Summary

The subject of this appendix is formulations and explicit equations for the total energy that properly take into account the long-range effects of Coulomb interactions. We emphasize the Kohn–Sham independent-particle equations and expressions for total energy; however, the ideas and many of the equations also apply to many-body calculations. There are three main issues:

• Identifying various convenient expressions that each yield properly the intrinsic total energy per formula unit for an extended bulk system.

• Understanding and calculating the effect upon the average potential in a bulk material due to dipole terms at surfaces and interfaces.

• Treating finite systems, where there is no essential difficulty, but where it is convenient to carry out the calculations in a periodic “supercell” geometry.

Basic issues

There is a simple set of guiding principles that must be followed to properly treat long-range Coulomb interactions in extended systems. If the calculations are carried out in a cell that represents an infinite system, i.e. the unit cell of a crystal, or a “supercell” constructed so that its limiting behavior represents a macroscopic system, then:

• The cell must be chosen to be neutral;

• The neutral cell can be used to define a proper thermodynamic “reference state” if in addition we require that there is no average (macroscopic) electric field;

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