Key points are not available for this paper at this time.
The energy functional expressed in terms of the Wannier function is varied with respect to the trial function in the region surrounding the lattice site on which the function is centred and with respect to parameters describing the exponential tail of the trial function for the remaining volume of the crystal. The differential equation for the trial function in the region surrounding the lattice site and the algebraic equation for the parameter describing the exponential tail of the function obtained in this way are simultaneously solved for the case of a one-dimensional crystal with a Mathieu potential. The band centre, the band width and the interaction integrals are compared with the exact values available for the crystal described by the Mathieu potential.
Modrak et al. (Wed,) studied this question.