Symmetries and independent components of Coulomb integrals
Résumé
The Coulomb integrals, i.e. the matrix elements of the Coulomb potential between one-electron orbitals, are basic objects of electronic structure calculations. In this paper Coulomb integrals are analyzed in terms of the symmetry of the crystal or molecule. We consider the case where the one-electron orbitals form the basis of a real, complex or pseudo-complex irreducible representation. We take into account both the symmetry group of the site and the permutation symmetries of Coulomb integrals. We provide explicit formulas to calculate all Coulomb integrals in terms of the minimum number of independent invariant integrals, that we enumerate. We give explicit formulas relating the Coulomb integrals of a symmetry group to the ones of a subgroup, taking the example of spherically symmetric orbitals. The group-theory techniques that we use include Clebsch-Gordan coefficients, co-representations of magnetic groups, recoupling formulas and the Racah factorization theorem.
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