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A new method for solving the problem of one electron in a periodic potential is presented; it is discussed in this paper mainly for k = 0, although it can be generalized to other k. The periodic potential is considered to be generated by spherically symmetric " atomic" potentials at each lattice site; this does not mean of course that the total potential near a lattice site need be spherically symmetric. The method has its origin in the observation that (for k = 0) the equation for C(K;), the Fourier coefficient of the wave function, becomes just the momentum-space Schrodinger equation when the lattice spacing becomes infinite. This latter equation is separable into a radial part, and an angledependent part expressible in spherical harmonics. This suggests that it would be advantageous to expand the C(Ki) for finite lattice spacing similarly, into radial functions C/sub 1/(K/sub m/), where K/sub m/ is the magnitude of the mth smallest reciprocal lattice vector, and into an angle-dependent part expressible (for cubic lattices) by Kubic harmonics. This is done and the Schrodinger equation for the system becomes a set of homogeneous linear equations for the C/sub 1/(K/sub m/), with a corresponding secular determinant for themore » eigenvalues. The method was tested numerically, as a function of lattice spacing and potential strength, for S-like states, when the " atomic" potentials are exponential ones, and the lattice is body-centered cubic. In many cases it turns out that one can solve the periodic potential case more easily and more accurately thnn one can solve for the isolated atom. This is because as the lattice spacing gets large the successive K/sub m/ became more and more closely spaced and this leads to larger and larger secular equations. The wave functions as well as energies are given for most lattice spacings to considerable accuracy (three to seven significant figures). When the lattice spacing gets large and the equations approach those for the isolated atom, it is shown how one can use the atomic momentum space functions as variational functions, in the same spirit as the usual tight-binding approximation (as applied for k == 0). The present method has the considerable advantage that it bypasses the usual difficulties with that approximationnear-neighbor approximations and calculation of overlap integrals-and permits an easy and accurate evaluation of the variational expression as a sum over the K/sub m/. (auth)« less |
