6.7 The pseudopotential method 153(^0) C 2 Gridsize 1
C 1
UC
Figure 6.14. Two-dimensional representation of the reciprocal grid. The dashed
sphereC1 contains the wave vectors of the wavefunctions – its periodic images
are also shown as solid circles. The bigger circleC2 contains the reciprocal wave
vectors for representing the electron density, and we need all those points for
accurately constructing the Hamiltonian. The cellUCis a unit cell of the reciprocal
lattice.
4 π^2 /L^2 , as two of these have energy 0 and the other two are divided over the second
level. In fact, in each of the six degenerate levels, we should put 1/3 electron because
of symmetry. This is an example of fractional filling resulting from degeneracy.
programming exercise
Write a program which diagonalises the Hamiltonian for a particle confined
to a box.
CheckFor the density in a box of size 5 a.u. we find in this case that the density
is homogeneous and equals 0.032= 4 /125. This is not surprising as we put four
electrons in a box of volume 5^3 =125.
6.7.5 Adding a pseudopotential
The pseudopotential is part of the total potential felt by the electrons. The pseudo-
potential consists of a local and a nonlocal part. Alocalpotential can be evaluated
as inEq. (6.71). The local pseudopotential potential depends only onr−r 0 , where
r 0 is the centre of an atom. We have:
VK,K′=V(K−K′)=e−i(K−K
′)·r 0
∫
e−i(K−K′)·r
V(r)d^3 r. (6.74)