About cookies on this site Our websites require some cookies to function properly (required). In addition, other cookies may be used with your consent to analyze site usage, improve the user experience and for advertising. For more information, please review your options. By visiting our website, you agree to our processing of information as described in IBM’sprivacy statement. To provide a smooth navigation, your cookie preferences will be shared across the IBM web domains listed here.
Publication
Physical Review
Paper
Exact matrix elements of a crystal hamiltonian between harmonic-oscillator wave functions
Abstract
A method is given for the exact calculation, including all anharmonic effects, of the matrix elements of the true Hamiltonian H of a crystalline solid between the eigenfunctions of any arbitrary harmonic Hamiltonian appropriate to the crystal structure. By the introduction of a variational scaling parameter a, a set of harmonic eigenfunctions can be generated from the eigenfunctions of a single such harmonic Hamiltonian. The ground-state energy W0 of the crystal and the optimum value a0 of a are then determined by a variational calculation which minimizes the expectation value of H between the ground-state harmonic eigenfunctions. The other diagonal matrix elements of H, calculated using the states determined by a0, then give first-order values for the energy of one-phonon and multiphonon states. In addition, the off-diagonal matrix elements can be obtained and used in a perturbation calculation to improve the energies. It is shown that the value of W0 will always be lower than energy obtained from a closely related variational calculation using a wave function constructed out of a product of Gaussian orbitals centered on the lattice sites. The decrease in energy is about 12% of the original kinetic energy, and is divided approximately equally between the kinetic- and potential-energy contributions. Numerical results are given for W0 and for a few phonon energies of a model bcc crystal in which each atom interacts with its first two shells of nearest neighbors through a realistic atomic potential. Possible applications of this theory to rare-gas solids are discussed. © 1966 The American Physical Society.