Some problems in density functional theory

Abstract

Though calculations based on density functional theory (DFT) are used remarkably widely in chemistry, physics, materials science, and biomolecular research and though the modern form of DFT has been studied for almost 60 years, some mathematical problems remain. From a physical science perspective, it is far from clear whether those problems are of major import. For context, we provide an outline of the basic structure of DFT as it is presented and used conventionally in physical sciences, note some unresolved mathematical difficulties with those conventional demonstrations, then pose several questions regarding both the time-independent and time-dependent forms of DFT that could benefit from attention in applied mathematics. Progress on any of these would aid in development of better approximate functionals and in interpretation of DFT.

Appendix 1: Generic overview of DFT

Though most of what follows is completely generic to any quantum mechanical single-species system, our focus is on many-electron systems. We note, for example, that there is active work on nuclear many-body DFT; see Ref. [77] and references therein.

In the simplest case, the Hamiltonian describes N particles interacting pairwise (e.g., Coulomb interactions) subject to an external single-particle potential that couples to each particle, $v_{\textrm{ext}}\varvec{(r)}$. (Any possible time dependence is left implicit to simplify the notation here.) The states of interest (wave function, density matrix) are “extremal” states (e.g., ground state, mechanical equilibrium) that are fixed by conditions involving the Hamiltonian. Hence, those states are functionals of $v_\mathrm{{ext}}$. Properties of interest (e.g., energy, free energy, magnetization, etc.) are expectation values of appropriate operators in these states; hence, they inherit a functional dependence on $v_\mathrm{{ext}}$. In particular, the number density $n\varvec{(r)}$ defined in this way is a functional of $v_{ext}$, denoted $n(\varvec{r}|{v}_\mathrm{{ext}})$.

For reasons of insight and computational accessibility mentioned in the main text, it typically is preferable to express properties of interest as functionals of n rather than of $v_\mathrm{{ext}}$. This change of variables can be implemented if there is a one-to-one relationship $v_\mathrm{{ext}}\varvec{(r)}\leftrightarrow n(\varvec{r}|v_\mathrm{{ext}})$, e.g., via a Legendre transformation (subject to certain conditions on the functional representing the property considered). The first task of DFT thus is to establish this bijective relationship of the density and external potential. Two complementary approaches have been used.

The first (historically) [9] is based on variational principles showing that the density associated with a given potential provides the extremum of a certain functional (or action in the time-dependent case). The convexity of the functional assures the uniqueness required. A second approach is based on the force balance for these states, or specifically, the conservation law for the local momentum density. Both approaches accomplish the goal formally, but without complete mathematical rigor. Specifically, the function space for $v_\mathrm{{ext}}\varvec{(r)}$ and that for $n\varvec{(r)}$ have not been fully characterized within the proof. As detailed in the main text, this deficiency remains an open problem for all states considered: time-independent, time-dependent, ground state, and finite temperature.

The variational approach also requires conditions on the associated functional to allow functional differentiation. Existence of such functionals in general remains an open problem, related to the above-mentioned characterization of function spaces. The force balance approach does not require functional differentiation and thus avoids this difficulty.

Important practical problems remain after the bijectivity is proved. The first is to know the functional $n(\varvec{r}|v_\mathrm{{ext}})$, and the second is to know the corresponding functional for the desired property (e.g., energy, free energy, etc.). These two tasks do not arise as separate problems in the variational approach because the extremum of the functional is identified as the primary property of interest (e.g., ground state energy or free energy). In the force balance approach, the density can be calculated, but the dependence of a property of interest upon that density remains to be fixed.

In practice, the calculation of the density for a given potential is accomplished by a mapping of the DFT for the interacting particle system to that for a non-interacting system, the Kohn–Sham representation. In the variational formulation, this is straightforward once the existence of the functional derivative (or its equivalent) can be established. Once again, this can be done in the force balance approach without need for the functional derivative.