Variational and diffusion quantum Monte Carlo calculations at nonzero wave vectors: Theory and application to diamond-structure germanium

A variational and diffusion quantum Monte Carlo study of germanium in the diamond structure is reported, in which local pseudopotentials are used to represent the ion cores. We calculate the energy of the free atom and the energy of the solid as a function of volume. The calculations for the solid are performed using a supercell method. We analyze the translational symmetry of the supercell Hamiltonian and show that the eigenstates can be labeled by two wave vectors ${\mathbf{k}}_{\mathit{s}}$ and ${\mathbf{k}}_{\mathit{p}}$. The wave vector ${\mathbf{k}}_{\mathit{s}}$ arises from the invariance of the Hamiltonian under the translation of any one electron coordinate by a supercell translation vector, while the wave vector ${\mathbf{k}}_{\mathit{p}}$, which is the crystal momentum of the wave function, arises from the invariance under the simultaneous translation of all electron coordinates by a translation vector of the crystal lattice. Our solid calculations are performed using wave functions with nonzero supercell wave vectors ${\mathbf{k}}_{\mathit{s}}$, which gives better convergence with the size of supercell than previous zero-wave-vector calculations. The relationship of this method to the special k-points techniques commonly used in band-structure calculations is discussed.