We review the concept of ultranonlocality in density functional theory and the relation between ultranonlocality, the derivative discontinuity of the exchange energy, and the static electric response in extended molecular systems. We present the construction of a new meta-generalized gradient approximation for exchange that captures the ultranonlocal response to a static electric field in very close correspondence to exact exchange, yet at a fraction of its computational cost. This functional, in particular, also captures the dependence of the response on the system size. The static electric polarizabilities of hydrogen chains and oligo-acetylene molecules calculated with this meta-GGA are quantitatively close to the ones obtained with exact exchange. The chances and challenges associated with the construction of meta-GGAs that are intended to combine a substantial derivative discontinuity and ultranonlocality with an accurate description of electronic binding are discussed.

Density functional theory (DFT) is a very successful approach to the electronic structure problem.1–4 This success rests on the highly non-trivial fact that the intricate quantum many-body effects of exchange and correlation (xc) can be captured with an accuracy that is good enough for many practical applications with relatively “simple” functionals of the density, such as the local density approximation (LDA) and generalized gradient approximations (GGAs). Such semilocal xc approximations provide an unrivaled computational efficiency. The constraint-guided construction strategy that has been advocated by Perdew, to whom this special issue is dedicated, has led to some of the most widely used semilocal functionals.5–7 

However, there are well-known classes of problems where LDA and typical GGAs fail even qualitatively. One of them is the electrical response of extended systems and long-range charge-transfer. The challenges, both conceptual and practical, that DFT can face in describing the electrical response of infinite periodic systems have been discussed from different perspectives in the past.8–17 Here, we focus on extended but finite systems. In these, the long-range charge-transfer problem manifests itself in both ground-state18,19 and time-dependent20,21 DFT. Our focus here is on ground-state DFT, and in this ground-state theory, the hallmark charge-transfer problem is the huge overestimation of the static electric longitudinal dipole polarizability of extended molecular systems that is observed with LDA and GGAs. The failure can be traced back19,22 to a well-known qualitative difference between the response of the exact exchange potential and the one of LDA and GGAs: the exact exchange response potential counteracts the externally applied dipole field, while the one of LDA and GGAs works with it. This feature of exact exchange has been termed “ultranonlocality.”19,22,23 Hydrogen chains and polyacetylene serve as the prime examples for studying this ultranonlocality and the ground-state charge-transfer problem,18,19,22,24–42 and they also set the stage for the present work. Since the terms local, semilocal, nonlocal, and ultranonlocal have been used in somewhat different contexts in the past, we first clarify in which sense we use these terms in the present work.

One context in which these terms are used in ground-state DFT is the definition of the xc energy
(1)
in terms of an xc energy density exc[n](r). An approximation to exc(r) is called nonlocal when evaluating it at the point r requires information from far-distant points r, e.g., via an integration over all space. The hallmark example of nonlocality is the exact exchange energy density
(2)
An approximation is local or semilocal in terms of Eq. (1) when the xc energy density can be evaluated based on quantities that are already available in a usual calculation, such as the density and the orbitals, and these quantities need to be evaluated only at r or an infinitesimal neighborhood of r, respectively, to find exc(r).

When the terms semilocal and nonlocal are used in this sense, they are also often associated with computational expense, in the sense that semilocal functionals are usually cheaper to evaluate than nonlocal ones. The term ultranonlocal, however, is difficult to define based on Eq. (1). For defining and understanding ultranonlocality, it is helpful to look at the terms local, semilocal, and nonlocal from a second, different perspective and think in terms of the xc potential vxc(r).

In this second perspective, the term “locality” refers to how the xc potential vxc at a given point r depends on the density n(r). That is, in a local approximation, it suffices to know the density n at r to compute vxc(r) at this point r. In a semilocal approximation, it suffices to know n and ∇n at the given point r to calculate vxc(r). Although calculating ∇n(r) requires information about the density beyond just r and is, therefore, not a strictly local procedure, the required additional information is restricted to an infinitesimally small neighborhood around r, i.e., semilocal. An approximation is called “nonlocal” when vxc(r) depends continuously on the value of the density in regions of space that are far from the point r.43 The Hartree potential is a typical example of a potential that is nonlocal in this sense.

Ultranonlocality is different from this usual nonlocality. We call a potential ultranonlocal when vxc(r) non-vanishingly depends on the density at points r′ that can be infinitely distant from r, or when an infinitesimally small change of the density can lead to a finite change of the potential. The finite “jumps” of the Kohn–Sham xc potential that are associated with the derivative discontinuity44,45 are a paradigm example of ultranonlocality. The field-counteracting terms mentioned in the second paragraph can be understood in terms of such potential steps induced by the derivative discontinuity:26,34 As the external field moves charge to a molecular unit, the potential “jumps up” on this unit and, thus, counteracts the polarizing field.

One should be aware that the three different contexts—energy density, computational cost, and xc potential—in which the terms local, semilocal, etc., are used, result in the fact that the classification of a functional can be non-obvious. We elucidate what we mean by this sentence with the help of two examples. The first is the Average Density Self-Interaction Correction (ADSIC) functional,46 which can be interpreted as a global average over the well-known orbital specific Perdew–Zunger SIC.47 In terms of the first definition via the energy density, ADSIC is nonlocal, as it features a Hartee-type integral in the xc approximation. In terms of computational cost, however, ADSIC is classified as local as it is hardly more expensive than the LDA. Furthermore, the ADSIC functional does not show48 the field-counteracting terms that are the hallmark of (ultra)nonlocality.

The second example is meta-GGAs. They are semilocal in terms of energy density and computational cost. However, their potential can show ultranonlocality, as demonstrated in Ref. 49. There is, however, an open question with respect to the degree of ultranonlocality that can be reached with a meta-GGA. While Ref. 49 demonstrated that a meta-GGA can show ultranonlocality and can thus also improve the static electric response of extended molecular systems, the degree of ultranonlocality did not match the one of exact exchange. Furthermore, the electrical response, while being much improved compared to usual semilocal functionals, did not fully capture the features that exact exchange shows, for example, with respect to the dependence of the response on the system size. This raises the question whether a meta-GGA can really reach the same degree of ultranonlocality as exact exchange or whether the electrical response that exact exchange yields incorporates features that decisively depend on the nonlocal Fock integrals and, thus, cannot be reproduced by a meta-GGA.

This is the question that this paper addresses. In Sec. II, we review existing meta-GGAs in view of a criterion that allows one to estimate the degree of ultranonlocality that one can expect from a meta-GGA. In Sec. III, we present a new meta-GGA that we construct non-empirically by focusing on constraints that guarantee important properties of the potential vx(r) and ultranonlocality. We demonstrate in Sec. IV that this meta-GGA reproduces the static electric response in close and quantitative similarity to exact exchange, despite using only quantities that can be evaluated at semilocal computational cost. The calculations also show that the functional can be evaluated without numerical problems. In the concluding Sec. V, we put these results into perspective with other requirements that xc approximations are typically expected to fulfill and discuss possible future developments.

By convention, semilocal functionals are often not given in the form of Eq. (1) but use a factorization with an enhancement factor Fx that indicates how strongly the functional differs from the LDA. We here focus on meta-GGAs for exchange that are written in the form
(3)
where Ax = −(3e2/4)(3/π)1/3, and the enhancement factor Fx(s, α) is parameterized in terms of the dimensionless variables s=|n|/(2(3π2)1/3n4/3) and α = (ττW)/τunif. Here, n=σ=,j=1Nσφjσ2 is the density, and the kinetic energy density τ,
(4)
is evaluated using the Kohn–Sham or generalized Kohn–Sham orbitals φ, depending on which framework of DFT one is working in. The von Weizsäcker kinetic energy density is τW = 2|∇n|2/(8mn), and τunif = Asn5/3 with As=(32/10m)(3π2)2/3 is the kinetic energy density of the homogeneous electron gas. e and m are the elementary charge and the electron mass, respectively. The parameterization of the enhancement factor in s and α is used in many meta-GGAs50 (although other variables are used with success as well51). In our experience, s and α are well suited for modeling xc approximations in which representing limiting cases, such as the homogenous electron gas limit (α → 1) or iso-orbital limit (α → 0), is part of the construction strategy.

In the following, we focus the formal discussion mostly on the derivative discontinuity and not on other manifestations of ultranonlocality, such as the field-counteracting terms. We can do so because, as explained in previous studies,20,26,49,52 a substantial derivative discontinuity will automatically translate into an improved description of the observables for which ultranonlocality is of interest, such as static charge-transfer properties and field-counteracting terms. Since exchange contributes dominant parts to the ultranonlocal response,19,24 as confirmed by the exact correlation contribution to the response in hydrogen chains,39 we further focus on exchange functionals.

The exchange derivative discontinuity for a system with N electrons is defined by
(5)
where |+ and | denote evaluation of the functional derivative at N + ϵ and Nϵ, with ϵ → 0, respectively. As the functional derivative of a meta-GGA is given by
(6)
the derivative discontinuity of a meta-GGA is given by
(7)
From this exact expression, one can derive an approximate one that allows one to easily develop a feeling for the expected nonlocality of a given meta-GGA by simple visual inspection of the enhancement factor. The key thought is the “system averaging approximation” that replaces ∂ex/∂τ in the integrand of Eq. (7) by its average over the integration region. This allows us to pull this average ex/τ̄ out of the integral. Interchanging the integration and the functional derivative, one can then integrate τ and obtain the approximate relation
(8)
The second factor on the right is just the Kohn–Sham eigenvalue gap53 Δs, and thus, one obtains ΔxmGGA(ex/τ̄)Δs in the system averaging approximation. Since Δs > 0, this equation shows that a meta-GGA that fulfills
(9)
will yield a positive exchange derivative discontinuity, as it should. Thus, Eq. (9) can be used as a guideline in meta-GGA constructions that aim at functionals that yield a pronounced derivative discontinuity Δx and related features, such as ultranonlocality.
For practical purposes, one prefers meta-GGAs that are parameterized in s and α; cf. Eq. (3). One can translate the condition of Eq. (9) to these variables by using the chain rule in combination with Eq. (3) and the definition of α. This shows49 that a positive ΔxmGGA is guaranteed when
(10)

The system averaging approximation is a non-trivial step, and its consequences have recently been discussed in detail in the context of band-gap prediction.54 However, based on it, Eqs. (9) and (10) allow us to gain a priori intuition for how much derivative discontinuity and (ultra)nonlocality to expect from a meta-GGA in a very simple way, namely by inspecting plots of the enhancement factor.

Figure 1 shows plots of the exchange enhancement factor Fx(s, α) as a function of α for different values of s for some paradigm meta-GGAs that we selected from the large number of meta-GGAs that are available in the literature. The left and middle panels in the top row show plots of the PKZB (Perdew, Kurth, Zupan, and Blaha55) and TPSS (Tao, Perdew, Staroverov, and Scuseria6) meta-GGAs, respectively. Evidently, their enhancement factors hardly show any slope, and the enhancement factor of PKZB for small values of s even slightly increases as a function of α. This observation, together with Eqs. (9) and (10), explains the earlier reported findings that, e.g., TPSS does not show non-locality,61 PKZB and TPSS show only a minute derivative discontinuity,52 and yield band gaps similar to usual GGAs.62 Many other meta-GGAs, e.g., the ones from Refs. 63–68, show similarly small derivatives ∂Fx/∂α, and one thus expects similarly little ultranonlocality from them.

FIG. 1.

Plots of the enhancement factor Fx(s, α) as a function of α for different values of s for the meta-GGAs (from left top to bottom right): PKZB,55 TPSS,6 local τ,56 M06-L,57 M11-L,58 MN15-L,59 SCAN,7 TASK,49 and PBE-GX.60 A negative slope in these plots leads to a positive exchange derivative discontinuity. For the local-τ approximation, not all the curves for the different values of s fall within the plotted range.

FIG. 1.

Plots of the enhancement factor Fx(s, α) as a function of α for different values of s for the meta-GGAs (from left top to bottom right): PKZB,55 TPSS,6 local τ,56 M06-L,57 M11-L,58 MN15-L,59 SCAN,7 TASK,49 and PBE-GX.60 A negative slope in these plots leads to a positive exchange derivative discontinuity. For the local-τ approximation, not all the curves for the different values of s fall within the plotted range.

Close modal

The rightmost panel in the first row depicts the enhancement factor of the local τ approximation.56 This non-empirical meta-GGA shows a very pronounced dependence of the enhancement factor on α, and one can, therefore, expect pronounced effects of ultranonlocality. However, all the curves for different values of s are monotonically increasing. Thus, Eq. (10) indicates that the local τ approximation56 will lead to a negative exchange derivative discontinuity, contrary to the sign that one finds in exact exchange. Based on general arguments,54 one expects that correlation will also contribute a negative sign to the derivative discontinuity. Therefore, the local τ approximation will yield an overall negative xc derivative discontinuity—which calls this approximation into question in view of the expectation of a positive derivative discontinuity.69 

The second row of panels depicts the enhancement factors of the M06-L,57 M11-L,58 and MN15-L59 meta-GGAs for exchange. Fitting parameters to large databases is part of the construction strategy of these functionals, and this leads to substantial variations in the enhancement factors. Consequently, because there is no universal trend in the slope of Fx, it is difficult to deduce general statements about the magnitude and the sign of the derivative discontinuity for M06-L and M11-L based on Eq. (10) and the plots of the enhancement factor: For some ranges of α, the slope is negative while it is positive for others, and for values of s ⪆ 2, the enhancement factor increases with increasing α, i.e., shows a trend similar to the local τ approximation. The same is true for the revM06-L functional,70 which is not shown in Fig. 1. Thus, the strength of the derivative discontinuity and ultranonlocal effects with these functionals can be very different for different systems, depending on which values of s and α are realized and which range of the enhancement factor is thus probed. The MN15-L functional, on the other hand, shows a more uniform enhancement factor. For most values of s, Fx decreases with increasing α, which translates into the proper sign for the exchange discontinuity according to Eq. (10). This general trend is, however, violated for s = 0 for values of α ⪅ 0.5, and therefore, some uncertainty about the strength of the exchange discontinuity in this functional remains.

The bottom line of panels in Fig. 1 finally shows three examples of functionals for which one can be sure to find a non-negative exchange discontinuity and at least some degree of ultranonlocality: The enhancement factors of the meta-GGAs SCAN,7 TASK,49 and PBE-GX60 decrease with increasing α for all values of s. SCAN shows more slope than, e.g., TPSS, and consequently, SCAN improves band gaps more than TPSS due to a larger contribution from the derivative discontinuity.71–73 There are several other functionals,74–78 not shown in Fig. 1 for the sake of space, whose enhancement factors show a similar or somewhat less negative derivative in α than SCAN. These functionals can, therefore, be expected to also show some degree of ultranonlocality, although not much. A larger degree of ultranonlocality can be expected from the TASK functional, which, while satisfying the same set of exact constraints as SCAN, has a more pronounced negative slope. Consequently, band gaps predicted with TASK reach a yet higher and remarkable accuracy49,62 due to a substantial contribution54 from the exchange discontinuity. In terms of the slope of the enhancement factor, cf. the bottom right panel in Fig. 1, one would expect that the PBE-GX functional shows a derivative discontinuity and ultranonlocality of a strength between SCAN and TASK. However, checking this in practice for systems of practically relevant complexity seems presently impossible, as the PBE-GX functional is numerically very ill-behaved.79 

The degree of ultranonlocality and the magnitude and sign of the derivative discontinuity that one expects from a meta-GGA can thus, in many cases, simply be estimated by analyzing the slope of the enhancement factor as a function of α. (We note in passing that this slope also determines the importance of the gauge-invariance restoring current-density correction when meta-GGAs are used in time-dependent DFT.80,81) To the best of our knowledge, among the presently existing meta-GGAs, the TASK functional is the one that shows the most pronounced effects of ultranonlocality.82,99 This, however, leads to the question that was mentioned toward the end of the introduction. In Ref. 49, the practical manifestation of ultranonlocality was tested for the TASK functional by using it to calculate the static electric polarizabilities of hydrogen chains, i.e., one of the paradigm test systems for the static charge-transfer problem.18,19,22,24,26,27,29–38,40,41 While TASK significantly improves the calculated polarizabilities compared to other semilocal functionals, it does not fully match exact exchange in terms of the strength of the field-counteracting terms. In particular, for the longer chains, the differences between TASK and exact exchange became noticeable. Thus, the question arises whether a meta-GGA can really capture the same degree of ultranonlocality as exact exchange or whether there is a fundamental limitation of the meta-GGA concept, e.g., resulting from the lack of exchange-like integrals, to yield full, exact-exchange-like ultranonlocality.

In the following, we demonstrate that one can construct a meta-GGA based solely on semilocal-cost functional ingredients that indeed yields an ultranonlocal static electric response in quantitative agreement with the exact exchange for hydrogen chains and in close similarity for real oligomers. The guidelines in our construction are the hydrogen atom as the paradigm localized one-electron system and the homogeneous electron gas as the paradigm extended many-electron system.

Our construction starts at the iso-orbital limit. In this limit, α ≡ 0, the exchange enhancement factor effectively depends only on s and can, therefore, be represented by a GGA-type enhancement factor,
(11)
The SCAN7 and TASK49 meta-GGAs choose
(12)
in order to obey the strongly tightened bound83 for two-electron densities,
(13)
and to enforce the correct nonuniform coordinate scaling of the exchange energy per particle to the true two-dimensional limit.84,85 With the choice7  c0 = 4.9479, SCAN and TASK obtain the exact hydrogen atom energy via spin scaling. Thus, for hydrogenic systems, the ansatz indirectly also minimizes the one-electron self-interaction energy. As our aim is to obtain a physical potential that is similar to the one of exact exchange, we want to avoid the divergences that GGA-type potentials typically show at a nucleus. Therefore, we generalize GSCAN(x) of Eq. (12) to
(14)
where Θ(x) is the Heaviside step function. Here, x0 ≥ 0 is an additional parameter that ensures G(x;x0)=hx0 for xx0. We choose x0=s02 with s0 = (6π)−1/3 being the minimal value of s realized in a doubly occupied, 1s-orbital like exponential density. This eliminates the spurious divergence in the exchange potential at the nucleus due to ∇2n contributions.6,86 With the choice c = 4.759 279, we ensure that the exact hydrogen atom energy is obtained again. This one-orbital limit enhancement factor is referred to as G(x) in the following.
We now generalize the one-orbital limit G(x) to a general meta-GGA enhancement factor Fx(s, α) by making use of the observation that in the one-orbital limit, the reduced kinetic energy density t = τ/τunif is proportional to s2, i.e., more precisely t53s2, because ττW.87 Furthermore, we make use of the general relation 3t/5 = s2 + 3α/5, which follows from the definition of α. Combining these steps, one can define a general enhancement factor by the linear combination
(15)
The idea of this linear combination is to define a family of enhancement factors as a function of the parameter k that all share the same one-orbital limit, i.e., respect the hydrogen atom limiting case, yet differ in their dependence on α. Furthermore, the strength of the α-dependence is directly controlled by the parameter k. As G(x) is a monotonically decreasing function, restricting k to positive values, i.e., k > 0, ensures that
(16)
Thus, the correct sign of the exchange derivative discontinuity, i.e., Δx > 0, is guaranteed for all enhancement factors of the type of Eq. (15).
In the following, we are interested in k > 1 to obtain an appreciable magnitude of the derivative discontinuity and associated ultranonlocal properties. Since α ≥ 0, the monotonicity of G implies that the enhancement factors of Eq. (15) for k > 1 are bounded from both below and above,
(17)
While the upper bound is reasonable as it imposes the conjectured strongly tightened bound,83 
(18)
the lower bound does not ensure the positivity of Fx and, thus, cannot guarantee the negativity of the exchange energy for any density. In practice, a negative Fx can arise from Eq. (15) for k > 1 when s is small and α is very large because then the negative second term can dominate over the positive first and third terms. This problem, however, can be avoided by a change of variables. We replace α with the monotonically increasing expression
(19)
This maintains the behavior for small values of α, i.e., α̃α for α → 0, while realizing an upper bound α0, i.e., limαα̃=α0.
The thus introduced parameter α0 has to be chosen large enough to guarantee that Eq. (19) provides a non-constant mapping from α to α̃ for the physically significant range of α-values, i.e., preserves the physically relevant behavior of α, and small enough to guarantee the positivity of Fx. We found that the choice of α0 = 3 respects both conditions.88 Therefore, we arrive at the final expression
(20)
The remaining parameter k is determined by enforcing the homogeneous electron gas limit, i.e.,
(21)
Imposing this limit yields
(22)
with k ≈ 51.5558 for α0 = 3. This concludes the construction of the exact-exchange-like response (EEL) functional, i.e., ExEEL is given by Eq. (3) with the Fx of Eq. (20) and the just mentioned values for the parameters k and α0.

Figure 2 shows a plot of this FxEEL as a function of α for different values of s, in analogy to the plots shown in Fig. 1. However, note that the scale of the vertical axis has been extended in Fig. 2 due to the EEL functional’s stronger dependence on α.

FIG. 2.

Plots of the enhancement factor FxEEL(s,α) from Eq. (20) as a function of α for different values of s.

FIG. 2.

Plots of the enhancement factor FxEEL(s,α) from Eq. (20) as a function of α for different values of s.

Close modal

This functional construction did not address covalent binding, and therefore, it should not be expected that it leads to a “general purpose” functional that would, for example, reliably describe electron bonds of different kinds. However, by constructing Eq. (20) such that it respects the hydrogen atom, the homogeneous electron gas, and the principle of a large derivative discontinuity, we aimed at a functional that is reasonable for 1s-orbital densities, for delocalized electrons, and yields sizeable field-counteracting terms. As a consequence, Eq. (20) should be well capable of describing the static electric response of systems such as hydrogen chains or conjugated polymers, whose electronic structure shows both atomic-like and delocalized features and in which field-counteracting terms are decisive for obtaining reasonable values for the electric polarizability. As previously explained, we thus want to clarify whether a meta-GGA that only uses semilocal ingredients can achieve an ultranonlocal response of the same magnitude as the fully non-local exact exchange.

Hydrogen chains with alternating bond lengths of 2 and 3 a0 are, as mentioned above, a well-established reference system for checking the ability of many-body methods to describe the electrical response of and static charge-transfer in extended systems. In fact, among the different molecular chains that have been studied in the context of ultranonlocality in DFT, hydrogen chains have even been identified as a particularly challenging test.23, Table I lists the results from our calculations of the longitudinal static electric dipole polarizability of hydrogen chains of increasing length. We here do not compare to many other previously published meta-GGAs, because such a comparison has been done in earlier work49 and it showed—confirming earlier findings52,61—that traditional meta-GGAs do not show much ultranonlocality and, therefore, lead to a substantial overestimation of the polarizabilities of extended, chain-like systems. Our focus is, therefore, on meta-GGAs that have been constructed to yield a substantial derivative discontinuity and, thus, field-counteracting terms, i.e., the PoC and TASK functionals from Ref. 49 and the EEL functional from this work. We compare these functionals to exact exchange (EXX) as the reference, and we also show results from the exchange LDA as the paradigm local approximation.

TABLE I.

Static electric longitudinal dipole polarizabilities in a03 for the hydrogen chains H2N for different exchange energy functionals. DFT calculations were performed self-consistently with the potentials of all orbital-dependent functionals evaluated in the Krieger–Li–Iafrate (KLI) approximation.44 

H2NxLDATASKPoCEELEXX
H2 13.2 12.1 12.1 12.0 12.0 
H4 39.6 34.7 31.9 33.7 33.2 
H6 76.4 64.5 55.5 60.9 60.3 
H8 120.6 98.7 81.0 91.9 90.9 
H10 169.6 135.5 107.4 123.7 123.4 
H12 221.8 173.7 134.3 158.0 156.9 
H14 275.9 212.9 161.4 190.8 191.1 
H16 331.2 252.5 188.6 226.4 225.5 
H18 387.3 292.4 215.9 259.4 260.3 
H20 444.0 332.6 243.3 295.8 295.1 
H24 558.1 413.2 298.1 365.4 365.0 
H28 672.9 494.1 353.1 435.3 435.1 
H32 788.1 575.2 408.1 505.2 505.3 
H36 903.6 656.4 463.2 575.3 575.6 
H40 1019.2 737.6 518.4 645.3 646.0 
H2NxLDATASKPoCEELEXX
H2 13.2 12.1 12.1 12.0 12.0 
H4 39.6 34.7 31.9 33.7 33.2 
H6 76.4 64.5 55.5 60.9 60.3 
H8 120.6 98.7 81.0 91.9 90.9 
H10 169.6 135.5 107.4 123.7 123.4 
H12 221.8 173.7 134.3 158.0 156.9 
H14 275.9 212.9 161.4 190.8 191.1 
H16 331.2 252.5 188.6 226.4 225.5 
H18 387.3 292.4 215.9 259.4 260.3 
H20 444.0 332.6 243.3 295.8 295.1 
H24 558.1 413.2 298.1 365.4 365.0 
H28 672.9 494.1 353.1 435.3 435.1 
H32 788.1 575.2 408.1 505.2 505.3 
H36 903.6 656.4 463.2 575.3 575.6 
H40 1019.2 737.6 518.4 645.3 646.0 

The rightmost column of Table I shows the exact exchange results, which are the reference numbers for our present purposes. The leftmost column shows the results from the local exchange, and these highlight the well-known problem of the local approximation: The polarizability is systematically overestimated, and the deviation has a pronounced size dependence: Errors increase significantly with increasing system size and reach almost 60% for H40. TASK, which is constructed to fulfill the gradient expansion and many other exact constraints, i.e., follows universal design criteria for overall accuracy,89 considerably improves over the LDA and yields much more realistic polarizabilities than other meta-GGAs.49 However, for large chain lengths, the deviation goes up to 14%, i.e., the difference with respect to exact exchange becomes noticeable. The PoC functional, which was presented in Ref. 49 as a very simple proof-of-concept with a focus on only the derivative discontinuity, shows deviations of up to 19% but of the opposite sign, i.e., it yields polarizabilities that are significantly too low. The EEL functional finally yields polarizabilities in very close agreement with exact exchange.

The trends in these results become yet clearer when one visualizes the data by plotting the relative polarizability with respect to exact exchange as a function of system size, as shown in Fig. 3. This shows that the smallest systems are reasonably well described by all exchange approximations, except for LDA, which shows a 10% deviation already for H2. However, the dependence on the system size is very different for the different functionals. LDA deviations increase significantly with system size and have not yet saturated at H40, the largest hydrogen system in our study. This size dependence is a hallmark sign of missing the required ultranonlocality.19,23 TASK also shows an increasing deviation, but the deviation saturates at about 15% already around H24. The simple PoC functional shows both a larger deviation with the opposite sign and a slower convergence of the deviation. The EEL functional, finally, stays very close to the exact exchange values for all system sizes, i.e., yields an almost ideal horizontal line at 1.00. This demonstrates that a meta-GGA can capture ultranonlocality in hydrogen chains in very close, quantitative agreement with exact exchange, despite being constructed from semilocal functional ingredients and without using non-local Fock exchange integrals.

FIG. 3.

Static electric longitudinal dipole polarizability from density functional exchange approximations divided by the polarizability obtained with exact exchange for different meta-GGAs and LDA as labeled in the figure. The discrete symbols denote the calculated data, while the continuous lines are fits that just serve as a guide to the eye.

FIG. 3.

Static electric longitudinal dipole polarizability from density functional exchange approximations divided by the polarizability obtained with exact exchange for different meta-GGAs and LDA as labeled in the figure. The discrete symbols denote the calculated data, while the continuous lines are fits that just serve as a guide to the eye.

Close modal

This is a truly encouraging result. However, the hydrogen atom was one of the guiderails in the construction of the EEL functional and one may, therefore, argue that it is to some extent “natural” that the EEL functional describes hydrogen chains well. Checking whether the EEL functional yields a reasonable response for other extended systems is, therefore, a relevant second test. Polyacetylene, or more precisely, C2NH2N+2 oligomers of increasing length, provide for a second established and challenging test of response properties. Figure 4 clarifies the geometry that we based our calculations on and which we chose as in earlier studies18,36 for ease of comparison of the results. Table II shows the longitudinal static electric dipole polarizabilities that one obtains for such acetylene oligomers. It is interesting to note that here, all functionals, including the PoC, yield larger polarizabilities than exact exchange. In order to clarify the trends, we again plot the polarizability with respect to the one from exact exchange as a function of the system size. Figure 5 reveals that the situation for polyacetylene differs from the one for the hydrogen chains in several respects. First, the trend with the system size is less uniform for all functionals, even the LDA. As a consequence, one cannot safely determine a saturation value for the deviation of any of the functionals, and we thus also refrain from showing fits. The data points themselves, however, still reveal the trends clearly enough. The LDA again shows the largest deviations, up to ∼35%, and also the steepest slope in the relative deviation. The PoC functional shows relatively large deviations for the small oligomers but has a negative slope in the relative deviation, i.e., the relative deviation decreases for the larger systems. For most systems, PoC yields deviations of about 20%. The deviation of the TASK functional increases with increasing system size, but at a lesser rate than LDA, and the deviations are between 9% and 19%. The best results are again obtained with the EEL functional: for most systems, the deviations are around 5% and the largest deviation is about 10%. It is also interesting to observe that for smaller oligomers, there seems to be very little variation in the deviation with the system size, which indicates that the EEL functional captures the size dependence of the response in great similarity to exact exchange. For large oligomers, there seems to be some increase in the deviation, although the situation is not uniform as C24H26 shows a relatively small deviation of 7%.

FIG. 4.

Sketch of the geometry18,36 that the polyacetylene calculations are based on; see the main text for discussion.

FIG. 4.

Sketch of the geometry18,36 that the polyacetylene calculations are based on; see the main text for discussion.

Close modal
TABLE II.

Static electric longitudinal dipole polarizabilities in a03 for the C2NH2N+2 oligomers of polyacetylene for different exchange energy functionals. DFT calculations were performed self-consistently with the potentials of all orbital-dependent functionals evaluated in the KLI approximation.44 

C2NH2N+2xLDATASKPoCEELEXX
C4H6 91 85 95 81 77 
C6H8 179 167 189 160 153 
C8H10 301 280 307 265 254 
C10H12 460 426 467 406 385 
C12H14 656 606 658 566 539 
C14H16 889 819 860 765 721 
C16H18 1158 1056 1103 990 917 
C20H22 1798 1621 1645 1509 1367 
C24H26 2564 2312 2282 2111 1970 
C28H30 3439 3071 2977 2824 2570 
C2NH2N+2xLDATASKPoCEELEXX
C4H6 91 85 95 81 77 
C6H8 179 167 189 160 153 
C8H10 301 280 307 265 254 
C10H12 460 426 467 406 385 
C12H14 656 606 658 566 539 
C14H16 889 819 860 765 721 
C16H18 1158 1056 1103 990 917 
C20H22 1798 1621 1645 1509 1367 
C24H26 2564 2312 2282 2111 1970 
C28H30 3439 3071 2977 2824 2570 
FIG. 5.

Longitudinal polarizabilities for the C2NH2N+2 oligomers of polyacetylene for different exchange energy functionals relative to the polarizability found with EXX.

FIG. 5.

Longitudinal polarizabilities for the C2NH2N+2 oligomers of polyacetylene for different exchange energy functionals relative to the polarizability found with EXX.

Close modal

These findings might appear somewhat unsystematic at first sight, but they can be understood when one analyzes the response potential in the way suggested by Ref. 19, i.e., by plotting the difference between the response potential with an external applied dipole field and without. This analysis is shown for the largest system in our study, C28H30, in Fig. 6. The full black line shows the exact exchange response, and the dashed line shows the constant potential of the polarizing dipole field. One clearly sees the field-counteracting effect of exact exchange: The response potential has an overall slope opposite to the slope of the external polarizing field. It is well known19,24,26 that functionals such as LDA and GGAs miss this field-counteracting term completely and, therefore, their polarizabilities are too large. Figure 6 also shows the response obtained from the three meta-GGAs, and this leads to an explanation of the above discussed findings.

FIG. 6.

Full lines: plots of the difference of the response potentials with and without an externally applied field19 for the molecule C28H30 for exact exchange as the reference and the meta-GGAs EEL, PoC, and TASK, as labeled. The dashed line indicates the potential associated with the polarizing uniform electric field. See the text for discussion.

FIG. 6.

Full lines: plots of the difference of the response potentials with and without an externally applied field19 for the molecule C28H30 for exact exchange as the reference and the meta-GGAs EEL, PoC, and TASK, as labeled. The dashed line indicates the potential associated with the polarizing uniform electric field. See the text for discussion.

Close modal

The first striking observation is that the PoC functional is more strongly field-counteracting than the exact exchange itself, as seen by the higher value of response potential on the far left and the lower value on the far right. However, along the chain, the PoC potential shows much more pronounced up- and down-spikes than exact exchange. To connect these observations to the above reported findings, one has to keep in mind that the response of an extended molecular system is strongly influenced by two different effects: On the one hand, the field-counteracting terms are important, and their relative importance increases with increasing system size. On the other hand, the shape of the potential itself, i.e., the wells and barriers that the Kohn–Sham potential features between the different segments of the molecular chain, also have a pronounced influence. If these wells and barriers are not of the correct height, then an externally applied field can shift electron density along the molecular backbone too easily or too hard, and this, too, leads to errors in the polarizability. This second effect, however, does not depend strongly on the system size, as the potential structure is similar in every repeat unit of the polymer. The pronounced spikes that one sees in Fig. 6 in the response potential of the PoC functional show that the potential structure differs pronouncedly from the one of exact exchange. Thus, the PoC functional gives a wrong response, yet the deviation decreases for larger systems because for the latter, the relative importance of the field-counteracting terms increases, and these are overestimated by PoC. The decrease in the relative deviation that is observed for PoC in Fig. 5 is thus a consequence of a lucky error cancellation of the two different effects for intermediate oligomer lengths. For systems longer than the ones that we study here, one thus expects from the PoC functional an underestimation of the relative polarizability in analogy to the trend that was observed for the hydrogen chains.

Looking at the potential structure of the TASK functional in Fig. 6 shows that it is much more similar to the exact exchange than the one of PoC. The field counteracting term, however, is underestimated: One clearly sees that there is a field-counteracting effect, i.e., the existence of the ultranonlocality effect is confirmed also for polyacetylene. The counteracting slope, however, is somewhat too small. This explains why TASK overestimates the polarizabilities of polyacetylene.

Finally, looking at the plot for the EEL functional shows that its response potential structure is also quite similar to exact exchange. Furthermore, it shows a significant field-counteracting term. The slope does not fully reach the one of exact exchange, but it is more pronounced than the one found with TASK and can thus keep the relative errors below 10% for all system sizes. These observations are in line with Table II and Fig. 5: They explain the relative accuracy of the different meta-GGAs and, more importantly, demonstrate that while the quantitative accuracy of the EEL functional is smaller for the polarizability of polyacetylene than for the one of hydrogen chains, the pronounced increase in the deviation with the system size is strongly reduced. This confirms that meta-GGAs can indeed capture the ultranonlocal response in great similarity to exact exchange.

The above results demonstrate that ultranonlocality of the Kohn–Sham exchange potential can be reached with meta-GGAs. This finding has the potential for a significant impact: Meta-GGAs offer the hope that material research in which a proper description of the derivative discontinuity and field-counteracting terms is decisive, e.g., because band gaps or long-range charge transfer and the electric response play an important role, can be done based on meta-GGAs. Up until now, such studies have usually had to resort to the different variants of hybrid functionals. Hybrids increase the computational cost enormously. This often limits studies to small, simplified models and prevents calculations that correspond to experimental reality. The much lower computational cost of meta-GGAs54 can enable calculations on larger scales, both in space and in time, and can thus bring first-principles simulations closer to material research reality.

Whether meta-GGAs can live up to this promise will, however, depend on further serious research efforts. Our construction of the EEL functional, which took into account just two paradigm systems, the hydrogen atom and the homogenous electron gas, and one construction principle, the negative slope of Fx in α, showed that a reasonable accuracy was obtained for polyacetylene, while extremely accurate results were obtained for the hydrogen chain response. The accuracy for the hydrogen chains is plausible in view of the hydrogen atom being taken into account in the functional construction. Following this logic, it seems reasonable to assume that a generally accurate functional needs to be constructed by taking into account more of the physics of, for example, covalent binding, which is important in molecules and solids. Thus, the challenge is to construct more general functionals that yield a sizeable derivative discontinuity and ultranonlocal response while at the same time yielding accurate binding energies.

There is hope that this will be possible because, so far, we have taken Eq. (10) literally: Functionals such as TASK and EEL fulfill this condition for all values of s and α, as visualized for the EEL functional in Fig. 7. This is sufficient for guaranteeing a positive derivative discontinuity. However, while Eq. (10) is a sufficient condition, it is not a necessary one. Equation (9) shows that violating Eq. (10) for some regions of s and α is acceptable as long as Eq. (9) is fulfilled for the energetically important region. The yet more general Eq. (7) widens the perspective further: Depending on which values of s and α are probed by a given system’s electronic structure, the integral of Eq. (7) can be positive even if the integrand locally takes on negative values in some regions of space. Thus, there is flexibility in constructing functionals with a pronounced ultranonlocality that has not yet been systematically explored in functionals such as TASK and EEL.

FIG. 7.

Visualization of FxEEL(s,α) as a three-dimensional contour. One can thus see that Eq. (10) is fulfilled at all points in (s, α)-space.

FIG. 7.

Visualization of FxEEL(s,α) as a three-dimensional contour. One can thus see that Eq. (10) is fulfilled at all points in (s, α)-space.

Close modal

These ideas are in line with Ref. 90, which argues that enforcing conditions locally to guarantee satisfaction of global bounds is typically excessive. Some guidance on which values of s are relevant for, for example, atomization energies or transition state barrier heights, is available from earlier studies.91,92 For α, and yet more for the combination of the variables, the situation is more complex. However, some steps to identify decisive parameter regions have recently been taken, e.g., by demonstrating the importance of the semicore region for bond lengths in solids,93 by analyzing which combinations of n, s, and α are important for determining the band gaps of solids,54 and by studying which part of the vast parameter space is probed by the densities of real Coulomb systems and the constraints that are relevant there.90 It is also clear that exploiting the freedom to locally deviate from Eq. (10) on the one hand, while, on the other hand, respecting the requirement of a negative derivative of Fx in α in a way that overall guarantees a positive derivative discontinuity, might require new ways of thinking about functional construction. In this light, it is reassuring to observe that promising new construction principles for meta-GGAs keep emerging.94–97 

Based on the demonstration that meta-GGAs, without doubt, can reach ultranonlocality, which has been given in this paper, there is thus hope that semi-local functionals can shed off some of their traditional qualitative shortcomings while retaining their computational efficiency. However, one must also be aware that meta-GGAs do not feature conventional nonlocality as in the Hartree-term. This puts some limits on what can be reached within the meta-GGA form. It seems unlikely, for example, that a meta-GGA will generally be able to meet the straight-line condition for the energy as a function of particle number.69,98 Furthermore, while meta-GGAs can easily be made self-correlation-free for one-electron densities by using iso-orbital indicators, the missing nonlocality prevents them from being generally free from one-electron self-interaction. The static charge transfer that we studied here is not too sensitive to these limitations. However, the limitations are more strongly seen in recent time-dependent DFT calculations based on meta-GGAs:81 While incorporating the derivative discontinuity does improve charge-transfer excitations to some extent, quantitative accuracy would require further improvements in the potential, e.g., to yield more strongly bound eigenvalues and the proper long-range asymptotics. It might, therefore, be a promising route for future functional development to combine the ultranonlocality that meta-GGAs naturally incorporate with the nonlocality that a one-electron self-interaction correction, e.g., of the Perdew–Zunger type,47 can bring.

We gratefully acknowledge the financial support from the Deutsche Forschungsgemeinschaft, DFG Project Grant No. KU 1410/4-1, from the Elite Study Program “Biological Physics” of the Elite Network of Bavaria, and from the Bavarian State Ministry of Science, Research, and the Arts for the Collaborative Research Network “Solar Technologies go Hybrid.” M.B. acknowledges the support from the “Studienstiftung des Deutschen Volkes.”

The authors have no conflicts to disclose.

T.A. wrote the routines for the EEL functional, did all the calculations, prepared first versions of most figures, developed the functional construction, and contributed to the text. T.L., M.B., and R.R. participated in discussions about functional construction. I.S. helped with the numerical implementation. S.K. conceptualized the work, prepared final versions of the figures, and wrote the manuscript. All authors discussed the results and read and discussed the manuscript.

Thilo Aschebrock: Conceptualization (supporting); Formal analysis (lead); Investigation (equal); Methodology (equal); Software (lead); Validation (lead); Visualization (lead); Writing – original draft (supporting); Writing – review & editing (equal). Timo Lebeda: Investigation (supporting); Writing – review & editing (supporting). Moritz Brütting: Investigation (supporting); Writing –review & editing (supporting). Rian Richter: Investigation (supporting); Writing – review & editing (supporting). Ingo Schelter: Software (supporting); Supervision (supporting); Writing – review & editing (supporting). Stephan Kümmel: Conceptualization (lead); Funding acquisition (lead); Investigation (equal); Project administration (lead); Resources (lead); Supervision (lead); Visualization (supporting); Writing – original draft (lead).

The data that support the findings of this study are available from the corresponding author upon reasonable request.

1.
J. P.
Perdew
and
S.
Kurth
, “
Density functionals for non-relativistic Coulomb systems in the new century
,” in
A Primer in Density Functional Theory
,
Lecture Notes in Physics
(
Springer Verlag Berlin Heidelberg, New York
,
2003
), Vol.
620
, pp.
1
55
.
2.
K.
Burke
, “
Perspective on density functional theory
,”
J. Chem. Phys.
136
,
150901
(
2012
).
3.
N.
Mardirossian
and
M.
Head-Gordon
, “
Thirty years of density functional theory in computational chemistry: An overview and extensive assessment of 200 density functionals
,”
Mol. Phys.
115
,
2315
2372
(
2017
).
4.
A. M.
Teale
,
T.
Helgaker
,
A.
Savin
,
C.
Adamo
,
B.
Aradi
,
A. V.
Arbuznikov
,
P. W.
Ayers
,
E. J.
Baerends
,
V.
Barone
,
P.
Calaminici
,
E.
Cancès
,
E. A.
Carter
,
P. K.
Chattaraj
,
H.
Chermette
,
I.
Ciofini
,
T. D.
Crawford
,
F.
De Proft
,
J. F.
Dobson
,
C.
Draxl
,
T.
Frauenheim
,
E.
Fromager
,
P.
Fuentealba
,
L.
Gagliardi
,
G.
Galli
,
J.
Gao
,
P.
Geerlings
,
N.
Gidopoulos
,
P. M. W.
Gill
,
P.
Gori-Giorgi
,
A.
Görling
,
T.
Gould
,
S.
Grimme
,
O.
Gritsenko
,
H. J. A.
Jensen
,
E. R.
Johnson
,
R. O.
Jones
,
M.
Kaupp
,
A. M.
Köster
,
L.
Kronik
,
A. I.
Krylov
,
S.
Kvaal
,
A.
Laestadius
,
M.
Levy
,
M.
Lewin
,
S.
Liu
,
P.-F.
Loos
,
N. T.
Maitra
,
F.
Neese
,
J. P.
Perdew
,
K.
Pernal
,
P.
Pernot
,
P.
Piecuch
,
E.
Rebolini
,
L.
Reining
,
P.
Romaniello
,
A.
Ruzsinszky
,
D. R.
Salahub
,
M.
Scheffler
,
P.
Schwerdtfeger
,
V. N.
Staroverov
,
J.
Sun
,
E.
Tellgren
,
D. J.
Tozer
,
S. B.
Trickey
,
C. A.
Ullrich
,
A.
Vela
,
G.
Vignale
,
T. A.
Wesolowski
,
X.
Xu
, and
W.
Yang
, “
DFT exchange: Sharing perspectives on the workhorse of quantum chemistry and materials science
,”
Phys. Chem. Chem. Phys.
24
,
28700
28781
(
2022
).
5.
J. P.
Perdew
,
K.
Burke
, and
M.
Ernzerhof
, “
Generalized gradient approximation made simple
,”
Phys. Rev. Lett.
77
,
3865
3868
(
1996
).
6.
J.
Tao
,
J. P.
Perdew
,
V. N.
Staroverov
, and
G. E.
Scuseria
, “
Climbing the density functional ladder: Nonempirical meta–generalized gradient approximation designed for molecules and solids
,”
Phys. Rev. Lett.
91
,
146401
(
2003
).
7.
J.
Sun
,
A.
Ruzsinszky
, and
J. P.
Perdew
, “
Strongly constrained and appropriately normed semilocal density functional
,”
Phys. Rev. Lett.
115
,
036402
(
2015
).
8.
X.
Gonze
,
P.
Ghosez
, and
R. W.
Godby
, “
Density-polarization functional theory of the response of a periodic insulating solid to an electric field
,”
Phys. Rev. Lett.
74
,
4035
4038
(
1995
).
9.
R.
Resta
, “
Density-polarization-functional theory and long-range correlation in dielectrics
,”
Phys. Rev. Lett.
77
,
2265
2267
(
1996
).
10.
R. M.
Martin
and
G.
Ortiz
, “
Comment on ‘density-polarization-functional theory and long-range correlation in dielectrics’
,”
Phys. Rev. Lett.
78
,
2028
(
1997
).
11.
R. M.
Martin
and
G.
Ortiz
, “
Functional theory of extended Coulomb systems
,”
Phys. Rev. B
56
,
1124
1140
(
1997
).
12.
X.
Gonze
,
P.
Ghosez
, and
R. W.
Godby
, “
Polarization dependence of the exchange energy
,”
Phys. Rev. Lett.
78
,
2029
(
1997
).
13.
X.
Gonze
,
P.
Ghosez
, and
R. W.
Godby
, “
Density-functional theory of polar insulators
,”
Phys. Rev. Lett.
78
,
294
297
(
1997
).
14.
P.
Ghosez
,
X.
Gonze
, and
R. W.
Godby
, “
Long-wavelength behavior of the exchange-correlation kernel in the Kohn-Sham theory of periodic systems
,”
Phys. Rev. B
56
,
12811
(
1997
).
15.
R.
Resta
, “
Resta replies:
,”
Phys. Rev. Lett.
78
,
2030
(
1997
).
16.
D.
Vanderbilt
, “
Nonlocality of Kohn-Sham exchange-correlation fields in dielectrics
,”
Phys. Rev. Lett.
79
,
3966
3969
(
1997
).
17.
G.
Ortiz
,
I.
Souza
, and
R. M.
Martin
, “
Exchange-correlation hole in polarized insulators: Implications for the microscopic functional theory of dielectrics
,”
Phys. Rev. Lett.
80
,
353
356
(
1998
).
18.
B.
Champagne
,
E. A.
Perpète
,
S. J. A.
van Gisbergen
,
E.-J.
Baerends
,
J. G.
Snijders
,
C.
Soubra-Ghaoui
,
K. A.
Robins
, and
B.
Kirtman
, “
Assessment of conventional density functional schemes for computing the polarizabilities and hyperpolarizabilities of conjugated oligomers: An ab initio investigation of polyacetylene chains
,”
J. Chem. Phys.
109
,
10489
(
1998
).
19.
S. J. A.
van Gisbergen
,
P. R. T.
Schipper
,
O. V.
Gritsenko
,
E. J.
Baerends
,
J. G.
Snijders
,
B.
Champagne
, and
B.
Kirtman
, “
Electric field dependence of the exchange-correlation potential in molecular chains
,”
Phys. Rev. Lett.
83
,
694
(
1999
).
20.
D.
Tozer
, “
Relationship between long-range charge-transfer excitation energy error and integer discontinuity in Kohn–Sham theory
,”
J. Chem. Phys.
119
,
12697
12699
(
2003
).
21.
A.
Dreuw
and
M.
Head-Gordon
, “
Failure of time-dependent density functional theory for long-range charge-transfer excited States: The zincbacteriochlorin−bacteriochlorin and bacteriochlorophyll−spheroidene complexes
,”
J. Am. Chem. Soc.
126
,
4007
(
2004
).
22.
M.
Grüning
,
O. V.
Gritsenko
, and
E. J.
Baerends
, “
Exchange potential from the common energy denominator approximation for the Kohn–Sham Green’s function: Application to (hyper)polarizabilities of molecular chains
,”
J. Chem. Phys.
116
,
6435
(
2002
).
23.
M.
van Faassen
,
P. L.
de Boeij
,
R.
van Leeuwen
,
J. A.
Berger
, and
J. G.
Snijders
, “
Ultranonlocality in time-dependent current-density-functional theory: Application to conjugated polymers
,”
Phys. Rev. Lett.
88
,
186401
(
2002
).
24.
P.
Mori-Sánchez
,
Q.
Wu
, and
W.
Yang
, “
Accurate polymer polarizabilities with exact exchange density-functional theory
,”
J. Chem. Phys.
119
,
11001
11004
(
2003
).
25.
S.
Kümmel
, “
Damped gradient iteration and multigrid relaxation: Tools for electronic structure calculations using orbital density-functionals
,”
J. Comput. Phys.
201
,
333
343
(
2004
).
26.
S.
Kümmel
,
L.
Kronik
, and
J. P.
Perdew
, “
Electrical response of molecular chains from density functional theory
,”
Phys. Rev. Lett.
93
,
213002
(
2004
).
27.
P.
Umari
,
A. J.
Willamson
,
G.
Galli
, and
N.
Marzari
, “
Dielectric response of periodic systems from quantum Monte Carlo calculations
,”
Phys. Rev. Lett.
95
,
207602
(
2005
).
28.
S.
Kümmel
and
L.
Kronik
, “
Hyperpolarizabilities of molecular chains: A real-space approach
,”
Comput. Mater. Sci.
35
,
321
326
(
2006
).
29.
H.
Sekino
,
Y.
Maeda
,
M.
Kamiya
, and
K.
Hirao
, “
Polarizability and second hyperpolarizability evaluation of long molecules by the density functional theory with long-range correction
,”
J. Chem. Phys.
126
,
014107
(
2007
).
30.
N.
Maitra
and
M.
van Faassen
, “
Improved exchange-correlation potential for polarizability and dissociation in density functional theory
,”
J. Chem. Phys.
126
,
191106
(
2007
).
31.
R.
Armiento
,
S.
Kümmel
, and
T.
Körzdörfer
, “
Electrical response of molecular chains in density functional theory: Ultranonlocal response from a semilocal functional
,”
Phys. Rev. B
77
,
165106
(
2008
).
32.
C. D.
Pemmaraju
,
S.
Sanvito
, and
K.
Burke
, “
Polarizability of molecular chains: A self-interaction correction approach
,”
Phys. Rev. B
77
,
121204
(
2008
).
33.
A.
Ruzsinszky
,
J. P.
Perdew
,
G. I.
Csonka
,
G. E.
Scuseria
, and
O. A.
Vydrov
, “
Understanding and correcting the self-interaction error in the electrical response of hydrogen chains
,”
Phys. Rev. A
77
,
060502
(
2008
).
34.
A.
Ruzsinszky
,
J. P.
Perdew
, and
G. I.
Csonka
, “
Simple charge-transfer model to explain the electrical response of hydrogen chains
,”
Phys. Rev. A
78
,
022513
(
2008
).
35.
T.
Körzdörfer
,
M.
Mundt
, and
S.
Kümmel
, “
Electrical response of molecular systems: The power of self-interaction corrected Kohn-Sham theory
,”
Phys. Rev. Lett.
100
,
133004
(
2008
).
36.
A.
Karolewski
,
R.
Armiento
, and
S.
Kümmel
, “
Polarizabilities of polyacetylene from a field-counteracting semilocal functional
,”
J. Chem. Theory Comput.
5
,
712
718
(
2009
).
37.
B.
Champagne
and
B.
Kirtman
, “
Polarizabilities and second hyperpolarizabilities of hydrogen chains using the spin-component-scaled Møller–Plesset second-order method
,”
Int. J. Quantum Chem.
109
,
3103
(
2009
).
38.
A.
Heßelmann
, “
Polarisabilities of long conjugated chain molecules with density functional response methods: The role of coupled and uncoupled response
,”
J. Chem. Phys.
142
,
164102
(
2015
).
39.
A.
Kaiser
and
S.
Kümmel
, “
Revealing the field-counteracting term in the exact Kohn-Sham correlation potential
,”
Phys. Rev. A
98
,
052505
(
2018
).
40.
Y.
Mei
,
N.
Yang
, and
W.
Yang
, “
Describing polymer polarizability with localized orbital scaling correction in density functional theory
,”
J. Chem. Phys.
154
,
054302
(
2021
).
41.
S.
Akter
,
J. A.
Vargas
,
K.
Sharkas
,
J. E.
Peralta
,
K. A.
Jackson
,
T.
Baruah
, and
R. R.
Zope
, “
How well do self-interaction corrections repair the overestimation of static polarizabilities in density functional calculations?
,”
Phys. Chem. Chem. Phys.
23
,
18678
18685
(
2021
).
42.
M.
Hellgren
and
L.
Baguet
, “
Strengths and limitations of the adiabatic exact-exchange kernel for total energy calculations
,”
J. Chem. Phys.
158
,
184107
(
2023
).
43.

To avoid confusion, one should realize that the term “nonlocal” is also used in other contexts of DFT with other meanings, e.g., in order to denote a non-multiplicative potential, or referring to the temporal dependence in time-dependent DFT.

44.
J. B.
Krieger
,
Y.
Li
, and
G. J.
Iafrate
, “
Construction and application of an accurate local spin-polarized Kohn-Sham potential with integer discontinuity: Exchange-only theory
,”
Phys. Rev. A
45
,
101
126
(
1992
).
45.
M.
Lein
and
S.
Kümmel
, “
Exact time-dependent exchange-correlation potentials for strong-field electron dynamics
,”
Phys. Rev. Lett.
94
,
143003
(
2005
).
46.
C.
Legrand
,
E.
Suraud
, and
P.-G.
Reinhard
, “
Comparison of self-interaction-corrections for metal clusters
,”
J. Phys. B: At. Mol. Opt. Phys.
35
,
1115
1128
(
2002
).
47.
J. P.
Perdew
and
A.
Zunger
, “
Self-interaction correction to density-functional approximations for many-electron systems
,”
Phys. Rev. B
23
,
5048
5079
(
1981
).
48.
J.
Kehrer
,
R.
Richter
,
J. M.
Foerster
,
I.
Schelter
, and
S.
Kümmel
, “
Self-interaction correction, electrostatic, and structural influences on time-dependent density functional theory excitations of bacteriochlorophylls from the light-harvesting complex 2
,”
J. Chem. Phys.
153
,
144114
(
2020
).
49.
T.
Aschebrock
and
S.
Kümmel
, “
Ultranonlocality and accurate band gaps from a meta-generalized gradient approximation
,”
Phys. Rev. Res.
1
,
033082
(
2019
).
50.
F.
Della Sala
,
E.
Fabiano
, and
L. A.
Constantin
, “
Kinetic-energy-density dependent semilocal exchange-correlation functionals
,”
Int. J. Quantum Chem.
116
,
1641
1694
(
2016
).
51.
A. D.
Becke
, “
Simulation of delocalized exchange by local density functionals
,”
J. Chem. Phys.
112
,
4020
4026
(
2000
).
52.
F. G.
Eich
and
M.
Hellgren
, “
Derivative discontinuity and exchange-correlation potential of meta-GGAs in density-functional theory
,”
J. Chem. Phys.
141
,
224107
(
2014
).
53.
R. M.
Dreizler
and
E. K. U.
Gross
,
Density Functional Theory: An Approach to the Quantum Many-Body Problem
(
Springer
,
Berlin
,
1990
).
54.
T.
Lebeda
,
T.
Aschebrock
,
J.
Sun
,
L.
Leppert
, and
S.
Kümmel
, “
Right band gaps for the right reason at low computational cost with a meta-GGA
,”
Phys. Rev. Mater.
7
,
093803
(
2023
).
55.
J. P.
Perdew
,
S.
Kurth
,
A.
Zupan
, and
P.
Blaha
,
Phys. Rev. Lett.
82
,
2544
(
1999
).
56.
M.
Ernzerhof
and
G. E.
Scuseria
, “
Kinetic energy density dependent approximations to the exchange energy
,”
J. Chem. Phys.
111
,
911
915
(
1999
).
57.
Y.
Zhao
and
D. G.
Truhlar
, “
A new local density functional for main-group thermochemistry, transition metal bonding, thermochemical kinetics, and noncovalent interactions
,”
J. Chem. Phys.
125
,
194101
(
2006
).
58.
R.
Peverati
and
D. G.
Truhlar
, “
M11-L: A local density functional that provides improved accuracy for electronic structure calculations in chemistry and physics
,”
J. Phys. Chem. Lett.
3
,
117
124
(
2012
).
59.
H. S.
Yu
,
X.
He
, and
D. G.
Truhlar
, “
MN15-L: A new local exchange-correlation functional for Kohn–Sham density functional theory with broad accuracy for atoms, molecules, and solids
,”
J. Chem. Theory Comput.
12
,
1280
1293
(
2016
).
60.
P.-F.
Loos
, “
Exchange functionals based on finite uniform electron gases
,”
J. Chem. Phys.
146
,
114108
(
2017
).
61.
V. U.
Nazarov
and
G.
Vignale
, “
Optics of semiconductors from meta-generalized-gradient-approximation-based time-dependent density-functional theory
,”
Phys. Rev. Lett.
107
,
216402
(
2011
).
62.
P.
Borlido
,
J.
Schmidt
,
A.
Huran
,
F.
Tran
,
M.
Marques
, and
S.
Botti
, “
Exchange-correlation functionals for band gaps of solids: Benchmark, reparametrization and machine learning
,”
npj Comput. Mater.
6
,
96
(
2020
).
63.
A. D.
Boese
and
N. C.
Handy
, “
New exchange-correlation density functionals: The role of the kinetic-energy density
,”
J. Chem. Phys.
116
,
9559
9569
(
2002
).
64.
J.
M del Campo
,
J. L.
Gàzquez
,
S.
Trickey
, and
A.
Vela
, “
A new meta-GGA exchange functional based on an improved constraint-based gga
,”
Chem. Phys. Lett.
543
,
179
183
(
2012
).
65.
J.
Tao
and
Y.
Mo
, “
Accurate semilocal density functional for condensed-matter physics and quantum chemistry
,”
Phys. Rev. Lett.
117
,
073001
(
2016
).
66.
L. A.
Constantin
,
E.
Fabiano
,
J. M.
Pitarke
, and
F.
Della Sala
, “
Semilocal density functional theory with correct surface asymptotics
,”
Phys. Rev. B
93
,
115127
(
2016
).
67.
L. A.
Constantin
,
E.
Fabiano
, and
F.
Della Sala
, “
Meta-GGA exchange-correlation functional with a balanced treatment of nonlocality
,”
J. Chem. Theory Comput.
9
,
2256
2263
(
2013
).
68.
J. P.
Perdew
,
A.
Ruzsinszky
,
G. I.
Csonka
,
L. A.
Constantin
, and
J.
Sun
, “
Workhorse semilocal density functional for condensed matter physics and quantum chemistry
,”
Phys. Rev. Lett.
103
,
026403
(
2009
).
69.
J. P.
Perdew
,
R. G.
Parr
,
M.
Levy
, and
J. L.
Balduz
, Jr.
, “
Density-functional theory for fractional particle number: Derivative discontinuities of the energy
,”
Phys. Rev. Lett.
49
,
1691
1694
(
1982
).
70.
Y.
Wang
,
X.
Jin
,
H. S.
Yu
,
D. G.
Truhlar
, and
X.
He
, “
Revised M06-L functional for improved accuracy on chemical reaction barrier heights, noncovalent interactions, and solid-state physics
,”
Proc. Natl. Acad. Sci. U. S. A.
114
,
8487
8492
(
2017
).
71.
Z.
Yang
,
H.
Peng
,
J.
Sun
, and
J. P.
Perdew
, “
More realistic band gaps from meta-generalized gradient approximations: Only in a generalized Kohn-Sham scheme
,”
Phys. Rev. B
93
,
205205
(
2016
).
72.
J. P.
Perdew
,
W.
Yang
,
K.
Burke
,
Z.
Yang
,
E. K. U.
Gross
,
M.
Scheffler
,
G. E.
Scuseria
,
T. M.
Henderson
,
I. Y.
Zhang
,
A.
Ruzsinszky
,
H.
Peng
,
J.
Sun
,
E.
Trushin
, and
A.
Görling
, “
Understanding band gaps of solids in generalized Kohn–Sham theory
,”
Proc. Natl. Acad. Sci. U. S. A.
114
,
2801
2806
(
2017
).
73.
W.
Yang
,
A. J.
Cohen
, and
P.
Mori-Sanchez
, “
Derivative discontinuity, bandgap and lowest unoccupied molecular orbital in density functional theory
,”
J. Chem. Phys.
136
,
204111
(
2012
).
74.
T.
Van Voorhis
and
G. E.
Scuseria
, “
A novel form for the exchange-correlation energy functional
,”
J. Chem. Phys.
109
,
400
410
(
1998
).
75.
J.
Sun
,
B.
Xiao
, and
A.
Ruzsinszky
, “
Communication: Effect of the orbital-overlap dependence in the meta generalized gradient approximation
,”
J. Chem. Phys.
137
,
051101
(
2012
).
76.
J.
Sun
,
R.
Haunschild
,
B.
Xiao
,
I. W.
Bulik
,
G. E.
Scuseria
, and
J. P.
Perdew
, “
Semilocal and hybrid meta-generalized gradient approximations based on the understanding of the kinetic-energy-density dependence
,”
J. Chem. Phys.
138
,
044113
(
2013
).
77.
J.
Sun
,
J. P.
Perdew
, and
A.
Ruzsinszky
, “
Semilocal density functional obeying a strongly tightened bound for exchange
,”
Proc. Natl. Acad. Sci. U. S. A.
112
,
685
689
(
2015
).
78.
J.
Wellendorff
,
K. T.
Lundgaard
,
K. W.
Jacobsen
, and
T.
Bligaard
, “
mBEEF: An accurate semi-local Bayesian error estimation density functional
,”
J. Chem. Phys.
140
,
144107
(
2014
).
79.
S.
Lehtola
and
M. A.
Marques
, “
Many recent density functionals are numerically ill-behaved
,”
J. Chem. Phys.
157
,
174114
(
2022
).
80.
R.
Grotjahn
,
F.
Furche
, and
M.
Kaupp
, “
Importance of imposing gauge invariance in time-dependent density functional theory calculations with meta-generalized gradient approximations
,”
J. Chem. Phys.
157
,
111102
(
2022
).
81.
R.
Richter
,
T.
Aschebrock
,
I.
Schelter
, and
S.
Kümmel
, “
Meta-generalized gradient approximations in time dependent generalized Kohn–Sham theory: Importance of the current density correction
,”
J. Chem. Phys.
159
,
124117
(
2023
).
82.

We note that the B97M-V functional,99 in which exchange and correlation have been designed together and which, therefore, falls out of our present discussion that focuses on exchange only, also shows a continuous non-positive slope of appreciable magnitude.

83.
J. P.
Perdew
,
A.
Ruzsinszky
,
J.
Sun
, and
K.
Burke
, “
Gedanken densities and exact constraints in density functional theory
,”
J. Chem. Phys.
140
,
18A533
(
2014
).
84.
M.
Levy
, “
Density-functional exchange correlation through coordinate scaling in adiabatic connection and correlation hole
,”
Phys. Rev. A
43
,
4637
4646
(
1991
).
85.
L.
Pollack
and
J. P.
Perdew
, “
Evaluating density functional performance for the quasi-two-dimensional electron gas
,”
J. Phys.: Condens. Matter
12
,
1239
1252
(
2000
).
86.
J. P.
Perdew
,
J.
Tao
,
V. N.
Staroverov
, and
G. E.
Scuseria
, “
Meta-generalized gradient approximation: Explanation of a realistic nonempirical density functional
,”
J. Chem. Phys.
120
,
6898
(
2004
).
87.

The equivalence of t and 53s2 for any single-orbital system does not only hold for the energy but also for the Kohn–Sham potential, as the corresponding Kohn–Sham potential of the meta-GGA reduces to the GGA potential associated with the energy functional one obtains by replacing t with 53s2. In the generalized Kohn–Sham scheme, the equivalence is only guaranteed for the occupied orbitals.

88.

We checked that the results of Sec. IV are not very sensitive to the precise value of α0 as long as the two conditions are met; for example, almost identical results are obtained with α0 = 2.

89.
T.
Lebeda
,
T.
Aschebrock
, and
S.
Kümmel
, “
First steps towards achieving both ultranonlocality and a reliable description of electronic binding in a meta-generalized gradient approximation
,”
Phys. Rev. Res.
4
,
023061
(
2022
).
90.
R.
Pederson
and
K.
Burke
, “
The difference between molecules and materials: Reassessing the role of exact conditions in density functional theory
,” arXiv:2303.01766 (
2023
).
91.
A.
Zupan
,
K.
Burke
,
M.
Ernzerhof
, and
J. P.
Perdew
, “
Distributions and averages of electron density parameters: Explaining the effects of gradient corrections
,”
J. Chem. Phys.
106
,
10184
10193
(
1997
).
92.
J. P.
Perdew
,
M.
Ernzerhof
,
A.
Zupan
, and
K.
Burke
, “
Nonlocality of the density functional for exchange and correlation: Physical origins and chemical consequences
,”
J. Chem. Phys.
108
,
1522
1531
(
1998
).
93.
P.
Kovács
,
F.
Tran
,
P.
Blaha
, and
G. K. H.
Madsen
, “
Comparative study of the PBE and SCAN functionals: The particular case of alkali metals
,”
J. Chem. Phys.
150
,
164119
(
2019
).
94.
S.
Dick
and
M.
Fernandez-Serra
, “
Highly accurate and constrained density functional obtained with differentiable programming
,”
Phys. Rev. B
104
,
L161109
(
2021
).
95.
C. M.
Horowitz
,
C. R.
Proetto
, and
J. M.
Pitarke
, “
Towards a universal exchange enhancement factor in density functional theory
,”
Phys. Rev. B
107
,
195120
(
2023
).
96.
C. M.
Horowitz
,
C. R.
Proetto
, and
J. M.
Pitarke
, “
Construction of a semilocal exchange density functional from a three-dimensional electron gas collapsing to two dimensions
,”
Phys. Rev. B
108
,
115119
(
2023
).
97.
S.
Jana
,
L. A.
Constantin
, and
P.
Samal
, “
Density functional applications of jellium with a local gap model correlation energy functional
,”
J. Chem. Phys.
159
,
114109
(
2023
).
98.
P.
Mori-Sánchez
,
A. J.
Cohen
, and
W.
Yang
, “
Many-electron self-interaction error in approximate density functionals
,”
J. Chem. Phys.
125
,
201102
(
2006
).
99.
N.
Mardirossian
and
M.
Head-Gordon
, “
Mapping the genome of meta-generalized gradient approximation density functionals: The search for B97M-V
,”
J. Chem. Phys.
142
,
074111
(
2015
).