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The question of whether the exact exchange-correlation (XC) potential in density functional theory can have discontinuities as a function of the number of electrons is discussed. For a very simple model the exact XC functional is constructed analytically using Levy's constrained-search method and shown to have a discontinuity in its derivative. Exact numerical calculations for the XC potential for a simple finite one-dimensional 'semiconductor' chain are performed. An extrapolation to infinite chain length shows the occurrence of a sizable discontinuity in certain parameter ranges, but the model has a small discontinuity for 'realistic' parameters. The question of how to calculate the value of the discontinuity for a general system is addressed using many-body theory. A formula proposed recently by other researchers is shown to be incorrect.
Schönhammer et al. (Sun,) studied this question.