Multireference quantum chemistry through a joint density matrix renormalization group and canonical transformation theory

Towards reliable and efficient modeling of [Cu2O2]2+-based compound electronic structures with the partially fixed reference space protocols

This work reports a computationally efficient approach for reliable modeling of complex electronic structures based on [Cu2O2]2+ moieties. Specifically, we explore the recently developed partially fixed reference space (PFRS) protocol to minimize the active space size, taking into account the double d-shell effects. We show that the ground-state electronic structure of the core [Cu2O2]2+ model system is dominated by the d9/d10 occupations. The PFRS-crafted active spaces are further used to generate the reference wave functions for the multi-reference coupled cluster, configuration interaction, and multi-reference perturbation theory calculations. Specifically, we demonstrate that the bare [Cu2O2]2+ core can be modeled qualitatively using active spaces as small as CAS(2,2)PFRS. To obtain quantitative agreement with the reference DMRG(32,62)CI calculations, the CAS(4,4) has to be used in conjunction with the MRCCSD correction on top of it. This reliable and computationally efficient protocol is further used to model the electronic structure and properties of ammonia coordinated [Cu2O2]2+ complexes. Finally, based on the large amount of available experimental data regarding the oxo-peroxo equilibrium of [Cu2O2]2+-based systems, it is possible to formulate educated guesses regarding the effect of each experimental variable over each d-occupancy-specific state. With a large sample size of d-occupancy-specific state dependence with ligands and solvents, it should be possible to propose new ligands with specific d-occupancy and, therefore, oxidative properties based on the d-occupancy energy gaps of relatively low-cost calculations.

MBE-CASSCF Approach for the Accurate Treatment of Large Active Spaces

We present a novel implementation of the complete active space self-consistent field (CASSCF) method that makes use of the many-body expanded full configuration interaction (MBE-FCI) method to incrementally approximate electronic structures within large active spaces. On the basis of a hybrid first-order algorithm employing both Super-CI and quasi-Newton strategies for the optimization of molecular orbitals, we demonstrate both computational efficacy and high accuracy of the resulting MBE-CASSCF method. We assess the performance of our implementation on a set of established numerical tests before applying MBE-CASSCF in the investigation of the triplet-quintet spin gap of iron(II) porphyrin with active spaces as large as 50 electrons in 50 orbitals.

Renormalized-Residue-Based Multireference Configuration Interaction Method for Strongly Correlated Systems

The implementation of multireference configuration interaction (MRCI) methods in quantum systems with large active spaces is hindered by the expansion of configuration bases or the intricate handling of reduced density matrices (RDMs). In this work, we present a spin-adapted renormalized-residue-based MRCI (RR-MRCI) approach that leverages renormalized residues to effectively capture the entanglement between active and inactive orbitals. This approach is reinforced by a novel efficient algorithm, which also facilitates an efficient deployment of spin-adapted matrix product state MRCI (MPS-MRCI). The RR-MRCI framework possesses several advantages: (1) It considers the orbital entanglement and utilizes highly compressed MPS structure, improving computational accuracy and efficiency compared with internally contracted (ic) MRCI. (2) Utilizing small-sized buffer environments of a few external orbitals as probes based on quantum information theory, it enhances computational efficiency over MPS-MRCI and offers potential application to large molecular systems. (3) The RR framework can be implemented in conjunction with ic-MRCI, eliminating the need for high-rank RDMs, by using distinct renormalized residues. We evaluated this method across nine diverse molecular systems, including Cu2O22+ with an active space of (24e,24o) and two complexes of lanthanide and actinide with active space (38e,36o), demonstrating the method's versatility and efficacy.

Block2: A comprehensive open source framework to develop and apply state-of-the-art DMRG algorithms in electronic structure and beyond

block2 is an open source framework to implement and perform density matrix renormalization group and matrix product state algorithms. Out-of-the-box it supports the eigenstate, time-dependent, response, and finite-temperature algorithms. In addition, it carries special optimizations for ab initio electronic structure Hamiltonians and implements many quantum chemistry extensions to the density matrix renormalization group, such as dynamical correlation theories. The code is designed with an emphasis on flexibility, extensibility, and efficiency and to support integration with external numerical packages. Here, we explain the design principles and currently supported features and present numerical examples in a range of applications.

Machine Learning the Quantum Mechanical Wave Function

Strongly correlated systems have been challenging to computational chemists for a long time. To solve these systems, multireference methods have been developed over the years. Recently, with the fast development of machine learning and artificial intelligence methods, these methods have also influenced the quest for optimal wave function ansatz. Machine learning approaches have been used in many different flavors. From this perspective, we will discuss the different milestones achieved in the use of machine learning for solving the quantum many body problem.

Dynamic Correlation on the Adaptive Sampling Configuration Interaction (ASCI) Reference Function: ASCI-DSRG-MRPT2

A balanced description of static and dynamic electron correlations is at the heart of quantum chemical methods. To obtain accurate results in strongly correlated systems using wave-function-based methods, a large active space is necessary to ensure correct descriptions of static correlations. Correcting the results for dynamic correlations is also necessary. In this work, we present implementations of second-order perturbation theory for dynamic correlations based on the adaptive sampling configuration interaction self-consistent field (ASCI-SCF) method. In particular, we implemented spin-free driven similarity renormalization group second-order multireference perturbation theory (DSRG-MRPT2). The extrapolation of the ASCI + PT2 energy based on the relaxed Hamiltonian in DSRG-MRPT2 gives a reasonable approximation of DSRG-MRPT2 based on CASSCF. We demonstrate the application of the ASCI-DSRG-MRPT2 method in evaluations of the spin-state energy gaps in iron porphyrins, polyacenes, and periacenes along with the reaction energies of methane oxidation by FeO+ and electrocyclic ring formation in cethrene.

New Density Matrix Renormalization Group Approaches for Strongly Correlated Systems Coupled with Large Environments

Thanks to the high compression of the matrix product state (MPS) form of the wave function and the efficient site-by-site iterative sweeping optimization algorithm, the density matrix normalization group (DMRG) and its time-dependent variant (TD-DMRG) have been established as powerful computational tools in accurately simulating the electronic structure and quantum dynamics of strongly correlated molecules with a large number (10^1-2) of quantum degrees of freedom (active orbitals or vibrational modes). However, the quantitative characterization of the quantum many-body behaviors of realistic strongly correlated systems requires a further consideration of the interaction between the embedded active subsystem and the remaining correlated environment, e.g., a larger number (10^2-3) of external orbitals in electronic structure or infinite condensed-phase phononic modes in nucleus dynamics. To this end, we introduced three new post-DMRG and TD-DMRG approaches, namely (1) DMRG2sCI-MRCI and DMRG2sCI-ENPT by the reconstruction of selected configuration interaction (sCI) type of compact reference function from DMRG coefficients and the use of externally contracted MRCI (multireference configuration interaction) and Epstein-Nesbet perturbation theory (ENPT), without recourse to the expensive high order n-electron reduced density matrices (n-RDMs). (2) DMRG combined with RR-MRCI (renormalized residue-based MRCI), which improves the computational accuracy and efficiency of internally contracted (ic) MRCI by renormalizing the contracted bases with small-sized buffer environment(s) of a few external orbitals as probes based on quantum information theory. (3) HM (hierarchical mapping)-TD-DMRG in which a large environment is reduced to a small number of renormalized environmental modes (which accounts for the most vital system-environment interactions) through stepwise mapping transformation. These advances extend the efficacy of highly accurate DMRG/TD-DMRG computations to the quantitative characterization of the electronic structure and quantum dynamics in realistic strongly correlated systems coupled with large environments and are reviewed in this paper.

Density Matrix Renormalization Group for Transcorrelated Hamiltonians: Ground and Excited States in Molecules

We present the theory of a density matrix renormalization group (DMRG) algorithm which can solve for both the ground and excited states of non-Hermitian transcorrelated Hamiltonians and show applications in molecular systems. Transcorrelation (TC) accelerates the basis set convergence rate by including known physics (such as, but not limited to, the electron-electron cusp) in the Jastrow factor used for the similarity transformation. It also improves the accuracy of approximate methods such as coupled cluster singles and doubles (CCSD) as shown by recent studies. However, the non-Hermiticity of the TC Hamiltonians poses challenges for variational methods like DMRG. Imaginary-time evolution on the matrix product state (MPS) in the DMRG framework has been proposed to circumvent this problem, but this is currently limited to treating the ground state and has lower efficiency than the time-independent DMRG (TI-DMRG) due to the need to eliminate Trotter errors. In this work, we show that with minimal changes to the existing TI-DMRG algorithm, namely, replacing the original Davidson solver with the general Davidson solver to solve the non-Hermitian effective Hamiltonians at each site for a few low-lying right eigenstates, and following the rest of the original DMRG recipe, one can find the ground and excited states with improved efficiency compared to the original DMRG when extrapolating to the infinite bond dimension limit in the same basis set. An accelerated basis set convergence rate is also observed, as expected, within the TC framework.

ipie: A Python-Based Auxiliary-Field Quantum Monte Carlo Program with Flexibility and Efficiency on CPUs and GPUs

We report the development of a python-based auxiliary-field quantum Monte Carlo (AFQMC) program, ipie, with preliminary timing benchmarks and new AFQMC results on the isomerization of [Cu₂O₂]²⁺. We demonstrate how implementations for both central and graphical processing units (CPUs and GPUs) are achieved in ipie. We show an interface of ipie with PySCF as well as a straightforward template for adding new estimators to ipie. Our timing benchmarks against other C++ codes, QMCPACK and Dice, suggest that ipie is faster or similarly performing for all chemical systems considered on both CPUs and GPUs. Our results on [Cu₂O₂]²⁺ using selected configuration interaction trials show that it is possible to converge the ph-AFQMC isomerization energy between bis(μ-oxo) and μ-η²:η² peroxo configurations to the exact known results for small basis sets with 10⁵-10⁶ determinants. We also report the isomerization energy with a quadruple-zeta basis set with an estimated error less than a kcal/mol, which involved 52 electrons and 290 orbitals with 10⁶ determinants in the trial wave function. These results highlight the utility of ph-AFQMC and ipie for systems with modest strong correlation and large-scale dynamic correlation.

Quantum Simulation of Molecular Electronic States with a Transcorrelated Hamiltonian: Higher Accuracy with Fewer Qubits

Simulation of electronic structure is one of the most promising applications on noisy intermediate-scale quantum (NISQ) era devices. However, NISQ devices suffer from a number of challenges like limited qubit connectivity, short coherence times, and sizable gate error rates. Thus, desired quantum algorithms should require shallow circuit depths and low qubit counts to take advantage of these devices. Here, we attempt to reduce quantum resource requirements for molecular simulations on a quantum computer while maintaining the desired accuracy with the help of classical quantum chemical theories of canonical transformation and explicit correlation. In this work, compact ab initio Hamiltonians are generated classically, in the second quantized form, through an approximate similarity transformation of the Hamiltonian with (a) an explicitly correlated two-body unitary operator with generalized pair excitations that remove the Coulombic electron-electron singularities from the Hamiltonian and (b) a unitary one-body operator to efficiently capture the orbital relaxation effects required for accurate description of the excited states. The resulting transcorrelated Hamiltonians are able to describe both the ground and the excited states of molecular systems in a balanced manner. Using the variational quantum eigensolver (VQE) method based on the unitary coupled cluster with singles and doubles (UCCSD) ansatz and only a minimal basis set (ANO-RCC-MB), we demonstrate that the transcorrelated Hamiltonians can produce ground state energies comparable to the reference CCSD energies with the much larger cc-pVTZ basis set. This leads to a reduction in the number of required CNOT gates by more than 3 orders of magnitude for the chemical species studied in this work. Furthermore, using the quantum equation of motion (qEOM) formalism in conjunction with the transcorrelated Hamiltonian, we are able to reduce the deviations in the excitation energies from the reference EOM-CCSD/cc-pVTZ values by an order of magnitude. The transcorrelated Hamiltonians developed here are Hermitian and contain only one- and two-body interaction terms and thus can be easily combined with any quantum algorithm for accurate electronic structure simulations.

Post-Density Matrix Renormalization Group Methods for Describing Dynamic Electron Correlation with Large Active Spaces

The ab initio density matrix renormalization group (DMRG) method has been well-established and has become one of the most accurate numerical methods for the precise electronic structure solution of large active spaces. In the past few years, to capture the missing dynamic correlation, various post-DMRG approaches have been proposed through the combination of DMRG and multireference quantum chemical methods or density functional theory. With this in mind, this work provides a brief overview of ab initio DMRG principles and the new developments within post-DMRG methods. For clarity, post-DMRG methods are classified into two main categories depending on whether high-order n-electron reduced density matrices are used, and their merits and disadvantages are properly discussed. Finally, we conclude by discussing unsolved bottlenecks and giving development perspectives of post-DMRG approaches, which are expected to yield quantitative descriptions of complex electronic structures in large strongly correlated molecules and materials.

Quantum-classical hybrid algorithm for the simulation of all-electron correlation

While chemical systems containing hundreds to thousands of electrons remain beyond the reach of quantum devices, hybrid quantum-classical algorithms present a promising pathway toward a quantum advantage. Hybrid algorithms treat the exponentially scaling part of the calculation-the static correlation-on the quantum computer and the non-exponentially scaling part-the dynamic correlation-on the classical computer. While a variety of algorithms have been proposed, the dependence of many methods on the total wave function limits the development of easy-to-use classical post-processing implementations. Here, we present a novel combination of quantum and classical algorithms, which computes the all-electron energy of a strongly correlated molecular system on the classical computer from the 2-electron reduced density matrix (2-RDM) evaluated on the quantum device. Significantly, we circumvent the wave function in the all-electron calculations by using density matrix methods that only require input of the statically correlated 2-RDM. Although the algorithm is completely general, we test it with two classical density matrix methods, the anti-Hermitian contracted Schrödinger equation (ACSE) and multiconfiguration pair-density functional theories, using the recently developed quantum ACSE method for simulating the statically correlated 2-RDM. We obtain experimental accuracy for the relative energies of all three benzyne isomers and thereby demonstrate the ability of the developed algorithm to achieve chemically relevant and accurate results on noisy intermediate-scale quantum devices.

Modern multireference methods and their application in transition metal chemistry

Transition metal chemistry is a challenging playground for quantum chemical methods owing to the simultaneous presence of static and dynamic electron correlation effects in many systems. Wavefunction based multireference (MR) methods constitute a physically sound and systematically improvable Ansatz to deal with this complexity but suffer from some conceptual difficulties and high computational costs. The latter problem partially arises from the unfavorable scaling of the Full Configuration Interaction (Full-CI) problem which in the majority of MR methods is solved for a subset of the molecular orbital space, the so-called active space. In the last years multiple methods such as modern variants of selected CI, Full-CI Quantum Monte Carlo (FCIQMC) and the density matrix renormalization group (DMRG) have been developed that solve the Full-CI problem approximately for a fraction of the computational cost required by conventional techniques thereby significantly extending the range of applicability of modern MR methods. This perspective review outlines recent advancements in the field of MR electronic structure methods together with the resulting chances and challenges for theoretical studies in the field of transition metal chemistry. In light of its emerging importance a special focus is put on the selection of adequate active spaces and the concomitant development of numerous selection aides in recent years.

Taming the Sign Problem in Auxiliary-Field Quantum Monte Carlo Using Accurate Wave Functions

We explore different ways of incorporating accurate trial wave functions into free projection auxiliary-field quantum Monte Carlo (fp-AFQMC). States employed include coupled-cluster singles and doubles, multi-Slater, and symmetry-projected mean-field wave functions. We adapt a recently proposed fast multi-Slater local energy evaluation algorithm for fp-AFQMC, making the use of long expansions from selected configuration interaction methods feasible. We demonstrate how these wave functions serve to mitigate the sign problem and accelerate convergence in quantum chemical problems, allowing the application of fp-AFQMC to systems of substantial sizes. Our calculations on the widely studied model Cu₂O₂²⁺ system show that many previously reported isomerization energies differ substantially from the near-exact fp-AFQMC value.

A framework for constructing linear free energy relationships to design molecular transition metal catalysts

A computational framework for ligand-driven design of transition metal complexes is presented in this work. We propose a general procedure for the construction of active site-specific linear free energy relationships (LFERs), which are inspired from Hammett and Taft correlations in organic chemistry and grounded in the activation strain model (ASM). Ligand effects are isolated and quantified in terms of their contribution to interaction and strain energy components of ASM. Scalar descriptors that are easily obtainable are then employed to construct the complete LFER. We successfully demonstrate proof-of-concept by constructing and applying an LFER to CH activation with enzyme-inspired [Cu2O2]2+ complexes. The key benefit of using ASM is a built-in compensation or error cancellation between LFER prediction of interaction and strain terms, resulting in accurate barrier predictions for 37 of the 47 catalysts examined in this study. The LFER is also transferable with respect to level of theory and flexible towards the choice of reference system. The absence of interaction-strain compensation or poor model performance for the remaining systems is a consequence of the approximate nature of the chosen interaction energy descriptor and LFER construction of the strain term, which focuses largely on trends in substrate and not catalyst strain.

Toward an Accurate Ab Initio Description of Low-Lying Singlet Excited States of Polyenes

The low-lying excited states of carotenoids play a crucial role in many important biophysical processes such as photosynthesis. Most of these excited states are strongly correlated, which makes them both challenging for a qualitative ab initio description and an engaging model system for trying out emerging multireference methods. Among these methods, driven similarity renormalization group (DSRG) and its perturbative version (DSRG-MRPT2) are especially attractive in terms of both accuracy and moderate numerical complexity. In this paper, we applied density matrix renormalization group (DMRG) followed by DSRG-MRPT2 for the calculation of vertical and adiabatic excitation energies into the 2A_g^-, 1B_u^-, and 1B_u⁺ electronic states of polyenes containing from 8 to 13 conjugating double bonds acting as a model for natural carotenoids. It was shown that the DSRG flow parameter should be adjusted to ensure both the energy convergence with respect to it and the agreement with the experimental data. With the increased flow parameter, the proposed combination of methods provides a reasonable agreement with the experiment. The deviations of the adiabatic excitation energies are less than 1000 cm^-1 for the 2A_g^- and less than 3000 cm^-1 for the excited states of the B_u symmetry, which in terms of accuracy significantly outperforms the N-electron valence state perturbation theory. At the same time, DSRG-MRPT2 is shown to be robust with respect to variation of quality of the DMRG reference wave function such as the orbital optimization or the number of electronic states in the averaging.

Externally Corrected CCSD with Renormalized Perturbative Triples (R-ecCCSD(T)) and the Density Matrix Renormalization Group and Selected Configuration Interaction External Sources

We investigate the renormalized perturbative triples correction together with the externally corrected coupled-cluster singles and doubles (ecCCSD) method. We use the density matrix renormalization group (DMRG) and heat-bath CI (HCI) as external sources for the ecCCSD equations. The accuracy is assessed for the potential energy surfaces of H₂O, N₂, and F₂. We find that the triples correction significantly improves upon ecCCSD, and we do not see any instability of the renormalized triples with respect to dissociation. We explore how to balance the cost of computing the external source amplitudes against the accuracy of the subsequent CC calculation. In this context, we find that very approximate wave functions (and their large amplitudes) serve as an efficient and accurate external source. Finally, we characterize the domain of correlation treatable using the ecCCSD and renormalized triples combination studied in this work via a well-known wave function diagnostic.

Approximations of density matrices in N-electron valence state second-order perturbation theory (NEVPT2). II. The full rank NEVPT2 (FR-NEVPT2) formulation

In Paper I, the performances of pre-screening (PS), extended PS (EPS), and cumulant (CU) approximations to the fourth-order density matrix were examined in the context of second-order N-electron valence state perturbation theory (NEVPT2). It has been found that the CU, PS, and even EPS approximations with loose thresholds may introduce intruder states. In the present work, the origin of these "false intruder" states introduced by approximated density matrices is discussed. Canonical NEVPT2 implementations employ a rank reduction trick. By analyzing its residual error, we find that the omission of the rank reduction leads to a more stable multireference perturbation theory for incomplete active space reference wave functions. Such a full rank (FR)-NEVPT2 formulation is equivalent to the conventional NEVPT2 method for the complete active space self-consistent field/complete active space configuration interaction reference wave function. A major drawback of the FR-NEVPT2 formulation is the necessity of the fifth-order density matrix. To avoid the construction of the high-order density matrices, the combination of the FR-NEVPT2 with the CU approximation is studied. However, we find that the CU approximation remains problematic as it still introduces intruder states. The question of how to robustly and efficiently perform internally contracted multireference perturbation theories with approximate densities remains a challenging field of investigation.

Spectroscopy and Scattering Studies Using Interpolated Ab Initio Potentials

The Born-Oppenheimer potential energy surface (PES) has come a long way since its introduction in the 1920s, both conceptually and in predictive power for practical applications. Nevertheless, nearly 100 years later-despite astonishing advances in computational power-the state-of-the-art first-principles prediction of observables related to spectroscopy and scattering dynamics is surprisingly limited. For example, the water dimer, (H₂O)₂, with only six nuclei and 20 electrons, still presents a formidable challenge for full-dimensional variational calculations of bound states and is considered out of reach for rigorous scattering calculations. The extremely poor scaling of the most rigorous quantum methods is fundamental; however, recent progress in development of approximate methodologies has opened the door to fairly routine high-quality predictions, unthinkable 20 years ago. In this review, in relation to the workflow of spectroscopy and/or scattering studies, we summarize progress and challenges in the component areas of electronic structure calculations, PES fitting, and quantum dynamical calculations.

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.

Portably parallel construction of a configuration-interaction wave function from a matrix-product state using the Charm++ framework

The construction of configuration-interaction (CI) expansions from a matrix product state (MPS) involves numerous matrix operations and the skillful sampling of important configurations in a large Hilbert space. In this work, we present an efficient procedure for constructing CI expansions from MPS employing the parallel object-oriented Charm++ programming framework, upon which automatic load-balancing and object migrating facilities can be employed. This procedure was employed in the MPS-to-CI utility (Moritz et al., J. Chem. Phys. 2007, 126, 224109), the sampling-reconstructed complete active-space algorithm (SR-CAS, Boguslawski et al., J. Chem. Phys. 2011, 134, 224101), and the entanglement-driven genetic algorithm (EDGA, Luo et al., J. Chem. Theory Comput. 2017, 13, 4699). It enhances productivity and allows the sampling programs to evolve to their population-expansion versions, for example, EDGA with population expansion (PE-EDGA). Further, examples of 1,2-dioxetanone and firefly dioxetanone anion (FDO^- ) molecules demonstrated the following: (a) parallel efficiencies can be persistently improved by simply by increasing the proportions of the asynchronous executions and (b) a sampled CAS-type CI wave function of a bi-radical-state FDO^- molecule utilizing the full valence (30e,26o) active space can be constructed within a few hours with using thousands of cores.

A multireference coupled-electron pair approximation combined with complete-active space perturbation theory in local pair-natural orbital framework

The Complete-Active Space Second-order Perturbation Theory (CASPT2) has been one of the most widely-used methods for reliably calculating electronic structures of multireference systems. Because of its lowest level treatment of dynamic correlation, it has a high computational feasibility; however, its accuracy in some cases falls short of needs. Here, as a simple yet higher-order alternative, we introduce a hybrid theory of the CASPT2 and a multireference variant of the Coupled-Electron Pair Approximation (CEPA), which is a class of high level correlation theory. A central feature of our theory (CEPT2) is to use the two underlying theories for describing different divisions of correlation components based on the full internal contraction framework. The external components, which usually give a major contribution to the dynamic correlation, are intensively described using the CEPA Ansatz, while the rests are treated at the CASPT2 level. Furthermore, to drastically reduce the computational demands, we have incorporated the pair-natural orbital (PNO) method into our multireference implementations. This development, thus, requires highly complex derivations and coding, while it has been largely facilitated with an automatic expression and code generation technique. To highlight the accuracy of the CEPT2 approach and to assess the errors caused by the PNO truncation, benchmark calculations are shown on small- to medium-size molecules, illustrating the high accuracy of the present CEPT2 model. By tightening the truncation thresholds, the PNO-CEPT2 energy converges toward the canonical counterpart and is more accurate than that of PNO-CASPT2 as long as the same truncation thresholds are used.

Elucidating the multi-configurational character of the firefly dioxetanone anion and its prototypes in the biradical region using full valence active spaces

We analyzed the near-degenerate states of the firefly dioxetanone anion (FDO-) and its prototypes, especially in the biradical region, using multi-configurational approaches. The importance of utilizing full valence active spaces by means of density-matrix renormalization group self-consistent field (DMRG-SCF) calculations was described. Our results revealed that the neglect of some valence orbitals can affect the quantitative accuracy in later multi-reference calculations or the qualitative conclusion when optimizing conical intersections. Using all of the relevant valence orbitals of FDO-, we confirmed that there were two conical intersections, as reported in previous work, and that the intersecting states were changed when the active space was enlarged. Beyond these, we found that there were strong interactions between states in the biradical regions, in which the changes in entanglements can be used to visualize the interacting state evolution.

Quantum simulation of electronic structure with a transcorrelated Hamiltonian: improved accuracy with a smaller footprint on the quantum computer

Molecular quantum computing simulations are currently limited by the use of minimal Gaussian bases, a problem we overcome using a canonical transcorrelated Hamiltonian to accelerate basis convergence, with unitary coupled cluster as an example.

Downfolding of many-body Hamiltonians using active-space models: Extension of the sub-system embedding sub-algebras approach to unitary coupled cluster formalisms

In this paper, we discuss the extension of the recently introduced subsystem embedding subalgebra coupled cluster (SES-CC) formalism to unitary CC formalisms. In analogy to the standard single-reference SES-CC formalism, its unitary CC extension allows one to include the dynamical (outside the active space) correlation effects in an SES induced complete active space (CAS) effective Hamiltonian. In contrast to the standard single-reference SES-CC theory, the unitary CC approach results in a Hermitian form of the effective Hamiltonian. Additionally, for the double unitary CC (DUCC) formalism, the corresponding CAS eigenvalue problem provides a rigorous separation of external cluster amplitudes that describe dynamical correlation effects-used to define the effective Hamiltonian-from those corresponding to the internal (inside the active space) excitations that define the components of eigenvectors associated with the energy of the entire system. The proposed formalism can be viewed as an efficient way of downfolding many-electron Hamiltonian to the low-energy model represented by a particular choice of CAS. In principle, this technique can be extended to any type of CAS representing an arbitrary energy window of a quantum system. The Hermitian character of low-dimensional effective Hamiltonians makes them an ideal target for several types of full configuration interaction type eigensolvers. As an example, we also discuss the algebraic form of the perturbative expansions of the effective DUCC Hamiltonians corresponding to composite unitary CC theories and discuss possible algorithms for hybrid classical and quantum computing. Given growing interest in quantum computing, we provide energies for H₂ and Be systems obtained with the quantum phase estimator algorithm available in the Quantum Development Kit for the approximate DUCC Hamiltonians.

Second-order MCSCF optimization revisited. I. Improved algorithms for fast and robust second-order CASSCF convergence

A new improved implementation of the second-order multiconfiguration self-consistent field optimization method of Werner and Knowles [J. Chem. Phys. 82, 5053 (1985)] is presented. It differs from the original method by more stable and efficient algorithms for minimizing the second-order energy approximation in the so-called microiterations. Conventionally, this proceeds by alternating optimizations of the orbitals and configuration (CI) coefficients and is linearly convergent. The most difficult part is the orbital optimization, which requires solving a system of nonlinear equations that are often strongly coupled. We present a much improved algorithm for solving this problem, using an iterative subspace method that includes part of the orbital Hessian explicitly, and discuss different strategies for performing the uncoupled optimization in a most efficient manner. Second, we present a new solver in which the orbital-CI coupling is treated explicitly. This leads to quadratic convergence of the microiterations but requires many additional evaluations of reduced (transition) density matrices. In difficult optimization problems with a strong coupling of the orbitals and CI coefficients, it leads to much improved convergence of both the macroiterations and the microiterations. Third, the orbital-CI coupling is treated approximately using a quasi-Newton approach with Broyden-Fletcher-Goldfarb-Shanno updates of the orbital Hessian. It is demonstrated that this converges almost as well as the explicitly coupled method but avoids the additional effort for computing many transition density matrices. The performance of the three methods is compared for a set of 21 aromatic molecules, an Fe(ii)-porphine transition metal complex, as well as for the [Cu₂O₂(NH₃)  ₆]²⁺, FeCl₃, Co₂(CO)₆C₂H₂, and Al₄O₂ complexes. In all cases, faster and more stable convergence than with the original implementation is achieved.

Can large active-space CASSCF calculation make sense to the reaction analysis of iron complex? A benchmark study of methane oxidation reaction by FeO. J Comput Chem 2019;40:414-420. [PMID: 30351477 DOI: 10.1002/jcc.25640]

A methane oxidation reaction by FeO⁺ cation was theoretically investigated based on the density functional theory (DFT) and the complete active-space self-consistent field (CASSCF) method as well as the coupled-cluster singles, doubles, and perturbative triples (CCSD(T)) to explore the active-space dependency to computational analyses in such strongly correlated reaction systems. A small active-space CASSCF(5e in 5o) calculation, which only includes five 3d orbitals of the Fe atom in the active-space, showed remarkable difference both in energy and geometry compared to those computed by the DFT and CCSD(T) methods. Interestingly, a large active-space CASSCF(17e in 17o) calculation, which includes almost all the valence orbitals gives a qualitative agreement with either the DFT or the CCSD(T) results in the first half part of the reaction, although it varies from them in the latter half part. Therefore, it is indicated that the active-space dependency is serious in some part of the reaction and the small active-space CASSCF might lead a wrong discussion. We further investigated the optimized geometry of the intermediate complex with the small and the large active-space CASSCF methods as well as the CCSD(T) method, and found that the CASSCF(5e in 5o)-optimized geometry is considerably different from the others. In consequence, a small active-space CASSCF/CASPT2 calculation does not really work for such a strongly correlated reaction system even qualitatively, and a sophisticated assessment using the large active-space CASSCF/CASPT2 method will be indispensable. © 2018 Wiley Periodicals, Inc.

Extending spin-symmetry projected coupled-cluster to large model spaces using an iterative null-space projection technique

Recently, we introduced an orbital-invariant approximate coupled-cluster (CC) method in the spin-projection manifold. The multi-determinantal property of spin-projection means that the parametrization in the spin-extended CC (ECC) ansatz is nonorthogonal and overcomplete. Therefore, the linear dependencies must be removed by an orthogonalization procedure to obtain meaningful solutions. Multi-reference methods often achieve this by diagonalizing a metric of the equation system, but this is not feasible with ECC because of the enormous size of the metric, a consequence of the incomplete active space of the spin-projected Hartree-Fock reference. As a result, the applicability of ECC has been limited to small benchmark systems, for which the ansatz was shown to be superior to the configuration interaction and linearized approximations. In this article, we provide a solution to this problem that completely avoids the metric diagonalization by iteratively projecting out its null-space from the working equations. As the additional computational cost required for this iterative projection is only marginal, it greatly expands the application range of ECC. We demonstrate the potential of approximate ECC by studying the complete basis set limit of F₂ and transition metal complexes such as NiO, Mn₂ , and [Cu₂ O₂ ]²⁺ , which have all been hindered by the prohibitively large metric size. We also identify the potential inadequacy of the molecular orbitals given by spin-projected Hartree-Fock in some cases, and propose possible solutions. © 2018 Wiley Periodicals, Inc.

Computational strategies to probe CH activation in dioxo-dicopper complexes

We employ density functional theory and energy decomposition analysis to probe the mechanism of CH activation in dioxo-dicopper complexes. The electrophilicity of monodentate N-donor ligands coordinated to Cu is systematically varied to examine the response of barriers to the two proposed pathways - one-step oxo-insertion and two-step radical recombination. Electron-withdrawing ligand stabilize the oxo-insertion transition state via charge transfer interactions, and therefore lead to lower barriers. On the other hand, barriers to the CH activation step in the radical recombination mechanism exhibit almost no dependence on N-donor electrophilicity. Based on the similarities between calculated and experimental Hammett relationships, the oxo-insertion pathway appears to be the preferred mechanism of CH activation in dioxo-dicopper catalysts.

Exchange Coupling Interactions from the Density Matrix Renormalization Group and N-Electron Valence Perturbation Theory: Application to a Biomimetic Mixed-Valence Manganese Complex

The accurate description of magnetic level energetics in oligonuclear exchange-coupled transition-metal complexes remains a formidable challenge for quantum chemistry. The density matrix renormalization group (DMRG) brings such systems for the first time easily within reach of multireference wave function methods by enabling the use of unprecedentedly large active spaces. But does this guarantee systematic improvement in predictive ability and, if so, under which conditions? We identify operational parameters in the use of DMRG using as a test system an experimentally characterized mixed-valence bis-μ-oxo/μ-acetato Mn(III,IV) dimer, a model for the oxygen-evolving complex of photosystem II. A complete active space of all metal 3d and bridge 2p orbitals proved to be the smallest meaningful starting point; this is readily accessible with DMRG and greatly improves on the unrealistic metal-only configuration interaction or complete active space self-consistent field (CASSCF) values. Orbital optimization is critical for stabilizing the antiferromagnetic state, while a state-averaged approach over all spin states involved is required to avoid artificial deviations from isotropic behavior that are associated with state-specific calculations. Selective inclusion of localized orbital subspaces enables probing the relative contributions of different ligands and distinct superexchange pathways. Overall, however, full-valence DMRG-CASSCF calculations fall short of providing a quantitative description of the exchange coupling owing to insufficient recovery of dynamic correlation. Quantitatively accurate results can be achieved through a DMRG implementation of second order N-electron valence perturbation theory (NEVPT2) in conjunction with a full-valence metal and ligand active space. Perspectives for future applications of DMRG-CASSCF/NEVPT2 to exchange coupling in oligonuclear clusters are discussed.

Analytic Energy Gradients for Variational Two-Electron Reduced-Density-Matrix-Driven Complete Active Space Self-Consistent Field Theory

Analytic energy gradients are presented for a variational two-electron reduced-density-matrix (2-RDM)-driven complete active space self-consistent field (CASSCF) method. The active-space 2-RDM is determined using a semidefinite programing (SDP) algorithm built upon an augmented Lagrangian formalism. Expressions for analytic gradients are simplified by the fact that the Lagrangian is stationary with respect to variations in both the primal and the dual solutions to the SDP problem. Orbital response contributions to the gradient are identical to those that arise in conventional CASSCF methods in which the electronic structure of the active space is described by a full configuration interaction (CI) wave function. We explore the relative performance of variational 2-RDM (v2RDM)- and CI-driven CASSCF for the equilibrium geometries of 20 small molecules. When enforcing two-particle N-representability conditions, full-valence v2RDM-CASSCF-optimized bond lengths display a mean unsigned error of 0.0060 Å and a maximum unsigned error of 0.0265 Å, relative to those obtained from full-valence CI-CASSCF. When enforcing partial three-particle N-representability conditions, the mean and maximum unsigned errors are reduced to only 0.0006 and 0.0054 Å, respectively. For these same molecules, full-valence v2RDM-CASSCF bond lengths computed in the cc-pVQZ basis set deviate from experimentally determined ones on average by 0.017 and 0.011 Å when enforcing two- and three-particle conditions, respectively, whereas CI-CASSCF displays an average deviation of 0.010 Å. The v2RDM-CASSCF approach with two-particle conditions is also applied to the equilibrium geometry of pentacene; optimized bond lengths deviate from those derived from experiment, on average, by 0.015 Å when using a cc-pVDZ basis set and a (22e,22o) active space.

Perspective: Quantum mechanical methods in biochemistry and biophysics

In this perspective article, I discuss several research topics relevant to quantum mechanical (QM) methods in biophysical and biochemical applications. Due to the immense complexity of biological problems, the key is to develop methods that are able to strike the proper balance of computational efficiency and accuracy for the problem of interest. Therefore, in addition to the development of novel ab initio and density functional theory based QM methods for the study of reactive events that involve complex motifs such as transition metal clusters in metalloenzymes, it is equally important to develop inexpensive QM methods and advanced classical or quantal force fields to describe different physicochemical properties of biomolecules and their behaviors in complex environments. Maintaining a solid connection of these more approximate methods with rigorous QM methods is essential to their transferability and robustness. Comparison to diverse experimental observables helps validate computational models and mechanistic hypotheses as well as driving further development of computational methodologies.

Multireference Perturbation Theory with Cholesky Decomposition for the Density Matrix Renormalization Group

We present a second-order N-electron valence state perturbation theory (NEVPT2) based on a density matrix renormalization group (DMRG) reference wave function that exploits a Cholesky decomposition of the two-electron repulsion integrals (CD-DMRG-NEVPT2). With a parameter-free multireference perturbation theory approach at hand, the latter allows us to efficiently describe static and dynamic correlation in large molecular systems. We demonstrate the applicability of CD-DMRG-NEVPT2 for spin-state energetics of spin-crossover complexes involving calculations with more than 1000 atomic basis functions. We first assess, in a study of a heme model, the accuracy of the strongly and partially contracted variant of CD-DMRG-NEVPT2 before embarking on resolving a controversy about the spin ground state of a cobalt tropocoronand complex.