A variational reformulation of molecular properties in electronic-structure theory.

Conventional quantum-mechanical calculations of molecular properties, such as dipole moments and electronic excitation energies, give errors that depend linearly on the error in the wave function. An exception is the electronic energy, whose error depends quadratically on the error in wave function. We here describe how all properties may be calculated with a quadratic error, by setting up a variational Lagrangian for the property of interest. Because the construction of the Lagrangian is less expensive than the calculation of the wave function, this approach substantially improves the accuracy of quantum-chemical calculations without increasing cost. As illustrated for excitation energies, this approach enables the accurate calculation of molecular properties for larger systems, with a short time-to-solution and in a manner well suited for modern computer architectures.

Cluster perturbation theory IX: Perturbation series for the coupled cluster singles and doubles ground state energy.

In this paper, we develop and analyze a number of perturbation series that target the coupled cluster singles and doubles (CCSD) ground state energy. We show how classical Møller-Plesset perturbation theory series can be restructured to target the CCSD energy based on a reference CCS calculation and how the corresponding cluster perturbation series differs from the classical Møller-Plesset perturbation series. Subsequently, we reformulate these series using the coupled cluster Lagrangian framework to obtain series, where fourth and fifth order energies are determined only using parameters through second order. To test the methods, we perform a series of test calculations on molecular photoswitches of both total energies and reaction energies. We find that the fifth order reaction energies are of CCSD quality and that they are of comparable accuracy to state-of-the-art approximations to the CCSD energy based on local pair natural orbitals. The advantage of the present approach over local correlation methods is the absence of user defined threshold parameters for neglecting or approximating contributions to the correlation energy. Fixed threshold parameters lead to discontinuous energy surfaces, although this effect is often small enough to be ignored, but the present approach has a differentiable energy that will facilitate derivation and implementation of gradients and higher derivatives. A further advantage is that the calculation of the perturbation correction is non-iterative and can, therefore, be calculated in parallel, leading to a short time-to-solution.

Corrigendum: Coupled cluster theory on modern heterogeneous supercomputers.

[This corrects the article DOI: 10.3389/fchem.2023.1154526.].

Coupled cluster theory on modern heterogeneous supercomputers.

This study examines the computational challenges in elucidating intricate chemical systems, particularly through ab-initio methodologies. This work highlights the Divide-Expand-Consolidate (DEC) approach for coupled cluster (CC) theory-a linear-scaling, massively parallel framework-as a viable solution. Detailed scrutiny of the DEC framework reveals its extensive applicability for large chemical systems, yet it also acknowledges inherent limitations. To mitigate these constraints, the cluster perturbation theory is presented as an effective remedy. Attention is then directed towards the CPS (D-3) model, explicitly derived from a CC singles parent and a doubles auxiliary excitation space, for computing excitation energies. The reviewed new algorithms for the CPS (D-3) method efficiently capitalize on multiple nodes and graphical processing units, expediting heavy tensor contractions. As a result, CPS (D-3) emerges as a scalable, rapid, and precise solution for computing molecular properties in large molecular systems, marking it an efficient contender to conventional CC models.

Massively parallel GPU enabled third-order cluster perturbation excitation energies for cost-effective large scale excitation energy calculations.

We present here a massively parallel implementation of the recently developed CPS(D-3) excitation energy model that is based on cluster perturbation theory. The new algorithm extends the one developed in Baudin et al. [J. Chem. Phys., 150, 134110 (2019)] to leverage multiple nodes and utilize graphical processing units for the acceleration of heavy tensor contractions. Furthermore, we show that the extended algorithm scales efficiently with increasing amounts of computational resources and that the developed code enables CPS(D-3) excitation energy calculations on large molecular systems with a low time-to-solution. More specifically, calculations on systems with over 100 atoms and 1000 basis functions are possible in a few hours of wall clock time. This establishes CPS(D-3) excitation energies as a computationally efficient alternative to those obtained from the coupled-cluster singles and doubles model.

Cluster perturbation theory. VIII. First order properties for a coupled cluster state.

We have extended cluster perturbation (CP) theory to comprehend the calculation of first order properties (FOPs). We have determined CP FOP series where FOPs are determined as a first energy derivative and also where the FOPs are determined as a generalized expectation value of the external perturbation operator over the coupled cluster state and its biorthonormal multiplier state. For S(D) orbital excitation spaces, we find that the CP series for FOPs that are determined as a first derivative, in general, in second order have errors of a few percent in the singles and doubles correlation contribution relative to the targeted coupled cluster (CC) results. For a SD(T) orbital excitation space, we find that the CP series for FOPs determined as a generalized expectation value in second order have errors of about ten percent in the triples correlation contribution relative to the targeted CC results. These second order models, therefore, constitute viable alternatives for determining high quality FOPs.

Cluster perturbation theory. VII. The convergence of cluster perturbation expansions.

The convergence of the recently developed cluster perturbation (CP) expansions [Pawlowski et al., J. Chem. Phys. 150, 134108 (2019)] is analyzed with the double purpose of developing the mathematical tools and concepts needed to describe these expansions at general order and to identify the factors that define the rate of convergence of CP series. To this end, the CP energy, amplitude, and Lagrangian multiplier equations as a function of the perturbation strength are developed. By determining the critical points, defined as the perturbation strengths for which the Jacobian becomes singular, the rate of convergence and the intruder and critical states are determined for five small molecules: BH, CO, H2O, NH3, and HF. To describe the patterns of convergence for these expansions at orders lower than the high-order asymptotic limit, a model is developed where the perturbation corrections arise from two critical points. It is shown that this model allows for rationalization of the behavior of the perturbation corrections at much lower order than required for the onset of the asymptotic convergence. For the H2O, CO, and HF molecules, the pattern and rate of convergence are defined by critical states where the Fock-operator underestimates the excitation energies, whereas the pattern and rate of convergence for BH are defined by critical states where the Fock-operator overestimates the excitation energy. For the NH3 molecule, both forms of critical points are required to describe the convergence behavior up to at least order 25.

Cluster perturbation theory. VI. Ground-state energy series using the Lagrangian.

We have extended cluster perturbation (CP) theory to comprehend the Lagrangian framework of coupled cluster (CC) theory and derived the CP Lagrangian energy series (LCP) where the 2n + 1/2n + 2 rules for the cluster amplitudes and multipliers are used to get the energy corrections. We have also developed the variational CP (LCP) series, where the total cluster amplitudes and multipliers are determined through the same orders as in the LCP series, but the energy is obtained by inserting the total cluster amplitudes and multipliers in the Lagrangian. The energies of the LCP series have errors that are bilinear in the errors of the total cluster amplitudes and multipliers. Test calculations have been performed for S(D) and SD(T) orbital excitation spaces. With the exception of molecular systems that have a low lying doubly excited state compared to the electronic ground state configuration, we find that the fourth order models LCPS (D-4), LCPSD (T-4), and LCPSD(T-4) give energies of CC target state quality. For the LCPS (D-4) model, CC target state quality is obtained as the LCPS (D-4) calculation determines more than 99.7% of the coupled cluster singles and doubles (CCSD) correlation energy as the numerical deviations of the LCPS (D-4) energy from the CCSD energy were more than an order of magnitude smaller than the triples correlation contribution. For the LCPSD (T-4) and LCPSD(T-4) models, CC target state quality was obtained, given that the LCPSD (T-4) and LCPSD(T-4) calculations recover more than 99% of the coupled cluster singles doubles and triples (CCSDT) correlation contribution and as the numerical deviations of the LCPSD (T-4) and LCPSD(T-4) energies from the CCSDT energy were nearly and order of magnitude smaller than the quadruples correlation contribution. We, thus, suggest that the fourth order models may replace the full target CC models with no or very limited loss of accuracy.

Convergence patterns and rates in two-state perturbation expansions.

A simple two-state model has previously been shown to be able to describe and rationalize the convergence of the most common perturbation method for including electron correlation, the Møller-Plesset expansion. In particular, this simple model has been able to predict the convergence rate and the form of the higher-order corrections for typical Møller-Plesset expansions of the correlation energy. In this paper, the convergence of nondegenerate perturbation expansions in the two-state model is analyzed in detail for a general form of two-state perturbation expansion by examining the analytic expressions of the corrections and series of the values of the corrections for various choices of the perturbation. The previous analysis that covered only a single form of the perturbation is thereby generalized to arbitrary forms of the perturbation. It is shown that the convergence may be described in terms of four characteristics: archetype, rate of convergence, length of recurring period, and sign pattern. The archetype defines the overall form of a plot of the energy-corrections, and the remaining characteristics specify details of the archetype. For symmetric (Hermitian) perturbations, five archetypes are observed: zigzag, interspersed zigzag, triadic, ripples, and geometric. Two additional archetypes are obtained for an asymmetric perturbation: zigzag-geometric and convex-geometric. For symmetric perturbations, each archetype has a distinctive pattern that recurs with a period which depends on the perturbation parameters, whereas no such recurrence exists for asymmetric perturbations from a series of numerical corrections. The obtained relations between the form of a two-state perturbation and the energy corrections allow us to obtain additional insights into the convergence behavior of the Møller-Plesset and other forms of perturbation expansions. This is demonstrated by analyzing several diverging or slowly converging perturbation expansions of ground state and excitation energies. It is demonstrated that the higher-order corrections of these expansions can be described using the two-state model and each expansion can therefore be described in terms of an archetype and the other three characteristics. Examples of all archetypes except the zigzag and convex-geometric archetypes are given. For each example, it is shown how the characteristics may be extracted from the higher-order corrections and used to identify the term in the perturbation that is the cause of the observed slow convergence or divergence.

Cluster perturbation theory. II. Excitation energies for a coupled cluster target state.

In cluster perturbation (CP) theory, we consider a target excitation space relative to a Hartree-Fock state and partition the target excitation space into a parent excitation space and an auxiliary excitation space. The zeroth-order state is in CP theory a coupled cluster (CC) state in the parent excitation space, and the target state is a CC state in the target excitation space. In this paper, we derive CP series for excitation energies in orders of the CC parent-state similarity-transformed fluctuation potential where the zeroth-order term in the series is an excitation energy for the CC parent state response eigenvalue equation and where the series formally converge to an excitation energy for the CC target state response eigenvalue equation. We give explicit expressions for the lowest-order excitation energy corrections. We also report calculations for CP excitation energy series for various parent and target excitation spaces and examine how well the lower-order corrections can reproduce the total excitation energies. Considering the fast local convergence we have observed for the CP excitation energy series, it becomes computationally attractive to use low-order corrections in CP series to obtain excitation energies of CC target state quality. For the CPS(D-n) series, the first-order correction vanishes, the second-order correction becomes the CIS(D) model, and for the CPS(D-3) model, our calculations suggest that excitation energies of CCSD quality are obtained. The numerical results also suggest that a similar behavior can be seen for the low-order excitation energy corrections for CP series where the parent state contains more than a singles excitation space, e.g., for the CPSD(T) model. We therefore expect the low-order excitation energy corrections in CP series soon to become state-of-the-art models for determining excitation energies of CC target state quality.

Cluster perturbation theory. I. Theoretical foundation for a coupled cluster target state and ground-state energies.

We introduce a new class of perturbation models-the cluster perturbation (CP) models-where the major drawbacks of Møller-Plesset perturbation theory and coupled cluster perturbation theory have been eliminated. In CP theory, we consider a target excitation space relative to the Hartree-Fock state and partition the target excitation space into a parent and an auxiliary excitation space. The zeroth-order state is a coupled cluster (CC) state in the parent excitation space, and the target state is either a cluster linear or a CC state in the target excitation space. In CP theory, perturbation series are determined in orders of the CC parent state similarity-transformed fluctuation potential for the energy and for a molecular property, where the zeroth-order term in the series is the energy or a molecular property for the CC parent state and where the series formally converge to the energy or a molecular property for the target state. In CP theory, we use a generalized order concept, where the zeroth-order component of the extended parent-state Jacobian contains a fluctuation potential contribution, and use this new generalized order to treat internal relaxation in the parent excitation space at zeroth order and hence remove it from the perturbation calculation. Even more importantly, using this new generalized order concept, CP series can be determined for molecular properties of ground and excited states and for transition properties between these states, including excitation energies and energies of the excited states. The applicability of CP theory to both the energy and molecular properties and numerical results for the CP energy and molecular property series demonstrate the superiority of CP theory compared to previous perturbation models. Low-order corrections in the CP perturbation series can be expected soon to become state-of-the-art electronic structure models for the determination of energies and molecular properties of target-state quality for single-configuration dominated molecular systems.

Cluster perturbation theory. IV. Convergence of cluster perturbation series for energies and molecular properties.

The theoretical foundation has been developed for establishing whether cluster perturbation (CP) series for the energy, molecular properties, and excitation energies are convergent or divergent and for using a two-state model to describe the convergence rate and convergence patterns of the higher-order terms in the CP series. To establish whether the perturbation series are convergent or divergent, a fictitious system is introduced, for which the perturbation is multiplied by a complex scaling parameter z. The requirement for convergent perturbation series becomes that the energy or molecular property, including an excitation energy, for the fictitious system is an analytic, algebraic function of z that has no singularities when the norm |z| is smaller than one. Examples of CP series for the energy and molecular properties, including excitation energies, are also presented, and the two-state model is used for the interpretation of the convergence rate and the convergence patterns of the higher-order terms in these series. The calculations show that the perturbation series effectively become a two-state model at higher orders.

Cluster perturbation theory. III. Perturbation series for coupled cluster singles and doubles excitation energies.

The cluster perturbation series, CPS(D), for coupled cluster singles and doubles excitation energies is considered. It is demonstrated that the second-order model CPS(D-2) is identical to the configuration interaction singles with perturbative doubles, CIS(D) model. The third-order model, CPS(D-3), provides excitation energies of coupled cluster singles and doubles (CCSD) quality in the sense that the difference between CPS(D-3) and CCSD excitation energies is of the same size or smaller than the effect of adding triples corrections to CCSD excitation energies. We further show that the third-order corrections can be efficiently implemented, in particular, when the resolution of the identity approximation is used for integrals. We also show that the CPS(D-3) excitation energies can be determined for system sizes that are far beyond what can be considered in conventional CCSD excitation energy calculations.

Cluster perturbation theory. V. Theoretical foundation for cluster linear target states.

Cluster perturbation (CP) theory was developed in Paper I [F. Pawlowski et al., J. Chem. Phys. 150, 134108 (2019)] for a coupled cluster (CC) target state and is extended in this paper to comprehend a cluster linear (CL) target state, for which the embedding of a CC parent state in the target excitation space is described using a linear parametrization. The theory is developed for determining the energy and molecular properties for a CL state. When CP theory is applied to a CL target state, a series of corrections is determined in orders of the CC parent-state similarity-transformed fluctuation potential, where the zeroth-order term is the energy or molecular property of the CC parent state and where the series formally converges to the energy or molecular property of the CL target state. The determination of energies and molecular properties is simpler for a CL state than for a CC state because the CL state is linearly parametrized. The amplitude equations are quadratic for a CL target state, while quartic for a CC target state, and molecular property expressions for a CL target state have the same simple structure as for a configuration interaction state. The linear parametrization introduces non-size-extensive contributions in the energy and molecular property expressions. However, since the linear parametrization describes the embedding of the CC parent state in the target excitation space, the energy and molecular properties for a CL state are weakly size-extensive. For the energy, weak size-extensivity means that non-size-extensive contributions enter in sixth and higher orders in the CP energy series, whereas for molecular properties, weak size-extensivity means that non-size-extensive contributions enter in second and higher orders. Weak size-extensivity therefore has a little or vanishing effect on calculated energies or molecular properties. The determination of the CP energy and molecular property corrections does not require that amplitude or response equations are solved explicitly for the target state and it becomes computationally tractable to use low-order corrections from these series to obtain energies and molecular properties of CL target state quality. For three simple molecules, HF, N2, and CH2, the accuracy of the CL approach for ground-state energies is tested using a parent state including single and double excitations (i.e., the CC singles-and-doubles state, CCSD) and a target state that includes triple excitations. It is found that the size-extensive fifth-order CL energies deviate by less than 0.0001 hartree from the energies of a target CC that includes triple excitations (i.e., the CC singles-doubles-and-triples state, CCSDT). CP theory with a CL target state therefore becomes a very attractive replacement of standard CC theory for high-accuracy energy and molecular property calculations, in which triple and higher excitation levels are considered.

Convergence of coupled cluster perturbation theory.

The convergence of a recently proposed coupled cluster (CC) family of perturbation series [J. J. Eriksen et al., J. Chem. Phys. 140, 064108 (2014)], in which the energetic difference between two CC models-a low-level parent and a high-level target model-is expanded in orders of the Møller-Plesset (MP) fluctuation potential, is investigated for four prototypical closed-shell systems (Ne, singlet CH2, distorted HF, and F-) in standard and augmented basis sets. In these investigations, energy corrections of the various series have been calculated to high orders and their convergence radii have been determined by probing for possible front- and back-door intruder states, the existence of which would make the series divergent. In summary, we conclude how it is primarily the choice of the target state, and not the choice of the parent state, which ultimately governs the convergence behavior of a given series. For example, restricting the target state to, say, triple or quadruple excitations might remove intruders present in series which target the full configuration interaction limit, such as the standard MP series. Furthermore, we find that whereas a CC perturbation series might converge within standard correlation consistent basis sets, it may start to diverge whenever these become augmented by diffuse functions, similar to the MP case. However, unlike for the MP case, such potential divergences are not found to invalidate the practical use of the low-order corrections of the CC perturbation series.

Orbital spaces in the divide-expand-consolidate coupled cluster method.

The theoretical foundation for solving coupled cluster singles and doubles (CCSD) amplitude equations to a desired precision in terms of independent fragment calculations using restricted local orbital spaces is reinvestigated with focus on the individual error sources. Four different error sources are identified theoretically and numerically and it is demonstrated that, for practical purposes, local orbital spaces for CCSD calculations can be identified from calculations at the MP2 level. The development establishes a solid theoretical foundation for local CCSD calculations for the independent fragments, and thus for divide-expand-consolidate coupled cluster calculations for large molecular systems with rigorous error control. Based on this theoretical foundation, we have developed an algorithm for determining the orbital spaces needed for obtaining the single fragment energies to a requested precision and numerically demonstrated the robustness and precision of this algorithm.

Assessment of the accuracy of coupled cluster perturbation theory for open-shell systems. I. Triples expansions.

The accuracy at which total energies of open-shell atoms and organic radicals may be calculated is assessed for selected coupled cluster perturbative triples expansions, all of which augment the coupled cluster singles and doubles (CCSD) energy by a non-iterative correction for the effect of triple excitations. Namely, the second- through sixth-order models of the recently proposed CCSD(T-n) triples series [J. J. Eriksen et al., J. Chem. Phys. 140, 064108 (2014)] are compared to the acclaimed CCSD(T) model for both unrestricted as well as restricted open-shell Hartree-Fock (UHF/ROHF) reference determinants. By comparing UHF- and ROHF-based statistical results for a test set of 18 modest-sized open-shell species with comparable RHF-based results, no behavioral differences are observed for the higher-order models of the CCSD(T-n) series in their correlated descriptions of closed- and open-shell species. In particular, we find that the convergence rate throughout the series towards the coupled cluster singles, doubles, and triples (CCSDT) solution is identical for the two cases. For the CCSD(T) model, on the other hand, not only its numerical consistency, but also its established, yet fortuitous cancellation of errors breaks down in the transition from closed- to open-shell systems. The higher-order CCSD(T-n) models (orders n > 3) thus offer a consistent and significant improvement in accuracy relative to CCSDT over the CCSD(T) model, equally for RHF, UHF, and ROHF reference determinants, albeit at an increased computational cost.

Assessment of the accuracy of coupled cluster perturbation theory for open-shell systems. II. Quadruples expansions.

We extend our assessment of the potential of perturbative coupled cluster (CC) expansions for a test set of open-shell atoms and organic radicals to the description of quadruple excitations. Namely, the second- through sixth-order models of the recently proposed CCSDT(Q-n) quadruples series [J. J. Eriksen et al., J. Chem. Phys. 140, 064108 (2014)] are compared to the prominent CCSDT(Q) and ΛCCSDT(Q) models. From a comparison of the models in terms of their recovery of total CC singles, doubles, triples, and quadruples (CCSDTQ) energies, we find that the performance of the CCSDT(Q-n) models is independent of the reference used (unrestricted or restricted (open-shell) Hartree-Fock), in contrast to the CCSDT(Q) and ΛCCSDT(Q) models, for which the accuracy is strongly dependent on the spin of the molecular ground state. By further comparing the ability of the models to recover relative CCSDTQ total atomization energies, the discrepancy between them is found to be even more pronounced, stressing how a balanced description of both closed- and open-shell species-as found in the CCSDT(Q-n) models-is indeed of paramount importance if any perturbative CC model is to be of chemical relevance for high-accuracy applications. In particular, the third-order CCSDT(Q-3) model is found to offer an encouraging alternative to the existing choices of quadruples models used in modern computational thermochemistry, since the model is still only of moderate cost, albeit markedly more costly than, e.g., the CCSDT(Q) and ΛCCSDT(Q) models.

A view on coupled cluster perturbation theory using a bivariational Lagrangian formulation.

We consider two distinct coupled cluster (CC) perturbation series that both expand the difference between the energies of the CCSD (CC with single and double excitations) and CCSDT (CC with single, double, and triple excitations) models in orders of the Møller-Plesset fluctuation potential. We initially introduce the E-CCSD(T-n) series, in which the CCSD amplitude equations are satisfied at the expansion point, and compare it to the recently developed CCSD(T-n) series [J. J. Eriksen et al., J. Chem. Phys. 140, 064108 (2014)], in which not only the CCSD amplitude, but also the CCSD multiplier equations are satisfied at the expansion point. The computational scaling is similar for the two series, and both are term-wise size extensive with a formal convergence towards the CCSDT target energy. However, the two series are different, and the CCSD(T-n) series is found to exhibit a more rapid convergence up through the series, which we trace back to the fact that more information at the expansion point is utilized than for the E-CCSD(T-n) series. The present analysis can be generalized to any perturbation expansion representing the difference between a parent CC model and a higher-level target CC model. In general, we demonstrate that, whenever the parent parameters depend upon the perturbation operator, a perturbation expansion of the CC energy (where only parent amplitudes are used) differs from a perturbation expansion of the CC Lagrangian (where both parent amplitudes and parent multipliers are used). For the latter case, the bivariational Lagrangian formulation becomes more than a convenient mathematical tool, since it facilitates a different and faster convergent perturbation series than the simpler energy-based expansion.

Characterization and Generation of Local Occupied and Virtual Hartree-Fock Orbitals.

The scope of this review article is to discuss the locality of occupied and virtual orthogonal Hartree-Fock orbitals generated by localization function optimization. Locality is discussed from the stand that an orbital is local if it is confined to a small region in space. Focusing on locality measures that reflects the spatial extent of the bulk of an orbital and the thickness of orbital tails, we discuss, with numerical illustrations, how the locality may be reported for individual orbitals as well as for sets of orbitals. Traditional and more recent orbital localization functions are reviewed, and the locality measures are used to compare the locality of the orbitals generated by the different localization functions, both for occupied and virtual orbitals. Numerical illustrations are given also for large molecular systems and for cases where diffuse functions are included in the atomic orbital basis. In addition, we have included a discussion on the physical and mathematical limitations on orbital locality.