Computational Physics

6.6 The linearised APW (LAPW) method 143

with

γl=R′l(R)R ̇l(R)[jl′(qR)jl(q′R)+jl(qR)jl′(q′R)]

−[R′l(R)R ̇′l(R)jl(qR)jl(q′R)+Rl(R)R ̇l(R)j′l(qR)jl′(q′R)]. (6.44b)

We see that a pleasing feature of these expressions is that we do not get APW-type
numerical inaccuracies due to radial solutions vanishing at the muffin tin radius and
occurring in the denominator of the expressions for the matrix elements.
Finally, we must find out how the energy derivative of the solution of the
radial Schrödinger equation,R ̇l, can be calculated. By differentiating the radial
Schrödinger equation
(H−E)Rl(r;E)= 0 (6.45)

with respect toE, we find thatR ̇lsatisfies the following differential equation:

(H−E)R ̇l(r;E)=Rl(r;E). (6.46)

This second order inhomogeneous differential equation needs two conditions to fix
the solution. The first condition is thatR ̇l(likeRl) is regular at the origin which
leaves the freedom of addingαRl(r)to it, for arbitraryα(Rlis the solution of
the homogeneous equation). The numberαis fixed by the requirement thatRlis
normalised: ∫
R

0

drr^2 R^2 l(r;E)= 1 (6.47)

which, after differentiation with respect toE, leads to
∫R

0

r^2 Rl(r)R ̇l(r)dr= 0 (6.48)

i.e.RlandR ̇lare orthogonal. The norm ofR ̇l,

Nl=

∫R

0

drr^2 |R ̇l(r)|^2 , (6.49)

which occurs in the definition of the overlap matrix, is therefore in general not
equal to one. It can be shown that the normalisation condition(6.47)leads to the
following boundary condition at the muffin tin bounday (r=R):

R^2 [R′l(R)R ̇l(R)−R(R)R ̇′l(R)]=1; (6.50)

see Problem 6.5.
The interested reader might try to write a program for calculating the band
structure of copper using this linearisation technique. The determination of the
eigenvalues will be found much more easily than in the case of the APW calculation,
as the Hamiltonian is energy-independent.