Active Space Selection Based on Natural Orbital Occupation Numbers from n-Electron Valence Perturbation Theory

Error Control and Automatic Detection of Reference Active Spaces in Many-Body Expanded Full Configuration Interaction

We present a wide-reaching revamp of the generalized many-body expanded full configuration interaction (MBE-FCI) method. First, we outline how to automatize the selection of reference active spaces, whereby the inherent bias introduced through a manual identification is reduced, also within the context of traditional complete active space methods. Second, we allow for the use of compact orbital clusters as expansion objects, which works to circumvent the unfavorable scaling with the number of orbitals included in the space complementary to the reference orbitals. Finally, we present a new algorithm for ensuring that many-body expansions can be efficiently terminated while conservatively accounting for resulting errors. These developments are all tested on a variety of molecular systems and different orbital representations to illustrate the abilities of our algorithm to produce correlation energies within predetermined error bounds, significantly broadening the overall applicability of the MBE-FCI method.

Strategies to Obtain Reliable Energy Landscapes from Embedded Multireference Correlated Wavefunction Methods for Surface Reactions

Embedded correlated wavefunction (ECW) theory is a powerful tool for studying ground- and excited-state reaction mechanisms and associated energetics in heterogeneous catalysis. Several factors are important to obtaining reliable ECW energies, critically the construction of consistent active spaces (ASs) along reaction pathways when using a multireference correlated wavefunction (CW) method that relies on a subset of orbital spaces in the configuration interaction expansion to account for static electron correlation, e.g., complete AS self-consistent field theory, in addition to the adequate partitioning of the system into a cluster and environment, as well as the choice of a suitable basis set and number of states included in excited-state simulations. Here, we conducted a series of systematic studies to develop best-practice guidelines for ground- and excited-state ECW theory simulations, utilizing the decomposition of NH3 on Pd(111) as an example. We determine that ECW theory results are relatively insensitive to cluster size, the aug-cc-pVDZ basis set provides an adequate compromise between computational complexity and accuracy, and that a fixed-clean-surface approximation holds well for the derivation of the embedding potential. Additionally, we demonstrate that a merging approach, which involves generating ASs from the molecular fragments at each configuration, is preferable to a creeping approach, which utilizes ASs from adjacent structures as an initial guess, for the generation of consistent potential energy curves involving open-d-shell metal surfaces, and, finally, we show that it is essential to include bands of excited states in their entirety when simulating excited-state reaction pathways.

The trials and triumphs of modelling X-ray absorption spectra of transition metal phthalocyanines

This study explores the electronic structure of Co, Fe and Mn phthalocyanines (TMPcs) as well as their perfluorinated counterparts through a series of electronic structure calculations utilizing multireference methods and by simulating their metal L-edge and ligand (nitrogen and fluorine) K-edge X-ray absorption spectra (XAS) in an angle-resolved manner. Simulations targeting different ground-state symmetries, where relevant, have been conducted to observe changes in the N K-edge lineshape. The applicability of the quasi-degenerate formulation of n-electron valence state perturbation theory (QD-NEVPT2) for L-edge X-ray absorption spectroscopy (XAS) is evaluated, alongside the use of a restricted active space (RAS) formalism to describe the final-state multiplets generated by L-shell X-ray processes. Our findings provide valuable insights into the electronic properties of TMPcs, in particular with respect to the effect of fluorination, and demonstrate the broad applicability of various formulations of NEVPT2 in spectral simulations. Moreover, this study highlights the utility of manual truncation of the configuration spaces in order to allow for large active orbital spaces in aforementioned calculations.

Physically Motivated Improvements of Variational Quantum Eigensolvers

The adaptive derivative-assembled pseudo-Trotter variational quantum eigensolver (ADAPT-VQE) has emerged as a pivotal promising approach for electronic structure challenges in quantum chemistry with noisy quantum devices. Nevertheless, to surmount existing technological constraints, this study endeavors to enhance ADAPT-VQE's efficacy. Leveraging insights from the electronic structure theory, we concentrate on optimizing state preparation without added computational burden and guiding ansatz expansion to yield more concise wave functions with expedited convergence toward exact solutions. These advancements culminate in shallower circuits and, as demonstrated, reduced measurement requirements. This research delineates these enhancements and assesses their performance across mono, di, and tridimensional arrangements of H4 models, as well as in the water molecule. Ultimately, this work attests to the viability of physically motivated strategies in fortifying ADAPT-VQE's efficiency, marking a significant stride in quantum chemistry simulations.

SCINE-Software for chemical interaction networks

The software for chemical interaction networks (SCINE) project aims at pushing the frontier of quantum chemical calculations on molecular structures to a new level. While calculations on individual structures as well as on simple relations between them have become routine in chemistry, new developments have pushed the frontier in the field to high-throughput calculations. Chemical relations may be created by a search for specific molecular properties in a molecular design attempt, or they can be defined by a set of elementary reaction steps that form a chemical reaction network. The software modules of SCINE have been designed to facilitate such studies. The features of the modules are (i) general applicability of the applied methodologies ranging from electronic structure (no restriction to specific elements of the periodic table) to microkinetic modeling (with little restrictions on molecularity), full modularity so that SCINE modules can also be applied as stand-alone programs or be exchanged for external software packages that fulfill a similar purpose (to increase options for computational campaigns and to provide alternatives in case of tasks that are hard or impossible to accomplish with certain programs), (ii) high stability and autonomous operations so that control and steering by an operator are as easy as possible, and (iii) easy embedding into complex heterogeneous environments for molecular structures taken individually or in the context of a reaction network. A graphical user interface unites all modules and ensures interoperability. All components of the software have been made available as open source and free of charge.

Chelate Complexes of 3d Transition Metal Ions─A Challenge for Electronic-Structure Methods?

Different electronic-structure methods were assessed for their ability to predict two important properties of the industrially relevant chelating agent nitrilotriacetic acid (NTA): its selectivity with respect to six different first-row transition metal ions and the spin-state energetics of its complex with Fe(III). The investigated methods encompassed density functional theory (DFT), the random phase approximation (RPA), coupled cluster (CC) theory, and the auxiliary-field quantum Monte Carlo (AFQMC) method, as well as the complete active space self-consistent field (CASSCF) method and the respective on-top methods: second-order N-electron valence state perturbation theory (NEVPT2) and multiconfiguration pair-density functional theory (MC-PDFT). Different strategies for selecting active spaces were explored, and the density matrix renormalization group (DMRG) approach was used to solve the largest active spaces. Despite somewhat ambiguous multi-reference diagnostics, most methods gave relatively good agreement with experimental data for the chemical reactions connected to the selectivity, which only involved transition-metal complexes in their high-spin state. CC methods yielded the highest accuracy followed by range-separated DFT and AFQMC. We discussed in detail that even higher accuracies can be obtained with NEVPT2, under the prerequisite that consistent active spaces along the entire chemical reaction can be selected, which was not the case for reactions involving Fe(III). A bigger challenge for electronic-structure methods was the prediction of the spin-state energetics, which additionally involved lower spin states that exhibited larger multi-reference diagnostics. Conceptually different, typically accurate methods ranging from CC theory via DMRG-NEVPT2 in combination with large active spaces to AFQMC agreed well that the high-spin state is energetically significantly favored over the other spin states. This was in contrast to most DFT functionals and RPA which yielded a smaller stabilization and some common DFT functionals and MC-PDFT even predicting the low-spin state to be energetically most favorable.

Frozen Natural Orbitals for the State-Averaged Driven Similarity Renormalization Group

We present a reduced-cost implementation of the state-averaged driven similarity renormalization group (SA-DSRG) based on the frozen natural orbital (FNO) approach. The natural orbitals (NOs) are obtained by diagonalizing the one-body reduced density matrix from SA-DSRG second-order perturbation theory (SA-DSRG-PT2). We consider three criteria to truncate the virtual NOs for the subsequent electron correlation treatment beyond SA-DSRG-PT2. An additive second-order correction is applied to the SA-DSRG Hamiltonian to reintroduce correlation effects from the discarded orbitals. The FNO SA-DSRG method is benchmarked on 35 small organic molecules in the QUEST database. When keeping 98-99% of the cumulative occupation numbers, the mean absolute error in the vertical transition energies due to FNO is less than 0.01 eV. Using the same FNO threshold, we observe a speedup of 9 times compared to the conventional SA-DSRG implementation for nickel carbonyl with a quadruple-ζ basis set. The FNO approach enables nonperturbative SA-DSRG computations on chloroiron corrole [FeCl(C19H11N4)] with more than 1000 basis functions, surpassing the current limit of a conventional implementation.

Identification, Characterization, and Electronic Structures of Interconvertible Cobalt-Oxygen TAML Intermediates

The reaction of Li[(TAML)CoIII]·3H2O (TAML = tetraamido macrocyclic tetraanionic ligand) with iodosylbenzene at 253 K in acetone in the presence of redox-innocent metal ions (Sc(OTf)3 and Y(OTf)3) or triflic acid affords a blue species 1, which is converted reversibly to a green species 2 upon cooling to 193 K. The electronic structures of 1 and 2 have been determined by combining advanced spectroscopic techniques (X-band electron paramagnetic resonance (EPR), electron nuclear double resonance (ENDOR), X-ray absorption spectroscopy/extended X-ray absorption fine structure (XAS/EXAFS), and magnetic circular dichroism (MCD)) with ab initio theoretical studies. Complex 1 is best represented as an S = 1/2 [(Sol)(TAML•+)CoIII---OH(LA)]- species (LA = Lewis/Brønsted acid and Sol = solvent), where an S = 1 Co(III) center is antiferromagnetically coupled to S = 1/2 TAML•+, which represents a one-electron oxidized TAML ligand. In contrast, complex 2, also with an S = 1/2 ground state, is found to be multiconfigurational with contributions of both the resonance forms [(H-TAML)CoIV═O(LA)]- and [(H-TAML•+)CoIII═O(LA)]-; H-TAML and H-TAML•+ represent the protonated forms of TAML and TAML•+ ligands, respectively. Thus, the interconversion of 1 and 2 is associated with a LA-associated tautomerization event, whereby H+ shifts from the terminal -OH group to TAML•+ with the concomitant formation of a terminal cobalt-oxo species possessing both singlet (SCo = 0) Co(III) and doublet (SCo = 1/2) Co(IV) characters. The reactivities of 1 and 2 at different temperatures have been investigated in oxygen atom transfer (OAT) and hydrogen atom transfer (HAT) reactions to compare the activation enthalpies and entropies of 1 and 2.

Small-Occupation Density Functional Correlation Energy Correction to Wave Function Approximations

In this work, we introduce a novel hybrid approach, termed WFT-soDFT, designed to seamlessly incorporate DFT correlation into wave function ansatzes. This is achieved through a partitioning of the orbital space, distinguishing between large and small natural occupation numbers associated with wave function theory (WFT) and DFT correlation, respectively. The method uses a novel criterion for partitioning the orbital space and mapping the electron density in natural orbitals with a small occupation with the correlation energy of fast electrons within the homogeneous electron gas. Central to our approach is the introduction of a separation parameter ν, the choice of the WFT approach, and the correlation functional. Here, we combine the RASCI wave function with hole and particle truncation with a local density correlation functional to only account for small-occupation correlation energy. We investigate the performance of the method in the study of small but challenging chemical systems, for which WFT-soDFT demonstrates notable improvements over pristine wave function calculations. These findings collectively highlight the potential of the WFT-soDFT approach as a computationally affordable strategy to improve the accuracy of WFT electronic structure calculations.

Molecular π-Orbital Construction for Non-Planar Conjugated Systems

We extend the π-orbital space (PiOS) method introduced for planar π-conjugated molecular systems [J. Chem. Theory Comput. 2019, 15, 1679] to also allow constructing efficient π-orbital active spaces for non-planar π-conjugated systems. We demonstrate the performance of this method with multiconfigurational and multireference calculations on prototypical non-planar π-systems: cycloacenes, short carbon nanotubes, various conformations of the 2,2-bipyridine anion, and C20 fullerenes.

A configuration-based heatbath-CI for spin-adapted multireference electronic structure calculations with large active spaces

This work reports on a spin-pure configuration-based implementation of the heatbath configuration interaction (HCI) algorithm for selective configuration interaction. Besides the obvious advantage of being spin-pure, the presented method combines the compactness of the configurational ansatz with the known efficiency of the HCI algorithm and a variety of algorithmic and conceptual ideas to achieve a high level of performance. In particular, through pruning of the selected configurational space after HCI selection by means of a more strict criterion, a more compact wavefunction representation is obtained. Moreover, the underlying logic of the method allows us to minimize the number of redundant matrix-matrix multiplications while making use of just-in-time compilation to achieve fast diagonalization of the Hamiltonian. The critical search for 2-electron connections within the configurational space is facilitated by a tree-based representation thereof as suggested previously by Gopal et al. Usage of a prefix-based parallelization and batching during the calculation of the PT2-correction leads to a good load balancing and significantly reduced memory requirements for these critical steps of the calculation. In this way, the need for a semistochastic approach to the PT2 correction is avoided even for large configurational spaces. Finally, several test-cases are discussed to demonstrate the strengths and weaknesses of the presented method.

Variational Active Space Selection with Multiconfiguration Pair-Density Functional Theory

The selection of an adequate set of active orbitals for modeling strongly correlated electronic states is difficult to automate because it is highly dependent on the states and molecule of interest. Although many approaches have shown some success, no single approach has worked well in all cases. In light of this, we present the "discrete variational selection" (DVS) approach to active space selection, in which one generates multiple trial wave functions from a diverse set of systematically constructed active spaces and then selects between these wave functions variationally. We apply this DVS approach to 207 vertical excitations of small-to-medium-sized organic and inorganic molecules (with 3 to 18 atoms) in the QUESTDB database by (i) constructing various sets of active space orbitals through diagonalization of parametrized operators and (ii) choosing the result with the lowest average energy among the states of interest. This approach proves ineffective when variationally selecting between wave functions using the density matrix renormalization group (DMRG) or complete active space self-consistent field (CASSCF) energy but is able to provide good results when variationally selecting between wave functions using the energy of the translated PBE (tPBE) functional from multiconfiguration pair-density functional theory (MC-PDFT). Applying this DVS-tPBE approach to selection among state-averaged DMRG wave functions, we obtain a mean unsigned error of only 0.17 eV using hybrid MC-PDFT. This result matches that of our previous benchmark without the need to filter out poor active spaces and with no further orbital optimization following active space selection of the SA-DMRG wave functions. Furthermore, we find that DVS-tPBE is able to robustly and effectively select between the new SA-DMRG wave functions and our previous SA-CASSCF results.

Automated, Consistent, and Even-Handed Selection of Active Orbital Spaces for Quantum Embedding

A widely used strategy to reduce the computational cost of quantum-chemical calculations is to partition the system into an active subsystem, which is the focus of the computational efforts, and an environment that is treated at a lower computational level. The system partitioning is mostly based on localized molecular orbitals. When reaction paths or energy differences are to be calculated, it is crucial to keep the orbital space consistent for all structures. Inconsistencies in orbital space can lead to unpredictable errors on the potential energy surface. While successful strategies to ensure this consistency have been established for organic and even metal-organic systems, these methods often fail for metal clusters or nanoparticles with a high density of near-degenerate and delocalized molecular orbitals. However, such systems are highly relevant for catalysis. Accurate yet feasible quantum-mechanical ab initio calculations are therefore highly desired. In this work, we present an approach based on the subsystem projected atomic orbital decomposition algorithm that allows us to ensure automated and consistent partitioning even for systems with delocalized and near-degenerate molecular orbitals and demonstrate the validity of this method for the binding energies of small molecules on transition-metal clusters.

High-Spin Blatter's Triradicals

Robust organic triradicals with high-spin quartet ground states provide promising applications in molecular magnets, spintronics, etc. In this context, a triradical based on Blatter's radical has been synthesized recently, having two low-lying non-degenerate doublet states with a quartet ground state. The traditional broken-symmetry (BS)-DFT computed doublet-quartet energy gaps are reported to be somewhat overestimated in comparison to the experimentally observed values. In this work, we have employed different ab initio methods on this prototypical system to obtain more accurate doublet-quartet energy gaps for this triradical. The spin-constraint broken-symmetry (CBS)-DFT method has been used to reduce the overestimation of energy gaps from BS-DFT. To address the issues of spin-contamination and the multireference nature of low-spin states affecting the DFT methods, we have computed the energy gaps using appropriately state-averaged CASSCF and NEVPT2 computations. Using a series of active spaces, our calculations are shown to provide quite accurate values in concordance with the experimentally observed results. Furthermore, we have proposed and modeled another two triradicals based on Blatter's radical, which are of interest for experimental synthesis and characterization. Our computations show that all these triradicals also have a quartet ground state with a similar energy difference between the excited doublet states.

Automated Active Space Selection with Dipole Moments

Multireference calculations can provide accurate information of systems with strong correlation, which have increasing importance in the development of new molecules and materials. However, selecting a suitable active space for multireference calculations is nontrivial, and the selection of an unsuitable active space can sometimes lead to results that are not physically meaningful. Active space selection often requires significant human input, and the selection that leads to reasonable results often goes beyond chemical intuition. In this work, we have developed and evaluated two protocols for automated selection of the active space for multireference calculations based on a simple physical observable, the dipole moment, for molecules with nonzero ground-state dipole moments. One protocol is based on the ground-state dipole moment, and the other is based on the excited-state dipole moments. To evaluate the protocols, we constructed a dataset of 1275 active spaces from 25 molecules, each with 51 active space sizes considered, and have mapped out the relationship between the active space, dipole moments, and vertical excitation energies. We have demonstrated that, within this dataset, our protocols allow one to choose among a number of accessible active spaces one that is likely to give reasonable vertical excitation energies, especially for the first three excitations, with no parameters manually decided by the user. We show that, with large active spaces removed from consideration, the accuracy is similar and the time-to-solution can be reduced by more than 10 fold. We also show that the protocols can be applied to potential energy surface scans and determining the spin states of transition metal oxides.

Corresponding Active Orbital Spaces along Chemical Reaction Paths

The accuracy of reaction energy profiles calculated with multiconfigurational electronic structure methods and corrected by multireference perturbation theory depends crucially on consistent active orbital spaces selected along the reaction path. However, it has been challenging to choose molecular orbitals that can be considered corresponding in different molecular structures. Here, we demonstrate how active orbital spaces can be selected consistently along reaction coordinates in a fully automatized way. The approach requires no structure interpolation between reactants and products. Instead, it emerges from a synergy of the Direct Orbital Selection orbital mapping ansatz combined with our fully automated active space selection algorithm autoCAS. We demonstrate our algorithm for the potential energy profile of the homolytic carbon-carbon bond dissociation and rotation around the double bond of 1-pentene in the electronic ground state. However, our algorithm also applies to electronically excited Born-Oppenheimer surfaces.

Quantum computing in pharma: A multilayer embedding approach for near future applications

Quantum computers are special purpose machines that are expected to be particularly useful in simulating strongly correlated chemical systems. The quantum computer excels at treating a moderate number of orbitals within an active space in a fully quantum mechanical manner. We present a quantum phase estimation calculation on F₂ in a (2,2) active space on Rigetti's Aspen-11 QPU. While this is a promising start, it also underlines the need for carefully selecting the orbital spaces treated by the quantum computer. In this work, a scheme for selecting such an active space automatically is described and simulated results obtained using both the quantum phase estimation (QPE) and variational quantum eigensolver (VQE) algorithms are presented and combined with a subtractive method to enable accurate description of the environment. The active occupied space is selected from orbitals localized on the chemically relevant fragment of the molecule, while the corresponding virtual space is chosen based on the magnitude of interactions with the occupied space calculated from perturbation theory. This protocol is then applied to two chemical systems of pharmaceutical relevance: the enzyme [Fe] hydrogenase and the photosenzitizer temoporfin. While the sizes of the active spaces currently amenable to a quantum computational treatment are not enough to demonstrate quantum advantage, the procedure outlined here is applicable to any active space size, including those that are outside the reach of classical computation.

Large-Scale Benchmarking of Multireference Vertical-Excitation Calculations via Automated Active-Space Selection

We have calculated state-averaged complete-active-space self-consistent-field (SA-CASSCF), multiconfiguration pair-density functional theory (MC-PDFT), hybrid MC-PDFT (HMC-PDFT), and n-electron valence state second-order perturbation theory (NEVPT2) excitation energies with the approximate pair coefficient (APC) automated active-space selection scheme for the QUESTDB benchmark database of 542 vertical excitation energies. We eliminated poor active spaces (20-40% of calculations) by applying a threshold to the SA-CASSCF absolute error. With the remaining calculations, we find that NEVPT2 performance is significantly impacted by the size of the basis set the wave functions are converged in, regardless of the quality of their description, which is a problem absent in MC-PDFT. Additionally, we find that HMC-PDFT is a significant improvement over MC-PDFT with the translated PBE (tPBE) density functional and that it performs about as well as NEVPT2 and second-order coupled cluster on a set of 373 excitations in the QUESTDB database. We optimized the percentage of SA-CASSCF energy to include in HMC-PDFT when using the tPBE on-top functional, and we find the 25% value used in tPBE0 to be optimal. This work is by far the largest benchmarking of MC-PDFT and HMC-PDFT to date, and the data produced in this work are useful as a validation of HMC-PDFT and of the APC active-space selection scheme. We have made all the wave functions produced in this work (orbitals and CI vectors) available to the public and encourage the community to utilize this data as a tool in the development of further multireference model chemistries.

InteraChem: Exploring Excited States in Virtual Reality with Ab Initio Interactive Molecular Dynamics

InteraChem is an ab initio interactive molecular dynamics (AI-IMD) visualizer that leverages recent advances in virtual reality hardware and software, as well as the graphical processing unit (GPU)-accelerated TeraChem electronic structure package, in order to render quantum chemistry in real time. We introduce the exploration of electronically excited states via AI-IMD using the floating occupation molecular orbital-complete active space configuration interaction method. The optimization tools in InteraChem enable identification of excited state minima as well as minimum energy conical intersections for further characterization of excited state chemistry in small- to medium-sized systems. We demonstrate that finite-temperature Hartree-Fock theory is an efficient method to perform ground state AI-IMD. InteraChem allows users to track electronic properties such as molecular orbitals and bond order in real time, resulting in an interactive visualization tool that aids in the interpretation of excited state chemistry data and makes quantum chemistry more accessible for both research and educational purposes.

Tuning the magnetic properties of a diamagnetic di-Blatter's zwitterion to antiferro- and ferromagnetically coupled diradicals

In the quest of obtaining organic molecular magnets based on stable diradicals, we have tuned the inherent zwitterionic ground state of tetraphenylhexaazaanthracene (TPHA), a molecule containing two Blatter's moieties, by adopting two different strategies. In the first strategy, we have increased the length of the coupler between the two radical moieties and observed a transition from the zwitterionic ground state to the diradicalized state. With a larger coupler, ferromagnetic interactions are realized based on density functional theory (DFT) and wave-function theory (WFT) based complete active space self-consistent field (CASSCF)-N-electron valence state perturbation theory (NEVPT2) methods. An analysis based on the extent of spin contamination, diradical character, CASSCF orbital occupation number, Head-Gordon's index, HOMO-LUMO and SOMOs energy gaps is demonstrated that marks the transition of the ground state in these systems. In another approach, we systematically explore the effect of push-pull substitution on the way to obtain molecules based on a TPHA skeleton with diradicaloid state and, in some cases, even a triplet ground state.

Machine Learning-Assisted Selection of Active Spaces for Strongly Correlated Transition Metal Systems

Active space quantum chemical methods could provide very accurate description of strongly correlated electronic systems, which is of tremendous value for natural sciences. The proper choice of the active space is crucial but a nontrivial task. In this article, we present a neural network-based approach for automatic selection of active spaces, focused on transition metal systems. The training set has been formed from artificial systems composed of one transition metal and various ligands, on which we have performed the density matrix renormalization group and calculated the single-site entropy. On the selected set of systems, ranging from small benchmark molecules up to larger challenging systems involving two metallic centers, we demonstrate that our machine learning models could predict the active space orbitals with reasonable accuracy. We also tested the transferability on out-of-the-model systems, including bimetallic complexes and complexes with ligands, which were not involved in the training set. Also, we tested the correctness of the automatically selected active spaces on a Fe(II)-porphyrin model, where we studied the lowest states at the DMRG level and compared the energy difference between spin states or the energy difference between conformations of ferrocene with recent studies.

Magnetic exchange coupling in Cu dimers studied with modern multireference methods and broken-symmetry coupled cluster theory

Spin-state energetics of exchange-coupled copper complexes pose a persistent challenge for applied quantum chemistry. Here, we provide a comprehensive comparison of all available theoretical approaches to the problem of exchange coupling in two antiferromagnetically coupled bis-μ-hydroxo Cu(II) dimers. The evaluated methods include multireference methods based on the density matrix renormalization group (DMRG), multireference methods that incorporate dynamic electron correlation either perturbatively, such as the N-electron valence state perturbation theory, or variationally, such as the difference-dedicated configuration interaction. In addition, we contrast the multireference results with those obtained using broken-symmetry approaches that utilize either density functional theory or, as demonstrated here for the first time in such systems, a local implementation of coupled cluster theory. The results show that the spin-state energetics of these copper dimers are dominated by dynamic electron correlation and represent an impossible challenge for multireference methods that rely on brute-force expansion of the active space to recover correlation energy. Therefore, DMRG-based methods even at the limit of their applicability cannot describe quantitatively the antiferromagnetic exchange coupling in these dimers, in contrast to dinuclear complexes of earlier transition metal ions. The convergence of the broken-symmetry coupled cluster approach is studied and shown to be a limiting factor for the practical application of the method. The advantages and disadvantages of all approaches are discussed, and recommendations are made for future developments.

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.

Active orbital preservation for multiconfigurational self-consistent field

We introduce Active Orbital Preservation for Multiconfigurational Self-Consistent Field (AOP-MCSCF), an automated approach to improving the consistency of active space orbitals over multiple molecular configurations. Our approach is based on maximum overlap with a reference set of active space orbitals taken from a single geometry of a chromophore in the gas phase and can be used to automatically preserve the appropriate orbitals of the chromophore across multiple thermally sampled configurations, even when the chromophore is solvated by quantum-mechanically treated water molecules. In particular, using the singular value decomposition of a Molecular Orbital (MO) overlap matrix between the system and reference, we rotate the MOs of the system to align with the reference active space orbitals and use the resulting rotated orbitals as an initial guess to a MCSCF calculation. We demonstrate the approach on aqueous p-hydroxybenzylidene-imidazolinone (HBI) and find that AOP-MCSCF converges to the "correct" orbitals for over 90% of 3000 thermally sampled configurations. In addition, we compute the linear absorption spectrum and find excellent agreement with new experimental measurements up to 5.4 eV (230 nm). We show that electrostatic contributions to the solvation energy of HBI largely explain the observed state-dependent solvatochromism.

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.

Auxiliary Atomic Relay Center Facilitates Enhanced Magnetic Couplings in Blatter's Radical

The recent accomplishments in obtaining strong ferromagnetic exchange interactions in organic diradicals have made the field quite fascinating and even more promising toward its technological applications. In this context, herein, we report a unique combination of remarkably strong ferromagnetic exchange interactions coupled with molecular rigidity, utilizing superstable Blatter's radical as a spin source. The planar analogues of the parent Blatter's radical obtained by annulation with a chalcogen coupled to nitronyl nitroxide (NN) are investigated using density functional theory along with the wave function-based multiconfigurational self-consistent field methods, for example, complete active space self-consistent field (CASSCF)-N-electron valence state perturbation theory (NEVPT2). The calculations reveal phenomenal modulation in exchange couplings upon annulation such that remarkably strong ferromagnetic interactions are realized especially for a certain class of the Blatter-NN diradicals. The modulation of spin-spin interactions is rationalized by variation in spin density distribution and molecular torsional angles. We demonstrate that annulation in OMMs opens an additional coupling pathway via auxiliary X-atom acting as the atomic relay center which strongly manipulates the magnitude of exchange couplings.

A Ranked-Orbital Approach to Select Active Spaces for High-Throughput Multireference Computation

The past decade has seen a great increase in the application of high-throughput computation to a variety of important problems in chemistry. However, one area which has been resistant to the high-throughput approach is multireference wave function methods, in large part due to the technicalities of setting up these calculations and in particular the not always intuitive challenge of active space selection. As we look toward a future of applying high-throughput computation to all areas of chemistry, it is important to prepare these methods for large-scale automation. Here, we propose a ranked-orbital approach to select active spaces with the goal of standardizing multireference methods for high-throughput computation. This method allows for the meaningful comparison of different active space selection schemes and orbital localizations, and we demonstrate the utility of this approach across 1120 multireference calculations for the excitation energies of small molecules. Our results reveal that it is helpful to distinguish the method used to generate orbitals from the method of ranking orbitals in terms of importance for the active space. Additionally, we propose our own orbital ranking scheme that estimates the importance of an orbital for the active space through a pair-interaction framework from orbital energies and features of the Hartree-Fock exchange matrix. We call this new scheme the "approximate pair coefficient" (APC) method and we show that it performs quite well for the test systems presented.

CAS without SCF-Why to use CASCI and where to get the orbitals

The complete active space self-consistent field (CASSCF) method has seen broad adoption due to its ability to describe the electronic structure of both the ground and excited states of molecules over a broader swath of the potential energy surface than is possible with the simpler Hartree-Fock approximation. However, it also has a reputation for being unwieldy, computationally costly, and un-black-box. Here, we discuss a class of alternatives, complete active space configuration interaction (CASCI) methods, paying particular attention to their application to electronic excited states. The goal of this Perspective is fourfold. First, we argue that CASCI is not merely an approximation to CASSCF, in that it can be designed to have important qualitative advantages over CASSCF. Second, we present several insights drawn from our experience experimenting with different schemes for computing orbitals to be employed in CASCI. Third, we argue that CASCI is well suited for application to nanomaterials. Finally, we reason that, with the rise in new low-scaling approaches for describing multireference systems, there is a greater need than ever to develop new methods for defining orbitals that provide an efficient and accurate description of both static correlation and electronic excitations in a limited active space.

Multicomponent CASSCF Revisited: Large Active Spaces Are Needed for Qualitatively Accurate Protonic Densities

Multicomponent methods seek to treat select nuclei, typically protons, fully quantum mechanically and equivalent to the electrons of a chemical system. In such methods, it is well-known that due to the neglect of electron-proton correlation, a Hartree-Fock (HF) description of the electron-proton interaction catastrophically fails leading to qualitatively incorrect protonic properties. In single-component quantum chemistry, the qualitative failure of HF is normally indicative of the need for multireference methods such as complete active space self-consistent field (CASSCF). While a multicomponent CASSCF method was implemented nearly 20 years ago, it is only able to perform calculations with very small active spaces (∼10⁵ multicomponent configurations). Therefore, in order to extend the realm of applicability of the multicomponent CASSCF method, this study derives and implements a new two-step multicomponent CASSCF method that uses multicomponent heat-bath configuration interaction for the configuration interaction step, enabling calculations with very large active spaces (up to 16 electrons in 48 orbitals). We find that large electronic active spaces are needed to obtain qualitatively accurate protonic densities for the HCN and FHF^- molecules. Additionally, the multicomponent CASSCF method implemented here should have further applications for double-well protonic potentials and systems that are inherently electronically multireference.

Organic Mixed-Valence Compounds and the Overhauser Effect in Insulating Solids

Recent experiments have shown that the organic free radical 1,3-bisdiphenylene-2-phenylallyl (BDPA) can induce an Overhauser effect dynamic nuclear polarization in insulating solids, a feat previously considered not to be possible. Here, we establish that this peculiar ability of the BDPA radical stems from its mixed-valence nature and the ensuing intramolecular charge transfer. Using state-of-the-art DMRGSCF calculations, we confirm the class II mixed-valence nature of BDPA with the characteristic double-well potential energy surface, and we investigate the mechanism of the consequent electron hopping. A two-component vibronic Hamiltonian is then employed to compute the rate of electron hopping from a quantum dynamical time-propagation of the density matrix. The predicted hyperfine coupling oscillations indeed fall within the frequency range required for an Overhauser effect. The paradigm of mixed-valence compounds as a mining source opens many possibilities for the development and fine tuning of novel polarizing agents.

Comparison of Methods for Active Orbital Selection in Multiconfigurational Calculations

Several methods of constructing the active orbital space for multiconfigurational wave functions are compared on typical moderately strongly or strongly correlated ground-state molecules. The relative merits of these methods and problems inherent in multiconfigurational calculations are discussed. Strong correlation in the ground electronic state is found typically in larger conjugated and in antiaromatic systems, transition states which involve bond breaking or formation, and transition metal complexes. Our examples include polyenes, polyacenes, the reactant, product and transition state of the Bergman cyclization, and two transition metal complexes: Hieber's anion [(CO)3FeNO]- and ferrocene. For the systems investigated, the simplest and oldest selection method, based on the fractional occupancy of unrestricted Hartree-Fock natural orbitals (the UNO criterion), yields the same active space as much more expensive approximate full CI methods. A disadvantage of this method used to be the difficulty of finding broken spin symmetry UHF solutions. However, our analytical method, accurate to fourth order in the orbital rotation angles (Tóth and Pulay J. Chem. Phys. 2016, 145, 164102.), has solved this problem. Two further advantages of the UNO criterion are that, unlike most other methods, it measures not only the energetic proximity to the Fermi level but also the magnitude of the exchange interaction with strongly occupied orbitals and therefore allows the estimation of the correlation strength for orbital selection in Restricted Active Space methods.

Modulating Effect of Ligand Charge on the Electronic Properties of 2Ni-2S Structures and Implications for Biological 2M-2S Sites

Sulfur-bridged bimetallic 2M-2S type structures are essential cofactors that participate in biological long-range electron transport and metabolism. Metal-sulfur bond covalency is a decisive property for inner sphere (through-bond) type electron transfer that dominates in buried or hydrophobic protein environments. This work reports on a combined experimental and computational study of the effect of ligand charge on the electronic structure of a 2Ni-2S model site that adopts the biologically relevant S = ¹/₂ redox state. Starting out from an isostructural dinickel(1.5+)-dithiophenolate platform with sulfur-bridged tetrahedral Ni sites, η²:η²-μ-coordination of the S = ¹/₂ [2Ni-2S]⁺ core to either a neutral π-system or strongly σ-donating cyclohexadienido renders its electronic structure substantially different. Density functional theory analysis corroborates pulse and continuous wave electron paramagnetic resonance data that associate co-ligand charge with the significant change in the mechanism and size of electron-³¹P nuclear spin hyperfine coupling to a phosphine reporter ligand at each nickel center. An increasing level of charge donation attenuates direct and through-bridge electronic coupling of the metal sites, resulting in a stronger electronic coupling of the 2Ni-2S core to its terminal phosphine donors. Drawing a connection to biological 2M-2S sites, our 2Ni-2S system indicates that a fine balance of intracore and core-protein electronic coupling is key to biological function for which the degree of charge donation by peripheral donors appears to be a significant parameter.

Extending the ASS1ST Active Space Selection Scheme to Large Molecules and Excited States

Multireference electronic structure methods based on the CAS (complete active space) ansatz are well-established as a means to provide reliable predictions of physical properties of strongly correlated systems. A critical aspect of every CAS calculation is the selection of an adequate active space, in particular as the boundaries for tractable active spaces have been shifted significantly with the emergence of efficient approximations to the Full-CI problem like the density matrix renormalization group and full-CI quantum Monte Carlo. Recently, we proposed an active space selection based on first-order perturbation theory (ASS1ST) that yields satisfactory results for the electronic ground state of a variety of strongly correlated systems. In this work, we present a state-averaged extension of ASS1ST (SA-ASS1ST) that determines suitable active spaces when electronically excited states are targeted. Furthermore, the computational costs of the single state and state-averaged variants are significantly reduced by a simple approximation that avoids the most expensive step of the original method, the evaluation of active space four-electron reduced density matrices, altogether. After the applicability of the approximation is established, test calculations on a biomimetic Mn₄O₄ cluster demonstrate the enhanced range of ASS1ST in terms of system size and complexity. Furthermore, calculations on [VOCl₄]₂^-, MeMn(CO)₃-α-diimine, and anthracene show that SA-ASS1ST suggests well-suited active spaces to describe d → d and charge-transfer excitations in transition-metal complexes as well as π → π* excitations in aryl compounds. Finally, the application of ASS1ST on multiple points of the potential energy surface of Cr₂ illustrates the applicability of the method even when extremely complicated bonding patterns are met. More importantly, however, it highlights the necessity to use special strategies when different points of a potential energy surface are investigated, e.g., during chemical reactions.

An ab initio multireference study of reductive eliminations from organoferrates(iii) in the gas-phase: it is all about the spin state

The reductive elimination reaction from organoferrates(iii) of the composition [FeR₃R′]⁻ is studied by state-of-the-art multireference electronic structure calculations.