Symmetry breaking with a proper approximate density functional: Exact ground-state energy as a close lower bound on the computed energy

In the widely-used Kohn-Sham density functional theory for the manyelectron ground state, only the density functional for the exchange-correlation energy is approximated. While the exact density functional minimizes at the exact ground-state energy for any system, widely used approximations can at best describe normally-correlated ground states. A “proper” approximation for a given problem is defined as one that well describes relevant normally-correlated states. Strong correlation, present in many quantum materials, can arise due to abnormally high degeneracy of the Kohn-Sham occupied and unoccupied one-electron energies near the Fermi level. In such situations, the approximate functional may lower its energy by spontaneously breaking symmetries that the ground-state density shares with the many-electron Hamiltonian, reducing the degeneracies that lead to strong correlation. Here the Levy constrained search is invoked to argue that the total energy of a “proper” functional with maximal spontaneous symmetry breaking is bounded from below by the exact total energy, and that the bound may be close to the extent that the abnormal degeneracies are removed. This is a consequence of a variational principle, and it may not hold in non-variational but more accurate approaches to normal correlation such as coupled cluster theory. Since the correlation energy of any one-electron state is zero, a widely “proper” approximate functional should be nearly or exactly self-interaction free, and progress toward a proper self-interaction correction is reviewed here. The time-centered physical interpretation of symmetry breaking is also discussed.