Ensemble Density Functional Theory of Neutral and Charged Excitations : Exact Formulations, Standard Approximations, and Open Questions

Extended N-centered ensemble density functional theory of double electronic excitations

A recent work (arXiv:2401.04685) has merged N-centered ensembles of neutral and charged electronic ground states with ensembles of neutral ground and excited states, thus providing a general and in-principle exact (so-called extended N-centered) ensemble density functional theory of neutral and charged electronic excitations. This formalism made it possible to revisit the concept of density-functional derivative discontinuity, in the particular case of single excitations from the highest occupied Kohn-Sham (KS) molecular orbital, without invoking the usual "asymptotic behavior of the density" argument. In this work, we address a broader class of excitations, with a particular focus on double excitations. An exact implementation of the theory is presented for the two-electron Hubbard dimer model. A thorough comparison of the true physical ground- and excited-state electronic structures with that of the fictitious ensemble density-functional KS system is also presented. Depending on the choice of the density-functional ensemble as well as the asymmetry of the dimer and the correlation strength, an inversion of states can be observed. In some other cases, the strong mixture of KS states within the true physical system makes the assignment "single excitation" or "double excitation" irrelevant.

All-Purpose Measure of Electron Correlation for Multireference Diagnostics

We present an analytical relationship between two natural orbital occupancy-based indices, I N D ¯ and INDmax, and two established electron correlation metrics: the leading term of a configuration interaction expansion, c0, and the D2 diagnostic. Numerical validation revealed that I N D ¯ and INDmax can effectively substitute for c0 and D2, respectively. These indices offer three distinct advantages: (i) they are universally applicable across all electronic structure methods, (ii) their interpretation is more intuitive, and (iii) they can be readily incorporated into the development of hybrid electronic structure methods. Additionally, we draw a distinction between correlation measures and correlation diagnostics, establishing MP2 and CCSD numerical thresholds for INDmax, which are to be used as a multireference diagnostic. Our findings further demonstrate that establishing thresholds for other electronic structure methods can be easily accomplished using small data sets.

Exact Excited-State Functionals of the Asymmetric Hubbard Dimer

The exact functionals associated with the (singlet) ground state and the two singlet excited states of the asymmetric Hubbard dimer at half-filling are calculated using both Levy's constrained search and Lieb's convex formulation. While the ground-state functional is, as is commonly known, a convex function with respect to the density, the functional associated with the doubly excited state is found to be concave. Also, because the density-potential mapping associated with the first excited state is noninvertible, its "functional" is a partial, multivalued function composed of one concave and one convex branch that correspond to two separate domains of the external potential. Remarkably, it is found that, although the one-to-one mapping between density and external potential may not apply (as in the case of the first excited state), each state-specific energy and corresponding universal functional are "functions" whose derivatives are each other's inverse, just as in the ground state formalism.

State-Specific Configuration Interaction for Excited States

We introduce and benchmark a systematically improvable route for excited-state calculations, labeled state-specific configuration interaction (ΔCI), which is a particular realization of multiconfigurational self-consistent field and multireference configuration interaction. Starting with a reference built from optimized configuration state functions, separate CI calculations are performed for each targeted state (hence, state-specific orbitals and determinants). Accounting for single and double excitations produces the ΔCISD model, which can be improved with second-order Epstein-Nesbet perturbation theory (ΔCISD+EN2) or a posteriori Davidson corrections (ΔCISD+Q). These models were gauged against a vast and diverse set of 294 reference excitation energies. We have found that ΔCI is significantly more accurate than standard ground-state-based CI, whereas close performances were found between ΔCISD and EOM-CC2 and between ΔCISD+EN2 and EOM-CCSD. For larger systems, ΔCISD+Q delivers more accurate results than EOM-CC2 and EOM-CCSD. The ΔCI route can handle challenging multireference problems, singly and doubly excited states, from closed- and open-shell species, with overall comparable accuracy and thus represents a promising alternative to more established methodologies. In its current form, however, it is reliable only for relatively low-lying excited states.

Fundamental Variable and Density Representation in Multistate DFT for Excited States

Complementary to the theorems of Hohenberg and Kohn for the ground state, Theophilou's subspace theory establishes a one-to-one relationship between the total eigenstate energy and density ρ_V(r) of the subspace spanned by the lowest N eigenstates. However, the individual eigenstate energies are not directly available from such a subspace density functional theory. Lu and Gao (J. Phys. Chem. Lett. 2022, 13, 7762) recently proved that the Hamiltonian projected on to this subspace is a matrix functional H[D] of the multistate matrix density D(r) and that variational optimization of the trace of the Hamiltonian matrix functional yields exactly the individual eigenstates and densities. This study shows that the matrix density D(r) is the necessary fundamental variable in order to determine the exact energies and densities of the individual eigenstates. Furthermore, two ways of representing the matrix density are introduced, making use of nonorthogonal and orthogonal orbitals. In both representations, a multistate active space of auxiliary states can be constructed to exactly represent D(r) with which an explicit formulation of the Hamiltonian matrix functional H[D] is presented. Importantly, the use of a common set of orthonormal orbitals makes it possible to carry out multistate self-consistent-field optimization of the auxiliary states with singly and doubly excited configurations (MS-SDSCF).

DFT exchange: sharing perspectives on the workhorse of quantum chemistry and materials science

In this paper, the history, present status, and future of density-functional theory (DFT) is informally reviewed and discussed by 70 workers in the field, including molecular scientists, materials scientists, method developers and practitioners. The format of the paper is that of a roundtable discussion, in which the participants express and exchange views on DFT in the form of 302 individual contributions, formulated as responses to a preset list of 26 questions. Supported by a bibliography of 777 entries, the paper represents a broad snapshot of DFT, anno 2022.

Excitations of Quantum Many-Body Systems via Purified Ensembles: A Unitary-Coupled-Cluster-Based Approach

State-average calculations based on a mixture of states are increasingly being exploited across chemistry and physics as versatile procedures for addressing excitations of quantum many-body systems. If not too many states should need to be addressed, calculations performed on individual states are also a common option. Here we show how the two approaches can be merged into one method, dealing with a generalized yet single pure state. Implications in electronic structure calculations are discussed and for quantum computations are pointed out.

Single Excitation Energies Obtained from the Ensemble "HOMO-LUMO Gap": Exact Results and Approximations

In calculations based on density functional theory, the "HOMO-LUMO gap" (difference between the highest occupied and lowest unoccupied molecular orbital energies) is often used as a low-cost, ad hoc approximation for the lowest excitation energy. Here we show that a simple correction based on rigorous ensemble density functional theory makes the HOMO-LUMO gap exact in principle and significantly more accurate in practice. The introduced perturbative ensemble density functional theory approach predicts different and useful values for singlet-singlet and singlet-triplet excitations, using semilocal and hybrid approximations. Excitation energies are similar in quality to time-dependent density functional theory, especially at high fractions of exact exchange. The approach therefore offers an easy-to-implement and low-cost route to robust prediction of molecular excitation energies.