Volume 13, Issue 2 e1634
Focus Article
Open Access

Density functionals based on the mathematical structure of the strong-interaction limit of DFT

Stefan Vuckovic

Corresponding Author

Stefan Vuckovic

Institute for Microelectronics and Microsystems (CNR-IMM), Lecce, Italy

Department of Chemistry & Pharmaceutical Sciences and Amsterdam Institute of Molecular and Life Sciences (AIMMS), Faculty of Science, Vrije Universiteit, Amsterdam, The Netherlands

Correspondence

Paola Gori-Giorgi, Department of Chemistry & Pharmaceutical Sciences and Amsterdam Institute of Molecular and Life Sciences (AIMMS), Faculty of Science, Vrije Universiteit, De Boelelaan 1083, 1081HV Amsterdam, The Netherlands.

Email: [email protected]

Stefan Vuckovic Institute for Microelectronics and Microsystems (CNR-IMM), 73100 Lecce Italy; Department of Chemistry & Pharmaceutical Sciences and Amsterdam Institute of Molecular and Life Sciences (AIMMS), Faculty of Science, Vrije Universiteit, De Boelelaan 1083, 1081HV Amsterdam, The Netherlands.

Email: [email protected]

Contribution: Conceptualization (lead), Formal analysis (equal), ​Investigation (equal), Methodology (equal), Supervision (equal), Validation (equal), Visualization (equal), Writing - original draft (lead), Writing - review & editing (lead)

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Augusto Gerolin

Augusto Gerolin

Department of Chemistry and Biomolecular Sciences, University of Ottawa, Ottawa, Ontario, Canada

Department of Mathematics and Statistics, University of Ottawa, Ottawa, Ontario, Canada

Contribution: Conceptualization (equal), Formal analysis (equal), Methodology (equal), Project administration (equal), Supervision (equal), Validation (equal), Visualization (equal), Writing - original draft (equal), Writing - review & editing (equal)

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Kimberly J. Daas

Kimberly J. Daas

Department of Chemistry & Pharmaceutical Sciences and Amsterdam Institute of Molecular and Life Sciences (AIMMS), Faculty of Science, Vrije Universiteit, Amsterdam, The Netherlands

Contribution: Data curation (equal), Validation (equal), Visualization (equal), Writing - original draft (equal), Writing - review & editing (equal)

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Hilke Bahmann

Hilke Bahmann

Physical and Theoretical Chemistry, University of Wuppertal, Wuppertal, Germany

Contribution: Data curation (equal), Methodology (equal), Validation (equal), Visualization (equal), Writing - review & editing (equal)

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Gero Friesecke

Gero Friesecke

Department of Mathematics, Technische Universität München, Munich, Germany

Contribution: Conceptualization (equal), Formal analysis (equal), Validation (equal), Writing - original draft (equal), Writing - review & editing (equal)

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Paola Gori-Giorgi

Corresponding Author

Paola Gori-Giorgi

Department of Chemistry & Pharmaceutical Sciences and Amsterdam Institute of Molecular and Life Sciences (AIMMS), Faculty of Science, Vrije Universiteit, Amsterdam, The Netherlands

Correspondence

Paola Gori-Giorgi, Department of Chemistry & Pharmaceutical Sciences and Amsterdam Institute of Molecular and Life Sciences (AIMMS), Faculty of Science, Vrije Universiteit, De Boelelaan 1083, 1081HV Amsterdam, The Netherlands.

Email: [email protected]

Stefan Vuckovic Institute for Microelectronics and Microsystems (CNR-IMM), 73100 Lecce Italy; Department of Chemistry & Pharmaceutical Sciences and Amsterdam Institute of Molecular and Life Sciences (AIMMS), Faculty of Science, Vrije Universiteit, De Boelelaan 1083, 1081HV Amsterdam, The Netherlands.

Email: [email protected]

Contribution: Conceptualization (equal), Supervision (lead), Writing - original draft (equal), Writing - review & editing (equal)

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First published: 29 August 2022
Citations: 11
Edited by: Peter R. Schreiner, Editor-in-Chief

Funding information: Canada Research Chairs, Grant/Award Number: RGPIN-2022-05207; Deutsche Forschungsgemeinschaft, Grant/Award Numbers: 418140043, CRC 109; Horizon Europe Marie Sklodowska-Curie Actions, Grant/Award Number: 101033630; Nederlandse Organisatie voor Wetenschappelijk Onderzoek, Grant/Award Number: 724.017.001

Abstract

While in principle exact, Kohn–Sham density functional theory—the workhorse of computational chemistry—must rely on approximations for the exchange–correlation functional. Despite staggering successes, present-day approximations still struggle when the effects of electron–electron correlation play a prominent role. The limit in which the electronic Coulomb repulsion completely dominates the exchange–correlation functional offers a well-defined mathematical framework that provides insight for new approximations able to deal with strong correlation. In particular, the mathematical structure of this limit, which is now well-established thanks to its reformulation as an optimal transport problem, points to the use of very different ingredients (or features) with respect to the traditional ones used in present approximations. We focus on strategies to use these new ingredients to build approximations for computational chemistry and highlight future promising directions.

This article is categorized under:

  • Electronic Structure Theory > Density Functional Theory

Graphical Abstract

Use of strongly-interacting limit (SIL) of DFT in different areas.

1 INTRODUCTION

Owing to its high accuracy-to-cost ratio, Kohn–Sham density functional theory (KS DFT) is presently the primary building block of the successes of quantum chemistry in disciplines that stretch from biochemistry to materials science.1-6 DFT calculations consume a significant fraction of the world's supercomputing power7 and tens of thousands of scientific papers report DFT calculations with the number ever growing.2 KS DFT is in principle exact, but in practice, it requires approximations to one piece of the total energy, the so-called exchange–correlation (XC) functional, which encodes the quantum, fermionic, and Coulombic nature of electrons.

The construction of modern XC approximations draws from different approaches. Some of them are based on forms fulfilling some known exact constraints,1, 8 some have been fitted to large databases,3, 5 and the most recent XC approximations are machine learned.9-11 Regardless of these differences in their design, nearly all current DFT approximations are constructed from the same ingredients (or features) that form the “Jacob's ladder.”12, 13

Despite the progress,10 state-of-the-art XC approximations have been greatly successful mainly in describing only weak and moderate electronic correlations.3, 4 The inability of state-of-the-art DFT to capture strong correlations hampers its reliability and predictive power.1, 2, 6, 14 Over the last two decades, the strongly interacting limit of DFT (SIL)15-19 has been explored and a rigorous theory has been established. This theory reveals mathematical objects that are very different from the ingredients that are used for building standard XC approximations (semilocal quantities and KS orbitals forming the Jacob's ladder). By offering building blocks for XC functionals tailored to describe strong correlations, the SIL has a potential to solve the long-standing problem of DFT simulations of strong electronic correlations.

Here we give a summary of the development of the SIL in different contexts: the development of the theory itself, its practical realization, and the development of approximations drawing from it. We discuss paths for using this limit in different ways to solve the problem of strong correlations within DFT and discuss how it has enabled the construction of a range of quantities that can guide the further development of DFT. We also give an overview of how the SIL has motivated the development of methods that go beyond DFT, such as wavefunction methods delivering highly accurate noncovalent interactions.20

2 EXCHANGE–CORRELATION FUNCTIONAL IN DFT

Using the Levy–Lieb (LL) constrained-search formalism21, 22 the ground state energy and density of a many-electron system in an external potential v : d can be obtained as
E GS v = min ρ F ρ + v r ρ r d r , (1)
where ρ r is the one-electron density, and where F[ρ] is the λ = 1 (physical) value of the generalized universal LL functional for arbitrary coupling constant λ,
F λ ρ = min Ψ ρ Ψ T ̂ + λ V ̂ ee Ψ , (2)
with T ̂ the kinetic energy operator and V ̂ ee the electron–electron (Coulomb) repulsion operator. The physical dimension d is 3 but, as we will see in the following, it is also interesting to consider models when d = 1 or d = 2.
In Equation (2), the minimization is performed over all antisymmetric wavefunctions that integrate to ρ r , whereas the minimization in Equation (1) is performed over all N-representable densities.21 In the Kohn–Sham formalism, the functional F[ρ] = Fλ=1[ρ] is partitioned as
F ρ = T s ρ + U ρ + E xc ρ , (3)
where T s ρ = F λ = 0 ρ is the KS non-interacting kinetic energy, U[ρ] is the Hartree (mean-field) energy, and Exc[ρ] is the exchange–correlation energy. The adiabatic connection (AC) formula for the XC functional reads23, 24
E xc ρ = 0 1 W λ ρ d λ , (4)
where Wλ[ρ] is the global AC integrand:
W λ ρ = Ψ λ ρ V ̂ ee Ψ λ ρ U ρ , (5)
and Ψλ[ρ] is the minimizing wavefunction in Equation (2). We can also write Wλ[ρ] as
W λ ρ = w λ r ρ r d r , (6)
where w λ r is a λ-dependent XC energy density (per particle) that is not uniquely defined. In the present work, we adhere to the definition in terms of the electrostatic potential of the XC hole (conventional DFT gauge),25-28
w λ r = 1 2 0 h xc λ r , u u 4 π u 2 d u , (7)
where h xc λ r , u is the spherical average (over directions of u = r r ) of the XC hole around a given position r. The XC hole, in turn, is determined by the pair density associated with the wavefunction Ψλ[ρ].

3 STRONGLY INTERACTING LIMIT OF DFT

In this section, we briefly review the physical ideas behind the strongly interacting limit of DFT. For a mathematically more rigorous and comprehensive overview, we recommend referring to Friesecke et al.19

The strongly interacting limit of DFT corresponds to the situation in which the electron–electron repulsion dominates in Fλ[ρ] of Equation (2), namely15, 16
lim λ 1 λ F λ ρ = V ee SCE ρ , (8)
where V ee SCE ρ is the strictly correlated electrons (SCE) functional defined by the minimization of the electronic repulsion over wavefunctions Ψ with density ρ r :
V ee SCE ρ = min Ψ ρ Ψ V ̂ ee Ψ . (9)
The limit in Equation (8) has been established rigorously,29-31 with the convergence of the “energies,” that is, the value of the functional Fλ[ρ] divided by λ tends to V ee SCE ρ , and qualitative convergence of the wave-functions squared. More precisely, for any Ψλ minimizing (2), the spatial part of the N-body density obtained by summing over the spin degrees of freedom, P λ N = spins Ψ λ 2 , in the λ →  limit, converges (after extraction of a subsequence) to a limiting N-body probability distribution P N ρ and convergence occurs in the sense that
lim λ + dN g s 1 , , s N Ψ λ 2 = dN g dP N , (10)
for all bounded continuous observables g : dN (mathematically: weak convergence of probability distributions). The limit state is concentrated on lower-dimensional sets, as discussed later.
The probability distribution P N ρ minimizes the following alternative definition of the SCE functional
V ee SCE ρ = min P N ρ i < j 1 r ij P N r 1 r N d r 1 d r N , (11)
where r ij = r i r j . Here the minimum is over all symmetric N-body probability distributions with one-body density ρ. Interestingly, unlike its absolute value squared, the wavefunction Ψλ itself does not converge to any meaningful limit. Since P N ρ minimizes only the electronic repulsion, one can think of it as a natural analog of the Kohn–Sham noninteracting state Ψ0[ρ], which minimizes the kinetic energy functional only. However, while with Ψ0[ρ] we can evaluate the expectation value of V ̂ ee , we cannot evaluate the kinetic energy expectation value with P N ρ . This can be done only through the next leading term of Fλ[ρ], which we discuss later.

3.1 Links to the XC functional

The functional V ee SCE ρ also corresponds to a well-defined limit of the XC functional. In fact, the λ →  limit of the AC integrand of Equation (5) is equal to
W ρ = V ee SCE ρ U ρ . (12)
Moreover, there is a well-known relationship32, 33 between scaling the coupling strength λ and performing uniform coordinate scaling on the density, ρ γ r = γ 3 ρ γ r (with γ > 0), which implies that the exact XC functional tends to W[ρ] in the low-density (γ → 0) limit. The SCE limit is thus complementary to exchange, which yields the high-density limit (γ → ) of Exc[ρ],
lim γ E xc ρ γ γ = E x ρ , lim γ 0 E xc ρ γ γ = W ρ . (13)

3.2 The SCE state

As a candidate for the wave-function squared P N ρ , Seidl and co-workers15, 16 proposed to restrict the minimization in (11) over singular distributions having the form:
P SCE N = 1 N ! P d s ρ s N δ r 1 f P 1 s × × δ r 2 f P 2 s × × δ r N f P N s , (14)
where f 1 , , f N are the so-called co-motion functions, with f 1 r = r , P is a permutation of {1,…N}, and δ r f i r denotes the delta function of r (alias Dirac measure) centered at f i r = f i ρ r . The singular distributions (14) are concentrated on the d-dimensional set Ω 0 dN ,
Ω 0 = r 1 = r r 2 = f 2 r r N = f N r , (15)
and its permutations. Intuitively speaking, such an N-body density describes a state in which the position of one of the electrons, say r d , can be freely chosen according to the density ρ, but this then uniquely fixes the position of all the other electrons through the co-motion maps f i r , that is, r 2 = f 2 r , , r N = f N r . Thus states of form (14) are called strictly correlated states, or SCE states for short. In other words, if a reference electron is at r, the other electrons in the SCE state can be found nowhere else, but at the f i r positions. Besides yielding minimal electronic repulsion, the co-motion functions need to satisfy group properties,15, 16, 34 accounting for the indistinguishability of electrons, and the push forward condition, ρ f i r d f i r = ρ r d r , which ensures that the density constraint is met.16, 34

Constructing the co-motion functions is not simple, except in some special cases such as one-dimensional and spherically symmetric systems.15, 16 In those cases, the co-motion maps are obtained from constrained integrals of the density. This is illustrated in Figure 1, which shows a simple one-dimensional example of the optimal solution for (9), which has the form (14), with strictly correlated positions separated by “chunks” of density that integrate into integers. We should stress that this solution has been rigorously proven to be exact for one-dimensional systems,34 which means that in this case the exact XC functional in the low-density limit is entirely determined by these constrained integrals rather than by any of the traditional Jacob's ladder ingredients.

Details are in the caption following the image
A pedagogical example: A sample of strictly correlated positions for a 1D density integrating to six electrons. A reference electron is placed at x = −2 (black point) and the SCE position of the other electron is determined by fi(x) and are represented by other colors. Notice that the shaded area between two adjacent SCE positions integrates to 1 and this is what defines fi(x) for 1D densities. Inset is showing the co-motion functions for the given density.
The SCE potential is defined as the functional derivative of the SCE functional V ee SCE ρ with respect to the density, v SCE r = δ V ee SCE ρ / δρ r , with the convention that v SCE r tends to zero as r for finite systems. Given an SCE state of Equation (14), the SCE functional and potential can be simply written in terms of the co-motion functions35-37:
V ee SCE ρ = ρ r 2 i = 2 N 1 r f i r d r (16)
v SCE r = i = 2 N r f i r r f i r 3 . (17)
Equation (17) has a simple physical interpretation: v SCE r is the one-body potential that corresponds to the net force exerted on an electron at position r by the other N − 1 electrons.

Is the SCE state of Equation (14) actually always the true minimizer of Equation (11)? It has been proven that this is true when N = 2 in any dimension d ≥ 118, 29 and when d = 1 for any number of electrons.34 In general, the SCE state of Equation (14) is not guaranteed to yield the absolute minimum for the electronic repulsion for a given arbitrary density ρ r .38, 39 This has been in-depth analyzed for spherically symmetric densities and it has been found that in the cases where the SCE solution is not optimal, V ee SCE ρ of Equation (16) is still very close to the true minimum of Equation (11).39

3.3 Other V ee SCE ρ formulations

In addition to the co-motion functions formulation (Equation (16)), there are other equivalent formulations for V ee SCE ρ arising from mass transportation theory. The link between the SCE functional and mass transportation (or optimal transport) theory was found, independently, by Buttazzo et al.18 and by Cotar et al.29 From the optimal transport viewpoint, the SCE functional defines a multimarginal problem, in which all the marginals are the same, so that the SCE mass-transportation problem corresponds to a reorganization of the “mass pieces” within the same density. From optimal transport theory, the dual Kantorovich formulation for V ee SCE ρ can be also deduced,18
V ee SCE ρ = max u u r ρ r d r : i = 1 N u r i i < j 1 r i r j , (18)
defining the Kantorovich potential u r as the maximizer of Equation (18). The same Equation (18) can be also reformulated40 as a nested optimization:
V ee SCE ρ = max v SCE v SCE r ρ r d r + g v SCE , (19)
where g v SCE is the minimum of the classical potential energy,
E SCE pot r 1 r N = i < j 1 r i r j i = 1 N v SCE r i , (20)
over r 1 , , r N . In Equations (17) and (20), v SCE r is defined up to a constant, which by convention is set such that v SCE r tends to 0 as r for finite systems. On the other hand, the constant in u r is fixed by the linear constraints of Equation (18). For finite systems, this constant (equal to u r v SCE r ) is exactly the strong-coupling limit of the Levy–Zahariev shift.41, 42

3.4 Next leading term

More information about the exact LL functional at low density can be gained by studying the next leading term in Equation (8). Under the assumption that the minimizer in (9) is of the SCE type (14), the classical potential energy (20) is minimum on the manifold Ω0 parametrized by the co-motion functions. The conjecture is then that the next leading term is given by zero-point oscillations in the directions perpendicular to the SCE manifold17
F λ ρ λ λ V ee SCE ρ + λ F ZPE ρ , (21)
where ZPE stands for zero-point electronic energies, and where,
F ZPE ρ = 1 2 ρ r N Tr r d r , (22)
and r is the hessian matrix composed of the second order derivatives of the SCE potential energy of Equation (20) evaluated at r 1 = r , r 2 = f 2 r , (i.e., on the manifold Ω0 parametrized by the co-motion functions). The intuition that this next term should be given by zero-point oscillations around the manifold parametrized by the co-motion functions appeared for the first time in Seidl's seminal work,15 and was later formalized, with calculations for small atoms (He to Ne) in Gori-Giorgi et al.17 A rigorous proof in the one-dimensional case for any N has been provided recently.43

3.5 The spin state

Besides the expansion of Equation (21) in terms of powers of λ, which is semiclassical in nature, it is conjectured17, 44 that the effect of the spin state will enter at large-λ through orders e λ , which corresponds to the overlap of Gaussians centered in different co-motion functions. This conjecture has been confirmed numerically for N = 2 electrons in 1D.45

3.6 Numerical realization of the SCE functional

The SCE functional cannot at the moment be accurately and efficiently computed for general 3D densities and large N. But accurate numerical methods are available for small N or special situations, and novel methods aimed at large N are under development. In particular, the very recent genetic column generation method46 appears in test examples to scale favorably with system size.

In Table 1, we give an overview of the proposed algorithms for computing the SCE functional and potential and refer to the book chapter19 for a more detailed review. From Table 1, we can see that for general 3D densities, numerical solutions were reported only for up to 10 electrons (the first method is limited to radial densities, whereas for the last method 3D tests are not yet available). This indicates again the level of complexity and ultra nonlocality of the SCE functional.

TABLE 1. An overview of proposed SCE algorithms
Algorithm References N max
SGS approach is based on co-motion functions (radial densities only) 50, see also 16 and 51 100
Linear programming applied to the N-body formulation (11) 39 and 52 3
Multi-marginal Sinkhorn algorithm 53-57 5
Algorithms based on the Kantorovich formulation (19) 40 and 42 6
Algorithm based on representability constraints for the pair density 58 10
Langevin dynamics with moment constraints 59 and 60 (To be assessed)
Genetic column generation (3D tests not yet available) 46 30
  • Note: The third column shows N max , which indicates up to how many electrons a given algorithm was applied to. This is not meant to be a direct comparison of methods as the reported results differ in the fineness of discretization and the accuracy achieved. Also, only the second, fourth and last algorithms are free of additional approximations beyond discretization.

In fact, in the worst-case scenario, the computational complexity of simple algorithms scales exponentially with the number N of electrons47 and computing the SCE functional may be NP-hard.46, 48, 49 In the discrete setting, where the single particle density ρ is supported on points, Equation (11) is equivalent to a linear programming problem with ℓN constraints and N variables.

Despite these limitations in solving the SCE problem exactly, rather accurate approximations, retaining some of the SCE nonlocality, have been recently proposed and they will be detailed in the next section.

4 APPROXIMATIONS TO THE SCE FUNCTIONAL

Existing approximations of the SCE functional are summarized in Table 2. They include the point-charge plus continuum (PC) model of Seidl and coworkers,61 which has both a local density approximation (LDA) version and a gradient expansion version (GEA), and the recent harmonium PC (hPC) model, which is a generalized gradient approximation (GGA).62 Notice that the PC model uses as LDA the idea of a spherical neutralizing cell around each electron, and its prefactor differs slightly from the exact SCE limit for a uniform density, given by the Madelung energy of the bcc Wigner crystal.66 The nonlocal radius functional (NLR) and the shell model retain some of the SCE nonlocality64, 65 through the integrals of the spherically averaged density, which is defined as:
ρ ˜ r , u = 1 4 π ρ r + u d Ω u . (23)
NLR approximates the XC hole in the strong coupling limit whose depth (nonlocal radius), u 1 r , is implicitly defined through the following integral, inspired by the exact SCE functional for 1D systems,
4 π 0 u 1 r u 2 ρ ˜ r , u d u = 1 . (24)
Once u 1 r is computed, the energy density from the electrostatic potential of the NLR XC hole is computed, which in turn, defines W[ρ] within NLR. The shell model is built upon NLR and makes it exact for the SCE limit of the uniform electron gas.65
TABLE 2. Approximate w r energy densities yielding W[ρ] from: W ρ = ρ r w r d r
Approximation w r form References
PC-LDA 9 10 4 π 3 1 / 3 ρ r 1 / 3 61
PC-GEA w PC - LDA r + 9 2 1 / 3 π 175 ρ r 1 / 3 s r 2 61
GGA (hPC) w PC - LDA r 1 + as r 2 1 + bs r 2 62
NLR 2 π 0 u 1 r ρ ˜ r , u u d u 64
Shell model 2 π 0 u s r ρ ˜ r , u u d u + 2 π u s r u c r ρ ˜ r , u u d u 65
  • Note: PC stands for point-charge plus continuum (PC) model,61 LDA stands for the local density approximation, GEA for the gradient expansion approximation, GGA for the generalized gradient approximation, and hPC62 stands for the harmonium PC based on a GGA form,63 whose parameters a and b are trained on the SCE energetics for the harmonium atom (for their numerical values see Ref. 62). The reduced density gradient, s, is given by s r = ρ r / 2 3 π 2 1 / 3 ρ r 4 / 3 . The nonlocal radius functional (NLR)64 approximates the strong coupling limit of the XC hole, whose depth is given by Equation (24) and is calculated from the integrals over the spherically averaged density (Equation (23)). The shell model6, 5 adds a positive shell to the NLR hole, and the radii of the negative and positive shell, u s r and u c r , respectively, are obtained at each r from the uniform electron gas constraint and the normalization constraint on the underlying XC hole. The approximate w r from PC-LDA, NLR, and shell model are in the gauge of Equation (7) and thereby directly approximate w r of Equation (26).

In Figure 2, we explore the accuracy of different SCE approximations for energy densities (see Equation (26) below). From this figure, we can see that the shell model is the most accurate approximation locally. The PC-GEA model is the best performer globally here, and generally, it gives a rather accurate W[ρ]. However, the functional derivative of the PC-GEA diverges in the exponentially decaying density tails,62, 67, 68 making self-consistent KS calculations impossible. This problem is solved by turning to GGA's.62, 68 In particular, the very recently proposed hPC functional62 preserves the accuracy of W[ρ] from PC-GEA, while making self-consistent KS calculations possible.

Details are in the caption following the image
Upper panel: Difference between the exact (SCE) and approximate strong coupling limit energy density for the beryllium atom, δ w r = w r w model r , as a function of the distance from the nucleus, r/a.u. The inset in the upper panel is focusing on the error of the shell and NLR model for larger r. Lower panel: The quantity from the upper panel multiplied by the density and the spherical volume element
In addition to the approximations for W[ρ], the approximations for the next leading term, F ZPE ρ (Equations (21) and (22)) have been also proposed. The most used61, 69-71 approximation to F ZPE ρ is the one from the PC-GEA model,61 which reads as:
F ZPE ρ F PC ZPE ρ = 2 r 3 / 2 + ρ r 2 ρ r 7 / 6 (25)
with C = 1.535 derived from the PC model,61 and D typically set to 2.8957 × 10−2 by ensuring that F PC ZPE ρ gives exact F ZPE ρ for the helium atom.17 In addition to PC-GEA, the very recent hPC model also provides a GGA form approximating F ZPE ρ .62

5 FROM SCE TO PRACTICAL METHODS

The bare SCE functional is not directly applicable in chemistry as it over-correlates electrons. If we take the dissociation curve of H2 as an example,42, 52 we can see that the SCE, unlike nearly all available XC approximations, dissociates the H2 correctly without artificially breaking any symmetries, but predicts far too low energies around equilibrium and too short bond lengths. For this reason, SCE is not directly applicable in chemistry. Instead one should devise smarter strategies for incorporating the SCE in an approximate XC functional. The challenge is then to use the SCE information to equip new functionals with the ability to capture strong electronic correlations, while maintaining the accuracy of the standard DFT for weakly and moderately correlated systems. These strategies and challenges that come along the way are discussed in the following sections.

5.1 Functionals via global interpolations between weak and strong coupling limit of DFT

XC approximations of different classes have been constructed from models to the global AC integrand (Equation (4)).72-76 A possible way to avoid bias toward the weakly correlated regime present in nearly all XC approximations is to also include the information from the strongly interacting limit of Wλ[ρ]. Such an approach, called the interaction strength interpolation (ISI), where Wλ[ρ] is obtained from an interpolation between its weakly and strongly interacting limits, has been proposed by Seidl and coworkers.77 Since the ISI approach has been proposed, different interpolation forms with different input ingredients have been tested.17, 26, 61, 65, 78-81 These approaches typically use the exact information from the weakly interacting limit [exact exchange and the correlation energy from the second-order Górling–Levy perturbation theory (GL2)82]. Except for some proof-of-principle calculations,26, 83, 84 the ISI scheme uses the approximate ingredients from the large-λ limit (W[ρ] and the next leading term described by F ZPE ρ ) and these are typically modeled at a semilocal level. In some cases,65, 80 the ISI forms have been tested in tandem with the W[ρ] approximations that retain some of the SCE nonlocality (see Section 10).

A potential problem of the ISI functionals is the lack of size consistency, which, however, can be easily corrected for interaction energies when there are no degeneracies.85 The ISI functionals have been tested on several chemical data sets and systems and they perform reasonably well for interaction energies (energy differences).69, 71, 85 When applied in the post-SCF fashion, the ISI approach seems more promising when used in tandem with Hartree–Fock (HF) than with semilocal Kohn–Sham orbitals. This finding has initiated the study of the strongly interacting limit in the Hartree–Fock theory70, 86 and the successes of approaches based on it will be briefly described in Section 15. Recently, the correlation potential from the ISI approach, which is needed for self-consistent ISI calculations to obtain the density and KS orbitals, has been computed.67 It has been shown that it is rather accurate for a set of small atoms and diatomic molecules (see Figure 3, where we show that the ISI correlation potential provides a substantial improvement over that from GL2 for the neon atom).67 The computed ISI correlation potentials have enabled fully self-consistent ISI calculations that have been recently reported in Śmiga et al.62

Details are in the caption following the image
Correlation potentials vc as a function of the distance from the nucleus (r) for the neon atom. The accurate correlation potential has been obtained from quantum Monte Carlo (QMC). All the data have been taken from Fabiano et al.67

5.2 Functionals via local interpolations between weak and strong coupling limit of DFT

In addition to building models for the XC energy via global interpolations between the strongly and weakly interacting limits of Wλ[ρ], one can also perform the interpolation locally (i.e., in each point of space).26, 81 This can be done by interpolating between the weakly and strongly interacting limits of w λ ρ r , which is the λ-dependent XC energy density of Equation (7). The main advantage of local interpolations26, 81, 84 over their global counterparts is that the former are size-consistent by construction if the interpolation ingredients are size-consistent.87, 88 Thereby, local interpolations, unlike their global counterparts, do not require size-consistency correction.28

As mentioned earlier, there is no unique definition for w λ ρ r . Vuckovic et al.28 explored the suitability of different definitions of the λ-dependent energy densities and it has been found that the energy densities definition of Equation (7) (electrostatic potential of the XC hole) is the best choice so far in this context. Within this definition, w λ ρ r reduces to the exact exchange energy density when λ = 0, whereas in the λ →  (within the SCE formulation), it is defined in terms of the co-motion functions35:
w ρ r = 1 2 i = 2 N 1 r f i r 1 2 v H r , (26)
where v H r is the Hartree potential. In addition to these two, a closed form expression for the local initial slope for w λ ρ r has been derived in Vuckovic et al.26 from second-order perturbation theory.

The accuracy of different local interpolation forms has been tested with both exact26, 84 and approximate26, 65, 80 ingredients. Relative to the global interpolations, local interpolations typically give improved results for tested small chemical systems,26 but usually do not fix the failures of global interpolations.81 Nevertheless, the accuracy of XC functionals based on the local interpolation is still underexplored. This local interpolation framework can also be used to improve the latest XC approximations, such as the deep learned local hybrids,10 especially when it comes to the treatment of strong electronic correlations.

5.3 Fully nonlocal multiple radii functional—inspired by the exact SCE form

The mathematical form of the SCE functional has inspired new fully nonlocal approximations, called the multiple radii functional (MRF).27, 89, 90 MRF approximates the XC energy densities of Equation (7) at arbitrary λ in the following way:
w λ MRF r = 1 2 i = 2 N 1 R i λ ρ r 1 2 v H r . (27)
Equation (27) can be thought of as the generalization of Equation (26), where starting from a reference electron at r, the remaining electrons are assigned effective radii or distances from r. The radii are then constructed from the integrals over the spherically averaged density and are implicitly defined by
4 π 0 R i λ r u 2 ρ ˜ r , u d u = i 1 + σ i λ r , (28)
where σ i λ r is the so-called fluctuation function. The construction of the XC functional within MRF essentially reduces to building σ i λ r . So far, the main focus has been on building approximations for the λ = 1 case, and already very simple forms for σ i λ = 1 r yield very accurate atomic w λ MRF r at the physical regime for atoms, while also accurately capturing the physics of stretched bonds. This shows that the forms inspired by the SCE can work for the physical regime if properly re-scaled. Furthermore, despite its full nonlocality, the cost of MRF is O(N3) within seminumerical schemes.91

By construction, MRF has very appealing properties: (1) it gives XC energies in the gauge of Equation (7) making it highly suitable to be used in the local interpolations described in Section 12; (2) these energy densities have the correct asymptotic behavior; (3) MRF captures the physics of bond breaking; (4) it is fully nonlocal so it can better describe the physics of strong electronic correlations that the usual semilocal DFT functionals; (5) its form is universal and does not change as dimensionality/interactions between particle changes as demonstrated in Gould and Vuckovic.90 All these features of MRF and its flexibility make it very promising for building the next-generation of DFT approximations. There are ongoing efforts to transform these appealing features into robust XC functionals by developing improved MRF forms and efficiently implementing the MRF package into standard quantum-chemical codes.

5.4 Other applications of SCE: Lower bounds to XC energies and correlation indicators

Besides being used to build XC approximations, the SCE approach has also proven very useful in understanding general features of the exact XC functional and the nature of electronic correlations. For example, the SCE limit is directly connected to the Lieb–Oxford (LO) inequality,92, 93 a key exact property used in the construction of XC approximations.8, 94 The LO inequality limits the value of the XC energy by bounding from below the AC integrand of Equation (5):
W λ ρ C LO ρ 4 / 3 r d r , (29)
where the optimal C LO is rigorously known to be between 1.4442 and 1.5765.66, 95, 96 More generally, C LO ρ 4 / 3 r d r bounds from below the indirect energy (electron–electron repulsion minus the Hartree energy) of any correctly normalized and antisymmetric Ψ[ρ]. Letting Ψ[ρ] be Ψλ[ρ], we obtain Equation (29). Since Wλ[ρ] monotonically decreases with λ, W[ρ] will be the smallest value for the l.h.s. of Equation (29). Thus, finding lower bounds for the optimal constant C LO is equivalent to searching for densities ρ that maximize the ratio between W[ρ] and ρ 4 / 3 r d r ,51, 97 a procedure that has been applied to both the optimal C LO for the general case and to the one for a specific number of electrons N.28, 51, 97, 98 An approach to tighten the lower bound to correlation energies for a given density has been also proposed by combining the adiabatic connection interpolation described in Section 11 and the SCE energies.81

In addition to provide tightened lower bounds for the XC energies, the SCE has also been used to define correlation indicators that quantify the ratio between dynamical and static correlation in a given system.81 This idea has been also generalized to local indicators, enabling to visualize the interplay of dynamical and static correlation at different points in space.81

5.5 Going beyond DFT—Large-λ limits in the Møller–Plesset adiabatic connection

The DFT AC introduced in Section 2, whose large λ limit is the focus of this article, defines the correlation energy in KS DFT. In a more traditional quantum-chemical sense, the correlation energy is defined as the difference between the true and Hartree–Fock (HF) energy. An exact expression for this correlation energy is given by the Møller–Plesset adiabatic connection (MPAC),70 which connects the HF and physical state and has the Møller–Plesset perturbation series as weak-interaction expansion. This AC is summarized and compared with the one of KS DFT used so far in this work in Figure 4. It has been recently shown that the large-λ limit of the MP AC is determined by functionals of the HF density.70, 86 Inequalities between the large-λ leading terms of the two AC's have been also established.70, 86

Details are in the caption following the image
Two different adiabatic connections, linking the physical (λ = 1) system to either the KS or the Hartree–Fock determinant (λ = 0). Left: The density-fixed adiabatic connection of KS DFT (see Section 2). The Hamiltonian H ̂ λ DFT corresponds to Equation (2), with the one-body potential v λ r enforcing the constraint Ψλ ↦ ρ, where ρ is the density of the physical system. The correlation part of the adiabatic connection integrand W c , λ DFT is equal to Wλ[ρ] of Equation (5) minus Wλ=0[ρ] = Ex[ρ]. Small and large-λ expansions for W c , λ DFT are also shown. Right: The adiabatic connection that has the Møller–Plesset (MP) series as a small perturbation expansion, considered in Section 15. The Hamiltonian H ̂ λ HF contains J ̂ ρ HF and K ̂ ϕ i HF , which are the standard Hartree and exchange HF operators, respectively. The λ-dependent Ψ minimizing H ̂ λ HF has a density that changes with λ: At λ = 0 is equal to the HF density, while at λ = 1 is equal to the physical density. The expectation W c , λ HF is the AC integrand defining the correlation energy in HF theory. Small and large-λ expansions for W c , λ HF are also shown. The operator V ̂ ext is the external (nuclear) potential

The MP AC theory has been used to construct a predictor for the accuracy of MP2 for noncovalent interactions.99 Methods that are based on the interpolation between the small and large λ limits of the MP AC have been also developed.20 They are analogous to the ISI methods outlined in Section 11, which are used in the DFT context. It has been shown that these interpolation methods for the MP AC give very accurate results for noncovalent interactions.20 We illustrate this in Figure 5, where we compare reference [CCSD(T)] to approximate dissociation curves from these interpolation approximations for the pyridine and argone dimers. The curves labeled SPL2 and MPACF-1 correspond to two new global interpolation forms20 constructed by adding more flexibility and empirical parameters to the existing interpolation forms used in DFT17, 77 to capture the known exact features of the MP AC. For both of these noncovalently bound dimers, and for many other cases,20 SPL2 and MPACF-1 show an excellent performance without using dispersion corrections. In general, they substantially improve over MP2 for noncovalent interactions, and are either on par with—or also improve—dispersion corrected (double) hybrids.20

Details are in the caption following the image
Dissociation curves of the pyridine (top) and argon dimers (bottom) obtained from MP2, SPL2, MPACF-1, B3LYP-D3, and B2PLYP with CCSD(T) (black line) as a reference

6 CONCLUSIONS AND OUTLOOK

Here we have reviewed the most important topics that the strongly interacting limit of DFT brings into focus. We have analyzed the development of different aspects of the underlying rigorous theory connecting DFT and optimal transport, and discussed how the SIL formulation influenced the development of different methods in DFT and beyond. Although this limit does not describe the physical regime, its mathematical structure contains essential elements pointing towards the real physics happening in molecular systems with strong correlations, whose description is one of the key unsolved problems in DFT. Thus, in the years and decades to come it will be very interesting to see how much the SIL ideas, formulations, and ensuing practical methods will be used to solve the strong correlation problem and to build the next generation of DFT methods. In particular, the new ingredients appearing in this limit can be used as new features to machine learn the XC functional.9-11

AUTHOR CONTRIBUTIONS

Stefan Vuckovic: Conceptualization (lead); formal analysis (equal); investigation (equal); methodology (equal); supervision (equal); validation (equal); visualization (equal); writing – original draft (lead); writing – review and editing (lead). Augusto Gerolin: Conceptualization (equal); formal analysis (equal); methodology (equal); project administration (equal); supervision (equal); validation (equal); visualization (equal); writing – original draft (equal); writing – review and editing (equal). Kimberly J. Daas: Data curation (equal); validation (equal); visualization (equal); writing – original draft (equal); writing – review and editing (equal). Hilke Bahmann: Data curation (equal); methodology (equal); validation (equal); visualization (equal); writing – review and editing (equal). Gero Friesecke: Conceptualization (equal); formal analysis (equal); validation (equal); writing – original draft (equal); writing – review and editing (equal). Paola Gori-Giorgi: Conceptualization (equal); supervision (lead); writing – original draft (equal); writing – review and editing (equal).

ACKNOWLEDGMENTS

Stefan Vuckovic acknowledges funding from the Marie Sklodowska–Curie grant 101033630 (EU's Horizon 2020 programme). Augusto Gerolin acknowledges the support of his research by the Canada Research Chairs Program and Natural Sciences and Engineering Research Council of Canada (NSERC), funding reference number RGPIN-2022-05207. Hilke Bahmann acknowledges funding by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) project no. 418140043. Kimberly J. Daas and Paola Gori-Giorgi were supported by the Netherlands Organization for Scientific Research (NWO) under Vici grant 724.017.001. Gero Friesecke was partially supported by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) through CRC 109.

    CONFLICT OF INTEREST

    All the authors declare to have no conflict of interest.

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