using the Kohn–Sham eigenfunctions
to evaluate all the contributions to the energy as in (5.17) one by one. We take the
second approach. First of all, thekinetic energyis given by

Ekin=

∑

j

f(Ej)

∑

K

|c(j)|^2 K^2 /2. (6.90)

The exchange-correlation energy is evaluated in real space:

Exc=

∑

r

εxc(r)n(r), (6.91)

where the sum is over the real-space lattice points, or in reciprocal space, where it
reads:
Exc=

∑

K

ε(K)n∗(K). (6.92)

Following Marx and Hutter [25], we combine the electrostatic contributions from
the electrons and the ion cores. Remember that the core part of the pseudopotential,

Vcore=−

Zn

|r−Rn|

erf

(

|r−Rn|

√

2 ξ

)

; (6.93)

derives from a Gaussian charge distribution:

ncore(r)=−

Zn

(

√

2 ξn)^3

π−^3 /^2 exp

[

−

1

2

(

r−Rn

ξ

) 2 ]

. (6.94)

The Fourier transform of the core density is

ncore(K)=−

Zn

exp

[

−

1

2

(ξK)^2

]

e−iK·Rn. (6.95)

For the total charge density we have

ntot(K)=nel(K)+ncore(K). (6.96)

The electrostatic energy resulting from the total charge density is

EES=

1

2

∫

nel(r)nel(r′)

|r−r′|

d^3 rd^3 r′+

∫

ncore(r)nel(r′)

|r−r′|

d^3 rd^3 r′

+

1

2

∑

n

=n′

ZnZn′

|Rn−R′n|

, (6.97)