The adiabatic connection formalism in DFT – theory and practice

Over the last half-century, density functional theory (DFT) has made an unparalleled impact on a variety of scientific and technological disciplines. Its remarkable balance between computational efficiency and predictive accuracy has rendered it the most widely-used electronic structure method with estimates suggesting that a staggering one-third of the world’s supercomputing power is deployed to run DFT simulations.¹  For a range of systems and chemical reactions, current DFT methodology is now capable of producing results that can rival in terms of the accuracy far more expensive wave function simulations. In the era of artificial intelligence, machine learning models trained on DFT results have significantly augmented the power of DFT. These advancements now enable simulations on larger time and length scales than were previously feasible.

A vast majority of DFT simulations employ the Kohn–Sham DFT (KS DFT) scheme. Within this practical framework, the exchange–correlation (XC) functional is the sole energy component that requires approximation. Even though it does not constitute the major component of the total electronic energy, the XC functional encodes the coulombic, fermionic, and quantum-mechanical nature of electrons. Approaches for developing XC approximations span a wide spectrum, from rigorous derivations firmly rooted in quantum mechanical principles to empirical fits and extensive databases. Even machine learning-derived functional forms for the XC have begun to compete with their human-designed counterparts.^2–9  In this chapter, we take a step back to the foundational concepts of the XC functional, centering our discussion around the density-fixed AC formalism – a framework that offers an exact formula for the XC functional. We note here that computing the exact XC functional via the AC formalism carries a heavier computational burden than solving the Schrödinger equation. Nevertheless, as a powerful tool for both, constructing and analysing XC approximations, the density-fixed AC formalism played a pivotal role in the development of DFT over the past 30 years. Specifically, this approach provides a number of valuable constraints for XC functional approximations and a range of approximations to the XC functional have been built by directly approximating the AC path.^10–15  The AC formalism has been also used to rationalize parameters used in hybrid^12,16,17  and double hybrid functionals.^18–20  Furthermore, by studying the strong interaction limit of the AC new ingredients for the construction of the XC approximation have been designed.

The idea of the adiabatic connection (AC) formalism is to employ a continuous path between the non-interacting KS system and the exact many-electron quantum system while keeping the ground-state electronic density function fixed.^21–23  This path is obtained by linking the non-interacting KS system with the physically interacting one through scaling of the electron–electron interaction. As a result, the AC formalism allows the derivation of the exact XC functional using only the electron–electron repulsion operator, even though the XC functional contains both, electron–electron repulsion component and a kinetic energy component.

In Section 2, we derive the AC based expression of the XC functional. It is an integral, where the integrand is a parameter dependent global density functional. In Section 3, we list the exact known conditions on the AC integrand. We provide an overview of constructing approximations to the XC functionals based on the AC framework in Section 4. Then, in Section 5, we elaborate briefly on the connection between the global and local AC integrands. Section 6 showcases the strong interaction limit in DFT and its link to the AC. Finally, in Section 7, we review the similarities and differences between the DFT’s adiabatic connection and that used in the Møller–Plesset theory.

It is interesting to observe the global AC integrand W_λ[n] at specific λ points. For instance, when λ = 0, then W₀[n] is equal to the exact exchange E_X[n]. When λ = 1, Ψ₁[n] is the minimizer for F₁[n] = F[n], which corresponds to the constrained full interacting ground-state wave function. Hence, W₁[n] is the electron–electron repulsion energy of the many-electron quantum system without Coulomb interaction, which is equivalent to the XC energy from which we substract the difference between the true and non-interacting kinetic energy.

Fig. 1 illustrates the nature of the global AC integrand. Integration between λ = 0 and λ = 1 corresponds to the shaded area below the function, which also stands for the exact XC energy according to eqn (4). The figure illustrates the monotonically decreasing nature of W_λ[n], and also illustrates the strong coupling limit when λ → ∞. More properties of W_λ[n] are listed in the following section.

The AC formalism provides a rigorous way for the development of new classes of density functional approximations. Thus, we want to emphasize in this chapter the exact features of the AC integrand, because they are most relevant for the construction of exchange–correlation functionals in DFT. The following is a list of several essential conditions:

• It has been proven that the AC integrand, W_λ[n], monotonically decreases with respect to λ,^25,26  i.e.

  $d W λ [ n ] d λ <0.$

  (13)

  Along with the assumption of a monotonically decreasing nature, it is generally taken that the function W_λ[n] always exhibits convexity. This assumption is supported by numerical evidence, but a formal proof is still lacking to the best of our knowledge. It is worth noting that, theoretically, convexity could be violated in cases of phase transitions along the adiabatic connection path, where the AC curves are likely piecewise convex. For instance, the AC corresponding to the uniform electron gas at a density lower than that of the ferromagnetic transition might exhibit a slight discontinuity. However, the density constraints imposed on W_λ[n] drastically reduce the likelihood of such occurrences.
• A specific scaling property is satisfied by the AC integrand linking AC with the uniform coordinate scaling:²⁷ 

  $W λ [n]=λ W 1 [ n 1/ λ ],$

  (14)

  where n_γ(r) is a scaled density:

  $n γ (r)= γ 3 n(γr)$

  (15)

  for γ > 0, which conserves the number of electrons. When γ increases, the density shrinks and when it decreases, the density spreads out. Using this uniform scaling relation one can also recover W_λ[n] from E_xc[n] with

  $W λ [n]= ∂ ∂ λ { λ 2 E xc [ n 1/ λ ( r ) ] } .$

  (16)

  The exact relation of eqn (16) applying to the exact XC functional has also been used to analyse the approximations and see how for small systems (e.g., hydrogen dimer along its dissociation curve) the AC curves arising from approximate XC functionals (e.g., PBE²⁸ ), deviate from the exact counterpart calculated directly from eqn (5).^15,29,30  Also, the behaviour of the XC functional in the high- (γ → 0) and low-density limits^25,31  (γ → ∞) are known:

  $lim γ → ∞ E xc [ n γ ] γ = E x [n], lim γ → 0 E xc [ n γ ] γ = W ∞ [n].$

  (17)

  Overall, from eqn (14), (16) and (17) we can see that the uniform density scaling and the adiabatic connection approaches in DFT are deeply interlinked.
• Using the differentiability with respect to λ allows an expansion of the global AC integrand in the non-interacting limit case (small λ). It is given by:³² 

  $W λ [n]= W 0 [n]+ W 0 ′ [n]λ+…(λ→0),$

  (18)

  where the AC integrand for λ = 0 corresponds to the exact exchange functional W₀[n] = E_x[n], which is given in terms of the occupied KS orbitals:

  $E x [n]=− 1 2 ∑ i j occ ∫ ϕ i ( r ′ ) ϕ j ( r ′ ) ϕ j ( r ) ϕ i ( r ) | r ′ − r | d r d r ′ .$

  (19)

  Furthermore, the slope of W_λ is given by two times the correlation energy of the second-order Görling–Levy perturbation theory (GL2):^32,33  $W 0 ′ [n]=2 E c GL2 [n]$ ⁠. This correlation energy can be also expressed in terms of occupied and unoccupied KS orbitals and their energies:

  $E c GL2 [n]=− 1 4 ∑ a b i j | 〈 ϕ i ϕ j | | ϕ a ϕ b 〉 | 2 ɛ a + ɛ b − ɛ i − ɛ j − ∑ i a | 〈 ϕ i | v x KS [ n ] − v x HF [ n ] | ϕ a 〉 | 2 ɛ a − ɛ i ,$

  (20)

  where the indices i, j and a, b run over occupied and non-occupied KS orbitals respectively. The second term in eqn (20) accounts for the difference between the local KS potential, $v x KS [n](r)=∫ n ( r ′ ) | r − r ′ | d r ′ ,$ and the non-local Hartree–Fock (HF) exchange potential, $v x HF [n](r)= ∂ E x [ n ] ∂ n ( r ) ,$ evaluated with KS orbitals.
• A similar expansion of the AC integrand exists in the strongly-interacting limit, i.e. λ → ∞:^34,35 

  $W λ [ n ] = W ∞ [ n ] + W ∞ ′ [ n ] λ − 1/2 + O ( λ − m ) ( λ → ∞ , m ≥ 5/4) .$

  (21)

  Thereby, W_∞[n] and $W ∞ ′ [n]$ have a completely different structure to the previous functionals, W₀[n] and $W 0 ′ [n]$ ⁠. They are highly non-local with respect to the density and a strictly-correlated electrons (SCE) approach³⁴  has been designed for their evaluation. Note that $W ∞ ′ [n]$ is denoted here as the second term in the expansion after W_∞[n], at large λ. In fact, $W ∞ ′ [n]$ does not correspond to the derivative of W_λ in the λ → ∞ limit. Its denotation resembles its role in the strong coupling limit, analogous to the one of $W 0 ′ [n]$ in the weak coupling limit.^34,36 
• The Lieb–Oxford inequality^31,37–40  provides a bound from below for the AC integrand:

  $W λ [n]≥− C LO ∫ n 4/3 ( r ) d r .$

  (22)

  The exact value for the C_LO constant is not known, but it is rigorously proven to lie in between 1.4442 and 1.5765.^41–43  More generally, $− C LO ∫ n 4/3 (r)dr$ bounds from below the indirect energy (electron–electron repulsion minus the Hartree energy) of any correctly normalized and antisymmetric Ψ[n]. Letting Ψ[n] be Ψ_λ[n], we obtain eqn (22). In view of the first AC constraint we listed here, W_∞[n] will be the smallest value for the l.h.s. of eqn (22). With this in mind, finding lower bounds for the optimal constant C_LO reduces to searching for a density n for which the ratio between W_∞[n] and $−∫ n 4/3 (r)dr$ ⁠,^44,45  will be the greatest. This procedure has been indeed applied for searching the optimal C_LO and understanding for what type of densities C_LO is large.^{39,40,44–46} 

Usually, the development of approximations to W_λ[n] aim to obey as many of the exact constraints listed in Section 3¹⁵  as possible, and then utilize the density functional approximations (DFAs) formulas for an evaluation. One of the first eminent AC-hybrid models was introduced by Becke who applied a linear model for the AC integrand.⁴⁷  In fact, Levy provided a proof showing the linearity of W_λ[n]²⁷  for small enough λ. There are many more approximation models for the AC integrand, such as the [1/1] Padé approximant,¹⁰  coming from the MCY family of functionals.⁴⁸  Teale et al. employed ab initio arguments to propose some models for W_λ[n].²⁶  Additionally, the interaction strength interpolation (ISI) class of functionals, firstly introduced by Seidl and co-workers^49,50  and later revised,^15,34,35  is a group of functionals based on an interpolation between the weak and strong coupling limit of AC DFT. By including the information from both weak- and strong-coupling regimes, the ISI framework circumvent the prevalent bias towards weak correlation, a characteristic feature of a vast majority of XC approximations. A potential problem of the ISI functionals is their lack of size consistency, which, however, can be easily corrected by using the size-consistency correction of ref. 51 when there are no degeneracies. Within the DFT realm, the ISI functionals have been tested against several chemical data sets, showing that they perform reasonably well for interaction energies (energy differences).^51–53  However, their application has proven to be far more successful outside the DFT realm, particularly when used in conjunction with Hartree–Fock, as opposed to DFT orbitals. These successes and underlying theoretical framework will be discussed in Section 7.

Approximations to XC functionals have been built by constructing ε_xc[n](r) through an interpolation at each point in space between w₀(r) and w_∞(r) (local interpolations).

Compared to global interpolations, these local interpolations usually deliver superior results for small chemical systems, as noted in previous studies.^15,30  However, they typically do not fix all the shortcomings found with global interpolations.³⁰  Despite these observations, the accuracy of XC functionals based on local interpolation between w₀(r) and w_∞(r) is largely unexplored and will likely benefit from the use of specific machine learning techniques.⁵⁹  Specifically, combining this local AC interpolation with deep learning-based ‘local hybrids’⁵⁹  holds promise for improving XC approximations, particularly when it comes to treating the long standing problem of DFT’s strong correlation.

In previous two sections we discussed strategies for building approximations for the XC exchange functional by using the information from both the weak- and strong-coupling limit of DFT [e.g., w₀(r) and w_∞(r) shown in the top panel of Fig. 2]. The key idea of this approach is that our description of the physical case (λ = 1) can be significantly improved if we have knowledge about both λ = 0 and λ = ∞ scenarios. This becomes particularly relevant when trying to describe systems characterised by strong correlation, whose treatment remained (one of) the biggest challenges in the field of DFT development.^60–63  In this section, we will briefly describe the main characteristics of the strong-interaction limit of DFT, which corresponds to the λ → ∞ limiting case of our density-fixed adiabatic connection (see ref. 64 for a more comprehensive review of the topic).

The construction of co-motion functions necessary for SCE solutions has introduced certain integrals of the density, which subsequently offer unique ingredients for the construction of the XC functional.⁶⁴  These new ingredients are very different from the conventional ones used for building XC approximations (i.e., semilocal ingredients and KS orbitals). Indeed, these novel definitions have been leveraged to develop new functionals that approximate both, λ → ∞ and physical regimes. All these new functionals strive to tackle the strong correlation problem in DFT. In the years and decades ahead, it will be interesting to see to what extent these new SCE-based mathematical features can address the strong correlation issue and to construct the next generation of DFT methods. Particularly, the new ingredients that emerge in the SCE limit could be used as novel features for machine learning the XC functional (Fig. 3).

Although the basic principle behind the two adiabatic connections is the same, there are some important differences. Firstly, the density constrain that characterizes the DFT AC is absent in the MP AC, which recovers the HF density, n^HF, at λ = 0 and the physical density, n, at λ = 1. Naively, one would expect the electrons to be unbounded, however the scaling with λ of the Hartree potential $( J ˆ [ n HF ] = v H [ n HF ] )$ creates an increasingly negative energy well which binds the electrons.⁷³ 

Similar to the functionals of Section 4, interpolation schemes have been derived for the MP AC.^78,79  The main difference is that the MP AC is not convex, because MP2 (only the first term of eqn (20)) underestimates the correlation energy forcing a concave curve for small λ, whereas at larger λ the curve was proven to be convex. To circumvent this issue, two SPL functionals,^49,78,79  called SPL2, were combined resulting in a functional with more freedom to approximate this feature correctly. To mimic the effect of the large corrections from subleading orders, two fitting parameters were added to the inequality of eqn (45). Afterwards, the point-charge-plus-continuum model³⁶  is used to approximate W_∞. Other interpolation schemes have also been introduced, but we will focus on SPL2.

Recently,⁷⁹  cost saving strategies and more accurate MP2 forms were used to not only decrease the cost but also increase the accuracy. In principle we are only allowed to use the standard MP2 correlation energy as the first order approximation, however it was shown that other forms of MP2 can improve on the previous iterations for a variety of different systems. As an example, Fig. 4 shows the dissocation curves of the benzene dimer for CCSD(T) (black), a SPL2 form (orange), MP2 (blue), and B2PLYP-D3 (red), where it shows that SPL2 is the only curve that improves MP2 towards CCSD(T) whereas the hybrid functional is significantly less accurate. Especially for halogen bonded complexes with strong charge transfer natures, the new SPL2 functional can improve by a factor of 3 to 5 on dispersion corrected hybrids and double hybrids, reaching a near CCSD(T) accuracy for a large range of non-covalant interactions.

With the help of very accurate wave function methods, the characteristic of the global AC curves for small systems is attainable.^80,81  A similar result is obtained for the local AC functions.^15,82  These observations are useful for assessing the accuracy of new approximation models for the XC functional. In addition to the density fixed AC reviewed here, several other alternatives have been proposed: the AC based on the range-separation of electronic interaction,⁸³  the kinetic-energy based adiabatic connection^84–86  and adiabatic connections for strictly correlated electrons.^87,88