Numerical Methods in Quantum Mechanics - Dipartimento di Fisica

such equations. Notice that the solutions having different k are by construction
orthogonal. Let us write the superposition integral between Bloch states for
different k:
∫ L/2
∫ L/2
ψk,n(x)ψ ∗ k ′ ,m(x)dx = e i(k′ −k)x u ∗ k,n(x)u k ′ ,m(x)dx (9.8)
−L/2
−L/2
( ) ∑ ∫ a/2
= e ip(k′ −k)a
e i(k′ −k)x u ∗ k,n(x)u k ′ ,m(x)dx,
p
−a/2
where the sum over p runs over all the N vectors of the lattice. The purely
geometric factor multiplying the integral differs from zero only if k and k ′
coincide:
∑
e ip(k′ −k)a = Nδ k,k ′. (9.9)
p
we have used Kronecker’s delta, not Dirac’s delta, because the k form a dense
but still finite set (there are N of them). We note that the latter orthogonality
relation holds for whatever periodic part, u(x), the Bloch states may have.
Nothing implies that the periodic parts of the Bloch states at different k are
orthogonal: only those for different Bloch states at the same k are orthogonal
(see Eq.(9.7)).
9.1.5 Plane-wave basis set
Let us come back to the numerical solution. We look for the solution using a
plane-wave basis set. This is especially appropriate for problems in which the
potential is periodic. We cannot choose any plane-wave set, though: the correct
choice is restricted by the Bloch vector and by the periodicity of the system.
Given the Bloch vector k, the “right” plane-wave basis set is the following:
b n,k (x) = √ 1 e i(k+Gn)x , G n = 2π n. (9.10)
L a
The “right” basis must in fact have a exp(ikx) behavior, like the Bloch states
with Bloch vector k; moreover the potential must have nonzero matrix elements
between plane waves. For a periodic potential like the one in Eq.(9.4), matrix
elements:
∫ L/2
〈b i,k |V |b j,k 〉 = 1 L
= 1 ( ∑
L
p
−L/2
V (x)e −iGx dx (9.11)
) ∫ a/2
e −ipGa V (x)e −iGx dx, (9.12)
−a/2
where G = G i − G j , are non-zero only for a discrete set of values of G. In fact,
the factor ∑ p e−ipGa is zero except when Ga is a multiple of 2π, i.e. only on
the reciprocal lattice vectors G n defined above. One finally finds
〈b i,k |V |b j,k ) = 1 a
∫ a/2
−a/2
V (x)e −i(G i−G j )x dx. (9.13)
The integral is calculated in a single unit cell and, if expressed as a sum of
atomic terms localized in each cell, for a single term in the potential. Note that
the factor N cancels and thus the N → ∞ thermodynamic limit is well defined.
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