Revisiting the Orbital Energy-Dependent Regularization of Orbital-Optimized Second-Order Møller-Plesset Theory

Second-Generation Energy Decomposition Analysis of Intermolecular Interaction Energies from the Second-Order Mo̷ller-Plesset Theory: An Extensible, Orthogonal Formulation with Useful Basis Set Convergence for All Terms

Energy decomposition analysis (EDA) based on density functional theory (DFT) and self-consistent field (SCF) calculations has become widely used for understanding intermolecular interactions. This work reports a new approach to EDA for post-SCF wave functions based on closed-shell restricted second-order Mo̷ller-Plesset (MP2) together with an efficient implementation that generalizes the successful SCF-level second-generation absolutely localized molecular orbital EDA approach, ALMO-EDA-II, and improves upon MP2 ALMO-EDA-I. The new MP2 ALMO-EDA-II provides distinct energy contributions for a frozen interaction energy containing permanent electrostatics and Pauli repulsions, polarized energy-yielding induced electrostatics, dispersion-corrected energy, and the fully relaxed energy, which describes charge transfer. All terms have useful complete basis set limits due to the design of the theory, corroborated by a range of test calculations on model systems, and the S22 and the Ionic43 data sets of weak and strong intermolecular interactions, respectively. Comparisons with the DFT-based ALMO-EDA-II suggest that the new MP2 EDA yields quite a consistent interpretation of intermolecular interactions when the total interaction energies are consistent. To begin to address the limitations of the MP2 theory itself, the MP2 ALMO-EDA-II was also implemented for κ-regularized MP2 and the size-consistent second-order Brillouin-Wigner (BW-s2) method, both of which are more accurate for dispersion-dominated interactions. The principal limitation of MP2 ALMO-EDA-II is associated with the need to obtain orthogonal fragment-localized virtual orbitals, which leads to clearly poorer results when using atomic orbital basis sets that contain diffuse functions. We therefore recommend using nonaugmented basis sets.

On the notion of strong correlation in electronic structure theory

Strong correlation has been said to have many faces, and appears to have many synonyms of questionable suitability. In this work we aim not to define the term once and for all, but to highlight one possibility that is both rigorously defined and physically transparent, and remains so in reference to molecules and quantum lattice models. We survey both molecular examples - hydrogen systems (Hn, n = 2, 4, 6), Be2, H-He-H, and benzene - and the half-filled Hubbard model over a range of correlation regimes. Various quantities are examined including the extent of spin symmetry breaking in correlated single-reference wave functions, energetic ratios inspired by the Hubbard model and the Virial theorem, and metrics derived from the one- and two-electron reduced density matrices (RDMs). The trace and the square norm of the cumulant of the two-electron reduced density matrix capture what may well be defined as strong correlation. Accordingly, strong correlation is understood as a statistical dependence between two electrons, and is distinct from the concepts of "correlation energy" and more general than entanglement quantities that require a partitioning of a quantum system into distinguishable subspaces. This work enables us to build a bridge between a rigorous and quantifiable regime of strong electron correlation and more familiar chemical concepts such as anti-aromaticity in the context of Baird's rule.

Chemical Potentials and the One-Electron Hamiltonian of the Second-Order Perturbation Theory from the Functional Derivative Approach

We develop a functional derivative approach to calculate the chemical potentials of second-order perturbation theory (MP2). In the functional derivative approach, the correlation part of the MP2 chemical potential, which is the derivative of the MP2 correlation energy with respect to the occupation number of frontier orbitals, is obtained from the chain rule via the noninteracting Green's function. First, the MP2 correlation energy is expressed in terms of the noninteracting Green's function, and its functional derivative to the noninteracting Green's function is the second-order self-energy. Then, the derivative of the noninteracting Green's function to the occupation number is obtained by including the orbital relaxation effect. We show that the MP2 chemical potentials obtained from the functional derivative approach agree with that obtained from the finite difference approach. The one-electron Hamiltonian, defined as the derivative of the MP2 energy with respect to the one particle density matrix, is also derived using the functional derivative approach, which can be used in the self-consistent calculations of MP2 and double-hybrid density functionals. The developed functional derivative approach is promising for calculating the chemical potentials and the one-electron Hamiltonian of approximate functionals and many-body perturbation approaches dependent explicitly on the noninteracting Green's function.

High-Performance Multi-GPU Analytic RI-MP2 Energy Gradients

This article presents a novel algorithm for the calculation of analytic energy gradients from second-order Møller-Plesset perturbation theory within the Resolution-of-the-Identity approximation (RI-MP2), which is designed to achieve high performance on clusters with multiple graphical processing units (GPUs). The algorithm uses GPUs for all major steps of the calculation, including integral generation, formation of all required intermediate tensors, solution of the Z-vector equation and gradient accumulation. The implementation in the EXtreme Scale Electronic Structure System (EXESS) software package includes a tailored, highly efficient, multistream scheduling system to hide CPU-GPU data transfer latencies and allows nodes with 8 A100 GPUs to operate at over 80% of theoretical peak floating-point performance. Comparative performance analysis shows a significant reduction in computational time relative to traditional multicore CPU-based methods, with our approach achieving up to a 95-fold speedup over the single-node performance of established software such as Q-Chem and ORCA. Additionally, we demonstrate that pairing our implementation with the molecular fragmentation framework in EXESS can drastically lower the computational scaling of RI-MP2 gradient calculations from quintic to subquadratic, enabling further substantial savings in runtime while retaining high numerical accuracy in the resulting gradients.

Reaching High Accuracy for Energetic Properties at Second-Order Perturbation Cost by Merging Self-Consistency and Spin-Opposite Scaling

Quantum chemical methods dealing with challenging systems while retaining low computational costs have attracted attention. In particular, many efforts have been devoted to developing new methods based on second-order perturbation that may be the simplest correlated method beyond Hartree-Fock. We have recently developed a self-consistent perturbation theory named one-body Møller-Plesset second-order perturbation theory (OBMP2) and shown that it can resolve issues caused by the noniterative nature of standard perturbation theory. In this work, we extend the method by introducing spin-opposite scaling to the double-excitation amplitudes, resulting in the O2BMP2 method. We assess the O2BMP2 performance on the triple-bond N2 dissociation, singlet-triplet gaps, and ionization potentials. O2BMP2 performs much better than standard MP2 and reaches the accuracy of coupled-cluster methods in all cases considered in this work.

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.

Regularized and Opposite Spin-Scaled Functionals from Møller-Plesset Adiabatic Connection─Higher Accuracy at Lower Cost

Noncovalent interactions (NCIs) play a crucial role in biology, chemistry, material science, and everything in between. To improve pure quantum-chemical simulations of NCIs, we propose a methodology for constructing approximate correlation energies by combining an interpolation along the Møller-Plesset adiabatic connection (MP AC) with a regularization and spin-scaling strategy applied to MP2 correlation energies. This combination yields cosκos-SPL2, which exhibits superior accuracy for NCIs compared to any of the individual strategies. With the N4 formal scaling, cosκos-SPL2 is competitive or often outperforms more expensive dispersion-corrected double hybrids for NCIs. The accuracy of cosκos-SPL2 particularly shines for anionic halogen bonded complexes, where it surpasses standard dispersion-corrected DFT by a factor of 3 to 5.

Toward Benchmark-Quality Ab Initio Predictions for 3d Transition Metal Electrocatalysts: A Comparison of CCSD(T) and ph-AFQMC

Generating accurate ab initio ionization energies for transition metal complexes is an important step toward the accurate computational description of their electrocatalytic reactions. Benchmark-quality data is required for testing existing theoretical methods and developing new ones but is complicated to obtain for many transition metal compounds due to the potential presence of both strong dynamical and static electron correlation. In this regime, it is questionable whether the so-called gold standard, coupled cluster with singles, doubles, and perturbative triples (CCSD(T)), provides the desired level of accuracy─roughly 1-3 kcal/mol. In this work, we compiled a test set of 28 3d metal-containing molecules relevant to homogeneous electrocatalysis (termed 3dTMV) and computed their vertical ionization energies (ionization potentials) with CCSD(T) and phaseless auxiliary-field quantum Monte Carlo (ph-AFQMC) in the def2-SVP basis set. A substantial effort has been made to converge away the phaseless bias in the ph-AFQMC reference values. We assess a wide variety of multireference diagnostics and find that spin-symmetry breaking of the CCSD wave function and the PBE0 density functional correlate well with our analysis of multiconfigurational wave functions. We propose quantitative criteria based on symmetry breaking to delineate correlation regimes inside of which appropriately performed CCSD(T) can produce mean absolute deviations from the ph-AFQMC reference values of roughly 2 kcal/mol or less and outside of which CCSD(T) is expected to fail. We also present a preliminary assessment of density functional theory (DFT) functionals on the 3dTMV set.

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.

Computational Studies of Rubber Ozonation Explain the Effectiveness of 6PPD as an Antidegradant and the Mechanism of Its Quinone Formation

The discovery that the commercial rubber antidegradant 6PPD reacts with ozone (O₃) to produce a highly toxic quinone (6PPDQ) spurred a significant research effort into nontoxic alternatives. This work has been hampered by lack of a detailed understanding of the mechanism of protection that 6PPD affords rubber compounds against ozone. Herein, we report high-level density functional theory studies into early steps of rubber and PPD (p-phenylenediamine) ozonation, identifying key steps that contribute to the antiozonant activity of PPDs. In this, we establish that our density functional theory approach can achieve chemical accuracy for many ozonation reactions, which are notoriously difficult to model. Using adiabatic energy decomposition analysis, we examine and dispel the notion that one-electron charge transfer initiates ozonation in these systems, as is sometimes argued. Instead, we find direct interaction between O₃ and the PPD aromatic ring is kinetically accessible and that this motif is more significant than interactions with PPD nitrogens. The former pathway results in a hydroxylated PPD intermediate, which reacts further with O₃ to afford 6PPD hydroquinone and, ultimately, 6PPDQ. This mechanism directly links the toxicity of 6PPDQ to the antiozonant function of 6PPD. These results have significant implications for development of alternative antiozonants, which are discussed.

Assessment of State-Averaged Driven Similarity Renormalization Group on Vertical Excitation Energies: Optimal Flow Parameters and Applications to Nucleobases

We present a comprehensive excited-state benchmark for the state-averaged (SA) driven similarity renormalization group (DSRG) [Li, C.; Evangelista, F. A. J. Chem. Phys. 2018, 148, 124106]. Following the QUEST database [Véril, M.; Scemama, A.; Caffarel, M.; Lipparini, F.; Boggio-Pasqua, M.; Jacquemin, D.; Loos, P.-F. Wiley Interdiscip. Rev. Comput. Mol. Sci. 2021, 11, e1517], 280 vertical transition energies of 35 medium-sized molecules are computed using the SA-DSRG derived second- and third-order perturbation theories (PT2/PT3) along with a nonperturbative approach [sq-LDSRG(2)]. Comparing to the theoretical best estimates, the optimal flow parameter is found to be 0.35 and 2.0 Eh-2 for SA-DSRG-PT2 and SA-DSRG-PT3, respectively. For SA-sq-LDSRG(2), a flow parameter of 1.5 Eh-2 provides converged equations without compromising the accuracy. We then assess the accuracy of the SA-DSRG hierarchy using these parameters. The SA-DSRG-PT2 scheme outperforms the level-shifted CASPT2 by 0.10 eV in mean absolute error (MAE), yet this accuracy is slightly inferior than that of CASPT2 with the ionization-potential-electron-affinity shift. Both SA-DSRG-PT3 and SA-sq-LDSRG(2) yield a MAE of 0.10 eV, which is comparable to that of CASPT3 (0.09 eV). Finally, we compute vertical excitation energies of several low-lying singlet states of nucleobases. The SA-sq-LDSRG(2) approach provides highly accurate results for π → π* excitations, while n → π* transitions are better described by SA-DSRG-PT3.