Abstract
When a system which contains a dipole, and whose dimensionality is less than three, is studied in a code which imposes periodic boundary conditions in all three dimensions, an artificial electric field arises which keeps the potential periodic. This has an impact on the total energy of the system, and on any other attribute which would respond to an electric field. Simple corrections are known for 0D systems embedded in a cubic geometry, and 2D slab systems. This paper shows how the 0D result can be extended to tetragonal geometries, and that for a particular c/a ratio the correction is zero. It also considers an exponential error term absent from the usual consideration of 2D slab geometries, and discusses an empirical form for this.
Highlights
Plane-wave electronic structure codes enforce periodicity in all three dimensions
This paper considers a limited generalisation of the cube result of Makov and Payne, a generalisation which avoids the need for summations, and it considers a particular geometry for which no correction is needed
When the slab geometry is considered, the additional correction proposed in equation 6 decays exponentially with cell length
Summary
Plane-wave electronic structure codes enforce periodicity in all three dimensions. Whilst this is ideal for studying bulk crystals, it can cause complications when the system of interest has lower dimensionality, such as a surface (2D), a nanowire (1D), or an isolated molecule (0D). For the 2D geometry of a slab of material, commonly used to study surfaces, a dipole moment perpendicular to the slab results in a compensating electric field in order to keep the potential periodic[1]. A 2D slab with a perpendicular dipole moment is effectively a charged parallel plate capacitor, and one would expect the potential on each side to differ, and the field outside the plates to be zero. Other methods have been considered to correct for unwanted dipole-dipole interactions These include the truncation of the Coulomb potential in real space[11], which was proven to be equivalent to the above self-consistent slab correction[12]. It finds a further correction term for the 2D slab geometry
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