Computational Physics

160 Solving the Schrödinger equation in periodic solids

Table 6.3.Contributions to electronic energy of

hydrogen atom.

Contribution Value

Kinetic 0.159 230 87

Short range part of pseudopotential −0.021 083 18

Local pseudopotential −0.244 116 10

Exchange correlation −0.210 599 25

Hartree energy 0.024 351 53

Nonlocal pseudopotential 0.000 000 00

Local core energy 1.12582002

Self-energy 0.92508958

Electrostatic overlap 0.000 000 00

Total energy −0.557 835 94

Finally, we must include the energy contributions due to the pseudopotential.
These do not depend on the charge distribution and they contain the local and the
nonlocal terms. The local contribution is easily evaluated using

Elocal=

∫ ∑

n

Vlocal,n(r−Rn)n(r)d^3 r

=

∑

n

∑

K

Vlocal,n(K)e−iK·Rnn∗(K). (6.104)

wherenruns over the atoms in the cell. The nonlocal energy reads:

Enonlocal=

∑

j

fj

∑

n

∑

l,mεn

(Fjlmn )∗hnlmFnjlm (6.105)

wherel,mεndenotes the orbital with quantum numbersl,mbelonging to atom
numbern, and
Fjlmn =

∑

K

e−iK·Rnc∗j(K)Ylm(Kˆ)plm(K). (6.106)

Now that you have everything in place, you can calculate the electronic energy
of the hydrogen atom. It is built up from the contributions shown inTable 6.3.

6.8 Extracting information from band structures

Apart from ground state energies, from which cohesion energies and lattice spacings
can be determined, and those energy levels that can be measured directly using
spectroscopy experiments, it is useful to determine the density of states,n(E),
which can also be determined experimentally. This is defined as the number of levels