New Theory of Lattice Dynamics at 0°K

Abstract

It has previously been shown that the optimum choice of harmonic Hamiltonian with which to approximate a crystal Hamiltonian is one is which the force constants are equal to the ground-state expectation value of the second derivative of the crystal potential, the expectation value being computed self-consistently with the ground-state eigenfunction of the harmonic Hamiltonian. It is shown here that the appearance of ground-state averages of various derivatives of the potential is due to an explicit or implicit expansion of the potential in a Hermite polynomial series, and that such an expansion is superior to the conventional Taylor-series expansion for anharmonic systems. In addition, it is shown that one can systematically treat very anharmonic systems by expanding the Hamiltonian in a set of polynomials orthogonalized with respect to weight function chosen to cut off the potential at short range, and that explicit incorporation of an easily optimized Gaussian factor in this weight function provides a computationally convenient way of introducing certain desirable features into the general expansion.