Properties of the exchange hole under an appropriate coordinate transformation

Under a general coordinate transformation, the exchange hole can be made localized, as shown by Springborg [Chem. Phys. Lett. 308, 83 (1999)]. While the conventional or untransformed hole is referenced to the position of an electron, the maximally localized one is referenced to the center of mass of an electron pair. The benefit of a more localized hole is that semilocal density functionals model it and the associated energy density more easily than they model the conventional ones. We show that, out of the class of general coordinate transformations, one can identify a subset (including the maximally localized case) which we call appropriate. Under an appropriate coordinate transformation, while the exchange hole is no longer always normalized, it retains other familiar and useful features such as the conventional on-top value and uniform-density limit. In particular, its system average remains invariant, retaining the normalization sum rule and the negativity property. Therefore, unlike the exchange energy density ex(r), the real-space analysis ẽx(u) of the exchange energy [into contributions from different electron–electron separations (u)] is uniquely defined. Thus the real-space analysis provides an alternative way to make simple and fair but detailed comparisons of approximate and exact exchange. As a byproduct, we show how to improve the accuracy of the Negele–Vautherin model for the density matrix expansion of the exchange energy by imposing negativity and sum rule constraints on the system average of its maximally localized hole.