Permutation symmetry in spin-adapted many-body wave functions.

In the domain of exchange-coupled polynuclear transition-metal (PNTM) clusters, local emergent symmetries exist which can be exploited to greatly increase the sparsity of the configuration interaction (CI) eigensolutions of such systems. Sparsity of the CI secular problem is revealed by exploring the site permutation space within spin-adapted many-body bases, and highly compressed wave functions may arise by finding optimal site orderings. However, the factorial cost of searching through the permutation space remains a bottleneck for clusters with a large number of metal centers. In this work, we explore ways to reduce the factorial scaling, by combining permutation and point group symmetry arguments, and using commutation relations between cumulative partial spin and the Hamiltonian operators, . Certain site orderings lead to commuting operators, from which more sparse wave functions arise. Two graphical strategies will be discussed, one to rapidly evaluate the commutators of interest, and one in the form of a tree search algorithm to predict how many and which distinct site permutations are to be analyzed, eliminating redundancies in the permutation space. Particularly interesting is the case of the singlet spin states for which an additional reversal symmetry can be utilized to further reduce the number of distinct site permutations.

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.

Magnetic Interactions in a [Co(II)3Er(III)(OR)4] Model Cubane through Forefront Multiconfigurational Methods.

Strong electron correlation effects are one of the major challenges in modern quantum chemistry. Polynuclear transition metal clusters are peculiar examples of systems featuring such forms of electron correlation. Multireference strategies, often based on but not limited to the concept of complete active space, are adopted to accurately account for strong electron correlation and to resolve their complex electronic structures. However, transition metal clusters already containing four magnetic centers with multiple unpaired electrons make conventional active space based strategies prohibitively expensive, due to their unfavorable scaling with the size of the active space. In this work, forefront techniques, such as density matrix renormalization group (DMRG), full configuration interaction quantum Monte Carlo (FCIQMC), and multiconfiguration pair-density functional theory (MCPDFT), are employed to overcome the computational limitation of conventional multireference approaches and to accurately investigate the magnetic interactions taking place in a [Co(II)3Er(III)(OR)4] (chemical formula [Co(II)3Er(III)(hmp)4(µ2-OAc)2(OH)3(H2O)], hmp = 2-(hydroxymethyl)-pyridine) model cubane water oxidation catalyst. Complete active spaces with up to 56 electrons in 56 orbitals have been constructed for the seven energetically lowest different spin states. Relative energies, local spin, and spin-spin correlation values are reported and provide crucial insights on the spin interactions for this model system, pivotal in the rationalization of the catalytic activity of this system in the water-splitting reaction. A ferromagnetic ground state is found with a very small, ∼50 cm-1, highest-to-lowest spin gap. Moreover, for the energetically lowest states, S = 3-6, the three Co(II) sites exhibit parallel aligned spins, and for the lower states, S = 0-2, two Co(II) sites retain strong parallel spin alignment.

Resolution of Electronic States in Heisenberg Cluster Models within the Unitary Group Approach.

In this work ground and excited electronic states of Heisenberg cluster models, in the form of configuration interaction many-body wave functions, are characterized within the spin-adapted Graphical Unitary Group Approach framework, and relying on a novel combined unitary and symmetric group approach. Finite-size cluster models of well-defined point-group symmetry and of general local-spin Slocal>12 are presented, including J1-J2 triangular and tetrahedral clusters, which are often used to describe magnetic interactions in biological and biomimetic polynuclear transition metal clusters with unique catalytic activity, such as nitrogen fixation and photosynthesis. We show that a unique block-diagonal structure of the underlying Hamiltonian matrix in the spin-adapted basis emerges when an optimal lattice site ordering is chosen that reflects the internal symmetries of the model investigated. The block-diagonal structure is bound to the commutation relations between cumulative spin operators and the Hamiltonian operator, that in turn depend on the geometry of the cluster investigated. The many-body basis transformation, in the form of the orbital/site reordering, exposes such commutation relations. These commutation relations represent a rigorous and formal demonstration of the block-diagonal structure in Hamiltonian matrices and the compression of the corresponding spin-adapted many-body wave functions. As a direct consequence of the block-diagonal structure of the Hamiltonian matrix, it is possible to selectively optimize electronic excited states without the overhead of calculating the lower-energy states by simply relying on the initial ansatz for the targeted wave function. Additionally, more compact many-body wave functions are obtained. In extreme cases, electronic states are precisely described by a single configuration state function, despite the curse of dimensionality of the corresponding Hilbert space. These findings are crucial in the electronic structure theory framework, for they offer a conceptual route toward wave functions of reduced multireference character, that can be optimized more easily by approximated eigensolvers and are of more facile physical interpretation. They open the way to study larger ab initio and model Hamiltonians of increasingly larger number of correlated electrons, while keeping the computational costs at their lowest. In particular, these elements will expand the potential of electronic structure methods in understanding magnetic interactions in exchange-coupled polynuclear transition metal clusters.

FCIQMC-Tailored Distinguishable Cluster Approach: Open-Shell Systems.

A recently proposed tailored approach based on the distinguishable cluster method and the stochastic FCI solver, FCIQMC [J. Chem. Theory Comput. 2020, 16, 5621], is extended to open-shell molecular systems. The method is employed to calculate spin gaps of various Fe(II) complexes, including a Fe(II) porphyrin model system. Both distinguishable cluster and fully relaxed CASSCF natural orbitals were used in this work as reference for the subsequent tailored distinguishable cluster calculations. The distinguishable cluster natural orbitals occupation numbers were also used as an aid to the selection of the active space. The effect of the active space sizes and of the explicit correlation correction (F12) onto the predicted spin gaps is investigated. The tailored distinguishable cluster with singles and doubles yields consistently more accurate results compared to the tailored coupled cluster with singles and doubles.

Spin Purification in Full-CI Quantum Monte Carlo via a First-Order Penalty Approach.

In this article, we demonstrate that a first-order spin penalty scheme can be efficiently applied to the Slater determinant based Full-CI Quantum Monte Carlo (FCIQMC) algorithm, as a practical route toward spin purification. Two crucial applications are presented to demonstrate the validity and robustness of this scheme: the 1Δg ← 3Σg vertical excitation in O2 and key spin gaps in a [Mn3(IV)O4] cluster. In the absence of a robust spin adaptation/purification technique, both applications would be unattainable by Slater determinant based ground state methods, with any starting wave function collapsing into the higher-spin ground state during the optimization. This strategy can be coupled to other algorithms that use the Slater determinant based FCIQMC algorithm as configuration interaction eigensolver, including the Stochastic Generalized Active Space, the similarity-transformed FCIQMC, the tailored-CC, and second-order perturbation theory approaches. Moreover, in contrast to the GUGA-FCIQMC technique, this strategy features both spin projection and total spin adaptation, making it appealing when solving anisotropic Hamiltonians. It also provides spin-resolved reduced density matrices, important for the investigation of spin-dependent properties in polynuclear transition metal clusters, such as the hyperfine-coupling constants.

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 3Eg over the 5A1g 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.

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.

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.

Quenched Lewis Acidity: Studies on the Medium Dependent Fluorescence of Zinc(II) Complexes.

Three new zinc(II) coordination units [Zn(1-3)] based on planar-directing tetradentate Schiff base-like ligands H2 (1-3) were synthesized. Their solid-state structures were investigated by single crystal X-ray diffraction, showing the tendency to overcome the square-planar coordination sphere by axial ligation. Affinity in solution towards axial ligation has been tested by extended spectroscopic studies, both in the absorption and emission mode. The electronic spectrum of the pyridine complex [Zn(1)(py)] has been characterized by MC-PDFT to validate the results of extended TD-DFT studies. Green emission of non-emissive solutions of [Zn(1-3)] in chloroform could be switched on in the presence of potent Lewis-bases. While interpretation in terms of an equilibrium of stacked/non-fluorescent and destacked/fluorescent species is in line with precedents from literature, the sensitivity of [Zn(1-3)] was greatly reduced. Results of a computation-based structure search allow to trace the hidden Lewis acidity of [Zn(1-3)] to a new stacking motif, resulting in a strongly enhanced stability of the dimers.

Resolution of Low-Energy States in Spin-Exchange Transition-Metal Clusters: Case Study of Singlet States in [Fe(III)4S4] Cubanes.

Polynuclear transition-metal (PNTM) clusters owe their catalytic activity to numerous energetically low-lying spin states and stable oxidation states. The characterization of their electronic structure represents one of the greatest challenges of modern chemistry. We propose a theoretical framework that enables the resolution of targeted electronic states with ease and apply it to two [Fe(III)4S4] cubanes. Through direct access to their many-body wave functions, we identify important correlation mechanisms and their interplay with the geometrical distortions observed in these clusters, which are core properties in understanding their catalytic activity. The simulated magnetic coupling constants predicted by our strategy allow us to make qualitative connections between spin interactions and geometrical distortions, demonstrating its predictive power. Moreover, despite its simplicity, the strategy provides magnetic coupling constants in good agreement with the available experimental ones. The complexes are intrinsically frustrated anti-ferromagnets, and the obtained spin structures together with the geometrical distortions represent two possible ways to release spin frustration (spin-driven Jahn-Teller distortion). Our paradigm provides a simple, yet rigorous, route to uncover the electronic structure of PNTM clusters and may be applied to a wide variety of such clusters.

Compression of Spin-Adapted Multiconfigurational Wave Functions in Exchange-Coupled Polynuclear Spin Systems.

We present a protocol based on unitary transformations of molecular orbitals to reduce the number of nonvanishing coefficients of spin-adapted configuration interaction expansions. Methods that exploit the sparsity of the Hamiltonian matrix and compactness of its eigensolutions, such as the full configuration interaction quantum Monte Carlo (FCIQMC) algorithm in its spin-adapted implementation, are well suited to this protocol. The wave function compression resulting from this approach is particularly attractive for antiferromagnetically coupled polynuclear spin systems, such as transition-metal cubanes in biocatalysis, and Mott and charge-transfer insulators in solid-state physics. Active space configuration interaction calculations on N2 and CN- at various bond lengths, the stretched square N4 compounds, the chromium dimer, and a [Fe2S2]2- model system are presented as a proof-of-concept. For the Cr2 case, large and intermediate bond distances are discussed, showing that the approach is effective in cases where static and dynamic correlations are equally important. The [Fe2S2]2- case shows the general applicability of the method.

OpenMolcas: From Source Code to Insight.

In this Article we describe the OpenMolcas environment and invite the computational chemistry community to collaborate. The open-source project already includes a large number of new developments realized during the transition from the commercial MOLCAS product to the open-source platform. The paper initially describes the technical details of the new software development platform. This is followed by brief presentations of many new methods, implementations, and features of the OpenMolcas program suite. These developments include novel wave function methods such as stochastic complete active space self-consistent field, density matrix renormalization group (DMRG) methods, and hybrid multiconfigurational wave function and density functional theory models. Some of these implementations include an array of additional options and functionalities. The paper proceeds and describes developments related to explorations of potential energy surfaces. Here we present methods for the optimization of conical intersections, the simulation of adiabatic and nonadiabatic molecular dynamics, and interfaces to tools for semiclassical and quantum mechanical nuclear dynamics. Furthermore, the Article describes features unique to simulations of spectroscopic and magnetic phenomena such as the exact semiclassical description of the interaction between light and matter, various X-ray processes, magnetic circular dichroism, and properties. Finally, the paper describes a number of built-in and add-on features to support the OpenMolcas platform with postcalculation analysis and visualization, a multiscale simulation option using frozen-density embedding theory, and new electronic and muonic basis sets.

Role of Valence and Semicore Electron Correlation on Spin Gaps in Fe(II)-Porphyrins.

The role of valence and semicore correlation in differentially stabilizing the intermediate spin state of Fe(II)-porphyrins is analyzed. For CASSCF treatments of valence correlation, a (32,34) active space containing metal 3 d, d' orbitals and the entire π system of the porphyrin is necessary to stabilize the intermediate spin state. Semicore correlation provides a minor (-1.6 kcal/mol) but quantitatively significant correction. Accounting for valence, semicore, and correlation beyond the active space enlarges the (3 E g-5 A1 g) spin gap to -5.7 kcal/mol.

Understanding the Mechanism Stabilizing Intermediate Spin States in Fe(II)-Porphyrin.

Spin fluctuations in Fe(II)-porphyrins are at the heart of heme-proteins functionality. Despite significant progress in porphyrin chemistry, the mechanisms that rule spin state stabilization remain elusive. Here, it is demonstrated by using multiconfigurational quantum chemical approaches, including the novel Stochastic-CASSCF method, that electron delocalization between the metal center and the π system of the macrocycle differentially stabilizes the triplet spin states over the quintet. This delocalization takes place via charge-transfer excitations, involving the π system of the macrocycle and the out-of-plane iron d orbitals, key linking orbitals between metal and macrocycle. Through a correlated breathing mechanism the 3d electrons can make transitions toward the π orbitals of the macrocycle. This guarantees a strong coupling between the on-site radial correlation on the metal and electron delocalization. Opposite-spin 3d electrons of the triplet can effectively reduce electron repulsion in this manner. Constraining the out-of-plane orbitals from breathing hinders delocalization and reverses the spin ordering. The breathing mechanism is made effective by strong electron correlation effects in the π system of the macrocycle. Reducing the correlation treatment on the macrocycle to second-order only also reverses the spin ordering. High order (beyond second-order) correlation on the macrocycle reduces the energetic cost of the additional electron to a sufficient extent to stabilize the triplet state. Our results find a qualitative analogue in six resonance structures involving the metal center in the Fe2+ and Fe3+ oxidation states.

Multiconfiguration Pair-Density Functional Theory: A New Way To Treat Strongly Correlated Systems.

The electronic energy of a system provides the Born-Oppenheimer potential energy for internuclear motion and thus determines molecular structure and spectra, bond energies, conformational energies, reaction barrier heights, and vibrational frequencies. The development of more efficient and more accurate ways to calculate the electronic energy of systems with inherently multiconfigurational electronic structure is essential for many applications, including transition metal and actinide chemistry, systems with partially broken bonds, many transition states, and most electronically excited states. Inherently multiconfigurational systems are called strongly correlated systems or multireference systems, where the latter name refers to the need for using more than one ("multiple") configuration state function to provide a good zero-order reference wave function. This Account describes multiconfiguration pair-density functional theory (MC-PDFT), which was developed as a way to combine the advantages of wave function theory (WFT) and density functional theory (DFT) to provide a better treatment of strongly correlated systems. First we review background material: the widely used Kohn-Sham DFT (which uses only a single Slater determinant as reference wave function), multiconfiguration WFT methods that treat inherently multiconfigurational systems based on an active space, and previous attempts to combine multiconfiguration WFT with DFT. Then we review the formulation of MC-PDFT. It is a generalization of Kohn-Sham DFT in that the electron kinetic energy and classical electrostatic energy are calculated from a reference wave function, while the rest of the energy is obtained from a density functional. However, there are two main differences with respent to Kohn-Sham DFT: (i) The reference wave function is multiconfigurational rather than being a single Slater determinant. (ii) The density functional is a function of the total density and the on-top pair density rather than being a function of the spin-up and spin-down densities. In work carried out so far, the multiconfigurational wave function is a multiconfiguration self-consistent-field wave function. The new formulation has the advantage that the reference wave function has the correct spatial and spin symmetry and can describe bond dissociation (of both single and multiple bonds) and electronic excitations in a formally and physically correct way. We then review the formulation of density functionals in terms of the on-top pair density. Finally we review successful applications of the theory to bond energies and bond dissociation potential energy curves of main-group and transition metal bonds, to barrier heights (including pericyclic reactions), to proton affinities, to the hydrogen bond energy of water dimer, to ground- and excited-state charge transfer, to valence and Rydberg excitations of molecules, and to singlet-triplet splittings of radicals. We find that that MC-PDFT can give accurate results not only with complete-active-space multiconfiguration wave functions but also with generalized-active-space multiconfiguration wave functions, which are practical for larger numbers of active electrons and active orbitals than are complete-active-space wave functions. The separated-pair approximation, which is a special case of generalized active space self-consistent-field theory, is especially promising. MC-PDFT, because it requires much less computer time and storage than pure WFT methods, has the potential to open larger and more complex strongly correlated systems to accurate simulation.

Second-Order Perturbation Theory for Generalized Active Space Self-Consistent-Field Wave Functions.

A multireference second-order perturbation theory approach based on the generalized active space self-consistent-field (GASSCF) wave function is presented. Compared with the complete active space (CAS) and restricted active space (RAS) wave functions, GAS wave functions are more flexible and can employ larger active spaces and/or different truncations of the configuration interaction expansion. With GASSCF, one can explore chemical systems that are not affordable with either CASSCF or RASSCF. Perturbation theory to second order on top of GAS wave functions (GASPT2) has been implemented to recover the remaining electron correlation. The method has been benchmarked by computing the chromium dimer ground-state potential energy curve. These calculations show that GASPT2 gives results similar to CASPT2 even with a configuration interaction expansion much smaller than the corresponding CAS expansion.

Comment on "Fe2: As simple as a Herculean labour. Neutral (Fe2), cationic (Fe2(+)), and anionic (Fe2(-)) species" [J. Chem. Phys. 142, 244304 (2015)].

A recent paper on Fe2 [A. Kalemos, J. Chem. Phys. 142, 244304 (2015)] critiqued our previous work on the system [Hoyer et al., J. Chem. Phys. 141, 204309 (2014)]. In this comment, we explain the nature of our previously reported potential energy curve for Fe2 and we discuss our computed properties for Fe2. Additionally, we fix a labeling error that was present in our previous work, although this error is unrelated to the main point of discussion.

Combining the Complete Active Space Self-Consistent Field Method and the Full Configuration Interaction Quantum Monte Carlo within a Super-CI Framework, with Application to Challenging Metal-Porphyrins.

A novel stochastic Complete Active Space Self-Consistent Field (CASSCF) method has been developed and implemented in the Molcas software package. A two-step procedure is used, in which the CAS configuration interaction secular equations are solved stochastically with the Full Configuration Interaction Quantum Monte Carlo (FCIQMC) approach, while orbital rotations are performed using an approximated form of the Super-CI method. This new method does not suffer from the strong combinatorial limitations of standard MCSCF implementations using direct schemes and can handle active spaces well in excess of those accessible to traditional CASSCF approaches. The density matrix formulation of the Super-CI method makes this step independent of the size of the CI expansion, depending exclusively on one- and two-body density matrices with indices restricted to the relatively small number of active orbitals. No sigma vectors need to be stored in memory for the FCIQMC eigensolver--a substantial gain in comparison to implementations using the Davidson method, which require three or more vectors of the size of the CI expansion. Further, no orbital Hessian is computed, circumventing limitations on basis set expansions. Like the parent FCIQMC method, the present technique is scalable on massively parallel architectures. We present in this report the method and its application to the free-base porphyrin, Mg(II) porphyrin, and Fe(II) porphyrin. In the present study, active spaces up to 32 electrons and 29 orbitals in orbital expansions containing up to 916 contracted functions are treated with modest computational resources. Results are quite promising even without accounting for the correlation outside the active space. The systems here presented clearly demonstrate that large CASSCF calculations are possible via FCIQMC-CASSCF without limitations on basis set size.