Systematic Expansion of Active Spaces beyond the CASSCF Limit: A GASSCF/SplitGAS Benchmark Study

Distinguishing homolytic vs heterolytic bond dissociation of phenylsulfonium cations with localized active space methods

Modeling chemical reactions with quantum chemical methods is challenging when the electronic structure varies significantly throughout the reaction and when electronic excited states are involved. Multireference methods, such as complete active space self-consistent field (CASSCF), can handle these multiconfigurational situations. However, even if the size of the needed active space is affordable, in many cases, the active space does not change consistently from reactant to product, causing discontinuities in the potential energy surface. The localized active space SCF (LASSCF) is a cheaper alternative to CASSCF for strongly correlated systems with weakly correlated fragments. The method is used for the first time to study a chemical reaction, namely the bond dissociation of a mono-, di-, and triphenylsulfonium cation. LASSCF calculations generate smooth potential energy scans more easily than the corresponding, more computationally expensive CASSCF calculations while predicting similar bond dissociation energies. Our calculations suggest a homolytic bond cleavage for di- and triphenylsulfonium and a heterolytic pathway for monophenylsulfonium.

Automatic State Interaction with Large Localized Active Spaces for Multimetallic Systems

The localized active space self-consistent field method factorizes a complete active space wave function into an antisymmetrized product of localized active space wave function fragments. Correlation between fragments is then reintroduced through localized active space state interaction (LASSI), in which the Hamiltonian is diagonalized in a model space of LAS states. However, the optimal procedure for defining the LAS fragments and LASSI model space is unknown. We here present an automated framework to explore systematically convergent sets of model spaces, which we call LASSI[r, q]. This method requires the user to select only r, the number of electron hops from one fragment to another, and q, the number of fragment basis functions per Hilbert space, which converges to CASCI in the limit of r, q → ∞. Numerical tests of this method on the trimetal oxo-centered complexes [Fe(III)Al(III)Fe(II)(μ3-O)(HCOO)6] and [Fe(III)2Fe(II)(μ3-O)(HCOO)6] show efficient convergence to the CASCI limit with 4-10 orders of magnitude fewer states than CASCI.

The overlooked role of excited anion states in NiO2- photodetachment

Photodetachment spectra of anionic species provide significant insights into the energies and nature of ground and excited states of both the anion and resultant neutral molecules. Direct detachment of the excess electron to the continuum may occur via formally allowed or forbidden transitions (perhaps as the result of intensity borrowing through vibronic coupling). However, alternate indirect pathways are also possible and often overlooked. Here, we report a two-dimensional photoelectron spectral study, combined with correlated electronic structure calculations, to elucidate the nature of photodetachment from NiO2-. The spectra are comprised of allowed and forbidden transitions, in excellent agreement with previously reported slow electron velocity mapped imaging spectra of the same system, which were interpreted in terms of direct detachment. In the current work, the contributions of indirect processes are revealed. Measured oscillations in the branching ratios of the spectral channels clearly indicate non-direct detachment processes, and the electronic structure calculations suggest that excited states of the appropriate symmetry and degeneracy lie slightly above the neutral ground state. Taken together, the results suggest that the origin of the observed forbidden transitions is the result of anion excited states mediating the electron detachment process.

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.

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.

Mn2 Dimers Encapsulated in Silicon Cages: A Complex Challenge to MC-SCF Theory

MC-SCF wavefunctions for three endohedral Mn/Si clusters, Mn2Si10, Mn2Si12, and [Mn2Si13]+, show evidence for strong static correlation, both in the Mn-Si bonds (’in–out correlation’) and between the two Mn centers (’up–down correlation’). We use both Restricted and Generalized Active Spaces (RAS and GAS) to place constraints on the configurations included in the trial wavefunction, showing that, particularly in the high-symmetry cases, the GAS approach captures more of the static correlation. The important correlating pairs are similar across the series, indicating that the electronic structure of the endohedral Mn2 unit is, to a first approximation, independent of the size of the silicon cage in which it is embedded.

Stochastic Generalized Active Space Self-Consistent Field: Theory and Application

An algorithm to perform stochastic generalized active space calculations, Stochastic-GAS, is presented, that uses the Slater determinant based FCIQMC algorithm as configuration interaction eigensolver. Stochastic-GAS allows the construction and stochastic optimization of preselected truncated configuration interaction wave functions, either to reduce the computational costs of large active space wave function optimizations, or to probe the role of specific electron correlation pathways. As for the conventional GAS procedure, the preselection of the truncated wave function is based on the selection of multiple active subspaces while imposing restrictions on the interspace excitations. Both local and cumulative minimum and maximum occupation number constraints are supported by Stochastic-GAS. The occupation number constraints are efficiently encoded in precomputed probability distributions, using the precomputed heat bath algorithm, which removes nearly all runtime overhead of GAS. This strategy effectively allows the FCIQMC dynamics to a priori exclude electronic configurations that are not allowed by GAS restrictions. Stochastic-GAS reduced density matrices are stochastically sampled, allowing orbital relaxations via Stochastic-GASSCF, and direct evaluation of properties that can be extracted from density matrices, such as the spin expectation value. Three test case applications have been chosen to demonstrate the flexibility of Stochastic-GAS: (a) the Stochastic-GASSCF [5·(6, 6)] optimization of a stack of five benzene molecules, that shows the applicability of Stochastic-GAS toward fragment-based chemical systems; (b) an uncontracted stochastic MRCISD calculation that correlates 96 electrons and 159 molecular orbitals, and uses a large (32, 34) active space reference wave function for an Fe(II)-porphyrin model system, showing how GAS can be applied to systematically recover dynamic electron correlation, and how in the specific case of the Fe(II)-porphyrin dynamic correlation further differentially stabilizes the ³E_g over the ⁵A_1g spin state; (c) the study of an Fe₄S₄ cluster's spin-ladder energetics via highly truncated stochastic-GAS [4·(5, 5)] wave functions, where we show how GAS can be applied to understand the competing spin-exchange and charge-transfer correlating mechanisms in stabilizing different spin-states.

Modeling magnetic interactions in high-valent trinuclear [Mn3(IV)O4]4+ complexes through highly compressed multi-configurational wave functions

In this work we apply a quantum chemical framework, recently designed in our laboratories, to rationalize the low-energy electronic spectrum and the magnetic properties of an homo-valent trinuclear [Mn₃^(IV)O₄]⁴⁺ model of the oxygen-evolving center in photosystem II. The method is based on chemically motivated molecular orbital unitary transformations, and the optimization of spin-adapted many-body wave functions, both for ground- and excited-states, in the transformed MO basis. In this basis, the configuration interaction Hamiltonian matrix of exchange-coupled multi-center clusters is extremely sparse and characterized by a unique block diagonal structure. This property leads to highly compressed wave functions (oligo- or single-reference) and crucially enables state-specific optimizations. This work is the first showing that compression and selective targeting of ground- and excited-states wave functions is possible for systems with three magnetic centers that are not exactly half-filled, and that potentially exhibit frustrated spin interactions. The reduced multi-reference character of the wave function greatly simplifies the interpretation of the ground- and excited-state electronic structures, and provides a route for the direct rationalization of magnetic interactions in these compounds, often considered a challenge in polynuclear transition-metal chemistry. In this study, strong electron correlation effects have explicitly been described by conventional and stochastic multiconfigurational methodologies, while dynamic correlation effects have been accounted for by multiconfigurational second order perturbation theory, CASPT2. Ab initio results for the [Mn₃^(IV)O₄]⁴⁺ system have been mapped to a three-site Heisenberg model with two magnetic coupling constants. The magnetic coupling constants and the temperature dependence of the effective magnetic moment predicted by the ab initio calculations are in good agreement with the available experimental data, and confirm the antiferromagnetic interaction among the three magnetic centers, while providing a simple and rigorous description of the noncollinearity of the local spins, that characterize most of the low-energy states for this system.

Spin-Pure Stochastic-CASSCF via GUGA-FCIQMC Applied to Iron-Sulfur Clusters

In this work, we demonstrate how to efficiently compute the one- and two-body reduced density matrices within the spin-adapted full configuration interaction quantum Monte Carlo (FCIQMC) method, which is based on the graphical unitary group approach (GUGA). This allows us to use GUGA-FCIQMC as a spin-pure configuration interaction (CI) eigensolver within the complete active space self-consistent field (CASSCF) procedure and hence to stochastically treat active spaces far larger than conventional CI solvers while variationally relaxing orbitals for specific spin-pure states. We apply the method to investigate the spin ladder in iron-sulfur dimer and tetramer model systems. We demonstrate the importance of the orbital relaxation by comparing the Heisenberg model magnetic coupling parameters from the CASSCF procedure to those from a CI-only (CASCI) procedure based on restricted open-shell Hartree-Fock orbitals. We show that the orbital relaxation differentially stabilizes the lower-spin states, thus enlarging the coupling parameters with respect to the values predicted by ignoring orbital relaxation effects. Moreover, we find that, while CASCI results are well fit by a simple bilinear Heisenberg Hamiltonian, the CASSCF eigenvalues exhibit deviations that necessitate the inclusion of biquadratic terms in the model Hamiltonian.

Model protein excited states: MRCI calculations with large active spaces vs CC2 method

Benchmarking calculations on excited states of models of phenylalanine protein chains are presented to assess the ability of alternative methods to the standard and most commonly used multiconfigurational wave function-based method, the complete active space self-consistent field (CASSCF), in recovering the non-dynamical correlation for systems that become not affordable by the CASSCF. The exploration of larger active spaces beyond the CASSCF limit is benchmarked through three strategies based on the reduction in the number of determinants: the restricted active space self-consistent field, the generalized active space self-consistent field (GASSCF), and the occupation-restricted multiple active space (ORMAS) schemes. The remaining dynamic correlation effects are then added by the complete active space second-order perturbation theory and by the multireference difference dedicated configuration interaction methods. In parallel, the approximate second-order coupled cluster (CC2), already proven to be successful for small building blocks of model proteins in one of our previous works [Ben Amor et al., J. Chem. Phys. 148, 184105 (2018)], is investigated to assess its performances for larger systems. Among the different alternative strategies to CASSCF, our results highlight the greatest efficiency of the GASSCF and ORMAS schemes in the systematic reduction of the configuration interaction expansion without loss of accuracy in both nature and excitation energies of both singlet ππ^* and nπ^* _CO excited states with respect to the equivalent CASSCF calculations. Guidelines for an optimum applicability of this scheme to systems requiring active spaces beyond the complete active space limit are then proposed. Finally, the extension of the CC2 method to such large systems without loss of accuracy is demonstrated, highlighting the great potential of this method to treat accurately excited states, mainly single reference, of very large systems.

iCAS: Imposed Automatic Selection and Localization of Complete Active Spaces

It is shown that in the spirit of "from fragments to molecule" for localizing molecular orbitals [J. Chem. Theory Comput. 2011, 7, 3643], a prechosen set of occupied/virtual valence/core atomic/fragmental orbitals can be transformed to an equivalent set of localized occupied/virtual pre-localized molecular orbitals (pre-LMO), which can then be taken as probes to select the same number of maximally matching localized occupied/virtual Hartree-Fock (HF) or restricted open-shell HF (ROHF) molecular orbitals as the initial local orbitals spanning the desired complete active space (CAS). In each cycle of the self-consistent field (SCF) calculation, the CASSCF orbitals can be localized by means of the noniterative "top-down least-change" algorithm for localizing ROHF orbitals [J. Chem. Phys. 2017, 146, 104104] such that the maximum matching between the orbitals of two adjacent iterations can readily be monitored, leading finally to converged localized CASSCF orbitals that overlap most the guess orbitals. Such an approach is to be dubbed as "imposed CASSCF" (iCASSCF or simply iCAS in short) for good reasons: (1) it has been assumed that only those electronic states that have largest projections onto the active space defined by the prechosen atomic/fragmental orbitals are to be targeted. This is certainly an imposed constraint but has wide applications in organic and transition metal chemistry where valence (or core) atomic/fragmental orbitals can readily be identified. (2) The selection of both initial and optimized local active orbitals is imposed from the very beginning by the pre-LMOs (which span the same space as the prechosen atomic/fragmental orbitals). Apart from the (imposed) automation and localization, iCAS has two additional merits: (a) the guess orbitals are guaranteed to be the same for all geometries, for the pre-LMOs do not change in character with geometry and (b) the use of localized orbitals facilitates the SCF convergence, particularly for large active spaces. Both organic molecules and transition-metal complexes are taken as showcases to reveal the efficacy of iCAS.

Near-Exact CASSCF-Level Geometry Optimization with a Large Active Space using Adaptive Sampling Configuration Interaction Self-Consistent Field Corrected with Second-Order Perturbation Theory (ASCI-SCF-PT2)

An accurate description of electron correlation is one of the most challenging problems in quantum chemistry. The exact electron correlation can be obtained by means of full configuration interaction (FCI). A simple strategy for approximating FCI at a reduced computational cost is selected CI (SCI), which diagonalizes the Hamiltonian within only the chosen configuration space. Recovery of the contributions of the remaining configurations is possible with second-order perturbation theory. Here, we apply adaptive sampling configuration interaction (ASCI) combined with molecular orbital optimizations (ASCI-SCF) corrected with second-order perturbation theory (ASCI-SCF-PT2) for geometry optimization by implementing the analytical nuclear gradient algorithm for ASCI-PT2 with the Z-vector (Lagrangian) formalism. We demonstrate that for phenalenyl radicals and anthracene, optimized geometries and the number of unpaired electrons can be obtained at nearly the CASSCF accuracy by incorporating PT2 corrections and extrapolating them. We demonstrate the current algorithm's utility for optimizing the equilibrium geometries and electronic structures of six-ring-fused polycyclic aromatic hydrocarbons and 4-periacene.

Open Shells in Endohedral Clusters: Structure and Bonding in the [Fe2@Ge16]4- Anion and Comparison to Isostructural [Co2@Ge16]4

The anionic cluster [Fe₂@Ge₁₆]^4- has been characterized and shown to be isostructural to the known D_2h-symmetric α isomer of the cobalt analogue [Co₂@Ge₁₆]^4-. Together with the known pair of compounds [Co@Ge₁₀]^3- and [Fe@Ge₁₀]^3-, the title compound completes a set of four closely related germanium clusters that allow us to explore how the metal-metal and metal-cage interactions evolve as a function of size and of the identity of the metal. The results of spin-unrestricted density functional theory (DFT) and multiconfigurational self-consistent field (MC-SCF) calculations present a consistent picture of the electronic structure where transfer of electron density from the metal to the cage is significant, particularly in the Fe clusters where the exchange stabilization of unpaired spin density is an important driving force.

Spin-Coupled Generalized Valence Bond Theory: New Perspectves on the Electronic Structure of Molecules and Chemical Bonds

Spin-Coupled Generalized Valence Bond (SCGVB) theory provides the foundation for a comprehensive theory of the electronic structure of molecules. SCGVB theory offers a compelling orbital description of the electronic structure of molecules as well as an efficient and effective zero-order wave function for calculations striving for quantitative predictions of molecular structures, energetics, and other properties. The orbitals in the SCGVB wave function are usually semilocalized, and for most molecules, they can be interpreted using concepts familiar to all chemists (hybrid orbitals, localized bond pairs, lone pairs, etc.). SCGVB theory also provides new perspectives on the nature of the bonds in molecules such as C₂, Be₂ and SF₄/SF₆. SCGVB theory contributes unparalleled insights into the underlying cause of the first-row anomaly in inorganic chemistry as well as the electronic structure of organic molecules and the electronic mechanisms of organic reactions. The SCGVB wave function accounts for nondynamical correlation effects and, thus, corrects the most serious deficiency in molecular orbital (RHF) wave functions. Dynamical correlation effects, which are critical for quantitative predictions, can be taken into account using the SCGVB wave function as the zero-order wave function for multireference configuration interaction or coupled cluster calculations.

Quantum Information and Algorithms for Correlated Quantum Matter

Discoveries in quantum materials, which are characterized by the strongly quantum-mechanical nature of electrons and atoms, have revealed exotic properties that arise from correlations. It is the promise of quantum materials for quantum information science superimposed with the potential of new computational quantum algorithms to discover new quantum materials that inspires this Review. We anticipate that quantum materials to be discovered and developed in the next years will transform the areas of quantum information processing including communication, storage, and computing. Simultaneously, efforts toward developing new quantum algorithmic approaches for quantum simulation and advanced calculation methods for many-body quantum systems enable major advances toward functional quantum materials and their deployment. The advent of quantum computing brings new possibilities for eliminating the exponential complexity that has stymied simulation of correlated quantum systems on high-performance classical computers. Here, we review new algorithms and computational approaches to predict and understand the behavior of correlated quantum matter. The strongly interdisciplinary nature of the topics covered necessitates a common language to integrate ideas from these fields. We aim to provide this common language while weaving together fields across electronic structure theory, quantum electrodynamics, algorithm design, and open quantum systems. Our Review is timely in presenting the state-of-the-art in the field toward algorithms with nonexponential complexity for correlated quantum matter with applications in grand-challenge problems. Looking to the future, at the intersection of quantum information science and algorithms for correlated quantum matter, we envision seminal advances in predicting many-body quantum states and describing excitonic quantum matter and large-scale entangled states, a better understanding of high-temperature superconductivity, and quantifying open quantum system dynamics.

Predicting Bond Dissociation Energies and Bond Lengths of Coordinatively Unsaturated Vanadium-Ligand Bonds

Understanding the electronic structure of coordinatively unsaturated transition-metal compounds and predicting their physical properties are of great importance for catalyst design. Bond dissociation energy D_e and bond length r_e are two of the fundamental quantities for which good predictions are important for a successful design strategy. In the present work, recent experimentally measured bond energies and bond lengths of VX diatomic molecules (X = C, N, S) are used as a gauge to consider the utility of a number of electronic structure methods. Single-reference methods are one focus because of their efficiency and utility in practical calculations, and multireference configuration interaction (MRCISD) methods and a composite coupled cluster (CCC) method are a second focus because of their potential high accuracy. The comparison is especially challenging because of the large multireference M diagnostics of these molecules, in the range 0.15-0.19. For the single-reference methods, Kohn-Sham density functional theory (KS-DFT) has been tested with a variety of approximate exchange-correlation functionals. Of these, MOHLYP provides the bond dissociation energies in best agreement with experiments, and BLYP provides the bond lengths that are in best agreement with experiments; but by requiring good performance for both the D_e and r_e of the vanadium compounds, MOHLYP, MN12-L, MGGA_MS1, MGGA_MS0, O3LYP, and M06-L are the most highly recommended functionals. The CCC calculations include up to connected pentuple excitations for the valence electrons and up to connected quadruple excitations for the core-valence terms; this results in highly accurate dissociation energies and good bond lengths. Averaged over the three molecules, the mean unsigned deviation of CCC bond energies from experimental ones is only 0.4 kcal/mol, demonstrating excellent convergence of theory and experiments.

Molecular Photochemistry: Recent Developments in Theory

Photochemistry is a fascinating branch of chemistry that is concerned with molecules and light. However, the importance of simulating light-induced processes is reflected also in fields as diverse as biology, material science, and medicine. This Minireview highlights recent progress achieved in theoretical chemistry to calculate electronically excited states of molecules and simulate their photoinduced dynamics, with the aim of reaching experimental accuracy. We focus on emergent methods and give selected examples that illustrate the progress in recent years towards predicting complex electronic structures with strong correlation, calculations on large molecules, describing multichromophoric systems, and simulating non-adiabatic molecular dynamics over long time scales, for molecules in the gas phase or in complex biological environments.

Computational Approach to Molecular Catalysis by 3d Transition Metals: Challenges and Opportunities

Computational chemistry provides a versatile toolbox for studying mechanistic details of catalytic reactions and holds promise to deliver practical strategies to enable the rational in silico catalyst design. The versatile reactivity and nontrivial electronic structure effects, common for systems based on 3d transition metals, introduce additional complexity that may represent a particular challenge to the standard computational strategies. In this review, we discuss the challenges and capabilities of modern electronic structure methods for studying the reaction mechanisms promoted by 3d transition metal molecular catalysts. Particular focus will be placed on the ways of addressing the multiconfigurational problem in electronic structure calculations and the role of expert bias in the practical utilization of the available methods. The development of density functionals designed to address transition metals is also discussed. Special emphasis is placed on the methods that account for solvation effects and the multicomponent nature of practical catalytic systems. This is followed by an overview of recent computational studies addressing the mechanistic complexity of catalytic processes by molecular catalysts based on 3d metals. Cases that involve noninnocent ligands, multicomponent reaction systems, metal-ligand and metal-metal cooperativity, as well as modeling complex catalytic systems such as metal-organic frameworks are presented. Conventionally, computational studies on catalytic mechanisms are heavily dependent on the chemical intuition and expert input of the researcher. Recent developments in advanced automated methods for reaction path analysis hold promise for eliminating such human-bias from computational catalysis studies. A brief overview of these approaches is presented in the final section of the review. The paper is closed with general concluding remarks.

Two-dimensional UV spectroscopy: a new insight into the structure and dynamics of biomolecules

Two-dimensional ultraviolet spectroscopy has the potential to deliver rich structural and dynamical information on biomolecules such as DNA and proteins.

Magnetic circular dichroism spectra of transition metal complexes calculated from restricted active space wavefunctions

Multiconfigurational restricted active space (RAS) self-consistent field (SCF) or configuration interaction (CI) approaches, augmented with a treatment of spin–orbit coupling by state interaction, were used to calculate the magnetic circular dichroism , , and/or for closed- and open-shell transition metal complexes.

Generalized Self-Energy Embedding Theory

Ab initio quantum chemistry calculations for systems with large active spaces are notoriously difficult and cannot be successfully tackled by standard methods. We generalize a Green's function QM/QM embedding method called self-energy embedding theory (SEET) that has the potential to be successfully employed to treat large active spaces. In generalized SEET, active orbitals are grouped into intersecting groups of a few orbitals, allowing us to perform multiple parallel calculations yielding results comparable to the full active-space treatment. We examine generalized SEET on a series of examples and discuss a hierarchy of systematically improvable approximations.

Separated-pair approximation and separated-pair pair-density functional theory

Multi-configuration pair-density functional theory (MC-PDFT) has proved to be a powerful way to combine the capabilities of multi-configuration self-consistent-field theory to represent the an electronic wave function with a highly efficient way to include dynamic correlation energy by density functional theory. All applications reported previously involved complete active space self-consistent-field (CASSCF) theory for the reference wave function. For treating large systems efficiently, it is necessary to ask whether good accuracy is retained when using less complete configuration interaction spaces. To answer this question, we present here calculations employing MC-PDFT with the separated pair (SP) approximation, which is a special case (defined in this article) of generalized active space self-consistent-field (GASSCF) theory in which no more than two orbitals are included in any GAS subspace and in which inter-subspace excitations are excluded. This special case of MC-PDFT will be called SP-PDFT. In SP-PDFT, the electronic kinetic energy and the classical Coulomb energy, the electronic density and its gradient, and the on-top pair density and its gradient are obtained from an SP approximation wave function; the electronic energy is then calculated from the first two of these quantities and an on-top density functional of the last four. The accuracy of the SP-PDFT method for predicting the structural properties and bond dissociation energies of twelve diatomic molecules and two triatomic molecules is compared to the SP approximation itself and to CASSCF, MC-PDFT based on CASSCF, CASSCF followed by second order perturbation theory (CASPT2), and Kohn-Sham density functional theory with the PBE exchange-correlation potential. We show that SP-PDFT reproduces the accuracy of MC-PDFT based on the corresponding CASSCF wave function for predicting C-H bond dissociation energies, the reaction barriers of pericyclic reactions and the properties of open-shell singlet systems, all at only a small fraction of the computational cost.