Regularized CASPT2: an Intruder-State-Free Approach

Perturbation theory in the complete degenerate active space (CDAS-PT2)

Methods based on the multireference perturbation theory (MRPT) with the one-electron zeroth-order Hamiltonian are widely used for the description of excited states, for example, due to their relatively low computational cost. However, current methods have a common drawback-use of a model space with low size. In this article, we propose the MRPT method with the model space extended to the complete active space. The one-electron zeroth-order Hamiltonian suitable for this extension is formulated. The proposed method was applied to common models, such as LiF, ethylene, and trans-butadiene. It was shown to have accuracy superior to XMCQDPT2 in most cases, especially in the case of the small active space.

Reference Energies for Double Excitations: Improvement and Extension

In the realm of photochemistry, the significance of double excitations (also known as doubly excited states), where two electrons are concurrently elevated to higher energy levels, lies in their involvement in key electronic transitions essential in light-induced chemical reactions as well as their challenging nature from the computational theoretical chemistry point of view. Based on state-of-the-art electronic structure methods (such as high-order coupled-cluster, selected configuration interaction, and multiconfigurational methods), we improve and expand our prior set of accurate reference excitation energies for electronic states exhibiting a substantial amount of double excitations [Loos et al. J. Chem. Theory Comput. 2019, 15, 1939]. This extended collection encompasses 47 electronic transitions across 26 molecular systems that we separate into two distinct subsets: (i) 28 "genuine" doubly excited states where the transitions almost exclusively involve doubly excited configurations and (ii) 19 "partial" doubly excited states which exhibit a more balanced character between singly and doubly excited configurations. For each subset, we assess the performance of high-order coupled-cluster (CC3, CCSDT, CC4, and CCSDTQ) and multiconfigurational methods (CASPT2, CASPT3, PC-NEVPT2, and SC-NEVPT2). Using as a probe the percentage of single excitations involved in a given transition (%T1) computed at the CC3 level, we also propose a simple correction that reduces the errors of CC3 by a factor of 3, for both sets of excitations. We hope that this more complete and diverse compilation of double excitations will help future developments of electronic excited-state methodologies.

Perspective: multi-configurational methods in bio-inorganic chemistry

Transition metal ions play crucial roles in the structure and function of numerous proteins, contributing to essential biological processes such as catalysis, electron transfer, and oxygen binding. However, accurately modeling the electronic structure and properties of metalloproteins poses significant challenges due to the complex nature of their electronic configurations and strong correlation effects. Multiconfigurational quantum chemistry methods are, in principle, the most appropriate tools for addressing these challenges, offering the capability to capture the inherent multi-reference character and strong electron correlation present in bio-inorganic systems. Yet their computational cost has long hindered wider adoption, making methods such as density functional theory (DFT) the method of choice. However, advancements over the past decade have substantially alleviated this limitation, rendering multiconfigurational quantum chemistry methods more accessible and applicable to a wider range of bio-inorganic systems. In this perspective, we discuss some of these developments and how they have already been used to answer some of the most important questions in bio-inorganic chemistry. We also comment on ongoing developments in the field and how the future of the field may evolve.

Understanding the Photoprocesses in Biological Systems: Need for Accurate Multireference Treatment

Light-matter interaction is crucial to life itself and revolves around many of the central processes in biology. The need for understanding these photochemical and photophysical processes cannot be overemphasized. Interaction of light with biological systems starts with the absorption of light and subsequent phenomena that occur in the excited states of the system. However, excited states are typically difficult to understand within the mean field approximation of quantum chemical methods. Therefore, suitable multireference methods and methodologies have been developed to understand these phenomena. In this Perspective, we will describe a few methods and methodologies suitable for these descriptions and discuss some persisting difficulties.

Analysis and Assessment of Knowles' Partitioning in Many-Body Perturbation Theory

A detailed analysis of a new partitioning in many-body perturbation theory recently proposed by Knowles (J. Chem. Phys. 156, 011101, 2022), termed "perturbation adapted partitioning" (PAPT), is presented. Level shift and orbital rotation effects are identified as gears of the zero-order Hamiltonian. These two components are examined separately, revealing that, in themselves, neither of the two is competitive with the combined effect. The success of PAPT can be attributed to determining a set of molecular orbitals and corresponding orbital energies that can systematically outperform the canonical orbitals and Koopmans' energy-based Møller-Plesset partitioning. The self-consistent version of the method is also tested in terms of energy and convergence. Previous numerical studies are further complemented with an application to an inherent multireference example and an investigation of van der Waals interaction energies. In addition, a rigorous mathematical analysis of the consequence of the linear dependence of projection functions on the solution of the Knowles' equations is provided.

New physical insights into the supporting subspace factorization of XMS-CASPT2 and generalization to multiple spin states via spin-free formulation

This paper introduces a spin-free formulation of the supporting subspace factorization [C. Song and T. J. Martínez, J. Chem. Phys. 149, 044108 (2018)], enabling a reduction in the computational scaling of the extended multi-state complete active space second-order perturbation (XMS-CASPT2) method for arbitrary spins. Compared to the original formulation that is defined in the spin orbitals and is limited to singlet states, the spin-free formulation in this work treats different spin states equivalently, thus naturally generalizing the idea beyond singlet states. In addition, we will present a new way of deriving the supporting subspace factorization with the purpose of understanding its physical interpretation. In this new derivation, we separate the sources that make CASPT2 difficult into the "same-site interactions" and "inter-site interactions." We will first show how the Kronecker sum can be used to remove the same-site interactions in the absence of inter-site interactions, leading to MP2 energy in dressed orbitals. We will then show how the inter-site interactions can be exactly recovered using Löwdin partition, where the supporting subspace concept will naturally arise. The new spin-free formulation maintains the main advantage of the supporting subspace factorization, i.e., allowing XMS-CASPT2 energies to be computed using highly optimized MP2 energy codes and Fock build codes, thus reducing the scaling of XMS-CASPT2 to the same scaling as MP2. We will present and discuss results that benchmark the accuracy and performance of the new method. To demonstrate how the new method can be useful in studying real photochemical systems, the supporting subspace XMS-CASPT2 is applied to a photoreaction sensitive to magnetic field effects. The new spin-free formulation makes it possible to calculate the doublet and quartet states required in this particular photoreaction mechanism.

Toward a Stochastic Complete Active Space Second-Order Perturbation Theory

In this work, an internally contracted stochastic complete active space second-order perturbation theory, stochastic-CASPT2, is reported. The method relies on stochastically sampled reduced density matrices (RDMs) up to rank four and contractions thereof with the generalized Fock matrix. A new protocol for calculating higher-order RDMs in full configuration interaction quantum Monte Carlo (FCIQMC) has been designed based on (1) restricting sampling of the corresponding excitations to a deterministic subspace, (2) averaging the RDMs from independent dynamics and (3) projecting them onto the closest positive semi-definite matrix. Our protocol avoids previously encountered numerical conditioning problems in the orthogonalization of the perturber overlap matrix stemming from numerical noise. The chromium dimer CASSCF(12,12)/CASPT2 binding curve is computed as a proof of concept.

Auger-Meitner and X-ray Absorption Spectra of Ethylene Cation: Insight into Conical Intersection Dynamics

We present a theoretical investigation of the near-edge X-ray absorption fine structure and the Auger-Meitner decay spectra of ethylene and its cation. Herein, we demonstrate that our method, coupled with the nuclear ensemble approach, successfully reproduces the natural bandwidth structure of the experimental resonant Auger-Meitner decay spectra of ethylene, which is not very well reproduced within the Franck-Condon approximation. Furthermore, we analyze the Auger-Meitner decay spectra of the ethylene cation in light of minimum energy conical intersection structures involving the two lowest cationic states (D1 and D0), providing valuable insights into the ultrafast D1/D0 relaxation dynamics. Our results suggest that Auger-Meitner electron spectroscopy can help elucidate the mechanism behind the initial 20 fs of the relaxation dynamics.

Performance Tests of the Second-Order Approximate Internally Contracted Multireference Coupled-Cluster Singles and Doubles Method icMRCC2

Benchmark results are presented for the second-order approximation of the internally contracted multireference coupled-cluster method with single and double excitations, icMRCC2 [Köhn, Bargholz, J. Chem. Phys. 2019, 151, 041106], which was designed as a multireference analogue of the single-reference second-order approximate coupled-cluster method CC2 [Christiansen, Koch, Jørgensen, Chem. Phys. Lett. 1995, 243, 409-418]. Vertical excitation energies of various small to medium-sized organic molecules are investigated based on established test sets from the literature. Additionally, the spectroscopic constants of ground and excited states of diatomics and the geometric parameters of excited triatomic molecules were determined and compared to the experimental data. The results show that the method clearly extends the applicability of single-reference CC2, including doubly excited states, and also artifacts of CC2 like too low Rydberg excitations and too weak multiple bonds are eliminated. The method is computationally more demanding than standard multireference second-order perturbation theories but improves significantly in accuracy, as shown by the benchmark results. In addition, it is demonstrated that small active spaces are often sufficient to obtain accurate energies with icMRCC2. Example applications like the automerization of cyclobutadiene, the deactivation pathway of ethylene, and the excited states of an iron complex with a noninnocent nitrosyl ligand demonstrate the potential of icMRCC2 in cases with strong multireference character.

Galván I, Alavi A, Aleotti F, Aquilante F, Autschbach J, Avagliano D, Baiardi A, Bao JJ, Battaglia S, Birnoschi L, Blanco-González A, Bokarev SI, Broer R, Cacciari R, Calio PB, Carlson RK, Carvalho Couto R, Cerdán L, Chibotaru LF, Chilton NF, Church JR, Conti I, Coriani S, Cuéllar-Zuquin J, Daoud RE, Dattani N, Decleva P, de Graaf C, Delcey M, De Vico L, Dobrautz W, Dong SS, Feng R, Ferré N, Filatov(Gulak) M, Gagliardi L, Garavelli M, González L, Guan Y, Guo M, Hennefarth MR, Hermes MR, Hoyer CE, Huix-Rotllant M, Jaiswal VK, Kaiser A, Kaliakin DS, Khamesian M, King DS, Kochetov V, Krośnicki M, Kumaar AA, Larsson ED, Lehtola S, Lepetit MB, Lischka H, López Ríos P, Lundberg M, Ma D, Mai S, Marquetand P, Merritt ICD, Montorsi F, Mörchen M, Nenov A, Nguyen VHA, Nishimoto Y, Oakley MS, Olivucci M, Oppel M, Padula D, Pandharkar R, Phung QM, Plasser F, Raggi G, Rebolini E, Reiher M, Rivalta I, Roca-Sanjuán D, Romig T, Safari AA, Sánchez-Mansilla A, Sand AM, Schapiro I, Scott TR, Segarra-Martí J, Segatta F, Sergentu DC, Sharma P, Shepard R, Shu Y, Staab JK, Straatsma TP, Sørensen LK, Tenorio BNC, Truhlar DG, Ungur L, Vacher M, Veryazov V, Voß TA, Weser O, Wu D, Yang X, Yarkony D, Zhou C, Zobel JP, Lindh R. The OpenMolcas Web: A Community-Driven Approach to Advancing Computational Chemistry

The developments of the open-source OpenMolcas chemistry software environment since spring 2020 are described, with a focus on novel functionalities accessible in the stable branch of the package or via interfaces with other packages. These developments span a wide range of topics in computational chemistry and are presented in thematic sections: electronic structure theory, electronic spectroscopy simulations, analytic gradients and molecular structure optimizations, ab initio molecular dynamics, and other new features. This report offers an overview of the chemical phenomena and processes OpenMolcas can address, while showing that OpenMolcas is an attractive platform for state-of-the-art atomistic computer simulations.

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.

A Regularized Second-Order Correlation Method from Green's Function Theory

We present a scalable single-particle framework to treat electronic correlation in molecules and materials motivated by Green's function theory. We derive a size-extensive Brillouin-Wigner perturbation theory from the single-particle Green's function by introducing the Goldstone self-energy. This new ground state correlation energy, referred to as Quasi-Particle MP2 theory (QPMP2), avoids the characteristic divergences present in both second-order Møller-Plesset perturbation theory and Coupled Cluster Singles and Doubles within the strongly correlated regime. We show that the exact ground state energy and properties of the Hubbard dimer are reproduced by QPMP2 and demonstrate the advantages of the approach for larger Hubbard models where the metal-to-insulator transition is qualitatively reproduced, contrasting with the complete failure of traditional methods. We apply this formalism to characteristic strongly correlated molecular systems and show that QPMP2 provides an efficient, size-consistent regularization of MP2.

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.

A Similarity Renormalization Group Approach to Green's Function Methods

The family of Green's function methods based on the GW approximation has gained popularity in the electronic structure theory thanks to its accuracy in weakly correlated systems combined with its cost-effectiveness. Despite this, self-consistent versions still pose challenges in terms of convergence. A recent study [Monino and Loos J. Chem. Phys. 2022, 156, 231101.] has linked these convergence issues to the intruder-state problem. In this work, a perturbative analysis of the similarity renormalization group (SRG) approach is performed on Green's function methods. The SRG formalism enables us to derive, from first-principles, the expression of a naturally static and Hermitian form of the self-energy that can be employed in quasiparticle self-consistent GW (qsGW) calculations. The resulting SRG-based regularized self-energy significantly accelerates the convergence of qsGW calculations, slightly improves the overall accuracy, and is straightforward to implement in existing code.

Analytic first-order derivatives of CASPT2 with IPEA shift

Complete active space second-order perturbation theory (CASPT2) is useful for accurately predicting properties of complex electronic structures, but it is well known that it systematically underestimates excitation energies. The underestimation can be corrected using the ionization potential-electron affinity (IPEA) shift. In this study, analytic first-order derivatives of CASPT2 with the IPEA shift are developed. CASPT2-IPEA is not invariant with respect to rotations among active molecular orbitals, and two additional constraint conditions are necessary in the CASPT2 Lagrangian to formulate analytic derivatives. The method developed here is applied to methylpyrimidine derivatives and cytosine, and minimum energy structures and conical intersections are located. By comparing energies relative to the closed-shell ground state, we find that the agreement with experiments and high-level calculations is indeed improved by the inclusion of the IPEA shift. The agreement of geometrical parameters with high-level calculations may also be improved in some cases.

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.

R2022: A DFT/MRCI Ansatz with Improved Performance for Double Excitations

A reformulation of the combined density functional theory and multireference configuration interaction method (DFT/MRCI) is presented. Expressions for ab initio matrix elements are used to derive correction terms for a new effective Hamiltonian. On the example of diatomic carbon, the correction terms are derived, focusing on the doubly excited ¹Δ_g state, which was problematic in previous formulations of the method, as were double excitations in general. The derivation shows that a splitting of the parameters for intra- and interorbital interactions is necessary for a concise description of the underlying physics. Results for ¹L_a and ¹L_b states in polyacenes and ¹A_u and ¹A_g states in mini-β-carotenoids suggest that the presented formulation is superior to former effective Hamiltonians. Furthermore, statistical analysis reveals that all the benefits of the previous DFT/MRCI Hamiltonians are retained. Consequently, the here presented formulation should be considered as the new standard for DFT/MRCI calculations.

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.

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.