142 Solving the Schrödinger equation in periodic solids
In comparison with the APW method, we have twice as many radial functions
inside the muffin tin sphere,RlandR ̇l, and we can match not only the value but
also the derivative of the plane wave exp(iq·r)across the sphere boundary. We
write the wave function inside as the expansion
(^) k+K(r)=
∑
lm[AlmRl(r;Ep)+BlmR ̇l(r;Ep)]Yml(θ,φ) (6.41)and the numbersAlmandBlmare fixed by the matching condition. There is no energy
dependence of the wave functions, and they are smooth across the sphere boundary,
but the price which is paid for this is giving up the exactness of the solution inside
the sphere for the range of energies we consider.
We end up with a generalised eigenvalue problem with energy-independent over-
lap and Hamiltonian matrices. These matrices are reliable for energies in some range
around the pivot energy. It turns out that the resulting wave functions have an inac-
curacy of(E−Ep)^2 as a result of the linearisation and that the energy eigenvalues
deviate as(E−Ep)^4 fromthoseevaluatedatthecorrectenergy–seeRef.[18].
The expressions for the matrix elements are again quite complicated. They
depend on the normalisations forRlandR ̇lwhich will be specified below. For
the coefficientsAlmandBlm, the matching conditions lead (withq =k+K,
q′=k+K′) to:
Alm(q)= 4 πR^2 il −^1 /^2 Yml
∗
(θq,φq)al; (6.42a)
al=jl′(qR)R ̇l(R)−jl(qR)R ̇′l(R); (6.42b)
Blm(q)= 4 πR^2 il −^1 /^2 Yml
∗
(θq,φq)bl; (6.42c)
bl=jl(qR)R′l(R)−jl′(qR)Rl(R). (6.42d)The matrix elements of the overlap matrix and the Hamiltonian can now be
calculated straightforwardly – the result for the overlap matrix is [18]
SK,K′=U(K−K′)+
4 πR^4∑
l( 2 l+ 1 )Pl(qˆ·qˆ′)slK,K′ with (6.43a)sKl,K′=al(q)al(q′)+bl(q)bl(q′)Nl and (6.43b)U(K)=δK,O−4 πR^2
jl(KR)
K. (6.43c)
Here,Nlis the norm of the energy derivative inside the muffin tin (see below). The
Hamiltonian is given by
HK,K′=(q·q′)U(K−K′)+
4 πR^2∑
l( 2 l+ 1 )Pl(ElslK,K′+γl) (6.44a)