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A: New Tools and Methods in Experiment and TheoryJanuary 9, 2025

Top-Down versus Bottom-Up Approaches for σ-Functionals Based on the Approximate Exchange Kernel
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Yannick Lemke
Yannick Lemke
Department of Chemistry, Ludwig-Maximilians-Universität München, Butenandtstr. 5-13, D-81377 Munich, Germany
More by Yannick Lemke
https://orcid.org/0000-0002-0930-5287
Christian Ochsenfeld*
Christian Ochsenfeld
Department of Chemistry, Ludwig-Maximilians-Universität München, Butenandtstr. 5-13, D-81377 Munich, Germany
Max-Planck-Institute for Solid State Research, Heisenbergstr. 1, D-70569 Stuttgart, Germany
*Email: [email protected]
More by Christian Ochsenfeld
https://orcid.org/0000-0002-4189-6558

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The Journal of Physical Chemistry A

Cite this: J. Phys. Chem. A 2025, 129, 3, 774–787

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https://doi.org/10.1021/acs.jpca.4c05289

Published January 9, 2025

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Abstract

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Recently, we investigated a number of so-called σ- and τ-functionals based on the adiabatic-connection fluctuation–dissipation theorem (ACFDT); particularly, extensions of the random phase approximation (RPA) with inclusion of an exchange kernel in the form of an antisymmetrized Hartree kernel. One of these functionals, based upon the approximate exchange kernel (AXK) of Bates and Furche, leads to a nonlinear contribution of the spline function used within σ-functionals, which we previously avoided through the introduction of a simplified “top-down” approach in which the σ-functional modification is inserted a posteriori following the analytic coupling strength integration within the framework of the ACFDT and which was shown to provide excellent performance for the GMTKN55 database when using hybrid PBE0 reference orbitals. In this work, we examine the analytic “bottom-up” approach in which the spline function is inserted a priori, i.e., before evaluation of the analytic coupling strength integral. The new bottom-up functionals, denoted σ↑AXK, considerably improve upon their top-down counterparts for problems dominated by self-interaction and delocalization errors. Despite a small loss of accuracy for noncovalent interactions, the σ↑AXK@PBE0 functionals comprehensively outperform regular σ-functionals, scaled σ-functionals, and the previously derived σ+SOSEX- and τ-functionals in the WTMAD-1 and WTMAD-2 metrics of the GMTKN55 database.

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Subjects

Special Issue

Published as part of The Journal of Physical Chemistry A special issue “Trygve Helgaker Festschrift”.

Introduction

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For decades, the search for electron correlation methods which are both highly accurate and computationally feasible has represented one of the focal points of quantum-chemical research. Besides continuous advancements in Kohn–Sham density functional theory (KS-DFT) (1,2) in the form of new density functional approximations (see ref (3) for a comprehensive review), there has also been a steady interest in methods residing on the fifth and highest rung of “Jacob’s ladder,” (4) including recent developments of double-hybrid functionals such as the DH23 functional of Becke et al. (5) as well as methods based on the adiabatic connection fluctuation–dissipation theorem (ACFDT) such as the direct random phase approximation (dRPA). For the latter, concentrated research efforts within the past decade in the groups of Kállay (6) and Ochsenfeld (7−10) have led to the development of dRPA and beyond-RPA methods with asymptotically linear scaling behavior, and reduced real-word memory demand through the application of optimized batching schemes. (11) However, despite promising results for noncovalent interactions (12−14) and metallic systems (15) – a central advantage over many-body perturbation theory approaches which fail due to a singularity for vanishing electronic gaps–the accuracy of dRPA has left much to be desired, especially compared to the much more affordable KS-DFT method when combined with empirical dispersion corrections. The use of a long-range dRPA correction on top of range-separated hybrid functionals leads to increased accuracy and faster basis set convergence than the regular dRPA ansatz, (16) but still does not offer a favorable cost-to-performance ratio compared to dispersion corrected hybrid functionals.

The recent development of σ-functionals in the Görling group (17,18) as an extension of the dRPA has served to remedy this issue in part, providing substantially better accuracies at virtually identical computational cost as the dRPA. The central idea of σ-functionals is to model the neglected exchange–correlation kernel in the ACFDT by inserting optimized, i.e., empirically fitted, cubic splines into the dRPA energy expression within the spectral representation of the noninteracting density–density response function. A recent study (19) on the basis set requirements for σ-functionals has concluded that a smaller auxiliary basis set can be chosen to compute the noninteracting response function without a considerable loss of accuracy, which further lowers the computational overhead of σ-functionals, especially when combined with the widely used frozen-core approximation. Further advances of σ-functionals include the computation of molecular properties such as nuclear gradients (20,21) and, most recently, the analytical computation of nuclear magnetic resonance (NMR) shieldings, (22−24) for which σ-functionals achieve similar levels of accuracy as the computationally much more demanding coupled-cluster singles and doubles (CCSD) method. (25)

Continuing on the hierarchies of post-Kohn–Sham methods toward the exact correlation energy, (26) the inclusion of an exchange kernel leads to so-called τ-functionals, for which approximations in the form of a power series approximation (27) and scaled σ-functionals (28) have been made. In a previous work, (29) we investigated the accuracy of τ-functionals obtained from the insertion of cubic splines in a similar fashion to σ-functionals, as well as approximations thereof based on second-order screened exchange (SOSEX) (30) and the approximate exchange kernel (AXK) of Bates and Furche, (31) both of which simplify to σ-functionals. In a first full evaluation of σ- and τ-functionals on the widely used GMTKN55 database, (29,32) we showed that in particular, the AXK-based σ-functionals with reference orbitals obtained from the PBE0 functional (33,34) – otherwise known as PBEh (33,35) – perform exceptionally well and provide similar levels of accuracy as the best dispersion-corrected double-hybrid functionals tested in the original GMTKN55 publication, (32) outperforming both the regular σ-functionals and scaled σ-functionals and thus highlighting the importance of the exchange kernel within the ACFDT energy expression. Despite this high degree of accuracy, however, the σ+AXK-functionals derived in ref (29) contained an additional approximation: To avoid a quadratic term of the inserted cubic splines, we introduced a “top-down” approach in which the σ-functional modification is added after carrying out the coupling strength integration and some rearrangement of terms. This approximation holds in the sense that if one does not modify the resulting energy expression through cubic splines, it exactly yields the regular dRPA+AXK energy. In this work, we derive and compare the alternative “bottom-up” approach in which the cubic splines are inserted before the coupling strength integration is performed, thus circumventing the approximation made in the “top-down” ansatz.

The manuscript is structured as follows: The Theory section provides a brief summary of the origins of σ-functionals in the context of the adiabatic connection fluctuation–dissipation theorem, and a more detailed derivation of both the top-down approach introduced in ref (29) and the new bottom-up ansatz. The Computational Details section provides an overview of the computational parameters used for the optimization and evaluation of the new σ-functionals, which are then analyzed and discussed in the Results and Discussion section. Finally, we provide some concluding remarks and recommendations in the Conclusion section.

Theory

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Throughout this work, we will consider the working equations for spin-saturated systems with real-valued atomic and molecular orbitals. The analogous equations for spin-polarized systems have been discussed in detail elsewhere, see, e.g., refs (17,28,and29).

Fluctuation–Dissipation Theorem

In the adiabatic connection formalism, the electronic energy for a Slater determinant |Φ⟩ is given by the adiabatic connection fluctuation–dissipation theorem (ACFDT) (36,37) as

E = ⟨ Φ | \hat{H} | Φ ⟩ + E_{c}^{ACFDT}

(1)

with the correlation energy

E_{c}^{ACFDT} = \frac{- 1}{2 π} \int_{0}^{\infty} d ω \int_{0}^{1} d λ \iint_{R^{6}} d r_{1} d r_{2} \times (χ_{λ} (i ω, r_{1}, r_{2}) - χ_{0} (i ω, r_{1}, r_{2})) f_{H} (r_{1}, r_{2})

(2)

where

f_{H} (r_{1}, r_{2}) = \frac{1}{| r_{1} - r_{2} |}

(3)

denotes the Hartree kernel and χ_λ(iω, r₁, r₂) is the frequency-dependent density–density response function for a given frequency ω and coupling strength λ. The special case λ = 0 represents the noninteracting response function χ₀(iω, r₁, r₂), for which a closed form in terms of the occupied and virtual molecular orbitals {φ_i} and {φ_a} and their respective orbital energies {ε_i} and {ε_a} is known and reads as

\begin{array}{l} χ_{0} (i ω, r_{1}, r_{2}) & = & \sum_{i a} (\frac{1}{(ε_{i} - ε_{a}) + i ω} + \frac{1}{(ε_{i} - ε_{a}) - i ω}) \times φ_{i} (r_{1}) φ_{a} (r_{1}) φ_{i} (r_{2}) φ_{a} (r_{2}) \\ = & 2 \sum_{i a} \frac{(ε_{i} - ε_{a})}{(ε_{i} - ε_{a})^{2} + ω^{2}} \times φ_{i} (r_{1}) φ_{a} (r_{1}) φ_{i} (r_{2}) φ_{a} (r_{2}), \end{array}

(4)

whereas for the more general case of λ ≠ 0, χ_λ is expressed through the Dyson-type equation (38)

χ_{λ} (i ω, r_{1}, r_{2}) = χ_{0} (i ω, r_{1}, r_{2}) + \iint_{R^{6}} d r_{3} d r_{4} χ_{0} (i ω, r_{1}, r_{3}) \times f_{Hxc}^{λ} (ω, r_{3}, r_{4}) χ_{λ} (i ω, r_{4}, r_{2})

(5)

where

f_{Hxc}^{λ} (ω, r_{3}, r_{4}) = λ f_{H} (r_{3}, r_{4}) + f_{xc}^{λ} (ω, r_{3}, r_{4})

(6)

comprises the Hartree kernel as well as the unknown coupling-strength- and frequency-dependent exchange–correlation kernel f_xc. Note that this formalism assumes a constant density along the path from λ = 0 to λ = 1, which is only strictly valid for local Kohn–Sham potentials. While an adiabatic connection formula for generalized Kohn–Sham theory (GKS), which represents the foundational backbone of hybrid density functional approximations, has been derived, (39) we would argue that the use of the ACFDT as formulated above is well-established in the context of the random phase approximation, even when hybrid functionals are applied for the generation of reference orbitals.

In typical implementations, the quantities in eqs 2–6 are expressed within the resolution-of-the-identity (RI) approximation, (40−45) which employs a set of auxiliary basis functions denoted by indices K, L, ··· to replace the costly four-center-two-electron integrals (4c2e) by simpler 3c2e and 2c2e integrals through so-called “density-fitting” as

\iint_{R^{6}} d r_{1} d r_{2} φ_{p} (r_{1}) φ_{q} (r_{1}) \frac{1}{| r_{1} - r_{2} |} φ_{r} (r_{2}) φ_{s} (r_{2}) = : (p q | r s) \approx \sum_{K L} (p q | K) (K | L)^{- 1} (L | r s)

(7)

Defining C_KL = (K|L)⁻¹ and the orthogonalized 3c2e integrals as

B_{p q}^{K} = \sum_{L} (p q | L) {(C^{1 / 2})}_{L K}

(8)

the correlation energy defined in eq 2 becomes

E_{c}^{ACFDT} = \frac{- 1}{2 π} \int_{0}^{\infty} d ω \int_{0}^{1} d λ Tr {(X_{λ} (i ω) - X_{0} (i ω)) F_{H}}

(9)

with F_H = 1,

X_{0, K L} (i ω) = 2 \sum_{i a} B_{i a}^{K} \frac{(ε_{i} - ε_{a})}{{(ε_{i} - ε_{a})}^{2} + ω^{2}} B_{i a}^{L}

(10)

for the noninteracting response function, and

X_{λ} (i ω) = X_{0} (i ω) + X_{0} (i ω) F_{Hxc}^{λ} (ω) X_{λ} (i ω)

(11)

\Leftrightarrow X_{λ} (i ω) = {(1 - X_{0} (i ω) F_{Hxc}^{λ} (ω))}^{- 1} X_{0} (i ω)

(12)

for the interacting response function.

Direct Random Phase Approximation and σ-Functionals

Since the exchange–correlation (xc) kernel remains unknown, further approximations must be made. The simplest such approximation comes in the form of the direct random phase approximation (dRPA), in which the xc kernel is neglected entirely, and which forms the basis for σ-functionals. Using the spectral representation of the negative-semidefinite X₀(iω),

- X_{0} = V σ V^{T}

(13)

(the explicit frequency dependence is omitted for the sake of legibility), the dRPA correlation energy is expressed as

E_{c}^{dRPA} = \frac{- 1}{2 π} \int_{0}^{\infty} d ω \int_{0}^{1} d λ Tr {({(1 - λ X_{0} F_{H})}^{- 1} - 1) \times X_{0} F_{H}}

(14)

= \frac{- 1}{2 π} \int_{0}^{\infty} d ω \int_{0}^{1} d λ Tr {(- {(1 + λ σ)}^{- 1} + 1) σ}

(15)

where in eq 15, we have exploited cyclic permutations of the trace as well as the relations

F_{H} = V V^{T} = V^{T} V = 1

. Since all matrices within the trace are now diagonal, one can define a simple function

h (x) = \frac{- 1}{1 + x} + 1

(16)

such that for σ = diag(σ₁, σ₂, ···), one has

h (λ σ) = diag (h (λ σ_{1}), h (λ σ_{2}), \cdot\cdot\cdot) = - {(1 + λ σ)}^{- 1} + 1

(17)

The idea of σ-functionals (17,18) is then to model the neglected xc kernel by modifying h(x), which is realized through the addition of piecewise cubic splines

\begin{aligned} h^{spline} (x) = {\begin{array}{l} 0 & if x \in I_{1} or x \in I_{M - 1} or x \in I_{M} \\ \sum_{k = 0}^{3} c_{m, k} {(x - x_{m})}^{k} & otherwise \end{array} \end{aligned}

(18)

over M intervals

I_{m} = [x_{m}, x_{m + 1})

(19)

with x_m < x_m+1, x₁ = 0, and x_M+1 = ∞. This ansatz offers vast flexibility, as one is free to choose (i) the underlying density functional approximation for the generation of reference orbitals, (ii) the number and placement of the abscissae {x_m}, (iii) the type of splines used to ensure h^spline is continuous and differentiable on [0, ∞), and (iv) how to optimize the spline coefficients {c_m,k}. All of these factors constitute a so-called “parametrization,” of which a considerable number have already been published. (17−19,28,29,46)

Inclusion of Exchange Kernels and τ-Functionals

Going beyond the direct RPA, one can approximate the exchange–correlation kernel by an exchange kernel that depends linearly on the coupling strength, i.e.,

f_{xc}^{λ} (ω, r_{1}, r_{2}) \approx λ f_{x}^{λ = 1} (ω, r_{1}, r_{2})

(20)

or equivalently

F_{xc}^{λ} (ω) \approx λ F_{x}^{λ = 1} (ω)

(21)

arriving at what is commonly referred to as “RPA with exchange.” While a variety of approaches which slightly differ in their definitions of f_x have been published under different names, (27,47−52) we shall herein adopt the electron–hole exchange kernel discussed in refs (51) and (52), which we will refer to as “RPAx.” The same-spin part of this exchange kernel is given as

F_{x} (ω) = - X_{0}^{- 1} (i ω) Y (i ω) X_{0}^{- 1} (i ω)

(22)

within the RI representation, with

Y_{K L} (i ω) = \sum_{i j a b} B_{i a}^{K} \frac{(ε_{i} - ε_{a})}{{(ε_{i} - ε_{a})}^{2} + ω^{2}} (i b | j a) \frac{(ε_{j} - ε_{b})}{{(ε_{j} - ε_{b})}^{2} + ω^{2}} \times B_{j b}^{L}

(23)

as discussed, e.g., in refs (10) and (52). Please note that this choice of F_x merely represents an implementation detail; the following derivations hold for any exchange kernel which can be expressed analytically within the RI approximation.

We note in passing that, given the approximation F_Hxc^λ(ω) ≈ λ (F_H + F_x(ω)), one can apply the idea of inserting spectral representations into eqs 9–12 to arrive at so-called τ-functionals, (26) which have been investigated using a power series approximation (27) and through cubic splines with either scaled σ values (28) or exact values of τ obtained from the electron–hole exchange kernel (29) and generally lead to increased accuracy, albeit at higher computational cost than the regular σ-functionals. In particular, the computation of the matrix Y was identified as the most time-consuming step, though efficient strategies to minimize the computational overhead have been developed. (10,53) Nevertheless, the cost of computing the exchange kernel significantly outweighs the simpler dRPA, σ-functionals, or scaled σ-functionals by 2–3 orders of magnitude, (29) and thus needs to be taken into consideration for sufficiently large systems.

Approximate Exchange Kernel

The approximate exchange kernel (AXK) introduced by Bates and Furche (31) is based on a truncated geometric series expansion of the interacting response function in the RPAx formalism, i.e.,

\begin{array}{l} X_{λ} (i ω) & = & (X_{0}^{- 1} (i ω) - λ F_{Hx} (ω))^{- 1} \\ = & (X_{0}^{- 1} (i ω) - λ F_{H} - λ F_{x} (ω))^{- 1} \\ = & ((1 - λ F_{x} (ω) X_{λ}^{dRPA} (i ω)) (X_{λ}^{dRPA} (i ω))^{- 1})^{- 1} \\ = & X_{λ}^{dRPA} (i ω) (1 - λ F_{x} (ω) X_{λ}^{dRPA} (i ω))^{- 1} \\ = & X_{λ}^{dRPA} (i ω) \sum_{k = 0}^{\infty} (λ F_{x} (ω) X_{λ}^{dRPA} (i ω))^{k} \\ \approx & X_{λ}^{dRPA} (i ω) + λ X_{λ}^{dRPA} (i ω) F_{x} (ω) X_{λ}^{dRPA} (i ω) . \end{array}

(24)

Inserting this expression into the ACFDT correlation energy, where we once again omit the frequency dependence for the sake of brevity, gives

\begin{array}{l} E_{c}^{dRPA + AXK} \\ = E_{c}^{dRPA} - \int_{0}^{\infty} \frac{d ω}{2 π} \int_{0}^{1} d λ Tr {λ X_{λ}^{dRPA} F_{x} X_{λ}^{dRPA} F_{H}} \\ = E_{c}^{dRPA} - \int_{0}^{\infty} \frac{d ω}{2 π} \int_{0}^{1} d λ Tr {λ (1 - λ X_{0} F_{H})^{- 1} X_{0} F_{x} \times (1 - λ X_{0} F_{H})^{- 1} X_{0} F_{H}} \\ = E_{c}^{dRPA} - \int_{0}^{\infty} \frac{d ω}{2 π} \int_{0}^{1} d λ Tr {λ (1 + λ σ)^{- 1} \times σ^{1 / 2} V^{T} F_{x} V σ^{1 / 2} (1 + λ σ)^{- 1} σ^{1 / 2} V^{T} F_{H} V σ^{1 / 2}} \\ = E_{c}^{dRPA} - \int_{0}^{\infty} \frac{d ω}{2 π} \int_{0}^{1} d λ Tr {λ σ (1 + λ σ)^{- 2} \times σ^{1 / 2} V^{T} F_{x} V σ^{1 / 2}} \\ = E_{c}^{dRPA} - \int_{0}^{\infty} \frac{d ω}{2 π} \int_{0}^{1} d λ Tr {h (λ σ) (1 - h (λ σ)) \times σ^{1 / 2} V^{T} F_{x} V σ^{1 / 2}} \end{array}

(25)

as we derived in ref (29). Clearly, this expression lends itself to the notion of σ-functionals as we can modify the same function h(x), though the presence of an h²(x) term in eq 25 makes the expressions slightly more complex than for regular σ-functionals. To avoid this term and achieve only a linear dependence on h(x), we introduced what we referred to as a “top-down” approach to the correlation energy, (29) which we denoted as “σ+AXK.” For the remainder of this work, we will refer to these top-down functionals as “σ↓AXK” for better differentiation–please note that the arrow refers to the distinction between the top-down and bottom-up ansatz and is in no way related to the notion of “spin-up” or “spin-down.”

Top-Down “σ↓AXK” Correlation Energy

For the “top-down” approach, we first carry out the integration over the coupling strength λ in eq 25 analytically to obtain

\begin{array}{l} E_{c}^{dRPA + AXK} & = & E_{c}^{dRPA} - \int_{0}^{\infty} \frac{d ω}{2 π} Tr {(\ln (1 + σ) σ^{- 1} - (1 + σ)^{- 1}) σ^{1 / 2} V^{T} F_{x} V σ^{1 / 2}} \\ = & \frac{- 1}{2 π} \int_{0}^{\infty} d ω Tr {(- \ln (1 + σ) σ^{- 1} + 1) σ + (\ln (1 + σ) σ^{- 1} - (1 + σ)^{- 1}) \times σ^{1 / 2} V^{T} F_{x} V σ^{1 / 2}} \\ = & \frac{- 1}{2 π} \int_{0}^{\infty} d ω Tr {(- \ln (1 + σ) σ^{- 1} + 1) \times (σ - σ^{1 / 2} V^{T} F_{x} V σ^{1 / 2}) + (1 - (1 + σ)^{- 1}) σ^{1 / 2} V^{T} F_{x} V σ^{1 / 2}} . \end{array}

(26)

Next, we realize that the first term in parentheses of eq 26 is precisely equal to the integral of h(λσ) over λ, that is,

- \ln (1 + σ) σ^{- 1} + 1 = \int_{0}^{1} d λ h (λ σ)

(27)

thereby motivating a reinsertion of the coupling strength integral for just the first term of eq 26 to obtain a σ-functional, i.e.,

(28)

This reformulation is exact in the sense that if we do not modify the function h, the dRPA+AXK energy is recovered exactly. However, if we choose to augment h through cubic splines, the top-down approach of eq 28 only represents an approximation to the analytic expression of eq 25.

The motivation for naming this approach “top-down” is as follows: If one pictures the sequence of steps to formulate the dRPA+AXK correlation energy (ACFDT → AXK

\to \int d λ \to \int d ω \to E_{c}^{dRPA+AXK}

) as “going up,” then this approach starts near the top and works downward to transform the dRPA+AXK energy into a σ-functional. In contrast, in the “bottom-up” approach discussed below, the σ-functional is inserted along the path from the bottom upward.

Bottom-Up “σ↑AXK” Correlation Energy

The new “bottom-up” approach, denoted σ↑AXK, foregoes the approximations of the top-down approach and instead uses the analytic expression of eq 25, in which we insert the spline function before the coupling strength integration is performed. Note that we also augment the dRPA correlation energy in eq 25 through the regular σ-functionals. To that extent, we write for the overall correlation energy

\begin{array}{l} E_{c}^{σ ↑ AXK} & = & \frac{- 1}{2 π} \int_{0}^{\infty} d ω \int_{0}^{1} d λ Tr {h (λ σ) σ + h (λ σ) \times (1 - h (λ σ)) σ^{1 / 2} V^{T} F_{x} V σ^{1 / 2}} \\ = & \frac{- 1}{2 π} \int_{0}^{\infty} d ω \int_{0}^{1} d λ Tr {h (λ σ) σ^{1 / 2} V^{T} F_{Hx} V σ^{1 / 2} - h^{2} (λ σ) σ^{1 / 2} V^{T} F_{x} V σ^{1 / 2}} . \end{array}

(29)

To evaluate the integrals over λ, we use the following straightforward substitution:

\int_{0}^{1} d λ f (λ σ) = \frac{1}{σ} \int_{0}^{σ} d x f (x)

(30)

With the usual decomposition h(λσ) = h^dRPA(λσ) + h^spline(λσ), the first term of eq 29 can easily be evaluated as was done in ref (17) to give

\frac{1}{σ} \int_{0}^{σ} d x h^{dRPA} (x) = - \frac{\ln (1 + σ)}{σ} + 1

(31)

for the dRPA contribution, whereas the spline contribution is computed as a piecewise integral,

\frac{1}{σ} \int_{0}^{σ} d x h^{spline} (x) = \frac{1}{σ} (\sum_{n = 1}^{m - 1} {I_{1} |}_{x_{n}}^{x_{n + 1}} + {I_{1} |}_{x_{m}}^{σ})

(32)

with

\begin{array}{l} {I_{1} |}_{a}^{b} & = & \int_{a}^{b} d x h^{spline} (x) \\ = & c_{0} (b - a) + \frac{c_{1}}{2} (b - a)^{2} + \frac{c_{2}}{3} (b - a)^{3} + \frac{c_{3}}{4} (b - a)^{4} \end{array}

(33)

and spline coefficients c₀, ···, c₃ belonging to the respective interval [a, b). For the second term of eq 29, we obtain three distinct integrals, namely

\begin{array}{l} \int_{0}^{1} d λ (h^{dRPA} (λ σ) + h^{spline} (λ σ))^{2} \\ = \int_{0}^{1} d λ (h^{dRPA} (λ σ))^{2} + \int_{0}^{1} d λ (h^{spline} (λ σ))^{2} + 2 \int_{0}^{1} d λ h^{dRPA} (λ σ) h^{spline} (λ σ) . \end{array}

(34)

Using eq 30, the first integral can be evaluated analytically as

\frac{1}{σ} \int_{0}^{σ} d x {(h^{dRPA} (x))}^{2} = - 2 \frac{\ln (1 + σ)}{σ} + \frac{1}{1 + σ} + 1

(35)

whereas the second and third integral are once again computed piecewise as

\frac{1}{σ} \int_{0}^{σ} d x h^{dRPA} (x) h^{spline} (x) = \frac{1}{σ} (\sum_{n = 1}^{m - 1} {I_{2} |}_{x_{n}}^{x_{n + 1}} + {I_{2} |}_{x_{m}}^{σ})

(36)

\frac{1}{σ} \int_{0}^{σ} d x {(h^{spline} (x))}^{2} = \frac{1}{σ} (\sum_{n = 1}^{m - 1} {I_{3} |}_{x_{n}}^{x_{n + 1}} + {I_{3} |}_{x_{m}}^{σ})

(37)

for σ ∈ [x_m, x_m+1) and with the auxiliary integrals

\begin{array}{l} {I_{2} |}_{a}^{b} & = & \int_{a}^{b} d x h^{dRPA} (x) h^{spline} (x) \\ = & c_{0} \int_{a}^{b} d x h^{dRPA} (x) + c_{1} \int_{a}^{b} d x h^{dRPA} (x) (x - a) + c_{2} \int_{a}^{b} d x h^{dRPA} (x) {(x - a)}^{2} + c_{3} \int_{a}^{b} d x h^{dRPA} (x) {(x - a)}^{3} \\ = & c_{0} {(b - a) - \ln (\frac{b + 1}{a + 1})} + c_{1} {\frac{1}{2} (b^{2} - a^{2}) - (a + 1) (b - a) + (a + 1) \ln (\frac{b + 1}{a + 1})} + c_{2} {\frac{1}{3} (b^{3} - a^{3}) - \frac{2 a + 1}{2} (b^{2} - a^{2}) + {(a + 1)}^{2} (b - a) - {(a + 1)}^{2} \ln (\frac{b + 1}{a + 1})} + c_{3} {\frac{1}{4} (b^{4} - a^{4}) - \frac{3 a + 1}{3} (b^{3} - a^{3}) + \frac{3 a^{2} + 3 a + 1}{2} (b^{2} - a^{2}) - {(a + 1)}^{3} (b - a) + {(a + 1)}^{3} \ln (\frac{b + 1}{a + 1})} \end{array}

(38)

and

\begin{array}{l} {I_{3} |}_{a}^{b} & = & \int_{a}^{b} d x (h^{spline} (x))^{2} \\ = & \frac{c_{3}^{2}}{7} (b - a)^{7} + \frac{c_{2} c_{3}}{3} (b - a)^{6} + \frac{2 c_{1} c_{3} + c_{2}^{2}}{5} (b - a)^{5} + \frac{c_{0} c_{3} + c_{1} c_{2}}{2} (b - a)^{4} + \frac{2 c_{0} c_{2} + c_{1}^{2}}{3} (b - a)^{3} + c_{0} c_{1} (b - a)^{2} + c_{0}^{2} (b - a), \end{array}

(39)

where c₀, ···, c₃ once again are the spline coefficients belonging to the respective interval [a, b). These expressions, while somewhat lengthy, are straightforward to implement on top of existing σ-functionals, along with the coefficient derivatives

\partial I_{1, 2, 3} / \partial c_{m, k}

which are needed for the optimization of the spline coefficients.

As previously mentioned, the computation of the exchange kernel presents the bottleneck in AXK-based calculations. Therefore, the cost of σ↓AXK-functionals and σ↑AXK-functionals is, for all intents and purposes, identical, and a differentiation can be made purely based on their respective accuracies.

Computational Details

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We have implemented the bottom-up σ↑AXK-functionals alongside their top-down variants in our C++ based FermiONs++ program package. (54,55) For all KS-DFT calculations, we employ the RI-J method (56) with the def2-universal-jfit auxiliary basis set (57) in combination with the sn-LinK method (58,59) on a gm4 grid (60) for the computation of exact exchange. Exchange–correlation contributions are obtained using the LibXC library (61) with numerical quadratures being evaluated on a gm5 grid. Throughout this work, we use basis sets of quadruple-ζ quality from the “def2” family (def2-QZVP, (62) def2-QZVPD, (63) and def2-QZVPPD (63,64)) in conjunction with their respective auxiliary basis sets (65−67) for density-fitted integrals within the dRPA+AXK calculation, for which the numerical frequency integration is performed on a 15-point minimax grid first introduced by the Kresse group. (68,69) Accordingly, we employ a frozen-core approximation and, if applicable, use the def2-ECP effective core potentials (70−73) to replace core electrons of heavy atoms.

Results and Discussion

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Optimization of Bottom-Up σ-Functionals

The new σ↑AXK-functionals were optimized on the 16 subsets of the ASCDB database, (74) which were grouped a posteriori by Morgante and Peverati following a statistical analysis of larger computational chemistry databases. (75) The optimizations were performed using piecewise cubic Hermite interpolating polynomials (PCHIPs) (76) and the A1 and A2 parametrizations for σ-functionals introduced in ref (29). The parametrizations are constructed as follows: A fixed set of interval end points x_m (cf. eq 19), taken from the W1 parametrization of ref (18), are distributed logarithmically within the range of 10^–5 to 10^4/3 ≈ 21.5 for PBE0 and 10^–5 to 10^3/2 ≈ 31.6 for PBE. The precise values of x_m are given in the Supporting Information. The corresponding ordinates h_m ≡ c_0,m are then determined to minimize an objective function, which for the A1 parametrization is the total MAE of the ASCDB database computed from the 16 subsets,

MAE (ASCDB) = \frac{1}{\sum_{i = 1}^{16} N_{i}} \sum_{i = 1}^{16} N_{i} {MAE}_{i}

(40)

where N_i and MAE_i are the number of entries and the computed MAE for subset i, respectively. The A2 parametrization aims to minimize the “weighted MAE” (wMAE) defined by Peverati as (77)

w MAE (ASCDB) = \frac{1}{\sum_{i = 1}^{16} \frac{1}{\bar{| Δ E |_{i}}}} \sum_{i = 1}^{16} \frac{{MAE}_{i}}{\bar{| Δ E |_{i}}}

(41)

with

\bar{| Δ E |_{i}}

being the average absolute reaction energy of subset i. The A1 parametrization is thus weighted more strongly toward reactions with large reaction energies, e.g., atomization reactions and “mindless benchmarks,” (32,78) whereas the A2 parametrization puts a larger emphasis on reactions with small reaction energies such as noncovalent interactions. The details of the optimization procedure employing the Broyden–Fletcher–Goldfarb–Shanno method (79−82) have been discussed at length in previous works (17,18,29,46) and shall therefore not be repeated here. All calculations on the ASCDB database were performed using the def2-QZVPPD basis set; two references from the NLMR11 subset containing the carbon dimer were excluded due to convergence problems in the preliminary SCF.

The resulting total MAE and wMAE for σ↑AXK@PBE and σ↑AXK@PBE0 are shown in Figure 1 alongside the previous σ↓AXK functionals, the plain dRPA and dRPA+AXK energies, and the overall best-performing regular σ-functional based on previous testing, σ(W1). (18,29) The advantages of the σ↓AXK and σ↑AXK approaches over the plain dRPA and dRPA+AXK approaches as well as regular σ-functionals are immediately apparent, whereas the differences between the top-down σ↓AXK and bottom-up σ↑AXK functionals are more nuanced: For the PBE reference, the new bottom-up functionals fall within less than 0.2 kcal mol^–1 of their top-down counterparts; σ↑AXK(A1)@PBE improves on both the total MAE and wMAE compared to σ↓AXK(A1)@PBE (4.61 vs 4.74 kcal mol^–1 and 3.05 vs 3.10 kcal mol^–1, respectively), whereas σ↑AXK(A2)@PBE slightly trails its top-down variant with a respective MAE and wMAE of 4.87 kcal mol^–1 and 3.05 kcal mol^–1 compared to 4.75 kcal mol^–1 and 2.95 kcal mol^–1 for σ↓AXK(A2)@PBE. In contrast, we obtain consistent improvements for the σ↑AXK@PBE0 functionals for MAE and wMAE regardless of the chosen parametrization. In particular, the respective total MAEs of 4.44 kcal mol^–1 and 4.49 kcal mol^–1 for σ↑AXK(A1)@PBE0 and σ↑AXK(A2)@PBE0 correspond to the lowest total MAEs for the ASCDB database of all σ-, scaled σ-, and τ-functionals investigated in this work and ref (29). In comparison, σ↓AXK(A1)@PBE0 and σ↓AXK(A2)@PBE0 achieved respective total MAEs of 4.86 kcal mol^–1 and 5.09 kcal mol^–1; the improvement of the bottom-up functionals is therefore particularly promising due to the fact that an evaluation of σ↓AXK@PBE0 on the GMTKN55 database (32) has shown the AXK-based σ-functionals to perform exceptionally well compared to similar methods. (29)

Figure 1. MAE (solid) and wMAE (striped) for the ASCDB database using different σ-functionals with PBE and PBE0 reference orbitals.

Evaluation Compared to Top-Down σ-Functionals

To further evaluate the new σ↑AXK functionals, we performed a full evaluation on the 55 subsets of the widely used GMTKN55 database for quantum chemistry, (32) which are grouped into four categories of chemical reactivity: (i) Basic properties and reaction energies for small systems, (ii) reaction energies for large systems and isomerization reactions, (iii) barrier heights, and (iv) noncovalent interactions, the latter of which is split further into categories centered around inter- and intramolecular noncovalent interactions.

To remain consistent with previous works, (19,28,29) we replaced the G21EA and G21IP subsets for electron affinities and ionization potentials with their W1-EA and W1-IP counterparts, which replace the reference energies with W1 results from ref (83). For subsets from the “intermolecular noncovalent interactions” category, we employ an averaging scheme between counterpoise-corrected and uncorrected results which we discussed in more detail in ref (29). Again, due to convergence issues, entries within the W4–11 and W4–17 benchmark sets containing either C₂ or ClOO were excluded, as well as entry 9 of the INV24 subset. The def2-QZVP basis set was used for all calculations with the following exceptions: For benchmark sets featuring anionic systems (AHB21, IL16, W1-EA, and WATER27), we used the augmented def2-QZVPD basis set, whereas the C60ISO and UPU23 subsets were evaluated with the def2-TZVPP basis set. (64,66)

The respective MAEs as well as the weighted total mean absolute deviations defined in ref (32) (WTMAD-1 and WTMAD-2) are shown in Figures 2 and 3 for PBE and PBE0 reference orbitals, respectively. In both cases, we observe a significant reduction of the total MAE in the “basic properties” category for the new σ↑AXK-functionals compared to both the regular σ(W1)-functionals as well as the previous σ↓AXK-functionals. Considering the best-performing functionals for each class, for σ↑AXK(A1)@PBE, the total MAE in this category is lowered to 2.81 kcal mol^–1 from 3.36 kcal mol^–1 for σ↓AXK(A2)@PBE and 5.46 kcal mol^–1 for σ(W1)@PBE. Using a PBE0 reference, the best results are also obtained with the σ↑AXK(A1) parametrization at 1.69 kcal mol^–1, improving by nearly 1 kcal mol^–1 over σ↓AXK(A2)@PBE0 with a total MAE of 2.60 kcal mol^–1 and even outperforming the best functional for this category in ref (29), τ(A2)@PBE0, which registered a total MAE of 1.87 kcal mol^–1.

Figure 2. (a) MAE, (b) WTMAD-1, and (c) WTMAD-2 for the GMTKN55 database and its subcategories using different σ-functionals with PBE reference orbitals.

Figure 3. (a) MAE, (b) WTMAD-1, and (c) WTMAD-2 for the GMTKN55 database and its subcategories using different σ-functionals with PBE0 reference orbitals.

A more detailed analysis of the subsets within this category (see Tables S3 and S4 in the Supporting Information) reveals that these improvements mainly stem from subsets commonly attributed to self-interaction and delocalization errors, namely those focused on atomization energies, electron and proton affinities, and ionization potentials. We therefore provide a more detailed overview of these subsets, along with the W4–17 benchmark set of atomization energies (84) and the autogenerated W4–11RE and W4–17RE benchmark sets (84−86) of reaction energies in Tables 1 and 2. In both cases, there are substantial improvements for the atomization energies of the W4–11 and W4–17 benchmark sets, as well as the ionization potentials and electron affinities of the W1-EA and W1-IP benchmark sets. For the DIPCS10 and PA26 subsets concerned with double ionization potentials, respectively, the trends are less clear-cut, and improvements of the bottom-up σ-functionals over their top-down counterparts (or lack thereof) seem to depend heavily on both the chosen parametrization and the underlying reference orbitals; the same is true for the SIE4 × 4 benchmark set which exposes a different quality of many-electron self-interaction error akin to left–right static correlation in three-electron bonds that are characterized mainly by dynamic correlation. (87,88)

Table 1. MAEs in kcal mol^–1 for Benchmark Sets Characterized by Self-Interaction and Delocalization Errors Using Top-Down and Bottom-Up σ-Functionals and PBE Reference Orbitals

	PBE
subset	dRPA+AXK	σ↓AXK(A1)	σ↓AXK(A2)	σ↑AXK(A1)	σ↑AXK(A2)
W4–11^a	14.54	4.11	4.02	2.30	2.63
W4–17^a	19.62	4.68	4.76	2.93	3.35
W1-EA^b	3.06	2.46	2.01	1.92	1.72
W1-IP^c	1.85	5.01	4.74	4.40	3.98
DIPCS10	3.26	8.67	7.55	9.74	8.36
PA26	2.03	4.48	4.34	1.72	1.54
SIE4 × 4	13.41	15.23	15.12	16.83	19.65
W4–11RE^a	2.25	1.74	1.68	1.54	2.03
W4–17RE^a	2.02	1.69	1.65	1.51	1.85

^{^a}

Excluding entries containing C₂ and ClOO.

^{^b}

G21EA with W1 reference energies from ref (83).

^{^c}

G21IP with W1 reference energies from ref (83).

Table 2. MAEs in kcal mol^–1 for Benchmark Sets Characterized by Self-Interaction and Delocalization Errors Using Top-Down and Bottom-Up σ-Functionals and PBE0 Reference Orbitals

	PBE0
subset	dRPA+AXK	σ↓AXK(A1)	σ↓AXK(A2)	σ↑AXK(A1)	σ↑AXK(A2)
W4–11^a	15.53	3.48	3.63	1.12	1.22
W4–17^a	19.22	3.65	4.04	1.25	1.32
W1-EA^b	4.35	1.59	1.42	1.20	1.07
W1-IP^c	2.27	3.42	3.13	1.95	2.22
DIPCS10	4.27	3.82	2.15	2.79	3.25
PA26	2.55	2.30	1.72	2.24	1.95
SIE4 × 4	6.42	13.17	13.68	11.81	12.29
W4–11RE^a	2.74	1.75	1.44	1.00	0.97
W4–17RE^a	2.52	1.68	1.42	1.02	0.99

^{^a}

Excluding entries containing C₂ and ClOO.

^{^b}

G21EA with W1 reference energies from ref (83).

^{^c}

G21IP with W1 reference energies from ref (83).

Finally, we consider the reaction energies of the W4–11RE and W4–17RE benchmark sets, which are neither part of our initial optimization nor of the GMTKN55 database, but have so far played an important role in the development and evaluation of σ-functionals due to their large number of data points and high-quality reference energies from W4 theory. (17,18,28,29,46) While for a PBE reference, we only observe small improvements in the case of σ↑AXK(A1)@PBE, improvements compared to the top-down variant using PBE0 reference orbitals are substantial: The σ↑AXK(A1)@PBE0 and σ↑AXK(A2)@PBE0 functionals achieve respective MAEs of 1.00/0.97 kcal mol^–1 for the W4–11RE benchmark set and 1.02/0.99 kcal mol^–1 for the W4–17RE benchmark set, which, to our knowledge, makes σ↑AXK(A2)@PBE0 the first σ-functional shown to produce MAEs of below 1 kcal mol^–1 for these two benchmark sets using the def2-QZVP basis set. These trends are also reflected in the error distributions for the W4–17RE subset, which we illustrate in Figure 4 for the A2 parametrizations using both PBE and PBE0 reference orbitals. For σ↑AXK(A2)@PBE0, the distribution is considerably narrower than for σ↓AXK(A2)@PBE0, which aligns with the overall lower MAE. Interestingly, the higher MAE of σ↑AXK(A2)@PBE compared to σ↓AXK(A2)@PBE does not appear to originate from a systemic issue which would manifest itself in the form of a broader error distribution. In fact, the error distributions for σ↑AXK(A2)@PBE and σ↓AXK(A2)@PBE nearly coincide, however, the bottom-up variant features a small number of outliers at comparatively large errors of ±10 kcal mol^–1 which lead to the overall larger MAE.

Figure 4. Error distribution for the W4–17RE benchmark set using the top-down σ↓AXK(A2) and bottom-up σ↑AXK(A2) functionals for (a) PBE and (b) PBE0 reference orbitals.

The substantial improvement for the total MAE for the “basic properties” category also carries over to the total MAE of the entire GMTKN55 database, where all bottom-up σ↑AXK variants outperform their top-down counterparts, even though we do observe a noticeable increase of the total MAE for the “large systems and isomerization reactions” category for σ↑AXK(A2)@PBE, σ↑AXK(A1)@PBE0, and σ↑AXK(A2)@PBE0 which is caused predominantly by higher errors for the challenging MB16–43 and C60ISO benchmark sets (cf. Tables S3 and S4 in the Supporting Information) with comparatively large average reaction energies of 414.73 and 98.25 kcal mol^–1, respectively. For the weighted WTMAD-1 and WTMAD-2 metrics, the trend of overall improvement for the full GMTKN55 database does not appear to hold, as no σ↑AXK functional consistently outperforms all σ↓AXK functionals. The reason for this lies mostly in a slight deterioration of the accuracy for the remaining categories, in particular, noncovalent interactions. While for the PBE0 reference, total MAEs for noncovalent interactions are increased by less than 0.15 kcal mol^–1 [Figure 3a] and for the PBE reference, we even observe improvements of the total MAE [Figure 2a], their comparatively large weights in the WTMAD-1 and WTMAD-2 metrics have a disproportionate impact on the overall results. This is especially true for the PBE reference: Whereas the total MAE of the σ↑AXK(A1)@PBE functional for intermolecular noncovalent interactions is substantially reduced compared to both top-down variants (0.80 vs 1.21 kcal mol^–1), the WTMAD-2 metric is worse by nearly 0.5 kcal mol^–1 (13.37 vs 12.90 kcal mol^–1 for σ↑AXK(A1)@PBE and σ↓AXK(A1)@PBE, respectively). This observation suggests that the bottom-up σ↑AXK-functionals predominantly perform worse than their top-down counterparts for subsets with particularly small average reaction energies, which therefore are assigned much larger weights in the computation of WTMAD-1 and WTMAD-2. Furthermore, it should be stressed that σ↓AXK-functionals ranked among the best performing functionals for noncovalent interactions in ref (29), especially those with a PBE0 reference, and both σ↑AXK(A1)@PBE0 and σ↑AXK(A2)@PBE0 still outperform all regular σ-functionals, scaled σ-functionals, σ+SOSEX-functionals, and τ-functionals investigated therein in terms of the WTMAD-1 and WTMAD-2 metrics for the full GMTKN55 database. Please note that these findings by no means imply that σ↑AXK-functionals are a universally better choice than, e.g., scaled σ-functionals (even if one were to ignore the important aspect of computational expense), since the vast number of degrees of freedom for both the parametrization and evaluation make a comprehensive ranking of functionals all but impossible. Nevertheless, within the confines of the same parametrizations and basis sets, there are noticeable improvements of our σ↑AXK(A1) and σ↑AXK(A2) functionals compared to, for instance, scσ(A1) and scσ(A2) from ref (29) for the considered benchmark sets. We therefore think the σ↑AXK functionals to present an excellent compromise for general-purpose computations of reaction energies and barrier heights, including systems for which regular σ-functionals and σ↓AXK-functionals suffer from delocalization errors.

Physical Interpretation of Top-Down versus Bottom-Up Approaches

For a better physical understanding of the differences between the top-down and bottom-up approaches, we herein consider the contributions of the Hartree and exchange kernels within our parametrizations. As discussed by Chen et al., (53) the AXK correction exhibits stronger screening compared to the Hartree kernel in the bare dRPA, especially at high coupling strengths λ and large eigenvalues σ, corresponding to long-range electronic separation. Consequently, the AXK correction is predominantly short-ranged in nature, and cancels the self-interaction of same-spin electrons exactly up to second order and approximately in higher orders. (53)

The insertion of spline functions for the generalization to σ-functionals has an obvious but nontrivial effect on the balance between Hartree and exchange contributions, which we wish to investigate in further detail. To that extent, we consider the coupling-strength-averaged weights of the Hartree and exchange kernels as a function of σ arising from eqs 28 and 29,

w_{H} (σ) = \frac{1}{σ} \int_{0}^{1} d λ h (λ σ)

(42)

w_{x}^{σ ↓ AXK} (σ) = \frac{1}{σ} \int_{0}^{1} d λ (- h (λ σ) + 1 - \frac{1}{1 + σ})

(43)

w_{x}^{σ ↑ AXK} (σ) = \frac{1}{σ} \int_{0}^{1} d λ h (λ σ) (1 - h (λ σ))

(44)

where the function h is partitioned into dRPA and σ-functional contributions, h(λσ) = h^dRPA(λσ) + h^spline(λσ). Using these definitions, the correlation energy can be recast as

E_{c} = \frac{- 1}{2 π} \int_{0}^{\infty} d ω Tr {X_{0} V w_{H} (σ) V^{T} X_{0} F_{H} + X_{0} V w_{x} (σ) V^{T} X_{0} F_{x}}

(45)

in analogy to ref (53). Figure 5 shows the functions w_H and w_x for the PBE0-parametrized σ↓AXK- and σ↑AXK-functionals, respectively; analogous plots for PBE reference orbitals can be found in the Supporting Information. A number of crucial observations are immediately apparent: First, due to the spline functions, neither w_H nor w_x are strictly positive at small values of σ compared to the unmodified dRPA+AXK ansatz; in fact, we observe strong undulations as σ approaches zero. However, it should be stressed that since the spline coefficients of the first and last intervals vanish by definition, the curves coincide in the limits, i.e., we have

\lim_{σ \to 0^{+}} w_{H}^{σ ↓ AXK} (σ) = \lim_{σ \to 0^{+}} w_{H}^{σ ↑ AXK} (σ) = \lim_{σ \to 0^{+}} w_{H}^{dRPA} (σ) = \frac{1}{2}

(46)

\lim_{σ \to \infty} w_{H}^{σ ↓ AXK} (σ) = \lim_{σ \to \infty} w_{H}^{σ ↑ AXK} (σ) = \lim_{σ \to \infty} w_{H}^{dRPA} (σ) = 0

(47)

\lim_{σ \to 0^{+}} w_{x}^{σ ↓ AXK} (σ) = \lim_{σ \to 0^{+}} w_{x}^{σ ↑ AXK} (σ) = \lim_{σ \to 0^{+}} w_{x}^{AXK} (σ) = \frac{1}{2}

(48)

\lim_{σ \to \infty} w_{x}^{σ ↓ AXK} (σ) = \lim_{σ \to \infty} w_{x}^{σ ↑ AXK} (σ) = \lim_{σ \to \infty} w_{x}^{AXK} (σ) = 0

(49)

and that the large magnitude of the oscillations originates from the prefactor if 1/σ in eqs 42–44, which is canceled by the multiplication with X₀ in eq 45. While the weights of the Hartree kernel w_H behave similarly for the top-down and bottom-up approaches, the same is not true for w_x: Whereas for the σ↑AXK-functionals [Figure 5c,d], the shape of w_x resembles that of w_H, the definition of the top-down correlation energy in eq 28 means that the leading h(λσ) term of w_x is negated and thus, for σ↓AXK-functionals, w_x [Figure 5b] behaves opposite to w_H [Figure 5a]. In particular, this behavior implies that in regions in which the σ-functionals amplify the contributions of F_H, e.g., σ ∼ 0.3, the exchange kernel is more strongly attenuated in σ↓AXK-functionals than in σ↑AXK-functionals, leading to higher contributions of same-spin self-correlation. This distinction likely explains the much better performance of σ↑AXK-functionals for problems dominated by self-interaction errors. Based on these findings, one might consider optimizing two separate sets of splines for the Hartree and exchange kernel, though some care would need to be taken to avoid the double-counting of (approximate) exchange contributions. However, such considerations, while interesting, exceed the scope of this work and have thus not been studied yet.

Figure 5. Weight functions of the Hartree (a, c) and exchange (b, d) kernels for σ↓AXK-functionals (a, b) and σ↑AXK-functionals (c, d) for PBE0 reference orbitals.

The severity of unphysical self-correlation can be estimated by considering hydrogen-like systems, for which the correlation energy should be identically equal to zero. In Table 3, we list correlation energies for the hydrogen atom computed with the aug-cc-pV6Z basis set. (89) As already discussed by Erhard et al., (28) the scaled σ-functionals reduce self-correlation by a factor of 10 for the H atom compared to the dRPA and plain σ-functionals (improvements for He⁺ or

H_{2}^{+}

are less pronounced, but still substantial). The inclusion of AXK appears to have a similar effect and drastically reduces the correlation energy–an improvement which is, in large part, undone by the σ↓AXK approach, which results in higher correlation energies than even dRPA for PBE reference orbitals and only modest improvements over dRPA for PBE0 reference orbitals. In contrast, the new σ↑AXK ansatz exhibits less unphysical self-correlation, which aligns with our observation that σ↑AXK-functionals are better suited for systems dominated by self-interaction errors, although the very low correlation energies of dRPA+AXK and scaled σ-functionals are not quite matched.

Table 3. Correlation Energies for the Hydrogen Atom in mE_h Computed Using the aug-cc-pV6Z Basis Set

	@PBE	@PBE0
dRPA	–20.768	–18.714
σ(W1)	–23.399	–16.080
scσ(S1)	–2.110	–1.512
dRPA+AXK	–2.219	–1.737
σ↓AXK(A1)	–22.174	–9.121
σ↓AXK(A2)	–24.143	–10.391
σ↑AXK(A1)	–8.851	–5.595
σ↑AXK(A2)	–7.602	–5.540

Finally, we gauge the quality of the predicted dispersion interactions on the neon dimer, which serves as a good model system for pure dispersive interactions without covalent or electrostatic contributions. As the well depth of this system is comparatively small, a high degree of accuracy with respect to integration grids and basis sets is paramount. Therefore, we performed a complete basis set (CBS) extrapolation to account for the basis set incompleteness error (BSIE) using the two-point formula of Helgaker et al., (90,91)

E (\infty) = \frac{X^{3} E (X) - Y^{3} E (Y)}{X^{3} - Y^{3}}

(50)

where we extrapolate using results computed with the aug-cc-pwCVQZ and aug-cc-pwCV5Z basis sets, (92−94) i.e., X = 5, Y = 4, with the appropriate auxiliary basis for the computation of the Coulomb and correlation energies. (65,95) In accordance with the basis sets, we correlate all electrons and do not apply a frozen-core approximation for these calculations. DFT and sn-LinK integrations were performed on the large gm7 grids, whereas the frequency integration used a 50-point Gauss–Legendre grid. To eliminate potential basis set superposition errors, we determined the equilibrium distance and dissociation energy based on counterpoise-corrected interaction energies. (96)

In Table 4, we report the equilibrium bond lengths R_e and dissociation energies D_e computed using PBE reference orbitals–an equivalent analysis with PBE0 reference orbitals could unfortunately not be performed due to some numerical instabilities in the

I_{2}

term (eq 38) arising for the quintuple-zeta basis set. While these instabilities are small in absolute terms (≲ 1 μE_h), they are substantial enough to prohibit a quantitative analysis for weakly bound systems such as Ne₂. We also list corrected CCSD(T) values from ref (97) in Table 4. A direct comparison reveals that all RPA-based methods significantly underestimate the well depth, whereas at the KS-DFT level, the dissociation energy is greatly overestimated. Note that at the KS-DFT level, we do not include any empirical dispersion correction, meaning that the observed interaction potential is highly functional-dependent: Whereas PBE or, for example, functionals including PW91 exchange (98,99) tend to predict a bound noble gas dimers (albeit not to quantitative accuracy), functionals built upon the exchange functional of Becke (100) often predict repulsive interactions. (101) Returning to Table 4, most correlated approaches overestimate the equilibrium bond length; in particular, dRPA@PBE and σ↓AXK(A2)@PBE deviate from the CCSD(T) value by more than 0.1 Å. While the scσ(S1) functional does predict a considerably higher dissociation energy, the bond length is practically unchanged compared to dRPA. In contrast, both σ↑AXK-functionals yield substantially shorter bond lengths which are in much better agreement with the CCSD(T) values. However, while the predicted dissociation energies are increased compared to the plain dRPA+AXK results, the σ-functionals still fall short of the reference value of 28.868 cm^–1 by a considerable margin.

Table 4. Dissociation Energies D_e and Equilibrium Bond Lengths R_e for the Neon Dimer Computed from CBS-Extrapolated Interaction Energies^a Using PBE Reference Orbitals

	ref^b	DFT	dRPA	σ	scσ	AXK	σ↓AXK		σ↑AXK
				W1	S1		A1	A2	A1	A2
D_e (cm^–1)	28.868	44.050	11.087	11.669	18.607	11.851	13.296	10.695	19.054	16.614
R_e (Å)	3.1007	3.0748	3.2029	3.1598	3.1927	3.1895	3.1672	3.2045	3.0790	3.0875

^{^a}

Counterpoise-corrected interaction energies with CBS extrapolation from aug-cc-pwCVQZ to aug-cc-pwCV5Z basis set according to eq 50.

^{^b}

CCSD(T)/aug-cc-pV5Z+bf interaction energies with corrections for core correlation, BSIE, full-CI extrapolation, and relativistic effects. Data from ref (97).

In addition, Figure 6 shows the complete dissociation curves for the considered functionals. As noted by Modrzejewski et al., (14) at the DFT level the binding energy decays too quickly due to lacking long-range correlation, which is generally corrected for by the RPA-based approaches with the exception of the scσ(S1)-functional, for which the interaction instead appears to decay too slowly compared to CCSD(T). Again, there is a clearly visible difference between the σ↓AXK-functionals, which more or less yield the same dissociation profile as the plain dRPA+AXK method, and the σ↑AXK-functionals, which shift the minimum toward lower bond lengths and a higher binding energy.

Figure 6. CBS-extrapolated neon dimer interaction energies computed using PBE reference orbitals for (a) KS-DFT, (b) σ-functionals and scaled σ-functionals, (c) top-down σ↓AXK-functionals, and (d) bottom-up σ↑AXK-functionals. For reference, corrected CCSD(T) values from ref (97) are also included; data between points has been interpolated using cubic splines.

It is interesting to note that the σ↑AXK-functionals show better agreement with the corrected CCSD(T) values than σ↓AXK even though they performed, on average, worse for the noncovalent interaction benchmarks of the GMTKN55 database; particularly for the RG18 test set, which includes noble gas oligomers. A potential explanation of this discrepancy lies in the basis set incompleteness of the def2-QZVP basis set used for the evaluation of the GMTKN55 database, as well as the smaller minimax grid we used for the numerical frequency integration. The slow basis set convergence of RPA-based methods for dispersion interactions is a well-documented issue (14,102) that warrants further investigation, however, an in-depth analysis of individual benchmarks clearly exceeds the scope of this work.

Conclusions

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We have derived and benchmarked so-called “bottom-up” σ↑AXK functionals in which the σ-functional modification is inserted before the coupling strength integration is performed, leading to nonlinear terms of the cubic splines. Following optimization on the ASCDB database, we have shown the new σ↑AXK functionals to drastically improve on the previous σ↓AXK functionals for benchmarks commonly connected to electronic self-interaction problems, such as atomization energies, electron affinities, and ionization potentials, while otherwise mostly retaining the excellent performance for the GMTKN55 database without the need for empirical dispersion corrections, particularly when using the hybrid PBE0 functional as reference. Since the computational cost is dominated by the exchange kernel, the use of hybrid functionals for the generation of reference orbitals is highly encouraged; furthermore, the A1 parametrization gives, on average, slightly better overall results than the A2 parametrization. Concerning the distinction between σ↓AXK and σ↑AXK, the precise ordering of the different methods depends on the error metric in question, as σ↓AXK functionals lead to slightly lower errors for noncovalent interactions which are weighted strongly in the WTMAD-1 and WTMAD-2 metrics. We therefore propose the usage of σ↓AXK@PBE0 for systems characterized predominantly by noncovalent interactions, whereas the σ↑AXK@PBE0 functionals seem well-suited for self-interaction-dominated problems, in particular, processes such as ionization which do not retain the particle number. These improvements may be connected to a stronger screening of the exchange kernel for small values of σ in the top-down approach which is remedied in the new bottom-up approach.

While the exploration of different parametrizations, further comparison to existing functionals, and especially the reduction of the large computational overhead arising from the exchange kernel are of course desirable, we expect the σ↑AXK functionals to be a useful supplement to the already existing σ-, scaled σ-, and τ-functionals for the high-quality simulation of chemical reactivity, and a viable alternative to other fifth-rung methods such as dispersion-corrected double-hybrid functionals.

Supporting Information

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The Supporting Information is available free of charge at https://pubs.acs.org/doi/10.1021/acs.jpca.4c05289.

MAEs for all subsets of the ASCDB and GMTKN55 databases for σ↑AXK functionals; weight functions w_H and w_x for PBE reference orbitals; and spline coefficients for σ↑AXK functionals (PDF)

jp4c05289_si_001.pdf (337.8 kb)

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Author Information

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Corresponding Author
- Christian Ochsenfeld - Department of Chemistry, Ludwig-Maximilians-Universität München, Butenandtstr. 5-13, D-81377 Munich, Germany; Max-Planck-Institute for Solid State Research, Heisenbergstr. 1, D-70569 Stuttgart, Germany; https://orcid.org/0000-0002-4189-6558; Email: [email protected]
Author
- Yannick Lemke - Department of Chemistry, Ludwig-Maximilians-Universität München, Butenandtstr. 5-13, D-81377 Munich, Germany; https://orcid.org/0000-0002-0930-5287
Author Contributions
Y.L.: Conceptualization; Investigation; Methodology; Software; Data curation; Formal analysis; Validation; Visualization; Writing–original draft. C.O.: Supervision; Project Administration; Funding acquisition; Writing–review and editing.
Funding
Open access funded by Max Planck Society.
Notes
The authors declare no competing financial interest.

Acknowledgments

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The authors thank Dr. Jörg Kussmann (LMU Munich) for providing access to a development version of the FermiONs++ program package. Financial support was provided by the Deutsche Forschungsgemeinschaft (DFG) through grant CRC 325 “Assembly Controlled Chemical Photocatalysis” (grant no. 444632635). CO acknowledges additional financial support as Max-Planck-Fellow at the Max Planck Institute for Solid State Research (MPI-FKF) Stuttgart.

References

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Kreppel, Andrea; Graf, Daniel; Laqua, Henryk; Ochsenfeld, Christian
Journal of Chemical Theory and Computation (2020), 16 (5), 2985-2994CODEN: JCTCCE; ISSN:1549-9618. (American Chemical Society)
A formulation of range-sepd. RPA based on our efficient ω-CDGD-RI-RPA [J. Chem. Theory Comput.2018, 14, 2505] method and a large scale benchmark study are presented. By application to the GMTKN55 data set, we obtain a comprehensive picture of the performance of range-sepd. RPA in general main group thermochem., kinetics, and noncovalent interactions. The results show that range-sepd. RPA performs stably over the broad range of mol. chem. included in the GMTKN55 set. It improves significantly over semilocal DFT but it is still less accurate than modern dispersion cor. double-hybrid functionals. Furthermore, range-sepd. RPA shows a faster basis set convergence compared to std. full-range RPA making it a promising applicable approach with only one empirical parameter.
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Trushin, E.; Thierbach, A.; Görling, A. Toward chemical accuracy at low computational cost: Density-functional theory with σ-functionals for the correlation energy. J. Chem. Phys. 2021, 154, 014104, DOI: 10.1063/5.0026849
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Toward chemical accuracy at low computational cost: Density-functional theory with σ-functionals for the correlation energy
Trushin, Egor; Thierbach, Adrian; Goerling, Andreas
Journal of Chemical Physics (2021), 154 (1), 014104CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics)
We introduce new functionals for the Kohn-Sham correlation energy that are based on the adiabatic-connection fluctuation-dissipation (ACFD) theorem and are named σ-functionals. Like in the well-established direct RPA (dRPA), σ-functionals require as input exclusively eigenvalues σ of the frequency-dependent KS response function. In the new functionals, functions of σ replace the σ-dependent dRPA expression in the coupling-const. and frequency integrations contained in the ACFD theorem. We optimize σ-functionals with the help of ref. sets for atomization, reaction, transition state, and non-covalent interaction energies. The optimized functionals are to be used in a post-self-consistent way using orbitals and eigenvalues from conventional Kohn-Sham calcns. employing the exchange-correlation functional of Perdew, Burke, and Ernzerhof. The accuracy of the presented approach is much higher than that of dRPA methods and is comparable to that of high-level wave function methods. Reaction and transition state energies from σ-functionals exhibit accuracies close to 1 kcal/mol and thus approach chem. accuracy. For the 10 966 reactions of the W4-11RE ref. set, the mean abs. deviation is 1.25 kcal/mol compared to 3.21 kcal/mol in the dRPA case. Non-covalent binding energies are accurate to a few tenths of a kcal/mol. The presented approach is highly efficient, and the post-self-consistent calcn. of the total energy requires less computational time than a d.-functional calcn. with a hybrid functional and thus can be easily carried out routinely. σ-Functionals can be implemented in any existing dRPA code with negligible programming effort. (c) 2021 American Institute of Physics.
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Fauser, S.; Trushin, E.; Neiss, C.; Görling, A. Chemical accuracy with σ-functionals for the Kohn–Sham correlation energy optimized for different input orbitals and eigenvalues. J. Chem. Phys. 2021, 155, 134111, DOI: 10.1063/5.0059641
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Chemical accuracy with σ-functionals for the Kohn-Sham correlation energy optimized for different input orbitals and eigenvalues
Fauser, Steffen; Trushin, Egor; Neiss, Christian; Goerling, Andreas
Journal of Chemical Physics (2021), 155 (13), 134111CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics)
Recently, a new type of orbital-dependent functional for the Kohn-Sham (KS) correlation energy, σ-functionals, was introduced. Tech., σ-functionals are closely related to the well-known direct RPA (dRPA). Within the dRPA, a function of the eigenvalues σ of the frequency-dependent KS response function is integrated over purely imaginary frequencies. In σ-functionals, this function is replaced by one that is optimized with respect to ref. sets of atomization, reaction, transition state, and non-covalent interaction energies. The previously introduced σ-functional uses input orbitals and eigenvalues from KS calcns. with the generalized gradient approxn. (GGA) exchange-correlation functional of Perdew, Burke, and Ernzerhof (PBE). Here, σ-functionals using input orbitals and eigenvalues from the meta-GGA TPSS and the hybrid-functionals PBE0 and B3LYP are presented and tested. The no. of ref. sets taken into account in the optimization of the σ-functionals is larger than in the first PBE based σ-functional and includes sets with 3d-transition metal compds. Therefore, also a reparameterized PBE based σ-functional is introduced. The σ-functionals based on PBE0 and B3LYP orbitals and eigenvalues reach chem. accuracy for main group chem. For the 10 966 reactions from the highly accurate W4-11RE ref. set, the B3LYP based σ-functional exhibits a mean av. deviation of 1.03 kcal/mol compared to 1.08 kcal/mol for the coupled cluster singles doubles perturbative triples method if the same valence quadruple zeta basis set is used. For 3d-transition metal chem., accuracies of about 2 kcal/mol are reached. The computational effort for the post-self-consistent evaluation of the σ-functional is lower than that of a preceding PBE0 or B3LYP calcn. for typical systems. (c) 2021 American Institute of Physics.
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Fauser, S.; Förster, A.; Redeker, L.; Neiss, C.; Erhard, J.; Trushin, E.; Görling, A. Basis Set Requirements of σ-Functionals for Gaussian- and Slater-Type Basis Functions and Comparison with Range-Separated Hybrid and Double Hybrid Functionals. J. Chem. Theory Comput. 2024, 20, 2404– 2422, DOI: 10.1021/acs.jctc.3c01132
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Drontschenko, V.; Graf, D.; Laqua, H.; Ochsenfeld, C. Efficient Method for the Computation of Frozen-Core Nuclear Gradients within the Random Phase Approximation. J. Chem. Theory Comput. 2022, 18, 7359– 7372, DOI: 10.1021/acs.jctc.2c00774
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Efficient Method for the Computation of Frozen-Core Nuclear Gradients within the Random Phase Approximation
Drontschenko, Viktoria; Graf, Daniel; Laqua, Henryk; Ochsenfeld, Christian
Journal of Chemical Theory and Computation (2022), 18 (12), 7359-7372CODEN: JCTCCE; ISSN:1549-9618. (American Chemical Society)
A method for the evaluation of anal. frozen-core gradients within the RPA is presented. We outline an efficient way to evaluate the response of the d. of active electrons arising only when introducing the frozen-core approxn. and constituting the main difficulty, together with the response of the std. Kohn-Sham d. The general framework allows to extend the outlined procedure to related electron correlation methods in the AO basis that require the evaluation of d. responses, such as second-order Moller-Plesset perturbation theory or coupled cluster variants. By using Cholesky decompd. densities - which reintroduce the occupied index in the time-detg. steps - we are able to achieve speedups of 20-30% (depending on the size of the basis set) by using the frozen-core approxn., which is of similar magnitude as for MO formulations. We further show that the errors introduced by the frozen-core approxn. are practically insignificant for mol. geometries.
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Neiss, C.; Fauser, S.; Görling, A. Geometries and vibrational frequencies with Kohn–Sham methods using σ-functionals for the correlation energy. J. Chem. Phys. 2023, 158, 044107, DOI: 10.1063/5.0129524
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Geometries and vibrational frequencies with Kohn-Sham methods using σ-functionals for the correlation energy
Neiss, Christian; Fauser, Steffen; Goerling, Andreas
Journal of Chemical Physics (2023), 158 (4), 044107CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics)
Recently, Kohn-Sham (KS) methods with new correlation functionals, called σ-functionals, have been introduced. Tech., σ-functionals are closely related to the well-known RPA; formally, σ-functionals are rooted in perturbation theory along the adiabatic connection. If employed in a post-SCF manner in a Gaussian basis set framework, then, σ-functional methods are computationally very efficient. Moreover, for main group chem., σ-functionals are highly accurate and can compete with high-level wave-function methods. For reaction and transition state energies, e.g., chem. accuracy of 1 kcal/mol is reached. Here, we show how to calc. first derivs. of the total energy with respect to nuclear coordinates for methods using σ-functionals and then carry out geometry optimizations for test sets of main group mols., transition metal compds., and non-covalently bonded systems. For main group mols., we addnl. calc. vibrational frequencies. σ-Functional methods are found to yield very accurate geometries and vibrational frequencies for main group mols. superior not only to those from conventional KS methods but also to those from RPA methods. For geometries of transition metal compds., not surprisingly, best geometries are found for RPA methods, while σ-functional methods yield somewhat less good results. This is attributed to the fact that in the optimization of σ-functionals, transition metal compds. could not be represented well due to the lack of reliable ref. data. For non-covalently bonded systems, σ-functionals yield geometries of the same quality as the RPA or as conventional KS schemes combined with dispersion corrections. (c) 2023 American Institute of Physics.
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Drontschenko, V.; Bangerter, F. H.; Ochsenfeld, C. Analytical Second-Order Properties for the Random Phase Approximation: Nuclear Magnetic Resonance Shieldings. J. Chem. Theory Comput. 2023, 19, 7542– 7554, DOI: 10.1021/acs.jctc.3c00542
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Fauser, S.; Drontschenko, V.; Ochsenfeld, C.; Görling, A. Accurate NMR Shieldings with σ-Functionals. J. Chem. Theory Comput. 2024, 20, 6028– 6036, DOI: 10.1021/acs.jctc.4c00512
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Drontschenko, V.; Ochsenfeld, C. Low-Scaling, Efficient and Memory Optimized Computation of Nuclear Magnetic Resonance Shieldings within the Random Phase Approximation using Cholesky-Decomposed Densities and an Attenuated Coulomb Metric. J. Phys. Chem. A 2024, 128, 7950– 7965, DOI: 10.1021/acs.jpca.4c02773
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Glasbrenner, M.; Graf, D.; Ochsenfeld, C. Benchmarking the Accuracy of the Direct Random Phase Approximation and σ-Functionals for NMR Shieldings. J. Chem. Theory Comput. 2022, 18, 192– 205, DOI: 10.1021/acs.jctc.1c00866
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Benchmarking the Accuracy of the Direct Random Phase Approximation and σ-Functionals for NMR Shieldings
Glasbrenner, Michael; Graf, Daniel; Ochsenfeld, Christian
Journal of Chemical Theory and Computation (2022), 18 (1), 192-205CODEN: JCTCCE; ISSN:1549-9618. (American Chemical Society)
A method for computing NMR shieldings with the direct RPA (RPA) and the closely related σ-functionals [Trushin, E.; Thierbach, A.; Gorling, A. Toward chem. accuracy at low computational cost: d. functional theory with σ-functionals for the correlation energy. J. Chem. Phys.2021,154, 014104] is presented, which is based on a finite-difference approach. The accuracy is evaluated in benchmark calcns. using high-quality coupled cluster values as a ref. Our results show that the accuracy of the computed NMR shieldings using direct RPA is strongly dependent on the d. functional theory ref. orbitals and improves with increasing amts. of exact Hartree-Fock exchange in the functional. NMR shieldings computed with the direct RPA with a Hartree-Fock ref. are significantly more accurate than MP2 shieldings and comparable to CCSD shieldings. Also, the basis set convergence is analyzed and it is shown that at least triple-zeta basis sets are required for reliable results.
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Görling, A. Hierarchies of methods towards the exact Kohn–Sham correlation energy based on the adiabatic-connection fluctuation-dissipation theorem. Phys. Rev. B 2019, 99, 235120, DOI: 10.1103/PhysRevB.99.235120
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Erhard, J.; Bleiziffer, P.; Görling, A. Power Series Approximation for the Correlation Kernel Leading to Kohn–Sham Methods Combining Accuracy, Computational Efficiency, and General Applicability. Phys. Rev. Lett. 2016, 117, 143002, DOI: 10.1103/PhysRevLett.117.143002
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Erhard, J.; Fauser, S.; Trushin, E.; Görling, A. Scaled σ-functionals for the Kohn–Sham correlation energy with scaling functions from the homogeneous electron gas. J. Chem. Phys. 2022, 157, 114105, DOI: 10.1063/5.0101641
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Scaled σ-functionals for Kohn-Sham correlation energy with scaling function from homogeneous electron gas
Erhard, Jannis; Fauser, Steffen; Trushin, Egor; Goerling, Andreas
Journal of Chemical Physics (2022), 157 (11), 114105CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics)
A review. The recently introduced σ-functionals constitute a new type of functionals for the Kohn-Sham (KS) correlation energy. The σ-functionals are based on the adiabatic-connection fluctuation-dissipation theorem, are computationally closely related to the well-known direct RPA (dRPA), and are formally rooted in many-body perturbation theory along the adiabatic connection. In σ-functionals, the function of the eigen values σ of the Kohn-Sham response matrix that enters the coupling const. and frequency integration in the dRPA is replaced by another function optimized with the help of ref. sets of atomization, reaction, transition state, and non-covalent interaction energies, and σ-Functionals are highly accurate and yield chem. accuracy of 1 kcal/mol in reaction or transition state energies, in main group chem. A shortcoming of σ-functionals is their inability to accurately describe processes involving a change of the electron no., such as ionization or electron attachment. This problem is attributed to unphys. self-interactions caused by the neglect of the exchange kernel in the dRPA and σ-functionals. Here, we tackle this problem by introducing a frequency- and σ-dependent scaling of the eigenvalues σ of the KS response function that models the effect of the exchange kernel. The scaling factors are detected with the help of the homogeneous electron gas. The resulting scaled σ-functionals retain the accuracy of their unscaled parent functionals but in addn. yield very accurate ionization potentials and electron affinities. Moreover, atomization and total energies are found to be exceptionally accurate. Scaled σ-functionals are computationally highly efficient like their unscaled counterparts. (c) 2022 American Institute of Physics.
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Lemke, Y.; Ochsenfeld, C. Highly accurate σ- and τ-functionals for beyond-RPA methods with approximate exchange kernels. J. Chem. Phys. 2023, 159, 194104, DOI: 10.1063/5.0173042
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Highly accurate σ- and τ-functionals for beyond-RPA methods with approximate exchange kernels
Lemke, Yannick; Ochsenfeld, Christian
Journal of Chemical Physics (2023), 159 (19), 194104CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics)
σ-Functionals are promising new developments for the Kohn-Sham correlation energy based upon the direct RPA (dRPA) within the adiabatic connection formalism, providing impressive improvements over dRPA for a broad range of benchmarks. However, σ-functionals exhibit a high amt. of self-interaction inherited from the approxns. made within dRPA. Inclusion of an exchange kernel in deriving the coupling-strength-dependent d.-d. response function leads to so-called τ-functionals, which - apart from a fourth-order Taylor series expansion - have only been realized in an approx. fashion so far to the best of our knowledge, most notably in the form of scaled σ-functionals. In this work, we derive, optimize, and benchmark three types of σ- and τ-functionals including approx. exchange effects in the form of an antisymmetrized Hartree kernel. These functionals, based on a second-order screened exchange type contribution in the adiabatic connection formalism, the electron-hole time-dependent Hartree-Fock kernel (eh-TDHF) otherwise known as RPA with exchange (RPAx), and an approxn. thereof known as approx. exchange kernel (AXK), are optimized on the ASCDB database using two new parametrizations named A1 and A2. In addn., we report a first full evaluation of σ- and τ-functionals on the GMTKN55 database, revealing our exchange-including functionals to considerably outperform existing σ-functionals while being highly competitive with some of the best double-hybrid functionals of the original GMTKN55 publication. In particular, the σ-functionals based on AXK and τ-functionals based on RPAx with PBE0 ref. stand out as highly accurate approaches for a wide variety of chem. relevant problems. (c) 2023 American Institute of Physics.
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Grüneis, A.; Marsman, M.; Harl, J.; Schimka, L.; Kresse, G. Making the random phase approximation to electronic correlation accurate. J. Chem. Phys. 2009, 131, 154115, DOI: 10.1063/1.3250347
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Making the random phase approximation to electronic correlation accurate
Grueneis, Andreas; Marsman, Martijn; Harl, Judith; Schimka, Laurids; Kresse, Georg
Journal of Chemical Physics (2009), 131 (15), 154115/1-154115/5CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics)
We show that the inclusion of second-order screened exchange to the RPA allows for an accurate description of electronic correlation in atoms and solids clearly surpassing the random phase approxn., but not yet approaching chem. accuracy. From a fundamental point of view, the method is self-correlation free for one-electron systems. From a practical point of view,the approach yields correlation energies for atoms, as well as for the jellium electron gas within a few kcal/mol of exact values, atomization energies within typically 2-3 kcal/mol of expt., and excellent lattice consts. for ionic and covalently bonded solids (0.2% error). The computational complexity is only O(N5), comparable to canonical second-order Moller-Plesset perturbation theory, which should allow for routine calcns. on many systems. (c) 2009 American Institute of Physics.
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Bates, J. E.; Furche, F. Communication: Random phase approximation renormalized many-body perturbation theory. J. Chem. Phys. 2013, 139, 171103, DOI: 10.1063/1.4827254
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Communication: Random phase approximation renormalized many-body perturbation theory
Bates, Jefferson E.; Furche, Filipp
Journal of Chemical Physics (2013), 139 (17), 171103/1-171103/4CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics)
We derive a renormalized many-body perturbation theory (MBPT) starting from the RPA. This RPA-renormalized perturbation theory extends the scope of single-ref. MBPT methods to small-gap systems without significantly increasing the computational cost. The leading correction to RPA, termed the approx. exchange kernel (AXK), substantially improves upon RPA atomization energies and ionization potentials without affecting other properties such as barrier heights where RPA is already accurate. Thus, AXK is more balanced than second-order screened exchange, which tends to overcorrect RPA for systems with stronger static correlation. Similarly, AXK avoids the divergence of second-order Moller-Plesset (MP2) theory for small gap systems and delivers a much more consistent performance than MP2 across the periodic table at comparable cost. RPA+AXK thus is an accurate, non-empirical, and robust tool to assess and improve semi-local d. functional theory for a wide range of systems previously inaccessible to first-principles electronic structure calcns. (c) 2013 American Institute of Physics.
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Goerigk, L.; Hansen, A.; Bauer, C.; Ehrlich, S.; Najibi, A.; Grimme, S. A look at the density functional theory zoo with the advanced GMTKN55 database for general main group thermochemistry, kinetics and noncovalent interactions. Phys. Chem. Chem. Phys. 2017, 19, 32184, DOI: 10.1039/C7CP04913G
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A look at the density functional theory zoo with the advanced GMTKN55 database for general main group thermochemistry, kinetics and noncovalent interactions
Goerigk, Lars; Hansen, Andreas; Bauer, Christoph; Ehrlich, Stephan; Najibi, Asim; Grimme, Stefan
Physical Chemistry Chemical Physics (2017), 19 (48), 32184-32215CODEN: PPCPFQ; ISSN:1463-9076. (Royal Society of Chemistry)
We present the GMTKN55 benchmark database for general main group thermochem., kinetics and noncovalent interactions. Compared to its popular predecessor GMTKN30, it allows assessment across a larger variety of chem. problems - with 13 new benchmark sets being presented for the first time - and it also provides ref. values of significantly higher quality for most sets. GMTKN55 comprises 1505 relative energies based on 2462 single-point calcns. and it is accessible to the user community via a dedicated website. Herein, we demonstrate the importance of better ref. values, and we re-emphasize the need for London-dispersion corrections in d. functional theory (DFT) treatments of thermochem. problems, including Minnesota methods. We assessed 217 variations of dispersion-cor. and -uncorrected d. functional approxns., and carried out a detailed anal. of 83 of them to identify robust and reliable approaches. Double-hybrid functionals are the most reliable approaches for thermochem. and noncovalent interactions, and they should be used whenever tech. feasible. These are, in particular, DSD-BLYP-D3(BJ), DSD-PBEP86-D3(BJ), and B2GPPLYP-D3(BJ). The best hybrids are ωB97X-V, M052X-D3(0), and ωB97X-D3, but we also recommend PW6B95-D3(BJ) as the best conventional global hybrid. At the meta-generalized-gradient (meta-GGA) level, the SCAN-D3(BJ) method can be recommended. Other meta-GGAs are outperformed by the GGA functionals revPBE-D3(BJ), B97-D3(BJ), and OLYP-D3(BJ). We note that many popular methods, such as B3LYP, are not part of our recommendations. In fact, with our results we hope to inspire a change in the user community's perception of common DFT methods. We also encourage method developers to use GMTKN55 for cross-validation studies of new methodologies.
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Perdew, J. P.; Burke, K.; Ernzerhof, M. Generalized Gradient Approximation Made Simple. Phys. Rev. Lett. 1996, 77, 3865– 3868, DOI: 10.1103/PhysRevLett.77.3865
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Generalized gradient approximation made simple
Perdew, John P.; Burke, Kieron; Ernzerhof, Matthias
Physical Review Letters (1996), 77 (18), 3865-3868CODEN: PRLTAO; ISSN:0031-9007. (American Physical Society)
Generalized gradient approxns. (GGA's) for the exchange-correlation energy improve upon the local spin d. (LSD) description of atoms, mols., and solids. We present a simple derivation of a simple GGA, in which all parameters (other than those in LSD) are fundamental consts. Only general features of the detailed construction underlying the Perdew-Wang 1991 (PW91) GGA are invoked. Improvements over PW91 include an accurate description of the linear response of the uniform electron gas, correct behavior under uniform scaling, and a smoother potential.
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Adamo, C.; Barone, V. Toward reliable density functional methods without adjustable parameters: The PBE0 model. J. Chem. Phys. 1999, 110, 6158– 6170, DOI: 10.1063/1.478522
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Toward reliable density functional methods without adjustable parameters: the PBE0 model
Adamo, Carlo; Barone, Vincenzo
Journal of Chemical Physics (1999), 110 (13), 6158-6170CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics)
We present an anal. of the performances of a parameter free d. functional model (PBE0) obtained combining the so called PBE generalized gradient functional with a predefined amt. of exact exchange. The results obtained for structural, thermodn., kinetic and spectroscopic (magnetic, IR and electronic) properties are satisfactory and not far from those delivered by the most reliable functionals including heavy parameterization. The way in which the functional is derived and the lack of empirical parameters fitted to specific properties make the PBE0 model a widely applicable method for both quantum chem. and condensed matter physics.
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Ernzerhof, M.; Scuseria, G. E. Assessment of the Perdew–Burke–Ernzerhof exchange-correlation functional. J. Chem. Phys. 1999, 110, 5029– 5036, DOI: 10.1063/1.478401
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Assessment of the Perdew-Burke-Ernzerhof exchange-correlation functional
Ernzerhof, Matthias; Scuseria, Gustavo E.
Journal of Chemical Physics (1999), 110 (11), 5029-5036CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics)
In order to discriminate between approxns. to the exchange-correlation energy EXC[ρ↑,ρ↓], we employ the criterion of whether the functional is fitted to a certain exptl. data set or if it is constructed to satisfy phys. constraints. We present extensive test calcns. for atoms and mols., with the nonempirical local spin-d. (LSD) and the Perdew-Burke-Ernzerhof (PBE) functional and compare our results with results obtained with more empirical functionals. For the atomization energies of the G2 set, we find that the PBE functional shows systematic errors larger than those of commonly used empirical functionals. The PBE ionization potentials, electron affinities, and bond lengths are of accuracy similar to those obtained from empirical functionals. Furthermore, a recently proposed hybrid scheme using exact exchange together with PBE exchange and correlation is investigated. For all properties studied here, the PBE hybrid gives an accuracy comparable to the frequently used empirical B3LYP hybrid scheme. Phys. principles underlying the PBE and PBE hybrid scheme are examd. and the range of their validity is discussed.
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Langreth, D. C.; Perdew, J. P. The exchange-correlation energy of a metallic surface. Solid State Commun. 1975, 17, 1425, DOI: 10.1016/0038-1098(75)90618-3
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Langreth, D. C.; Perdew, J. P. Exchange-correlation energy of a metallic surface: Wave-vector analysis. Phys. Rev. B 1977, 15, 2884, DOI: 10.1103/PhysRevB.15.2884
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Petersilka, M.; Gossmann, U. J.; Gross, E. K. U. Excitation Energies from Time-Dependent Density-Functional Theory. Phys. Rev. Lett. 1996, 76, 1212– 1215, DOI: 10.1103/PhysRevLett.76.1212
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Excitation energies from time-dependent density-functional theory
Petersilka, M.; Gossmann, U. J.; Gross, E. K. U.
Physical Review Letters (1996), 76 (8), 1212-15CODEN: PRLTAO; ISSN:0031-9007. (American Physical Society)
A new d.-functional approach to calc. the excitation spectrum of many-electron systems is proposed. It is shown that the full linear d. response of the interacting system, which has poles at the exact excitation energies, can rigorously be expressed in terms of the response function of the noninteracting (Kohn-Sham) system and a frequency-dependent exchange-correlation kernel. Using this expression, the poles of the full response function are obtained by systematic improvement upon the poles of the Kohn-Sham response function. Numerical results are presented for Be, Mg, Ca, Zn, Sr, and Cd atoms.
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Garrick, R.; Natan, A.; Gould, T.; Kronik, L. Exact Generalized Kohn–Sham Theory for Hybrid Functionals. Phys. Rev. X 2020, 10, 021040, DOI: 10.1103/PhysRevX.10.021040
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40
Whitten, J. L. Coulombic potential energy integrals and approximations. J. Chem. Phys. 1973, 58, 4496– 4501, DOI: 10.1063/1.1679012
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Coulombic potential energy integrals and approximations
Whitten, J. L.
Journal of Chemical Physics (1973), 58 (10), 4496-501CODEN: JCPSA6; ISSN:0021-9606.
Theorems are derived which establish a method of approxg. 2-particle Coulombic potential energy integrals, [.vphi.a(1)|r12-1|.vphi.b-(2)], in terms of approx. charge ds. .vphi.a' and .vphi.b'. Rigorous error bounds, |[.vphi.a(1)|r12-1|.vphi.b(2)] - [.vphi.a'(1)|r12-1|.vphi.b'(2)]| ≤ δ, are simply expressed in terms of information calcd. sep. for the pair of ds. .vphi.a and .vphi.b' and the pair .vphi.b and .vphi.b'. From the structure of the bound, a simple method of optimizing charge d. approxns. such that δ is minimized is derived. The framework of the theory appears to be well suited for application to the approxn. of electron repulsion integrals which occur in mol. structure theory, and applications to the approxn. of integrals over Slater orbitals or grouped Gaussian functions are discussed.
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Dunlap, B. I.; Connolly, J. W. D.; Sabin, J. R. On some approximations in applications of X_α theory. J. Chem. Phys. 1979, 71, 3396– 3402, DOI: 10.1063/1.438728
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On some approximations in applications of X α theory
Dunlap, B. I.; Connolly, J. W. D.; Sabin, J. R.
Journal of Chemical Physics (1979), 71 (8), 3396-402CODEN: JCPSA6; ISSN:0021-9606.
An approx. Xα functional is proposed from which the charge d. fitting equations follow variationally. LCAO Xα calcns. on at. Ni and H2 show the method independent of the fitting (auxiliary) bases to within 0.02 eV. Variational properties assocd. with both orbital and auxiliary basis set incompleteness are used to approach within 0.2 eV the Xα total energy limit for the N mol.
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Vahtras, O.; Almlöf, J.; Feyereisen, M. W. Integral approximations for LCAO-SCF calculations. Chem. Phys. Lett. 1993, 213, 514– 518, DOI: 10.1016/0009-2614(93)89151-7
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Integral approximations for LCAO-SCF calculations
Vahtras, O.; Almloef, J.; Feyereisen, M. W.
Chemical Physics Letters (1993), 213 (5-6), 514-18CODEN: CHPLBC; ISSN:0009-2614.
Three-center approxns. to the four-center integrals occurring in ab initio LCAO calcns. are investigated. Significant gains in computer time can be obtained without sacrificing accuracy, if a suitable expansion basis is chosen.
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Furche, F. Developing the random phase approximation into a practical post-Kohn–Sham correlation model. J. Chem. Phys. 2008, 129, 114105, DOI: 10.1063/1.2977789
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43
Developing the random phase approximation into a practical post-Kohn-Sham correlation model
Furche, Filipp
Journal of Chemical Physics (2008), 129 (11), 114105/1-114105/8CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics)
The RPA (RPA) to the d. functional correlation energy systematically improves upon many limitations of present semilocal functionals, but was considered too computationally expensive for widespread use in the past. Here a phys. appealing reformulation of the RPA correlation model is developed that substantially reduces its computational complexity. The d. functional RPA correlation energy is shown to equal one-half times the difference of all RPA electronic excitation energies computed at full and first order coupling. Thus, the RPA correlation energy may be considered as a difference of electronic zero point vibrational energies, where each eigenmode corresponds to an electronic excitation. This surprisingly simple result is intimately related to plasma theories of electron correlation. Differences to electron pair correlation models underlying popular correlated wave function methods are discussed. The RPA correlation energy is further transformed into an explicit functional of the Kohn-Sham orbitals. The only nontrivial ingredient to this functional is the sign function of the response operator. A stable iterative algorithm to evaluate this sign function based on the Newton-Schulz iteration is presented. Integral direct implementations scale as the fifth power of the system size, similar to second order Moeller-Plesset calcns. With these improvements, RPA may become the long-sought robust and efficient zero order post-Kohn-Sham correlation model. (c) 2008 American Institute of Physics.
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Eshuis, H.; Yarkony, J.; Furche, F. Fast computation of molecular random phase approximation correlation energies using resolution of the identity and imaginary frequency integration. J. Chem. Phys. 2010, 132, 234114, DOI: 10.1063/1.3442749
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Fast computation of molecular random phase approximation correlation energies using resolution of the identity and imaginary frequency integration
Eshuis, Henk; Yarkony, Julian; Furche, Filipp
Journal of Chemical Physics (2010), 132 (23), 234114/1-234114/9CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics)
The RPA is an increasingly popular post-Kohn-Sham correlation method, but its high computational cost has limited mol. applications to systems with few atoms. Here we present an efficient implementation of RPA correlation energies based on a combination of resoln. of the identity (RI) and imaginary frequency integration techniques. We show that the RI approxn. to four-index electron repulsion integrals leads to a variational upper bound to the exact RPA correlation energy if the Coulomb metric is used. Auxiliary basis sets optimized for second-order Moller-Plesset (MP2) calcns. are well suitable for RPA, as is demonstrated for the HEAT and MOLEKEL benchmark sets. Using imaginary frequency integration rather than diagonalization to compute the matrix square root necessary for RPA, evaluation of the RPA correlation energy requires O(N4logN) operations and O(N3) storage only; the price for this dramatic improvement over existing algorithms is a numerical quadrature. We propose a numerical integration scheme that is exact in the two-orbital case and converges exponentially with the no. of grid points. For most systems, 30-40 grid points yield μH accuracy in triple zeta basis sets, but much larger grids are necessary for small gap systems. The lowest-order approxn. to the present method is a post-Kohn-Sham frequency-domain version of opposite-spin Laplace-transform RI-MP2. Timings for polyacenes with up to 30 atoms show speed-ups of two orders of magnitude over previous implementations. The present approach makes it possible to routinely compute RPA correlation energies of systems well beyond 100 atoms, as is demonstrated for the octapeptide angiotensin II. (c) 2010 American Institute of Physics.
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Bleiziffer, P.; Heßelmann, A.; Görling, A. Resolution of identity approach for the Kohn–Sham correlation energy within the exact-exchange random-phase approximation. J. Chem. Phys. 2012, 136, 134102, DOI: 10.1063/1.3697845
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Lemke, Y.; Graf, D.; Kussmann, J.; Ochsenfeld, C. An assessment of orbital energy corrections for the direct random phase approximation and explicit σ-functionals. Mol. Phys. 2023, 121, e2098862, DOI: 10.1080/00268976.2022.2098862
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Heßelmann, A.; Görling, A. Random phase approximation correlation energies with exact Kohn–Sham exchange. Mol. Phys. 2010, 108, 359– 372, DOI: 10.1080/00268970903476662
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Random phase approximation correlation energies with exact Kohn-Sham exchange
Hesselmann, Andreas; Goerling, Andreas
Molecular Physics (2010), 108 (3-4), 359-372CODEN: MOPHAM; ISSN:0026-8976. (Taylor & Francis Ltd.)
The RPA correlation energy is expressed in terms of the exact local Kohn-Sham (KS) exchange potential and corresponding adiabatic and nonadiabatic exchange kernels for d.-functional ref. determinants. The approach naturally extends the RPA method in which, conventionally, only the Coulomb kernel is included. By comparison with the coupled cluster singles doubles with perturbative triples method it is shown for a set of small mols. that the new RPA method based on KS exchange yields correlation energies more accurate than RPA on the basis of Hartree-Fock exchange.
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Ángyán, J. G.; Liu, R.-F.; Toulouse, J.; Jansen, G. Correlation Energy Expressions from the Adiabatic-Connection Fluctuation–Dissipation Theorem Approach. J. Chem. Theory Comput. 2011, 7, 3116– 3130, DOI: 10.1021/ct200501r
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Correlation Energy Expressions from the Adiabatic-Connection Fluctuation-Dissipation Theorem Approach
Angyan, Janos G.; Liu, Ru-Fen; Toulouse, Julien; Jansen, Georg
Journal of Chemical Theory and Computation (2011), 7 (10), 3116-3130CODEN: JCTCCE; ISSN:1549-9618. (American Chemical Society)
We explore several RPA correlation energy variants within the adiabatic-connection fluctuation-dissipation theorem approach. These variants differ in the way the exchange interactions are treated. One of these variants, named dRPA-II, is original to this work and closely resembles the second-order screened exchange (SOSEX) method. We discuss and clarify the connections among different RPA formulations. We derive the spin-adapted forms of all the variants for closed-shell systems and test them on a few at. and mol. systems with and without range sepn. of the electron-electron interaction.
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Heßelmann, A. Random-phase-approximation correlation method including exchange interactions. Phys. Rev. A 2012, 85, 012517, DOI: 10.1103/PhysRevA.85.012517
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Random-phase-approximation correlation method including exchange interactions
Hesselmann, Andreas
Physical Review A: Atomic, Molecular, and Optical Physics (2012), 85 (1-A), 012517/1-012517/10CODEN: PLRAAN; ISSN:1050-2947. (American Physical Society)
Two random-phase-approxn. correlation methods are introduced that take into account exchange interactions. The first one, termed RPAX, is obtained from a simple modification of the ring coupled-cluster doubles amplitude equation, while the second, termed RPAX2, is based on the first method using a slightly modified update equation for the amplitudes. It is shown that this second RPAX2 method can be implemented with a computational algorithm that scales only with the fifth power of the mol. size with the aid of d. fitting or the Cholesky decompn. of two-electron integrals. It is thus not much more costly than std. second-order perturbation theory methods and can be applied to quite large mol. systems. Moreover, numerical tests for chem. reaction energies and intermol. interaction energies have shown that the RPAX2 method, if based on a Perdew-Burke-Ernzerhof exchange Kohn-Sham ref. determinant, yields results which are very close to coupled-cluster with single, double, and perturbative triple excitations ref. results.
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Eshuis, H.; Bates, J. E.; Furche, F. Electron correlation methods based on the random phase approximation. Theor. Chem. Acc. 2012, 131, 1084, DOI: 10.1007/s00214-011-1084-8
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51
Mussard, B.; Rocca, D.; Jansen, G.; Ángyán, J. G. Dielectric Matrix Formulation of Correlation Energies in the Random Phase Approximation: Inclusion of Exchange Effects. J. Chem. Theory Comput. 2016, 12, 2191– 2202, DOI: 10.1021/acs.jctc.5b01129
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Dielectric Matrix Formulation of Correlation Energies in the Random Phase Approximation: Inclusion of Exchange Effects
Mussard, Bastien; Rocca, Dario; Jansen, Georg; Angyan, Janos G.
Journal of Chemical Theory and Computation (2016), 12 (5), 2191-2202CODEN: JCTCCE; ISSN:1549-9618. (American Chemical Society)
Starting from the general expression for the ground state correlation energy in the adiabatic-connection fluctuation-dissipation theorem (ACFDT) framework, it is shown that the dielec. matrix formulation, which is usually applied to calc. the direct RPA (dRPA) correlation energy, can be used for alternative RPA expressions including exchange effects. Within this famework, the ACFDT analog of the second order screened exchange (SOSEX) approxn. leads to a logarithmic formula for the correlation energy similar to the direct RPA expression. Alternatively, the contribution of the exchange can be included in the kernel used to evaluate the response functions. In this case, the use of an approx. kernel is crucial to simplify the formalism and to obtain a correlation energy in logarithmic form. Tech. details of the implementation of these methods are discussed, and it is shown that one can take advantage of d. fitting or Cholesky decompn. techniques to improve the computational efficiency; a discussion on the numerical quadrature made on the frequency variable is also provided. A series of test calcns. on at. correlation energies and mol. reaction energies shows that exchange effects are instrumental for improvement over direct RPA results.
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Dixit, A.; Ángyán, J. G.; Rocca, D. Improving the accuracy of ground-state correlation energies within a plane-wave basis set: The electron-hole exchange kernel. J. Chem. Phys. 2016, 145, 104105, DOI: 10.1063/1.4962352
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Improving the accuracy of ground-state correlation energies within a plane-wave basis set: The electron-hole exchange kernel
Dixit, Anant; Angyan, Janos G.; Rocca, Dario
Journal of Chemical Physics (2016), 145 (10), 104105/1-104105/12CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics)
A new formalism was recently proposed to improve RPA correlation energies by including approx. exchange effects. Within this framework, by keeping only the electron-hole contributions to the exchange kernel, two approxns. can be obtained: An adiabatic connection analog of the second order screened exchange (AC-SOSEX) and an approx. electron-hole time-dependent Hartree-Fock (eh-TDHF). Here we show how this formalism is suitable for an efficient implementation within the plane-wave basis set. The response functions involved in the AC-SOSEX and eh-TDHF equations can indeed be compactly represented by an auxiliary basis set obtained from the diagonalization of an approx. dielec. matrix. Addnl., the explicit calcn. of unoccupied states can be avoided by using d. functional perturbation theory techniques and the matrix elements of dynamical response functions can be efficiently computed by applying the Lanczos algorithm. As shown by several applications to reaction energies and weakly bound dimers, the inclusion of the electron-hole kernel significantly improves the accuracy of ground-state correlation energies with respect to RPA and semi-local functionals. (c) 2016 American Institute of Physics.
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Chen, G. P.; Agee, M. M.; Furche, F. Performance and Scope of Perturbative Corrections to Random-Phase Approximation Energies. J. Chem. Theory Comput. 2018, 14, 5701– 5714, DOI: 10.1021/acs.jctc.8b00777
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Performance and Scope of Perturbative Corrections to Random-Phase Approximation Energies
Chen, Guo P.; Agee, Matthew M.; Furche, Filipp
Journal of Chemical Theory and Computation (2018), 14 (11), 5701-5714CODEN: JCTCCE; ISSN:1549-9618. (American Chemical Society)
It has been suspected since the early days of the RPA that corrections to RPA correlation energies result mostly from short-range correlation effects and are thus amenable to perturbation theory. Here we test this hypothesis by analyzing formal and numerical results for the most common beyond-RPA perturbative corrections, including the bare second-order exchange (SOX), second-order screened exchange (SOSEX), and approx. exchange kernel (AXK) methods. Our anal. is facilitated by efficient and robust algorithms based on the resoln.-of-the-identity (RI) approxn. and numerical frequency integration, which enable benchmark beyond-RPA calcns. on medium- and large-size mols. with size-independent accuracy. The AXK method systematically improves upon RPA, SOX, and SOSEX for reaction barrier heights, reaction energies, and noncovalent interaction energies of main-group compds. The improved accuracy of AXK compared with SOX and SOSEX is attributed to stronger screening of bare SOX in AXK. For reactions involving transition-metal compds., particularly 3d transition-metal dimers, the AXK correction is too small and can even have the wrong sign. These observations are rationalized by a measure ‾α of the effective coupling strength for beyond-RPA correlation. When the effective coupling strength increases beyond a crit. ‾α value of approx. 0.5, the RPA errors increase rapidly and perturbative corrections become unreliable. Thus, perturbation theory can systematically correct RPA but only for systems and properties qual. well captured by RPA, as indicated by small ‾α values.
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Kussmann, J.; Ochsenfeld, C. Pre-selective screening for matrix elements in linear-scaling exact exchange calculations. J. Chem. Phys. 2013, 138, 134114, DOI: 10.1063/1.4796441
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Pre-selective screening for matrix elements in linear-scaling exact exchange calculations
Kussmann, Joerg; Ochsenfeld, Christian
Journal of Chemical Physics (2013), 138 (13), 134114/1-134114/7CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics)
We present a simple but accurate preselection method based on Schwarz integral ests. to det. the significant elements of the exact exchange matrix before its evaluation, thus providing an asymptotical linear-scaling behavior for non-metallic systems. Our screening procedure proves to be highly suitable for exchange matrix calcns. on massively parallel computing architectures, such as graphical processing units, for which we present a first linear-scaling exchange matrix evaluation algorithm. (c) 2013 American Institute of Physics.
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Kussmann, J.; Ochsenfeld, C. Preselective Screening for Linear-Scaling Exact Exchange-Gradient Calculations for Graphics Processing Units and General Strong-Scaling Massively Parallel Calculations. J. Chem. Theory Comput. 2015, 11, 918, DOI: 10.1021/ct501189u
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Preselective Screening for Linear-Scaling Exact Exchange-Gradient Calculations for Graphics Processing Units and General Strong-Scaling Massively Parallel Calculations
Kussmann, Joerg; Ochsenfeld, Christian
Journal of Chemical Theory and Computation (2015), 11 (3), 918-922CODEN: JCTCCE; ISSN:1549-9618. (American Chemical Society)
We present an extension of our recently presented PreLinK scheme (J. Chem. Phys. 2013, 138, 134114) for the exact exchange contribution to nuclear forces. The significant contributions to the exchange gradient are detd. by preselection based on accurate shell-pair contributions to the SCF exchange energy prior to the calcn. Therefore, our method is highly suitable for massively parallel electronic structure calcns. because of an efficient load balancing of the significant contributions only and an unhampered control flow. The efficiency of our method is shown for several illustrative calcns. on single GPU servers, as well as for hybrid MPI/CUDA parallel calcns. with the largest system comprising 3369 atoms and 26952 basis functions.
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Kussmann, J.; Laqua, H.; Ochsenfeld, C. Highly Efficient Resolution-of-Identity Density Functional Theory Calculations on Central and Graphics Processing Units. J. Chem. Theory Comput. 2021, 17, 1512, DOI: 10.1021/acs.jctc.0c01252
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Highly Efficient Resolution-of-Identity Density Functional Theory Calculations on Central and Graphics Processing Units
Kussmann, Joerg; Laqua, Henryk; Ochsenfeld, Christian
Journal of Chemical Theory and Computation (2021), 17 (3), 1512-1521CODEN: JCTCCE; ISSN:1549-9618. (American Chemical Society)
We present an efficient method to evaluate Coulomb potential matrixes using the resoln. of identity approxn. and semilocal exchange-correlation potentials on central (CPU) and graphics processing units (GPU). The new GPU-based RI-algorithm shows a high performance and ensures the favorable scaling with increasing basis set size as the conventional CPU-based method. Furthermore, our method is based on the J-engine algorithm [White, Head-Gordon, J. Chem. Phys., 1996, 7, 2620], which allows for further optimizations that also provide a significant improvement of the corresponding CPU-based algorithm. Due to the increased performance for the Coulomb evaluation, the calcn. of the exchange-correlation potential of d. functional theory on CPUs quickly becomes a bottleneck to the overall computational time. Hence, we also present a GPU-based algorithm to evaluate the exchange-correlation terms, which results in an overall high-performance method for d. functional calcns. The algorithms to evaluate the potential and nuclear deriv. terms are discussed, and their performance on CPUs and GPUs is demonstrated for illustrative calcns.
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Weigend, F. Accurate Coulomb-fitting basis sets for H to Rn. Phys. Chem. Chem. Phys. 2006, 8, 1057, DOI: 10.1039/b515623h
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Accurate Coulomb-fitting basis sets for H to Rn
Weigend, Florian
Physical Chemistry Chemical Physics (2006), 8 (9), 1057-1065CODEN: PPCPFQ; ISSN:1463-9076. (Royal Society of Chemistry)
A series of auxiliary basis sets to fit Coulomb potentials for the elements H to Rn (except lanthanides) is presented. For each element only one auxiliary basis set is needed to approx. Coulomb energies in conjunction with orbital basis sets of split valence, triple zeta valence and quadruple zeta valence quality with errors of typically below ca. 0.15 kJ mol-1 per atom; this was demonstrated in conjunction with the recently developed orbital basis sets of types def2-SV(P), def2-TZVP and def2-QZVPP for a large set of small mols. representing (nearly) each element in all of its common oxidn. states. These auxiliary bases are slightly more than three times larger than orbital bases of split valence quality. Compared to non-approximated treatments, computation times for the Coulomb part are reduced by a factor of ca. 8 for def2-SV(P) orbital bases, ca. 25 for def2-TZVP and ca. 100 for def2-QZVPP orbital bases.
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Laqua, H.; Kussmann, J.; Ochsenfeld, C. Efficient and Linear-Scaling Seminumerical Method for Local Hybrid Density Functionals. J. Chem. Theory Comput. 2018, 14, 3451– 3458, DOI: 10.1021/acs.jctc.8b00062
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Efficient and Linear-Scaling Seminumerical Method for Local Hybrid Density Functionals
Laqua, Henryk; Kussmann, Joerg; Ochsenfeld, Christian
Journal of Chemical Theory and Computation (2018), 14 (7), 3451-3458CODEN: JCTCCE; ISSN:1549-9618. (American Chemical Society)
Local hybrid functionals, i.e., functionals with local dependence on the exact exchange energy d., generalize the popular class of global hybrid functionals and extend the applicability of d. functional theory to electronic structures that require an accurate description of static correlation. However, the higher computational cost compared to conventional Kohn-Sham d. functional theory restrained their widespread application. Here, we present a low-prefactor, linear-scaling method to evaluate the local hybrid exchange-correlation potential as well as the corresponding nuclear forces by employing a seminumerical integration scheme. In the seminumerical scheme, one integration in the electron repulsion integrals is carried out anal. and the other one is carried out numerically, employing an integration grid. A high computational efficiency is achieved by combining the preLinK method with explicit screening of integrals for batches of grid points to minimize the screening overhead. This new method, denoted as preLinX, provides an 8-fold performance increase for a DNA fragment contg. four base pairs as compared to existing S- and P-junction-based methods. In this way, our method allows for the evaluation of local hybrid functionals at a cost similar to that of global hybrid functionals. The linear-scaling behavior, efficiency, accuracy, and multi-node parallelization of the presented method is demonstrated for large systems contg. more than 1000 atoms.
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Laqua, H.; Thompson, T. H.; Kussmann, J.; Ochsenfeld, C. Highly Efficient, Linear-Scaling Seminumerical Exact-Exchange Method for Graphic Processing Units. J. Chem. Theory Comput. 2020, 16, 1456, DOI: 10.1021/acs.jctc.9b00860
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Highly Efficient, Linear-Scaling Seminumerical Exact-Exchange Method for Graphic Processing Units
Laqua, Henryk; Thompson, Travis H.; Kussmann, Joerg; Ochsenfeld, Christian
Journal of Chemical Theory and Computation (2020), 16 (3), 1456-1468CODEN: JCTCCE; ISSN:1549-9618. (American Chemical Society)
We present a highly efficient and asymptotically linear-scaling graphic processing unit accelerated seminumerical exact-exchange method (sn-LinK). We go beyond our previous central processing unit-based method (H. Laqua et al., 2018) by employing our recently developed integral bounds (T.H. Thomson and C. Ochsenfeld, 2019) and high-accuracy numerical integration grid (H. Laqua et al., 2018). The accuracy is assessed for several established test sets, providing errors significantly below 1mEh for the smallest grid. Moreover, a comprehensive performance anal. for large mols. between 62 and 1347 atoms is provided, revealing the outstanding performance of our method, in particular, for large basis sets such as the polarized quadruple-zeta level with diffuse functions.
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Laqua, H.; Kussmann, J.; Ochsenfeld, C. An improved molecular partitioning scheme for numerical quadratures in density functional theory. J. Chem. Phys. 2018, 149, 204111, DOI: 10.1063/1.5049435
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An improved molecular partitioning scheme for numerical quadratures in density functional theory
Laqua, Henryk; Kussmann, Joerg; Ochsenfeld, Christian
Journal of Chemical Physics (2018), 149 (20), 204111/1-204111/8CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics)
We present a modification to Becke's mol. partitioning scheme [A. D. Becke, J. Chem. Phys. 88, 2547 (1988)] that provides substantially better accuracy for weakly bound complexes and allows for a faster and linear scaling grid generation without introducing a cutoff error. We present the accuracy of our new partitioning scheme for atomization energies of small mols. and for interaction energies of van der Waals complexes. Furthermore, the efficiency and scaling behavior of the grid generation are demonstrated for large mol. systems with up to 1707 atoms. (c) 2018 American Institute of Physics.
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Lehtola, S.; Steigemann, C.; Oliveira, M. J. T.; Marques, M. A. L. Recent developments in libxc–A comprehensive library of functionals for density functional theory. Software X 2018, 7, 1– 5, DOI: 10.1016/j.softx.2017.11.002
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Weigend, F.; Furche, F.; Ahlrichs, R. Gaussian basis sets of quadruple zeta valence quality for atoms H–Kr. J. Chem. Phys. 2003, 119, 12753– 12762, DOI: 10.1063/1.1627293
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Gaussian basis sets of quadruple zeta valence quality for atoms H-Kr
Weigend, Florian; Furche, Filipp; Ahlrichs, Reinhart
Journal of Chemical Physics (2003), 119 (24), 12753-12762CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics)
We present Gaussian basis sets of quadruple zeta valence quality with a segmented contraction scheme for atoms H to Kr. This extends earlier work on segmented contracted split valence (SV) and triple zeta valence (TZV) basis sets. Contraction coeffs. and orbital exponents are fully optimized in at. Hartree-Fock (HF) calcns. As opposed to other quadruple zeta basis sets, the basis set errors in at. ground-state HF energies are less than 1 mEh and increase smoothly across the Periodic Table, while the no. of primitives is comparably small. Polarization functions are taken partly from previous work, partly optimized in at. MP2 calcns., and for a few cases detd. at the HF level for excited at. states nearly degenerate with the ground state. This leads to basis sets denoted QZVP for HF and d. functional theory (DFT) calcns., and for some atoms to a larger basis recommended for correlated treatments, QZVPP. We assess the performance of the basis sets in mol. HF, DFT, and MP2 calcns. for a sample of diat. and small polyat. mols. by a comparison of energies, bond lengths, and dipole moments with results obtained numerically or using very large basis sets. It is shown that basis sets of quadruple zeta quality are necessary to achieve an accuracy of 1 kcal/mol per bond in HF and DFT atomization energies. For compds. contg. third row as well as alk. and earth alk. metals it is demonstrated that the inclusion of high-lying core orbitals in the active space can be necessary for accurate correlated treatments. The QZVPP basis sets provide sufficient flexibility to polarize the core in those cases. All test calcns. indicate that the new basis sets lead to consistent accuracies in HF, DFT, or correlated treatments even in crit. cases where other basis sets may show deficiencies.
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Rappoport, D.; Furche, F. Property-optimized Gaussian basis sets for molecular response calculations. J. Chem. Phys. 2010, 133, 134105, DOI: 10.1063/1.3484283
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Property-optimized Gaussian basis sets for molecular response calculations
Rappoport, Dmitrij; Furche, Filipp
Journal of Chemical Physics (2010), 133 (13), 134105/1-134105/11CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics)
With recent advances in electronic structure methods, first-principles calcns. of electronic response properties, such as linear and nonlinear polarizabilities, have become possible for mols. with more than 100 atoms. Basis set incompleteness is typically the main source of error in such calcns. since traditional diffuse augmented basis sets are too costly to use or suffer from near linear dependence. To address this problem, we construct the first comprehensive set of property-optimized augmented basis sets for elements H-Rn except lanthanides. The new basis sets build on the Karlsruhe segmented contracted basis sets of split-valence to quadruple-zeta valence quality and add a small no. of moderately diffuse basis functions. The exponents are detd. variationally by maximization of at. Hartree-Fock polarizabilities using anal. deriv. methods. The performance of the resulting basis sets is assessed using a set of 313 mol. static Hartree-Fock polarizabilities. The mean abs. basis set errors are 3.6%, 1.1%, and 0.3% for property-optimized basis sets of split-valence, triple-zeta, and quadruple-zeta valence quality, resp. D. functional and second-order Moller-Plesset polarizabilities show similar basis set convergence. We demonstrate the efficiency of our basis sets by computing static polarizabilities of icosahedral fullerenes up to C720 using hybrid d. functional theory. (c) 2010 American Institute of Physics.
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Weigend, F.; Ahlrichs, R. Balanced basis sets of split valence, triple zeta valence and quadruple zeta valence quality for H to Rn: Design and assessment of accuracy. Phys. Chem. Chem. Phys. 2005, 7, 3297, DOI: 10.1039/b508541a
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Balanced basis sets of split valence, triple zeta valence and quadruple zeta valence quality for H to Rn: Design and assessment of accuracy
Weigend, Florian; Ahlrichs, Reinhart
Physical Chemistry Chemical Physics (2005), 7 (18), 3297-3305CODEN: PPCPFQ; ISSN:1463-9076. (Royal Society of Chemistry)
Gaussian basis sets of quadruple zeta valence quality for Rb-Rn are presented, as well as bases of split valence and triple zeta valence quality for H-Rn. The latter were obtained by (partly) modifying bases developed previously. A large set of more than 300 mols. representing (nearly) all elements-except lanthanides-in their common oxidn. states was used to assess the quality of the bases all across the periodic table. Quantities investigated were atomization energies, dipole moments and structure parameters for Hartree-Fock, d. functional theory and correlated methods, for which we had chosen Moller-Plesset perturbation theory as an example. Finally recommendations are given which type of basis set is used best for a certain level of theory and a desired quality of results.
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Hättig, C. Optimization of auxiliary basis sets for RI-MP2 and RI-CC2 calculations: Core–valence and quintuple-ζ basis sets for H to Ar and QZVPP basis sets for Li to Kr. Phys. Chem. Chem. Phys. 2005, 7, 59– 66, DOI: 10.1039/B415208E
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Hellweg, A.; Hättig, C.; Höfener, S.; Klopper, W. Optimized accurate auxiliary basis sets for RI-MP2 and RI-CC2 calculations for the atoms Rb to Rn. Theor. Chem. Acc. 2007, 117, 587– 597, DOI: 10.1007/s00214-007-0250-5
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Optimized accurate auxiliary basis sets for RI-MP2 and RI-CC2 calculations for the atoms Rb to Rn
Hellweg, Arnim; Haettig, Christof; Hoefener, Sebastian; Klopper, Wim
Theoretical Chemistry Accounts (2007), 117 (4), 587-597CODEN: TCACFW; ISSN:1432-881X. (Springer GmbH)
The introduction of the resoln.-of-the-identity (RI) approxn. for electron repulsion integrals in quantum chem. calcns. requires in addn. to the orbital basis so-called auxiliary or fitting basis sets. We report here such auxiliary basis sets optimized for second-order Moller-Plesset perturbation theory for the recently published (Weigend and Ahlrichs Phys. Chem. Chem. Phys., 2005, 7, 3297-3305) segmented contracted Gaussian basis sets of split, triple-ζ and quadruple-ζ valence quality for the atoms Rb-Rn (except lanthanides). These basis sets are designed for use in connection with small-core effective core potentials including scalar relativistic corrections. Hereby accurate resoln.-of-the-identity calcns. with second-order Moller-Plesset perturbation theory (MP2) and related methods can now be performed for mols. contg. elements from H to Rn. The error of the RI approxn. has been evaluated for a test set of 385 small and medium sized mols., which represent the common oxidn. states of each element, and is compared with the one-electron basis set error, estd. based on highly accurate explicitly correlated MP2-R12 calcns. With the reported auxiliary basis sets the RI error for MP2 correlation energies is typically two orders of magnitude smaller than the one-electron basis set error, independent on the position of the atoms in the periodic table.
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Hellweg, A.; Rappoport, D. Development of new auxiliary basis functions of the Karlsruhe segmented contracted basis sets including diffuse basis functions (def2-SVPD, def2-TZVPPD, and def2-QZVPPD) for RI-MP2 and RI-CC calculations. Phys. Chem. Chem. Phys. 2015, 17, 1010– 1017, DOI: 10.1039/C4CP04286G
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Kaltak, M.; Klimeš, J.; Kresse, G. Low Scaling Algorithms for the Random Phase Approximation: Imaginary Time and Laplace Transformations. J. Chem. Theory Comput. 2014, 10, 2498, DOI: 10.1021/ct5001268
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Low Scaling Algorithms for the Random Phase Approximation: Imaginary Time and Laplace Transformations
Kaltak, Merzuk; Klimes, Jiri; Kresse, Georg
Journal of Chemical Theory and Computation (2014), 10 (6), 2498-2507CODEN: JCTCCE; ISSN:1549-9618. (American Chemical Society)
We det. efficient imaginary frequency and imaginary time grids for second-order Moller-Plesset (MP) perturbation theory. The least-squares and Minimax quadratures are compared for periodic systems, finding that the Minimax quadrature performs slightly better for the considered materials. We show that the imaginary frequency grids developed for second order also perform well for the correlation energy in the direct RPA. Furthermore, we show that the polarizabilities on the imaginary time axis can be Fourier-transformed to the imaginary frequency domain, since the time and frequency Minimax grids are dual to each other. The same duality is obsd. for the least-squares grids. The transformation from imaginary time to imaginary frequency allows one to reduce the time complexity to cubic (in system size), so that RPA correlation energies become accessible for large systems.
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Kaltak, M.; Kresse, G. Minimax isometry method: A compressive sensing approach for Matsubara summation in many-body perturbation theory. Phys. Rev. B 2020, 101, 205145, DOI: 10.1103/PhysRevB.101.205145
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Minimax isometry method: A compressive sensing approach for Matsubara summation in many-body perturbation theory
Kaltak, Merzuk; Kresse, Georg
Physical Review B (2020), 101 (20), 205145CODEN: PRBHB7; ISSN:2469-9969. (American Physical Society)
We present a compressive sensing approach for the long-standing problem of Matsubara summation in many-body perturbation theory. By constructing low-dimensional, almost isometric subspaces of the Hilbert space we obtain optimum imaginary time and frequency grids that allow for extreme data compression of fermionic and bosonic functions in a broad temp. regime. The method is applied to the random phase and self-consistent GW approxn. of the grand potential. Integration and transformation errors are investigated for Si and SrVO3.
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Andrae, D.; Häußermann, U.; Dolg, M.; Stoll, H.; Preuß, H. Energy-adjusted ab initio pseudopotentials for the second and third row transition elements. Theor. Chim. Acta 1990, 77, 123– 141, DOI: 10.1007/BF01114537
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Energy-adjusted ab initio pseudopotentials for the second and third row transition elements
Andrae, D.; Haeussermann, U.; Dolg, M.; Stoll, H.; Preuss, H.
Theoretica Chimica Acta (1990), 77 (2), 123-41CODEN: TCHAAM; ISSN:0040-5744.
Nonrelativistic and quasirelativistic ab initio pseudopotentials substituting the M(z-28)+-core orbitals of the second row transition elements and M(z-60)+-core orbitals of the third row transition elements, resp., and optimized (8s7p6d)/[6s5p3d]-GTO valence basis sets for use in mol. calcns. have been generated. Addnl., corresponding spin-orbit operators have also been derived. At. excitation and ionization energies from numerical HF as well as from SCF pseudopotential calcns. using the derived basis sets differ in most cases by less than 0.1 eV from corresponding numerical all-electron results. Spin-orbit splittings for low-lying states are in reasonable agreement with corresponding all-electron Dirac-Fock (DF) results.
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Metz, B.; Stoll, H.; Dolg, M. Small-core multiconfiguration-Dirac–Hartree–Fock-adjusted pseudopotentials for post-d main group elements: Application to PbH and PbO. J. Chem. Phys. 2000, 113, 2563– 2569, DOI: 10.1063/1.1305880
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Small-core multiconfiguration-Dirac-Hartree-Fock-adjusted pseudopotentials for post-d main group elements: Application to PbH and PbO
Metz, Bernhard; Stoll, Hermann; Dolg, Michael
Journal of Chemical Physics (2000), 113 (7), 2563-2569CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics)
Relativistic pseudopotentials (PPs) of the energy-consistent variety have been generated for the post-d group 13-15 elements, by adjustment to multiconfiguration Dirac-Hartree-Fock data based on the Dirac-Coulomb-Breit Hamiltonian. The outer-core (n-1)spd shells are explicitly treated together with the nsp valence shell, with these PPs, and the implications of the small-core choice are discussed by comparison to a corresponding large-core PP, in the case of Pb. Results from valence ab initio one- and two-component calcns. using both PPs are presented for the fine-structure splitting of the ns2np2 ground-state configuration of the Pb atom, and for spectroscopic consts. of PbH (X 2Π1/2, 2Π3/2) and PbO (X 1Σ+). In addn., a combination of small-core and large-core PPs has been explored in spin-free-state shifted calcns. for the above mols.
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Peterson, K. A. Systematically convergent basis sets with relativistic pseudopotentials. I. Correlation consistent basis sets for the post-d group 13–15 elements. J. Chem. Phys. 2003, 119, 11099– 11112, DOI: 10.1063/1.1622923
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Systematically convergent basis sets with relativistic pseudopotentials. I. Correlation consistent basis sets for the post-d group 13-15 elements
Peterson, Kirk A.
Journal of Chemical Physics (2003), 119 (21), 11099-11112CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics)
New correlation consistent-like basis sets have been developed for the post-d group 13-15 elements (Ga-As, In-Sb, Tl-Bi) employing accurate, small-core relativistic pseudopotentials. The resulting basis sets, which are denoted cc-pVnZ-PP, are appropriate for valence electron correlation and range in size from (8s7p7d)/[4s3p2d] for the cc-pVDZ-PP to (16s13p12d3f2g1h)/[7s7p5d3f2g1h] for the cc-pV5Z-PP sets. Benchmark calcns. on selected diat. mols. (As2, Sb2, Bi2, AsN, SbN, BiN, GeO, SnO, PbO, GaCl, InCl, TlCl, GaH, InH, and TlH) are reported using these new basis sets at the coupled cluster level of theory. Much like their all-electron counterparts, the cc-pVnZ-PP basis sets yield systematic convergence of total energies and spectroscopic consts. In several cases all-electron benchmark calcns. were also carried out for comparison. The results from the pseudopotential and all-electron calcns. were nearly identical when scalar relativity was accurately included in the all-electron work. Diffuse-augmented basis sets, aug-cc-pVnZ-PP, have also been developed and have been used in calcns. of the at. electron affinities.
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Peterson, K. A.; Figgen, D.; Goll, E.; Stoll, H.; Dolg, M. Systematically convergent basis sets with relativistic pseudopotentials. II. Small-core pseudopotentials and correlation consistent basis sets for the post-d group 16–18 elements. J. Chem. Phys. 2003, 119, 11113– 11123, DOI: 10.1063/1.1622924
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Systematically convergent basis sets with relativistic pseudopotentials. II. Small-core pseudopotentials and correlation consistent basis sets for the post-d group 16-18 elements
Peterson, Kirk A.; Figgen, Detlev; Goll, Erich; Stoll, Hermann; Dolg, Michael
Journal of Chemical Physics (2003), 119 (21), 11113-11123CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics)
A series of correlation consistent basis sets have been developed for the post-d group 16-18 elements in conjunction with small-core relativistic pseudopotentials of the energy-consistent variety. The latter were adjusted to multiconfiguration Dirac-Hartree-Fock data based on the Dirac-Coulomb-Breit Hamiltonian. The outer-core (n-1)spd shells are explicitly treated together with the nsp valence shell with these PPs. The accompanying cc-pVnZ-PP and aug-cc-pVnZ-PP basis sets range in size from DZ to 5Z quality and yield systematic convergence of both Hartree-Fock and correlated total energies. In addn. to the calcn. of at. electron affinities and dipole polarizabilities of the rare gas atoms, numerous mol. benchmark calcns. (HBr, HI, HAt, Br2, I2, At2, SiSe, SiTe, SiPo, KrH+, XeH+, and RnH+) are also reported at the coupled cluster level of theory. For the purposes of comparison, all-electron calcns. using the Douglas-Kroll-Hess Hamiltonian have also been carried out for the halogen-contg. mols. using basis sets of 5Z quality.
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Morgante, P.; Peverati, R. Statistically representative databases for density functional theory via data science. Phys. Chem. Chem. Phys. 2019, 21, 19092– 19103, DOI: 10.1039/C9CP03211H
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Statistically representative databases for density functional theory via data science
Morgante, Pierpaolo; Peverati, Roberto
Physical Chemistry Chemical Physics (2019), 21 (35), 19092-19103CODEN: PPCPFQ; ISSN:1463-9076. (Royal Society of Chemistry)
The amt. of data and no. of databases for the assessment and parameterization of d. functional theory methods has grown substantially in the past two decades. In this work, we introduce a novel cluster anal. technique for d. functional theory calcns. of the electronic structure of atoms and mols. with the goal of creating new statistically significant databases with broad chem. scope, and a manageable no. of data-points. By analyzing without a priori chem. assumptions a population of almost 350k data-points, we create a new database called ASCDB contg. only 200 data-points. This new database holds the same chem. information as the larger population of data from which it is obtained, but with a computational cost that is reduced by several orders of magnitude. The labeling of the significant chem. properties is performed a posteriori on the resulting 16 subsets, classifying them into four areas of chem. importance: non-covalent interactions, thermochem., non-local effects, and unbiased calcns. The anal. of the results and their transferability shows that ASCDB is capable of providing the same information as that of the larger collection of data-such as GMTKN55, MGCDB84, and Minnesota 2015B-for several d. functional theory methods and basis sets. In light of these results, we suggest the use of this new small database as a first inexpensive tool for the evaluation and parameterization of electronic structure theory methods.
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Morgante, P.; Peverati, R. ACCDB: A collection of chemistry databases for broad computational purposes. J. Comput. Chem. 2019, 40, 839– 848, DOI: 10.1002/jcc.25761
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ACCDB: A collection of chemistry databases for broad computational purposes
Morgante, Pierpaolo; Peverati, Roberto
Journal of Computational Chemistry (2019), 40 (6), 839-848CODEN: JCCHDD; ISSN:0192-8651. (John Wiley & Sons, Inc.)
The importance of databases of reliable and accurate data in chem. has substantially increased in the past two decades. Their main usage is to parametrize electronic structure theory methods, and to assess their capabilities and accuracy for a broad set of chem. problems. The collection we present here-ACCDB-includes data from 16 different research groups, for a total of 44,931 unique ref. data points, all at a level of theory significantly higher than d. functional theory, and covering most of the periodic table. It is composed of five databases taken from literature (GMTKN, MGCDB84, Minnesota2015, DP284, and W4-17), two newly developed reaction energy databases (W4-17-RE and MN-RE), and a new collection of databases contg. transition metals. A set of expandable software tools for its manipulation is also presented here for the first time, as well as a case study where ACCDB is used for benchmarking com. CPUs for chem. calcns. © 2018 Wiley Periodicals, Inc.
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Fritsch, F. N.; Butland, J. A Method for Constructing Local Monotone Piecewise Cubic Interpolants. SIAM J. Sci. Stat. Comp. 1984, 5, 300, DOI: 10.1137/0905021
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Peverati, R. Fitting elephants in the density functionals zoo: Statistical criteria for the evaluation of density functional theory methods as a suitable replacement for counting parameters. Int. J. Quantum Chem. 2021, 121, e26379, DOI: 10.1002/qua.26379
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Fitting elephants in the density functionals zoo: Statistical criteria for the evaluation of density functional theory methods as a suitable replacement for counting parameters
Peverati, Roberto
International Journal of Quantum Chemistry (2021), 121 (1), e26379CODEN: IJQCB2; ISSN:0020-7608. (John Wiley & Sons, Inc.)
Counting parameters has become customary in the d. functional theory community as a way to infer the transferability of popular approxns. to the exchange-correlation functionals. Recent work in data science, however, has demonstrated that the no. of parameters of a fitted model is not related to the complexity of the model itself, nor to its eventual overfitting. Using similar arguments, here, we show that it is possible to represent every modern exchange-correlation functional approxns. using just one single parameter. This procedure proves the futility of the no. of parameters as a measure of transferability. To counteract this shortcoming, we introduce and analyze the performance of three statistical criteria for the evaluation of the transferability of exchange-correlation functionals. The three criteria are called Akaike information criterion, Vapnik-Chervonenkis criterion, and cross-validation criterion and are used in a preliminary assessment to rank 60 exchange-correlation functional approxns. using the ASCDB database of chem. data.
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Korth, M.; Grimme, S. Mindless DFT Benchmarking. J. Chem. Theory Comput. 2009, 5, 993– 1003, DOI: 10.1021/ct800511q
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"Mindless" DFT Benchmarking
Korth, Martin; Grimme, Stefan
Journal of Chemical Theory and Computation (2009), 5 (4), 993-1003CODEN: JCTCCE; ISSN:1549-9618. (American Chemical Society)
A diversity-oriented approach for the generation of thermochem. benchmark sets is presented. Test sets consisting of randomly generated "artificial mols." (AMs) are proposed that rely on systematic constraints rather than uncontrolled chem. biases. In this way, the narrow structural space of chem. intuition is opened up and electronically difficult cases can be produced in an unforeseeable manner. For the calcn. of chem. meaningful relative energies, AMs are systematically decompd. into small mols. (hydrides and diatomics). Two different example test sets contg. eight-atom, single-ref., main group AMs with chem. very diverse and unusual structures are generated. Highly accurate all-electron, estd. CCSD(T)/complete basis set ref. energies are also provided. They are used to benchmark the d. functionals S-VWN, BP86, B-LYP, B97-D, PBE, TPSS, PBEh, BH-LYP, B3-PW91, B3-LYP, B2-PLYP, B2GP-PLYP, BMK, MPW1B95, M05, M05-2X, PW6B95, M06, M06-L, and M06-2X. In selected cases, an empirical dispersion correction (DFT-D) has been applied. Due to the compn. of the sets, it is expected that a good performance indicates "robustness" in many different chem. applications. The results of a statistical anal. of the errors for the entire set with 165 entries (av. reaction energy of 117 kcal/mol, dubbed as the MB08-165 set) perfectly fit to the "Jacob's ladder" metaphor for the ordering of d. functionals according to their theor. complexity. The mean abs. deviation (MAD) decreases very strongly from LDA (20 kcal/mol) to GGAs (MAD of about 10 kcal/mol) but then was less pronounced to hybrid-GGAs (MAD of about 6-8 kcal/mol). The best performance (MAD of 4.1-4.2 kcal/mol) is found for the (fifth-rung) double-hybrid functionals B2-PLYP-D and B2GP-PLYP-D, followed by the M06-2X meta-hybrid (MAD of 4.8 kcal/mol). The significance of the proposed approach for thermodn. benchmarking is discussed and related to the obsd. performance ranking also regarding wave function based methods.
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Broyden, C. G. The Convergence of a Class of Double-rank Minimization Algorithms 1. General Considerations. IMA J. Appl. Math. 1970, 6, 76, DOI: 10.1093/imamat/6.1.76
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Fletcher, R. A new approach to variable metric algorithms. Comput. J. 1970, 13, 317, DOI: 10.1093/comjnl/13.3.317
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Goldfarb, D. A family of variable-metric methods derived by variational methods. Math. Comput. 1970, 24, 23, DOI: 10.1090/S0025-5718-1970-0258249-6
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Shanno, D. F. Conditioning of quasi-Newton methods for function minimization. Math. Comput. 1970, 24, 647, DOI: 10.1090/S0025-5718-1970-0274029-X
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Parthiban, S.; Martin, J. M. L. Assessment of W1 and W2 theories for the computation of electron affinities, ionization potentials, heats of formation, and proton affinities. J. Chem. Phys. 2001, 114, 6014– 6029, DOI: 10.1063/1.1356014
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Assessment of W1 and W2 theories for the computation of electron affinities, ionization potentials, heats of formation, and proton affinities
Parthiban, Srinivasan; Martin, Jan M. L.
Journal of Chemical Physics (2001), 114 (14), 6014-6029CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics)
The performance of two recent ab initio computational thermochem. schemes, W1 and W2 theory [J. M. L. Martin and G. de Oliveira, J. Chem. Phys. 111, 1843 (1999)], is assessed for an enlarged sample of thermochem. data consisting of the ionization potentials and electron affinities in the G2-1 and G2-2 sets, as well as the heats of formation in the G2-1 and a subset of the G2-2 set. We find W1 theory to be several times more accurate for ionization potentials and electron affinities than commonly used (and less expensive) computational thermochem. schemes such as G2, G3, and CBS-QB3: W2 theory represents a slight improvement for electron affinities but no significant one for ionization potentials. The use of a two-point A+B/L5 rather than a three-point A+B/CL extrapolation for the SCF component greatly enhances the numerical stability of the W1 method for systems with slow basis set convergence. Inclusion of first-order spin-orbit coupling is essential for accurate ionization potentials and electron affinities involving degenerate electronic states: Inner-shell correlation is somewhat more important for ionization potentials than for electron affinities, while scalar relativistic effects are required for the highest accuracy. The mean deviation from expt. for the G2-1 heats of formation is within the av. exptl. uncertainty. W1 theory appears to be a valuable tool for obtaining benchmark quality proton affinities.
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Karton, A.; Sylvetsky, N.; Martin, J. M. L. W4–17: A diverse and high-confidence dataset of atomization energies for benchmarking high-level electronic structure methods. J. Comput. Chem. 2017, 38, 2063, DOI: 10.1002/jcc.24854
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W4-17: A diverse and high-confidence dataset of atomization energies for benchmarking high-level electronic structure methods
Karton, Amir; Sylvetsky, Nitai; Martin, Jan M. L.
Journal of Computational Chemistry (2017), 38 (24), 2063-2075CODEN: JCCHDD; ISSN:0192-8651. (John Wiley & Sons, Inc.)
Atomization reactions are among the most challenging tests for electronic structure methods. We use the first-principles Weizmann-4 (W4) computational thermochem. protocol to generate the W4-17 dataset of 200 total atomization energies (TAEs) with 3σ confidence intervals of 1 kJ mol-1. W4-17 is an extension of the earlier W4-11 dataset; it includes first- and second-row mols. and radicals with up to eight non-hydrogen atoms. These cover a broad spectrum of bonding situations and multireference character, and as such are an excellent benchmark for the parameterization and validation of highly accurate ab initio methods (e.g., CCSD(T) composite procedures) and double-hybrid d. functional theory (DHDFT) methods. The W4-17 dataset contains two subsets (i) a non-multireference subset of 183 systems characterized by dynamical or moderate nondynamical correlation effects (denoted W4-17-nonMR) and (ii) a highly multireference subset of 17 systems (W4-17-MR). We use these databases to evaluate the performance of a wide range of CCSD(T) composite procedures (e.g., G4, G4(MP2), G4(MP2)-6X, ROG4(MP2)-6X, CBS-QB3, ROCBS-QB3, CBS-APNO, ccCA-PS3, W1, W2, W1-F12, W2-F12, W1X-1, and W2X) and DHDFT methods (e.g., B2-PLYP, B2GP-PLYP, B2K-PLYP, DSD-BLYP, DSD-PBEP86, PWPB95, ωB97X-2(LP), and ωB97X-2(TQZ)). © 2017 Wiley Periodicals, Inc.
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Karton, A.; Daon, S.; Martin, J. M. W4–11: A high-confidence benchmark dataset for computational thermochemistry derived from first-principles W4 data. Chem. Phys. Lett. 2011, 510, 165, DOI: 10.1016/j.cplett.2011.05.007
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W4-11: A high-confidence benchmark dataset for computational thermochemistry derived from first-principles W4 data
Karton, Amir; Daon, Shauli; Martin, Jan M. L.
Chemical Physics Letters (2011), 510 (4-6), 165-178CODEN: CHPLBC; ISSN:0009-2614. (Elsevier B.V.)
We show that the purely first-principles Weizmann-4 (W4) computational thermochem. method developed in our group can reproduce available Active Thermochem. Tables atomization energies for 35 mols. with a 3σ uncertainty of under 1 kJ/mol. We then employ this method to generate the W4-11 dataset of 140 total atomization energies of small first-and second-row mols. and radicals. These cover a broad spectrum of bonding situations and multireference character, and as such are an excellent, quasi-automated benchmark (available electronically as Supporting Information) for parametrization and validation of more approx. methods (such as DFT functionals and composite methods). Secondary contributions such as relativity can be included or omitted at will, unlike with exptl. data. A broad variety of more approx. methods is assessed against the W4-11 benchmark and recommendations are made.
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Margraf, J. T.; Ranasinghe, D. S.; Bartlett, R. J. Automatic generation of reaction energy databases from highly accurate atomization energy benchmark sets. Phys. Chem. Chem. Phys. 2017, 19, 9798, DOI: 10.1039/C7CP00757D
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86
Automatic generation of reaction energy databases from highly accurate atomization energy benchmark sets
Margraf, Johannes T.; Ranasinghe, Duminda S.; Bartlett, Rodney J.
Physical Chemistry Chemical Physics (2017), 19 (15), 9798-9805CODEN: PPCPFQ; ISSN:1463-9076. (Royal Society of Chemistry)
In this contribution, we discuss how reaction energy benchmark sets can automatically be created from arbitrary atomization energy databases. As an example, over 11 000 reaction energies derived from the W4-11 database, as well as some relevant subsets are reported. Importantly, there is only very modest computational overhead involved in computing >11 000 reaction energies compared to 140 atomization energies, since the rate-detg. step for either benchmark is performing the same 140 quantum chem. calcns. The performance of commonly used electronic structure methods for the new database is analyzed. This allows investigating the relationship between the performances for atomization and reaction energy benchmarks based on an identical set of mols. The atomization energy is found to be a weak predictor for the overall usefulness of a method. The performance of d. functional approxns. in light of the no. of empirically optimized parameters used in their design is also discussed.
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Henderson, T. M.; Scuseria, G. E. The connection between self-interaction and static correlation: A random phase approximation perspective. Mol. Phys. 2010, 108, 2511, DOI: 10.1080/00268976.2010.507227
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The connection between self-interaction and static correlation: a random phase approximation perspective
Henderson, Thomas M.; Scuseria, Gustavo E.
Molecular Physics (2010), 108 (19-20), 2511-2517CODEN: MOPHAM; ISSN:0026-8976. (Taylor & Francis Ltd.)
Semi-local d. functional theory suggests a connection between static correlation and self-interaction. It is difficult to make such a connection from the wave function theory perspective, since few wave function methods permit self-interaction error. However, the RPA for ground-state correlation, which has a wave function derivation, does include self-interaction in its direct (Hartree) variant. This variant also describes left-right correlation. The self-interaction can be removed by means of second-order screened exchange; however, this also has neg. consequences for the description of static correlation. This paper discusses the connection between the two concepts (static correlation and self-interaction) from the perspective provided by the RPA.
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Grüning, M.; Gritsenko, O. V.; van Gisbergen, S. J. A.; Baerends, E. J. The Failure of Generalized Gradient Approximations (GGAs) and Meta-GGAs for the Two-Center Three-Electron Bonds in He₂⁺, (H₂O)₂⁺, and (NH₃)₂⁺. J. Phys. Chem. A 2001, 105, 9211, DOI: 10.1021/jp011239k
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The Failure of Generalized Gradient Approximations (GGAs) and Meta-GGAs for the Two-Center Three-Electron Bonds in He2+, (H2O)2+, and (NH3)2+
Gruening, M.; Gritsenko, O. V.; van Gisbergen, S. J. A.; Baerends, E. J.
Journal of Physical Chemistry A (2001), 105 (40), 9211-9218CODEN: JPCAFH; ISSN:1089-5639. (American Chemical Society)
The radical cations He2+, (H2O)2+, and (NH3)2+ with two-center three-electron A-A bonds are investigated at the CI, accurate Kohn-Sham (KS), generalized gradient approxn. (GGA), and meta-GGA levels. Assessment of seven different GGA and six meta-GGA methods shows that the A2+ systems remain a difficult case for d. functional theory (DFT). All methods tested consistently overestimate the stability of A2+: the corresponding De errors decrease for more diffuse valence densities in the series He2+ > (H2O)2+ > (NH3)2+. Upon comparison to the energy terms of the accurate Kohn-Sham solns., the approx. exchange functionals are found to be responsible for the errors of GGA-type methods, which characteristically overestimate the exchange in A2+. These so-called exchange functionals implicitly use localized holes. Such localized holes do occur if there is left-right correlation, i.e., the exchange functionals then also describe non-dynamical correlation. However, in the hemibonded A2+ systems the typical mol. (left-right, non-dynamical) correlation of the two-electron pair bond is absent. The non-dynamical correlation built into the exchange functionals is then spurious and yields too low energies.
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Peterson, K. A.; Woon, D. E.; Dunning, T. H. Benchmark calculations with correlated molecular wave functions. IV. The classical barrier height of the H+H₂→H₂+H reaction. J. Chem. Phys. 1994, 100, 7410– 7415, DOI: 10.1063/1.466884
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Benchmark calculations with correlated molecular wave functions. IV. The classical barrier height of the H + H2 → H2 + H reaction
Peterson, Kirk A.; Woon, David E.; Dunning, Thom H., Jr.
Journal of Chemical Physics (1994), 100 (10), 7410-15CODEN: JCPSA6; ISSN:0021-9606.
Using systematic sequences of correlation consistent Gaussian basis sets from double to sextuple zeta quality, the classical barrier height of the H + H2 exchange reaction has been calcd. by multireference CI (MRCI) methods. The MRCI calcns. for collinear H3 have also been calibrated against large basis set full CI (FCI) results, which demonstrate that the MRCI treatment leads to energies less than 1 μhartree (<0.001 kcal/mol) above the FCI energies. The dependence of both the H2 and H3 total energies on the basis set is found to be very regular, and this behavior has been used to extrapolate to the complete basis set (CBS) limits. The resulting est. of the H-H-H- CBS limit yields a classical barrier height, relative to exact H + H2, of 9.60 ± 0.02 kcal/mol; the best directly calcd. value for the barrier is equal to 9.62 kcal/mol. These results are in excellent agreement with recent quantum Monte Carlo calcns.
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Helgaker, T.; Klopper, W.; Koch, H.; Noga, J. Basis-set convergence of correlated calculations on water. J. Chem. Phys. 1997, 106, 9639– 9646, DOI: 10.1063/1.473863
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Basis-set convergence of correlated calculations on water
Helgaker, Trygve; Klopper, Wim; Koch, Henrik; Noga, Jozef
Journal of Chemical Physics (1997), 106 (23), 9639-9646CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics)
The basis-set convergence of the electronic correlation energy in the water mol. is investigated at the second-order Moller-Plesset level and at the coupled-cluster singles-and-doubles level with and without perturbative triples corrections applied. The basis-set limits of the correlation energy are established to within 2mEh by means of (1) extrapolations from sequences of calcns. using correlation-consistent basis sets and (2) from explicitly correlated calcns. employing terms linear in the inter-electronic distances rij. For the extrapolations to the basis-set limit of the correlation energies, fits of the form a + bX-3 (where X is two for double-zeta sets, three for triple-zeta sets, etc.) are found to be useful. CCSD(T) calcns. involving as many as 492 AOs are reported.
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Halkier, A.; Helgaker, T.; Jo̷rgensen, P.; Klopper, W.; Koch, H.; Olsen, J.; Wilson, A. K. Basis-set convergence in correlated calculations on Ne, N₂, and H₂O. Chem. Phys. Lett. 1998, 286, 243– 252, DOI: 10.1016/S0009-2614(98)00111-0
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Basis-set convergence in correlated calculations on Ne, N2, and H2O
Halkier, Asger; Helgaker, Trygve; Jorgensen, Poul; Klopper, Wim; Koch, Henrik; Olsen, Jeppe; Wilson, Angela K.
Chemical Physics Letters (1998), 286 (3,4), 243-252CODEN: CHPLBC; ISSN:0009-2614. (Elsevier Science B.V.)
Valence and all-electron correlation energies of Ne, N2, and H2O at fixed exptl. geometries are computed at the levels of second-order perturbation theory (MP2) and coupled cluster theory with singles and doubles excitations (CCSD), and singles and doubles excitations with a perturbative triples correction (CCSD(T)). Correlation-consistent polarized valence and core-valence basis sets up to sextuple zeta quality are employed. Guided by basis-set limits established by rij-dependent methods, a no. of extrapolation schemes for use with the correlation-consistent basis sets are investigated. Among the schemes considered here, a linear least-squares procedure applied to the quintuple and sextuple zeta results yields the most accurate extrapolations.
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Dunning, J.; Thom, H. Gaussian basis sets for use in correlated molecular calculations. I. The atoms boron through neon and hydrogen. J. Chem. Phys. 1989, 90, 1007– 1023, DOI: 10.1063/1.456153
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Gaussian basis sets for use in correlated molecular calculations. I. The atoms boron through neon and hydrogen
Dunning, Thom H., Jr.
Journal of Chemical Physics (1989), 90 (2), 1007-23CODEN: JCPSA6; ISSN:0021-9606.
Guided by the calcns. on oxygen in the literature, basis sets for use in correlated at. and mol. calcns. were developed for all of the first row atoms from boron through neon, and for hydrogen. As in the oxygen atom calcns., the incremental energy lowerings, due to the addn. of correlating functions, fall into distinct groups. This leads to the concept of correlation-consistent basis sets, i.e., sets which include all functions in a given group as well as all functions in any higher groups. Correlation-consistent sets are given for all of the atoms considered. The most accurate sets detd. in this way, [5s4p3d2f1g], consistently yield 99% of the correlation energy obtained with the corresponding at.-natural-orbital sets, even though the latter contains 50% more primitive functions and twice as many primitive polarization functions. It is estd. that this set yields 94-97% of the total (HF + 1 + 2) correlation energy for the atoms neon through boron.
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Kendall, R. A.; Dunning, T. H.; Harrison, R. J. Electron affinities of the first-row atoms revisited. Systematic basis sets and wave functions. J. Chem. Phys. 1992, 96, 6796– 6806, DOI: 10.1063/1.462569
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Electron affinities of the first-row atoms revisited. Systematic basis sets and wave functions
Kendall, Rick A.; Dunning, Thom H., Jr.; Harrison, Robert J.
Journal of Chemical Physics (1992), 96 (9), 6796-806CODEN: JCPSA6; ISSN:0021-9606.
The authors describe a reliable procedure for calcg. the electron affinity of an atom and present results for H, B, C, O, and F (H is included for completeness). This procedure involves the use of the recently proposed correlation-consistent basis sets augmented with functions to describe the more diffuse character of the at. anion coupled with a straightforward, uniform expansion of the ref. space for multireference singles and doubles configuration-interaction (MRSD-CI) calcns. A comparison is given with previous results and with corresponding full CI calcns. The most accurate EAs obtained from the MRSD-CI calcns. are (with exptl. values in parentheses): H 0.740 eV (0.754), B 0.258 (0.277), C 1.245 (1.263), O 1.384 (1.461), and F 3.337 (3.401). The EAs obtained from the MR-SDCI calcns. differ by less than 0.03 eV from those predicted by the full CI calcns.
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Peterson, K. A.; Dunning, T. H. Accurate correlation consistent basis sets for molecular core–valence correlation effects: The second row atoms Al–Ar, and the first row atoms B–Ne revisited. J. Chem. Phys. 2002, 117, 10548– 10560, DOI: 10.1063/1.1520138
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Accurate correlation consistent basis sets for molecular core-valence correlation effects: The second row atoms Al-Ar, and the first row atoms B-Ne revisited
Peterson, Kirk A.; Dunning, Thom H., Jr.
Journal of Chemical Physics (2002), 117 (23), 10548-10560CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics)
Correlation consistent basis sets for accurately describing core-core and core-valence correlation effects in atoms and mols. have been developed for the second row atoms Al-Ar. Two different optimization strategies were investigated, which led to two families of core-valence basis sets when the optimized functions were added to the std. correlation consistent basis sets (cc-pVnZ). In the first case, the exponents of the augmenting primitive Gaussian functions were optimized with respect to the difference between all-electron and valence-electron correlated calcns., i.e., for the core-core plus core-valence correlation energy. This yielded the cc-pCVnZ family of basis sets, which are analogous to the sets developed previously for the first row atoms [D. E. Woon and T. H. Dunning, Jr., J. Chem. Phys. 103, 4572 (1995)]. Although the cc-pCVnZ sets exhibit systematic convergence to the all-electron correlation energy at the complete basis set limit, the intershell (core-valence) correlation energy converges more slowly than the intrashell (core-core) correlation energy. Since the effect of including the core electrons on the calcn. of mol. properties tends to be dominated by core-valence correlation effects, a second scheme for detg. the augmenting functions was investigated. In this approach, the exponents of the functions to be added to the cc-pVnZ sets were optimized with respect to just the core-valence (intershell) correlation energy, except that a small amt. of core-core correlation energy was included in order to ensure systematic convergence to the complete basis set limit. These new sets, denoted weighted core-valence basis sets (cc-pwCVnZ), significantly improve the convergence of many mol. properties with n. Optimum cc-pwCVnZ sets for the first-row atoms were also developed and show similar advantages. Both the cc-pCVnZ and cc-pwCVnZ basis sets were benchmarked in coupled cluster [CCSD(T)] calcns. on a series of second row homonuclear diat. mols. (Al2, Si2, P2, S2, and Cl2), as well as on selected diat. mols. involving first row atoms (CO, SiO, PN, and BCl). For the calcn. of core correlation effects on energetic and spectroscopic properties, the cc-pwCVnZ basis sets are recommended over the cc-pCVnZ ones.
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Weigend, F.; Köhn, A.; Hättig, C. Efficient use of the correlation consistent basis sets in resolution of the identity MP2 calculations. J. Chem. Phys. 2002, 116, 3175– 3183, DOI: 10.1063/1.1445115
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Efficient use of the correlation consistent basis sets in resolution of the identity MP2 calculations
Weigend, Florian; Kohn, Andreas; Hattig, Christof
Journal of Chemical Physics (2002), 116 (8), 3175-3183CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics)
The convergence of the second-order Moller-Plesset perturbation theory (MP2) correlation energy with the cardinal no. X is investigated for the correlation consistent basis-set series cc-pVXZ and cc-pV(X+d)Z. For the aug-cc-pVXZ and aug-cc-pV(X+d)Z series the convergence of the MP2 correlation contribution to the dipole moment is studied. It is found that, when d-shell electrons cannot be frozen, the cc-pVXZ and aug-cc-pVXZ basis sets converge much slower for third-row elements then they do for first- and second-row elements. Based on the results of these studies criteria are deduced for the accuracy of auxiliary basis sets used in the resoln. of the identity (RI) approxn. for electron repulsion integrals. Optimized auxiliary basis sets for RI-MP2 calcns. fulfilling these criteria are reported for the sets cc-pVXZ, cc-pV(X+d)Z, aug-cc-pVXZ, and aug-cc-pV(X+d)Z with X=D, T, and Q. For all basis sets the RI error in the MP2 correlation energy is more than two orders of magnitude smaller than the usual basis-set error. For the auxiliary aug-cc-pVXZ and aug-cc-pV(X+d)Z sets the RI error in the MP2 correlation contribution to the dipole moment is one order of magnitude smaller than the usual basis set error. Therefore extrapolations towards the basis-set limit are possible within the RI approxn. for both energies and properties. The redn. in CPU time obtained with the RI approxn. increases rapidly with basis set size. For the cc-pVQZ basis an acceleration by a factor of up to 170 is obsd.
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Boys, S. F.; Bernardi, F. The calculation of small molecular interactions by the differences of separate total energies. Some procedures with reduced errors. Mol. Phys. 1970, 19, 553– 566, DOI: 10.1080/00268977000101561
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The calculation of small molecular interactions by the differences of separate total energies. Some procedures with reduced errors
Boys, S. F.; Bernardi, F.
Molecular Physics (1970), 19 (4), 553-566CODEN: MOPHAM; ISSN:0026-8976. (Taylor & Francis Ltd.)
A new direct difference method for the computation of mol. interactions has been based on a bivariational transcorrelated treatment, together with special methods for the balancing of other errors. It appears that these new features can give a strong redn. in the error of the interaction energy, and they seem to be particularly suitable for computations in the important region near the min. energy. It has been generally accepted that this problem is dominated by unresolved difficulties and the relation of the new methods of these apparent difficulties is analyzed here.
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Gdanitz, R. J. An accurate interaction potential for neon dimer (Ne₂). Chem. Phys. Lett. 2001, 348, 67– 74, DOI: 10.1016/S0009-2614(01)01088-0
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Perdew, J. P. Proceedings of the 75. WE-Heraeus-Seminar and 21st Annual International Symposium on Electronic Structure of Solids; Akademie Verlag: Berlin, 1991; p 11.
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There is no corresponding record for this reference.
99
Perdew, J. P.; Chevary, J. A.; Vosko, S. H.; Jackson, K. A.; Pederson, M. R.; Singh, D. J.; Fiolhais, C. Atoms, molecules, solids, and surfaces: Applications of the generalized gradient approximation for exchange and correlation. Phys. Rev. B 1992, 46, 6671– 6687, DOI: 10.1103/PhysRevB.46.6671
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Atoms, molecules, solids, and surfaces: applications of the generalized gradient approximation for exchange and correlation
Perdew, John P.; Chevary, J. A.; Vosko, S. H.; Jackson, Koblar A.; Pederson, Mark R.; Singh, D. J.; Fiolhais, Carlos
Physical Review B: Condensed Matter and Materials Physics (1992), 46 (11), 6671-87CODEN: PRBMDO; ISSN:0163-1829.
Generalized gradient approxns. (GGA's) seek to improve upon the accuracy of the local-spin-d. (LSD) approxn. in electronic-structure calcns. Perdew and Wang have developed a GGA based on real-space cutoff of the spurious long-range components of the second-order gradient expansion for the exchange-correlation hole. Authors have found that this d. functional performs well in numerical tests for a variety of systems: Total energies of 30 atoms are highly accurate. Ionization energies and electron affinities are improved in a statistical sense, although significant interconfigurational and interterm errors remain. Accurate atomization energies are found for seven hydrocarbon mols., with a rms error per bond of 0.1 eV, compared with 0.7 eV for the LSD approxn. and 2.4 eV for the Hartree-Fock approxn. For atoms and mols., there is a cancellation of error between d. functionals for exchange and correlation, which is most striking whenever the Hartree-Fock result is furthest from expt. The surprising LSD underestimation of the lattice consts. of Li and Na by 3-4% is cor., and the magnetic ground state of solid Fe is restored. The work function, surface energy (neglecting the long-range contribution), and curvature energy of a metallic surface are all slightly reduced in comparison with LSD. Taking account of the pos. long-range contribution, authors find surface and curvature energies in good agreement with exptl. or exact values. Finally, a way is found to visualize and understand the nonlocality of exchange and correlation, its origins, and its phys. effects.
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100
Becke, A. D. Density-functional exchange-energy approximation with correct asymptotic behavior. Phys. Rev. A 1988, 38, 3098– 3100, DOI: 10.1103/PhysRevA.38.3098
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Density-functional exchange-energy approximation with correct asymptotic behavior
Becke, A. D.
Physical Review A: Atomic, Molecular, and Optical Physics (1988), 38 (6), 3098-100CODEN: PLRAAN; ISSN:0556-2791.
Current gradient-cor. d.-functional approxns. for the exchange energies of at. and mol. systems fail to reproduce the correct 1/r asymptotic behavior of the exchange-energy d. A gradient-cor. exchange-energy functional is given with the proper asymptotic limit. This functional, contg. only one parameter, fits the exact Hartree-Fock exchange energies of a wide variety of at. systems with remarkable accuracy, surpassing the performance of previous functionals contg. two parameters or more.
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101
Zhang, Y.; Pan, W.; Yang, W. Describing van der Waals interaction in diatomic molecules with generalized gradient approximations: The role of the exchange functional. J. Chem. Phys. 1997, 107, 7921– 7925, DOI: 10.1063/1.475105
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Describing van der Waals Interaction in diatomic molecules with generalized gradient approximations: the role of the exchange functional
Zhang, Yingkai; Pan, Wei; Yang, Weitao
Journal of Chemical Physics (1997), 107 (19), 7921-7925CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics)
Generalized gradient approxns. have been used to calc. the potential energy curves for six rare gas diat. mols. (He2, Ne2, Ar2, HeNe, ArHe, ArNe). Several generalized gradient approxns. are found to provide a good description of binding in these diat. mols. and show a significant improvement over the local d. approxn. in the prediction of bond lengths and dissocn. energies. It is shown here that the behavior of an exchange functional in the region of small d. and large d. gradient plays a very important role in the ability of the functional to describe this type of van der Waals attraction.
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Eshuis, H.; Furche, F. Basis set convergence of molecular correlation energy differences within the random phase approximation. J. Chem. Phys. 2012, 136, 084105, DOI: 10.1063/1.3687005
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Basis set convergence of molecular correlation energy differences within the random phase approximation
Eshuis, Henk; Furche, Filipp
Journal of Chemical Physics (2012), 136 (8), 084105/1-084105/6CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics)
The basis set convergence of energy differences obtained from the RPA to the correlation energy is investigated for a wide range of mol. interactions. For dispersion bound systems the basis set incompleteness error is most pronounced, as shown for the S22 benchmark. The use of very large basis sets (> quintuple-zeta) or extrapolation to the complete basis set (CBS) limit is necessary to obtain a reliable est. of the binding energy for these systems. Counterpoise cor. results converge to the same CBS limit, but counterpoise correction without extrapolation is insufficient. Core-valence correlations do not play a significant role. For medium- and short-range correlation, quadruple-zeta results are essentially converged, as demonstrated for relative alkane conformer energies, reaction energies dominated by intramol. dispersion, isomerization energies, and reaction energies of small org. mols. Except for weakly bound systems, diffuse augmentation almost universally slows down basis set convergence. For most RPA applications, quadruple-zeta valence basis sets offer a good balance between accuracy and efficiency. (c) 2012 American Institute of Physics.
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Abstract
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Figure 1
Figure 1. MAE (solid) and wMAE (striped) for the ASCDB database using different σ-functionals with PBE and PBE0 reference orbitals.
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Figure 2
Figure 2. (a) MAE, (b) WTMAD-1, and (c) WTMAD-2 for the GMTKN55 database and its subcategories using different σ-functionals with PBE reference orbitals.
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Figure 3
Figure 3. (a) MAE, (b) WTMAD-1, and (c) WTMAD-2 for the GMTKN55 database and its subcategories using different σ-functionals with PBE0 reference orbitals.
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Figure 4
Figure 4. Error distribution for the W4–17RE benchmark set using the top-down σ↓AXK(A2) and bottom-up σ↑AXK(A2) functionals for (a) PBE and (b) PBE0 reference orbitals.
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Figure 5
Figure 5. Weight functions of the Hartree (a, c) and exchange (b, d) kernels for σ↓AXK-functionals (a, b) and σ↑AXK-functionals (c, d) for PBE0 reference orbitals.
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Figure 6
Figure 6. CBS-extrapolated neon dimer interaction energies computed using PBE reference orbitals for (a) KS-DFT, (b) σ-functionals and scaled σ-functionals, (c) top-down σ↓AXK-functionals, and (d) bottom-up σ↑AXK-functionals. For reference, corrected CCSD(T) values from ref (97) are also included; data between points has been interpolated using cubic splines.
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References
This article references 102 other publications.
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  Thirty years of density functional theory in computational chemistry: an overview and extensive assessment of 200 density functionals
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  In the past 30 years, Kohn-Sham d. functional theory has emerged as the most popular electronic structure method in computational chem. To assess the ever-increasing no. of approx. exchange-correlation functionals, this review benchmarks a total of 200 d. functionals on a mol. database (MGCDB84) of nearly 5000 data points. The database employed, provided as Supplemental Data, is comprised of 84 data-sets and contains non-covalent interactions, isomerisation energies, thermochem., and barrier heights. In addn., the evolution of non-empirical and semi-empirical d. functional design is reviewed, and guidelines are provided for the proper and effective use of d. functionals. The most promising functional considered is ωB97M-V, a range-sepd. hybrid meta-GGA with VV10 nonlocal correlation, designed using a combinatorial approach. From the local GGAs, B97-D3, revPBE-D3, and BLYP-D3 are recommended, while from the local meta-GGAs, B97M-rV is the leading choice, followed by MS1-D3 and M06-L-D3. The best hybrid GGAs are ωB97X-V, ωB97X-D3, and ωB97X-D, while useful hybrid meta-GGAs (besides ωB97M-V) include ωM05-D, M06-2X-D3, and MN15. Ultimately, today's state-of-the-art functionals are close to achieving the level of accuracy desired for a broad range of chem. applications, and the principal remaining limitations are assocd. with systems that exhibit significant self-interaction/delocalisation errors and/or strong correlation effects.
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  Jacob's ladder of density functional approximations for the exchange-correlation energy
  Perdew, John P.; Schmidt, Karla
  AIP Conference Proceedings (2001), 577 (Density Functional Theory and Its Application to Materials), 1-20CODEN: APCPCS; ISSN:0094-243X. (American Institute of Physics)
  The ground-state energy and d. of a many-electron system are often calcd. by Kohn-Sham d. functional theory. We describe a ladder of approxns. for the exchange-correlation energy as a functional of the electron d. At the lowest rung of this ladder, the contribution to the energy from a vol. element of 3-dimensional space is detd. by the local d. there. Higher rungs or levels incorporate increasingly complex ingredients constructed from the d. or the Kohn-Sham orbitals in or around this vol. element. We identify which addnl. exact conditions can be satisfied at each level, and discuss the extent to which the functionals at each level may be constructed without empirical input. We also discuss the research that remains to be done at the exact-exchange level, and present our "dreams of a final theory". "Jacob left Beer-sheba and went toward Haran. He came to a certain place and stayed there for the night, because the sun had set. Taking one of the stones of the place, he put it under his head and lay down in that place. And he dreamed that there was a ladder set up on the earth, the top of it reaching to heaven; and the angels of God were ascending and descending on it.".
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  Becke, A. D.; Santra, G.; Martin, J. M. L. A double-hybrid density functional based on good local physics with outstanding performance on the GMTKN55 database. J. Chem. Phys. 2023, 158, 151103, DOI: 10.1063/5.0141238
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  Kállay, M. Linear-scaling implementation of the direct random-phase approximation. J. Chem. Phys. 2015, 142, 204105, DOI: 10.1063/1.4921542
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  Linear-scaling implementation of the direct random-phase approximation
  Kallay, Mihaly
  Journal of Chemical Physics (2015), 142 (20), 204105/1-204105/16CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics)
  We report the linear-scaling implementation of the direct RPA (dRPA) for closed-shell mol. systems. As a bonus, linear-scaling algorithms are also presented for the second-order screened exchange extension of dRPA as well as for the second-order Moller-Plesset (MP2) method and its spin-scaled variants. Our approach is based on an incremental scheme which is an extension of our previous local correlation method [Rolik et al., J. Chem. Phys. 139, 094105 (2013)]. The approach extensively uses local natural orbitals to reduce the size of the MO basis of local correlation domains. In addn., we also demonstrate that using natural auxiliary functions [M. Kallay, J. Chem. Phys. 141, 244113 (2014)], the size of the auxiliary basis of the domains and thus that of the three-center Coulomb integral lists can be reduced by an order of magnitude, which results in significant savings in computation time. The new approach is validated by extensive test calcns. for energies and energy differences. Our benchmark calcns. also demonstrate that the new method enables dRPA calcns. for mols. with more than 1000 atoms and 10 000 basis functions on a single processor. (c) 2015 American Institute of Physics.
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  Schurkus, H. F.; Ochsenfeld, C. Communication: An effective linear-scaling atomic-orbital reformulation of the random-phase approximation using a contracted double-Laplace transformation. J. Chem. Phys. 2016, 144, 031101, DOI: 10.1063/1.4939841
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  Communication: An effective linear-scaling atomic-orbital reformulation of the random-phase approximation using a contracted double-Laplace transformation
  Schurkus, Henry F.; Ochsenfeld, Christian
  Journal of Chemical Physics (2016), 144 (3), 031101/1-031101/5CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics)
  An at.-orbital (AO) reformulation of the RPA correlation energy is presented allowing to reduce the steep computational scaling to linear, so that large systems can be studied on simple desktop computers with fully numerically controlled accuracy. Our AO-RPA formulation introduces a contracted double-Laplace transform and employs the overlap-metric resoln.-of-the-identity. First timings of our pilot code illustrate the reduced scaling with systems comprising up to 1262 atoms and 10 090 basis functions. (c) 2016 American Institute of Physics.
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  Luenser, A.; Schurkus, H. F.; Ochsenfeld, C. Vanishing-Overhead Linear-Scaling Random Phase Approximation by Cholesky Decomposition and an Attenuated Coulomb-Metric. J. Chem. Theory Comput. 2017, 13, 1647, DOI: 10.1021/acs.jctc.6b01235
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  Vanishing-Overhead Linear-Scaling Random Phase Approximation by Cholesky Decomposition and an Attenuated Coulomb-Metric
  Luenser, Arne; Schurkus, Henry F.; Ochsenfeld, Christian
  Journal of Chemical Theory and Computation (2017), 13 (4), 1647-1655CODEN: JCTCCE; ISSN:1549-9618. (American Chemical Society)
  A reformulation of the RPA within the Resoln.-of-the-Identity (RI) scheme is presented, that is competitive to canonical MO RI-RPA already for small- to medium-sized mols. For electronically sparse systems drastic speedups due to the reduced scaling behavior compared to the MO formulation are demonstrated. Our reformulation is based on two ideas which are independently useful: First, a Cholesky decompn. of d. matrixes that reduces the scaling with basis set size for a fixed-size mol. by one order, leading to massive performance improvements. Second, replacement of the overlap RI metric used in the original AO-RPA by an attenuated Coulomb metric. Accuracy is significantly improved compared to the overlap metric, while locality and sparsity of the integrals are retained, as is the effective linear scaling behavior.
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9. 9
  Graf, D.; Beuerle, M.; Schurkus, H. F.; Luenser, A.; Savasci, G.; Ochsenfeld, C. Accurate and Efficient Parallel Implementation of an Effective Linear-Scaling Direct Random Phase Approximation Method. J. Chem. Theory Comput. 2018, 14, 2505, DOI: 10.1021/acs.jctc.8b00177
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  Accurate and Efficient Parallel Implementation of an Effective Linear-Scaling Direct Random Phase Approximation Method
  Graf, Daniel; Beuerle, Matthias; Schurkus, Henry F.; Luenser, Arne; Savasci, Goekcen; Ochsenfeld, Christian
  Journal of Chemical Theory and Computation (2018), 14 (5), 2505-2515CODEN: JCTCCE; ISSN:1549-9618. (American Chemical Society)
  An efficient algorithm for calcg. the RPA correlation energy is presented that is as accurate as the canonical MO resoln.-of-the-identity RPA (RI-RPA) with the important advantage of an effective linear-scaling behavior (instead of quartic) for large systems due to a formulation in the local AO space. The high accuracy is achieved by utilizing optimized minimax integration schemes and the local Coulomb metric attenuated by the complementary error function for the RI approxn. The memory bottleneck of former AO (AO)-RI-RPA implementations is addressed by precontraction of the large 3-center integral matrix with the Cholesky factors of the ground state d. reducing the memory requirements of that matrix by a factor of (Nbasis/Nocc). Furthermore, we present a parallel implementation of our method, which not only leads to faster RPA correlation energy calcns. but also to a scalable decrease in memory requirements, opening the door for investigations of large mols. even on small- to medium-sized computing clusters. Although it is known that AO methods are highly efficient for extended systems, where sparsity allows for reaching the linear-scaling regime, we show that our work also extends the applicability when considering highly delocalized systems for which no linear scaling can be achieved. As an example, the interlayer distance of two covalent org. framework pore fragments (comprising 384 atoms in total) is analyzed.
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10. 10
  Beuerle, M.; Graf, D.; Schurkus, H. F.; Ochsenfeld, C. Efficient calculation of beyond RPA correlation energies in the dielectric matrix formalism. J. Chem. Phys. 2018, 148, 204104, DOI: 10.1063/1.5025938
  10
  Efficient calculation of beyond RPA correlation energies in the dielectric matrix formalism
  Beuerle, Matthias; Graf, Daniel; Schurkus, Henry F.; Ochsenfeld, Christian
  Journal of Chemical Physics (2018), 148 (20), 204104/1-204104/11CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics)
  We present efficient methods to calc. beyond RPA correlation energies for mol. systems with up to 500 atoms. To reduce the computational cost, we employ the resoln.-of-the-identity and a double-Laplace transform of the non-interacting polarization propagator in conjunction with an AO formalism. Further improvements are achieved using integral screening and the introduction of Cholesky decompd. densities. Our methods are applicable to the dielec. matrix formalism of RPA including second-order screened exchange (RPA-SOSEX), the RPA electron-hole time-dependent Hartree-Fock (RPA-eh-TDHF) approxn., and RPA renormalized perturbation theory using an approx. exchange kernel (RPA-AXK). We give an application of our methodol. by presenting RPA-SOSEX benchmark results for the L7 test set of large, dispersion dominated mols., yielding a mean abs. error below 1 kcal/mol. The present work enables calcg. beyond RPA correlation energies for significantly larger mols. than possible to date, thereby extending the applicability of these methods to a wider range of chem. systems. (c) 2018 American Institute of Physics.
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11. 11
  Drontschenko, V.; Graf, D.; Laqua, H.; Ochsenfeld, C. Lagrangian-Based Minimal-Overhead Batching Scheme for the Efficient Integral-Direct Evaluation of the RPA Correlation Energy. J. Chem. Theory Comput. 2021, 17, 5623, DOI: 10.1021/acs.jctc.1c00494
  11
  Lagrangian-Based Minimal-Overhead Batching Scheme for the Efficient Integral-Direct Evaluation of the RPA Correlation Energy
  Drontschenko, Viktoria; Graf, Daniel; Laqua, Henryk; Ochsenfeld, Christian
  Journal of Chemical Theory and Computation (2021), 17 (9), 5623-5634CODEN: JCTCCE; ISSN:1549-9618. (American Chemical Society)
  A highly memory-efficient integral-direct RPA method based on our ω-CDGD-RI-RPA method is presented that completely alleviates the memory bottleneck of storing the multidimensional three-center integral tensor, which severely limited the tractable system sizes. Based on a Lagrangian formulation, we introduce an optimized batching scheme over the auxiliary and basis-function indexes, which allows to compute the optimal no. of batches for a given amt. of system memory, while minimizing the batching overhead. Thus, our optimized batching constitutes the best tradeoff between program runtime and memory demand. Within this batching scheme, the half-transformed three-center integral tensor BiμM is recomputed for each batch of auxiliary and basis functions. This allows the computation of systems that were out of reach before. The largest system within this work consists of a DNA fragment comprising 1052 atoms and 11 230 basis functions calcd. on a single node, which emphasizes the new possibilities of our integral-direct RPA method.
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12. 12
  Harl, J.; Kresse, G. Cohesive energy curves for noble gas solids calculated by adiabatic connection fluctuation-dissipation theory. Phys. Rev. B 2008, 77, 045136, DOI: 10.1103/PhysRevB.77.045136
  12
  Cohesive energy curves for noble gas solids calculated by adiabatic connection fluctuation-dissipation theory
  Harl, Judith; Kresse, Georg
  Physical Review B: Condensed Matter and Materials Physics (2008), 77 (4), 045136/1-045136/8CODEN: PRBMDO; ISSN:1098-0121. (American Physical Society)
  We present first-principles calcns. for the fcc noble gas solids Ne, Ar, and Kr applying the adiabatic connection fluctuation-dissipation theorem (ACFDT) to evaluate the correlation energy. The ACFDT allows us to describe long-range correlation effects including London dispersion or van der Waals interaction on top of conventional d. functional theory calcns. Even within the RPA, the typical 1/V2 vol. dependence for the cohesive energy of the noble gas solids is reproduced, and equil. cohesive energies and lattice consts. are improved compared to d. functional theory calcns. Furthermore, we present atomization energies for H2, N2, and O2 within the same post-d.-functional-theory framework, finding an excellent agreement with previously published data.
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13. 13
  Su, H.; Wu, Q.; Wang, H.; Wang, H. An assessment of the random-phase approximation functional and characteristics analysis for noncovalent cation−π interactions. Phys. Chem. Chem. Phys. 2017, 19, 26014– 26021, DOI: 10.1039/C7CP04504B
  There is no corresponding record for this reference.
14. 14
  Modrzejewski, M.; Yourdkhani, S.; Klimeš, J. Random Phase Approximation Applied to Many-Body Noncovalent Systems. J. Chem. Theory Comput. 2020, 16, 427– 442, DOI: 10.1021/acs.jctc.9b00979
  14
  Random Phase Approximation Applied to Many-Body Noncovalent Systems
  Modrzejewski Marcin; Yourdkhani Sirous; Klimes Jiri; Modrzejewski Marcin
  Journal of chemical theory and computation (2020), 16 (1), 427-442 ISSN:.
  The random phase approximation (RPA) has received considerable interest in the field of modeling systems where noncovalent interactions are important. Its advantages over widely used density functional theory (DFT) approximations are the exact treatment of exchange and the description of long-range correlation. In this work, we address two open questions related to RPA. First, we demonstrate how accurately RPA describes nonadditive interactions encountered in many-body expansion of a binding energy. We consider three-body nonadditive energies in molecular and atomic clusters. Second, we address how the accuracy of RPA depends on input provided by different DFT models, without resorting to self-consistent RPA procedure, which is currently impractical for calculations employing periodic boundary conditions. We find that RPA based on the SCAN0 and PBE0 models, that is, hybrid DFT, achieves an overall accuracy between CCSD and MP3 on a data set of molecular trimers from Rezac et al. ( J. Chem. Theory. Comput. 2015 , 11 , 3065 ). Finally, many-body expansion for molecular clusters and solids often leads to a large number of small contributions that need to be calculated with high precision. We therefore present a cubic-scaling (or self-consistent field (SCF)-like) implementation of RPA in atomic basis set, which is designed for calculations with high numerical precision.
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15. 15
  Harl, J.; Kresse, G. Accurate Bulk Properties from Approximate Many-Body Techniques. Phys. Rev. Lett. 2009, 103, 056401, DOI: 10.1103/PhysRevLett.103.056401
  15
  Accurate Bulk Properties from Approximate Many-Body Techniques
  Harl, Judith; Kresse, Georg
  Physical Review Letters (2009), 103 (5), 056401/1-056401/4CODEN: PRLTAO; ISSN:0031-9007. (American Physical Society)
  For ab initio electronic structure calcns., the RPA to the correlation energy is supposed to be a suitable complement to the exact exchange energy. We show that lattice consts., atomization energies of solids, and adsorption energies on metal surfaces evaluated using this approxn. are in very good agreement with expt. Since the method is fairly efficient and handles ionic, metallic, and van der Waals bonded systems equally well, it is a very promising choice to improve upon d. functional theory calcns., without resorting to more demanding diffusion Monte Carlo or quantum chem. methods.
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16. 16
  Kreppel, A.; Graf, D.; Laqua, H.; Ochsenfeld, C. Range-Separated Density-Functional Theory in Combination with the Random Phase Approximation: An Accuracy Benchmark. J. Chem. Theory Comput. 2020, 16, 2985– 2994, DOI: 10.1021/acs.jctc.9b01294
  16
  Range-Separated Density-Functional Theory in Combination with the Random Phase Approximation: An Accuracy Benchmark
  Kreppel, Andrea; Graf, Daniel; Laqua, Henryk; Ochsenfeld, Christian
  Journal of Chemical Theory and Computation (2020), 16 (5), 2985-2994CODEN: JCTCCE; ISSN:1549-9618. (American Chemical Society)
  A formulation of range-sepd. RPA based on our efficient ω-CDGD-RI-RPA [J. Chem. Theory Comput.2018, 14, 2505] method and a large scale benchmark study are presented. By application to the GMTKN55 data set, we obtain a comprehensive picture of the performance of range-sepd. RPA in general main group thermochem., kinetics, and noncovalent interactions. The results show that range-sepd. RPA performs stably over the broad range of mol. chem. included in the GMTKN55 set. It improves significantly over semilocal DFT but it is still less accurate than modern dispersion cor. double-hybrid functionals. Furthermore, range-sepd. RPA shows a faster basis set convergence compared to std. full-range RPA making it a promising applicable approach with only one empirical parameter.
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17. 17
  Trushin, E.; Thierbach, A.; Görling, A. Toward chemical accuracy at low computational cost: Density-functional theory with σ-functionals for the correlation energy. J. Chem. Phys. 2021, 154, 014104, DOI: 10.1063/5.0026849
  17
  Toward chemical accuracy at low computational cost: Density-functional theory with σ-functionals for the correlation energy
  Trushin, Egor; Thierbach, Adrian; Goerling, Andreas
  Journal of Chemical Physics (2021), 154 (1), 014104CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics)
  We introduce new functionals for the Kohn-Sham correlation energy that are based on the adiabatic-connection fluctuation-dissipation (ACFD) theorem and are named σ-functionals. Like in the well-established direct RPA (dRPA), σ-functionals require as input exclusively eigenvalues σ of the frequency-dependent KS response function. In the new functionals, functions of σ replace the σ-dependent dRPA expression in the coupling-const. and frequency integrations contained in the ACFD theorem. We optimize σ-functionals with the help of ref. sets for atomization, reaction, transition state, and non-covalent interaction energies. The optimized functionals are to be used in a post-self-consistent way using orbitals and eigenvalues from conventional Kohn-Sham calcns. employing the exchange-correlation functional of Perdew, Burke, and Ernzerhof. The accuracy of the presented approach is much higher than that of dRPA methods and is comparable to that of high-level wave function methods. Reaction and transition state energies from σ-functionals exhibit accuracies close to 1 kcal/mol and thus approach chem. accuracy. For the 10 966 reactions of the W4-11RE ref. set, the mean abs. deviation is 1.25 kcal/mol compared to 3.21 kcal/mol in the dRPA case. Non-covalent binding energies are accurate to a few tenths of a kcal/mol. The presented approach is highly efficient, and the post-self-consistent calcn. of the total energy requires less computational time than a d.-functional calcn. with a hybrid functional and thus can be easily carried out routinely. σ-Functionals can be implemented in any existing dRPA code with negligible programming effort. (c) 2021 American Institute of Physics.
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18. 18
  Fauser, S.; Trushin, E.; Neiss, C.; Görling, A. Chemical accuracy with σ-functionals for the Kohn–Sham correlation energy optimized for different input orbitals and eigenvalues. J. Chem. Phys. 2021, 155, 134111, DOI: 10.1063/5.0059641
  18
  Chemical accuracy with σ-functionals for the Kohn-Sham correlation energy optimized for different input orbitals and eigenvalues
  Fauser, Steffen; Trushin, Egor; Neiss, Christian; Goerling, Andreas
  Journal of Chemical Physics (2021), 155 (13), 134111CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics)
  Recently, a new type of orbital-dependent functional for the Kohn-Sham (KS) correlation energy, σ-functionals, was introduced. Tech., σ-functionals are closely related to the well-known direct RPA (dRPA). Within the dRPA, a function of the eigenvalues σ of the frequency-dependent KS response function is integrated over purely imaginary frequencies. In σ-functionals, this function is replaced by one that is optimized with respect to ref. sets of atomization, reaction, transition state, and non-covalent interaction energies. The previously introduced σ-functional uses input orbitals and eigenvalues from KS calcns. with the generalized gradient approxn. (GGA) exchange-correlation functional of Perdew, Burke, and Ernzerhof (PBE). Here, σ-functionals using input orbitals and eigenvalues from the meta-GGA TPSS and the hybrid-functionals PBE0 and B3LYP are presented and tested. The no. of ref. sets taken into account in the optimization of the σ-functionals is larger than in the first PBE based σ-functional and includes sets with 3d-transition metal compds. Therefore, also a reparameterized PBE based σ-functional is introduced. The σ-functionals based on PBE0 and B3LYP orbitals and eigenvalues reach chem. accuracy for main group chem. For the 10 966 reactions from the highly accurate W4-11RE ref. set, the B3LYP based σ-functional exhibits a mean av. deviation of 1.03 kcal/mol compared to 1.08 kcal/mol for the coupled cluster singles doubles perturbative triples method if the same valence quadruple zeta basis set is used. For 3d-transition metal chem., accuracies of about 2 kcal/mol are reached. The computational effort for the post-self-consistent evaluation of the σ-functional is lower than that of a preceding PBE0 or B3LYP calcn. for typical systems. (c) 2021 American Institute of Physics.
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19. 19
  Fauser, S.; Förster, A.; Redeker, L.; Neiss, C.; Erhard, J.; Trushin, E.; Görling, A. Basis Set Requirements of σ-Functionals for Gaussian- and Slater-Type Basis Functions and Comparison with Range-Separated Hybrid and Double Hybrid Functionals. J. Chem. Theory Comput. 2024, 20, 2404– 2422, DOI: 10.1021/acs.jctc.3c01132
  There is no corresponding record for this reference.
20. 20
  Drontschenko, V.; Graf, D.; Laqua, H.; Ochsenfeld, C. Efficient Method for the Computation of Frozen-Core Nuclear Gradients within the Random Phase Approximation. J. Chem. Theory Comput. 2022, 18, 7359– 7372, DOI: 10.1021/acs.jctc.2c00774
  20
  Efficient Method for the Computation of Frozen-Core Nuclear Gradients within the Random Phase Approximation
  Drontschenko, Viktoria; Graf, Daniel; Laqua, Henryk; Ochsenfeld, Christian
  Journal of Chemical Theory and Computation (2022), 18 (12), 7359-7372CODEN: JCTCCE; ISSN:1549-9618. (American Chemical Society)
  A method for the evaluation of anal. frozen-core gradients within the RPA is presented. We outline an efficient way to evaluate the response of the d. of active electrons arising only when introducing the frozen-core approxn. and constituting the main difficulty, together with the response of the std. Kohn-Sham d. The general framework allows to extend the outlined procedure to related electron correlation methods in the AO basis that require the evaluation of d. responses, such as second-order Moller-Plesset perturbation theory or coupled cluster variants. By using Cholesky decompd. densities - which reintroduce the occupied index in the time-detg. steps - we are able to achieve speedups of 20-30% (depending on the size of the basis set) by using the frozen-core approxn., which is of similar magnitude as for MO formulations. We further show that the errors introduced by the frozen-core approxn. are practically insignificant for mol. geometries.
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21. 21
  Neiss, C.; Fauser, S.; Görling, A. Geometries and vibrational frequencies with Kohn–Sham methods using σ-functionals for the correlation energy. J. Chem. Phys. 2023, 158, 044107, DOI: 10.1063/5.0129524
  21
  Geometries and vibrational frequencies with Kohn-Sham methods using σ-functionals for the correlation energy
  Neiss, Christian; Fauser, Steffen; Goerling, Andreas
  Journal of Chemical Physics (2023), 158 (4), 044107CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics)
  Recently, Kohn-Sham (KS) methods with new correlation functionals, called σ-functionals, have been introduced. Tech., σ-functionals are closely related to the well-known RPA; formally, σ-functionals are rooted in perturbation theory along the adiabatic connection. If employed in a post-SCF manner in a Gaussian basis set framework, then, σ-functional methods are computationally very efficient. Moreover, for main group chem., σ-functionals are highly accurate and can compete with high-level wave-function methods. For reaction and transition state energies, e.g., chem. accuracy of 1 kcal/mol is reached. Here, we show how to calc. first derivs. of the total energy with respect to nuclear coordinates for methods using σ-functionals and then carry out geometry optimizations for test sets of main group mols., transition metal compds., and non-covalently bonded systems. For main group mols., we addnl. calc. vibrational frequencies. σ-Functional methods are found to yield very accurate geometries and vibrational frequencies for main group mols. superior not only to those from conventional KS methods but also to those from RPA methods. For geometries of transition metal compds., not surprisingly, best geometries are found for RPA methods, while σ-functional methods yield somewhat less good results. This is attributed to the fact that in the optimization of σ-functionals, transition metal compds. could not be represented well due to the lack of reliable ref. data. For non-covalently bonded systems, σ-functionals yield geometries of the same quality as the RPA or as conventional KS schemes combined with dispersion corrections. (c) 2023 American Institute of Physics.
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22. 22
  Drontschenko, V.; Bangerter, F. H.; Ochsenfeld, C. Analytical Second-Order Properties for the Random Phase Approximation: Nuclear Magnetic Resonance Shieldings. J. Chem. Theory Comput. 2023, 19, 7542– 7554, DOI: 10.1021/acs.jctc.3c00542
  There is no corresponding record for this reference.
23. 23
  Fauser, S.; Drontschenko, V.; Ochsenfeld, C.; Görling, A. Accurate NMR Shieldings with σ-Functionals. J. Chem. Theory Comput. 2024, 20, 6028– 6036, DOI: 10.1021/acs.jctc.4c00512
  There is no corresponding record for this reference.
24. 24
  Drontschenko, V.; Ochsenfeld, C. Low-Scaling, Efficient and Memory Optimized Computation of Nuclear Magnetic Resonance Shieldings within the Random Phase Approximation using Cholesky-Decomposed Densities and an Attenuated Coulomb Metric. J. Phys. Chem. A 2024, 128, 7950– 7965, DOI: 10.1021/acs.jpca.4c02773
  There is no corresponding record for this reference.
25. 25
  Glasbrenner, M.; Graf, D.; Ochsenfeld, C. Benchmarking the Accuracy of the Direct Random Phase Approximation and σ-Functionals for NMR Shieldings. J. Chem. Theory Comput. 2022, 18, 192– 205, DOI: 10.1021/acs.jctc.1c00866
  25
  Benchmarking the Accuracy of the Direct Random Phase Approximation and σ-Functionals for NMR Shieldings
  Glasbrenner, Michael; Graf, Daniel; Ochsenfeld, Christian
  Journal of Chemical Theory and Computation (2022), 18 (1), 192-205CODEN: JCTCCE; ISSN:1549-9618. (American Chemical Society)
  A method for computing NMR shieldings with the direct RPA (RPA) and the closely related σ-functionals [Trushin, E.; Thierbach, A.; Gorling, A. Toward chem. accuracy at low computational cost: d. functional theory with σ-functionals for the correlation energy. J. Chem. Phys.2021,154, 014104] is presented, which is based on a finite-difference approach. The accuracy is evaluated in benchmark calcns. using high-quality coupled cluster values as a ref. Our results show that the accuracy of the computed NMR shieldings using direct RPA is strongly dependent on the d. functional theory ref. orbitals and improves with increasing amts. of exact Hartree-Fock exchange in the functional. NMR shieldings computed with the direct RPA with a Hartree-Fock ref. are significantly more accurate than MP2 shieldings and comparable to CCSD shieldings. Also, the basis set convergence is analyzed and it is shown that at least triple-zeta basis sets are required for reliable results.
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26. 26
  Görling, A. Hierarchies of methods towards the exact Kohn–Sham correlation energy based on the adiabatic-connection fluctuation-dissipation theorem. Phys. Rev. B 2019, 99, 235120, DOI: 10.1103/PhysRevB.99.235120
  There is no corresponding record for this reference.
27. 27
  Erhard, J.; Bleiziffer, P.; Görling, A. Power Series Approximation for the Correlation Kernel Leading to Kohn–Sham Methods Combining Accuracy, Computational Efficiency, and General Applicability. Phys. Rev. Lett. 2016, 117, 143002, DOI: 10.1103/PhysRevLett.117.143002
  There is no corresponding record for this reference.
28. 28
  Erhard, J.; Fauser, S.; Trushin, E.; Görling, A. Scaled σ-functionals for the Kohn–Sham correlation energy with scaling functions from the homogeneous electron gas. J. Chem. Phys. 2022, 157, 114105, DOI: 10.1063/5.0101641
  28
  Scaled σ-functionals for Kohn-Sham correlation energy with scaling function from homogeneous electron gas
  Erhard, Jannis; Fauser, Steffen; Trushin, Egor; Goerling, Andreas
  Journal of Chemical Physics (2022), 157 (11), 114105CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics)
  A review. The recently introduced σ-functionals constitute a new type of functionals for the Kohn-Sham (KS) correlation energy. The σ-functionals are based on the adiabatic-connection fluctuation-dissipation theorem, are computationally closely related to the well-known direct RPA (dRPA), and are formally rooted in many-body perturbation theory along the adiabatic connection. In σ-functionals, the function of the eigen values σ of the Kohn-Sham response matrix that enters the coupling const. and frequency integration in the dRPA is replaced by another function optimized with the help of ref. sets of atomization, reaction, transition state, and non-covalent interaction energies, and σ-Functionals are highly accurate and yield chem. accuracy of 1 kcal/mol in reaction or transition state energies, in main group chem. A shortcoming of σ-functionals is their inability to accurately describe processes involving a change of the electron no., such as ionization or electron attachment. This problem is attributed to unphys. self-interactions caused by the neglect of the exchange kernel in the dRPA and σ-functionals. Here, we tackle this problem by introducing a frequency- and σ-dependent scaling of the eigenvalues σ of the KS response function that models the effect of the exchange kernel. The scaling factors are detected with the help of the homogeneous electron gas. The resulting scaled σ-functionals retain the accuracy of their unscaled parent functionals but in addn. yield very accurate ionization potentials and electron affinities. Moreover, atomization and total energies are found to be exceptionally accurate. Scaled σ-functionals are computationally highly efficient like their unscaled counterparts. (c) 2022 American Institute of Physics.
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29. 29
  Lemke, Y.; Ochsenfeld, C. Highly accurate σ- and τ-functionals for beyond-RPA methods with approximate exchange kernels. J. Chem. Phys. 2023, 159, 194104, DOI: 10.1063/5.0173042
  29
  Highly accurate σ- and τ-functionals for beyond-RPA methods with approximate exchange kernels
  Lemke, Yannick; Ochsenfeld, Christian
  Journal of Chemical Physics (2023), 159 (19), 194104CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics)
  σ-Functionals are promising new developments for the Kohn-Sham correlation energy based upon the direct RPA (dRPA) within the adiabatic connection formalism, providing impressive improvements over dRPA for a broad range of benchmarks. However, σ-functionals exhibit a high amt. of self-interaction inherited from the approxns. made within dRPA. Inclusion of an exchange kernel in deriving the coupling-strength-dependent d.-d. response function leads to so-called τ-functionals, which - apart from a fourth-order Taylor series expansion - have only been realized in an approx. fashion so far to the best of our knowledge, most notably in the form of scaled σ-functionals. In this work, we derive, optimize, and benchmark three types of σ- and τ-functionals including approx. exchange effects in the form of an antisymmetrized Hartree kernel. These functionals, based on a second-order screened exchange type contribution in the adiabatic connection formalism, the electron-hole time-dependent Hartree-Fock kernel (eh-TDHF) otherwise known as RPA with exchange (RPAx), and an approxn. thereof known as approx. exchange kernel (AXK), are optimized on the ASCDB database using two new parametrizations named A1 and A2. In addn., we report a first full evaluation of σ- and τ-functionals on the GMTKN55 database, revealing our exchange-including functionals to considerably outperform existing σ-functionals while being highly competitive with some of the best double-hybrid functionals of the original GMTKN55 publication. In particular, the σ-functionals based on AXK and τ-functionals based on RPAx with PBE0 ref. stand out as highly accurate approaches for a wide variety of chem. relevant problems. (c) 2023 American Institute of Physics.
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30. 30
  Grüneis, A.; Marsman, M.; Harl, J.; Schimka, L.; Kresse, G. Making the random phase approximation to electronic correlation accurate. J. Chem. Phys. 2009, 131, 154115, DOI: 10.1063/1.3250347
  30
  Making the random phase approximation to electronic correlation accurate
  Grueneis, Andreas; Marsman, Martijn; Harl, Judith; Schimka, Laurids; Kresse, Georg
  Journal of Chemical Physics (2009), 131 (15), 154115/1-154115/5CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics)
  We show that the inclusion of second-order screened exchange to the RPA allows for an accurate description of electronic correlation in atoms and solids clearly surpassing the random phase approxn., but not yet approaching chem. accuracy. From a fundamental point of view, the method is self-correlation free for one-electron systems. From a practical point of view,the approach yields correlation energies for atoms, as well as for the jellium electron gas within a few kcal/mol of exact values, atomization energies within typically 2-3 kcal/mol of expt., and excellent lattice consts. for ionic and covalently bonded solids (0.2% error). The computational complexity is only O(N5), comparable to canonical second-order Moller-Plesset perturbation theory, which should allow for routine calcns. on many systems. (c) 2009 American Institute of Physics.
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31. 31
  Bates, J. E.; Furche, F. Communication: Random phase approximation renormalized many-body perturbation theory. J. Chem. Phys. 2013, 139, 171103, DOI: 10.1063/1.4827254
  31
  Communication: Random phase approximation renormalized many-body perturbation theory
  Bates, Jefferson E.; Furche, Filipp
  Journal of Chemical Physics (2013), 139 (17), 171103/1-171103/4CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics)
  We derive a renormalized many-body perturbation theory (MBPT) starting from the RPA. This RPA-renormalized perturbation theory extends the scope of single-ref. MBPT methods to small-gap systems without significantly increasing the computational cost. The leading correction to RPA, termed the approx. exchange kernel (AXK), substantially improves upon RPA atomization energies and ionization potentials without affecting other properties such as barrier heights where RPA is already accurate. Thus, AXK is more balanced than second-order screened exchange, which tends to overcorrect RPA for systems with stronger static correlation. Similarly, AXK avoids the divergence of second-order Moller-Plesset (MP2) theory for small gap systems and delivers a much more consistent performance than MP2 across the periodic table at comparable cost. RPA+AXK thus is an accurate, non-empirical, and robust tool to assess and improve semi-local d. functional theory for a wide range of systems previously inaccessible to first-principles electronic structure calcns. (c) 2013 American Institute of Physics.
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32. 32
  Goerigk, L.; Hansen, A.; Bauer, C.; Ehrlich, S.; Najibi, A.; Grimme, S. A look at the density functional theory zoo with the advanced GMTKN55 database for general main group thermochemistry, kinetics and noncovalent interactions. Phys. Chem. Chem. Phys. 2017, 19, 32184, DOI: 10.1039/C7CP04913G
  32
  A look at the density functional theory zoo with the advanced GMTKN55 database for general main group thermochemistry, kinetics and noncovalent interactions
  Goerigk, Lars; Hansen, Andreas; Bauer, Christoph; Ehrlich, Stephan; Najibi, Asim; Grimme, Stefan
  Physical Chemistry Chemical Physics (2017), 19 (48), 32184-32215CODEN: PPCPFQ; ISSN:1463-9076. (Royal Society of Chemistry)
  We present the GMTKN55 benchmark database for general main group thermochem., kinetics and noncovalent interactions. Compared to its popular predecessor GMTKN30, it allows assessment across a larger variety of chem. problems - with 13 new benchmark sets being presented for the first time - and it also provides ref. values of significantly higher quality for most sets. GMTKN55 comprises 1505 relative energies based on 2462 single-point calcns. and it is accessible to the user community via a dedicated website. Herein, we demonstrate the importance of better ref. values, and we re-emphasize the need for London-dispersion corrections in d. functional theory (DFT) treatments of thermochem. problems, including Minnesota methods. We assessed 217 variations of dispersion-cor. and -uncorrected d. functional approxns., and carried out a detailed anal. of 83 of them to identify robust and reliable approaches. Double-hybrid functionals are the most reliable approaches for thermochem. and noncovalent interactions, and they should be used whenever tech. feasible. These are, in particular, DSD-BLYP-D3(BJ), DSD-PBEP86-D3(BJ), and B2GPPLYP-D3(BJ). The best hybrids are ωB97X-V, M052X-D3(0), and ωB97X-D3, but we also recommend PW6B95-D3(BJ) as the best conventional global hybrid. At the meta-generalized-gradient (meta-GGA) level, the SCAN-D3(BJ) method can be recommended. Other meta-GGAs are outperformed by the GGA functionals revPBE-D3(BJ), B97-D3(BJ), and OLYP-D3(BJ). We note that many popular methods, such as B3LYP, are not part of our recommendations. In fact, with our results we hope to inspire a change in the user community's perception of common DFT methods. We also encourage method developers to use GMTKN55 for cross-validation studies of new methodologies.
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33. 33
  Perdew, J. P.; Burke, K.; Ernzerhof, M. Generalized Gradient Approximation Made Simple. Phys. Rev. Lett. 1996, 77, 3865– 3868, DOI: 10.1103/PhysRevLett.77.3865
  33
  Generalized gradient approximation made simple
  Perdew, John P.; Burke, Kieron; Ernzerhof, Matthias
  Physical Review Letters (1996), 77 (18), 3865-3868CODEN: PRLTAO; ISSN:0031-9007. (American Physical Society)
  Generalized gradient approxns. (GGA's) for the exchange-correlation energy improve upon the local spin d. (LSD) description of atoms, mols., and solids. We present a simple derivation of a simple GGA, in which all parameters (other than those in LSD) are fundamental consts. Only general features of the detailed construction underlying the Perdew-Wang 1991 (PW91) GGA are invoked. Improvements over PW91 include an accurate description of the linear response of the uniform electron gas, correct behavior under uniform scaling, and a smoother potential.
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34. 34
  Adamo, C.; Barone, V. Toward reliable density functional methods without adjustable parameters: The PBE0 model. J. Chem. Phys. 1999, 110, 6158– 6170, DOI: 10.1063/1.478522
  34
  Toward reliable density functional methods without adjustable parameters: the PBE0 model
  Adamo, Carlo; Barone, Vincenzo
  Journal of Chemical Physics (1999), 110 (13), 6158-6170CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics)
  We present an anal. of the performances of a parameter free d. functional model (PBE0) obtained combining the so called PBE generalized gradient functional with a predefined amt. of exact exchange. The results obtained for structural, thermodn., kinetic and spectroscopic (magnetic, IR and electronic) properties are satisfactory and not far from those delivered by the most reliable functionals including heavy parameterization. The way in which the functional is derived and the lack of empirical parameters fitted to specific properties make the PBE0 model a widely applicable method for both quantum chem. and condensed matter physics.
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35. 35
  Ernzerhof, M.; Scuseria, G. E. Assessment of the Perdew–Burke–Ernzerhof exchange-correlation functional. J. Chem. Phys. 1999, 110, 5029– 5036, DOI: 10.1063/1.478401
  35
  Assessment of the Perdew-Burke-Ernzerhof exchange-correlation functional
  Ernzerhof, Matthias; Scuseria, Gustavo E.
  Journal of Chemical Physics (1999), 110 (11), 5029-5036CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics)
  In order to discriminate between approxns. to the exchange-correlation energy EXC[ρ↑,ρ↓], we employ the criterion of whether the functional is fitted to a certain exptl. data set or if it is constructed to satisfy phys. constraints. We present extensive test calcns. for atoms and mols., with the nonempirical local spin-d. (LSD) and the Perdew-Burke-Ernzerhof (PBE) functional and compare our results with results obtained with more empirical functionals. For the atomization energies of the G2 set, we find that the PBE functional shows systematic errors larger than those of commonly used empirical functionals. The PBE ionization potentials, electron affinities, and bond lengths are of accuracy similar to those obtained from empirical functionals. Furthermore, a recently proposed hybrid scheme using exact exchange together with PBE exchange and correlation is investigated. For all properties studied here, the PBE hybrid gives an accuracy comparable to the frequently used empirical B3LYP hybrid scheme. Phys. principles underlying the PBE and PBE hybrid scheme are examd. and the range of their validity is discussed.
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36. 36
  Langreth, D. C.; Perdew, J. P. The exchange-correlation energy of a metallic surface. Solid State Commun. 1975, 17, 1425, DOI: 10.1016/0038-1098(75)90618-3
  There is no corresponding record for this reference.
37. 37
  Langreth, D. C.; Perdew, J. P. Exchange-correlation energy of a metallic surface: Wave-vector analysis. Phys. Rev. B 1977, 15, 2884, DOI: 10.1103/PhysRevB.15.2884
  There is no corresponding record for this reference.
38. 38
  Petersilka, M.; Gossmann, U. J.; Gross, E. K. U. Excitation Energies from Time-Dependent Density-Functional Theory. Phys. Rev. Lett. 1996, 76, 1212– 1215, DOI: 10.1103/PhysRevLett.76.1212
  38
  Excitation energies from time-dependent density-functional theory
  Petersilka, M.; Gossmann, U. J.; Gross, E. K. U.
  Physical Review Letters (1996), 76 (8), 1212-15CODEN: PRLTAO; ISSN:0031-9007. (American Physical Society)
  A new d.-functional approach to calc. the excitation spectrum of many-electron systems is proposed. It is shown that the full linear d. response of the interacting system, which has poles at the exact excitation energies, can rigorously be expressed in terms of the response function of the noninteracting (Kohn-Sham) system and a frequency-dependent exchange-correlation kernel. Using this expression, the poles of the full response function are obtained by systematic improvement upon the poles of the Kohn-Sham response function. Numerical results are presented for Be, Mg, Ca, Zn, Sr, and Cd atoms.
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39. 39
  Garrick, R.; Natan, A.; Gould, T.; Kronik, L. Exact Generalized Kohn–Sham Theory for Hybrid Functionals. Phys. Rev. X 2020, 10, 021040, DOI: 10.1103/PhysRevX.10.021040
  There is no corresponding record for this reference.
40. 40
  Whitten, J. L. Coulombic potential energy integrals and approximations. J. Chem. Phys. 1973, 58, 4496– 4501, DOI: 10.1063/1.1679012
  40
  Coulombic potential energy integrals and approximations
  Whitten, J. L.
  Journal of Chemical Physics (1973), 58 (10), 4496-501CODEN: JCPSA6; ISSN:0021-9606.
  Theorems are derived which establish a method of approxg. 2-particle Coulombic potential energy integrals, [.vphi.a(1)|r12-1|.vphi.b-(2)], in terms of approx. charge ds. .vphi.a' and .vphi.b'. Rigorous error bounds, |[.vphi.a(1)|r12-1|.vphi.b(2)] - [.vphi.a'(1)|r12-1|.vphi.b'(2)]| ≤ δ, are simply expressed in terms of information calcd. sep. for the pair of ds. .vphi.a and .vphi.b' and the pair .vphi.b and .vphi.b'. From the structure of the bound, a simple method of optimizing charge d. approxns. such that δ is minimized is derived. The framework of the theory appears to be well suited for application to the approxn. of electron repulsion integrals which occur in mol. structure theory, and applications to the approxn. of integrals over Slater orbitals or grouped Gaussian functions are discussed.
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41. 41
  Dunlap, B. I.; Connolly, J. W. D.; Sabin, J. R. On some approximations in applications of X_α theory. J. Chem. Phys. 1979, 71, 3396– 3402, DOI: 10.1063/1.438728
  41
  On some approximations in applications of X α theory
  Dunlap, B. I.; Connolly, J. W. D.; Sabin, J. R.
  Journal of Chemical Physics (1979), 71 (8), 3396-402CODEN: JCPSA6; ISSN:0021-9606.
  An approx. Xα functional is proposed from which the charge d. fitting equations follow variationally. LCAO Xα calcns. on at. Ni and H2 show the method independent of the fitting (auxiliary) bases to within 0.02 eV. Variational properties assocd. with both orbital and auxiliary basis set incompleteness are used to approach within 0.2 eV the Xα total energy limit for the N mol.
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42. 42
  Vahtras, O.; Almlöf, J.; Feyereisen, M. W. Integral approximations for LCAO-SCF calculations. Chem. Phys. Lett. 1993, 213, 514– 518, DOI: 10.1016/0009-2614(93)89151-7
  42
  Integral approximations for LCAO-SCF calculations
  Vahtras, O.; Almloef, J.; Feyereisen, M. W.
  Chemical Physics Letters (1993), 213 (5-6), 514-18CODEN: CHPLBC; ISSN:0009-2614.
  Three-center approxns. to the four-center integrals occurring in ab initio LCAO calcns. are investigated. Significant gains in computer time can be obtained without sacrificing accuracy, if a suitable expansion basis is chosen.
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43. 43
  Furche, F. Developing the random phase approximation into a practical post-Kohn–Sham correlation model. J. Chem. Phys. 2008, 129, 114105, DOI: 10.1063/1.2977789
  43
  Developing the random phase approximation into a practical post-Kohn-Sham correlation model
  Furche, Filipp
  Journal of Chemical Physics (2008), 129 (11), 114105/1-114105/8CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics)
  The RPA (RPA) to the d. functional correlation energy systematically improves upon many limitations of present semilocal functionals, but was considered too computationally expensive for widespread use in the past. Here a phys. appealing reformulation of the RPA correlation model is developed that substantially reduces its computational complexity. The d. functional RPA correlation energy is shown to equal one-half times the difference of all RPA electronic excitation energies computed at full and first order coupling. Thus, the RPA correlation energy may be considered as a difference of electronic zero point vibrational energies, where each eigenmode corresponds to an electronic excitation. This surprisingly simple result is intimately related to plasma theories of electron correlation. Differences to electron pair correlation models underlying popular correlated wave function methods are discussed. The RPA correlation energy is further transformed into an explicit functional of the Kohn-Sham orbitals. The only nontrivial ingredient to this functional is the sign function of the response operator. A stable iterative algorithm to evaluate this sign function based on the Newton-Schulz iteration is presented. Integral direct implementations scale as the fifth power of the system size, similar to second order Moeller-Plesset calcns. With these improvements, RPA may become the long-sought robust and efficient zero order post-Kohn-Sham correlation model. (c) 2008 American Institute of Physics.
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44. 44
  Eshuis, H.; Yarkony, J.; Furche, F. Fast computation of molecular random phase approximation correlation energies using resolution of the identity and imaginary frequency integration. J. Chem. Phys. 2010, 132, 234114, DOI: 10.1063/1.3442749
  44
  Fast computation of molecular random phase approximation correlation energies using resolution of the identity and imaginary frequency integration
  Eshuis, Henk; Yarkony, Julian; Furche, Filipp
  Journal of Chemical Physics (2010), 132 (23), 234114/1-234114/9CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics)
  The RPA is an increasingly popular post-Kohn-Sham correlation method, but its high computational cost has limited mol. applications to systems with few atoms. Here we present an efficient implementation of RPA correlation energies based on a combination of resoln. of the identity (RI) and imaginary frequency integration techniques. We show that the RI approxn. to four-index electron repulsion integrals leads to a variational upper bound to the exact RPA correlation energy if the Coulomb metric is used. Auxiliary basis sets optimized for second-order Moller-Plesset (MP2) calcns. are well suitable for RPA, as is demonstrated for the HEAT and MOLEKEL benchmark sets. Using imaginary frequency integration rather than diagonalization to compute the matrix square root necessary for RPA, evaluation of the RPA correlation energy requires O(N4logN) operations and O(N3) storage only; the price for this dramatic improvement over existing algorithms is a numerical quadrature. We propose a numerical integration scheme that is exact in the two-orbital case and converges exponentially with the no. of grid points. For most systems, 30-40 grid points yield μH accuracy in triple zeta basis sets, but much larger grids are necessary for small gap systems. The lowest-order approxn. to the present method is a post-Kohn-Sham frequency-domain version of opposite-spin Laplace-transform RI-MP2. Timings for polyacenes with up to 30 atoms show speed-ups of two orders of magnitude over previous implementations. The present approach makes it possible to routinely compute RPA correlation energies of systems well beyond 100 atoms, as is demonstrated for the octapeptide angiotensin II. (c) 2010 American Institute of Physics.
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45. 45
  Bleiziffer, P.; Heßelmann, A.; Görling, A. Resolution of identity approach for the Kohn–Sham correlation energy within the exact-exchange random-phase approximation. J. Chem. Phys. 2012, 136, 134102, DOI: 10.1063/1.3697845
  There is no corresponding record for this reference.
46. 46
  Lemke, Y.; Graf, D.; Kussmann, J.; Ochsenfeld, C. An assessment of orbital energy corrections for the direct random phase approximation and explicit σ-functionals. Mol. Phys. 2023, 121, e2098862, DOI: 10.1080/00268976.2022.2098862
  There is no corresponding record for this reference.
47. 47
  Heßelmann, A.; Görling, A. Random phase approximation correlation energies with exact Kohn–Sham exchange. Mol. Phys. 2010, 108, 359– 372, DOI: 10.1080/00268970903476662
  47
  Random phase approximation correlation energies with exact Kohn-Sham exchange
  Hesselmann, Andreas; Goerling, Andreas
  Molecular Physics (2010), 108 (3-4), 359-372CODEN: MOPHAM; ISSN:0026-8976. (Taylor & Francis Ltd.)
  The RPA correlation energy is expressed in terms of the exact local Kohn-Sham (KS) exchange potential and corresponding adiabatic and nonadiabatic exchange kernels for d.-functional ref. determinants. The approach naturally extends the RPA method in which, conventionally, only the Coulomb kernel is included. By comparison with the coupled cluster singles doubles with perturbative triples method it is shown for a set of small mols. that the new RPA method based on KS exchange yields correlation energies more accurate than RPA on the basis of Hartree-Fock exchange.
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48. 48
  Ángyán, J. G.; Liu, R.-F.; Toulouse, J.; Jansen, G. Correlation Energy Expressions from the Adiabatic-Connection Fluctuation–Dissipation Theorem Approach. J. Chem. Theory Comput. 2011, 7, 3116– 3130, DOI: 10.1021/ct200501r
  48
  Correlation Energy Expressions from the Adiabatic-Connection Fluctuation-Dissipation Theorem Approach
  Angyan, Janos G.; Liu, Ru-Fen; Toulouse, Julien; Jansen, Georg
  Journal of Chemical Theory and Computation (2011), 7 (10), 3116-3130CODEN: JCTCCE; ISSN:1549-9618. (American Chemical Society)
  We explore several RPA correlation energy variants within the adiabatic-connection fluctuation-dissipation theorem approach. These variants differ in the way the exchange interactions are treated. One of these variants, named dRPA-II, is original to this work and closely resembles the second-order screened exchange (SOSEX) method. We discuss and clarify the connections among different RPA formulations. We derive the spin-adapted forms of all the variants for closed-shell systems and test them on a few at. and mol. systems with and without range sepn. of the electron-electron interaction.
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49. 49
  Heßelmann, A. Random-phase-approximation correlation method including exchange interactions. Phys. Rev. A 2012, 85, 012517, DOI: 10.1103/PhysRevA.85.012517
  49
  Random-phase-approximation correlation method including exchange interactions
  Hesselmann, Andreas
  Physical Review A: Atomic, Molecular, and Optical Physics (2012), 85 (1-A), 012517/1-012517/10CODEN: PLRAAN; ISSN:1050-2947. (American Physical Society)
  Two random-phase-approxn. correlation methods are introduced that take into account exchange interactions. The first one, termed RPAX, is obtained from a simple modification of the ring coupled-cluster doubles amplitude equation, while the second, termed RPAX2, is based on the first method using a slightly modified update equation for the amplitudes. It is shown that this second RPAX2 method can be implemented with a computational algorithm that scales only with the fifth power of the mol. size with the aid of d. fitting or the Cholesky decompn. of two-electron integrals. It is thus not much more costly than std. second-order perturbation theory methods and can be applied to quite large mol. systems. Moreover, numerical tests for chem. reaction energies and intermol. interaction energies have shown that the RPAX2 method, if based on a Perdew-Burke-Ernzerhof exchange Kohn-Sham ref. determinant, yields results which are very close to coupled-cluster with single, double, and perturbative triple excitations ref. results.
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50. 50
  Eshuis, H.; Bates, J. E.; Furche, F. Electron correlation methods based on the random phase approximation. Theor. Chem. Acc. 2012, 131, 1084, DOI: 10.1007/s00214-011-1084-8
  There is no corresponding record for this reference.
51. 51
  Mussard, B.; Rocca, D.; Jansen, G.; Ángyán, J. G. Dielectric Matrix Formulation of Correlation Energies in the Random Phase Approximation: Inclusion of Exchange Effects. J. Chem. Theory Comput. 2016, 12, 2191– 2202, DOI: 10.1021/acs.jctc.5b01129
  51
  Dielectric Matrix Formulation of Correlation Energies in the Random Phase Approximation: Inclusion of Exchange Effects
  Mussard, Bastien; Rocca, Dario; Jansen, Georg; Angyan, Janos G.
  Journal of Chemical Theory and Computation (2016), 12 (5), 2191-2202CODEN: JCTCCE; ISSN:1549-9618. (American Chemical Society)
  Starting from the general expression for the ground state correlation energy in the adiabatic-connection fluctuation-dissipation theorem (ACFDT) framework, it is shown that the dielec. matrix formulation, which is usually applied to calc. the direct RPA (dRPA) correlation energy, can be used for alternative RPA expressions including exchange effects. Within this famework, the ACFDT analog of the second order screened exchange (SOSEX) approxn. leads to a logarithmic formula for the correlation energy similar to the direct RPA expression. Alternatively, the contribution of the exchange can be included in the kernel used to evaluate the response functions. In this case, the use of an approx. kernel is crucial to simplify the formalism and to obtain a correlation energy in logarithmic form. Tech. details of the implementation of these methods are discussed, and it is shown that one can take advantage of d. fitting or Cholesky decompn. techniques to improve the computational efficiency; a discussion on the numerical quadrature made on the frequency variable is also provided. A series of test calcns. on at. correlation energies and mol. reaction energies shows that exchange effects are instrumental for improvement over direct RPA results.
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52. 52
  Dixit, A.; Ángyán, J. G.; Rocca, D. Improving the accuracy of ground-state correlation energies within a plane-wave basis set: The electron-hole exchange kernel. J. Chem. Phys. 2016, 145, 104105, DOI: 10.1063/1.4962352
  52
  Improving the accuracy of ground-state correlation energies within a plane-wave basis set: The electron-hole exchange kernel
  Dixit, Anant; Angyan, Janos G.; Rocca, Dario
  Journal of Chemical Physics (2016), 145 (10), 104105/1-104105/12CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics)
  A new formalism was recently proposed to improve RPA correlation energies by including approx. exchange effects. Within this framework, by keeping only the electron-hole contributions to the exchange kernel, two approxns. can be obtained: An adiabatic connection analog of the second order screened exchange (AC-SOSEX) and an approx. electron-hole time-dependent Hartree-Fock (eh-TDHF). Here we show how this formalism is suitable for an efficient implementation within the plane-wave basis set. The response functions involved in the AC-SOSEX and eh-TDHF equations can indeed be compactly represented by an auxiliary basis set obtained from the diagonalization of an approx. dielec. matrix. Addnl., the explicit calcn. of unoccupied states can be avoided by using d. functional perturbation theory techniques and the matrix elements of dynamical response functions can be efficiently computed by applying the Lanczos algorithm. As shown by several applications to reaction energies and weakly bound dimers, the inclusion of the electron-hole kernel significantly improves the accuracy of ground-state correlation energies with respect to RPA and semi-local functionals. (c) 2016 American Institute of Physics.
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53. 53
  Chen, G. P.; Agee, M. M.; Furche, F. Performance and Scope of Perturbative Corrections to Random-Phase Approximation Energies. J. Chem. Theory Comput. 2018, 14, 5701– 5714, DOI: 10.1021/acs.jctc.8b00777
  53
  Performance and Scope of Perturbative Corrections to Random-Phase Approximation Energies
  Chen, Guo P.; Agee, Matthew M.; Furche, Filipp
  Journal of Chemical Theory and Computation (2018), 14 (11), 5701-5714CODEN: JCTCCE; ISSN:1549-9618. (American Chemical Society)
  It has been suspected since the early days of the RPA that corrections to RPA correlation energies result mostly from short-range correlation effects and are thus amenable to perturbation theory. Here we test this hypothesis by analyzing formal and numerical results for the most common beyond-RPA perturbative corrections, including the bare second-order exchange (SOX), second-order screened exchange (SOSEX), and approx. exchange kernel (AXK) methods. Our anal. is facilitated by efficient and robust algorithms based on the resoln.-of-the-identity (RI) approxn. and numerical frequency integration, which enable benchmark beyond-RPA calcns. on medium- and large-size mols. with size-independent accuracy. The AXK method systematically improves upon RPA, SOX, and SOSEX for reaction barrier heights, reaction energies, and noncovalent interaction energies of main-group compds. The improved accuracy of AXK compared with SOX and SOSEX is attributed to stronger screening of bare SOX in AXK. For reactions involving transition-metal compds., particularly 3d transition-metal dimers, the AXK correction is too small and can even have the wrong sign. These observations are rationalized by a measure ‾α of the effective coupling strength for beyond-RPA correlation. When the effective coupling strength increases beyond a crit. ‾α value of approx. 0.5, the RPA errors increase rapidly and perturbative corrections become unreliable. Thus, perturbation theory can systematically correct RPA but only for systems and properties qual. well captured by RPA, as indicated by small ‾α values.
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54. 54
  Kussmann, J.; Ochsenfeld, C. Pre-selective screening for matrix elements in linear-scaling exact exchange calculations. J. Chem. Phys. 2013, 138, 134114, DOI: 10.1063/1.4796441
  54
  Pre-selective screening for matrix elements in linear-scaling exact exchange calculations
  Kussmann, Joerg; Ochsenfeld, Christian
  Journal of Chemical Physics (2013), 138 (13), 134114/1-134114/7CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics)
  We present a simple but accurate preselection method based on Schwarz integral ests. to det. the significant elements of the exact exchange matrix before its evaluation, thus providing an asymptotical linear-scaling behavior for non-metallic systems. Our screening procedure proves to be highly suitable for exchange matrix calcns. on massively parallel computing architectures, such as graphical processing units, for which we present a first linear-scaling exchange matrix evaluation algorithm. (c) 2013 American Institute of Physics.
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55. 55
  Kussmann, J.; Ochsenfeld, C. Preselective Screening for Linear-Scaling Exact Exchange-Gradient Calculations for Graphics Processing Units and General Strong-Scaling Massively Parallel Calculations. J. Chem. Theory Comput. 2015, 11, 918, DOI: 10.1021/ct501189u
  55
  Preselective Screening for Linear-Scaling Exact Exchange-Gradient Calculations for Graphics Processing Units and General Strong-Scaling Massively Parallel Calculations
  Kussmann, Joerg; Ochsenfeld, Christian
  Journal of Chemical Theory and Computation (2015), 11 (3), 918-922CODEN: JCTCCE; ISSN:1549-9618. (American Chemical Society)
  We present an extension of our recently presented PreLinK scheme (J. Chem. Phys. 2013, 138, 134114) for the exact exchange contribution to nuclear forces. The significant contributions to the exchange gradient are detd. by preselection based on accurate shell-pair contributions to the SCF exchange energy prior to the calcn. Therefore, our method is highly suitable for massively parallel electronic structure calcns. because of an efficient load balancing of the significant contributions only and an unhampered control flow. The efficiency of our method is shown for several illustrative calcns. on single GPU servers, as well as for hybrid MPI/CUDA parallel calcns. with the largest system comprising 3369 atoms and 26952 basis functions.
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56. 56
  Kussmann, J.; Laqua, H.; Ochsenfeld, C. Highly Efficient Resolution-of-Identity Density Functional Theory Calculations on Central and Graphics Processing Units. J. Chem. Theory Comput. 2021, 17, 1512, DOI: 10.1021/acs.jctc.0c01252
  56
  Highly Efficient Resolution-of-Identity Density Functional Theory Calculations on Central and Graphics Processing Units
  Kussmann, Joerg; Laqua, Henryk; Ochsenfeld, Christian
  Journal of Chemical Theory and Computation (2021), 17 (3), 1512-1521CODEN: JCTCCE; ISSN:1549-9618. (American Chemical Society)
  We present an efficient method to evaluate Coulomb potential matrixes using the resoln. of identity approxn. and semilocal exchange-correlation potentials on central (CPU) and graphics processing units (GPU). The new GPU-based RI-algorithm shows a high performance and ensures the favorable scaling with increasing basis set size as the conventional CPU-based method. Furthermore, our method is based on the J-engine algorithm [White, Head-Gordon, J. Chem. Phys., 1996, 7, 2620], which allows for further optimizations that also provide a significant improvement of the corresponding CPU-based algorithm. Due to the increased performance for the Coulomb evaluation, the calcn. of the exchange-correlation potential of d. functional theory on CPUs quickly becomes a bottleneck to the overall computational time. Hence, we also present a GPU-based algorithm to evaluate the exchange-correlation terms, which results in an overall high-performance method for d. functional calcns. The algorithms to evaluate the potential and nuclear deriv. terms are discussed, and their performance on CPUs and GPUs is demonstrated for illustrative calcns.
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57. 57
  Weigend, F. Accurate Coulomb-fitting basis sets for H to Rn. Phys. Chem. Chem. Phys. 2006, 8, 1057, DOI: 10.1039/b515623h
  57
  Accurate Coulomb-fitting basis sets for H to Rn
  Weigend, Florian
  Physical Chemistry Chemical Physics (2006), 8 (9), 1057-1065CODEN: PPCPFQ; ISSN:1463-9076. (Royal Society of Chemistry)
  A series of auxiliary basis sets to fit Coulomb potentials for the elements H to Rn (except lanthanides) is presented. For each element only one auxiliary basis set is needed to approx. Coulomb energies in conjunction with orbital basis sets of split valence, triple zeta valence and quadruple zeta valence quality with errors of typically below ca. 0.15 kJ mol-1 per atom; this was demonstrated in conjunction with the recently developed orbital basis sets of types def2-SV(P), def2-TZVP and def2-QZVPP for a large set of small mols. representing (nearly) each element in all of its common oxidn. states. These auxiliary bases are slightly more than three times larger than orbital bases of split valence quality. Compared to non-approximated treatments, computation times for the Coulomb part are reduced by a factor of ca. 8 for def2-SV(P) orbital bases, ca. 25 for def2-TZVP and ca. 100 for def2-QZVPP orbital bases.
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58. 58
  Laqua, H.; Kussmann, J.; Ochsenfeld, C. Efficient and Linear-Scaling Seminumerical Method for Local Hybrid Density Functionals. J. Chem. Theory Comput. 2018, 14, 3451– 3458, DOI: 10.1021/acs.jctc.8b00062
  58
  Efficient and Linear-Scaling Seminumerical Method for Local Hybrid Density Functionals
  Laqua, Henryk; Kussmann, Joerg; Ochsenfeld, Christian
  Journal of Chemical Theory and Computation (2018), 14 (7), 3451-3458CODEN: JCTCCE; ISSN:1549-9618. (American Chemical Society)
  Local hybrid functionals, i.e., functionals with local dependence on the exact exchange energy d., generalize the popular class of global hybrid functionals and extend the applicability of d. functional theory to electronic structures that require an accurate description of static correlation. However, the higher computational cost compared to conventional Kohn-Sham d. functional theory restrained their widespread application. Here, we present a low-prefactor, linear-scaling method to evaluate the local hybrid exchange-correlation potential as well as the corresponding nuclear forces by employing a seminumerical integration scheme. In the seminumerical scheme, one integration in the electron repulsion integrals is carried out anal. and the other one is carried out numerically, employing an integration grid. A high computational efficiency is achieved by combining the preLinK method with explicit screening of integrals for batches of grid points to minimize the screening overhead. This new method, denoted as preLinX, provides an 8-fold performance increase for a DNA fragment contg. four base pairs as compared to existing S- and P-junction-based methods. In this way, our method allows for the evaluation of local hybrid functionals at a cost similar to that of global hybrid functionals. The linear-scaling behavior, efficiency, accuracy, and multi-node parallelization of the presented method is demonstrated for large systems contg. more than 1000 atoms.
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59. 59
  Laqua, H.; Thompson, T. H.; Kussmann, J.; Ochsenfeld, C. Highly Efficient, Linear-Scaling Seminumerical Exact-Exchange Method for Graphic Processing Units. J. Chem. Theory Comput. 2020, 16, 1456, DOI: 10.1021/acs.jctc.9b00860
  59
  Highly Efficient, Linear-Scaling Seminumerical Exact-Exchange Method for Graphic Processing Units
  Laqua, Henryk; Thompson, Travis H.; Kussmann, Joerg; Ochsenfeld, Christian
  Journal of Chemical Theory and Computation (2020), 16 (3), 1456-1468CODEN: JCTCCE; ISSN:1549-9618. (American Chemical Society)
  We present a highly efficient and asymptotically linear-scaling graphic processing unit accelerated seminumerical exact-exchange method (sn-LinK). We go beyond our previous central processing unit-based method (H. Laqua et al., 2018) by employing our recently developed integral bounds (T.H. Thomson and C. Ochsenfeld, 2019) and high-accuracy numerical integration grid (H. Laqua et al., 2018). The accuracy is assessed for several established test sets, providing errors significantly below 1mEh for the smallest grid. Moreover, a comprehensive performance anal. for large mols. between 62 and 1347 atoms is provided, revealing the outstanding performance of our method, in particular, for large basis sets such as the polarized quadruple-zeta level with diffuse functions.
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60. 60
  Laqua, H.; Kussmann, J.; Ochsenfeld, C. An improved molecular partitioning scheme for numerical quadratures in density functional theory. J. Chem. Phys. 2018, 149, 204111, DOI: 10.1063/1.5049435
  60
  An improved molecular partitioning scheme for numerical quadratures in density functional theory
  Laqua, Henryk; Kussmann, Joerg; Ochsenfeld, Christian
  Journal of Chemical Physics (2018), 149 (20), 204111/1-204111/8CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics)
  We present a modification to Becke's mol. partitioning scheme [A. D. Becke, J. Chem. Phys. 88, 2547 (1988)] that provides substantially better accuracy for weakly bound complexes and allows for a faster and linear scaling grid generation without introducing a cutoff error. We present the accuracy of our new partitioning scheme for atomization energies of small mols. and for interaction energies of van der Waals complexes. Furthermore, the efficiency and scaling behavior of the grid generation are demonstrated for large mol. systems with up to 1707 atoms. (c) 2018 American Institute of Physics.
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61. 61
  Lehtola, S.; Steigemann, C.; Oliveira, M. J. T.; Marques, M. A. L. Recent developments in libxc–A comprehensive library of functionals for density functional theory. Software X 2018, 7, 1– 5, DOI: 10.1016/j.softx.2017.11.002
  There is no corresponding record for this reference.
62. 62
  Weigend, F.; Furche, F.; Ahlrichs, R. Gaussian basis sets of quadruple zeta valence quality for atoms H–Kr. J. Chem. Phys. 2003, 119, 12753– 12762, DOI: 10.1063/1.1627293
  62
  Gaussian basis sets of quadruple zeta valence quality for atoms H-Kr
  Weigend, Florian; Furche, Filipp; Ahlrichs, Reinhart
  Journal of Chemical Physics (2003), 119 (24), 12753-12762CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics)
  We present Gaussian basis sets of quadruple zeta valence quality with a segmented contraction scheme for atoms H to Kr. This extends earlier work on segmented contracted split valence (SV) and triple zeta valence (TZV) basis sets. Contraction coeffs. and orbital exponents are fully optimized in at. Hartree-Fock (HF) calcns. As opposed to other quadruple zeta basis sets, the basis set errors in at. ground-state HF energies are less than 1 mEh and increase smoothly across the Periodic Table, while the no. of primitives is comparably small. Polarization functions are taken partly from previous work, partly optimized in at. MP2 calcns., and for a few cases detd. at the HF level for excited at. states nearly degenerate with the ground state. This leads to basis sets denoted QZVP for HF and d. functional theory (DFT) calcns., and for some atoms to a larger basis recommended for correlated treatments, QZVPP. We assess the performance of the basis sets in mol. HF, DFT, and MP2 calcns. for a sample of diat. and small polyat. mols. by a comparison of energies, bond lengths, and dipole moments with results obtained numerically or using very large basis sets. It is shown that basis sets of quadruple zeta quality are necessary to achieve an accuracy of 1 kcal/mol per bond in HF and DFT atomization energies. For compds. contg. third row as well as alk. and earth alk. metals it is demonstrated that the inclusion of high-lying core orbitals in the active space can be necessary for accurate correlated treatments. The QZVPP basis sets provide sufficient flexibility to polarize the core in those cases. All test calcns. indicate that the new basis sets lead to consistent accuracies in HF, DFT, or correlated treatments even in crit. cases where other basis sets may show deficiencies.
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63. 63
  Rappoport, D.; Furche, F. Property-optimized Gaussian basis sets for molecular response calculations. J. Chem. Phys. 2010, 133, 134105, DOI: 10.1063/1.3484283
  63
  Property-optimized Gaussian basis sets for molecular response calculations
  Rappoport, Dmitrij; Furche, Filipp
  Journal of Chemical Physics (2010), 133 (13), 134105/1-134105/11CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics)
  With recent advances in electronic structure methods, first-principles calcns. of electronic response properties, such as linear and nonlinear polarizabilities, have become possible for mols. with more than 100 atoms. Basis set incompleteness is typically the main source of error in such calcns. since traditional diffuse augmented basis sets are too costly to use or suffer from near linear dependence. To address this problem, we construct the first comprehensive set of property-optimized augmented basis sets for elements H-Rn except lanthanides. The new basis sets build on the Karlsruhe segmented contracted basis sets of split-valence to quadruple-zeta valence quality and add a small no. of moderately diffuse basis functions. The exponents are detd. variationally by maximization of at. Hartree-Fock polarizabilities using anal. deriv. methods. The performance of the resulting basis sets is assessed using a set of 313 mol. static Hartree-Fock polarizabilities. The mean abs. basis set errors are 3.6%, 1.1%, and 0.3% for property-optimized basis sets of split-valence, triple-zeta, and quadruple-zeta valence quality, resp. D. functional and second-order Moller-Plesset polarizabilities show similar basis set convergence. We demonstrate the efficiency of our basis sets by computing static polarizabilities of icosahedral fullerenes up to C720 using hybrid d. functional theory. (c) 2010 American Institute of Physics.
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64. 64
  Weigend, F.; Ahlrichs, R. Balanced basis sets of split valence, triple zeta valence and quadruple zeta valence quality for H to Rn: Design and assessment of accuracy. Phys. Chem. Chem. Phys. 2005, 7, 3297, DOI: 10.1039/b508541a
  64
  Balanced basis sets of split valence, triple zeta valence and quadruple zeta valence quality for H to Rn: Design and assessment of accuracy
  Weigend, Florian; Ahlrichs, Reinhart
  Physical Chemistry Chemical Physics (2005), 7 (18), 3297-3305CODEN: PPCPFQ; ISSN:1463-9076. (Royal Society of Chemistry)
  Gaussian basis sets of quadruple zeta valence quality for Rb-Rn are presented, as well as bases of split valence and triple zeta valence quality for H-Rn. The latter were obtained by (partly) modifying bases developed previously. A large set of more than 300 mols. representing (nearly) all elements-except lanthanides-in their common oxidn. states was used to assess the quality of the bases all across the periodic table. Quantities investigated were atomization energies, dipole moments and structure parameters for Hartree-Fock, d. functional theory and correlated methods, for which we had chosen Moller-Plesset perturbation theory as an example. Finally recommendations are given which type of basis set is used best for a certain level of theory and a desired quality of results.
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65. 65
  Hättig, C. Optimization of auxiliary basis sets for RI-MP2 and RI-CC2 calculations: Core–valence and quintuple-ζ basis sets for H to Ar and QZVPP basis sets for Li to Kr. Phys. Chem. Chem. Phys. 2005, 7, 59– 66, DOI: 10.1039/B415208E
  There is no corresponding record for this reference.
66. 66
  Hellweg, A.; Hättig, C.; Höfener, S.; Klopper, W. Optimized accurate auxiliary basis sets for RI-MP2 and RI-CC2 calculations for the atoms Rb to Rn. Theor. Chem. Acc. 2007, 117, 587– 597, DOI: 10.1007/s00214-007-0250-5
  66
  Optimized accurate auxiliary basis sets for RI-MP2 and RI-CC2 calculations for the atoms Rb to Rn
  Hellweg, Arnim; Haettig, Christof; Hoefener, Sebastian; Klopper, Wim
  Theoretical Chemistry Accounts (2007), 117 (4), 587-597CODEN: TCACFW; ISSN:1432-881X. (Springer GmbH)
  The introduction of the resoln.-of-the-identity (RI) approxn. for electron repulsion integrals in quantum chem. calcns. requires in addn. to the orbital basis so-called auxiliary or fitting basis sets. We report here such auxiliary basis sets optimized for second-order Moller-Plesset perturbation theory for the recently published (Weigend and Ahlrichs Phys. Chem. Chem. Phys., 2005, 7, 3297-3305) segmented contracted Gaussian basis sets of split, triple-ζ and quadruple-ζ valence quality for the atoms Rb-Rn (except lanthanides). These basis sets are designed for use in connection with small-core effective core potentials including scalar relativistic corrections. Hereby accurate resoln.-of-the-identity calcns. with second-order Moller-Plesset perturbation theory (MP2) and related methods can now be performed for mols. contg. elements from H to Rn. The error of the RI approxn. has been evaluated for a test set of 385 small and medium sized mols., which represent the common oxidn. states of each element, and is compared with the one-electron basis set error, estd. based on highly accurate explicitly correlated MP2-R12 calcns. With the reported auxiliary basis sets the RI error for MP2 correlation energies is typically two orders of magnitude smaller than the one-electron basis set error, independent on the position of the atoms in the periodic table.
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67. 67
  Hellweg, A.; Rappoport, D. Development of new auxiliary basis functions of the Karlsruhe segmented contracted basis sets including diffuse basis functions (def2-SVPD, def2-TZVPPD, and def2-QZVPPD) for RI-MP2 and RI-CC calculations. Phys. Chem. Chem. Phys. 2015, 17, 1010– 1017, DOI: 10.1039/C4CP04286G
  There is no corresponding record for this reference.
68. 68
  Kaltak, M.; Klimeš, J.; Kresse, G. Low Scaling Algorithms for the Random Phase Approximation: Imaginary Time and Laplace Transformations. J. Chem. Theory Comput. 2014, 10, 2498, DOI: 10.1021/ct5001268
  68
  Low Scaling Algorithms for the Random Phase Approximation: Imaginary Time and Laplace Transformations
  Kaltak, Merzuk; Klimes, Jiri; Kresse, Georg
  Journal of Chemical Theory and Computation (2014), 10 (6), 2498-2507CODEN: JCTCCE; ISSN:1549-9618. (American Chemical Society)
  We det. efficient imaginary frequency and imaginary time grids for second-order Moller-Plesset (MP) perturbation theory. The least-squares and Minimax quadratures are compared for periodic systems, finding that the Minimax quadrature performs slightly better for the considered materials. We show that the imaginary frequency grids developed for second order also perform well for the correlation energy in the direct RPA. Furthermore, we show that the polarizabilities on the imaginary time axis can be Fourier-transformed to the imaginary frequency domain, since the time and frequency Minimax grids are dual to each other. The same duality is obsd. for the least-squares grids. The transformation from imaginary time to imaginary frequency allows one to reduce the time complexity to cubic (in system size), so that RPA correlation energies become accessible for large systems.
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69. 69
  Kaltak, M.; Kresse, G. Minimax isometry method: A compressive sensing approach for Matsubara summation in many-body perturbation theory. Phys. Rev. B 2020, 101, 205145, DOI: 10.1103/PhysRevB.101.205145
  69
  Minimax isometry method: A compressive sensing approach for Matsubara summation in many-body perturbation theory
  Kaltak, Merzuk; Kresse, Georg
  Physical Review B (2020), 101 (20), 205145CODEN: PRBHB7; ISSN:2469-9969. (American Physical Society)
  We present a compressive sensing approach for the long-standing problem of Matsubara summation in many-body perturbation theory. By constructing low-dimensional, almost isometric subspaces of the Hilbert space we obtain optimum imaginary time and frequency grids that allow for extreme data compression of fermionic and bosonic functions in a broad temp. regime. The method is applied to the random phase and self-consistent GW approxn. of the grand potential. Integration and transformation errors are investigated for Si and SrVO3.
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70. 70
  Andrae, D.; Häußermann, U.; Dolg, M.; Stoll, H.; Preuß, H. Energy-adjusted ab initio pseudopotentials for the second and third row transition elements. Theor. Chim. Acta 1990, 77, 123– 141, DOI: 10.1007/BF01114537
  70
  Energy-adjusted ab initio pseudopotentials for the second and third row transition elements
  Andrae, D.; Haeussermann, U.; Dolg, M.; Stoll, H.; Preuss, H.
  Theoretica Chimica Acta (1990), 77 (2), 123-41CODEN: TCHAAM; ISSN:0040-5744.
  Nonrelativistic and quasirelativistic ab initio pseudopotentials substituting the M(z-28)+-core orbitals of the second row transition elements and M(z-60)+-core orbitals of the third row transition elements, resp., and optimized (8s7p6d)/[6s5p3d]-GTO valence basis sets for use in mol. calcns. have been generated. Addnl., corresponding spin-orbit operators have also been derived. At. excitation and ionization energies from numerical HF as well as from SCF pseudopotential calcns. using the derived basis sets differ in most cases by less than 0.1 eV from corresponding numerical all-electron results. Spin-orbit splittings for low-lying states are in reasonable agreement with corresponding all-electron Dirac-Fock (DF) results.
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71. 71
  Metz, B.; Stoll, H.; Dolg, M. Small-core multiconfiguration-Dirac–Hartree–Fock-adjusted pseudopotentials for post-d main group elements: Application to PbH and PbO. J. Chem. Phys. 2000, 113, 2563– 2569, DOI: 10.1063/1.1305880
  71
  Small-core multiconfiguration-Dirac-Hartree-Fock-adjusted pseudopotentials for post-d main group elements: Application to PbH and PbO
  Metz, Bernhard; Stoll, Hermann; Dolg, Michael
  Journal of Chemical Physics (2000), 113 (7), 2563-2569CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics)
  Relativistic pseudopotentials (PPs) of the energy-consistent variety have been generated for the post-d group 13-15 elements, by adjustment to multiconfiguration Dirac-Hartree-Fock data based on the Dirac-Coulomb-Breit Hamiltonian. The outer-core (n-1)spd shells are explicitly treated together with the nsp valence shell, with these PPs, and the implications of the small-core choice are discussed by comparison to a corresponding large-core PP, in the case of Pb. Results from valence ab initio one- and two-component calcns. using both PPs are presented for the fine-structure splitting of the ns2np2 ground-state configuration of the Pb atom, and for spectroscopic consts. of PbH (X 2Π1/2, 2Π3/2) and PbO (X 1Σ+). In addn., a combination of small-core and large-core PPs has been explored in spin-free-state shifted calcns. for the above mols.
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72. 72
  Peterson, K. A. Systematically convergent basis sets with relativistic pseudopotentials. I. Correlation consistent basis sets for the post-d group 13–15 elements. J. Chem. Phys. 2003, 119, 11099– 11112, DOI: 10.1063/1.1622923
  72
  Systematically convergent basis sets with relativistic pseudopotentials. I. Correlation consistent basis sets for the post-d group 13-15 elements
  Peterson, Kirk A.
  Journal of Chemical Physics (2003), 119 (21), 11099-11112CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics)
  New correlation consistent-like basis sets have been developed for the post-d group 13-15 elements (Ga-As, In-Sb, Tl-Bi) employing accurate, small-core relativistic pseudopotentials. The resulting basis sets, which are denoted cc-pVnZ-PP, are appropriate for valence electron correlation and range in size from (8s7p7d)/[4s3p2d] for the cc-pVDZ-PP to (16s13p12d3f2g1h)/[7s7p5d3f2g1h] for the cc-pV5Z-PP sets. Benchmark calcns. on selected diat. mols. (As2, Sb2, Bi2, AsN, SbN, BiN, GeO, SnO, PbO, GaCl, InCl, TlCl, GaH, InH, and TlH) are reported using these new basis sets at the coupled cluster level of theory. Much like their all-electron counterparts, the cc-pVnZ-PP basis sets yield systematic convergence of total energies and spectroscopic consts. In several cases all-electron benchmark calcns. were also carried out for comparison. The results from the pseudopotential and all-electron calcns. were nearly identical when scalar relativity was accurately included in the all-electron work. Diffuse-augmented basis sets, aug-cc-pVnZ-PP, have also been developed and have been used in calcns. of the at. electron affinities.
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73. 73
  Peterson, K. A.; Figgen, D.; Goll, E.; Stoll, H.; Dolg, M. Systematically convergent basis sets with relativistic pseudopotentials. II. Small-core pseudopotentials and correlation consistent basis sets for the post-d group 16–18 elements. J. Chem. Phys. 2003, 119, 11113– 11123, DOI: 10.1063/1.1622924
  73
  Systematically convergent basis sets with relativistic pseudopotentials. II. Small-core pseudopotentials and correlation consistent basis sets for the post-d group 16-18 elements
  Peterson, Kirk A.; Figgen, Detlev; Goll, Erich; Stoll, Hermann; Dolg, Michael
  Journal of Chemical Physics (2003), 119 (21), 11113-11123CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics)
  A series of correlation consistent basis sets have been developed for the post-d group 16-18 elements in conjunction with small-core relativistic pseudopotentials of the energy-consistent variety. The latter were adjusted to multiconfiguration Dirac-Hartree-Fock data based on the Dirac-Coulomb-Breit Hamiltonian. The outer-core (n-1)spd shells are explicitly treated together with the nsp valence shell with these PPs. The accompanying cc-pVnZ-PP and aug-cc-pVnZ-PP basis sets range in size from DZ to 5Z quality and yield systematic convergence of both Hartree-Fock and correlated total energies. In addn. to the calcn. of at. electron affinities and dipole polarizabilities of the rare gas atoms, numerous mol. benchmark calcns. (HBr, HI, HAt, Br2, I2, At2, SiSe, SiTe, SiPo, KrH+, XeH+, and RnH+) are also reported at the coupled cluster level of theory. For the purposes of comparison, all-electron calcns. using the Douglas-Kroll-Hess Hamiltonian have also been carried out for the halogen-contg. mols. using basis sets of 5Z quality.
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74. 74
  Morgante, P.; Peverati, R. Statistically representative databases for density functional theory via data science. Phys. Chem. Chem. Phys. 2019, 21, 19092– 19103, DOI: 10.1039/C9CP03211H
  74
  Statistically representative databases for density functional theory via data science
  Morgante, Pierpaolo; Peverati, Roberto
  Physical Chemistry Chemical Physics (2019), 21 (35), 19092-19103CODEN: PPCPFQ; ISSN:1463-9076. (Royal Society of Chemistry)
  The amt. of data and no. of databases for the assessment and parameterization of d. functional theory methods has grown substantially in the past two decades. In this work, we introduce a novel cluster anal. technique for d. functional theory calcns. of the electronic structure of atoms and mols. with the goal of creating new statistically significant databases with broad chem. scope, and a manageable no. of data-points. By analyzing without a priori chem. assumptions a population of almost 350k data-points, we create a new database called ASCDB contg. only 200 data-points. This new database holds the same chem. information as the larger population of data from which it is obtained, but with a computational cost that is reduced by several orders of magnitude. The labeling of the significant chem. properties is performed a posteriori on the resulting 16 subsets, classifying them into four areas of chem. importance: non-covalent interactions, thermochem., non-local effects, and unbiased calcns. The anal. of the results and their transferability shows that ASCDB is capable of providing the same information as that of the larger collection of data-such as GMTKN55, MGCDB84, and Minnesota 2015B-for several d. functional theory methods and basis sets. In light of these results, we suggest the use of this new small database as a first inexpensive tool for the evaluation and parameterization of electronic structure theory methods.
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75. 75
  Morgante, P.; Peverati, R. ACCDB: A collection of chemistry databases for broad computational purposes. J. Comput. Chem. 2019, 40, 839– 848, DOI: 10.1002/jcc.25761
  75
  ACCDB: A collection of chemistry databases for broad computational purposes
  Morgante, Pierpaolo; Peverati, Roberto
  Journal of Computational Chemistry (2019), 40 (6), 839-848CODEN: JCCHDD; ISSN:0192-8651. (John Wiley & Sons, Inc.)
  The importance of databases of reliable and accurate data in chem. has substantially increased in the past two decades. Their main usage is to parametrize electronic structure theory methods, and to assess their capabilities and accuracy for a broad set of chem. problems. The collection we present here-ACCDB-includes data from 16 different research groups, for a total of 44,931 unique ref. data points, all at a level of theory significantly higher than d. functional theory, and covering most of the periodic table. It is composed of five databases taken from literature (GMTKN, MGCDB84, Minnesota2015, DP284, and W4-17), two newly developed reaction energy databases (W4-17-RE and MN-RE), and a new collection of databases contg. transition metals. A set of expandable software tools for its manipulation is also presented here for the first time, as well as a case study where ACCDB is used for benchmarking com. CPUs for chem. calcns. © 2018 Wiley Periodicals, Inc.
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76. 76
  Fritsch, F. N.; Butland, J. A Method for Constructing Local Monotone Piecewise Cubic Interpolants. SIAM J. Sci. Stat. Comp. 1984, 5, 300, DOI: 10.1137/0905021
  There is no corresponding record for this reference.
77. 77
  Peverati, R. Fitting elephants in the density functionals zoo: Statistical criteria for the evaluation of density functional theory methods as a suitable replacement for counting parameters. Int. J. Quantum Chem. 2021, 121, e26379, DOI: 10.1002/qua.26379
  77
  Fitting elephants in the density functionals zoo: Statistical criteria for the evaluation of density functional theory methods as a suitable replacement for counting parameters
  Peverati, Roberto
  International Journal of Quantum Chemistry (2021), 121 (1), e26379CODEN: IJQCB2; ISSN:0020-7608. (John Wiley & Sons, Inc.)
  Counting parameters has become customary in the d. functional theory community as a way to infer the transferability of popular approxns. to the exchange-correlation functionals. Recent work in data science, however, has demonstrated that the no. of parameters of a fitted model is not related to the complexity of the model itself, nor to its eventual overfitting. Using similar arguments, here, we show that it is possible to represent every modern exchange-correlation functional approxns. using just one single parameter. This procedure proves the futility of the no. of parameters as a measure of transferability. To counteract this shortcoming, we introduce and analyze the performance of three statistical criteria for the evaluation of the transferability of exchange-correlation functionals. The three criteria are called Akaike information criterion, Vapnik-Chervonenkis criterion, and cross-validation criterion and are used in a preliminary assessment to rank 60 exchange-correlation functional approxns. using the ASCDB database of chem. data.
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78. 78
  Korth, M.; Grimme, S. Mindless DFT Benchmarking. J. Chem. Theory Comput. 2009, 5, 993– 1003, DOI: 10.1021/ct800511q
  78
  "Mindless" DFT Benchmarking
  Korth, Martin; Grimme, Stefan
  Journal of Chemical Theory and Computation (2009), 5 (4), 993-1003CODEN: JCTCCE; ISSN:1549-9618. (American Chemical Society)
  A diversity-oriented approach for the generation of thermochem. benchmark sets is presented. Test sets consisting of randomly generated "artificial mols." (AMs) are proposed that rely on systematic constraints rather than uncontrolled chem. biases. In this way, the narrow structural space of chem. intuition is opened up and electronically difficult cases can be produced in an unforeseeable manner. For the calcn. of chem. meaningful relative energies, AMs are systematically decompd. into small mols. (hydrides and diatomics). Two different example test sets contg. eight-atom, single-ref., main group AMs with chem. very diverse and unusual structures are generated. Highly accurate all-electron, estd. CCSD(T)/complete basis set ref. energies are also provided. They are used to benchmark the d. functionals S-VWN, BP86, B-LYP, B97-D, PBE, TPSS, PBEh, BH-LYP, B3-PW91, B3-LYP, B2-PLYP, B2GP-PLYP, BMK, MPW1B95, M05, M05-2X, PW6B95, M06, M06-L, and M06-2X. In selected cases, an empirical dispersion correction (DFT-D) has been applied. Due to the compn. of the sets, it is expected that a good performance indicates "robustness" in many different chem. applications. The results of a statistical anal. of the errors for the entire set with 165 entries (av. reaction energy of 117 kcal/mol, dubbed as the MB08-165 set) perfectly fit to the "Jacob's ladder" metaphor for the ordering of d. functionals according to their theor. complexity. The mean abs. deviation (MAD) decreases very strongly from LDA (20 kcal/mol) to GGAs (MAD of about 10 kcal/mol) but then was less pronounced to hybrid-GGAs (MAD of about 6-8 kcal/mol). The best performance (MAD of 4.1-4.2 kcal/mol) is found for the (fifth-rung) double-hybrid functionals B2-PLYP-D and B2GP-PLYP-D, followed by the M06-2X meta-hybrid (MAD of 4.8 kcal/mol). The significance of the proposed approach for thermodn. benchmarking is discussed and related to the obsd. performance ranking also regarding wave function based methods.
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79. 79
  Broyden, C. G. The Convergence of a Class of Double-rank Minimization Algorithms 1. General Considerations. IMA J. Appl. Math. 1970, 6, 76, DOI: 10.1093/imamat/6.1.76
  There is no corresponding record for this reference.
80. 80
  Fletcher, R. A new approach to variable metric algorithms. Comput. J. 1970, 13, 317, DOI: 10.1093/comjnl/13.3.317
  There is no corresponding record for this reference.
81. 81
  Goldfarb, D. A family of variable-metric methods derived by variational methods. Math. Comput. 1970, 24, 23, DOI: 10.1090/S0025-5718-1970-0258249-6
  There is no corresponding record for this reference.
82. 82
  Shanno, D. F. Conditioning of quasi-Newton methods for function minimization. Math. Comput. 1970, 24, 647, DOI: 10.1090/S0025-5718-1970-0274029-X
  There is no corresponding record for this reference.
83. 83
  Parthiban, S.; Martin, J. M. L. Assessment of W1 and W2 theories for the computation of electron affinities, ionization potentials, heats of formation, and proton affinities. J. Chem. Phys. 2001, 114, 6014– 6029, DOI: 10.1063/1.1356014
  83
  Assessment of W1 and W2 theories for the computation of electron affinities, ionization potentials, heats of formation, and proton affinities
  Parthiban, Srinivasan; Martin, Jan M. L.
  Journal of Chemical Physics (2001), 114 (14), 6014-6029CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics)
  The performance of two recent ab initio computational thermochem. schemes, W1 and W2 theory [J. M. L. Martin and G. de Oliveira, J. Chem. Phys. 111, 1843 (1999)], is assessed for an enlarged sample of thermochem. data consisting of the ionization potentials and electron affinities in the G2-1 and G2-2 sets, as well as the heats of formation in the G2-1 and a subset of the G2-2 set. We find W1 theory to be several times more accurate for ionization potentials and electron affinities than commonly used (and less expensive) computational thermochem. schemes such as G2, G3, and CBS-QB3: W2 theory represents a slight improvement for electron affinities but no significant one for ionization potentials. The use of a two-point A+B/L5 rather than a three-point A+B/CL extrapolation for the SCF component greatly enhances the numerical stability of the W1 method for systems with slow basis set convergence. Inclusion of first-order spin-orbit coupling is essential for accurate ionization potentials and electron affinities involving degenerate electronic states: Inner-shell correlation is somewhat more important for ionization potentials than for electron affinities, while scalar relativistic effects are required for the highest accuracy. The mean deviation from expt. for the G2-1 heats of formation is within the av. exptl. uncertainty. W1 theory appears to be a valuable tool for obtaining benchmark quality proton affinities.
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84. 84
  Karton, A.; Sylvetsky, N.; Martin, J. M. L. W4–17: A diverse and high-confidence dataset of atomization energies for benchmarking high-level electronic structure methods. J. Comput. Chem. 2017, 38, 2063, DOI: 10.1002/jcc.24854
  84
  W4-17: A diverse and high-confidence dataset of atomization energies for benchmarking high-level electronic structure methods
  Karton, Amir; Sylvetsky, Nitai; Martin, Jan M. L.
  Journal of Computational Chemistry (2017), 38 (24), 2063-2075CODEN: JCCHDD; ISSN:0192-8651. (John Wiley & Sons, Inc.)
  Atomization reactions are among the most challenging tests for electronic structure methods. We use the first-principles Weizmann-4 (W4) computational thermochem. protocol to generate the W4-17 dataset of 200 total atomization energies (TAEs) with 3σ confidence intervals of 1 kJ mol-1. W4-17 is an extension of the earlier W4-11 dataset; it includes first- and second-row mols. and radicals with up to eight non-hydrogen atoms. These cover a broad spectrum of bonding situations and multireference character, and as such are an excellent benchmark for the parameterization and validation of highly accurate ab initio methods (e.g., CCSD(T) composite procedures) and double-hybrid d. functional theory (DHDFT) methods. The W4-17 dataset contains two subsets (i) a non-multireference subset of 183 systems characterized by dynamical or moderate nondynamical correlation effects (denoted W4-17-nonMR) and (ii) a highly multireference subset of 17 systems (W4-17-MR). We use these databases to evaluate the performance of a wide range of CCSD(T) composite procedures (e.g., G4, G4(MP2), G4(MP2)-6X, ROG4(MP2)-6X, CBS-QB3, ROCBS-QB3, CBS-APNO, ccCA-PS3, W1, W2, W1-F12, W2-F12, W1X-1, and W2X) and DHDFT methods (e.g., B2-PLYP, B2GP-PLYP, B2K-PLYP, DSD-BLYP, DSD-PBEP86, PWPB95, ωB97X-2(LP), and ωB97X-2(TQZ)). © 2017 Wiley Periodicals, Inc.
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85. 85
  Karton, A.; Daon, S.; Martin, J. M. W4–11: A high-confidence benchmark dataset for computational thermochemistry derived from first-principles W4 data. Chem. Phys. Lett. 2011, 510, 165, DOI: 10.1016/j.cplett.2011.05.007
  85
  W4-11: A high-confidence benchmark dataset for computational thermochemistry derived from first-principles W4 data
  Karton, Amir; Daon, Shauli; Martin, Jan M. L.
  Chemical Physics Letters (2011), 510 (4-6), 165-178CODEN: CHPLBC; ISSN:0009-2614. (Elsevier B.V.)
  We show that the purely first-principles Weizmann-4 (W4) computational thermochem. method developed in our group can reproduce available Active Thermochem. Tables atomization energies for 35 mols. with a 3σ uncertainty of under 1 kJ/mol. We then employ this method to generate the W4-11 dataset of 140 total atomization energies of small first-and second-row mols. and radicals. These cover a broad spectrum of bonding situations and multireference character, and as such are an excellent, quasi-automated benchmark (available electronically as Supporting Information) for parametrization and validation of more approx. methods (such as DFT functionals and composite methods). Secondary contributions such as relativity can be included or omitted at will, unlike with exptl. data. A broad variety of more approx. methods is assessed against the W4-11 benchmark and recommendations are made.
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86. 86
  Margraf, J. T.; Ranasinghe, D. S.; Bartlett, R. J. Automatic generation of reaction energy databases from highly accurate atomization energy benchmark sets. Phys. Chem. Chem. Phys. 2017, 19, 9798, DOI: 10.1039/C7CP00757D
  86
  Automatic generation of reaction energy databases from highly accurate atomization energy benchmark sets
  Margraf, Johannes T.; Ranasinghe, Duminda S.; Bartlett, Rodney J.
  Physical Chemistry Chemical Physics (2017), 19 (15), 9798-9805CODEN: PPCPFQ; ISSN:1463-9076. (Royal Society of Chemistry)
  In this contribution, we discuss how reaction energy benchmark sets can automatically be created from arbitrary atomization energy databases. As an example, over 11 000 reaction energies derived from the W4-11 database, as well as some relevant subsets are reported. Importantly, there is only very modest computational overhead involved in computing >11 000 reaction energies compared to 140 atomization energies, since the rate-detg. step for either benchmark is performing the same 140 quantum chem. calcns. The performance of commonly used electronic structure methods for the new database is analyzed. This allows investigating the relationship between the performances for atomization and reaction energy benchmarks based on an identical set of mols. The atomization energy is found to be a weak predictor for the overall usefulness of a method. The performance of d. functional approxns. in light of the no. of empirically optimized parameters used in their design is also discussed.
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87. 87
  Henderson, T. M.; Scuseria, G. E. The connection between self-interaction and static correlation: A random phase approximation perspective. Mol. Phys. 2010, 108, 2511, DOI: 10.1080/00268976.2010.507227
  87
  The connection between self-interaction and static correlation: a random phase approximation perspective
  Henderson, Thomas M.; Scuseria, Gustavo E.
  Molecular Physics (2010), 108 (19-20), 2511-2517CODEN: MOPHAM; ISSN:0026-8976. (Taylor & Francis Ltd.)
  Semi-local d. functional theory suggests a connection between static correlation and self-interaction. It is difficult to make such a connection from the wave function theory perspective, since few wave function methods permit self-interaction error. However, the RPA for ground-state correlation, which has a wave function derivation, does include self-interaction in its direct (Hartree) variant. This variant also describes left-right correlation. The self-interaction can be removed by means of second-order screened exchange; however, this also has neg. consequences for the description of static correlation. This paper discusses the connection between the two concepts (static correlation and self-interaction) from the perspective provided by the RPA.
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88. 88
  Grüning, M.; Gritsenko, O. V.; van Gisbergen, S. J. A.; Baerends, E. J. The Failure of Generalized Gradient Approximations (GGAs) and Meta-GGAs for the Two-Center Three-Electron Bonds in He₂⁺, (H₂O)₂⁺, and (NH₃)₂⁺. J. Phys. Chem. A 2001, 105, 9211, DOI: 10.1021/jp011239k
  88
  The Failure of Generalized Gradient Approximations (GGAs) and Meta-GGAs for the Two-Center Three-Electron Bonds in He2+, (H2O)2+, and (NH3)2+
  Gruening, M.; Gritsenko, O. V.; van Gisbergen, S. J. A.; Baerends, E. J.
  Journal of Physical Chemistry A (2001), 105 (40), 9211-9218CODEN: JPCAFH; ISSN:1089-5639. (American Chemical Society)
  The radical cations He2+, (H2O)2+, and (NH3)2+ with two-center three-electron A-A bonds are investigated at the CI, accurate Kohn-Sham (KS), generalized gradient approxn. (GGA), and meta-GGA levels. Assessment of seven different GGA and six meta-GGA methods shows that the A2+ systems remain a difficult case for d. functional theory (DFT). All methods tested consistently overestimate the stability of A2+: the corresponding De errors decrease for more diffuse valence densities in the series He2+ > (H2O)2+ > (NH3)2+. Upon comparison to the energy terms of the accurate Kohn-Sham solns., the approx. exchange functionals are found to be responsible for the errors of GGA-type methods, which characteristically overestimate the exchange in A2+. These so-called exchange functionals implicitly use localized holes. Such localized holes do occur if there is left-right correlation, i.e., the exchange functionals then also describe non-dynamical correlation. However, in the hemibonded A2+ systems the typical mol. (left-right, non-dynamical) correlation of the two-electron pair bond is absent. The non-dynamical correlation built into the exchange functionals is then spurious and yields too low energies.
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89. 89
  Peterson, K. A.; Woon, D. E.; Dunning, T. H. Benchmark calculations with correlated molecular wave functions. IV. The classical barrier height of the H+H₂→H₂+H reaction. J. Chem. Phys. 1994, 100, 7410– 7415, DOI: 10.1063/1.466884
  89
  Benchmark calculations with correlated molecular wave functions. IV. The classical barrier height of the H + H2 → H2 + H reaction
  Peterson, Kirk A.; Woon, David E.; Dunning, Thom H., Jr.
  Journal of Chemical Physics (1994), 100 (10), 7410-15CODEN: JCPSA6; ISSN:0021-9606.
  Using systematic sequences of correlation consistent Gaussian basis sets from double to sextuple zeta quality, the classical barrier height of the H + H2 exchange reaction has been calcd. by multireference CI (MRCI) methods. The MRCI calcns. for collinear H3 have also been calibrated against large basis set full CI (FCI) results, which demonstrate that the MRCI treatment leads to energies less than 1 μhartree (<0.001 kcal/mol) above the FCI energies. The dependence of both the H2 and H3 total energies on the basis set is found to be very regular, and this behavior has been used to extrapolate to the complete basis set (CBS) limits. The resulting est. of the H-H-H- CBS limit yields a classical barrier height, relative to exact H + H2, of 9.60 ± 0.02 kcal/mol; the best directly calcd. value for the barrier is equal to 9.62 kcal/mol. These results are in excellent agreement with recent quantum Monte Carlo calcns.
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90. 90
  Helgaker, T.; Klopper, W.; Koch, H.; Noga, J. Basis-set convergence of correlated calculations on water. J. Chem. Phys. 1997, 106, 9639– 9646, DOI: 10.1063/1.473863
  90
  Basis-set convergence of correlated calculations on water
  Helgaker, Trygve; Klopper, Wim; Koch, Henrik; Noga, Jozef
  Journal of Chemical Physics (1997), 106 (23), 9639-9646CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics)
  The basis-set convergence of the electronic correlation energy in the water mol. is investigated at the second-order Moller-Plesset level and at the coupled-cluster singles-and-doubles level with and without perturbative triples corrections applied. The basis-set limits of the correlation energy are established to within 2mEh by means of (1) extrapolations from sequences of calcns. using correlation-consistent basis sets and (2) from explicitly correlated calcns. employing terms linear in the inter-electronic distances rij. For the extrapolations to the basis-set limit of the correlation energies, fits of the form a + bX-3 (where X is two for double-zeta sets, three for triple-zeta sets, etc.) are found to be useful. CCSD(T) calcns. involving as many as 492 AOs are reported.
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91. 91
  Halkier, A.; Helgaker, T.; Jo̷rgensen, P.; Klopper, W.; Koch, H.; Olsen, J.; Wilson, A. K. Basis-set convergence in correlated calculations on Ne, N₂, and H₂O. Chem. Phys. Lett. 1998, 286, 243– 252, DOI: 10.1016/S0009-2614(98)00111-0
  91
  Basis-set convergence in correlated calculations on Ne, N2, and H2O
  Halkier, Asger; Helgaker, Trygve; Jorgensen, Poul; Klopper, Wim; Koch, Henrik; Olsen, Jeppe; Wilson, Angela K.
  Chemical Physics Letters (1998), 286 (3,4), 243-252CODEN: CHPLBC; ISSN:0009-2614. (Elsevier Science B.V.)
  Valence and all-electron correlation energies of Ne, N2, and H2O at fixed exptl. geometries are computed at the levels of second-order perturbation theory (MP2) and coupled cluster theory with singles and doubles excitations (CCSD), and singles and doubles excitations with a perturbative triples correction (CCSD(T)). Correlation-consistent polarized valence and core-valence basis sets up to sextuple zeta quality are employed. Guided by basis-set limits established by rij-dependent methods, a no. of extrapolation schemes for use with the correlation-consistent basis sets are investigated. Among the schemes considered here, a linear least-squares procedure applied to the quintuple and sextuple zeta results yields the most accurate extrapolations.
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92. 92
  Dunning, J.; Thom, H. Gaussian basis sets for use in correlated molecular calculations. I. The atoms boron through neon and hydrogen. J. Chem. Phys. 1989, 90, 1007– 1023, DOI: 10.1063/1.456153
  92
  Gaussian basis sets for use in correlated molecular calculations. I. The atoms boron through neon and hydrogen
  Dunning, Thom H., Jr.
  Journal of Chemical Physics (1989), 90 (2), 1007-23CODEN: JCPSA6; ISSN:0021-9606.
  Guided by the calcns. on oxygen in the literature, basis sets for use in correlated at. and mol. calcns. were developed for all of the first row atoms from boron through neon, and for hydrogen. As in the oxygen atom calcns., the incremental energy lowerings, due to the addn. of correlating functions, fall into distinct groups. This leads to the concept of correlation-consistent basis sets, i.e., sets which include all functions in a given group as well as all functions in any higher groups. Correlation-consistent sets are given for all of the atoms considered. The most accurate sets detd. in this way, [5s4p3d2f1g], consistently yield 99% of the correlation energy obtained with the corresponding at.-natural-orbital sets, even though the latter contains 50% more primitive functions and twice as many primitive polarization functions. It is estd. that this set yields 94-97% of the total (HF + 1 + 2) correlation energy for the atoms neon through boron.
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93. 93
  Kendall, R. A.; Dunning, T. H.; Harrison, R. J. Electron affinities of the first-row atoms revisited. Systematic basis sets and wave functions. J. Chem. Phys. 1992, 96, 6796– 6806, DOI: 10.1063/1.462569
  93
  Electron affinities of the first-row atoms revisited. Systematic basis sets and wave functions
  Kendall, Rick A.; Dunning, Thom H., Jr.; Harrison, Robert J.
  Journal of Chemical Physics (1992), 96 (9), 6796-806CODEN: JCPSA6; ISSN:0021-9606.
  The authors describe a reliable procedure for calcg. the electron affinity of an atom and present results for H, B, C, O, and F (H is included for completeness). This procedure involves the use of the recently proposed correlation-consistent basis sets augmented with functions to describe the more diffuse character of the at. anion coupled with a straightforward, uniform expansion of the ref. space for multireference singles and doubles configuration-interaction (MRSD-CI) calcns. A comparison is given with previous results and with corresponding full CI calcns. The most accurate EAs obtained from the MRSD-CI calcns. are (with exptl. values in parentheses): H 0.740 eV (0.754), B 0.258 (0.277), C 1.245 (1.263), O 1.384 (1.461), and F 3.337 (3.401). The EAs obtained from the MR-SDCI calcns. differ by less than 0.03 eV from those predicted by the full CI calcns.
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94. 94
  Peterson, K. A.; Dunning, T. H. Accurate correlation consistent basis sets for molecular core–valence correlation effects: The second row atoms Al–Ar, and the first row atoms B–Ne revisited. J. Chem. Phys. 2002, 117, 10548– 10560, DOI: 10.1063/1.1520138
  94
  Accurate correlation consistent basis sets for molecular core-valence correlation effects: The second row atoms Al-Ar, and the first row atoms B-Ne revisited
  Peterson, Kirk A.; Dunning, Thom H., Jr.
  Journal of Chemical Physics (2002), 117 (23), 10548-10560CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics)
  Correlation consistent basis sets for accurately describing core-core and core-valence correlation effects in atoms and mols. have been developed for the second row atoms Al-Ar. Two different optimization strategies were investigated, which led to two families of core-valence basis sets when the optimized functions were added to the std. correlation consistent basis sets (cc-pVnZ). In the first case, the exponents of the augmenting primitive Gaussian functions were optimized with respect to the difference between all-electron and valence-electron correlated calcns., i.e., for the core-core plus core-valence correlation energy. This yielded the cc-pCVnZ family of basis sets, which are analogous to the sets developed previously for the first row atoms [D. E. Woon and T. H. Dunning, Jr., J. Chem. Phys. 103, 4572 (1995)]. Although the cc-pCVnZ sets exhibit systematic convergence to the all-electron correlation energy at the complete basis set limit, the intershell (core-valence) correlation energy converges more slowly than the intrashell (core-core) correlation energy. Since the effect of including the core electrons on the calcn. of mol. properties tends to be dominated by core-valence correlation effects, a second scheme for detg. the augmenting functions was investigated. In this approach, the exponents of the functions to be added to the cc-pVnZ sets were optimized with respect to just the core-valence (intershell) correlation energy, except that a small amt. of core-core correlation energy was included in order to ensure systematic convergence to the complete basis set limit. These new sets, denoted weighted core-valence basis sets (cc-pwCVnZ), significantly improve the convergence of many mol. properties with n. Optimum cc-pwCVnZ sets for the first-row atoms were also developed and show similar advantages. Both the cc-pCVnZ and cc-pwCVnZ basis sets were benchmarked in coupled cluster [CCSD(T)] calcns. on a series of second row homonuclear diat. mols. (Al2, Si2, P2, S2, and Cl2), as well as on selected diat. mols. involving first row atoms (CO, SiO, PN, and BCl). For the calcn. of core correlation effects on energetic and spectroscopic properties, the cc-pwCVnZ basis sets are recommended over the cc-pCVnZ ones.
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95. 95
  Weigend, F.; Köhn, A.; Hättig, C. Efficient use of the correlation consistent basis sets in resolution of the identity MP2 calculations. J. Chem. Phys. 2002, 116, 3175– 3183, DOI: 10.1063/1.1445115
  95
  Efficient use of the correlation consistent basis sets in resolution of the identity MP2 calculations
  Weigend, Florian; Kohn, Andreas; Hattig, Christof
  Journal of Chemical Physics (2002), 116 (8), 3175-3183CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics)
  The convergence of the second-order Moller-Plesset perturbation theory (MP2) correlation energy with the cardinal no. X is investigated for the correlation consistent basis-set series cc-pVXZ and cc-pV(X+d)Z. For the aug-cc-pVXZ and aug-cc-pV(X+d)Z series the convergence of the MP2 correlation contribution to the dipole moment is studied. It is found that, when d-shell electrons cannot be frozen, the cc-pVXZ and aug-cc-pVXZ basis sets converge much slower for third-row elements then they do for first- and second-row elements. Based on the results of these studies criteria are deduced for the accuracy of auxiliary basis sets used in the resoln. of the identity (RI) approxn. for electron repulsion integrals. Optimized auxiliary basis sets for RI-MP2 calcns. fulfilling these criteria are reported for the sets cc-pVXZ, cc-pV(X+d)Z, aug-cc-pVXZ, and aug-cc-pV(X+d)Z with X=D, T, and Q. For all basis sets the RI error in the MP2 correlation energy is more than two orders of magnitude smaller than the usual basis-set error. For the auxiliary aug-cc-pVXZ and aug-cc-pV(X+d)Z sets the RI error in the MP2 correlation contribution to the dipole moment is one order of magnitude smaller than the usual basis set error. Therefore extrapolations towards the basis-set limit are possible within the RI approxn. for both energies and properties. The redn. in CPU time obtained with the RI approxn. increases rapidly with basis set size. For the cc-pVQZ basis an acceleration by a factor of up to 170 is obsd.
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96. 96
  Boys, S. F.; Bernardi, F. The calculation of small molecular interactions by the differences of separate total energies. Some procedures with reduced errors. Mol. Phys. 1970, 19, 553– 566, DOI: 10.1080/00268977000101561
  96
  The calculation of small molecular interactions by the differences of separate total energies. Some procedures with reduced errors
  Boys, S. F.; Bernardi, F.
  Molecular Physics (1970), 19 (4), 553-566CODEN: MOPHAM; ISSN:0026-8976. (Taylor & Francis Ltd.)
  A new direct difference method for the computation of mol. interactions has been based on a bivariational transcorrelated treatment, together with special methods for the balancing of other errors. It appears that these new features can give a strong redn. in the error of the interaction energy, and they seem to be particularly suitable for computations in the important region near the min. energy. It has been generally accepted that this problem is dominated by unresolved difficulties and the relation of the new methods of these apparent difficulties is analyzed here.
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97. 97
  Gdanitz, R. J. An accurate interaction potential for neon dimer (Ne₂). Chem. Phys. Lett. 2001, 348, 67– 74, DOI: 10.1016/S0009-2614(01)01088-0
  There is no corresponding record for this reference.
98. 98
  Perdew, J. P. Proceedings of the 75. WE-Heraeus-Seminar and 21st Annual International Symposium on Electronic Structure of Solids; Akademie Verlag: Berlin, 1991; p 11.
  There is no corresponding record for this reference.
99. 99
  Perdew, J. P.; Chevary, J. A.; Vosko, S. H.; Jackson, K. A.; Pederson, M. R.; Singh, D. J.; Fiolhais, C. Atoms, molecules, solids, and surfaces: Applications of the generalized gradient approximation for exchange and correlation. Phys. Rev. B 1992, 46, 6671– 6687, DOI: 10.1103/PhysRevB.46.6671
  99
  Atoms, molecules, solids, and surfaces: applications of the generalized gradient approximation for exchange and correlation
  Perdew, John P.; Chevary, J. A.; Vosko, S. H.; Jackson, Koblar A.; Pederson, Mark R.; Singh, D. J.; Fiolhais, Carlos
  Physical Review B: Condensed Matter and Materials Physics (1992), 46 (11), 6671-87CODEN: PRBMDO; ISSN:0163-1829.
  Generalized gradient approxns. (GGA's) seek to improve upon the accuracy of the local-spin-d. (LSD) approxn. in electronic-structure calcns. Perdew and Wang have developed a GGA based on real-space cutoff of the spurious long-range components of the second-order gradient expansion for the exchange-correlation hole. Authors have found that this d. functional performs well in numerical tests for a variety of systems: Total energies of 30 atoms are highly accurate. Ionization energies and electron affinities are improved in a statistical sense, although significant interconfigurational and interterm errors remain. Accurate atomization energies are found for seven hydrocarbon mols., with a rms error per bond of 0.1 eV, compared with 0.7 eV for the LSD approxn. and 2.4 eV for the Hartree-Fock approxn. For atoms and mols., there is a cancellation of error between d. functionals for exchange and correlation, which is most striking whenever the Hartree-Fock result is furthest from expt. The surprising LSD underestimation of the lattice consts. of Li and Na by 3-4% is cor., and the magnetic ground state of solid Fe is restored. The work function, surface energy (neglecting the long-range contribution), and curvature energy of a metallic surface are all slightly reduced in comparison with LSD. Taking account of the pos. long-range contribution, authors find surface and curvature energies in good agreement with exptl. or exact values. Finally, a way is found to visualize and understand the nonlocality of exchange and correlation, its origins, and its phys. effects.
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100. 100
  Becke, A. D. Density-functional exchange-energy approximation with correct asymptotic behavior. Phys. Rev. A 1988, 38, 3098– 3100, DOI: 10.1103/PhysRevA.38.3098
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  Density-functional exchange-energy approximation with correct asymptotic behavior
  Becke, A. D.
  Physical Review A: Atomic, Molecular, and Optical Physics (1988), 38 (6), 3098-100CODEN: PLRAAN; ISSN:0556-2791.
  Current gradient-cor. d.-functional approxns. for the exchange energies of at. and mol. systems fail to reproduce the correct 1/r asymptotic behavior of the exchange-energy d. A gradient-cor. exchange-energy functional is given with the proper asymptotic limit. This functional, contg. only one parameter, fits the exact Hartree-Fock exchange energies of a wide variety of at. systems with remarkable accuracy, surpassing the performance of previous functionals contg. two parameters or more.
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101. 101
  Zhang, Y.; Pan, W.; Yang, W. Describing van der Waals interaction in diatomic molecules with generalized gradient approximations: The role of the exchange functional. J. Chem. Phys. 1997, 107, 7921– 7925, DOI: 10.1063/1.475105
  101
  Describing van der Waals Interaction in diatomic molecules with generalized gradient approximations: the role of the exchange functional
  Zhang, Yingkai; Pan, Wei; Yang, Weitao
  Journal of Chemical Physics (1997), 107 (19), 7921-7925CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics)
  Generalized gradient approxns. have been used to calc. the potential energy curves for six rare gas diat. mols. (He2, Ne2, Ar2, HeNe, ArHe, ArNe). Several generalized gradient approxns. are found to provide a good description of binding in these diat. mols. and show a significant improvement over the local d. approxn. in the prediction of bond lengths and dissocn. energies. It is shown here that the behavior of an exchange functional in the region of small d. and large d. gradient plays a very important role in the ability of the functional to describe this type of van der Waals attraction.
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102. 102
  Eshuis, H.; Furche, F. Basis set convergence of molecular correlation energy differences within the random phase approximation. J. Chem. Phys. 2012, 136, 084105, DOI: 10.1063/1.3687005
  102
  Basis set convergence of molecular correlation energy differences within the random phase approximation
  Eshuis, Henk; Furche, Filipp
  Journal of Chemical Physics (2012), 136 (8), 084105/1-084105/6CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics)
  The basis set convergence of energy differences obtained from the RPA to the correlation energy is investigated for a wide range of mol. interactions. For dispersion bound systems the basis set incompleteness error is most pronounced, as shown for the S22 benchmark. The use of very large basis sets (> quintuple-zeta) or extrapolation to the complete basis set (CBS) limit is necessary to obtain a reliable est. of the binding energy for these systems. Counterpoise cor. results converge to the same CBS limit, but counterpoise correction without extrapolation is insufficient. Core-valence correlations do not play a significant role. For medium- and short-range correlation, quadruple-zeta results are essentially converged, as demonstrated for relative alkane conformer energies, reaction energies dominated by intramol. dispersion, isomerization energies, and reaction energies of small org. mols. Except for weakly bound systems, diffuse augmentation almost universally slows down basis set convergence. For most RPA applications, quadruple-zeta valence basis sets offer a good balance between accuracy and efficiency. (c) 2012 American Institute of Physics.
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Supporting Information
Supporting Information
The Supporting Information is available free of charge at https://pubs.acs.org/doi/10.1021/acs.jpca.4c05289.
- MAEs for all subsets of the ASCDB and GMTKN55 databases for σ↑AXK functionals; weight functions w_H and w_x for PBE reference orbitals; and spline coefficients for σ↑AXK functionals (PDF)
- jp4c05289_si_001.pdf (337.8 kb)
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Top-Down versus Bottom-Up Approaches for σ-Functionals Based on the Approximate Exchange KernelClick to copy article linkArticle link copied!

The Journal of Physical Chemistry A

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Abstract

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Introduction

Theory

Fluctuation–Dissipation Theorem

Direct Random Phase Approximation and σ-Functionals

Inclusion of Exchange Kernels and τ-Functionals

Approximate Exchange Kernel

Top-Down “σ↓AXK” Correlation Energy

Bottom-Up “σ↑AXK” Correlation Energy

Computational Details

Results and Discussion

Optimization of Bottom-Up σ-Functionals

Figure 1

Evaluation Compared to Top-Down σ-Functionals

Figure 2

Figure 3

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Physical Interpretation of Top-Down versus Bottom-Up Approaches

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Conclusions

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Top-Down versus Bottom-Up Approaches for σ-Functionals Based on the Approximate Exchange Kernel
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