Reaching Full Correlation through Nonorthogonal Configuration Interaction: A Second-Order Perturbative Approach

Selected Nonorthogonal Configuration Interaction with Compressed Single and Double Excitations

Addressing both dynamic and static correlations accurately is a primary goal in electronic structure theory. Nonorthogonal configuration interaction (NOCI) is a versatile tool for treating static correlation, offering chemical insights by combining diverse reference states. Nevertheless, achieving quantitative accuracy requires the inclusion of the missing dynamic correlation. This work introduces a framework for compressing orthogonal single and double excitations into a NOCI of a much smaller dimension. This compression is repeated with each Slater determinant in a reference NOCI, resulting in another NOCI that includes all of its single and double excitations (NOCISD), effectively recovering the missing dynamic correlations from the reference. This compressed NOCISD is further refined through a selection process using metric and energy tests (SNOCISD). We validate the effectiveness of SNOCISD through its application to the dissociation of the nitrogen molecule and the hole-doped two-dimensional Hubbard model at various interaction strengths.

Non-orthogonal Configuration Interaction Study on the Effect of Thermal Distortions on the Singlet Fission Process in Photoexcited Pure and B,N-Doped Pentacene Crystals

The present computational work analyzes singlet fission (SF) as a pathway for multiplication of photogenerated excitons in crystalline polyacenes. Our study explores the well-known crystalline pentacene (C22H14) and the prospective and potentially interesting doped B,N-pentacene (BC20NH14). At the molecular level, the singlet fission process involves a pair of neighboring molecules and is based on the coupling between an excited singlet state (S1S0) and two singlet-coupled triplets (1T1T1), which, typically, is influenced by an intermolecular charge transfer state. Taking data from periodic density functional theory and ab initio wave function calculations, we applied the non-orthogonal configuration interaction method to determine electronic coupling parameters. The comparison of the results for both equilibrium structures reveal smaller SF coupling for pentacene than for B,N-pentacene by a factor of ∼5. Introduction of the dynamic behavior to the crystals (vibrations, thermal motion) provides a more realistic picture of the effect of the disorder at the molecular level on the SF efficiency. The coupling values associated to out-of-equilibrium structures show that most of the S1S0/1T1T1 couplings remain virtually constant or slightly increase for pentacene when molecular disorder is introduced. Homologous calculations on B,N-pentacene show a generalized decrease in the coupling values, notably if large phonon displacements are considered. A few of the structures analyzed feature much larger SF coupling if some distortion results in (nearly) degenerate charge transfer and excited singlet and triplet states. For these particular situations, an acceleration of the SF process could occur although in competition with electron-hole separation as an alternative pathway.

Excited States, Symmetry Breaking, and Unphysical Solutions in State-Specific CASSCF Theory

State-specific electronic structure theory provides a route toward balanced excited-state wave functions by exploiting higher-energy stationary points of the electronic energy. Multiconfigurational wave function approximations can describe both closed- and open-shell excited states and avoid the issues associated with state-averaged approaches. We investigate the existence of higher-energy solutions in complete active space self-consistent field (CASSCF) theory and characterize their topological properties. We demonstrate that state-specific approximations can provide accurate higher-energy excited states in H₂ (6-31G) with more compact active spaces than would be required in a state-averaged formalism. We then elucidate the unphysical stationary points, demonstrating that they arise from redundant orbitals when the active space is too large or symmetry breaking when the active space is too small. Furthermore, we investigate the singlet-triplet crossing in CH₂ (6-31G) and the avoided crossing in LiF (6-31G), revealing the severity of root flipping and demonstrating that state-specific solutions can behave quasi-diabatically or adiabatically. These results elucidate the complexity of the CASSCF energy landscape, highlighting the advantages and challenges of practical state-specific calculations.

On the potentially transformative role of auxiliary-field quantum Monte Carlo in quantum chemistry: A highly accurate method for transition metals and beyond

Approximate solutions to the ab initio electronic structure problem have been a focus of theoretical and computational chemistry research for much of the past century, with the goal of predicting relevant energy differences to within "chemical accuracy" (1 kcal/mol). For small organic molecules, or in general, for weakly correlated main group chemistry, a hierarchy of single-reference wave function methods has been rigorously established, spanning perturbation theory and the coupled cluster (CC) formalism. For these systems, CC with singles, doubles, and perturbative triples is known to achieve chemical accuracy, albeit at O(N7) computational cost. In addition, a hierarchy of density functional approximations of increasing formal sophistication, known as Jacob's ladder, has been shown to systematically reduce average errors over large datasets representing weakly correlated chemistry. However, the accuracy of such computational models is less clear in the increasingly important frontiers of chemical space including transition metals and f-block compounds, in which strong correlation can play an important role in reactivity. A stochastic method, phaseless auxiliary-field quantum Monte Carlo (ph-AFQMC), has been shown to be capable of producing chemically accurate predictions even for challenging molecular systems beyond the main group, with relatively low O(N3 - N4) cost and near-perfect parallel efficiency. Herein, we present our perspectives on the past, present, and future of the ph-AFQMC method. We focus on its potential in transition metal quantum chemistry to be a highly accurate, systematically improvable method that can reliably probe strongly correlated systems in biology and chemical catalysis and provide reference thermochemical values (for future development of density functionals or interatomic potentials) when experiments are either noisy or absent. Finally, we discuss the present limitations of the method and where we expect near-term development to be most fruitful.

Generalized nonorthogonal matrix elements. II: Extension to arbitrary excitations

Electronic structure methods that exploit nonorthogonal Slater determinants face the challenge of efficiently computing nonorthogonal matrix elements. In a recent publication [H. G. A. Burton, J. Chem. Phys. 154, 144109 (2021)], I introduced a generalized extension to the nonorthogonal Wick’s theorem that allows matrix elements to be derived between excited configurations from a pair of reference determinants with a singular nonorthogonal orbital overlap matrix. However, that work only provided explicit expressions for one- and two-body matrix elements between singly- or doubly-excited configurations. Here, this framework is extended to compute generalized nonorthogonal matrix elements between higher-order excitations. Pre-computing and storing intermediate values allows one- and two-body matrix elements to be evaluated with an [Formula: see text] scaling relative to the system size, and the LIBGNME computational library is introduced to achieve this in practice. These advances make the evaluation of all nonorthogonal matrix elements almost as easy as their orthogonal counterparts, facilitating a new phase of development in nonorthogonal electronic structure theory.

GronOR: Scalable and Accelerated Nonorthogonal Configuration Interaction for Molecular Fragment Wave Functions

GronOR is a program package for nonorthogonal configuration interaction calculations. Electronic wave functions are constructed in terms of antisymmetrized products of multiconfiguration molecular fragment wave functions. The computational complexity of the nonorthogonal methodologies implemented in GronOR applied to large molecular assemblies requires a design that takes full advantage of massively parallel supercomputer architectures and accelerator technologies. This work describes the implementation strategy and resulting performance characteristics. In addition to parallelization and acceleration, the software development strategy includes aspects of fault resiliency and heterogeneous computing. The program was designed for large-scale supercomputers but also runs effectively on small clusters and workstations for small molecular systems. GronOR is available as open source to the scientific community.

On the role of dynamic electron correlation in non-orthogonal configuration interaction with fragments

Two different approaches have been implemented to include the effect of dynamic electron correlation in the Non-Orthogonal Configuration Interaction for Fragments (NOCI-F) method. The first is based on shifting the diagonal matrix elements of the NOCI matrix, while the second incorporates the dynamic correlation explicitly in the fragment wave functions used to construct the many-electron basis functions of the NOCI. The two approaches are illustrated for the calculation of the electronic coupling relevant in singlet fission and the coupling of spin moments in organic radicals. Comparison of the calculated diabatic couplings, the NOCI energies and wave functions shows that dynamic electron correlation is not only efficiently but also effectively incorporated by the shifting approach and can largely affect the coupling between electronic states. Also, it brings the NOCI coupling of the spin moments in close agreement with benchmark calculations.

Relativistic nonorthogonal configuration interaction: application to L2,3-edge X-ray spectroscopy

In this article, we develop a relativistic exact-two-component nonorthogonal configuration interaction (X2C-NOCI) for computing L-edge X-ray spectra. This article to our knowledge is the first time NOCI has been used for relativistic wave functions. A set of molecular complexes, including SF₆, SiCl₄ and [FeCl₆]^3-, are used to demonstrate the accuracy and computational scaling of the X2C-NOCI method. Our results suggest that X2C-NOCI is able to satisfactorily capture the main features of the L_2,3-edge X-ray absorption spectra. Excitations from the core require a large amount of orbital relaxation to yield reasonable energies and X2C-NOCI allows us to treat orbital optimization explicitly. However, the cost of computing the nonorthogonal coupling is higher than in conventional CI. Here, we propose an improved integral screening using overlap-scaled density combined with a continuous measure of the generalized Slater-Condon rules that allows us to estimate if an element is zero before attempting a two-electron integral contraction.

Calculation of Metallocene Ionization Potentials via Auxiliary Field Quantum Monte Carlo: Toward Benchmark Quantum Chemistry for Transition Metals

The accurate ab initio prediction of ionization energies is essential to understanding the electrochemistry of transition metal complexes in both materials science and biological applications. However, such predictions have been complicated by the scarcity of gas phase experimental data, the relatively large size of the relevant molecules, and the presence of strong electron correlation effects. In this work, we apply all-electron phaseless auxiliary-field quantum Monte Carlo (ph-AFQMC) utilizing multideterminant trial wave functions to six metallocene complexes to compare the computed adiabatic and vertical ionization energies with experimental results. We find that ph-AFQMC yields mean absolute errors (MAEs) of 1.69 ± 1.02 kcal/mol for the adiabatic energies and 2.85 ± 1.13 kcal/mol for the vertical energies. We also carry out density functional theory (DFT) calculations using a variety of functionals, which yields MAEs of 3.62-6.98 kcal/mol and 3.31-9.88 kcal/mol, as well as one variant of localized coupled cluster calculations (DLPNO-CCSD(T0) with moderate PNO cutoffs), which has MAEs of 4.96 and 6.08 kcal/mol, respectively. We also test the reliability of DLPNO-CCSD(T0) and DFT on acetylacetonate (acac) complexes for adiabatic energies measured in the same manner experimentally, and we find higher MAEs, ranging from 4.56 to 10.99 kcal/mol (with a different ordering) for DFT and 6.97 kcal/mol for DLPNO-CCSD(T0). Finally, by utilizing experimental solvation energies, we show that accurate reduction potentials in solution for the metallocene series can be obtained from the AFQMC gas phase results.

Localized Spin Rotations: A Size-Consistent Approach to Nonorthogonal Configuration Interaction

Current nonorthogonal configuration interaction (NOCI) methods often use a set of self-consistent field (SCF) states selected based on chemical intuition. However, it may be challenging to track these SCF states across a dissociation profile and the NOCI states recovered may be spin contaminated. In this Article, we propose a method of applying spin rotation on symmetry broken unrestricted Hartree-Fock (sb-UHF) states to generate a basis for NOCI. The dissociation of ethene was examined by localizing spin rotation on each resulting carbene fragment. We show that this gives a size-consistent description of its dissociation and results in spin-pure states at all geometries. The dissociation was also studied with different orbitals, namely, canonical UHF and absolutely localized molecular orbitals (ALMO). Furthermore, we demonstrate that the method can be used to restore spin symmetry of symmetry broken SCF wave functions for molecules of various sizes, marking an improvement over existing NOCI methods.

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.

Orbital optimization in nonorthogonal multiconfigurational self-consistent field applied to the study of conical intersections and avoided crossings

Nonorthogonal approaches to electronic structure methods have recently received renewed attention, with the hope that new forms of nonorthogonal wavefunction Ansätze may circumvent the computational bottleneck of orthogonal-based methods. The basis in which nonorthogonal configuration interaction is performed defines the compactness of the wavefunction description and hence the efficiency of the method. Within a molecular orbital approach, nonorthogonal configuration interaction is defined by a "different orbitals for different configurations" picture, with different methods being defined by their choice of determinant basis functions. However, identification of a suitable determinant basis is complicated, in practice, by (i) exponential scaling of the determinant space from which a suitable basis must be extracted, (ii) possible linear dependencies in the determinant basis, and (iii) inconsistent behavior in the determinant basis, such as disappearing or coalescing solutions, as a result of external perturbations, such as geometry change. An approach that avoids the aforementioned issues is to allow for basis determinant optimization starting from an arbitrarily constructed initial determinant set. In this work, we derive the equations required for performing such an optimization, extending previous work by accounting for changes in the orthogonality level (defined as the dimension of the orbital overlap kernel between two determinants) as a result of orbital perturbations. The performance of the resulting wavefunction for studying avoided crossings and conical intersections where strong correlation plays an important role is examined.

Improved Description and Efficient Implementation of Spin-Projected Perturbation Theory for Practical Applications

In this study, we continue to develop the recently proposed second-order perturbation theory for the spin-projected Hartree-Fock method [Tsuchimochi, T.; Ten-no, S. L. J. Chem. Theory Comput. 2019, 15, 6688] in various aspects. A new, stable imaginary level-shift scheme is derived to obtain a well-conditioned equation, enabling a significantly faster convergence. To achieve a further speed-up, we propose a preconditioning scheme considering the pair character on a spin-projected basis. We also eliminate the computational memory bottleneck in solving the linear equation for large systems using a distributed memory parallel implementation. Finally, for the description of open-shell molecules, several modified zeroth-order Hamiltonians are introduced and tested using the Mn₂O₂(NHCHCO₂)₄ complex. These developments enable practical calculations of a second-order perturbation theory with improved accuracy at a reduced computational cost.

Generalized nonorthogonal matrix elements: Unifying Wick's theorem and the Slater-Condon rules

Matrix elements between nonorthogonal Slater determinants represent an essential component of many emerging electronic structure methods. However, evaluating nonorthogonal matrix elements is conceptually and computationally harder than their orthogonal counterparts. While several different approaches have been developed, these are predominantly derived from the first-quantized generalized Slater-Condon rules and usually require biorthogonal occupied orbitals to be computed for each matrix element. For coupling terms between nonorthogonal excited configurations, a second-quantized approach such as the nonorthogonal Wick's theorem is more desirable, but this fails when the two reference determinants have a zero many-body overlap. In this contribution, we derive an entirely generalized extension to the nonorthogonal Wick's theorem that is applicable to all pairs of determinants with nonorthogonal orbitals. Our approach creates a universal methodology for evaluating any nonorthogonal matrix element and allows Wick's theorem and the generalized Slater-Condon rules to be unified for the first time. Furthermore, we present a simple well-defined protocol for deriving arbitrary coupling terms between nonorthogonal excited configurations. In the case of overlap and one-body operators, this protocol recovers efficient formulas with reduced scaling, promising significant computational acceleration for methods that rely on such terms.

The Shape of Full Configuration Interaction to Come

We present a Perspective on what the future holds for full configuration interaction (FCI) theory, with an emphasis on conceptual rather than technical details. Upon revisiting the early history of FCI, a number of its key contemporary approximations are compared on as equal a footing as possible, using a recent blind challenge on the benzene molecule as a testbed [Eriksen et al., J. Phys. Chem. Lett., 2020 11, 8922]. In the process, we review the scope of applications for which FCI continues to prove indispensable, and the required traits in terms of robustness, efficacy, and reliability its modern approximations must satisfy are discussed. We close by conveying a number of general observations on the merits offered by the state-of-the-art alongside some of the challenges still faced to this day. While the field has altogether seen immense progress over the years-the past decade, in particular-it remains clear that our community as a whole has a substantial way to go in enhancing the overall applicability of near-exact electronic structure theory for systems of general composition and increasing size.