The expression for the gap energy in Eq.() is based on the change in the ground state energy of the bulk solid when a single electron is added or removed. In all our QMC calculations for bulk solids, we choose to model this system using a finite simulation cell to which we apply periodic boundary conditions. When an additional electron is added to this simulation cell, the periodic boundary conditions effectively introduce an equivalent extra electron into each of the periodic images of the simulation cell. This adds an extra electrostatic energy into the system, which when combined with a compensating background charge is just the Madelung energy of an electron (hole) crystal with the periodicity of the simulation cell. A similar effect has been observed in LDA calculations[105, 106, 107, 108]. Leslie and Gillan proposed a correction term to the Hartree energy of the system to account for the additional electrostatic energy due to an array of charged defects,
where is the Madelung constant for the supercell geometry, L is the length of the simulation cell, is a dielectric constant for the material, and q is the charge on the defect. The problem with this correction is how to choose the dielectric constant, . In general experimental values have been used and these have not been found to work particularly well.
It has also been speculated by Engel et al. that similar effects may be present in their VMC calculations of the band structure of a two-dimensional model crystal. In their calculations, an extra electron was added into an orbital in the conduction band. However, this orbital is actually spread throughout the simulation cell and so can be regarded as contributing a much smaller term to the Hartree energy than a point defect plus background would. In the limit of infinite simulation cell size L, any additional energy due to interactions between the array of additional electrons would disappear. Therefore Engel et al. treat this as a finite size effect and deal with it by fitting results for a series of VMC calculations at different system sizes to the expression
where L is the length of the simulation cell, and is a parameter that represents a reduced due to the screening of the other valence electrons.
In our QMC calculations[110, 3, 111] we no longer use the Ewald interaction to evaluate the electron-electron interaction between pairs of electrons and therefore we are not necessarily restricted to including all the periodic images of the additional electron(hole) in our system in the same way as Engel et al.
Figure: Addition of a single electron to the simulation cell. Figure (a) shows an N electron simulation cell periodically repeated. Figure (b) shows the same bulk system with an additional electron added only to the simulation cell (red).
Consider the two systems illustrated in figure . Figure (a) schematically represents the standard simulation cell for the N electron system and a few of the periodic images of the simulation cell. The electron-electron energy associated with this system can be defined as in section by the new electron-electron energy expression for N electron systems,
In Figure (b) the same system is shown with an extra electron added only to the actual simulation cell, not to any of its periodic images. We can represent the change in the charge density of the whole system due to the additional electron by , and we would like to confine to within the central simulation cell, i.e. there should be no additional electrons in the periodic images of the simulation cell. This effect can be achieved by altering the interaction so that each electron `feels' the full 1/r interaction with all N+1 electrons within the simulation cell surrounding it, but only feels the Hartree interaction with the charge density due to N electrons in each periodic image outside the simulation cell. We re-write Eq.() to take account of the extra electron and the change in the charge density, , which is confined to the central simulation cell, as
and expand out the product
in the second term. We can discard the term in
which is small as
is a short ranged function and
is a small for small. This can be understood physically in the following way; represents the change in the charge density due to adding an electron. The term represents the interaction of this change with itself. In an infinite system is virtually zero and so this term should disappear. Removing this term yields
As in Eq.(), the first term describes the full Hartree and exchange/correlation interaction between all X electrons in the simulation cell. The second two terms can be interpreted as representing the Hartree interaction between the charge density due to X electrons inside the simulation cell and the charge density due to N electrons outside the simulation cell. Therefore, as far as the electron-electron interaction is concerned there is only one extra electron present in the system rather than the whole periodic array which is normally introduced. The use of this new energy expression removes the need for ad hoc corrections to the finite size effects such as those used by Engel et al. in Eq.(). Note, when using either of the two energy expressions, Eq.() and Eq.(), we include background charges so there is no contribution to the total energy or any gap energies from the component of the f interaction. This is equivalent to ensuring that each cell is neutral, as would be the case when a single electron is added to the infinite system.