Third-Order Møller-Plesset Perturbation Theory Made Useful? Choice of Orbitals and Scaling Greatly Improves Accuracy for Thermochemistry, Kinetics, and Intermolecular Interactions

Doubles Connected Moments Expansion: A Tractable Approximate Horn-Weinstein Approach for Quantum Chemistry

Ab initio methods based on the second-order and higher connected moments, or cumulants, of a reference function have seen limited use in the determination of correlation energies of chemical systems over the years. Moment-based methods have remained unattractive relative to more ubiquitous methods, such as perturbation theory and coupled cluster theory, due in part to the intractable cost of assembling moments of high-order and poor performance of low-order expansions. Many of the traditional quantum chemical methodologies can be recast as a selective summation of perturbative contributions to their energy; using this familiar structure as a guide in selecting terms, we develop a scheme to approximate connected moments limited to double excitations. The tractable Doubles Connected Moments [DCM(N)] approximation is developed and tested against a multitude of common single-reference methods to determine its efficacy in the determination of the correlation energy of model systems and small molecules. The DCM(N) sequence of energies exhibits smooth convergence toward limiting values in the range of N = 11-14, with compute costs that scale as a noniterative O(M6) with molecule size, M. Numerical tests on correlation energy recovery for 55 small molecules comprising the G1 test set in the cc-pVDZ basis show that DCM(N) strongly outperforms MP2 and even CCD with a Hartree-Fock reference. When using an approximate Brueckner reference from orbital-optimized (oo) MP2, the resulting oo:DCM(N) energies converge to values more accurate than CCSD for 49 of 55 molecules. The qualitative success of the method in regions where strong correlation effects begin to dominate, even while maintaining spin purity, suggests this may be a good starting point in the development of methodologies for the description of strongly correlated or spin-contaminated systems while maintaining a tractable single-reference formalism.

Repartitioned Brillouin-Wigner perturbation theory with a size-consistent second-order correlation energy

Second-order Møller-Plesset perturbation theory (MP2) often breaks down catastrophically in small-gap systems, leaving much to be desired in its performance for myriad chemical applications such as noncovalent interactions, thermochemistry, and dative bonding in transition metal complexes. This divergence problem has reignited interest in Brillouin-Wigner perturbation theory (BWPT), which is regular at all orders but lacks size consistency and extensivity, severely limiting its application to chemistry. In this work, we propose an alternative partitioning of the Hamiltonian that leads to a regular BWPT perturbation series that, through the second order, is size-extensive, size-consistent (provided its Hartree-Fock reference is also), and orbital invariant. Our second-order size-consistent Brillouin-Wigner (BW-s2) approach can describe the exact dissociation limit of H2 in a minimal basis set, regardless of the spin polarization of the reference orbitals. More broadly, we find that BW-s2 offers improvements relative to MP2 for covalent bond breaking, noncovalent interaction energies, and metal/organic reaction energies, although rivaling coupled-cluster with single and double substitutions for thermochemical properties.

An in-silico NMR laboratory for nuclear magnetic shieldings computed via finite fields: Exploring nucleus-specific renormalizations of MP2 and MP3

We developed and implemented a method-independent, fully numerical, finite difference approach to calculating nuclear magnetic resonance shieldings, using gauge-including atomic orbitals. The resulting capability can be used to explore non-standard methods, given only the energy as a function of finite-applied magnetic fields and nuclear spins. For example, standard second-order Møller-Plesset theory (MP2) has well-known efficacy for 1H and 13C shieldings and known limitations for other nuclei such as 15N and 17O. It is, therefore, interesting to seek methods that offer good accuracy for 15N and 17O shieldings without greatly increased compute costs, as well as exploring whether such methods can further improve 1H and 13C shieldings. Using a small molecule test set of 28 species, we assessed two alternatives: κ regularized MP2 (κ-MP2), which provides energy-dependent damping of large amplitudes, and MP2.X, which includes a variable fraction, X, of third-order correlation (MP3). The aug-cc-pVTZ basis was used, and coupled cluster with singles and doubles and perturbative triples [CCSD(T)] results were taken as reference values. Our κ-MP2 results reveal significant improvements over MP2 for 13C and 15N, with the optimal κ value being element-specific. κ-MP2 with κ = 2 offers a 30% rms error reduction over MP2. For 15N, κ-MP2 with κ = 1.1 provides a 90% error reduction vs MP2 and a 60% error reduction vs CCSD. On the other hand, MP2.X with a scaling factor of 0.6 outperformed CCSD for all heavy nuclei. These results can be understood as providing renormalization of doubles amplitudes to partially account for neglected triple and higher substitutions and offer promising opportunities for future applications.

Cluster Amplitudes and Their Interplay with Self-Consistency in Density Functional Methods

Density functional theory (DFT) provides convenient electronic structure methods for the study of molecular systems and materials. Regular Kohn-Sham DFT calculations rely on unitary transformations to determine the ground-state electronic density, ground state energy, and related properties. However, for dissociation of molecular systems into open-shell fragments, due to the self-interaction error present in a large number of density functional approximations, the self-consistent procedure based on the this type of transformation gives rise to the well-known charge delocalization problem. To avoid this issue, we showed previously that the cluster operator of coupled-cluster theory can be utilized within the context of DFT to solve in an alternative and approximate fashion the ground-state self-consistent problem. This work further examines the application of the singles cluster operator to molecular ground state calculations. Two approximations are derived and explored: i) A linearized scheme of the quadratic equation used to determine the cluster amplitudes. ii) The effect of carrying the calculations in a non-self-consistent field fashion. These approaches are found to be capable of improving the energy and density of the system and are quite stable in either case. The theoretical framework discussed in this work could be used to describe, with an added flexibility, quantum systems that display challenging features and require expanded theoretical methods.

Accurate Property Prediction by Second Order Perturbation Theory: The REMP and OO-REMP Hybrids

The prediction of molecular properties such as equilibrium structures or vibrationalwavenumbers is a routine task in computational chemistry. If very high accuracy is required, however, the use of computationally demanding ab initio wavefunction methods is mandatory. We present property calculations utilizing the REMP and OO-REMP hybrid perturbation theories showing that with the latter approach, very accurate results are obtained at second order in perturbation theory. Specifically, equilibrium structures and harmonic vibrational wavenumbers as well as dipole moments of closed and open shell molecules were calculated and compared to the best available experimental results or very accurate calculations.OO-REMP is capable of predicting bond lengths of small closed and open shell molecules with an accuracy of 0.2 pm and 0.5 pm, respectively, often within the range of experimental uncertainty. Equilibrium harmonic vibrational wavenumbers are predicted with an accuracy better than 20 cm−1 . Dipole moments of small closed and open shell molecules are reproduced with a relative error of less than 3 %. Across all investigated properties it turns out that a 20 %:80 % MP:RE mixing ratio consistently provides the best results. This is in line with our previous findings featuring closed and open shell reaction energies.

Elucidating the molecular orbital dependence of the total electronic energy in multireference problems

The accurate resolution of the chemical properties of strongly correlated systems, such as biradicals, requires the use of electronic structure theories that account for both multi-reference and dynamic correlation effects. A variety of methods exist that aim to resolve the dynamic correlation in multi-reference problems, commonly relying on an exponentially scaling complete-active-space self-consistent-field (CASSCF) calculation to generate reference molecular orbitals (MOs). However, while CASSCF orbitals provide the optimal solution for a selected set of correlated (active) orbitals, their suitability in the quest for the resolution of the total correlation energy has not been thoroughly investigated. Recent research has shown the ability of Kohn-Shan density functional theory to provide improved orbitals for coupled cluster (CC) and Møller-Plesset perturbation theory (MP) calculations. Here, we extend the search for optimal and more cost effective MOs to post-configuration-interaction [post-(CI)] methods, surveying the ability of the MOs obtained with various density functional theory (DFT) functionals, as well as Hartree-Fock and CC and MP calculations to accurately capture the total electronic correlation energy. Applying the anti-Hermitian contracted Schrödinger equation to the dissociation of N₂, the calculation of biradical singlet-triplet gaps, and the transition states of bicylobutane isomerization, we demonstrate that DFT provides a cost-effective alternative to CASSCF in providing reference orbitals for post-CI dynamic correlation calculations.

UREMP, RO-REMP, and OO-REMP: Hybrid perturbation theories for open-shell electronic structure calculations

An accurate description of the electron correlation energy in closed- and open-shell molecules is shown to be obtained by a second-order perturbation theory (PT) termed REMP. REMP is a hybrid of the Retaining the Excitation degree (RE) and the Møller–Plesset (MP) PTs. It performs particularly encouragingly in an orbital-optimized variant (OO-REMP) where the reference wavefunction is given by an unrestricted Slater determinant whose spin orbitals are varied such that the total energy becomes a minimum. While the approach generally behaves less satisfactorily with unrestricted Hartree–Fock references, reasonable performance is observed for restricted Hartree–Fock and restricted open-shell Hartree–Fock references. Inclusion of single excitations to OO-REMP is investigated and found—as in similar investigations—to be dissatisfying as it deteriorates performance. For the non-multireference subset of the accurate W4-11 benchmark set of Karton et al. [Chem. Phys. Lett. 510, 165–178 (2011)], OO-REMP predicts most atomization and reaction energies with chemical accuracy (1 kcal mol−1) if complete-basis-set extrapolation with augmented and core-polarized basis sets is used. For the W4-11 related test-sets, the error estimates obtained with the OO-REMP method approach those of coupled-cluster with singles, doubles and perturbative triples [CCSD(T)] within 20%–35%. The best performance of OO-REMP is found for a mixing ratio of 20%:80% MP:RE, which is essentially independent of whether radical stabilization energies, barrier heights, or reaction energies are investigated. Orbital optimization is shown to improve the REMP approach for both closed and open shell cases and outperforms coupled-cluster theory with singles and doubles (CCSD), spin-component scaled Møller-Plesset theory at second order (SCS-MP2), and density functionals, including double hybrids in all the cases considered.

Regularized second-order correlation methods for extended systems

Second-order Møller-Plesset perturbation theory (MP2) constitutes the simplest form of many-body wavefunction theory and often provides a good compromise between efficiency and accuracy. There are, however, well-known limitations to this approach. In particular, MP2 is known to fail or diverge for some prototypical condensed matter systems like the homogeneous electron gas (HEG) and to overestimate dispersion-driven interactions in strongly polarizable systems. In this paper, we explore how the issues of MP2 for metallic, polarizable, and strongly correlated periodic systems can be ameliorated through regularization. To this end, two regularized second-order methods (including a new, size-extensive Brillouin-Wigner approach) are applied to the HEG, the one-dimensional Hubbard model, and the graphene-water interaction. We find that regularization consistently leads to improvements over the MP2 baseline and that different regularizers are appropriate for the various systems.

Regularized Second-Order Møller-Plesset Theory: A More Accurate Alternative to Conventional MP2 for Noncovalent Interactions and Transition Metal Thermochemistry for the Same Computational Cost

Second-order Møller-Plesset theory (MP2) notoriously breaks down for π-driven dispersion interactions and dative bonds in transition metal complexes. Herein, we investigate three physically justified forms of single-parameter, energy-gap dependent regularization which can yield high and transferable accuracy for a variety of noncovalent interactions (including S22, S66, and L7 test sets) and (mostly closed shell) transition metal thermochemistry. Regularization serves to damp overestimated pairwise additive contributions, renormalizing first-order amplitudes such that the effects of higher-order correlations are incorporated. The optimal parameter values for the noncovalent and transition metal sets are 1.1, 0.7, and 0.4 for κ, σ, and σ² regularizers, respectively. However, such regularization slightly degrades the accuracy of conventional MP2 for some small-molecule test sets, most of which have relatively large average frontier energy gaps. Our results suggest that appropriately regularized MP2 models may improve double hybrid density functionals, at no additional cost over conventional MP2.

Improving Perturbation Theory for Open-Shell Molecules via Self-Consistency

We present an extension of our one-body Møller-Plesset second-order perturbation (OBMP2) method for open-shell systems. We derived the OBMP2 Hamiltonian through the canonical transformation followed by the cumulant approximation to reduce many-body operators into one-body ones. The resulting Hamiltonian consists of an uncorrelated Fock (unperturbed Hamiltonian) and a one-body correlation potential (perturbed Hamiltonian) composed of only double excitations. Molecular orbitals and associated energy levels are then relaxed via self-consistency, similar to Hartree-Fock, in the presence of the correlation at the MP2 level. We demonstrate the OBMP2 performance by considering two examples well-known for requiring orbital optimization: bond breaking and isotropic hyperfine coupling constants. In contrast to noniterative MP2, we show that OBMP2 can yield a smooth transition through the unrestriction point and accurately predict isotropic hyperfine coupling constants.

Assessing Density Functional Theory for Chemically Relevant Open-Shell Transition Metal Reactions

Due to the principle lack of systematic improvement possibilities of density functional theory, careful assessment of the performance of density functional approximations (DFAs) on well-designed benchmark sets, for example, for reaction energies and barrier heights, is crucial. While main-group chemistry is well covered by several available sets, benchmark data for transition metal chemistry is sparse. This is especially the case for larger, chemically relevant molecules. Addressing this issue, we recently introduced the MOR41 benchmark which covers chemically relevant reactions of closed-shell complexes. In this work, we extend these efforts to single-reference open-shell systems and introduce the "reactions of open-shell single-reference transition metal complexes" (ROST61) benchmark set. ROST61 includes accurate coupled-cluster reference values for 61 reaction energies with a mean reaction energy of -42.8 kcal mol^-1. Complexes with 13-93 atoms covering 20 d-block elements are included, but due to the restriction to single-reference open-shell systems, important elements such as iron or platinum could not be taken into account, or only to a small extent. We assess the performance of 31 DFAs in combination with three London dispersion (LD) correction schemes. Further, DFT-based composite methods, MP2, and a few semiempirical quantum chemical methods are evaluated. Consistent with the results for the MOR41 closed-shell benchmark, we find that the ordering of DFAs according to Jacob's ladder is preserved and that adding an LD correction is crucial, clearly improving almost all tested methods. The recently introduced r²SCAN-3c composite method stands out with a remarkable mean absolute deviation (MAD) of only 2.9 kcal mol^-1, which is surpassed only by hybrid DFAs with low amounts of Fock exchange (e.g., 2.3 kcal mol^-1 for TPSS0-D4/def2-QZVPP) and double-hybrid (DH) DFAs but at a significantly higher computational cost. The lowest MAD of only 1.6 kcal mol^-1 is obtained with the DH DFA PWPB95-D4 in the def2-QZVPP basis set approaching the estimated accuracy of the reference method. Overall, the ROST61 set adds important reference data to a sparsely sampled but practically relevant area of chemistry. At this point, it provides valuable orientation for the application and development of new DFAs and electronic structure methods in general.

Efficient implementations of the symmetric and asymmetric triple excitation corrections for the orbital-optimized coupled-cluster doubles method with the density-fitting approximation

Efficient implementations of the symmetric and asymmetric triple excitation corrections for the orbital-optimized coupled-cluster doubles (OCCD) method with the density-fitting approach, denoted by DF-OCCD(T) and DF-OCCD(T)_Λ, are presented. The computational cost of the DF-OCCD(T) method is compared with that of the conventional OCCD(T). In the conventional OCCD(T) and OCCD(T)_Λ methods, one needs to perform four-index integral transformations at each coupled-cluster doubles iterations, which limits its applications to large chemical systems. Our results demonstrate that DF-OCCD(T) provides dramatically lower computational costs compared to OCCD(T), and there are more than 68-fold reductions in the computational time for the C₅H₁₂ molecule with the cc-pVTZ basis set. Our results show that the DF-OCCD(T) and DF-OCCD(T)_Λ methods are very helpful for the study of single bond-breaking problems. Performances of the DF-OCCD(T) and DF-OCCD(T)_Λ methods are noticeably better than that of the coupled-cluster singles and doubles with perturbative triples [CCSD(T)] method for the potential energy surfaces of the molecules considered. Specifically, the DF-OCCD(T)_Λ method provides dramatic improvements upon CCSD(T), and there are 8-14-fold reductions in nonparallelity errors. Overall, we conclude that the DF-OCCD(T)_Λ method is very promising for the study of challenging chemical systems, where the CCSD(T) fails.

Exploring the Limits of Second- and Third-Order Møller-Plesset Perturbation Theories for Noncovalent Interactions: Revisiting MP2.5 and Assessing the Importance of Regularization and Reference Orbitals

This work systematically assesses the influence of reference orbitals, regularization, and scaling on the performance of second- and third-order Møller-Plesset perturbation theory wave function methods for noncovalent interactions (NCIs). Testing on 19 data sets (A24, DS14, HB15, HSG, S22, X40, HW30, NC15, S66, AlkBind12, CO2Nitrogen16, HB49, Ionic43, TA13, XB18, Bauza30, CT20, XB51, and Orel26rad) covers a wide range of different NCIs including hydrogen bonding, dispersion, and halogen bonding. Inclusion of potential energy surfaces from different hydrogen bonds and dispersion-bound complexes gauges accuracy for nonequilibrium geometries. Fifteen methods are tested. In notation where nonstandard choices of orbitals are denoted as methods:orbitals, these are MP2, κ-MP2, SCS-MP2, OOMP2, κ-OOMP2, MP3, MP2.5, MP3:OOMP2, MP2.5:OOMP2, MP3:κ-OOMP2, MP2.5:κ-OOMP2, κ-MP3:κ-OOMP2, κ-MP2.5:κ-OOMP2, MP3:ωB97X-V, and MP2.5:ωB97X-V. Furthermore, we compare these methods to the ωB97M-V and B3LYP-D3 density functionals, as well as CCSD. We find that the κ-regularization (κ = 1.45 au was used throughout) improves the energetics in almost all data sets for both MP2 (in 17 out of 19 data sets) and OOMP2 (16 out of 19). The improvement is significant (e.g., the root-mean-square deviation (RMSD) for the S66 data set is 0.29 kcal/mol for κ-OOMP2 versus 0.67 kcal/mol for MP2) and for interactions between stable closed-shell molecules, not strongly dependent on the reference orbitals. Scaled MP3 (with a factor of 0.5) using κ-OOMP2 reference orbitals (MP2.5:κ-OOMP2) provides significantly more accurate results for NCIs across all data sets with noniterative O(N⁶) scaling (S66 data set RMSD: 0.10 kcal/mol). Across the entire data set of 356 points, the improvement over standard MP2.5 is approximately a factor of 2: RMSD for MP3:κ-OOMP2 is 0.25 vs 0.50 kcal/mol for MP2.5. The use of high-quality density functional reference orbitals (ωB97X-V) also significantly improves the results of MP2.5 for NCI over a Hartree-Fock orbital reference. All our assessments and conclusions are based on the use of the medium-sized aug-cc-pVTZ basis to yield results that are directly compared against complete basis set limit reference values.

Software for the frontiers of quantum chemistry: An overview of developments in the Q-Chem 5 package

This article summarizes technical advances contained in the fifth major release of the Q-Chem quantum chemistry program package, covering developments since 2015. A comprehensive library of exchange-correlation functionals, along with a suite of correlated many-body methods, continues to be a hallmark of the Q-Chem software. The many-body methods include novel variants of both coupled-cluster and configuration-interaction approaches along with methods based on the algebraic diagrammatic construction and variational reduced density-matrix methods. Methods highlighted in Q-Chem 5 include a suite of tools for modeling core-level spectroscopy, methods for describing metastable resonances, methods for computing vibronic spectra, the nuclear-electronic orbital method, and several different energy decomposition analysis techniques. High-performance capabilities including multithreaded parallelism and support for calculations on graphics processing units are described. Q-Chem boasts a community of well over 100 active academic developers, and the continuing evolution of the software is supported by an "open teamware" model and an increasingly modular design.

OO-REMP: Approaching Chemical Accuracy with Second-Order Perturbation Theory

We present a perturbation theory (PT) providing second-order energies that reproduce main group chemistry benchmark sets for reaction energies, barrier heights, and atomization energies with mean absolute deviations below 1 kcal mol^-1. The PT is defined as a constrained mixture of the unperturbed Hamiltonians of the Retaining the Excitation degree (RE) and the Møller-Plesset (MP) PTs. The orbitals of the reference wave function, a single unrestricted Slater determinant, are iteratively optimized to minimize the total energy. For all benchmark sets, good and near optimal performance of OO-REMP was observed for an unperturbed Hamiltonian consisting of 25% MP and 75% RE contributions.

Perturbation theory in the complex plane: exceptional points and where to find them

We explore the non-Hermitian extension of quantum chemistry in the complex plane and its link with perturbation theory. We observe that the physics of a quantum system is intimately connected to the position of complex-valued energy singularities, known as exceptional points. After presenting the fundamental concepts of non-Hermitian quantum chemistry in the complex plane, including the mean-field Hartree-Fock approximation and Rayleigh-Schrödinger perturbation theory, we provide a historical overview of the various research activities that have been performed on the physics of singularities. In particular, we highlight seminal work on the convergence behaviour of perturbative series obtained within Møller-Plesset perturbation theory, and its links with quantum phase transitions. We also discuss several resummation techniques (such as Padé and quadratic approximants) that can improve the overall accuracy of the Møller-Plesset perturbative series in both convergent and divergent cases. Each of these points is illustrated using the Hubbard dimer at half filling, which proves to be a versatile model for understanding the subtlety of analytically-continued perturbation theory in the complex plane.

Accurate Electronic Excitation Energies in Full-Valence Active Space via Bootstrap Embedding

Fragment embedding has been widely used to circumvent the high computational scaling of using accurate electron correlation methods to describe the electronic ground states of molecules and materials. However, similar applications that utilize fragment embedding to treat electronic excited states are comparably less reported in the literature. The challenge here is twofold. First, most fragment embedding methods are most effective when the property of interest is local, but the change of the wave function upon excitation is nonlocal in general. Second, even for local excitations, an accurate estimate of, for example, the excitation energy can still be challenging owing to the need for a balanced treatment of both the ground and the excited states. In this work, we show that bootstrap embedding (BE), a fragment embedding method developed recently by our group, is promising toward describing general electronic excitations. Numerical simulations show that the excitation energies in full-valence active space (FVAS) can be well-estimated by BE to an error of ∼0.05 eV using relatively small fragments, for both local excitations and the excitations of some large dye molecules that exhibit strong charge-transfer characters. We hence anticipate BE to be a promising solution to accurately describing the excited states of large chemical systems.

Polishing the Gold Standard: The Role of Orbital Choice in CCSD(T) Vibrational Frequency Prediction

While CCSD(T) with spin-restricted Hartree-Fock (RHF) orbitals has long been lauded for its ability to accurately describe closed-shell interactions, the performance of CCSD(T) on open-shell species is much more erratic, especially when using a spin-unrestricted HF (UHF) reference. Previous studies have shown improved treatment of open-shell systems when a non-HF set of molecular orbitals, like Brueckner or Kohn-Sham density functional theory (DFT) orbitals, is used as a reference. Inspired by the success of regularized orbital-optimized second-order Møller-Plesset perturbation theory (κ-OOMP2) orbitals as reference orbitals for MP3, we investigate the use of κ-OOMP2 orbitals and various DFT orbitals as reference orbitals for CCSD(T) calculations of the corrected ground-state harmonic vibrational frequencies of a set of 36 closed-shell (29 neutrals, 6 cations, 1 anion) and 59 open-shell diatomic species (38 neutrals, 15 cations, 6 anions). The aug-cc-pwCVTZ basis set is used for all calculations. The use of κ-OOMP2 orbitals in this context alleviates difficult cases observed for both UHF orbitals and OOMP2 orbitals. Removing two multireference systems and 12 systems with ambiguous experimental data leaves a pruned data set. Overall performance on the pruned data set highlights CCSD(T) with a B97 orbital reference (CCSD(T):B97), CCSD(T) with a κ-OOMP2 orbital reference (CCSD(T):κ-OOMP2), and CCSD(T) with a B97M-rV orbital reference (CCSD(T):B97M-rV) with RMSDs of 8.48 cm^-1, and 8.50 cm^-1, and 8.75 cm^-1 respectively, outperforming CCSD(T):UHF by nearly a factor of 5. Moreover, the performance on the closed- and open-shell subsets shows these methods are able to treat open-shell and closed-shell systems with comparable accuracy and robustness. CCSD(T) with RHF orbitals is seen to improve upon UHF for the closed-shell species, while spatial symmetry breaking in a number of restricted open-shell HF (ROHF) references leads CCSD(T) with ROHF reference orbitals to exhibit the poorest statistical performance of all methods surveyed for open-shell species. The use of κ-OOMP2 orbitals has also proven useful in diagnosing multireference character that can hinder the reliability of CCSD(T).

Energy and analytic gradients for the orbital-optimized coupled-cluster doubles method with the density-fitting approximation: An efficient implementation

Efficient implementations of the orbital-optimized coupled-cluster doubles (or simply "optimized CCD," OCCD, for short) method and its analytic energy gradients with the density-fitting (DF) approach, denoted by DF-OCCD, are presented. In addition to the DF approach, the Cholesky-decomposed variant (CD-OCCD) is also implemented for energy computations. The computational cost of the DF-OCCD method (available in a plugin version of the DFOCC module of PSI4) is compared with that of the conventional OCCD (from the Q-CHEM package). The OCCD computations were performed with the Q-CHEM package in which OCCD are denoted by OD. In the conventional OCCD method, one needs to perform four-index integral transformations at each of the CCD iterations, which limits its applications to large chemical systems. Our results demonstrate that DF-OCCD provides dramatically lower computational costs compared to OCCD, and there are almost eightfold reductions in the computational time for the C₆H₁₄ molecule with the cc-pVTZ basis set. For open-shell geometries, interaction energies, and hydrogen transfer reactions, DF-OCCD provides significant improvements upon DF-CCD. Furthermore, the performance of the DF-OCCD method is substantially better for harmonic vibrational frequencies in the case of symmetry-breaking problems. Moreover, several factors make DF-OCCD more attractive compared to CCSD: (1) for DF-OCCD, there is no need for orbital relaxation contributions in analytic gradient computations; (2) active spaces can readily be incorporated into DF-OCCD; (3) DF-OCCD provides accurate vibrational frequencies when symmetry-breaking problems are observed; (4) in its response function, DF-OCCD avoids artificial poles; hence, excited-state molecular properties can be computed via linear response theory; and (5) symmetric and asymmetric triples corrections based on DF-OCCD [DF-OCCD(T)] have a significantly better performance in near degeneracy regions.

Third-Order Møller-Plesset Theory Made More Useful? The Role of Density Functional Theory Orbitals

The practical utility of Møller-Plesset (MP) perturbation theory is severely constrained by the use of Hartree-Fock (HF) orbitals. It has recently been shown that the use of regularized orbital-optimized MP2 orbitals and scaling of MP3 energy could lead to a significant reduction in MP3 error [Bertels, L. W.; J. Phys. Chem. Lett. 2019, 10, 4170 4176]. In this work, we examine whether density functional theory (DFT)-optimized orbitals can be similarly employed to improve the performance of MP theory at both the MP2 and MP3 levels. We find that the use of DFT orbitals leads to significantly improved performance for prediction of thermochemistry, barrier heights, noncovalent interactions, and dipole moments relative to the standard HF-based MP theory. Indeed, MP3 (with or without scaling) with DFT orbitals is found to surpass the accuracy of coupled-cluster singles and doubles (CCSD) for several data sets. We also found that the results are not particularly functional sensitive in most cases (although range-separated hybrid functionals with low delocalization error perform the best). MP3 based on DFT orbitals thus appears to be an efficient, noniterative O(N⁶) scaling wave-function approach for single-reference electronic structure computations. Scaled MP2 with DFT orbitals is also found to be quite accurate in many cases, although modern double hybrid functionals are likely to be considerably more accurate.

Remarkable Accuracy of an O(N6) Perturbative Correction to Opposite-Spin CCSD: Are Triples Necessary for Chemical Accuracy in Coupled Cluster?

The focus of this work is OS-CCSD-SPT(2), which is a second-order similarity transformed perturbation theory correction to opposite spin coupled cluster singles doubles, where in the latter the same-spin amplitudes are removed and the opposite-spin ones are solved self-consistently. OS-CCSD-SPT(2) is free of empirical parameters, has an instrinsic scaling of O(N⁶), and makes no use of triples. We demonstrate that, for non-multireference molecules, OS-CCSD-SPT(2) produces relative energies whose accuracy is significantly higher than what is generally expected of a triples-free model. For example, using PBE0 orbitals in the reference, OS-CCSD-SPT(2) exhibits a mean absolute deviation (MAD) of 1.13 kcal/mol with respect to CCSD(2_F) benchmark values for the non-multireference subset of W4-08 atomization energies (cf. a MAD > 6.5 kcal/mol for CCSD) and a MAD of 0.68 kcal/mol for the energies of reactions generated from the W4-08 molecules. These MADs are reduced to 0.61 and 0.63 kcal/mol, respectively, by a simple one-parameter spin-component scaling of the OS-CCSD-SPT(2) same-spin correlation energy. OS-CCSD is also naturally amenable to higher order corrections: the associated third-order correction, OS-CCSD-SPT(3), which does involve connected triples and quadruples, exhibits a MAD of 0.44 kcal/mol for the same atomization-energy benchmark.