Computational Physics

6.7 The pseudopotential method 159

where the first term is the Hartree electrostatic energy due to the electrons, the
second term is the interaction between the electrons and the core, and the last term
is the core–core interaction.
This last term causes problems as we must sum it over all periodic images of the
unit cell for which we are performing the calculations (for a discussion concerning
convergence of this type of expression, see Section 8.7.1). These problems can
be avoided by replacing the last term by the electrostatic interaction of the core
charges. This is done by adding and subtracting a term

Ecc=

1

2

∫

ncore(r)ncore(r′)

|r−r′|

d^3 rd^3 r′ (6.98)

to the expression for the total energy,Eq. (6.97). The added term, together with the
first two terms in that equation, yields a contribution

1

2

∫

ntot(r)ntot(r′)

|r−r′|

d^3 rd^3 r′ (6.99)

The remaining terms can be written as a convergent sum[25], leading to

EES=

1

2

∫

ntot(r)ntot(r′)

|r−r′|

d^3 rd^3 r′

+

1

2

∑

n,n′

ZnZn′

|Rn−Rn′|

erfc







|Rn−Rn′|

√

2 (ξn^2 +ξn^2 ′)





−

∑

n

Zn^2

2

√

πξn

. (6.100)

The second term on the right hand side is due to the overlap of the core distributions,
and the third term corrects for the self-energy (that is, the energy of a core with
itself). Both of these are contained in the first term.
For periodic boundaries, we can reformulate this expression in Fourier space,
where it reads:

EES= 2 π

∑

K

= 000

|ntot(K)|^2

K^2

+Eovrl−Eself, (6.101)

wherentot(K)is given above(Eq. (6.95))and where

Eovrl=

∑

L

∑

n,n′

′ ZnZn′

|Rn−Rn′−L|

erfc







|Rn−Rn′−L|

√

2 (ξn^2 +ξn^2 ′)





, (6.102)

whereLis an integer linear combination of the sides of the unit cell, the second
sum is restricted ton<n′forL= 00 0, and

Eself=

∑

n

Zn^2

2

√

πξn

. (6.103)