How electrons guard the space: Shape optimization with probability distribution criteria

Efficient formulas for computing the probabilities of finding exactly v electrons in an arbitrarily chosen volume Ω ⊂ ℝ³ for Hartree-Fock wavefunctions are presented. These formulas allow the use of shape optimization techniques, such as level set methods, for optimizing with respect to Q various criteria involving such probabilities. The criterion defined as the difference between the Hartree-Fock and the independent-particle model probabilities of finding v electrons in Q stresses the quantum effects due to the Pauli principle. We have implemented a 2D level set method for optimizing this criterion in order to study spatial separation of electron pairs in linear molecules. The method is described and the illustrative example of the BH molecule is reported.