6.5 Augmented plane wave methods 137
Summarising the results so far, we can say that in the APW method the wave
function is approximated in the interstitial region by plane waves, whereas in the
core region the rapid oscillations are automatically incorporated via direct integra-
tion of the Schrödinger equation. The basis functions are continuous at the sphere
boundaries, but their derivative is not. The APW functions are not exact solutions
to the Schrödinger equation, but they are appropriate basis functions for expanding
the actual wave function:
ψk(r)=∑
KCKψkAPW+K(r). (6.31)The muffin tin parts of theψAPWin this expansion are all evaluated at the same
energyE. The coefficientsCK are given by the lowest energy solution of the
generalised eigenvalue equation:
HC=ESC (6.32)where the matrix elements ofHandSare given by quite complicated expressions.
In the resulting solution, the mismatch in the derivative across the sphere boundary
is minimised.
Before giving the matrix elements of the Hamiltonian and the overlap matrix,
we must point out that (6.32) differs from usual generalised eigenvalue equa-
tions in that the matrix elements of the Hamiltoniandepend on energy. This
dependence is caused by the fact that they are calculated as matrix elements of
energy-dependent wave functions (remember the radial wave functions depend
on the energy). Straightforward application of the matrix methods for generalised
eigenvalue problems is therefore impossible.
In order to obtain the spectrum, we rewrite Eq. (6.32) in the following form:
(H−E)C= 0 (6.33)whereH=H−ES+EI(Iis the unit matrix). Although the form(6.33)suggests that
we are dealing with an ordinary eigenvalue problem, this is not the case: the overlap
matrix has been moved intoH. To find the eigenvalues we calculate the determinant
|H−EI|on a fine energy mesh and see where the zeroes are. It is sometimes
possible to use root-finding algorithms (see Appendix A3) to locate the zeroes of
the determinant, but these often fail because the energy levels may be calculated
along symmetry lines in the Brillouin zone and in that case the determinant may
vary quadratically about the zero, so that it becomes impossible to locate this point
by detecting a change of sign (this problem does not arise in Green’s function
approaches, where one evaluates the Green’s function at a definite energy).
For crystals with one atom per unit cell, the matrix elements of the Hamiltonian
for APW basis functions with wave vectorsqi =k+Kiandqj =k+Kjare