Numerical Methods in Quantum Mechanics - Dipartimento di Fisica

The origin of the coordinate system is arbitrary; typical choices are one of
the atoms, or the middle point between two neighboring atoms. We use the
latter choice because it yields inversion symmetry. The two atoms in the unit
cell are thus at positions d 1 = −d, d 2 = +d, where
( 1
d = a 0
8 , 1 8 , 1 )
. (10.10)
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The lattice parameter for Si is a 0 = 10.26 a.u.. 3
Let us re-examine the matrix elements between plane waves of a potential
V , given by a sum of spherical symmetric potentials V µ centered around atomic
positions:
V (r) = ∑ ∑
V µ (|r − d µ − R n |) (10.13)
n µ
With some algebra, one finds:
〈b i,k |V |b j,k 〉 = 1 ∫
V (r)e −iG·r dr = V Si (G) cos(G · d), (10.14)
Ω
Ω
where G = G i − G j . The cosine term is a special case of a geometrical factor
known as structure factor, while V Si (G) is known as the atomic form factor:
V Si (G) = 1 ∫
V Si (r)e −iG·r dr. (10.15)
Ω
Ω
Cohen-Bergstresser pseudopotentials are given as atomic form factors for a few
values of |G|, corresponding to the smallest allowed modules: G 2 = 0, 3, 4, 8, 11, ...,
in units of (2π/a 0 ) 2 .
The code requires on input the cutoff (in Ry) for the kinetic energy of plane
waves, and a list of vectors k in the Brillouin Zone. Traditionally these points
are chosen along high-symmetry lines, joining high-symmetry points shown in
figure and listed below:
Γ = (0, 0, 0),
X = 2π
a 0
(1, 0, 0),
W = 2π
a 0
(1, 1 2 , 0),
K = 2π
a 0
( 3 4 , 3 4 , 0),
L = 2π
a 0
( 1 2 , 1 2 , 1 2 ),
3 We remark that the face-centered cubic lattice can also be described as a simple-cubic
lattice:
a 1 = a 0 (1, 0, 0) , a 2 = a 0 (0, 1, 0) , a 3 = a 0 (0, 0, 1) (10.11)
with four atoms in the unit cell, at positions:
(
d 1 = a 0 0, 1 2 , 1 ) ( 1
, d 2 = a 0
2
2 , 0, 1 ) ( 1
, d 3 = a 0
2
2 , 1 )
2 , 0 , d 4 = (0, 0, 0) . (10.12)
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