Set of supplementary notes.

72 CHAPTER 4. ELECTRONIC STRUCTURE
are plane wave like, but in the vicinity of the core they oscillate rapidly so as to be orthogonal to the
core levels.
We can now use the OPW’s as basis states for the diagonalisation in the same way that we used
plane waves in the NFE, viz
|ψ k >= ∑ α k−G |χ k−G > . (4.34)
G
This turns out to converge very rapidly, with very few coefficients, and only a few reciprocal lattice
vectors are included in the sum. The following discussion explains why.
Suppose we have solved our problem exactly and determined the coefficients α. Now consider the
sum of plane waves familiar from the plane-wave expansion, but using the same coefficients, i.e.
|φ k >= ∑ G
α k−G |k − G > , (4.35)
and then 4 it is easily shown that
|ψ >= |φ > − ∑ n
< f n |φ > |f n > . (4.36)
Then substitute into the Schrodinger equation H|ψ >= E|ψ >, which gives us
H|φ > + ∑ n
(E − E n ) < f n |φ > |f n >= E|φ > (4.37)
We may look upon this as a new Schrödinger equation with a pseudopotential defined by the operator
V s |φ >= U|φ > + ∑ n
(E − E n ) < f n |φ > |f n > (4.38)
which may be written as a non-local operator in space
∫
(V s − U)φ(r) = V R (r, r ′ )φ(r ′ ) dr ′ ,
where
V R (r, r ′ ) = ∑ n
(E − E n )f n (r)f ∗ n(r ′ ) . (4.39)
The pseudopotential acts on the smooth pseudo-wavefunctions |φ >, whereas the bare Hamiltonian acts
on the highly oscillating wavefunctions |ψ >.
One can see in (4.38) that there is cancellation between the two terms. The bare potential is large and
attractive, especially near the atomic core at r ≈ 0; the second term V R is positive, and this cancellation
reduces the total value of V s especially near the core. Away from the core, the pseudopotential approaches
the bare potential.
4 Saving more notation by dropping the index k