Degeneracy in density functional theory: topology in the v and n spaces

Stretching Bonds without Breaking Symmetries in Density Functional Theory

Kohn-Sham density functional theory (KS-DFT) stands out among electronic structure methods due to its balance of accuracy and computational efficiency. However, to achieve chemically accurate energies, standard density functional approximations in KS-DFT often need to break underlying symmetries, a long-standing "symmetry dilemma". By employing fragment spin densities as the main variables in calculations (rather than total molecular densities, as in KS-DFT), we present an embedding framework in which this symmetry dilemma is understood and partially resolved. The spatial overlap between fragment densities is used as the main ingredient to construct a simple, physically motivated approximation to a universal functional of the fragment densities. This "overlap approximation" is shown to significantly improve semilocal KS-DFT binding energies of molecules without artificially breaking either charge or spin symmetries. The approach is shown to be applicable to covalently bonded molecules and to systems of the "strongly correlated" type.

Density-functional theory on graphs

The principles of density-functional theory are studied for finite lattice systems represented by graphs. Surprisingly, the fundamental Hohenberg-Kohn theorem is found void, in general, while many insights into the topological structure of the density-potential mapping can be won. We give precise conditions for a ground state to be uniquely v-representable and are able to prove that this property holds for almost all densities. A set of examples illustrates the theory and demonstrates the non-convexity of the pure-state constrained-search functional.

Inverse Kohn-Sham Density Functional Theory: Progress and Challenges

Inverse Kohn-Sham (iKS) methods are needed to fully understand the one-to-one mapping between densities and potentials on which density functional theory is based. They can contribute to the construction of empirical exchange-correlation functionals and to the development of techniques for density-based embedding. Unlike the forward Kohn-Sham problems, numerical iKS problems are ill-posed and can be unstable. We discuss some of the fundamental and practical difficulties of iKS problems with constrained-optimization methods on finite basis sets. Various factors that affect the performance are systematically compared and discussed, both analytically and numerically, with a focus on two of the most practical methods: the Wu-Yang method (WY) and the partial differential equation constrained optimization (PDE-CO). Our analysis of the WY and PDE-CO highlights the limitation of finite basis sets. We introduce new ideas to make iKS problems more tractable, provide an overall strategy for performing numerical density-to-potential inversions, and discuss challenges and future directions.

Ground-state degeneracies leave recognizable topological scars in the electronic density

In Kohn-Sham density functional theory (KS DFT) a fictitious system of noninteracting particles is constructed having the same ground-state (GS) density as the physical system of interest. A fundamental open question in DFT concerns the ability of an exact KS calculation to spot and characterize the GS degeneracies in the physical system. In this Letter we provide theoretical evidence suggesting that the GS density, as a function of position on a 2D manifold of parameters affecting the external potential, is "topologically scarred" in a distinct way by degeneracies. These scars are sufficiently detailed to enable determination of the positions of degeneracies and even the associated Berry phases. We conclude that an exact KS calculation can spot and characterize the degeneracies of the physical system.

Orbital currents in the Colle-Salvetti correlation energy functional and the degeneracy problem

Popular density functionals for the exchange-correlation energy typically fail to reproduce the degeneracy of different ground states of open-shell atoms. As a remedy, functionals which explicitly depend on the current density have been suggested. We present an analysis of this problem by investigating functionals that explicitly depend on the Kohn-Sham orbitals. Going beyond the exact-exchange approximation by adding correlation in the form of the Colle-Salvetti functional, we show how current-dependent terms enter the Colle-Salvetti expression and their relevance is evaluated. A very good description of the degeneracy of ground states for atoms of the first and second rows of the Periodic Table is obtained.

On the degeneracy of atomic states within exact-exchange (spin-) density functional theory

The problem of degenerate ground states of open-shell atoms is investigated in spin-restricted and spin-unrestricted density functional theories using the exact-exchange energy functional. For the spin-unrestricted case, spurious energy splittings of the order of 2-3 kcal/mol are found for atoms of the second and third periods which are larger than the splittings obtained from recently proposed approximate exchange functionals depending explicitly on the current density. In remarkable contrast, for spin-restricted calculations the degeneracy of different atomic ground states is recovered to within less than 0.6 kcal/mol.

Exchange-correlation potentials for high-electron-density ions in the Be isoelectronic series

Accurate reference wave functions and their densities have been used to obtain accurate exchange-correlation potentials for the beryllium isoelectronic sequence Be, Ne(+6), Ar(+14), and Kr(+32). The exchange-correlation potentials for the four-electron cations with high Z in this sequence exhibit structure prior to the intershell peak that is not present in neutral atoms. The kinetic energy contribution to the exchange-correlation potential contributes to the early structure as well as to the intershell peak. The near degeneracy in this four-electron sequence plays a significant role in the structure prior to the intershell peak. Several of the quantities on which the Perdew-Burke-Ernzerhoff and Tao-Perdew-Staroverov-Scuseria functionals are dependent are examined. The generalized gradient approximations appear not to account for the near degeneracy in this series.

Lack of Hohenberg-Kohn theorem for excited states

For a given excited state there exist densities that arise from more than one external potential. This is due to a qualitatively different energy-density relationship from that of the ground state and is related to positive eigenvalues in the nonlocal susceptibility for excited states. Resulting problems with the generalization of the density functional methodology to excited states are discussed.

Quantal density functional theory of degenerate states

The treatment of degenerate states within Kohn-Sham density functional theory is a problem of long-standing and current interest. We propose a solution to this mapping from the interacting degenerate system to that of the noninteracting fermion model whereby the equivalent density and energy are obtained via the unifying physical framework of quantal density functional theory. We describe the quantal theory of both ground and excited degenerate states, and for the cases of both pure state and ensemble v-representable densities. The quantal description further provides a rigorous physical interpretation of the corresponding Kohn-Sham energy functionals of the density, ensemble density, bidensity and ensemble bidensity, and of their respective functional derivatives. We conclude with examples of the mappings within the quantal theory.