Symmetry dependence and universality of practical algebraic functionals in density-matrix-functional theory

Abstract

When dealing with a fully symmetrical ground state, the symmetry dependence of the universal Hohenberg-Kohn energy functional $F[\ensuremath{\gamma}]$ of the first-order reduced density matrix (RDM) $\ensuremath{\gamma}$ can be conveniently neglected. The situation changes drastically in the case of the dissociation of a symmetrical molecule with the state crossing, in the course of which the potential energy curve of the initial non-fully symmetrical ground state is eventually crossed with that of the fully symmetrical state. In this case, as is demonstrated in the present paper, the second-order RDM ${\mathrm{\ensuremath{\Gamma}}}_{ij,kl}$ in the representation of the natural orbitals (NOs) is symmetry dependent. Since ${\mathrm{\ensuremath{\Gamma}}}_{ij,kl}$ is the goal in the design of ${\mathrm{\ensuremath{\Gamma}}}_{ij,kl}(n)$ as a functional of NO occupations ${n}$, which is part of a practical density matrix functional $F[\ensuremath{\gamma}],\phantom{\rule{0.16em}{0ex}}{\mathrm{\ensuremath{\Gamma}}}_{ij,kl}(n)$ must also depend on the symmetry, especially the irreducible representation of the symmetry group. The result has immediate implications for study of structural (or phase) transitions based on a single symmetry-independent functional. The demonstration is given in the minimal-base model of the dissociation of the prototype ${\mathrm{H}}_{4}$ molecule in the rhombic structure.