Multiconfiguration Self-Consistent Field and Multireference Configuration Interaction Methods and Applications

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.

Development and application of a complete active space spin‐orbit configuration interaction program designed for molecule magnets

We present a spin‐orbit configuration interaction program which has been tailored for the description of the magnetic properties of polynuclear metal complexes with partially filled d‐ and f‐shells. The spin‐orbit operators are directly included in the configuration interaction program based on Slater‐determinants. The lowest states are obtained by a Block‐Davidson‐type diagonalisation. The usage of localised active orbitals enables the construction of start vectors from tensor products of single‐center wave functions that already include spin‐orbit interaction. This allows for an analysis of the role and the interplay of the different metal centres. Furthermore, in case of weak coupling of the metal centres these tensor products are already close to the final wave functions ensuring fast convergence. In combination with a two‐layer hybrid parallelisation, this makes the program highly efficient. Based on the spin‐orbit coupled wave functions, magnetic D‐tensors, g‐tensors and temperature‐dependent susceptibilities can be calculated. The applicability and performance of the program is shown exemplarily on a trinuclear transition metal (Co^IIV^IICo^II) complex.

Machine Learning for Electronically Excited States of Molecules

Electronically excited states of molecules are at the heart of photochemistry, photophysics, as well as photobiology and also play a role in material science. Their theoretical description requires highly accurate quantum chemical calculations, which are computationally expensive. In this review, we focus on not only how machine learning is employed to speed up such excited-state simulations but also how this branch of artificial intelligence can be used to advance this exciting research field in all its aspects. Discussed applications of machine learning for excited states include excited-state dynamics simulations, static calculations of absorption spectra, as well as many others. In order to put these studies into context, we discuss the promises and pitfalls of the involved machine learning techniques. Since the latter are mostly based on quantum chemistry calculations, we also provide a short introduction into excited-state electronic structure methods and approaches for nonadiabatic dynamics simulations and describe tricks and problems when using them in machine learning for excited states of molecules.

Combining Machine Learning and Computational Chemistry for Predictive Insights Into Chemical Systems

Machine learning models are poised to make a transformative impact on chemical sciences by dramatically accelerating computational algorithms and amplifying insights available from computational chemistry methods. However, achieving this requires a confluence and coaction of expertise in computer science and physical sciences. This Review is written for new and experienced researchers working at the intersection of both fields. We first provide concise tutorials of computational chemistry and machine learning methods, showing how insights involving both can be achieved. We follow with a critical review of noteworthy applications that demonstrate how computational chemistry and machine learning can be used together to provide insightful (and useful) predictions in molecular and materials modeling, retrosyntheses, catalysis, and drug design.

Machine Learning Force Fields

In recent years, the use of machine learning (ML) in computational chemistry has enabled numerous advances previously out of reach due to the computational complexity of traditional electronic-structure methods. One of the most promising applications is the construction of ML-based force fields (FFs), with the aim to narrow the gap between the accuracy of ab initio methods and the efficiency of classical FFs. The key idea is to learn the statistical relation between chemical structure and potential energy without relying on a preconceived notion of fixed chemical bonds or knowledge about the relevant interactions. Such universal ML approximations are in principle only limited by the quality and quantity of the reference data used to train them. This review gives an overview of applications of ML-FFs and the chemical insights that can be obtained from them. The core concepts underlying ML-FFs are described in detail, and a step-by-step guide for constructing and testing them from scratch is given. The text concludes with a discussion of the challenges that remain to be overcome by the next generation of ML-FFs.

Machine Learning for Electronically Excited States of Molecules

Electronically excited states of molecules are at the heart of photochemistry, photophysics, as well as photobiology and also play a role in material science. Their theoretical description requires highly accurate quantum chemical calculations, which are computationally expensive. In this review, we focus on not only how machine learning is employed to speed up such excited-state simulations but also how this branch of artificial intelligence can be used to advance this exciting research field in all its aspects. Discussed applications of machine learning for excited states include excited-state dynamics simulations, static calculations of absorption spectra, as well as many others. In order to put these studies into context, we discuss the promises and pitfalls of the involved machine learning techniques. Since the latter are mostly based on quantum chemistry calculations, we also provide a short introduction into excited-state electronic structure methods and approaches for nonadiabatic dynamics simulations and describe tricks and problems when using them in machine learning for excited states of molecules.

The Quest to Simulate Excited-State Dynamics of Transition Metal Complexes

This Perspective describes current computational efforts in the field of simulating photodynamics of transition metal complexes. We present the typical workflows and feature the strengths and limitations of the different contemporary approaches. From electronic structure methods suitable to describe transition metal complexes to approaches able to simulate their nuclear dynamics under the effect of light, we give particular attention to build a bridge between theory and experiment by critically discussing the different models commonly adopted in the interpretation of spectroscopic experiments and the simulation of particular observables. Thereby, we review all the studies of excited-state dynamics on transition metal complexes, both in gas phase and in solution from reduced to full dimensionality.

An orbital picture extracted from correlated electronic wavefunctions and application to forbidden reactions: 70 years of the frontier orbital theory

The one-electron picture in molecular electronic state theory, particularly the molecular orbital (MO) theory with the Hartree-Fock approximation, has set a foundation to develop chemical science. Frontier orbital theory, or the theory of HOMO (highest occupied MO)-LUMO (lowest unoccupied MO) interaction, and the conservation rule of orbital symmetry are among the brightest achievements in a molecular orbital picture. After 70 years from the birth of frontier orbital theory, however, electronic wavefunctions treated in current quantum chemistry are often highly correlated and consist of extensive scales of electronic configurations to be more accurate and to cope with far more complicated reactions than concerted reactions. Under such circumstances, the MO approximation itself readily loses its validity, let alone the utter dominance of the HOMO-LUMO interaction. Recently, we have proposed an invariant method to extract general orbitals from such correlated electronic wavefunctions, which we refer to as Energy Natural Orbitals (ENOs) [K. Takatsuka and Y. Arasaki, J. Chem. Phys. 154, 094103 (2021)]. The energies of ENOs are summed exactly to the total electronic energy. The topological (symmetry) properties of a total wavefunction are represented by the relative phases of ENOs along with the continuity and crossing (avoided and conical intersection) among them. Only a small number of ENOs often dominate and characterize chemical reactions. With these properties of ENO, we explore a couple of simple and typical symmetry forbidden reactions, illustrating the effects of electron correlation and degeneracy in relevant ENOs. We propose the notion of "internal conical intersection" among ENOs, which leads to Jahn-Teller effect, pseudo-Jahn-Teller effect, and so on. We dare to explain the primary origin of elementary conical intersections and multidimensional avoided crossing in chemical reactions with the use of the notion of orbital crossing between those of HOMO-HOMO and LUMO-LUMO interactions and so on.

Electronic Structure Methods for the Description of Nonadiabatic Effects and Conical Intersections

Nonadiabatic effects are ubiquitous in photophysics and photochemistry, and therefore, many theoretical developments have been made to properly describe them. Conical intersections are central in nonadiabatic processes, as they promote efficient and ultrafast nonadiabatic transitions between electronic states. A proper theoretical description requires developments in electronic structure and specifically in methods that describe conical intersections between states and nonadiabatic coupling terms. This review focuses on the electronic structure aspects of nonadiabatic processes. We discuss the requirements of electronic structure methods to describe conical intersections and nonadiabatic couplings, how the most common excited state methods perform in describing these effects, and what the recent developments are in expanding the methodology and implementing nonadiabatic couplings.

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.

Ab initio study of the O3-N2 complex: Potential energy surface and rovibrational states

The formation and destruction of O₃ within the Chapman cycle occurs as a result of inelastic collisions with a third body. Since N₂ is the most abundant atmospheric molecule, it can be considered as the most typical candidate when modeling energy-transfer dynamics. We report a new ab initio potential energy surface (PES) of the O₃-N₂ van der Waals complex. The interaction energies were calculated using the explicitly correlated single- and double-excitation coupled cluster method with a perturbative treatment of triple excitations [CCSD(T)-F12a] with the augmented correlation-consistent triple-zeta aug-cc-pVTZ basis set. The five-dimensional PES was analytically represented by an expansion in spherical harmonics up to eighth order inclusive. Along with the global minimum of the complex (D_e = 348.88 cm^-1), with N₂ being perpendicular to the O₃ plane, six stable configurations were found with a smaller binding energy. This PES was employed to calculate the bound states of the O₃-N₂ complex with both ortho- and para-N₂ for total angular momentum J = 0 and 1, as well as dipole transition probabilities. The nature of the bound states of the O₃-oN₂ and O₃-pN₂ species is discussed based on their rovibrational wave functions.

Description of Sudden Polarization in the Excited Electronic States with an Ensemble Density Functional Theory Method

Sudden polarization (SP) is one of the manifestations of electron transfer in the electronically excited states of molecules. Proposed initially to explain the unusual reactivity of photoexcited olefins, SP often occurs in the excited states of molecules possessing strongly correlated diradical ground state. Theoretical description of SP involves mixing between the singly excited and the doubly excited zwitterionic states, which makes it inaccessible with the use of the popular linear-response time-dependent density functional theory methods. In this work, an extended variant of the state-interaction state-averaged spin-restricted ensemble-referenced Kohn-Sham (SI-SA-REKS, or SSR) method is applied to study SP in a number of organic diradical systems. To this end, the analytical derivative formalism is derived and implemented for the SSR(3,2) method (see the main text for explanation of the acronym), which enables the automatic geometry optimization and obtains the relaxed density matrices as well as the electron binding energies and respective Dyson's orbitals. Application of the new method to SP in the lowest singlet excited state of ethylene agrees with the results obtained previously with the use of multireference methods of wavefunction theory. A number of interesting manifestations of SP are observed, such as the charge transfer in photoexcited tetramethyleneethene (TME) diradical mediated by the vibrational motion and conductivity switching in the excited state of a donor-acceptor dyad placed in an external electric field.

Exploration of interlacing and avoided crossings in a manifold of potential energy curves by a unitary group adapted state specific multi-reference perturbation theory (UGA-SSMRPT)

The Unitary Group Adapted State-Specific Multi-Reference Perturbation Theory (UGA-SSMRPT2) developed by Mukherjee et al. [J. Comput. Chem. 36, 670 (2015)] has successfully realized the goal of studying bond dissociation in a numerically stable, spin-preserving, and size-consistent manner. We explore and analyze here the efficacy of the UGA-SSMRPT2 theory in the description of the avoided crossings and interlacings between a manifold of potential energy curves for states belonging to the same space-spin symmetry. Three different aspects of UGA-SSMRPT2 have been studied: (a) We introduce and develop the most rigorous version of UGA-SSMRPT2 that emerges from the rigorous version of UGA-SSMRCC utilizing a linearly independent virtual manifold; we call this the "projection" version of UGA-SSMRPT2 (UGA-SSMRPT2 scheme P). We compare and contrast this approach with our earlier formulation that used extra sufficiency conditions via amplitude equations (UGA-SSMRPT2 scheme A). (b) We present the results for a variety of electronic states of a set of molecules, which display the striking accuracy of both the two versions of UGA-SSMRPT2 with respect to three different situations involving weakly avoided crossings, moderate/strongly avoided crossings, and interlacing in a manifold of potential energy curves (PECs) of the same symmetry. Accuracy of our results has been benchmarked against IC-MRCISD + Q.

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

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

Approximations of density matrices in N-electron valence state second-order perturbation theory (NEVPT2). I. Revisiting the NEVPT2 construction

Over the last decade, the second-order N-electron valence state perturbation theory (NEVPT2) has developed into a widely used multireference perturbation method. To apply NEVPT2 to systems with large active spaces, the computational bottleneck is the construction of the fourth-order reduced density matrix. Both its generation and storage become quickly problematic beyond the usual maximum active space of about 15 active orbitals. To reduce the computational cost of handling fourth-order density matrices, the cumulant approximation (CU) has been proposed in several studies. A more conventional strategy to address the higher-order density matrices is the pre-screening approximation (PS), which is the default one in the ORCA program package since 2010. In the present work, the performance of the CU, PS, and extended PS (EPS) approximations for the fourth-order density matrices is compared. Following a pedagogical introduction to NEVPT2, contraction schemes, as well as the approximations to density matrices, and the intruder state problem are discussed. The CU approximation, while potentially leading to large computational savings, virtually always leads to intruder states. With the PS approximation, the computational savings are more modest. However, in conjunction with conservative cutoffs, it produces stable results. The EPS approximation to the fourth-order density matrices can reproduce very accurate NEVPT2 results without any intruder states. However, its computational cost is not much lower than that of the canonical algorithm. Moreover, we found that a good indicator of intrude states problems in any approximation to high order density matrices is the eigenspectra of the Koopmans matrices.

Analytic gradients for restricted active space second-order perturbation theory (RASPT2)

The computational cost of analytic derivatives in multireference perturbation theory is strongly affected by the size of the active space employed in the reference self-consistent field calculation. To overcome previous limits on the active space size, the analytic gradients of single-state restricted active space second-order perturbation theory (RASPT2) and its complete active space second-order perturbation theory (CASPT2) have been developed and implemented in a local version of OpenMolcas. Similar to previous implementations of CASPT2, the RASPT2 implementation employs the Lagrangian or Z-vector method. The numerical results show that restricted active spaces with up to 20 electrons in 20 orbitals can now be employed for geometry optimizations.

Localized Active Space Pair-Density Functional Theory

Accurate quantum chemical methods for the prediction of spin-state energy gaps for strongly correlated systems are computationally expensive and scale poorly with the size of the system. This makes calculations for many experimentally interesting molecules impractical even with abundant computational resources. Previous work has shown that the localized active space (LAS) self-consistent field (SCF) method can be an efficient way to obtain multiconfiguration SCF wave functions of comparable quality to the corresponding complete active space (CAS) ones. To obtain quantitative results, a post-SCF method is needed to estimate the complete correlation energy. One such method is multiconfiguration pair-density functional theory (PDFT), which calculates the energy based on the density and on-top pair density obtained from a multiconfiguration wave function. In this work, we introduce localized-active-space PDFT, which uses a LAS wave function for subsequent PDFT calculations. The method is tested by computing spin-state energies and gaps in conjugated organic molecules and a bimetallic compound and comparing to the corresponding CAS-PDFT values.

Improved Description and Efficient Implementation of Spin-Projected Perturbation Theory for Practical Applications

In this study, we continue to develop the recently proposed second-order perturbation theory for the spin-projected Hartree-Fock method [Tsuchimochi, T.; Ten-no, S. L. J. Chem. Theory Comput. 2019, 15, 6688] in various aspects. A new, stable imaginary level-shift scheme is derived to obtain a well-conditioned equation, enabling a significantly faster convergence. To achieve a further speed-up, we propose a preconditioning scheme considering the pair character on a spin-projected basis. We also eliminate the computational memory bottleneck in solving the linear equation for large systems using a distributed memory parallel implementation. Finally, for the description of open-shell molecules, several modified zeroth-order Hamiltonians are introduced and tested using the Mn₂O₂(NHCHCO₂)₄ complex. These developments enable practical calculations of a second-order perturbation theory with improved accuracy at a reduced computational cost.

Intermediate Valence States in Lanthanide Compounds

Over more than 50 years, intermediate valence states in lanthanide compounds have often resulted in unexpected or puzzling spectroscopic and magnetic properties. Such experimental singularities could not be rationalised until new theoretical models involving multiconfigurational electronic ground states were established. In this minireview, the different singularities that have been observed among lanthanide complexes are highlighted, the models used to rationalise them are detailed and how such electronic effects may be adjusted depending on energy and symmetry considerations is considered. Understanding and tuning the ground-state multiconfigurational behaviour in lanthanide complexes may open new doors to modular and unusual reactivities.

How the Donor/Acceptor Spin States Affect the Electronic Couplings in Molecular Charge-Transfer Processes?

The electronic coupling matrix element H_AB is an essential ingredient of most electron-transfer theories. H_AB depends on the overlap between donor and acceptor wave functions and is affected by the involved states’ spin. We classify the spin-state effects into three categories: orbital occupation, spin-dependent electron density, and density delocalization. The orbital occupancy reflects the diverse chemical nature and reactivity of the spin states of interest. The effect of spin-dependent density is related to a more compact electron density cloud at lower spin states due to decreased exchange interactions between electrons. Density delocalization is strongly connected with the covalency concept that increases the spatial extent of the diabatic state’s electron density in specific directions. We illustrate these effects with high-level ab initio calculations on model direct donor–acceptor systems relevant to metal oxide materials and biological electron transfer. Obtained results can be used to benchmark existing methods for H_AB calculations in complicated cases such as spin-crossover materials or antiferromagnetically coupled systems.

Optimization of Three State Conical Intersections by Adaptive Penalty Function Algorithm in Connection with the Mixed-Reference Spin-Flip Time-Dependent Density Functional Theory Method (MRSF-TDDFT)

A new adaptive algorithm for penalty function optimization for minimum-energy three-states conical intersections (ME3CI) is suggested. The new algorithm differs from the original penalty function algorithm by (a) removing the redundancy in the target function, (b) using an adaptive increment for the penalty function weighting factor, and (c) using tighter convergence criteria for the energy gap. The latter was introduced to guarantee convergence to a true conical intersection rather than to a narrowly avoided crossing geometry. The new algorithm was tested in the optimization of the ME3CI geometries in butadiene and malonaldehyde, where all of the previously found true ME3CI geometries were recovered. The previously found butadiene's CI_3/2/1 turned out to be a narrowly avoided crossing. For butadiene, seven new ME3CI geometries have been located. Because of the removal of the redundancy and the use of the adaptive weighting factor, the convergence rate of the new algorithm is noticeably improved as compared to that of the previously proposed penalty function algorithm. The application to malonaldehyde and butadiene demonstrates that the three-state conical intersections may be more abundant and hence more involved in the photochemistry than previously thought. The recently developed mixed-reference spin flip (MRSF)-TDDFT method yields ME3CI geometries and relative energies quantitatively consistent with the previously reported calculations at a much reduced computational cost.

Further Development of iCIPT2 for Strongly Correlated Electrons

The efficiency of the recently proposed iCIPT2 [iterative configuration interaction (iCI) with selection and second-order perturbation theory (PT2); J. Chem. Theory Comput. 2020, 16, 2296] for strongly correlated electrons is further enhanced (by up to 20×) by using (1) a new ranking criterion for configuration selection, (2) a new particle-hole algorithm for Hamiltonian construction over randomly selected configuration state functions (CSF), and (3) a new data structure for the quick sorting of the variational and first-order interaction spaces. Meanwhile, the memory requirement is also significantly reduced. As a result, this improved implementation of iCIPT2 can handle 1 order of magnitude more CSFs than the previous version, as revealed by taking the chromium dimer and an iron-sulfur cluster, [Fe₂S₂(SCH₃)]₄^2-, as examples.

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.

DMRG on Top of Plane-Wave Kohn-Sham Orbitals: A Case Study of Defected Boron Nitride

In this paper, we analyze the numerical aspects of the inherent multireference density matrix renormalization group (DMRG) calculations on top of the periodic Kohn-Sham density functional theory using the complete active space approach. The potential of the framework is illustrated by studying hexagonal boron nitride nanoflakes embedding a charged single boron vacancy point defect by revealing a vertical energy spectrum with a prominent multireference character. We investigate the consistency of the DMRG energy spectrum from the perspective of sample size, basis size, and active space selection protocol. Results obtained from standard quantum chemical atom-centered basis calculations and plane-wave based counterparts show excellent agreement. Furthermore, we also discuss the spectrum of the periodic sheet which is in good agreement with extrapolated data of finite clusters. These results pave the way toward applying the DMRG method in extended correlated solid-state systems, such as point defect qubit in wide band gap semiconductors.

Interpretation of Exchange Interaction through Orbital Entanglement

Recently, the analysis of single-orbital entropy and mutual information has been introduced as a tool for the investigation of contributions to the exchange (J) coupling between open-shell metal ions [Stein et al. J. Phys. Chem. Lett. 2019, 10, 6762-6770]. Here, we show that this analysis may lead to an incorrect interpretation of the J-coupling mechanism. Instead, we propose an orbital-entanglement analysis that is based on the two-electron density and that provides a coherent picture of the contributing exchange pathways, which seems fully consistent with the available J values. For this purpose, we used a prototypical bis-μ-oxo binuclear manganese complex ([Mn₂O₂(NH₃)₈]⁴⁺) and demonstrated that its antiferromagnetism (J < 0), calculated by using the active space composed of all valence p_O and d_Mn orbitals, correlates well with the largest elements in the differential low-spin vs high-spin entanglement map. These elements correspond to interactions between the pairs of d_Mn orbitals mediated by the oxo-bridging out-of-plane p orbitals, representing the π superexchange pathway. We also show that the reduction of active space to manifold of the singly occupied magnetic orbitals does not lead to discrepancy between the calculated J values and entanglement maps. This contrasts to analysis of mutual information, which suggests the "direct" d_Mn-d_Mn interactions to play a dominant role for the J coupling, irrespective of the size of active space as well as of the antiferromagnetism expected. The failure is attributed to the large contribution of spin entanglement contained in the mutual information of the low-spin state, which may be regarded as the origin of the different complexity of its wave function and electron density.

Forecasting System of Computational Time of DFT/TDDFT Calculations under the Multiverse Ansatz via Machine Learning and Cheminformatics

With the view of achieving a better performance in task assignment and load-balancing, a top-level designed forecasting system for predicting computational times of density-functional theory (DFT)/time-dependent DFT (TDDFT) calculations is presented. The computational time is assumed as the intrinsic property for the molecule. Based on this assumption, the forecasting system is established using the "reinforced concrete", which combines the cheminformatics, several machine-learning (ML) models, and the framework of many-world interpretation (MWI) in multiverse ansatz. Herein, the cheminformatics is used to recognize the topological structure of molecules, the ML models are used to build the relationships between topology and computational cost, and the MWI framework is used to hold various combinations of DFT functionals and basis sets in DFT/TDDFT calculations. Calculated results of molecules from the DrugBank dataset show that (1) it can give quantitative predictions of computational costs, typical mean relative errors can be less than 0.2 for DFT/TDDFT calculations with derivations of ±25% using the exactly pretrained ML models and (2) it can also be employed to various combinations of DFT functional and basis set cases without exactly pretrained ML models, while only slightly enlarge predicting errors.

Unconventional singlet fission materials

Singlet fission (SF) is a photophysical downconversion pathway, in which a singlet excitation transforms into two triplet excited states. As such, it constitutes an exciton multiplication generation process, which is currently at the focal point for future integration into solar energy conversion devices. Beyond this, various other exciting applications were proposed, including quantum cryptography or organic light emitting diodes. Also, the mechanistic understanding evolved rapidly during the last year. Unfortunately, the number of suitable SF-chromophores is still limited. This is per se problematic, considering the wide range of envisaged applicability. With that in mind, we emphasize uncommon SF-scaffolds and outline requirements as well as strategies to expand the chromophore pool of SF-materials.

Single-Root Multireference Brillouin-Wigner Perturbative Approach to Excitation Energies

The state-specific Brillouin-Wigner multireference perturbation theory [which employs Jeziorski-Monkhorst parametrization of the wave function] using improved virtual orbitals, denoted as IVO-BWMRPT, is applied to calculate excitation energies (EEs) for methylene, ethylene, trimethylenemethane, and benzyne systems exhibiting various degrees of diradical character. In IVO-BWMRPT, all of the parameters appearing in the wave function ansatz are optimized for a specific electronic state. For these systems, the IVO-BWMRPT method provides EEs that are in close agreement with the benchmark results and experiments, where available, indicating that the method does not introduce imbalance in the target-specific treatment of closed- and open-shell states involved. The good performance of the present methodology is primarily related to structural compactness of the formalism. Overall, present findings are encouraging for both further development of the approach and chemical applications on the energy differences of strongly correlated systems.

Diffusion Monte Carlo method on small boron clusters using single- and multi- determinant-Jastrow trial wavefunctions

Multireference character in some small boron clusters could be significant, and a previous all-electron fixed-node diffusion quantum Monte Carlo (FN-DMC) calculation with the single-determinant-Jastrow (SDJ) trial wavefunction shows that the atomization energy (AE) of B₄ ⁺ is overestimated by about 1.4 eV compared with the coupled cluster method with single, doubles, and perturbative triples [CCSD(T)] results. All-electron FN-DMC calculations and those with the pseudopotential (PP) using SDJ and multi-determinant-Jastrow (MDJ) trial wavefunctions with B3LYP orbitals as well as CC calculations at different levels are carried out on B_n ^Q (n = 1-5, Q = -1, 0, 1) clusters. The obtained FN-DMC energies indicate that the node error of the employed SDJ trial wavefunction in all-electron calculations is different from that with the PP for some clusters. The error of AEs and dissociation energies (DEs) from all-electron FN-DMC calculations is larger than that with the PP when the SDJ trial wavefunction is employed, while errors of CC methods do not depend on whether the PP is used. AEs and DEs of the boron clusters are improved significantly when MDJ trial wavefunctions are used in both all-electron calculations and those with the PP, and their error is similar to that of CCSD(T) compared with CCSDT(Q) results. On the other hand, reasonable adiabatic electron detachment energies (ADEs) and ionization potentials (AIPs) are achieved with FN-DMC using SDJ trial wavefunctions and MDJ is less effective on ADEs and AIPs. Furthermore, the relative energy between two structures of B₉ ^- is predicted reliably with FN-DMC using the SDJ trial wavefunction and the effect of MDJ is negligible, while density functional theory results using different exchange-correlation functionals differ significantly.

Spin-state energetics of metallocenes: How do best wave function and density functional theory results compare with the experimental data?

We benchmark the accuracy of quantum-chemical methods, including wave function theory methods [coupled cluster theory at the CCSD(T) level, multiconfigurational perturbation-theory (CASPT2, NEVPT2) and internally contracted multireference configuration interaction (MRCI)] and 30 density functional theory (DFT) approximations, in reproducing the spin-state splittings of metallocenes. The reference values of the electronic energy differences are derived from the experimental spin-crossover enthalpy for manganocene and the spectral data of singlet-triplet transitions for ruthenocene, ferrocene, and cobaltocenium. For ferrocene and cobaltocenium we revise the previous experimental interpretations regarding the lowest triplet energy; our argument is based on the comparison with the lowest singlet excitation energy and herein reported, carefully determined absorption spectrum of ferrocene. When deriving vertical energies from the experimental band maxima, we go beyond the routine vertical energy approximation by introducing vibronic corrections based on simulated vibrational envelopes. The benchmarking result confirms the high accuracy of the CCSD(T) method (in particular, for UCCSD(T) based on Hartree-Fock orbitals we find for our dataset: maximum error 0.12 eV, weighted mean absolute error 0.07 eV, weighted mean signed error 0.01 eV). The high accuracy of the single-reference method is corroborated by the analysis of a multiconfigurational character of the complete active space wave function for the triplet state of ferrocene. On the DFT side, our results confirm the non-universality problem with approximate functionals. The present study is an important step toward establishing an extensive and representative benchmark set of experiment-derived spin-state energetics for transition metal complexes.

Computational Methods in Heterogeneous Catalysis

The unprecedented ability of computations to probe atomic-level details of catalytic systems holds immense promise for the fundamentals-based bottom-up design of novel heterogeneous catalysts, which are at the heart of the chemical and energy sectors of industry. Here, we critically analyze recent advances in computational heterogeneous catalysis. First, we will survey the progress in electronic structure methods and atomistic catalyst models employed, which have enabled the catalysis community to build increasingly intricate, realistic, and accurate models of the active sites of supported transition-metal catalysts. We then review developments in microkinetic modeling, specifically mean-field microkinetic models and kinetic Monte Carlo simulations, which bridge the gap between nanoscale computational insights and macroscale experimental kinetics data with increasing fidelity. We finally review the advancements in theoretical methods for accelerating catalyst design and discovery. Throughout the review, we provide ample examples of applications, discuss remaining challenges, and provide our outlook for the near future.

Index of multi-determinantal and multi-reference character in coupled-cluster theory

A full configuration interaction calculation (FCI) ultimately defines the innate molecular orbital description of a molecule. Its density matrix and the natural orbitals obtained from it quantify the difference between having N-dominantly occupied orbitals in a reference determinant for a wavefunction to describe N-correlated electrons and how many of those N-electrons are left to the remaining virtual orbitals. The latter provides a measure of the multi-determinantal character (MDC) required to be in a wavefunction. MDC is further split into a weak correlation part and a part that indicates stronger correlation often called multi-reference character (MRC). If several virtual orbitals have high occupation numbers, then one might argue that these additional orbitals should be allowed to have a larger role in the calculation, as in MR methods, such as MCSCF, MR-CI, or MR-coupled-cluster (MR-CC), to provide adequate approximations toward the FCI. However, there are problems with any of these MR methods that complicate the calculations compared to the uniformity and ease of application of single-reference CC calculations (SR-CC) and their operationally single-reference equation-of-motion (EOM-CC) extensions. As SR-CC theory is used in most of today's "predictive" calculations, an assessment of the accuracy of SR-CC at some truncation of the cluster operator would help to quantify how large an issue MRC actually is in a calculation, and how it might be alleviated while retaining the convenient SR computational character of CC/EOM-CC. This paper defines indices that identify MRC situations and help assess how reliable a given calculation is.

Spin-state dependence of exchange-correlation holes

Applications of density-functional theory (DFT) in computational chemistry rely on an approximate exchange-correlation (xc) functional. However, existing approximations can fail dramatically for open-shell molecules, in particular for transition-metal complexes or radicals. Most importantly, predicting energy differences between different spin-states with approximate exchange-correlation functionals remains extremely challenging. Formally, it is known that the exact xc functional should be spin-state dependent, but none of the available approximations feature such an explicit spin-state dependence [C. R. Jacob and M. Reiher, Int. J. Quantum Chem., 2012, 112, 3661-3684]. Thus, to find novel approximations for the xc functional for open-shell systems, the development of spin-state dependent xc functionals appears to be a promising avenue. Here, we set out to shed light on the spin-state dependence of the xc functional by investigating the underlying xc holes, which we extract from configuration interaction calculations for model systems. We analyze the similarities and differences between the xc holes of the lowest-energy singlet and triplet states of the dihydrogen molecule, the helium atom, and the lithium dimer. To shed further light on the spin-state dependence of these xc holes we also discuss exact conditions that can be derived from the spin structure of the reduced two-electron density matrix. Altogether, our results suggest several possible routes towards the construction of explicitly spin-state dependent approximations for the xc functional.

Analytic gradients and derivative couplings for configuration interaction with all single excitations and one double excitation-En route to nonadiabatic dynamics

We present analytic gradients and derivative couplings for the simplest possible multireference configuration interaction method, CIS-1D, an electronic structure Ansatz that includes all single excitations and one lone double excitation on top of a Hartree-Fock reference state. We show that the resulting equations are numerically stable and require the evaluation of a similar number of integrals as compared to standard CIS theory; one can easily differentiate the required frontier orbitals (h and l) with minimal cost. The resulting algorithm has been implemented within the Q-Chem electronic structure package and should be immediately useful for understanding photochemistry with S₀-S₁ crossings.

Neural Network Potential Energy Surfaces for Small Molecules and Reactions

We review progress in neural network (NN)-based methods for the construction of interatomic potentials from discrete samples (such as ab initio energies) for applications in classical and quantum dynamics including reaction dynamics and computational spectroscopy. The main focus is on methods for building molecular potential energy surfaces (PES) in internal coordinates that explicitly include all many-body contributions, even though some of the methods we review limit the degree of coupling, due either to a desire to limit computational cost or to limited data. Explicit and direct treatment of all many-body contributions is only practical for sufficiently small molecules, which are therefore our primary focus. This includes small molecules on surfaces. We consider direct, single NN PES fitting as well as more complex methods that impose structure (such as a multibody representation) on the PES function, either through the architecture of one NN or by using multiple NNs. We show how NNs are effective in building representations with low-dimensional functions including dimensionality reduction. We consider NN-based approaches to build PESs in the sums-of-product form important for quantum dynamics, ways to treat symmetry, and issues related to sampling data distributions and the relation between PES errors and errors in observables. We highlight combinations of NNs with other ideas such as permutationally invariant polynomials or sums of environment-dependent atomic contributions, which have recently emerged as powerful tools for building highly accurate PESs for relatively large molecular and reactive systems.

Photochemical Ring Opening of Oxirane Modeled by Constrained Density Functional Theory

A constrained density functional theory/classical trajectory surface hopping study of the photochemical dissociation of oxirane (CH₂)₂O is presented. The calculations confirm the Gomer-Noyes mechanism for the initial reaction and agree largely with experimental photolysis data including reaction yields. The calculated yields, however, depend both on temperature and its modeling. The timescales of the various reaction steps are well below 100 fs, similar to previous time-dependent density functional calculations. At variance with those, however, the present calculations obey Kasha's rule, i.e., the photoreaction is initiated in the energetically lowest excited state.

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.

Exploiting chemistry and molecular systems for quantum information science

The power of chemistry to prepare new molecules and materials has driven the quest for new approaches to solve problems having global societal impact, such as in renewable energy, healthcare and information science. In the latter case, the intrinsic quantum nature of the electronic, nuclear and spin degrees of freedom in molecules offers intriguing new possibilities to advance the emerging field of quantum information science. In this Perspective, which resulted from discussions by the co-authors at a US Department of Energy workshop held in November 2018, we discuss how chemical systems and reactions can impact quantum computing, communication and sensing. Hierarchical molecular design and synthesis, from small molecules to supramolecular assemblies, combined with new spectroscopic probes of quantum coherence and theoretical modelling of complex systems, offer a broad range of possibilities to realize practical quantum information science applications.

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.