Contemporary DFT: learning from traditional and recent trends for the development and assessment of accurate exchange–correlation functionals

Abstract

Density functional theory (DFT) is the most widely and accepted model for calculating the electronic structure of physical systems, but practical applications rely on approximations to the (still) unknown exact exchange–correlation functional. We revisit here the main advances, families, and tipping points for the development of accurate expressions for the exchange–correlation functional, focusing on the historical evolution followed by DFT but also on the underlying reasons for that, while emphasizing both the theoretical foundations and the last methodological and technical developments (exemplified by deep-learned models). The latter are built by training a neural network on a very extended collection of molecular data, with the DM21, the aPBE0-ML, and the Skala functionals now available using such a strategy. This degree of development could have not been possible without the knowledge achieved so far, after more than half a century of investigations and applications of DFT to all kind of systems, and it is rooted on the traditional local (LDA), semi-local (GGA or meta-GGA) and non-local (hybrid and double-hybrid) functionals, which are still important pillars of DFT and are expected to coexist with deep-learned models, as well as on the creation of large and diverse datasets of nearly-exact reference results, which are also needed to train any of the deep-learned models envisioned.

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1 Introduction

It is no exaggeration to say that density functional theory (DFT) forms part now of the core of quantum and computational chemistry, electronic structure theories, atomic and molecular physics, theoretical spectroscopy and light-matter interactions, finite-size or periodic materials science, and condensed matter physics, to name just a few of scientific fields more proned to use this theory widely and routinely. Evidently, given the historical and current importance of these fields for the whole scientific knowledge achieved so far in Chemistry, Physics, and Materials Science, this statement can be scaled up to imagine applications and developments comprising systems and properties from all the elements of the periodic table and from all states of matter. Since the main ingredient of any DFT calculation is the exchange–correlation functional, and actually the key for its impactful success and universal adoption by the scientific community, we will focus on its evolution in this perspective, which is summarized in Table 1: the first models were proposed in the late 50s, while the most modern machine-learned models date from the early 2020s, thus covering more than half a century (seven decades in fact) of continuous improvements, studies, and validations.

Table 1 Summary of the developments and families of functionals (with some examples) of DFT

Jacob's ladder rung	Scaling	Examples	Availability
LDA	O(N^3)	SVWN	>1950s
GGA	O(N^3)	BLYP (semiempirical), PBE (non-empirical),	>1980s
meta-GGA	O(N^3)	M06-L (semiempirical), r^2SCAN (non-empirical)	>1990s
Hybrid	O(N^4)	B3LYP (semiempirical), PBE0 (non-empirical), ωB97X (semiempirical), RSX-PBE0 (non-empirical)	>1990s
Double-hybrid	O(N^5)	B2-PLYP (semiempirical), PBE-QIDH (non-empirical), ωB97X-2 (semiempirical), RSX-PBE-QIDH (non-empirical)	>2000s
Deep-learned		DM21, CF22D, aPBE0-ML, Skala, etc.	>2020s

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Therefore, considering a context focused on the development and assessment of exchange–correlation functionals, DFT could be now experiencing a “new age”, with the developers of modern functionals based on Machine Learning (ML) advances invoking a “game change”, after a set of deep-learned exchange–correlation functionals have recently appeared with the hope of reaching a competitive accuracy, and at a computational cost similar or even lower, compared with the best among the more traditional models. To better put these new models into the due perspective, we will first summarize here the main theoretical frameworks successfully employed so far for the development of exchange–correlation functionals, with the most modern expression being sufficiently accurate and of general applicability to drive such an immense success of DFT in recent times. We will restrict ourselves to ground-state formulations, not only for simplicity but also because deep-learned expressions are not extensively applied to excited states yet. We refer the readers to literature reviews^1–7 on excited-state methods such as TD-DFT, GW, or DFT+DMFT for a deeper discussion. We will also follow a chronological order to also inform the readers about the timeline followed by DFT to reach its current state-of-the-art.

2 A route to DFT through its bibliographic and historical importance

We will first illustrate the importance of DFT along the history by resorting first to bibliographic arguments. In 2014, a literature survey was conducted to disclose the top-10 most cited scientific publications for modern Science,⁸ among a larger collection of top-100 studies which was also recently updated.⁹ Entering into that top-100 list required more than 12 000 citations in 2014. However, that threshold was increased to 30 000 in 2025, which also evidences this increase as a byproduct of the large volume of bibliographic records occurred in the last decade. The top-10 list from 2014 included two of the expressions most used for the exchange–correlation functionals: the correlation functional developed¹⁰ by Lee–Yang–Parr (LYP) in 1988 and the first hybrid functional¹¹ (named later as B3LYP) developed by Becke in 1993. It is important to remark that about half of the papers in the former (2014) top-100 list have changed due to the irruption of newer papers in the updated (2025) list. However, the update made in 2025 of the top-10 still contains two expressions for exchange–correlation functionals: the LYP correlation functional again, together with the Perdew–Burke–Erzenhof (PBE) exchange and correlation functionals developed¹² in 1996 (the previous B3LYP expression ranks now 13). Interestingly, comparing both LYP and PBE models one can already observe another difference which has always permeated the field: while both expressions respond to a solid and deep mathematical formulation, the LYP expression (or the B3LYP one) includes a very reduced set of to-be-fitted parameters, while the PBE expression does not, with the values of the constants of the latter adjusted to reproduce known conditions and numerical limits. Actually, those DFT-based publications are gaining citations at an accelerating rate, and are accompanied into that top-10 by another manuscript dealing with the extension of DFT calculations from molecular to solid-state situations.¹³ Furthermore, the foundational manuscripts for DFT are also included into both (2014 and 2025) top-100 lists: the Hohenberg–Kohn theorems¹⁴ and the Kohn–Sham (KS) orbitalic (or one-electron) model¹⁵ in positions 39 and 34 (2014) and 57 and 41 (2025), which also shows the importance of the theoretical framework general to any DFT modern calculation (and not only the importance of the particular model chosen for practical calculations). To conclude this bibliographic analysis, we also notice that the exchange functional developed¹⁶ by Becke in 1988 is also listed in positions 25 (2014) and 48 (2025) and that an extension of DFT to mimic non-covalent interactions¹⁷ is also part of the last top-100 list at position 92. Those numbers are certainly an impressive achievement for DFT from the bibliometric point of view.

Another milestone for DFT was the awarding in 1998 of the Nobel Prize in Chemistry to Walter Kohn¹⁸ for “his development of DFT”, shared with John A. Pople “for his development of computational methods in quantum chemistry”, recognizing in such a way how DFT was a perfect companion for wavefunction-based theories. To bracket the importance of DFT from a different point of view, it is pertinent to comment that the libxc library^19–21 of density functional approximations, which is also an example of the state-of-the-art of the field, includes over 600 expressions for the exchange–correlation functional and is being used or interfaced with more than 30 codes. Note that this holds independently of the type of software (proprietary or commercial) or the basis set nature (Slater-type, Gaussian-type, plane waves, or grid-free) used, to name just a pair of the existing differences between the set of massively adopted codes nowadays. Indeed, this proliferation of expressions, implementations, and codes also explains the rise in scientific publications during the last decades:²² the yearly publications citing DFT in the 90s was approximately proportional to O(10³) but increased an order of magnitude the next decade, to O(10⁴), and is now doubling its number every 5 years.²³ This slope is only comparable to the rate of growth of publications for e.g. climate change, which is somehow surprising for a theory, with the due approximations, having the goal of understanding the electronic structure of matter after calculating the electronic energy and its changes according to the microscopic environment. The growing number of DFT expressions is also accompanied by a higher (formal) computational cost, but this higher scaling with the system size has not precluded the large number of applications leading to this impressive situation. The reasons for that can be roughly traced back to better algorithms and techniques^24–27 together with the exponential improvement in scientific computing along the last decades.

3 The form of the exchange–correlation functional as a timeline of DFT advances

A general, but unfortunately unknown, expression for any exchange–correlation functional, E_xc[ρ(r)], arises from the contribution of the following terms:

E_xc[ρ(r)] = (V_ee[ρ(r)] − J[ρ(r)]) + (T[ρ(r)] − T_s[ρ(r)]), (1)

with the density functionals entering the above equation being, respectively, the exact kinetic energy, T[ρ(r)], the kinetic energy of the non-interacting particle system taken as reference, T_s[ρ(r)], the electron–electron interaction energy, V_ee[ρ(r)], and the classical charge density interaction, J[ρ(r)]. In other words, the exchange–correlation functional incorporates the non-classical effects of the type: (i) the (Fermi) correlation between same-spin electrons (or exchange energy), (ii) the (Coulomb) correlation between opposite-spin electrons (or correlation energy), and (iii) the difference between the kinetic energy of the interacting and the (fictitious or KS) non-interacting particle system. The explicit calculation of the latter term is omitted in practice and thus assumed to be part of the E_xc[ρ(r)] total values. Note that the explicit (closed-shell, for simplicity) expressions of all the terms are known, since:but not the corresponding (explicit) form of E_xc[ρ(r)] which needs to be thus approximated. Note also that a set of self-consistently obtained orbitals ({ϕ_i}) is also needed as part of the calculations, with T_s[ρ(r)] using the KS orbitals for that, and from which the corresponding density is obtained as . On the other hand, ρ_xc(r,r′) is the exchange–correlation hole of an electron at r, or the reduction in probability around each of the electrons to find another electron at r′. One can thus assume that the possible (general) ingredients of the E_xc[ρ(r)] expression can be the density at the reference (or local) point, ρ(r), the density at another (non-local) spatially separated point, ρ(r′), the mathematical form adopted or simplified for the exchange–correlation hole, ρ_xc(r, r′), and the occupied (or even virtual) orbitals, {ϕ_i}.

We should also comment at this point some of the specific errors made in practical DFT calculations, arising from approximating this formal (and exact up to now) theoretical framework. First of all, it should be noticed that for the exact E_xc[ρ(r)] expression, the V_ee[ρ(r)] term is not affected by the so-called Self-Interaction Error (SIE). However, when approximated expressions are introduced for the exchange–correlation functional, SIE is not zero anymore, thus leading to one of the intrinsic problems of DFT^28,29 and its multiple analysis and approaches to understand and then reduce the impact on practical calculations.^30–35 On the other hand, a density-driven error has been recently identified and examined, providing a simple procedure for improving the results of DFT calculations in many situations upon using the Hartree–Fock density in place of the exact density.^36,37

Of course, an outcome such as eqn (1) obeys to a solid theoretical framework, rooted first on the Hohenberg–Kohn theorems, proving the existence of an energy density functional for all the electronic systems which is also variationally well-defined, and on the Kohn–Sham approximation, simplifying the treatment after introducing a non-interacting particle systems whose associated density matches that of the exact system formed by interacting particles.¹⁵ These KS orbitals, {ϕ_i}, and their self-energies, , are self-consistently obtained as part of any DFT calculation after solving iteratively the corresponding one-particle equations:

with v(r) the external potential of the electronic system, and v_xc(r) the exchange–correlation potential or . However, the exchange–correlation functional also enters into this equation through the corresponding potential, explaining why its initial choice is so critical to the performance and accuracy of any DFT calculation, still deserving theoretical and computational efforts after more than half a century of developments. Finally, within this formalism, the exact electronic energy is therefore written as:for which any prediction and comparison can be done further.

3.1 Former but pedagogical models

Arguably, the first time (1927) in which an electronic energy term could be written as an explicit functional of the density was part of the Thomas-Fermi (TF) model,^38,39 a semiclassical theory developed for the electronic structure of many-body systems arising shortly after the Schrödinger equation, circumventing for the first time in history the use of the wavefunction as the primary variable to obtain the energy and associated properties of an N-electron system. The total kinetic energy of the electrons in this model is given by with C_TF a constant, which can be thus added to the potential energy of the electrons. However, that model neglected any difference for electron–electron interactions with respect to spin, which was later added thanks to the work⁴⁰ of Dirac (1930) and again in the simple form with C_x a constant, which also connects with the X_α method developed later (1951) by Slater. Despite the limitations of these early theories, it might bring the tipping point needed to understand the origin of all the later efforts. Simply speaking, E_xc[ρ(r)] is thought to be modelled as with e_xc the exchange–correlation energy per unit volume, with the possible separability of terms e_xc = e_x + e_c for convenience. Later on, the successful treatment of the system⁴¹ known as the “Uniform Electron Gas” corroborated this framework, leading to the same functional form for T[ρ(r)] and E_x[ρ(r)], but with and . The correlation energy of this model was also numerically modelled⁴² by Vosko–Wilk–Nusair (VWN), giving rise to a functional form for E_c[ρ(r)] too. Since both exchange and correlation functionals only depend on ρ(r), the approach was consequently called Local Density Approximation (LDA) as it is known today.

3.2 GGA and meta-GGA models

The form of the LDA exchange and kinetic energies can also be derived from the coordinate scaling of the density, ρ_λ(r) = λ³ρ(λr), which is one of the known conditions to be satisfied by any model, and where the λ³ factor ensures that the scaled density still integrates to N electrons as it should be. However, the poor predictions of LDA in molecular systems motivated to go a step further considering the inhomogeneity of matter around any spatial reference point. This was done by introducing the dimensionless variable into the integrand of the e.g. exchange functional, which also guarantees the correct coordinate scaling for this kind of expressions:giving thus rise to the generalized gradient approximation (GGA). The enhancement factor F_x (or F_xc if the total exchange–correlation expression is instead considered) can take different mathematical forms depending on the route followed by the developers.^43–46 Note that if ρ(r) tends to a constant value, the enhancement factor should vanish approaching the LDA limit, which can be thus viewed as another known condition to be imposed for developing models. If that factor is intended to bring some inhomogeneity, the simplest gradient expansion would thus be F_x(x(r)) = 1 + x² + O(x⁴) but it unfortunately diverges. Therefore, one of the most famous solutions¹⁶ was proposed in 1988 by A. Becke as which also fits to the long-range behaviour of the energy density, another known condition to be fulfilled by approximate expressions. The introduction of β also allows for some numerical flexibility of the model, and it is actually fitted to reproduce a small set of atomic exchange energies as close as possible. Overall, GGA is not only numerically more accurate than LDA but it also permits to introduce some of the mathematical conditions and exact constraints known for the exact functional. Considering higher-order derivatives of the density, as ∇²ρ(r) by itself or through the kinetic energy density has also been explored successfully^47,48 leading to the next level of functionals known as meta-GGA. Additionally, this set of elements (i.e., the density, its gradient, and higher-order derivatives) introduced step-by-step into the exchange–correlation integrand motivated their hierarchization, with LDA, GGA, and meta-GGA forming the three (lowest) ladders of a hierarchy known as Jacob's ladder⁴⁹ (not only symbolically but also considering the timeline of DFT).

On the other hand, there are also examples of GGA exchange–correlation functionals free of adjustable parameters, such as the prominent case of PBE, which introduced in 1996 an enhancement factor for the exchange functional of the form with s = x/x₀ and x₀ = 2(3π²)^1/3 able to satisfy the correct uniform gas limit, F_x(0) = 1, the spin-scaling relationship of the exact exchange energy, the expected behaviour at small density variations, F_x(x(r)) = 1 + μx² if x → 0, and the Lieb-Oxford bound, . Note that the values of κ and μ are not longer fitted to any external or reference data but obtained during the own derivation of the model, thus motivating the non-empirical adjective added to these models. The extension of this philosophy to the meta-GGA case is also possible, with the Tao–Perdew–Staroverov–Scuseria (TPSS) functional one of the step further in the hierarchy of models⁵⁰ thanks to the progressive introduction of more exact constraints without empirical parameters. The most recent example of meta-GGA models is probably the SCAN (strongly constrained and appropriately normed) family of functionals,^51,52 with the latest regularized-restored SCAN (or r²SCAN) functional⁵³ including most of the exact constraints known for (meta-)GGAs. It is important to remark that the division between semiempirical and non-empirical functionals still impregnates the field.

3.3 Hybrid and double-hybrid models

All the LDA, GGA, and meta-GGA functionals share the formal (asymptotic) scaling of O(N³) with respect to the system size N, although differing in their numerical quadrature and possible algorithmic speedups, but also share some of the limitations imposed by the functional form chosen as starting point, see eqn (7). GGA and meta-GGA are also considered as semi-local functionals, in the sense that some degree of non-locality is introduced thanks to that dependence on ∇ρ(r) and ∇²ρ(r), but the importance of non-locality has prompted other developments. For instance, it is well-known that local or semi-local functionals fail to deliver accurate band gaps of extended systems and key materials (i.e., insulators or semiconductors) due to the self-interaction error,⁵⁴ and that the incorporation of a fraction of non-local exchange reduces the systematic underestimation of band gaps.^55,56 On the other hand, the introduction of non-local exchange is also known to benefit thermochemistry and thermochemical kinetics calculations.^57,58 Finally, we also mention their importance for accurate predictions for electron transport, photocurrent generation, and photovoltaics applications.^59–61 Overall, global hybrid functionals add a relatively low-to-medium fraction of exact-exchange energy to a GGA or meta-GGA exchange functional, improving considerably their accuracy, which have therefore prompted its wide use in Chemistry, Materials Science, or Condensed Matter Physics.⁶²

Actually, the adiabatic connection method (ACM) allows to go a step further with respect to GGA or meta-GGA models, representing the majority of the calculations performed today at least at the molecular and supramolecular scales, thus briefly revisiting here the foundations and formal framework. The motivation is to find a mathematically admissible connection^63–65 between the non-interacting (fictitious) and interacting (real) particle system, thanks to a λ-dependent electron–electron interaction term written as . If λ ∈ [0,1], both systems are defined at the limits λ = 0 (fictitious) and λ = 1 (real). The exchange–correlation functional therefore represents the interpolation path between them, expressed as:

with , with Ψ_λ the wavefunction minimizing 〈T̂ + λV̂_ee〉 whilst giving the exact density. The pending task is thus to guess the form of with the simplest (linear⁶⁶ and quadratic⁶⁷) expressions being and respectively. The information needed to infer the final functional form of E_xc[ρ(r)] is again based on a set of known conditions, such as:with the EXact-Exchange (EXX) energy, E^EXX_x[{ϕ_i}], the exact exchange–correlation energy, E^exact_xc[ρ(r)], and the 2nd-order Perturbation Theory, E^PT2_c[{ϕ_i, ϕ_a}], depending now on the set of occupied ({ϕ_i}) and virtual ({ϕ_a}) KS orbitals, respectively. The implications of this dependence on wavefunction-based energy terms cannot be underemphasized: any exchange–correlation functional might include exact-exchange and/or correlation energies additionally to the exchange and correlation energies given by a density functional, thus representing the next ladders of the hierarchy of models, fourth and fifth respectively, at the price of increasing the formal computational scaling from O(N³) (LDA, GGA, and mega-GGA) to O(N⁴) (hybrid), and O(N⁵) (double-hybrid).

Interestingly, the combination of E^EXX_x[{ϕ_i}], E^PT2_c[{ϕ_i,ϕ_a}], and E_xc[ρ(r)] energy terms can be easily controlled and tuned step-by-step by introducing a linear combination of them:

E_xc[ρ(r)] = λ_xE^EXX_x[{ϕ_i}] + (1 − λ_x)E_x[ρ(r)] + E_c[ρ(r)], (12)

E_xc[ρ(r)] = λ_xE^EXX_x[{ϕ_i}] + (1 − λ_x)E_x[ρ(r)] + λ_cE^PT2_c[{ϕ_i,ϕ_a}] + (1 − λ_c)E_c[ρ(r)], (13)

leading to the hybrid and double-hybrid approximations, depending on λ_c being zero or not, respectively. Note the universality of this particular yet simple combination of energy terms, in the sense that in principle any form for the E_x[ρ(r)] and the E_c[ρ(r)] functionals can be chosen for that, and the flexibility, in the sense that the values of λ_x and λ_c are not initially restricted. We also note other non-linear forms^68,69 as alternative to eqn (13) also motivated by the ACM.

The preference for the fitting procedure of those values has led to the proliferation of expressions such as the original B3PW91¹¹ or the later popularized B3LYP,⁷⁰ for which the particular combination of energy terms is:

with a = 0.20, b = 0.72, and c = 0.81 values obtained from their fitting to a set of thermochemical data taken as reference. It is worth to mention that since the first applications of the B3LYP functional,⁷¹ the used coefficients are those proposed by Becke in its original work, obtained for the B3PW91 and thus not specifically optimized for B3LYP. Their optimization lead to a functional close to the so-called B1LYP.⁷² However, it is also known that those values are functional-dependent and/or training-set dependent; for instance, using the OPTX exchange functional⁷³ instead, the coefficients of the corresponding O3LYP model become a = 0.12, b = 0.93, and c = 0.81. Another of the most widely applied hybrid functionals is M06-2X,⁷⁴ incorporating more than 30 parameters optimized simultaneously and including λ_x in the fit. The corresponding extension to doubly-hybrid functionals was pioneered by S. Grimme in 2006, with the B2-PLYP model,⁷⁵ for which the optimal fit to thermochemical data led to λ_x = 0.53 and λ_c = 0.27, without reoptimizing the parameters entering into the E_x[ρ(r)] o E_c[ρ(r)] functionals in line with the B3LYP model. One can easily observe how BLYP, B3LYP, and B2-PLYP would form a hierarchy of models (i.e., any step up would ideally lead to more accurate values) according to the Jacob's ladder (see Fig. 1) as it has been thoroughly illustrated along the decades^76–79 with some exceptions too.^80–82

Fig. 1 Illustration of the Jacob's ladder in DFT, where any upper rung of the hierarchy adds more physical information and complexity, thus allowing to reach a more accurate representation of chemical systems.

The advantages of developing and working with non-empirical expressions (i.e., PBE) can also be extended to these (double-)hybrid expressions too,⁸³ as it is exemplified by the parameter-free PBE0-n models:

with n = 4 corresponding to the PBE0-1/4 (better known as PBE0 or PBE1PBE⁸⁴) model,⁸⁵ which has also been updated more recently to the PBE0-1/3 (n = 3) functional.⁸⁶ Furthermore, a set of non-empirical double-hybrid expressions is also available, with an additional constraint imposed^87,88 such as λ_x = λ_c² (1-DH models) or λ_x = λ_c³ (LS1-DH models) with examples of the latter such as PBE0-DH,⁸⁹ or PBE-QIDH⁶⁷ if the ACM is concomitantly employed to derive the final expression. These non-empirical hybridization schemes are shown to be very robust, as it has also been systematically demonstrated,⁹⁰ since the results are not largely modified depending on the underlying expressions used for E_xc[ρ(r)], or in other words, the linear combination of terms given by eqn (13) plays a more important role for the final accuracy of the results than the functional itself. The hierarchy constituted by e.g. PBE, PBE0, and PBE-QIDH could thus serve as an indication of the errors provided by the lower levels, and the physical reasons for that, bracketing conveniently the influence of each of the terms in the final results according again to the Jacob's ladder typical performance.

Actually, the large number of exchange–correlation functionals available, belonging to the categories briefly discussed above, motivated the compilation of the libxc library,¹⁹ encompassing all the developments along the last 50 years and rapidly being interfaced to main codes.²⁰

3.4 Other variants and corrections

Complementarily to these LDA, GGA, meta-GGA, hybrid or double-hybrid approximations, there have also been transversal developments transferable to all the models, such as the range-separation of the two-electron operator, or the use of a spatial-dependent EXX weight, in the hope of correcting some of the identified deficiencies of DFT. As a matter of illustration, hybrid and double-hybrid functionals still formally miss the correct long-range behaviour of the exchange potential, which should asymptotically go as . A possible solution to the problem, since λ_x → 1 is known to deteriorate the accuracy of the corresponding expressions for e.g. thermochemical calculations, is to force that the amount of exact exchange increases as the interaction becomes more long-ranged, by splitting the electron–electron interaction term:with erf being the error function, and where the first (second) term gives the short-range (long-range) contribution.⁹¹ Another strategy⁹² consists on a partition respecting the weight of the hybrid functional at short-range:with erf being the error function and α, β, and ω to be defined and/or fitted to reproduce the chosen reference data (highly employed range-separated hybrid functionals are LC-ωPBE,⁹³ CAM-B3LYP⁹² or the ωB97X family^94,95). These range-separated functionals are now considered a practical remedy for dealing with the (total or partial) missing of long-range interactions of (mostly) semi-local and hybrid functionals, which is known to largely impact on charge-transfer processes and excitations, among other applications.^96–100

Note that any range-separated hybrid functional can also be made non-empirical thanks to a recent development disclosing^101,102 the relationship between λ_x and ω, and that range-separated semiempirical (e.g. ωB97X-2¹⁰³) and non-empirical (e.g. RSX-PBE-QIDH¹⁰⁴) double-hybrid functionals are also available. Another degree of empiricism is introduced if the value of ω is adapted to any of the system and/or geometries considered, which defines the optimally-tuned range-separated functionals.¹⁰⁵ On the other hand, local hybrid functionals¹⁰⁶ involve position-dependent ratio of EXX and an exchange (GGA or meta-GGA) functional, with the latter models¹⁰⁷ employing τ_w(r) and the conventional KS kinetic-energy density τ(r) in the form .

Particular to the case of double-hybrid functionals, there is also the possibility of allowing the splitting of the PT2 term in Eq. (13) into the same-spin and opposite-spin correlation contributions,¹⁰⁸ scaled by the corresponding coefficients c_ss and c_os:

E^PT2_c = c_ssE^ss-PT2_c + c_osE^os-PT2_c, (18)

with again coexisting the two approaches (semiempirical and non-empirical) to find the c_ss and c_os values: by fitting the c_ss and c_os values to reference data (e.g. the DSD-based family of expressions¹⁰⁹) or by imposing some universal-like constraints (e.g. the SOS1-PBE-QIDH functional¹¹⁰) to obtain the best possible values (c_ss = 0, c_os = 4/3).

Finally, it is also known that non-covalent (long-range) electronic effects are not completely introduced by any rung of the Jacob's ladder, although this omission is attenuated by the correlation energy introduced by the PT2 term of double-hybrid functionals but it is also largely dependent on the λ_c weight. However, the flexibility (and additivity) of DFT energy terms also allows to add a posteriori this energy, once the densities ρ(r) and ρ(r′) are available, thanks to the e.g. VV10¹¹¹ correlation energy functional, taken here as example of a modern and widely used non-local correlation functional:

which correctly decays thanks to the form of the function . The parameter b can be optimized to merge this correction to any of the available GGA, meta-GGA, hybrid, and double-hybrid functionals.^112,113 The VV10 expression is part of a broader family of density functional methods for van der Waals interactions (vdW-DF), which can be found as part of the libvdwxc library¹¹⁴ (vdW-DF1, vdW-DF2, vdW-DF-C09, vdW-DF-cx, vdW-DF-optb86, rev-vdW-DF2, vdW-DF-optPBE, vdW-DF-optB88, BEEF-vdW, and mBEEF-vdW) where the non-local correlation forces are captured through a formal analysis of screened response in the electron gas.

Other cost-efficient alternatives also exist, such as the widely used Dn methods^17,115 (D2, D3, and D4) of S. Grimme et al., the eXchange-hole Dipole Moment (XDM) formalism,¹¹⁶ the density-dependent energy correction¹¹⁷ (dDsC), or the tailored DH-SVPD basis set¹¹⁸ for double-hybrid functionals such as PBE-QIDH. Due to its widespread use to efficiently (but more empirically) incorporate the non-covalent interactions at a negligible computational cost, the expression for the modern versions of those Dn methods is:

which is built on the pairwise (two-body) interactions between atom centers A and B (R_AB is the distance between atoms pairs), C^AB_n are the corresponding nth-order dispersion coefficients, and s₆ and s₈ are fitting parameters, with fⁿ(R_AB) the (Becke–Johnson) damping function:based on R^AB₀ (an atom-pair-specific cutoff radii) and introducing two new parameters a₁ and a₂. A three-body term from the geometrical orientation of all atom triples A, B, C is also added. As it happens with other variants, the set of parameters is fitted for each functional using high-level reference interaction energies of weakly bound systems.

4 Optimal fitting and datasets in DFT

All the advances in DFT over the last decades have been fostered by the extensive benchmarking done in parallel for all the models, either by the own developers or as a part of larger and comparative studies, using extended datasets for that and providing guidance to any user about what functionals are recommended in general or for some specific or tailored application. Computational protocols,¹¹⁹ tutorial reviews,¹²⁰ and best practices guidelines^121,122 for a wise use of DFT are rooted on the robust and general performance of the models on these datasets after the adequate and step-by-step calibration studies done. On the other hand, in the case of semiempirical functionals, these datasets are also very often used for the fitting of the parameters entering into the model. Therefore, the importance of designing and using datasets for past, current, and future studies and developments cannot be underrated. The design of the reference datasets (not specifically for DFT) has reached a high degree of maturity,¹²³ which also means to include chemically relevant systems of varied sizes and composition together with a wide variety of properties. We here mention some of the most general-purpose employed datasets, such as the GMTKN55,¹²⁴ the W4,¹²⁵ the MGCDB84,¹²¹ or the ACCDB,¹²⁶ together with some other more suited to a particular electronic structure problem, such as the treatment of non-covalent interactions (e.g. S66 × 8^127,128), the treatment of very large systems (e.g. CiM13¹²⁹), the extension to excited-states (e.g. QUEST¹³⁰), etc. to name just a few of the most known examples nowadays. Regarding the Materials Science field, and thus the large-scale datasets available, we can mention Materials Project,^131,132 JARVIS,^133,134 and Materials Cloud.¹³⁵

Of particular importance for DFT was also the derivation of the Minnesota datasets,¹³⁶ created along several years of efforts and refinements and used to develop some of the most used exchange–correlation functionals, and the generalized use of the GMTKN55 dataset in comparison studies, basically covering basic properties and reaction energies for small systems, reaction energies for large systems and isomerization reactions, reaction barrier heights, and intra- and intermolecular non-covalent interactions. Another effort to create meta-data is the NCIAtlas collection^137–141 of non-covalent interactions of several types. Finally, we also emphasize the large-scale dataset Open Molecules 2025 (OMol25),¹⁴² recently launched as an example of the culmination of the work around compilation of datasets. OMol25 is composed of small molecules, biomolecules, metal complexes, and electrolytes, covering 83 elements of the periodic table and systems up to a size of 350 atoms, and includes more than 100 million of DFT calculations at the ωB97M-V/def2-TZVPD level of theory. The motivation behind that huge dataset was to create a comprehensive data for training of machine learning models performing accurately for molecular chemistry.

Note also that these larger and modern datasets are rooted on historical developments mostly done for wavefunction-based theories, such as W4,¹⁴³ HEAT,¹⁴⁴ FPA,¹⁴⁵ FPD,¹⁴⁶ Gn theories,¹⁴⁷ ccCA,¹⁴⁸ to name just a few of them, also learning about the minimum level of theory needed to deliver sub-kcal mol⁻¹ or even sub-kJ mol⁻¹ accuracy for e.g. thermochemical calculations. That means that the reference (or nearly-exact) results needed for modern datasets are usually obtained at the (DLPNO-)CCSD(T)/CBS level of theory; i.e., using CCSD(T) (Coupled-Cluster Singles, Doubles, and perturbatively estimated Triples) possibly assisted by DLPNO (Domain Local Pair Natural Orbitals¹⁴⁹), a technique applied in recent years to extend the size of the system tackled,¹⁵⁰ and with basis sets large enough to reach the Complete Basis Set (CBS) limit. This model chemistry represents the best trade-off between accuracy and computational cost, although higher levels of CC theory can also be employed if those results are believed to be not sufficiently converged.¹⁵¹

Despite the emergence of lower cost methods able to deliver reference results, such as those provided by DLPNO, databases of reactions remain poorly diverse and are mostly focused on small molecular systems, not often representative of systems in use for chemical (real-world) applications. Following this line of research, some efforts were recently done to develop chemical databases gathering reactions more and more representative of current trend in chemistry, and containing new systems not yet used to parameterized semi empirical approaches, opening thus the road to unbiased benchmarking.¹⁵² As as example of this, we mention the BH9 database¹⁵³ gathering a total of 449 real-life organic chemistry reactions developed with the purpose of filling this gap. It is also worth noting that machine-learning techniques are an open road to improve the diversity of these databases,¹⁵⁴ with some new datasets (e.g. CYCLO70) already available built under this principle, even if yet marginally applied for that purpose.

Delving into the use of datasets for the optimization of the parameters entering into widely used models, also recognizing that the following division is necessarily arbitrary, users have qualitatively divided the functionals into marginally parameterized (if the exchange and correlation functionals contains very few parameters, between 5–10, and the reference data were limited but chemically meaningful, as e.g. BLYP^10,16 or B3LYP¹¹), moderately parameterized (if more parameters are introduced into the models, but not exceeding 10–20, using extended datasets for that, as e.g. B97M-V¹⁵⁵), and heavily or highly parameterized (if a relatively large number of parameters is used, in the range of 30–50, and intensively trained, as e.g. M06-L and M06-2X⁷⁴). Note that this rough classification also basically obeys to the temporal timeline followed for the development of DFT models, since the number and quality of the reference data and datasets in the 80s and 90s were necessarily more modest and smaller than in the next decades. Note also that the final number of parameters of an exchange–correlation functional, specially true for those heavily parameterized, is not critically determined by the range-separation (i.e., only ω is added), or the hybrid (i.e., only λ_x is needed) or double-hybrid (i.e., only λ_c is required, and possibly but not always c_ss and c_os) schemes, although their ultimate performance it is. On the other hand, non-empirical (or minimally empirical) functionals are obviously not parameterized, neither being proned to properties or systems covered as part of the training sets, so that a genuine use of the datasets for assessment purposes is done. A different consideration deserves the coupling of any correction for non-covalent interactions with the exchange–correlation functional, independently of semiempirical or non-empirical models, since the corresponding parameterization cannot be neglected.

Motivated by the difficulties to express the exchange–correlation functional in a compact and mathematical meaningful analytical form, some authors^156–158 searched in the late 90s and early 00s for a more systematic yet optimal functional form based on extensive parameterization of polynomial or other continuum expansions. We show just an example of this approach, which additionally avoided the splitting of the exchange and correlation expressions:

with , and where ω_abcd are parameters, with the remaining functions being A^a = ρ^a_↑ + ρ^a_↓, , , and , with in principle no limits imposed for the set of (a, b, c, d) exponents and corresponding parameters. However, this line of research was more an exploration of the limit of accuracy (and redundancy of parameters and/or training data) that is ultimately possible with GGA or meta-GGA inspired expressions, but it constitutes a first step towards more systematic and modern approaches. Overall, the intense parameterization done in recent times has also disclosed some bottlenecks of semiempirical approaches, such as the bias-variance tradeoff,¹⁵⁹ the numerical instabilities possibly caused by an overfitting,¹⁶⁰ and the importance of the quality of the data used for the fitting.¹⁶¹

Another recent approach¹⁹⁵ circumvents the need to develop new exchange–correlation functionals, and their optimization, and simply weights the total (ground-state) energy by functional-specific weights, in the form with [small omega, Greek, circumflex]_i the normalized weights and E_i the total energies calculated with the X density functionals entering the ensemble. This procedure guarantees the size-consistency of the results and the easy calculation of forces and potentials, and it could be also extended to densities with . The GMTKN55 data are used to train the ensemble with up to 77 functionals belonging to 2nd–5th rungs of Jacob's ladder included into the regression procedure, to reach an optimal functional (DENS24) achieving an error as small as 1.6 kcal mol⁻¹ for that GMTKN55 dataset. The transferability of the ensemble and the basis set dependence of the model was also tested, as well as ways to reduce the associated computational cost to get each of the E_i energies.

5 Deep-learned DFT models

We have summarized in previous sections how the conscious and meticulous work for many decades is behind the current success of DFT, regardless some generalized flaws of the theory¹⁶² such as the (many-electron) self-interaction error, or some limits imposed by the search and optimization of the parameter space. On the other hand, the recent rise of machine learning for the design of exchange–correlation functionals¹⁶³ has consequently led to a new paradigm of work, which we will also try to summarize here with three examples of functionals fully machine-learned, but rooted on the prescriptions and knowledge so far achieved and belonging to the 3rd–4th rungs of the Jacob's ladder: the DeepMind (DM21),¹⁶⁴ the “adaptative” PBE0 (aPBE0-ML)¹⁶⁵ and the recently presented Skala model.¹⁶⁶ Interestingly, two of these models (DM21 and Skala) are developed by private companies, Google Deepmind and Microsoft (Microsoft Research – AI for Science, and Microsoft Quantum), respectively. A common point of all these ML-based approaches is the interest to replace traditional, empirically or physically motivated approximations, with a neural-network mapping from electron density (and related magnitudes) to exchange–correlation energies. Note also that prior to DM21, there was some attempts to develop machine-learned exchange–correlation functionals (e.g. ML-ωPBE¹⁶⁷) but not for such a general purpose as those developed later. Additionally, the use of neural network had not demonstrated any comparable or superior performance with respect to traditional (or analytical, from a mathematical point of view) functionals, as those summarized before, before the level of accuracy achieved by DM21, aPBE0-ML, or Skala functionals. For other pioneering approaches, we also mention OrbNet,¹⁶⁸ NeuralXC,¹⁶⁹ DeePHF,¹⁷⁰ or CF22D.¹⁷¹

5.1 The deepmind functional

In the DM21 model¹⁷² the proportion of exact-exchange varies locally in space, thus belonging to the family of local hybrid functionals, and also incorporates their range-separated values, which has the advantage of (formally) removing the delocalization error.¹⁷³ The explicit form of the DM21 functional integrates local energies calculated by a MultiLayer Perceptron (MLP) which took as input both local and non-local features of the occupied KS orbitals:where f_θ(x(r)) represents a single set of weights, θ, accepting a vector, x(r), at each grid point r, and returning a vector of 3 enhancement factors before computation of the integral. The MLP part is complemented by the typical D3(BJ) term, and the DM21 expression is just made E^DM21_xc[ρ(r)] = E^MLP_xc[ρ(r)] + E_D3(BJ). The x(r) values supplied at each point of the grid are computed from a spin-indexed (σ ∈ {↑,↓}) set of typical DFT variables, based on the density matrix, D^σ_ij, the density itself, the square norm of the gradient of the spin densities and of the total density, |∇ρ^σ(r)|² and |∇ρ(r)|², and the kinetic energy density, . The local energies are:

e^LDA_x(r) = −C_x[ρ_↑(r) + ρ_↓(r)]^4/3 (24)

with ω = 0.4 chosen based on validation studies according to the own developers; e^EXX_x(r) is calculated just setting ω = 0 in the above equation. Note that all the ingredients entering into the model are part of the main features of the DFT advances gained in the last decades. The model was trained on extensive atomic and molecular datasets (more than 1000 points), but also considering exact (integer-charge) mathematical constraints, and fractional-charge and fractional-spin information (more than 1000 points), complemented by a D3(BJ) two-body correction as it is usually done in traditional models. Note also that the hard-to-control piecewise linear variation of the total energy with respect to fractional electron number was forced to be part of those constraints. Despite some initial criticism of the model,^174,175 DM21 was found to be more accurate than a set of (more standard) functionals such as revPBE-D3(BJ), MN15-L-D3(0), SCAN-D3(0), B3LYP-D3(BJ), ωB97X-V, M06-2X-D3(0), and PW6B95-D3(0) by looking at the WTMAD-2 error of the GMTKN55 dataset, and only inferior to the DSD-PBEP86-D3(BJ) double-hybrid functional, among the traditional functionals considered. One notable performance of the DM21 model was the correct dissociation limit of challenging systems such as H₂, H₂⁺, and N₂, the good performance for the SIE4x4 dataset, as well as for other complex situations like a charged adenine–thymine base pair, a compressed hydrogen H_n chain, and diradical transition states. The developers of DM21 caution against its application to atoms heavier than Kr because the training set does not include heavy elements or pseudopotentials.

DM21 has recently been applied to transition metal chemistry,¹⁷⁶ which can be seen as a proof-of-concept for its transferability since DM21 was exclusively trained on main-group chemistry as said before; more particularly, it was applied to the TMC151 dataset, featuring transition metal dimer dissociation energies, metal–organic reaction energies, and barriers of complexes of second- and third-row transition metals. The authors concluded that despite a similar performance to B3LYP, the model faced convergence issues most of the times, thus complicating its self-consistent implementation; however, the use of B3LYP densities instead (DM21@B3LYP) improved the results, although at the price of a higher computational cost comparable even to i.e. B2-PLYP, a double-hybrid exchange–correlation functional. One possible solution for this, already pointed out by the authors of the study, would be to incorporate transition metal reactions into the machine-learned functional, with the due care to not deteriorate the performance of main-group chemistry, borrowing again one of the strategies followed historically, that is, extending and extending the size and diversity of the datasets used for the development and assessment of the models.

DM21 has also been applied to neutral and charged water clusters,¹⁷⁷ with a varying degree of accuracy as a function of the water cluster size: (H₂O)_n (n = 2–10), H₃O⁺·(H₂O)_n and OH⁻·(H₂O)_n clusters with n = 2–6. The functional performed similarly to one the best range-separated meta-GGA functionals (ωB97M-V) but the errors for the normalized interaction energies (i.e., interaction energies divided by n) increased with the cluster size, thus being size-dependent. The authors also introduced a many-body potential (MB-DM21) to correct for missing many-body non-covalent interactions, and applied the corrected model to calculate some properties of liquid water with partial success, recommending to improve the functional form and physical content of machine-learned density functionals. Besides energy calculations, DM21 has also been extended to geometry optimizations and potential energy surfaces (PES). As another example of application of DM21, we mention the DM21-based calculations¹⁷⁸ of some PES upon variation of bond angles (e.g., H–O–H in water) and bond lengths (e.g., C–H or C–C bond length in common organic molecules such as C₄H₈). The authors found a very good agreement between CCSD(T) and DM21-based results for all the PES tackled, including the stretched H₂ and H₂⁺ PES at long H–H distances, comparable or better than results obtained at the hybrid PW6B95/cc-pVnZ level.

5.2 The Skala functional

The first step of the developers of the Skala functional¹⁶⁶ was the compilation of an unprecedented dataset of physically meaningful values (149 079) to train the model, including a set of Total Atomization Energies (TAE) as the largest subset, which is a property known to be difficult to calculate and has always been part of both early and modern datasets. The TAE dataset is composed by 78 650 values with an expected accuracy of ±1 kcal mol⁻¹ relative to experiments, and achieved by a suited wavefunction-based protocol such as CCSD(T)/CBS. The TAE were calculated for molecules containing up to five non-H atoms (Li–F, Na–Cl) and the total number of points was complemented by other datasets. The latter included atomic (total energies, electron affinities, and ionization potentials), molecular (conformational energies, proton affinities, and ionization potentials), reaction kinetics, and both intra- and intermolecular non-covalent interactions from the NCIAtlas collection (D442x10, SH250x10, R739x5, and HB300SPXx10).

The chosen form of the Skala functional was also motivated by satisfying energetically relevant constraints such as the high-density uniform coordinate scaling, the Lieb-Oxford lower bound, and the LDA limit if the enhancement factor is made f_θ(x(r)) = 1, which can be achieved by the form:

with f_θ(x(r)) a learnable enhancement factor, whose modelling is done by a deep neural network taken as input a set of semi-local, density-dependent features x(r), from a meta-GGA model. Actually, those features considered are again the density, the square norm of the gradient of the density and of the total density, and the kinetic energy density. The model was also complemented by a standard D3(BJ) correction. The authors chosen to work with discretized models, by evaluating the expression on a set of points {r_i ∈ R³, i = 1,…,M}:with the associated weights w_i ∈ R for those points r_i. The set of x(r_i) features are ρ^↑(r_i), ρ^↓(r_i), |∇ρ^↑(r_i)|², |∇ρ^↓(r_i)|², |∇ρ^↑(r_i) + ∇ρ^↓(r_i)|², τ^↑(r_i), τ^↓(r_i), with the input thus of the neural network being a tensor R^M+7. These spatial fields are projected onto grids in a way compatible with typical DFT integration schemes. The model learns non-local correlation effects directly from these fields (i.e., message-passing mechanisms to encode long-range electron correlation) thus improving the description given by semi-local functionals. Among other interesting features of the Skala model, its accuracy was demonstrated to improve systematically with adding more training data by including step-by-step more datasets into the model. Concerning the computational cost of Skala, a pair of implementations were tested: a GPU-based version integrated into Accelerated DFT,¹⁷⁹ and a CPU-based version implemented in PySCF.¹⁸⁰ Skala showed to be very competitive particularly for the GPU-based version, with a cost comparable to that of r²SCAN and ten times lower than that of M06-2X. The CPU-based implementation was slightly less successful but still competitive with those of r²SCAN, B97M-V, M06-2X, and ωB97M-V.

5.2.1 Comparison between DM21 and Skala for some established datasets and subsets. Compared to the WTMAD-2 value dropped by DM21 for the GMTKN55 dataset, 3.97 kcal mol⁻¹, Skala gave a very close value of 3.89 kcal mol⁻¹ despite an almost two order of magnitude increase in the number of data points used to train the model. To put these numbers in a due context, the lowest achieved WTMAD-2 value nowadays is obtained by the DH23 model,^181,182 a 12-parameter double-hybrid density functional leading to an error or 1.7 kcal mol⁻¹, with some non-empirical double-hybrid density functionals^183,184 also giving errors as low as 3–5 kcal mol⁻¹. Skala performed remarkably accurate on the W4–17 dataset¹²⁵ of TAE, with a mean absolute error of 1 kcal mol⁻¹ for the 200 values included, roughly speaking, situating the model as the best performer for this dataset among all the functionals considered (revPBE, r²SCAN, B97M-V, B3LYP, M06-2X, ωB97X-V, and ωB97M-V). The W4–17 was part of the training of DM21, and it was thus not analyzed separately in the corresponding publication.

Looking more in detail at the performance of Skala for the different subsets of the families of the GMTKN55 dataset, the largest error was found for the SIE4x4 (self-interaction-error related systems, with 13.6 kcal mol⁻¹) subset, the DIE60 (relative energies between C₆₀ isomers, 8.54 kcal mol⁻¹), BHPERI (barrier heights of pericyclic reactions, 3.13 kcal mol⁻¹), WATER27 (Binding energies in (H₂O)_n, H⁺·(H₂O)_n and OH⁻·(H₂O)_n, 2.47 kcal mol⁻¹), and IDISP (intramolecular dispersion interactions, 3.46 kcal mol⁻¹). On the other hand, the errors given by DM21 for those subsets are 4.92, 11.35, 0.77, 1.96, and 1.37 kcal mol⁻¹, respectively, evidencing the large discrepancies between both models when looking at their performances for the individual subsets. Actually, the largest error dropped by DM21 were found for the SIE4x4 (4.92 kcal mol⁻¹) and DIE60 (11.35 kcal mol⁻¹) for the families of basic properties and reaction energies for small systems, and reaction energies for large systems and isomerization reactions, respectively. For the family of reaction barrier heights, the DM21 largest error was found for the BH76 dataset (barrier heights of various reaction types, 2.08 kcal mol⁻¹). Considering the intra- and intermolecular families of non-covalent interactions, the DM21 largest errors are again found for the WATER27 (1.96 kcal mol⁻¹) and IDISP (1.37 kcal mol⁻¹) datasets.

Additionally, Skala was also tested¹⁶⁶ for equilibrium geometries with a deviation of 0.012 Å for the bond lengths of the CCse21 dataset,¹⁸⁵ a typical deviation by semi-local functionals¹⁸⁶ and larger than the values (mostly around 0.04–0.05 Å) provided by the rest of the functionals (r²SCAN, B97M-V, B3LYP, M06-2X, ωB97X-V, and ωB97M-V) chosen with the exception of revPBE which also leads to a deviation of 0.012 Å. By perusing now¹⁸⁷ the LMGB35 dataset¹⁷ of light atoms main group bond lengths, Skala gave a mean absolute error of 0.014 Å, compared with 0.006 Å by DM21 and thus competitive with PBE0 results. However, except BO, F₂⁺, N₂⁺, NF, NH⁺, O₂⁺, and OH⁺, those molecules belonging to LMGB35 are also included into the W4–17 used for training the DM21. Skala was also applied to the HMGB11 dataset of heavy atoms main group bond length, providing larger deviations (of 0.03 Å) than for light atoms, but comparable to those achieved by revPBE or ωB97X-V functionals.

Finally, taken from the original study,¹⁶⁶ the authors also compared Skala and DM21 cost. We remind that the former deep-learned form (Skala) avoids a position-dependent fraction of exact-exchange (DM21) with a clear benefit for the (CPU-based) computational cost which is substantially lower by two orders of magnitude. For this comparison, the authors used a sample of molecules of varying sizes, from 50 to 900 atoms and from 1000 to 20 000 orbitals, roughly speaking, taken also from established datasets such as S30L¹⁸⁸ HS13L¹⁸⁹ and NCI16L.¹⁹⁰

5.3 The adaptative PBE0 functional

The “adaptative” PBE0 functional (aPBE0) was also recently developed¹⁶⁵ by von Lilienfeld et al., based on the hypothesis that a system-dependent optimal λ^opt_x value exists, which is actually a system-specific scalar label leading to vanishing error for the model if compared with reference of high-level electronic structures methods, in the form:

E_xc[ρ(r)] = λ^opt_x{(Z_I, R_I)_I, S}E^EXX_x[{ϕ_i}] + (1 − λ^opt_x{(Z_I, R_I)_I, S})E_x[ρ(r)] + E_c[ρ(r)], (28)

with Z_I and R_I denoting the nuclear charge and coordinates, respectively, with S specifying the electronic spin state; or, in other words, the optimal value depends on both the external potential and the electronic spin state. In a further step, by setting λ^opt_x = 1 the correlation term can also be made system-dependent to find the corresponding γ^opt_c optimal values, in the form γ^opt_c{(Z_I, R_I)_I, S}E_c[ρ(r)]. The authors first calculated at the CCSD(T) level the TAE of 1169 small molecular fragments (containing 1–5 heavy atoms) reported in ref. 191 to find the optimal λ^opt_x values for those systems. The normal distribution plot of the individual 1169 values was centered at λ_x = 0.42 approximately. The mean absolute error was impressively reduced from 3.46 (PBE0) to 0.02 (aPBE0) kcal mol⁻¹ upon this adaptation of values. To assess the transferability of the model, they calculate TAE for a set of 50 new molecules, largely reducing again the mean absolute error with respect to nearly-exact reference results from 4.68 (PBE0) to 0.02 (aPBE0) kcal mol⁻¹. The procedure was also applied to the spin gap of a large set of carbenes (≈3000) in their singlet and triplet states and taken from the QMSpin dataset,¹⁹² with excellent results too going from a mean absolute error of 10.2 (PBE0) to 0.1 (aPBE0) kcal mol⁻¹. Note also that λ^opt,triplet_x = 1.102λ^opt,singlet_x − 0.118. The results by aPBE0, be for TAE or for the singlet–triplet spin gaps, were always better than those provided by semi-local (PBE, BLYP or r²SCAN), hybrid (PBE0, B3LYP, and M06-2X), and ωB97XD functionals.

After having confirmed the hypothesis that a significant improvement of the results can be obtained by a system-dependent mixing of semi-local and exact-exchange, the authors explored the possibility of deep-learned λ^opt_x values: using the generated λ^opt_x values as training data, they use a predictive model to infer values for any other system of interest, naming this variant as aPBE0-ML consequently. They predicted on-the-fly the λ^opt_x values for the revised QM9 dataset¹⁹³ (revQM9) consisting on 130 000 molecules, roughly speaking. The deep-learned λ^opt_x values followed again a Gaussian-like distribution, peaking at λ_x = 0.44, with a standard deviation of 0.037, and thus very close to the λ_x = 0.42 value disclosed before. The TAE of the reduced set of 50 molecules showed now an error of 1.32 kcal mol⁻¹ by aPBE0-ML, and thus slightly larger than that by aPBE0 itself. HOMO–LUMO gaps errors, with respect to nearly-exact GW values for 100 organic molecules from the QM7b dataset,¹⁹⁴ are also reduced from 3.52 (PBE0) to 0.86 (aPBE0-ML) eV, with aPBE0-ML again on top-tier of the models tested (PBE, BLYP, r²SCAN, PBE0, B3LYP, M06-2X, LC-ωPBE, and ωB97XD) with the exception of M06-2X for which a slightly lower error (0.67 eV) was found.

6 Conclusions

The field of DFT has been living a golden age for decades, as it is clearly evidenced by the large volume of scientific publications, advances, and exchange–correlation functionals expressions implemented in codes and exploited in endless applications. Generally speaking, the main advances and developments of DFT has gone in parallel with those of modern Science. For instance, the deployment of large computing facilities and hardware/software improvements has allowed the application of DFT to systems of all sizes and chemical composition, thus going a step further with respect to earlier studies. However, other issues have also arised. Among them, we mention some of the most critical ones, such as the increasing cost of the calculations with rungs of Jacob's ladder and how to alleviate it, the dependence and nature of the errors with respect to the system size and composition, the different performance (and the reasons for that) between the class of exchange–correlation functionals employed, etc. That continuous validation and assessment of models was also possible by preserving the accuracy of wavefunction-based results, which are very often taken as reference, at all size scales, thanks to the introduction of extended and chemically diverse datasets, from both the composition of the systems and the properties included in them.

This rich and very fruitful context is now also experiencing another step forward due to deep-learned expressions for the exchange–correlation functional. However, the underlying hypothesis and expressions are deeply rooted on the lessons achieved so far, as well as on the key role played by the ingredients defining the exchange–correlation integrand. Furthermore, in a kind of virtuous circle, the size and availability of datasets has also been invigorated by the need to train those deep-learned expressions, which can also be seen as a source of opportunities for assessing and comparing more traditional expressions. With the due efforts in both directions, the field of DFT is expected to still grow strongly in the next decades, full of opportunities and success.

Conflicts of interest

There are no conflicts to declare.

Data availability

The data that supports the findings of this study are available within the article or are available from the corresponding authors upon reasonable request.

Acknowledgements

The work in Alicante is supported by grant PID2023-152372NB-I00 funded by MICIU/AEI/10.13039/501100011033. E. B. thanks ANR (Agence Nationale de la Recherche) and CGI (Commissariat à l'Investissement d'Avenir) for their financial support to this work through Labex SEAM (Science and Engineering for Advanced Materials and devices), Grant no. ANR-10-LABX-096, ANR-18-IDEX-0001, and ANR-21-CE29-0003. Funded by the European Union (ERC, MaMa, no. 101097351). Views and opinions expressed are however those of the author(s) only and do not necessarily reflect those of the European Union or the European Research Council Executive Agency. Neither the European Union nor the granting authority can be held responsible for them.

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