"Ensemblization" of density functional theory

TL;DR

The work develops ensemble density functional theory (EDFT) as a rigorous extension of standard DFT to handle degenerate ground states, fractional electron numbers, and excited states, by introducing the concept of ensemblization. It builds a coherent five-step framework that starts from explicit state averaging, moves through a unified Hartree–exchange derivation via a fluctuation-dissipation theorem, and culminates in a practical, orbital-optimized implementation using state-resolved, symmetry-adapted CSFs. A key contribution is the decomposition of Hxc into Hartree and exchange components via first-principles definitions, and the identification of two types of correlation in ensembles: state-driven (sd) and density-driven (dd), with tailored approximations for each. The approach is validated with illustrative examples (e.g., fractional electrons, excited states, and double excitations) and extended to recent advances like pEDFT and GX24, demonstrating significant improvements over traditional DFT and TDDFT for challenging cases, with practical pathways toward applications in chemistry and materials science.

Abstract

Density functional theory (DFT) has transformed our ability to investigate and understand electronic ground states. In its original formulation, however, DFT is not suited to addressing (e.g.) degenerate ground states, mixed states with different particle numbers, or excited states. All these issues can be handled, in principle exactly, via ensemble DFT (EDFT). This Perspective provides a detailed introduction to and analysis of EDFT, in an in-principle exact framework that is constructed to avoid uncontrolled errors and inconsistencies that may be associated with {\it ad hoc} extensions of conventional DFT. In particular, it focuses on the "ensemblization" of both exact and approximate density functionals, a term we coin to describe a rigorous approach that lends itself to the construction of novel approximations consistent with the general ensemble framework, yet applicable to practical problems where traditional DFT tends to fail or does not apply at all. Specifically, symmetry considerations and ensemble properties are shown to enable each other in shaping a practical DFT-based methodology that extends beyond the ground state and, in doing so, highlights the need to look outside the standard ground state Kohn-Sham treatment.

Figures (13)

• Figure 1: Real-valued ($2p_x$, $2p_y$, $2p_z$) and complex-valued ($2p_{1}$, $2p_{-1}$, $2p_{0}$) orbitals. Surface levels correspond to points in space with constant absolute values, whose color represents the phase of the orbitals. Each $p$ orbital has cylindrical symmetry, whereas combining them as denoted with equal weights, yields an ensemble that retains spherical symmetry.
• Figure 2: KS energy levels of Li in different exact KS approaches. Energy levels of spin-polarized DFT are inconsistent with the aufbau principle, whereas the ensemble is fully consistent with aufbau. IP indicates the first ionisation potential and $^3$IP indicates ionisation to a triplet state, i.e. removal of a $1s$ electron, (with experimental numbers given in the figure).
• Figure 3: Solid line: Illustration of piecewise linearity (based on atomic Cl), with energies at arbitrary $N$ shown relative to that of the neutral atom and points indicating integer particle numbers. Gradients are related to electron removal (IP) or addition (EA) energies. The dotted lines illustrate the concept of piecewise convexity, showing the increasing slope of each linear segment. The dashed curves illustrate (convex or concave) deviation from linear behaviour that is typical to naive (non-ensemble) use of approximate density functionals.
• Figure 4: Left and right Fukui functions of water (on the bonding plane) compared against HOMO and LUMO densities, respectively. The Fukui functions were computed using CCSD densities and the KS results were obtained from exact inversion of these densities.Gould2023-JCP Contour lines indicate values of zero (solid) and $\pm 10^{-3}$ Bohr$^{-3}$ (dash-dot/dashes); HOMO is exactly zero on the plane. In the left-most plots, colors range from navy (essentially no density) through blue to light green (maximal density) and are graded logarithmically.
• Figure 5: Energy derivative with respect to electron number, $M=N+\omega$, for the H$_2$ molecule, computed using the LDA as adapted via naive [blue solid line, Eq. \ref{['eqn:EDFA_trivial']}] and via weighted average [orange dashed line, Eq. \ref{['eqn:EDFA_wavg']}] approaches. The ideal piecewise behaviour is shown as a black line. Data taken from Kraisler and Kronik, Ref. Kraisler2013, used with permission.