Matching the boundary conditions

We have defined our APW as:

$$\psi_{\bf k}({\bf r}) = \left\{ \begin{array}{cc} e^{i{\bf k... ... \vert{\bf r} - {\bf R}\vert< r_0 \\ \end{array}\right.$$ (243)

If an eigenfunction would be discontinuous, its kinetic energy would not be well-defined. Such a situation can therefore never happen, and we have to require that the plane wave outside the sphere matches the function inside the sphere over the complete surface of the sphere (in value, not in slope). That seems a weird thing to do: a plane wave is oscillating and has a unique direction built in, how can it match another function based on spherical harmonics over the entire surface of a sphere? To see how this is possible, we expand the plane wave in spherical harmonics:

$$\exp{(i{\bf k}\cdot {\bf r})} = 4\pi \sum_{l=0}^\infty \sum_... ...l(kr)y_{lm}^*(\theta_{\bf k},\phi_{\bf k}) Y_{lm}(\theta,\phi)$$ (244)

where $\theta$,$\phi$,$r$ correspond to the polar representation of ${\bf r}$ and $\theta_{\bf k}$,$\phi_{\bf k}$,$k$ to ${\bf k}$. To keep the problem tractable, we cut all the expansions in $lm$ to a finite value of $l$. $j_l (r)$ is the Bessel function of order $l$. Requiring this at the sphere boundary means that all the coefficients of $Y_{lm}$ have to be equal for both parts of the function at the boundary. Thsi conditiono fixes the $A_{lm}$ and we obtain:

$$\psi_{\bf k}^{APW}({\bf r}) = 4 \pi \sum_{lm} i^l\left[\frac... ...R_l(r)Y_{lm}^*(\theta_{\bf k},\phi_{\bf k})Y_{lm}(\theta,\phi)$$ (245)

for the function inside the sphere.

The APW function is no a solution to the Scrödinger equation, but they are appropriate for expanding the actual wave function. The APW method tries to approximate the correct solution to the crystal by a superposition of APW's, all with the same energy. For any reciprocal lattice vector ${\bf K}$, the APW satisfies the Bloch condition with wave vector k, but for the entire wavefunctioon to be of the Bloch form we need the expansion of $\psi_{\bf k}({\bf r})$

$$\psi_{\bf k}({\bf r}) = \sum_{\bf K} c_{\bf K}\psi_{{\bf k}+{\bf K},\epsilon({\bf k})}^{APW}({\bf r})$$ (246)

where the sum is over all the reciprocal lattice vectors.

The hope is that we swill need only a small number of APW's to approximate the full Schrödinger euqation in the interstitial region and at the boundary. In practice, as many as a several hundreeds can be used. By the time we do this, the energy does not change much, as more APW's are added, and we achieve convergence.

All the APW have to be evaluated at the same energy. The coefficients are given, again, by solving the generalized eigenvalue equation

$$Hc=ESc$$ (247)

where the elements of $H$ and $S$ have very complicated expressions.

The most remarkable aspect of this equation is, that even though is looks like an ordinary eigenvalue problem, the marix elements depend on energy! To solve the problem it is convenient to work a fixed energy, and look for the $k$'s at which the following secular expresion is satisfied:

$$(H-ES)c=0$$ (248)

Another possibility, is to fix the momentum ${\bf k}$, and define a fine energy mesh, and look for the zeros of the determinant $\vert H-ES\vert$.