Exchange-only virial relation from the adiabatic connection

Despite the long history of density functional theory (DFT) and research concerning its foundation, DFT still has many mathematical challenges remaining.^1–4 Addressing these contributes to the development of improved approximate functionals and enhances the overall understanding of DFT. The aim of this work is to investigate the exchange-only virial relation of Perdew and Levy from a mathematical standpoint, motivated by some recent developments in mathematical DFT.^5–9 It is with great enthusiasm we submit this contribution to the special issue honoring the great achievements of John Perdew in the field of quantum chemistry in general and DFT in particular.

Exact constraints play an important role in the development and testing of density functional approximations,^10–14 bearing in mind that semiempirical functionals can fail outside their training set.¹⁵ These constraints are based on conditions that the exact exchange–correlation functional E_xc[ρ] or its constituent exchange E_x[ρ] and correlation E_c[ρ] parts satisfy. For example, the second-order gradient expansion, aimed at improving on the local density approximation (LDA), can perform worse than LDA for real systems because LDA satisfies certain exact constraints that finite-order gradient expansions break.^16–18 In the generalized gradient approximation (GGA), these constraints are restored, leading to an overall better performance of the functionals.

There exists an alternative, equivalent definition for E_x[ρ] that employs the adiabatic connection in DFT. This extremely useful concept not only relates the noninteracting system (with interaction strength λ = 0) with the fully interacting one (λ = 1) for a fixed, given density ρ, as it is customary in Kohn–Sham DFT, but also reveals the properties of the functionals for all intermediate values of the coupling strength λ. In the following, we study F[ρ] as a function the interaction strength for a fixed, given density ρ, using the notation λ ↦ F(λ); see Eq. (8). The theory of convex analysis then provides multiple tools to study the adiabatic connection. Already from the definition of the universal density functional, many practical properties can be derived when considering it as a function of λ. The intuitive understanding of the exchange energy in this setting is then as the linear component of λE_xc(λ) = F(λ) − F(0) − λE_H, where E_H is the Hartree energy [see Eq. (32)].

Here, all derivations will be performed for general, mixed N-electron states, thus allowing for degeneracy at all coupling strengths, with the overall assumption of a v-representable density ρ at all coupling strengths $λ∈R$ (a property henceforth called v^λ-representability). This comes as a form of minimal assumption, since, without v^λ-representability, the selected density cannot be assigned to a valid ground state of the system under consideration. On the other hand, it cannot be assumed that the universal density functional is functionally differentiable with respect to the density, nor does this need to hold for its individual parts such as the exchange functional. Equation (1) would then be the direct consequence of a functionally differentiable E_x[ρ] (see Appendix B in the work of Tancogne-Dejean et al.⁹ in the same special issue of this journal), but, unfortunately, the universal density functional is everywhere discontinuous.⁵ 

In the pure-state formulation, the functionals are defined as expectation values with Ψ(λ) ↦ ρ as the minimizer in the constrained-search functional. It is possible that this minimizing state is then uniquely given, and so the functionals are already defined unambiguously, but this does not seem to be guaranteed. Consequently, by the same argument as above, we rely on pure-state v^λ-representability to make sure that $⟨Ψ(λ)|T̂|Ψ(λ)⟩$ and $⟨Ψ(λ)|Ŵ|Ψ(λ)⟩$ have values that only depend on ρ for λ >0. Only if Ψ(0) is determined uniquely—for example, if ρ is noninteracting pure-state v-representable by a nondegenerate ground state—can also $W(0)=⟨Ψ(0)|Ŵ|Ψ(0)⟩$ be given, as it is the case in most presentations on the subject. Here, we aim to avoid such restrictions, only demanding ensemble v-representability of ρ, defining W(0) by a limit procedure that is laid out in Sec. IV.

It is worth noting that for a ρ that is v-representable and has a certain regularity, the treatment in Sec. 4 of Lammert⁵ shows that F[ρ_γ] is differentiable with respect to γ and thus also F(λ) for that ρ is differentiable with respect to λ for λ > 0. This would be beneficial, since then the supergradient would always be unique for λ > 0, but we will continue here without any such assumption.

A second remark concerns Remark 3.1 in the work of Lewin et al.,³³ which states that not only W(λ), but also the right-derivative and the left-derivative of F(λ), and thus by convex combination all values in-between, come from $TrŴΓ(λ)$ with different minimizers Γ(λ) from Eq. (8). This is equivalent to saying that the interval in Eq. (22) agrees with the one from Eq. (30). Since for v^λ-representable densities $TrŴΓ(λ)$ has a fixed value for all minimizers, as was noted in Sec. II, this would imply that the right-derivative and left-derivative agree and thus F(λ) would be differentiable at λ. Nevertheless, we have note been able to provide a proof for this statement at this point, so we do not treat F(λ) as differentiable here.

The adiabatic connection relates the different energy contributions of a quantum many-body system in its ground state with variable coupling strength λ and fixed one-particle density ρ. Central to our work is the assumption of v^λ-representability of ρ, which means that ρ is v-representable at each coupling strength λ ≥ 0. This allowed us to assign unique values to the kinetic-energy function T(λ) and the interaction-energy function W(λ), see Sec. II. Various convex-analytic properties of the adiabatic connection were studied in Secs. III and IV. We found that W(λ) is exactly a supergradient of F(λ) at λ ≠ 0 and if defined as W(0) = lim_λ↘0 W(λ) at λ = 0, then W(0) is given by the right-derivative ∂⁺F(0) of F(λ) at λ = 0. We then confirmed the usual relation E_x = W(0) − E_H, which now alternatively allows a definition of the exchange energy through the adiabatic connection as ∂⁺F(0) − E_H = E_x. The last equation connects the zero-coupling limit to the high-density limit from coordinate scaling, since E_x is here defined through the latter.

In this context, it is natural to also ask for the virial relation (1) relating the exchange energy with an exchange potential and we examined this possibility in Sec. VII. While the zero-coupling limits in Eq. (43) can indeed be guaranteed to exist, this is unlikely to be the case for the associated exchange–correlation potential $vxcλ$ without further assumptions. Although one would usually like to set $vx=limλ↘0vxcλ$⁠, only the limit of the corresponding virial integral in Eq. (45) is proven to exist to the exchange energy. This means that while we have a well-defined exchange energy E_x[ρ] for a v^λ-representable density ρ from Eq. (36), there might not be a general well-defined local-exchange potential from a similar limit or a functional derivative. This realization is in line with the results from the work of Tancogne-Dejean et al.⁹ in the same special issue of this journal. In that work, following a definition of the exchange contribution in terms of force densities, one needs an additional vector potential to fulfill the exchange-only virial relation (1). That the exchange-only virial relation does not hold in general can also be seen from numerical evidence.⁹ Of course, techniques like the optimized-effective potential (OEP) method^34–36 can still be used to get an approximate local-exchange potential.

A.L., M.A.C., and M.P. have received funding from the ERC-2021-STG under Grant Agreement No. 101041487 REGAL. A.R., N.T.-D., and M.R. were supported by the European Research Council (ERC-2015-AdG694097) and by the Cluster of Excellence “CUI: Advanced Imaging of Matter” of the Deutsche Forschungsgemeinschaft (DFG)—EXC 2056—Project ID 390715994, and the Grupos Consolidados (IT1249-19). T.H. was supported by the Research Council of Norway through “Magnetic Chemistry” Grant No. 287950. A.L. and M.A.C. have received funding from the Research Council of Norway through CCerror Grant No. 287906. A.L., M.A.C., and T.H. were funded by the Research Council of Norway through CoE Hylleraas Centre for Quantum Molecular Sciences Grant No. 262695.

The authors have no conflicts to disclose.

Andre Laestadius: Conceptualization (equal); Formal analysis (equal); Funding acquisition (equal); Project administration (equal); Supervision (equal); Validation (equal); Visualization (equal); Writing – original draft (equal); Writing – review & editing (equal). Mihály A. Csirik: Conceptualization (equal); Formal analysis (equal); Validation (equal); Writing – original draft (equal); Writing – review & editing (equal). Markus Penz: Conceptualization (equal); Formal analysis (equal); Validation (equal); Visualization (equal); Writing – original draft (equal); Writing – review & editing (equal). Nicolas Tancogne-Dejean: Conceptualization (equal); Validation (equal); Writing – review & editing (equal). Michael Ruggenthaler: Conceptualization (equal); Validation (equal); Writing – review & editing (equal). Angel Rubio: Conceptualization (equal); Validation (equal); Writing – review & editing (equal). Trygve Helgaker: Conceptualization (equal); Formal analysis (equal); Funding acquisition (equal); Validation (equal); Visualization (equal); Writing – original draft (equal); Writing – review & editing (equal).

Data sharing is not applicable to this article as no new data were created or analyzed in this study.