The LAPW method

The traditional way of linearizing the APW method is the LAPW method, which was developed in the beginning of the 1970s. In this approach the basis functions are expanded in the same way as in Eq. (245) in the interstitial, but inside the muffin- tin the basis functions do not only depend on $R_l(r,\epsilon)$, but also on its derivative $\dot{R}_l(r,\epsilon) \equiv \partial R_l /\partial \epsilon$.

The idea is that the radial wavefunction can be approximated well around an energy of interest by a linearization of the form:

$$R(r,\epsilon) = R(r,\epsilon_p) + (\epsilon-\epsilon_p)\dot{R}(r,\epsilon_p),$$ (253)

where $\epsilon_p$ is some reference energy, or pivot energy.

The LAPW wavefunction then reads

$$\psi_{\bf k}({\bf r}) = \left\{ \begin{array}{cc} e^{i{\bf k... ... \vert{\bf r} - {\bf R}\vert< r_0 \\ \end{array}\right.$$ (254)

The remarkable aspect of this expression is that the wavefunction no longer depends on the energy. The price we pay is in the accuracy of the wavefunction inside the MT sphere.

We end up with a generalized eigenvalue problem with energy-independent overlap and Hamiltonian matrices. these matrices are reliables within some range around the pivot energy.