Computational Physics

136 Solving the Schrödinger equation in periodic solids

where the functionsRl(r)are the solutions of the radial Schrödinger equation with
energyE:

−

1

2 r^2

d

dr

[

r^2

dRl(r)

dr

]

+

[

l(l+ 1 )

2 r^2

+V(r)

]

Rl(r)=ERl(r) (6.27)

which can be solved with high accuracy, and theYml(θ,φ)are the spherical har-
monics. The expansion coefficientsAlmare found by matching the solution inside
the muffin tin to the plane wave outside.
The Bloch wave exp(iq·x)in the interstitial region is an exact solution to the
Schrödinger equation at energyq^2 /2 (in atomic units). The muffin tin solutions
are numerically exact solutions of the Schrödinger equation at the energyEfor
which the radial Schrödinger equation has been solved. However, if we take this
energy equal toq^2 /2, the two solutions do not match perfectly. The reason is that
thegeneralsolution in the interstitial region for some energyE=q^2 /2 includes
allwave vectorsqwith the same length. In the Bloch solution we take only one of
these. If we want to solve the Schrödinger equation inside the muffin tins with the
boundary condition imposed by this single plane wave solution then we would have
to include solutions that diverge at the nucleus, which is physically not allowed.
An APW basis function contains a muffin tin solution with a definite energyE,
and a Bloch wave exp(iq·x)in the interstitial region. It turns out to be possible to
match the amplitude of the wave function across the muffin tin sphere boundary.
In order to carry out the matching procedure at the muffin tin boundary we expand
the plane wave in spherical harmonics[13]:

exp(iq·r)= 4 π

∑∞

l= 0

∑l

m=−l

iljl(qr)Yml

∗

(θq,φq)Yml(θ,φ) (6.28)

wherer,θandφare the polar coordinates ofrandq,θqandφqthose ofq. To keep
the problem tractable, we cut all expansions inlmoff at a finite value forl:

∑∞

l= 0

∑l

m=−l

→

∑lmax

l= 0

∑l

m=−l

(6.29)

From now on, we shall denote these sums by

∑

lm.
The matching condition implies that the coefficients of theYml must be equal for
both parts of the basis function,(6.26)and(6.28), as theYml form an orthogonal
set over the spherical coordinates. This condition fixes the coefficientsAlm, and we
arrive at

ψqAPW(r)= 4 π

∑

lm

il

[

jl(qR)

Rl(R)

]

Rl(r)Yml

∗

(θq,φq)Yml(θ,φ) (6.30)

for the APW basis function inside the sphere.