Accurate Total Energies from the Adiabatic-Connection Fluctuation-Dissipation Theorem

The origin of the ACFDT in the context of inhomogenous systems dates back around five decades to the series of articles given in Refs. [54, 55, 56]. Since then, a number of resources have covered the derivation of the ACFDT [57, 4, 5, 1, 28, 58, 59, 60], a brief review is given here. The adiabatic connection establishes a link between the interacting many-body system and its corresponding non-interacting Kohn-Sham system, ultimately leading to an alternate expression for the xc energy. Toward this end, a one-parameter family of many-body Hamiltonians is defined,

such that the deus ex machina potential [61] $\hat{v}_{\text{dxm}}(\lambda)$ is the unique [62] potential that ensures the ground-state density at all values of $\lambda\in[0,1]$ is equal to the ground-state electron density at $\lambda=1$, labeled $n(x)$ – this is the adiabatic connection. Hence, $\hat{v}_{\text{dxm}}(\lambda=1)=0$ and $\hat{v}_{\text{dxm}}(\lambda=0)=\hat{v}_{\text{H}}+\hat{v}_{\text{xc}}$ where $\hat{v}_{\text{H}}$ is the Hartree potential. The $\lambda$-interacting ground state of $H(\lambda)$ is denoted $|\Psi^{\lambda}\rangle$, allowing the total energy to be expressed as such,

where the latter two formulae constitute the conventional definition of the Kohn-Sham system [63], i.e. $T_{0}$ is the non-interacting kinetic energy, $E_{\text{H}}$ is the Hartree energy, $E_{\text{ext}}$ is the external energy, and $E_{\text{xc}}$ is the xc energy. Moreover, $|\Psi^{\lambda=0}\rangle$ represents the Kohn-Sham Slater determinant ground state.

Rearrangement of the above expressions yields an alternate form for the Hxc energy $E_{\text{Hxc}}=E_{\text{H}}+E_{\text{xc}}$,

which becomes, upon use of the fundamental theorem of calculus and the Hellmann-Feynmann theorem,

The expression in Eq. (3) contrasts with the conventional one Eq. (2) as it does not involve the kinetic operator at the seemingly steep price of having to know the xc potential energy [5],

at each value of $\lambda$ along the adiabatic connection ¹¹1The $\lambda$-dependent xc energy (and xc kernel) is zero when $\lambda=0$ because the particles are non-interacting..

However, knowledge of the challenging expectation value in Eq. (3) is tantamount to knowledge of the static (equal-time) two-point correlator $\langle\Psi^{\lambda}|\hat{n}(x)\hat{n}(x^{\prime})|\Psi^{\lambda}\rangle$, which describes quantum statistical fluctuations in the density inherent to the state $|\Psi^{\lambda}\rangle$ [65]. The fluctuation-dissipation theorem [66] provides a relationship between the response of a system to these spontaneous internal changes (fluctuations) in its density, and the response of that same system to external perturbations in its density. The latter is described with the density-density linear response function,

i.e. the first-order change in the density due to a perturbation in the external potential within a system of $\lambda$-interacting particles described by $H(\lambda)$ in Eq. (1). In the present context, the fluctuation-dissipation theorem takes the form [67]

thus connecting the ground-state xc energy with linear response theory.

The above derivation outlines an in principle exact reformulation of conventional Kohn-Sham DFT [5], meaning the total energy functional

has the correct minimum, i.e. the exact ground-state energy, and this minimum is attained at the interacting ground-state density. Having performed the rearrangements given from the fluctuation-dissipation theorem, the ACFDT xc energy functional $E_{\text{xc}}^{\text{ACFD}}[n]=E_{\text{x}}^{\text{ACFD}}[n]+E_{\text{c}}^{\text{ACFD}}[n]$ becomes

where $E_{\text{x}}[\{|\phi_{i}\rangle\}]$ is the exact-exchange functional evaluated at the Kohn-Sham orbitals $\{|\phi_{i}\rangle\}$, and $\chi_{0}[n]=\delta n/\delta v_{\text{KS}}$ is the associated response function of the Kohn-Sham system. These are the central expressions around which the remainder of this paper is based. Namely, we evaluate the total energy functional Eq. (6) with its exact definition, and then introduce approximations into the functional through $f^{\lambda}_{\text{xc}}[n]$ (i.e. $\chi^{\lambda}[n]$).

Linear response time-dependent DFT establishes a unique map [68, 69] from a tractable non-interacting (Kohn-Sham) response function $\chi_{0}[n]$ to an otherwise intractable $\lambda$-interacting response function $\chi^{\lambda}[n]$ through the xc kernel,

i.e. the first-order change in the $\lambda$-dependent xc potential due to a perturbation in the density oscillating in time with frequency $\omega$. This map constitutes the Dyson equation of linear response time-dependent DFT,

where $A*B=\int A(x,x^{\prime})B(x^{\prime},x^{\prime\prime})dx^{\prime}$, and $f_{\text{H}}=\delta v_{\text{H}}/\delta n$ is the Hartree kernel (the electron-electron interaction).

The xc kernel $f^{\lambda}_{\text{xc}}[n]$ is the central subject of approximation in linear response time-dependent DFT [1], and therefore the principal ingredients in an ACFDT total energy calculation are an approximation to the xc kernel functional $f^{\lambda}_{\text{xc}}[n]$ together with a prescription for determining the density $n$ at which to evaluate the ACFDT total energy functional $E^{\text{ACFD}}[n]$. In regard to the latter concern: there are two main approaches that are applied in practice to determine this density. The first is the most common, and is often referred to as a one-shot ACFDT calculation, wherein the density $n$ is obtained from a self-consistent solution of the ground-state Kohn-Sham equations with an approximate xc potential $v_{\text{xc}}[n]$ (see Ref. [2] for a recent review). The second, often referred to as a self-consistent ACFDT calculation [70, 16, 71, 36, 40, 72, 73], solves

where the equations that yield a stationary (presumed to be minimizing) density are known [70, 74, 16, 32]. In the instance that both $v_{\text{xc}}[n]$ and $f^{\lambda}_{\text{xc}}[n]$ are exact, the output of a one-shot and self-consistent ACFDT calculation coincide, however this is not true when approximations are involved, as we shall explore in Section III.

In practice, an approximate xc kernel functional $f^{\lambda}_{\text{xc}}[n]$ that is also parameterized with respect to $\lambda$ is not required as it is possible to exploit a curious relationship between ground-state wavefunctions along the adiabatic connection and ground-state wavefunctions with scaled spatial coordinates, see Refs. [52, 5]. This relationship permits us to specify a conventional functional $f_{\text{xc}}[n]\coloneqq f_{\text{xc}}^{\lambda=1}[n]$ and obtain its $\lambda$ dependence with little-to-no additional expense. Such an observation is central to practical ACFDT calculations, although we are unable to exploit it here due to using the softened Coulomb interaction.

This work involves finite systems in one dimension interacting with a softened Coulomb electron-electron interaction

where $\alpha$ is the softening parameter; $\alpha=1$ a.u. is used in this work. A real-space grid of dimension $N$ discretizes the spatial domain $[-L,L]$ subject to Dirichlet boundary conditions. We consider four prototype systems, each of which include two like-spin electrons ²²2Electrons of like spin obey the Pauli exclusion principle, and exhibit features that would need a larger number of spin-half electrons to become apparent, e.g. two like-spin electrons experience the exchange effect, which is not the case for two spin-half electrons in an $S=0$ state. in the external potentials described below.

The central complication when implementing the exact ACFDT total energy functional is evaluation of the ACFDT correlation functional Eq. (8), which we now proceed to elaborate, see also Fig. 1.

Having chosen some external potential $v_{\text{ext}}(x)$, the exact interacting density $n(x)$ is obtained through solution of the time-independent Schrödinger equation. The corresponding unique Kohn-Sham potential $v_{\text{KS}}(x)$ is then reverse-engineered by applying preconditioned root-finding techniques to an appropriate fixed-point map [76]. The Kohn-Sham orbitals and energies $\{|\phi_{i}\rangle,\varepsilon_{i}\}$ are used to construct the non-interacting response function $\chi_{0}$ along $i\omega$ in the Lehmann representation [1].

In order to calculate the final ingredient $\chi^{\lambda}[n]$, we first obtain the $\lambda$-dependent wavefunctions $\{|\Psi^{\lambda}_{i}\rangle\}$ along the adiabatic connection, as defined in Section II.1 – this grossly impractical step is performed for investigative reasons, and constitutes the primary computational expense. For each value of the coupling constant on some discrete grid, the potential $v_{\text{dxm}}(\lambda,x)$ is obtained by yet again using root-finding techniques to target the $\lambda=1$ interacting density, $n(x)$ (see supplemental material for an example $v_{\text{dxm}}(\lambda,x)$ ³³3URL to be inserted). The full set of $\lambda$-dependent wavefunctions and energies $\{|\Psi_{i}^{\lambda}\rangle,E_{i}^{\lambda}\}$ is then used to calculate the $\lambda$-interacting response functions in the Lehmann representation,

where $\Omega^{\lambda}_{n}=E^{\lambda}_{n}-E^{\lambda}_{0}$ is the $n-$th excitation energy of the $\lambda$-interacting Hamiltonian along the adiabatic connection.

At this stage it is possible to construct the exact $\lambda$-dependent xc kernel using the expression

which comes from inspection of the Dyson equation Eq. (10) (note that superscript ${-1}$ signifies the matrix inverse in a finite spatial basis). Construction of $f^{\lambda}_{\text{xc}}$ in this fashion is an intricate matter that has been dealt with in prior work [7]. For our purposes, it suffices to observe that in all the cases presented below the exact $\lambda$-dependent response function is reconstructed to within machine precision when the exact $f^{\lambda}_{\text{xc}}$ and exact $\chi_{0}$ are used to solve the Dyson equation. This step is critical as ultimately we shall isolate certain features of the exact $f_{\text{xc}}^{\lambda}$ in order to examine their impact on the correlation energy, for example utilizing only the $\omega=0$ component of $f^{\lambda}_{\text{xc}}$ here defines the AE approximation.

The final step toward obtaining $E_{c}^{\text{ACFD}}$ involves evaluating the coupling constant and frequency integrals in Eq. (8). For the systems presented in Section III, the $\lambda$-dependent integrand does not deviate much from a linear form, and thus Gauss-Legendre integration is able to reach machine precision with $N_{\lambda}=10$ grid points. However, the $\omega$-dependent integrand is not suited to the traditional Gauss-Legendre scheme, and so it is convention to utilize a change of coordinates in order to reduce the number of frequency grid points required to reach a desired accuracy ⁴⁴4We note that the Gauss-Legendre scheme with $N$ grid points is able to exactly integrate a polynomial of degree less than or equal to $2N+1$. Therefore, an integral change of coordinates aims to transform the integrand into a low-degree polynomial.. Upon careful comparison with a variety of methods from literature [60], we find the novel substitution

performs best, where $a$ is a numerical parameter and $\tilde{\omega}$ is the new coordinate. (The traditional Gauss-Legendre scheme is then applied to the transformed $\tilde{\omega}$-dependent integral.) This approach is able to reach more than sufficient accuracies with $N_{\omega}=30$ grid points – a comprehensive motivation and derivation can be found in the supplemental material.

The algorithm that has just been outlined captures the exact correlation energy to within $\mathcal{O}(10^{-10})$ a.u. across the systems studied in this work. This procedure can be suitably adapted, see Fig. 1, to include an approximate $f_{\text{xc}}[n]$ and/or an approximate $v_{\text{xc}}[n]$, where we recall the latter is used to determine the density at which $E^{\text{ACFD}}[n]$ is evaluated in a one-shot calculation (rather than evaluating $E^{\text{ACFD}}[n]$ at the exact interacting density as is done in the first and second panels of Fig. 1).

On the other hand, a self-consistent ACFDT total energy calculation comprises first specifying an initial guess Kohn-Sham potential $v_{\text{KS}}(x)$ in place of the first and second panels in Fig. 1. Note that since there is a one-to-one correspondence between the density and the Kohn-Sham potential, it is sufficient to minimize over variations in $v_{\text{KS}}$ in order to minimize the ACFDT total energy functional as in Eq. (11). We are then able to iterate the initial guess toward the minimizing Kohn-Sham potential by utilizing the BFGS optimization algorithm – this involves looping over the flow chart in Fig. 1. In the event that the ACFDT total energy functional is specified with the exact $f_{\text{xc}}[n]$, the minimization procedure terminates at the exact Kohn-Sham potential $v_{\text{KS}}(x)$/the exact interacting density $n(x)$ without their explicit inclusion. In general, iterations are terminated when the Jacobian norm is $\mathcal{O}(10^{-6})$, meaning the BFGS algorithm is making energy variations $\mathcal{O}(10^{-8})$ a.u. The calculations are parallelized over $\lambda$ grid points using dask [79].

Minimizing the ACFDT total energy functional is able to circumvent the troublesome ‘starting-point dependence’ inherent to a one-shot calculation. In practice, the minimization is accomplished by solving a set of optimized effective potential equations in order to generate the minimizing density [70, 16, 71, 36, 40, 72, 73], rather than direct minimization of the functional as is considered in this work. Despite the latter being much more expensive, we are required to take these measures as the AE kernel $f_{\text{xc}}[n](\omega=0)$ has no analytic representation in terms of the density/orbitals. In either case, self-consistent calculations are more computationally demanding than one-shot calculations. However, the accuracy of self-consistent ACFDT total energies is entirely determined by the approximate $f_{\text{xc}}[n]$, which in certain circumstances can be advantageous, as we shall examine in Section III.

We shall now consider two like-spin electrons confined in an atom-like potential $v_{\text{ext}}=-1/(|0.05x|+1)$ within the domain $[-15,15]$ a.u. which is discretized over $N_{x}=121$ grid points – see upper panel of Fig. 2. The so-called exact adiabatic connection curve [5, 81] is given in the lower panel of Fig. 2 which provides a geometric interpretation of the $\lambda$-dependent ACFDT integrand,

i.e. the $\lambda$-dependent xc potential energy, see Eq. (3) and Eq. (4). The slight convex bend in the adiabatic connection curve that is observed here implies a modest static correlation [5]. In this instance, the correlation energy is $1.3\%$ of the xc energy, and the xc energy is $24\%$ of the total energy, $E_{\text{tot}}=-1.510$ a.u.

The relative error in the atomic total energy is illustrated in Fig. 3 across the whole range of approximate ground-state xc potentials and xc kernels considered herein.

With the exception of the energies calculated utilizing the notoriously poor Hartree orbitals, the predominant clustering in error appears to be according to the approximate $f_{\text{xc}}[n]$, rather than the approximate $v_{\text{xc}}[n]$ used to generate the input density. This suggests that there is a fairly general insensitivity to the density at which $E^{\text{ACFD}}[n]$ is evaluated after having been specified with some $f_{\text{xc}}[n]$. Arguments have been made that this must be the case in the context of the RPA [6], and we are now able to demonstrate that it is also the case when using the adiabatic LDA and the AE approximations. This conclusion translates to all other systems studied here as can be seen in the supplemental material, wherein similar figures can be found for each of the three remaining systems: an infinite potential well, a slab, and a double well.

Secondly, the AE kernel, i.e. ignoring the frequency dependence in the otherwise exact functional $f_{\text{xc}}[n](x,x^{\prime},\omega=0)$, reduces the error in the total energy by orders of magnitude when compared to the RPA and adiabatic LDA kernels. For example, the absolute error in the AE-ACFDT total energy is $0.0006$ a.u. when ground-state LDA density/orbitals are used in a one-shot calculation – significantly better than the usual measure of chemical accuracy, $0.0016$ a.u. (1 kcal/mol). Moreover, we find that this level of accuracy is reliably achieved across differing ground-state orbitals (with the exception of the Hartree orbitals) and differing systems.

There is a sense in which this performance can be said to be inherent to the AE approximation $f_{\text{xc}}[n](x,x^{\prime},\omega=0)$. Namely, in using the exact Kohn-Sham potential $v_{\text{KS}}(x)$ to generate the one-shot ground-state orbitals, we are able to isolate error coming solely from the fact that the approximate xc kernel is not exact. Furthermore, minimizing the ACFDT total energy functional serves a similar purpose, i.e. error is entirely due to the approximate $f_{\text{xc}}[n]$. It can be observed in Fig. 3 and in its corresponding tabulated data (see supplemental material) that these two methods produce relative errors in the total energy to within $\mathcal{O}(10^{-6})$ of each other. In other words, the implicit (minimizing) potential/orbitals that are contained within the self-consistent AE-ACFDT calculation are close to the exact Kohn-Sham potential/orbitals, both of which yield ACFDT total energies well below chemical accuracy.

This similarity between the self-consistent AE-ACFDT orbitals and the exact Kohn-Sham orbitals is manifest when comparing the associated minimizing density with the exact interacting density, Fig. 4. These observations point toward a central conclusion of this work: the spatial structure in the exact xc kernel at $\omega=0$, see Fig. 5, is sufficient to almost entirely determine the exact total/correlation energy, and in turn the exact interacting density and exact Kohn-Sham potential. Hence, the intricate difference between the non-local spatial structure in Fig. 5 versus the local spatial structure in, for example, an adiabatic LDA kernel gives rise to a significant difference in the respective total energies, the reasons for which we shall now explore.

Turning now toward the RPA and adiabatic LDA kernels, inspection of Fig. 3 and Fig. 4 leads one to conjecture that these approximate kernels share the same fundamental issues, with the adiabatic LDA suffering slightly less. Both approximations are in serious error when it comes to the correlation energy: an order of magnitude too negative in the case of the atom, and more so in the case of the infinite potential well and double well. Therefore, minimizing the corresponding ACFDT total energy functionals necessarily makes matters worse – the RPA and adiabatic LDA minimizing densities deviate significantly from the interacting ground-state density, and even deviate from most approximations to it, Fig. 4.

The source of this error is understood here in the context of the so-called $\lambda$-averaged xc hole. In conventional wavefunction-based theories, the statistical hole $n^{\text{hole}}(x,x^{\prime})$ describes how the probability distribution of particle positions $n(x)$ changes upon measurement of a particle at $x^{\prime}$ ⁵⁵5The probability that a particle resides at position $x$ given that a particle has been measured at $x^{\prime}$ is thus given by the conditional probability $n(x|x^{\prime})=n(x)-n^{\text{hole}}(x,x^{\prime})$.. In density-based theories, however, the xc hole is redefined and is the object with which the density undergoes Coulomb interaction to produce the xc energy [5, 19, 20, 54, 83],

where $n^{\text{hole},\lambda}$ is the traditional statistical hole of the $\lambda$-interacting systems along the adiabatic connection. Comparing Eq. (18) with the ACFDT xc energy expression provides a unique definition of the xc hole in the present context, and moreover it defines an approximate xc hole in terms of $f_{\text{xc}}[n]$ [54]. Nevertheless, the two definitions are closely related, and it is instructive to view the $\lambda$-averaged xc hole in both contexts, not least because certain important sum rules are shared, e.g. $\int n^{\text{hole}}_{\text{xc}}(x,x^{\prime})\ dx=-1$.

As discussed above, in order to isolate deficiencies in the approximate $f_{\text{xc}}[n]$, we shall hereafter consider xc holes that utilize the exact Kohn-Sham orbitals and density in the corresponding one-shot ACFDT calculations ⁶⁶6This means that the exchange hole will be exact.. The upper panel in Fig. 6 demonstrates that the so-called on-top xc hole is far too deep in the case of the RPA and the adiabatic LDA. This can be interpreted as the second particle having negative probability to be measured at the position of the first – an artifact due to the original particle interacting with itself at $x^{\prime}=2.5$ a.u. This self-interaction at the level of the xc kernel plagues both the RPA and adiabatic LDA similarly, which can be seen more clearly in the correlation hole [5] (lower panel of Fig. 6). In fact, the RPA and adiabatic LDA correlation holes reach a minimum at $x=2.5$ a.u. where they should be identically zero, that is to say, exchange is entirely responsible for the fact that two fermions cannot be measured at the same position. Minimizing the ACFDT energy functional defined with $f_{\text{xc}}^{\text{ALDA}}$ or $f_{\text{xc}}^{\text{RPA}}=0$ accentuates the self-interaction thereby making the on-top correlation hole even deeper.

Note that the traditional on-top LDA xc hole, that is, the on-top hole defined with the exact LDA pair-correlation function [54], is known to be accurate and therefore central to the success of conventional ground-state LDA calculations [19]. However, the ALDA-ACFDT approximate xc hole, which utilizes an adiabatic xc kernel derived from an LDA functional, is distinct from the traditional LDA xc hole, leading to errors of the kind discussed in the previous paragraph.

A simple quantitative measure of self-interaction is given using one-particle calculations, wherein any correlation present is necessarily due to self-interaction. For the atomic system, the one-particle adiabatic LDA correlation energy is $-0.016$ a.u., whereas the corresponding adiabatic LDA two-particle correlation energy is $-0.05$ a.u. Doubling the spurious one-particle energy reveals that the majority of the two-particle correlation energy constitutes self-interaction. The AE approximation has no spurious one-particle correlation energy, because the exact one-particle $f_{\text{xc}}[n]$ is itself adiabatic. Whilst this line of reasoning suggests that the AE approximation is self-interaction free, this is not quite the case: the exact exchange kernel $f_{\text{x}}[n](x,x^{\prime},\omega)$ provides a more rigorous definition of a self-interaction-free kernel, and this includes an $\omega$ dependence [35, 31]. Nonetheless, it can be seen in Fig. 6 that the non-local spatial structure in the exact $f_{\text{xc}}[n]$ at $\omega=0$ is able to largely remedy the deficiencies due to self-interaction – Section III.3 provides more insight on this matter.

We now examine a system containing long-range correlations, i.e. van der Waals correlations [67]: a double well, see Fig. 7. This system is also known to exhibit a step-like feature in the Kohn-Sham potential [85, 86, 87] in order to localize one electron in each well – without such a step, the non-interacting particles would spuriously collapse into the same well, as is the case in Hartree theory.

The wells are separated at distance $R=14$ a.u., meaning the correlation energy is low $\mathcal{O}(10^{-6})$ a.u., and as such the RPA and adiabatic LDA suffer even more as a result of self-interaction: the relative error in $E_{\text{c}}$ is now $\mathcal{O}(10^{5})$. However, one might expect such error is alleviated to a degree when computing quantities that involve energy differences, e.g. dissociation curves. In the context of dissociation curves, it is imperative that the total energy method in question provides at least some account of long-range correlations, and it is this phenomenon toward which we now focus our attention.

The correlation hole for the double well is given in Fig. 8. As expected, upon measuring an electron in the right-hand well at $x^{\prime}=8$ a.u., there is a strong erroneous contribution to RPA and adiabatic LDA correlation holes in the same well where the electron was measured due to the electron Coulomb interacting with itself – the correlation hole should largely reside in the opposite well. Inclusion of the non-local spatial structure in the exact xc kernel at $\omega=0$ is able to almost entirely correct this issue, as discussed in the previous section.

On the other hand, it can also be observed in the lower panel of Fig. 8 that the correlation hole in the opposing well, a necessarily long-range feature, is modeled successfully in all three of the approximations considered. This observation represents the systematic non-locality introduced via the Dyson equation, and constitutes the principal advantage of the ACFDT formalism over conventional Kohn-Sham calculations. We expect that these considerations account for the fact that, even in the case of the RPA, the limit of large separation in molecules can be described [14]. We further note that the AE approximation $f_{\text{xc}}[n](\omega=0)$ is able to quantitatively capture the long-range correlation hole, thus further establishing the notion that the majority of both short-range and long-range correlations reside in the spatial dependence of $f_{\text{xc}}[n](x,x^{\prime},\omega=0)$ when using the ACFDT framework.

Despite self-consistent RPA-ACFDT and adiabatic LDA-ACFDT calculations accentuating self-interaction error by making the on-top correlation hole even deeper, we find that the long-range correlation hole is improved by minimizing the ACFDT functional. Therefore, even though these approximate self-consistent ACFDT calculations yield poorer absolute energies than their one-shot counterparts, it appears that the long-range properties are improved.

The exact double well $f_{\text{xc}}$ computed here contains the step-like features discussed in [88, 89, 90] that relate to the derivative discontinuity (see supplemental material). In fact, all three approximations to $f_{\text{xc}}[n]$ recover the step in the Kohn-Sham potential when minimizing the corresponding ACFDT energy functional. This is due to the fact that the exact-exchange optimized effective potential contains the step, and the ACFDT total energy functionals comprise in large part the Hartree-Fock functional.

The central aim of this section is to carefully examine the reasons behind the exceptional performance of the AE approximation $f_{\text{xc}}[n](\omega=0)$ as demonstrated in the previous sections. The slab system, see supplemental material, is used to illustrate the forthcoming conclusions. First, it is possible to observe that the exact $f_{\text{xc}}[n](x,x^{\prime},i\omega)$ along $i\omega$ undergoes considerable change away from its adiabatic $\omega=0$ limit, see Fig. 9. (Note, however, that this change is much more regular than along the real $\omega$-axis due to circumventing the singularities described in [7]). Therefore, the performance of the AE approximation cannot be attributed to xc kernels generally possessing modest frequency dependence along $i\omega$.

Since we have access to the exact $\lambda$-interacting adiabatic connection wavefunctions/energies $\{|\Psi_{i}^{\lambda}\rangle,E_{i}^{\lambda}\}$ and the exact Kohn-Sham single-particle wavefunctions/energies $\{|\Phi_{i}\rangle,\varepsilon_{i}\}$, the $\omega$-dependent integral in $E_{\text{c}}^{\text{ACFD}}[n]$ can be evaluated analytically up to some arbitrary $\omega_{\text{max}}$, see supplemental material. It is thus possible to isolate error whose exclusive source is a finite $\omega_{\text{max}}$, as depicted in Fig. 10 – another perspective on this is that the exact $f_{\text{xc}}[n]$ is used for $\omega\leq\omega_{\text{max}}$, and $f_{\text{xc}}[n]=-\lambda f_{\text{H}}$, i.e. $\chi=\chi_{0}$, is used for $\omega>\omega_{\text{max}}$.

The absolute error in the correlation energy is defined,

where $E_{\text{c}}^{\text{ACFD}}(\omega_{\text{max}})$ is the ACFDT correlation energy whose analytic $\omega$ integration has been terminated at $\omega_{\text{max}}$ ($E^{\text{ACFD}}_{c}(\infty)$ is therefore the exact correlation energy). We find that chemical accuracy is surpassed prior to $\omega=1$ a.u., i.e. the capacity for an approximate $f_{\text{xc}}[n]$ to yield accurate correlation energies predominantly resides in its structure below some characteristic $\omega_{\text{max}}$ ($\omega_{\text{max}}\approx 1$ in this case) – see supplemental material for a plot. In fact, beyond this $\omega_{\text{max}}$, the AE approximation $f_{\text{xc}}[n](\omega=0)$ produces approximate interacting response functions upon solution of the Dyson equation that are as poor as the RPA, adiabatic LDA, or simply setting $\chi^{\lambda}=\chi_{0}$.

In all systems studied here, whilst the exact $f_{\text{xc}}[n]$ is not adiabatic in general, it is adiabatic over the most relevant $\omega$ range, and therefore the AE approximation performs accordingly. Such an observation is commensurate with prior findings [7]: in the context of the optical spectrum, $f_{\text{xc}}[n](\omega)$ is required at the $\omega$ corresponding to a transition energy, and thus at frequencies beyond the lowest lying excitations, the AE approximation ceases to perform.

In light of these observations, it is imperative that practical approximate integration schemes target the relevant $\omega$ region, whereas the long-range tail of the $\omega$-dependent integrand is less important. Whilst the integration scheme proposed in this work Eq. (15) compresses the domain $[0,\infty]$ to $[0,\sqrt{\pi/a}]$, thus allowing us to capture the tail, its central advantage instead comes from an explicit treatment of the terms responsible for the low-$\omega$ structure in $g(\omega)$, see Fig. 10. Namely, the scheme defines an integral change of coordinates such that a single term in the Lehmann representation of $\chi$ (or $\chi_{0}$) Eq. (13) is linear, meaning Gauss-Legendre quadrature is exact with just one $\omega$ grid point, see supplemental material. The characteristic range $\omega_{\text{max}}$ within which most of the correlation energy resides will depend on a number of factors in practice, such as the size of the interacting and Kohn-Sham gap, and number of excitations clustered around this gap [60].