The general Hamiltonian of Eq.() can be adapted for a supercell calculation in the following way
where is the set of translation vectors of the supercell lattice, the potential has the periodicity of , and N is the number of electrons in the supercell. In the case where the supercell is constructed from integer multiples of the primitive unit cells, as is the case for all the calculations described here, also then has the periodicity of the set of translation vectors of the underlying crystal lattice.
Trial wavefunctions for this supercell Hamiltonian are based on the general trial wavefunction introduced in Eq.(),
The Slater determinants are constructed from one-electron orbitals obtained from an LDA calculation. The -point sampling in the LDA calculation is chosen to produce the desired number of one-electron orbitals for constructing the Slater determinant in the QMC trial/guiding wavefunction. For example, if an n=1, 1x1x1 supercell is chosen for the QMC calculation then the LDA calculation is also performed on a single unit cell and the wavefunctions are sampled at one -point. If an n=2, 2x2x2 supercell is chosen for the QMC calculation then the LDA calculation is again performed on a single unit cell but now the wavefunctions are sampled from a 2x2x2 mesh of -points in the Brillouin Zone of the primitive lattice.
Recently, new insights have been made into the best choice for the -points at which the one-electron wavefunctions should be calculated [33, 50] for use in QMC calculations. To understand these, first one should consider the translational symmetries of the above Hamiltonian.
Symmetry (1) implies that the wavefunction can only change by a phase factor when any single electron is translated by a supercell lattice vector. The indistinguishability of the electrons ensures that this phase factor must be the same no matter which electron is moved. This can be demonstrated by applying Bloch's theorem separately to the first and second arguments of the wavefunction and, for the moment, assuming that the two -vectors are different;
We can then apply the permutation symmetry to Eq.():
We now translate the second argument by ,
and then apply permutation symmetry once more, giving
It therefore follows that
where is the set of vectors reciprocal to . The vectors and can be reduced into the first Brillouin Zone (BZ) of the supercell reciprocal lattice, therefore we can choose without loss on generality. The wavefunction can therefore be written in the form
where is invariant under the translation of any electron coordinate by a vector in , and is antisymmetric under particle exchange.
Now consider the second symmetry of which states that the wavefunction can only change by a phase factor when all the electrons are translated by a vector in . This allows us to write
where is the crystal momentum of the wavefunction and can be reduced into the first BZ of the lattice reciprocal to . It therefore follows that can be written in the alternative form
where in invariant under the simultaneous translation of all electron coordinates by a vector in and is antisymmetric under particle exchange.
The operators which translate all the electrons by a vector in and the operators which translate a single electrons by a vector in commute with each other and with the Hamiltonian, i.e. they form a complete set of commuting operators. Therefore the eigenfunctions of the Hamiltonian in Eq.() can be chosen to satisfy both the above symmetries at the same time. We can obtain a relationship between the values of and by translating all the electrons by a vector in (which is a subset of ), and using Eq.() we find
This must agree with Eq.(), which yields
In a QMC trial wavefunction, the value of is determined by the Slater determinant. If all the one-electron wavefunctions making up the determinant reduce to the same value of in the supercell BZ, then the overall determinant and hence the wavefunction will have that value of . The value of for a QMC trial wavefunction is determined by the sum of all the values of the one-electron wavefunctions making up the determinant. Applications of QMC prior to Refs.[33, 50] used the conceptually simplest choice of and , namely . This is achieved by choosing the -values for the one-electron orbitals on a uniform grid or mesh centred on the origin in reciprocal space, with a grid spacing .
In the limit of an infinite simulation cell, the value of must tend to zero. However, for a finite simulation cell, the groundstate does not always take the values . In the following section, we consider the specific systems of diamond structure germanium and silicon and explore what is the best choice of values of and for these systems.