Density functionals based on the mathematical structure of the strong-interaction limit of DFT
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 power^{7} 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
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}
3.1 Links to the XC functional
3.2 The SCE state
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.
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 ≥ 1^{18, 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 $\rho \left(\mathbf{r}\right)$.^{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, ${V}_{\mathrm{ee}}^{\mathrm{SCE}}\left[\rho \right]$ of Equation (16) is still very close to the true minimum of Equation (11).^{39}
3.3 Other ${V}_{\mathrm{ee}}^{\mathrm{SCE}}\left[\rho \right]$ formulations
3.4 Next leading term
3.5 The spin state
Besides the expansion of Equation (21) in terms of powers of λ, which is semiclassical in nature, it is conjectured^{17, 44} that the effect of the spin state will enter at large-λ through orders ${e}^{-\sqrt{\lambda}}$, 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 method^{46} 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 chapter^{19} 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.
Algorithm | References | ${N}_{\mathrm{max}}$ |
---|---|---|
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 ${N}_{\mathrm{max}}$, 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 electrons^{47} 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 ℓ points, Equation (11) is equivalent to a linear programming problem with ℓN constraints and ℓ^{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
Approximation | ${w}_{\infty}\left(\mathbf{r}\right)$ form | References |
---|---|---|
PC-LDA | $-\frac{9}{10}{\left(\frac{4\pi}{3}\right)}^{1/3}\rho {\left(\mathbf{r}\right)}^{1/3}$ | 61 |
PC-GEA | ${w}_{\infty}^{\mathrm{PC}-\mathrm{LDA}}\left(\mathbf{r}\right)+9\frac{{2}^{1/3}\pi}{175}\rho {\left(\mathbf{r}\right)}^{1/3}s{\left(\mathbf{r}\right)}^{2}$ | 61 |
GGA (hPC) | ${w}_{\infty}^{\mathrm{PC}-\mathrm{LDA}}\left(\mathbf{r}\right)\frac{1+\mathit{as}{\left(\mathbf{r}\right)}^{2}}{1+\mathit{bs}{\left(\mathbf{r}\right)}^{2}}$ | 62 |
NLR | $-2\pi {\int}_{0}^{{u}_{1}\left(\mathbf{r}\right)}\tilde{\rho}\left(\mathbf{r},u\right)u\phantom{\rule{0.25em}{0ex}}\mathrm{d}u$ | 64 |
Shell model | $-2\pi {\int}_{0}^{{u}_{s}\left(\mathbf{r}\right)}\tilde{\rho}\left(\mathbf{r},u\right)u\phantom{\rule{0.25em}{0ex}}\mathrm{d}u+2\pi {\int}_{{u}_{s}\left(\mathbf{r}\right)}^{{u}_{c}\left(\mathbf{r}\right)}\tilde{\rho}\left(\mathbf{r},u\right)u\phantom{\rule{0.25em}{0ex}}\mathrm{d}u$ | 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 hPC^{62} 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 $s\left(\mathbf{r}\right)=\left|\nabla \rho \left(\mathbf{r}\right)\right|/\left(2{\left(3{\pi}^{2}\right)}^{1/3}\rho {\left(\mathbf{r}\right)}^{4/3}\right)$. 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 model^{6, 5} adds a positive shell to the NLR hole, and the radii of the negative and positive shell, ${u}_{s}\left(\mathbf{r}\right)$ and ${u}_{c}\left(\mathbf{r}\right)$, respectively, are obtained at each r from the uniform electron gas constraint and the normalization constraint on the underlying XC hole. The approximate ${w}_{\infty}\left(\mathbf{r}\right)$ from PC-LDA, NLR, and shell model are in the gauge of Equation (7) and thereby directly approximate ${w}_{\infty}\left(\mathbf{r}\right)$ 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 functional^{62} preserves the accuracy of W_{∞}[ρ] from PC-GEA, while making self-consistent KS calculations possible.
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 H_{2} as an example,^{42, 52} we can see that the SCE, unlike nearly all available XC approximations, dissociates the H_{2} 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 W_{λ}[ρ]. Such an approach, called the interaction strength interpolation (ISI), where W_{λ}[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 ${F}^{\mathrm{ZPE}}\left[\rho \right]$) 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 theory^{70, 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}
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 W_{λ}[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 ${w}_{\lambda}\left(\left[\rho \right];\mathbf{r}\right)$, which is the λ-dependent XC energy density of Equation (7). The main advantage of local interpolations^{26, 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}
The accuracy of different local interpolation forms has been tested with both exact^{26, 84} and approximate^{26, 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
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
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}
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 forms^{20} constructed by adding more flexibility and empirical parameters to the existing interpolation forms used in DFT^{17, 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}
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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