Multiconfiguration Pair-Density Functional Theory

Multiconfigurational Correlation at DFT + U Cost: On-Site Electron-Electron Interactions Yield a Block-Localized Configuration Interaction Hamiltonian

This work presents a first-principles wavefunction-in-DFT approach based on the Hubbard density functional theory (DFT) + U method. This approach begins with the standard DFT reference system of noninteracting electrons and introduces an electron-electron interaction projected onto DFT+U-type atomic states. The reference system's configuration interaction Hamiltonian is block-localized to these states and can be expressed in terms of state occupation numbers, state self-energies (which correspond to unscreened Hubbard U values), and the promotion energies of doubly excited Slater determinants. Simple approximations for the promotion energies provide multiconfigurational correlation energies without requiring explicit orbital localization/transform. Numerical results for fractionally occupied chromium atom, bonded chromium dimer, dissociating covalent bonds, and large active spaces show that the approach provides beyond-zero-sum accuracy at computational cost comparable to standard DFT+U.

Perspective: multi-configurational methods in bio-inorganic chemistry

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

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

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

A Merger of the Spin-Flip ORMAS Approach and the MC-PDFT Method

The SF-ORMAS-PDFT (spin-flip occupation restricted multiple active space-pair density functional theory) approach combines the SF-ORMAS-CI method with the MC-PDFT method to treat both static and dynamic correlation in multiconfigurational systems. The static correlation description is generated via the spin-flip approach, which uses a high-spin single reference determinant to treat excited states with multiconfigurational characters. The on-top pair density functional theory uses a translation scheme applied to GGA density functionals. The SF-ORMAS-PDFT scheme has also been combined with virtual valence orbitals (VVO), a well-defined subspace of the virtual molecular orbitals, giving rise to significant speedups relative to the use of the full virtual space. The accuracy of the SF-ORMAS-PDFT method is tested by calculating 65 vertical excitation energies of 12 small- and medium-sized organic molecules. The SF-ORMAS-PDFT vertical excitation energies calculated with VVOs are comparable to those calculated with the full virtual space. The SF-ORMAS-PDFT/6-31G(d) level of theory predicts the rotational barrier of ethylene to be 65.5 and 65.9 kcal/mol, with full virtual space and VVOs, respectively. These predicted barrier heights compare well with the experimental value of 65 kcal/mol.

Quantum Information Orbitals (QIO): Unveiling Intrinsic Many-Body Complexity by Compressing Single-Body Triviality

The simultaneous treatment of static and dynamic correlations in strongly correlated electron systems is a critical challenge. In particular, finding a universal scheme for identifying a single-particle orbital basis that minimizes the representational complexity of the many-body wave function is a formidable and longstanding problem. As a contribution toward its solution, we show that the total orbital correlation actually reveals and quantifies the intrinsic complexity of the wave function, once it is minimized via orbital rotations. To demonstrate the power of this concept in practice, an iterative scheme is proposed to optimize the orbitals by minimizing the total orbital correlation calculated by the tailored coupled cluster singles and doubles (TCCSD) ansatz. The optimized orbitals enable the limited TCCSD ansatz to capture more nontrivial information on the many-body wave function, indicated by the improved wave function and energy. An initial application of this scheme shows great improvement of TCCSD in predicting the singlet ground state potential energy curves of the strongly correlated C2 and Cr2 molecule.

Approximate Functionals for Multistate Density Functional Theory

Recently Lu and Gao [J. Phys. Chem. Lett. 2022, 13 (33), 7762-7769] published a new, rigorous density functional theory for excited states and proved that the projection of the kinetic and electron-repulsion operators into the subspace of the lowest electronic states are a universal functional of the matrix density D(r). This is the first attempt to find an approximation to the multistate universal functional F [ D ( r ) ] . It is shown that F (i) does not explicitly depend on the number of electronic states and (ii) is an analytic matrix functional. The Thomas-Fermi-Dirac-von Weizsäcker model and the correlation energy of the homogeneous electron gas are turned into matrix functionals guided by two principles: that each matrix functional should transform properly under basis set transformations and that the ground state functional should be recovered for a single electronic state. Lieb-Oxford-like bounds on the average kinetic and electron-repulsion energies in the subspace are given. When evaluated on the numerically exact matrix density of LiF, this simple approximation reproduces the matrix elements of the electron-repulsion operator in the basis of the exact eigenstates accurately for all bond lengths. In particular the off-diagonal elements of the effective Hamiltonian that come from the interactions of different electronic states can be calculated with the same or better accuracy than the diagonal elements. Unsurprisingly, the largest error comes from the kinetic energy functional. More exact conditions that constrain the functional form of F are needed to go beyond the local density approximation.

Desulfurdioxidative N-N Coupling of N-Arylhydroxylamines and N-Sulfinylanilines: Reaction Development and Mechanism

A highly efficient and chemoselective approach for the divergent assembling of unsymmetrical hydrazines through an unprecedented intermolecular desulfurdioxidative N-N coupling is developed. This metal free protocol employs readily accessible N-arylhydroxylamines and N-sulfinylanilines to provide highly valuable hydrazine products with good reaction yields and excellent functional group tolerance under simple conditions. Computational studies suggest that the in situ generated O-sulfenylated arylhydroxylamine intermediate undergoes a retro-[2π+2σ] cycloaddition via a stepwise diradical mechanism to form the N-N bond and release SO2.

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.

Automatic State Interaction with Large Localized Active Spaces for Multimetallic Systems

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

Contemporary Assessment of Energy Degeneracy in Orbital Mixing with Tetravalent f-Block Compounds

The f block is a comparatively understudied group of elements that find applications in many areas. Continued development of technologies involving the lanthanides (Ln) and actinides (An) requires a better fundamental understanding of their chemistry. Specifically, characterizing the electronic structure of the f elements presents a significant challenge due to the spatially core-like but energetically valence-like nature of the f orbitals. This duality led f-block scientists to hypothesize for decades that f-block chemistry is dominated by ionic metal-ligand interactions with little covalency because canonical covalent interactions require both spatial orbital overlap and orbital energy degeneracy. Recent studies on An compounds have suggested that An ions can engage in appreciable orbital mixing between An 5f and ligand orbitals, which was attributed to "energy-degeneracy-driven covalency". This model of bonding has since been a topic of debate because different computational methods have yielded results that support and refute the energy-degeneracy-driven covalency model. In this Viewpoint, literatures concerning the metal- and ligand-edge X-ray absorption near-edge structure (XANES) of five tetravalent f-block systems─MO2 (M = Ln, An), LnF4, MCl62-, and [Ln(NP(pip)3)4]─are compiled and discussed to explore metal-ligand bonding in f-block compounds through experimental metrics. Based on spectral assignments from a variety of theoretical models, covalency is seen to decrease from CeO2 and PrO2 to TbO2 through weaker ligand-to-metal charge-transfer (LMCT) interactions, while these LMCT interactions are not observed in the trivalent Ln sesquixodes until Yb. In comparison, while XANES characterization of AnO2 compounds is scarce, computational modeling of available X-ray absorption spectra suggests that covalency among AnO2 reaches a maximum between Am and Cm. Moreover, a decrease in covalency is observed upon changing ligands while maintaining an isostructural coordination environment from CeO2 to CeF4. These results could allude to the importance of orbital energy degeneracy in f-block bonding, but there are a variety of data gaps and conflicting results from different modeling techniques that need to be addressed before broad conclusions can be drawn.

Multireference Correlated Oscillator Strengths from Adiabatic Connection Approaches Based on Extended Random Phase Approximation

We show that accurate oscillator strengths can be obtained from adiabatic connection (AC) approaches based on the extended random phase approximation (ERPA) combined with multireference (complete active space, CAS) wave functions. The oscillator strengths calculated using the perturbation-corrected ERPA transition density matrices, proposed in this work, and the excitation energies calculated with recently introduced AC correlation energy methods, AC0 and AC0D, compete with accuracy in the perturbational CASPT2 approach and require less computational effort. AC0 and AC0D methods scale more favorably with the number of active orbitals than multiconfigurational perturbation approaches like CASPT2 and NEVPT2 thanks to their dependence on reduced density matrices up to the order of 2. Importantly, the newly developed approach for computing correlated transition dipole moments does not entail any additional costs, as all intermediate quantities become available when AC0 energies are being computed. We also test the performance of the recently proposed AC method corrected for the negative-transition contributions to the correlation energy, AC0D, for triplet excitation energies. Similarly, as for the singlet excitations, the correction improves the performance of the AC0 method, particularly for the low-lying excited states.

Analytic Nuclear Gradients for Complete Active Space Linearized Pair-Density Functional Theory

Accurately modeling photochemical reactions is difficult due to the presence of conical intersections and locally avoided crossings, as well as the inherently multiconfigurational character of excited states. As such, one needs a multistate method that incorporates state interaction in order to accurately model the potential energy surface at all nuclear coordinates. The recently developed linearized pair-density functional theory (L-PDFT) is a multistate extension of multiconfiguration PDFT, and it has been shown to be a cost-effective post-MCSCF method (as compared to more traditional and expensive multireference many-body perturbation methods or multireference configuration interaction methods) that can accurately model potential energy surfaces in regions of strong nuclear-electronic coupling in addition to accurately predicting Franck-Condon vertical excitations. In this paper, we report the derivation of analytic gradients for L-PDFT and their implementation in the PySCF-forge software, and we illustrate the utility of these gradients for predicting ground- and excited-state equilibrium geometries and adiabatic excitation energies for formaldehyde, s-trans-butadiene, phenol, and cytosine.

Influence of Selective Carbon 1s Excitation on Auger-Meitner Decay in the ESCA Molecule

Two-dimensional spectral mapping is used to visualize how resonant Auger-Meitner spectra are influenced by the site of the initial core-electron excitation and the symmetry of the core-excited state in the trifluoroethyl acetate molecule (ESCA). We observe a significant enhancement of electron yield for excitation of the COO 1s → π* and CF3 1s → σ* resonances unlike excitation at resonances involving the CH3 and CH2 sites. The CF3 1s → π* and CF3 1s → σ* resonance spectra are very different from each other, with the latter populating most valence states equally. Two complementary electronic structure calculations for the photoelectron cross section and Auger-Meitner intensity are shown to effectively reproduce the site- and state-selective nature of the resonant enhancement features. The site of the core-electron excitation and the respective final state hole locality increase the sensistivity of the photoelectron signal at specific functional group sites. This showcases resonant Auger-Meitner decay as a potentially powerful tool for selectively probing structural changes at specific functional group sites of polyatomic molecules.

Organic Reactivity Made Easy and Accurate with Automated Multireference Calculations

In organic reactivity studies, quantum chemical calculations play a pivotal role as the foundation of understanding and machine learning model development. While prevalent black-box methods like density functional theory (DFT) and coupled-cluster theory (e.g., CCSD(T)) have significantly advanced our understanding of chemical reactivity, they frequently fall short in describing multiconfigurational transition states and intermediates. Achieving a more accurate description necessitates the use of multireference methods. However, these methods have not been used at scale due to their often-faulty predictions without expert input. Here, we overcome this deficiency with automated multiconfigurational pair-density functional theory (MC-PDFT) calculations. We apply this method to 908 automatically generated organic reactions. We find 68% of these reactions present significant multiconfigurational character in which the automated multiconfigurational approach often provides a more accurate and/or efficient description than DFT and CCSD(T). This work presents the first high-throughput application of automated multiconfigurational methods to reactivity, enabled by automated active space selection algorithms and the computation of electronic correlation with MC-PDFT on-top functionals. This approach can be used in a black-box fashion, avoiding significant active space inconsistency error in both single- and multireference cases and providing accurate multiconfigurational descriptions when needed.

Analogies between photochemical reactions and ground-state post-transition-state bifurcations shed light on dynamical origins of selectivity

Revealing the origins of kinetic selectivity is one of the premier tasks of applied theoretical organic chemistry, and for many reactions, doing so involves comparing competing transition states. For some reactions, however, a single transition state leads directly to multiple products, in which case non-statistical dynamic effects influence selectivity control. The selectivity of photochemical reactions-where crossing between excited-state and ground-state surfaces occurs near ground-state transition structures that interconvert competing products-also should be controlled by the momentum of the reacting molecules as they return to the ground state in addition to the shape of the potential energy surfaces involved. Now, using machine-learning-assisted non-adiabatic molecular dynamics and multiconfiguration pair-density functional theory, these factors are examined for a classic photochemical reaction-the deazetization of 2,3-diazabicyclo[2.2.2]oct-2-ene-for which we demonstrate that momentum dominates the selectivity for hexadiene versus [2.2.2] bicyclohexane products.

Variational Pair-Density Functional Theory: Dealing with Strong Correlation at the Protein Scale

Multiconfigurational pair-density functional theory (MC-PDFT) offers a promising solution to the challenges faced by traditional density functional theory (DFT) in addressing molecular systems containing transition metals, open-shells, or strong correlations in general. By utilizing both the density and on-top pair-density, MC-PDFT can make use of a more flexible multiconfigurational wave function to capture the necessary static correlation, while the pair-density functional also includes the effect of dynamic correlation. So far, MC-PDFT has been used after a multiconfigurational self-consistent field (MCSCF) step, using the orbitals and configuration interaction coefficients from the converged MCSCF wave function to compute PDFT energies and properties. Here, instead, we propose to perform a direct optimization of the wave function using the pair-density functionals, resulting in a variational formulation of MC-PDFT. We derive the expressions for the wave function gradient and illustrate their similarity to standard MCSCF equations. Furthermore, we illustrate the accuracy on a set of singlet-triplet gaps as well as dissociation curves. Our findings highlight one of MC-PDFT's standout features: a reduced dependency on the active space size compared to conventional multiconfigurational wave function methodologies. Additionally, we show that the computational cost of MC-PDFT is potentially lower than MCSCF and often on-par with standard Kohn-Sham DFT, which is demonstrated by performing a MC-PDFT calculation of the entire ferredoxin protein with 1447 atoms and nearly 12 000 basis functions.

Minimum-Energy Conical Intersections by Compressed Multistate Pair-Density Functional Theory

Compressed multistate pair-density functional theory (CMS-PDFT) is a multistate version of multiconfiguration pair-density functional theory that can capture the correct topology of coupled potential energy surfaces (PESs) around conical intersections. In this work, we develop interstate coupling vectors (ISCs) for CMS-PDFT in the OpenMolcas and PySCF/mrh electronic structure packages. Yet, the main focus of this work is using ISCs to calculate minimum-energy conical intersections (MECIs) by CMS-PDFT. This is performed using the projected constrained optimization method in OpenMolcas, which uses ISCs to restrain the iterations to the conical intersection seam. We optimize the S1/S0 MECIs for ethylene, butadiene, and benzene and show that CMS-PDFT gives smooth PESs in the vicinities of the MECIs. Furthermore, the CMS-PDFT MECIs are in good agreement with the MECI calculated by the more expensive XMS-CASPT2 method.

How well do one-electron self-interaction-correction methods perform for systems with fractional electrons?

Recently developed locally scaled self-interaction correction (LSIC) is a one-electron SIC method that, when used with a ratio of kinetic energy densities (zσ) as iso-orbital indicator, performs remarkably well for both thermochemical properties as well as for barrier heights overcoming the paradoxical behavior of the well-known Perdew-Zunger self-interaction correction (PZSIC) method. In this work, we examine how well the LSIC method performs for the delocalization error. Our results show that both LSIC and PZSIC methods correctly describe the dissociation of H2+ and He2+ but LSIC is overall more accurate than the PZSIC method. Likewise, in the case of the vertical ionization energy of an ensemble of isolated He atoms, the LSIC and PZSIC methods do not exhibit delocalization errors. For the fractional charges, both LSIC and PZSIC significantly reduce the deviation from linearity in the energy vs number of electrons curve, with PZSIC performing superior for C, Ne, and Ar atoms while for Kr they perform similarly. The LSIC performs well at the endpoints (integer occupations) while substantially reducing the deviation. The dissociation of LiF shows both LSIC and PZSIC dissociate into neutral Li and F but only LSIC exhibits charge transfer from Li+ to F- at the expected distance from the experimental data and accurate ab initio data. Overall, both the PZSIC and LSIC methods reduce the delocalization errors substantially.

Assessing the importance of multireference correlation in predicting reversed conductance decay

In a classical electronic resistor, conductance decays as the device length increases according to Ohm's Law. While most molecular series display a comparable exponential decay in conductance with increasing molecular length, a class of single-molecule device series exists where conductance instead increases with molecular/device length, a phenomenon called reversed conductance decay. While reversals of conductance decay have been repeatedly theoretically predicted, they have been far more difficult to demonstrate experimentally. Previous studies have suggested that theoretical multi-reference(static) correlation errors may be a major cause of this discrepancy, yet most single-molecule transport methods are unable to treat multireference correlation. Using our unique multireference transport method based on non-equilibrium Green's function and multiconfigurational pair-density functional theory (NEGF-MCPDFT), we examined a previously predicted case of reversed conductance decay in systems of linear chains of phenyl rings with varying lengths and electrode designs. We compare our NEGF-MCPDFT results to those of non-multireference NEGF methods to quantify the exact role of static correlation in conductance decay reversals and clarify their relative importance to geometric and electrode design/coupling considerations.

Accurate Electronic and Optical Properties of Organic Doublet Radicals Using Machine Learned Range-Separated Functionals

Luminescent organic semiconducting doublet-spin radicals are unique and emergent optical materials because their fluorescent quantum yields (Φfl) are not compromised by the spin-flipping intersystem crossing (ISC) into a dark high-spin state. The multiconfigurational nature of these radicals challenges their electronic structure calculations in the framework of single-reference density functional theory (DFT) and introduces room for method improvement. In the present study, we extended our earlier development of ML-ωPBE [J. Phys. Chem. Lett., 2021, 12, 9516-9524], a range-separated hybrid (RSH) exchange-correlation (XC) functional constructed using the stacked ensemble machine learning (SEML) algorithm, from closed-shell organic semiconducting molecules to doublet-spin organic semiconducting radicals. We assessed its performance for a new test set of 64 doublet-spin radicals from five categories while placing all previously compiled 3926 closed-shell molecules in the new training set. Interestingly, ML-ωPBE agrees with the nonempirical OT-ωPBE functional regarding the prediction of the molecule-dependent range-separation parameter (ω), with a small mean absolute error (MAE) of 0.0197 a0-1, but saves the computational cost by 2.46 orders of magnitude. This result demonstrates an outstanding domain adaptation capacity of ML-ωPBE for diverse organic semiconducting species. To further assess the predictive power of ML-ωPBE in experimental observables, we also applied it to evaluate absorption and fluorescence energies (Eabs and Efl) using linear-response time-dependent DFT (TDDFT), and we compared its behavior with nine popular XC functionals. For most radicals, ML-ωPBE reproduces experimental measurements of Eabs and Efl with small MAEs of 0.299 and 0.254 eV, only marginally different from those of OT-ωPBE. Our work illustrates a successful extension of the SEML framework from closed-shell molecules to doublet-spin radicals and will open the venue for calculating optical properties for organic semiconductors using single-reference TDDFT.

Hybrid Density Functional Valence Bond Method with Multistate Treatment

Recently, a hybrid density functional valence bond (VB) method, λ-DFVB(U), has been proposed and shown to give accuracy that is comparable to that of CASPT2 in calculations of atomization energies, atomic excitation energies, and reaction barriers, while its computational cost is approximately the same as the valence bond self-consistent-field (VBSCF) method. However, the interaction between electronic states is not included in λ-DFVB(U) since the last step of λ-DFVB(U) is not a diagonalization of the Hamiltonian matrix on the electronic state basis. Therefore, λ-DFVB(U) gives the wrong topology of the potential energy surfaces (PESs) near the conical intersection region. In the present paper, we propose a novel hybrid density functional VB method with multistate treatment, named λ-DFVB(MS), in which an effective Hamiltonian matrix is constructed on the basis of the diabatic states obtained by the valence-bond-based compression approach for the diabatization scheme, and the interaction between electronic states can be included through the diagonalization of the effective Hamiltonian matrix. Test calculations show that λ-DFVB(MS) gives the correct topology of the PESs near the conical intersection region. We also show that the VBSCF wave function with selected VB structures can be applied as a reference in λ-DFVB(MS).

Stretching Bonds without Breaking Symmetries in Density Functional Theory

Kohn-Sham density functional theory (KS-DFT) stands out among electronic structure methods due to its balance of accuracy and computational efficiency. However, to achieve chemically accurate energies, standard density functional approximations in KS-DFT often need to break underlying symmetries, a long-standing "symmetry dilemma". By employing fragment spin densities as the main variables in calculations (rather than total molecular densities, as in KS-DFT), we present an embedding framework in which this symmetry dilemma is understood and partially resolved. The spatial overlap between fragment densities is used as the main ingredient to construct a simple, physically motivated approximation to a universal functional of the fragment densities. This "overlap approximation" is shown to significantly improve semilocal KS-DFT binding energies of molecules without artificially breaking either charge or spin symmetries. The approach is shown to be applicable to covalently bonded molecules and to systems of the "strongly correlated" type.

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.

Toward a Stochastic Complete Active Space Second-Order Perturbation Theory

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

My life in science: Lessons for yours?

Because of an acquired obsession to understand as much as possible in a limited but important area of science and because of optimism, luck, and help from others, my life in science turned out to be much better than I or others could have expected or planned. This is the story of how that happened, and also the story of the groundstate density functional theory of electronic structure, told from a personal perspective.

Universal Generalization of Density Functional Theory for Static Correlation

A major challenge for density functional theory (DFT) is its failure to treat static correlation, yielding errors in predicted charges, band gaps, van der Waals forces, and reaction barriers. Here we combine one- and two-electron reduced density matrix (1- and 2-RDM) theories with DFT to obtain a universal O(N^{3}) generalization of DFT for static correlation. Using the lowest unitary invariant of the cumulant 2-RDM, we generate a 1-RDM functional theory that corrects the convexity of any DFT functional to capture static correlation in its fractional orbital occupations. Importantly, the unitary invariant yields a predictive theory by revealing the dependence of the correction's strength upon the trace of the two-electron repulsion matrix. We apply the theory to the barrier to rotation in ethylene, the relative energies of the benzynes, as well as an 11-molecule, dissociation benchmark. By inheriting the computational efficiency of DFT without sacrificing the treatment of static correlation, the theory opens new possibilities for the prediction and interpretation of significant quantum molecular effects and phenomena.

What Controls the Quality of Photodynamical Simulations? Electronic Structure Versus Nonadiabatic Algorithm

The field of nonadiabatic dynamics has matured over the last decade with a range of algorithms and electronic structure methods available at the moment. While the community currently focuses more on developing and benchmarking new nonadiabatic dynamics algorithms, the underlying electronic structure controls the outcome of nonadiabatic simulations. Yet, the electronic-structure sensitivity analysis is typically neglected. In this work, we present a sensitivity analysis of the nonadiabatic dynamics of cyclopropanone to electronic structure methods and nonadiabatic dynamics algorithms. In particular, we compare wave function-based CASSCF, FOMO-CASCI, MS- and XMS-CASPT2, density-functional REKS, and semiempirical MRCI-OM3 electronic structure methods with the Landau-Zener surface hopping, fewest switches surface hopping, and ab initio multiple spawning with informed stochastic selection algorithms. The results clearly demonstrate that the electronic structure choice significantly influences the accuracy of nonadiabatic dynamics for cyclopropanone even when the potential energy surfaces exhibit qualitative and quantitative similarities. Thus, selecting the electronic structure solely on the basis of the mapping of potential energy surfaces can be misleading. Conversely, we observe no discernible differences in the performance of the nonadiabatic dynamics algorithms across the various methods. Based on the above results, we discuss the present-day practice in computational photodynamics.

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.

Linearized Pair-Density Functional Theory for Vertical Excitation Energies

Multiconfiguration pair-density functional theory (MC-PDFT) is a computationally efficient method that computes the energies of electronic states in a state specific or state average framework via an on-top functional. However, MC-PDFT does not include state interaction among these states since the final energies do not come from the diagonalization of an effective model-space Hamiltonian. Recently, multistate extensions such as linearized PDFT (L-PDFT) have been developed to accurately model the potentials near conical intersections and avoided crossings. However, there has not been any systematic study evaluating their performance for predicting vertical excitations at the equilibrium geometry of a molecule, when the excited states are generally well separated. In this paper, we report the performance of L-PDFT on the extensive QUESTDB data set of vertical excitations using a database of automatically selected active spaces. We show that L-PDFT performs well on all these excitations and successfully reproduces the performance of MC-PDFT. These results further demonstrate the potential of L-PDFT, as its scaling is constant with the number of states included in the state-average manifold, whereas MC-PDFT scales linearly in this regard.

Chemically accurate singlet-triplet gaps of organic chromophores and linear acenes by the random phase approximation and σ-functionals

Predicting the energy differences between different spin-states is challenging for many widely used ab initio electronic structure methods. We here assess the ability of the direct random phase approximation (dRPA), dRPA plus two different screened second-order exchange (SOX) corrections, and σ-functionals to predict adiabatic singlet-triplet gaps. With mean absolute deviations of below 0.1 eV to experimental reference values, independent of the Kohn-Sham starting point, dRPA and σ-functionals accurately predict singlet-triplet gaps of 18 organic chromophores. The addition of SOX corrections to dRPA considerably worsens agreement with experiment, adding to the mounting evidence that dRPA+SOX methods are not generally applicable beyond-RPA methods. Also for a series of linear acene chains with up to ten fused rings, dRPA, and σ-functionals are in excellent agreement with coupled-cluster single double triple reference data. In agreement with advanced multi-reference methods, dRPA@PBE and σ-functional@PBE predict a singlet ground state for all chain lengths, while dRPA@PBE0 and σ-functional@PBE0 predict a triplet ground state for longer acenes. Our work shows dRPA and σ-functionals to be reliable methods for calculating singlet-triplet gaps in aromatic molecules.

Multiconfigurational Pair-Density Functional Theory Is More Complex than You May Think

Multiconfigurational pair-density functional theory (MC-PDFT) is a promising way to describe both strong and dynamic correlations in an inexpensive way. The functionals in MC-PDFT are often "translated" from standard spin density functionals. However, these translated functionals can in principle lead to "translated spin densities" with a nonzero imaginary component. Current developments so far neglect this imaginary part by simply setting it to zero. In this work, we show how this imaginary component is actually needed to reproduce the correct physical behavior in a range of cases, especially low-spin open shells. We showcase the resulting formalism on both local density approximation and generalized gradient approximation functionals and illustrate the numerical behavior by benchmarking a number of singlet-triplet splittings (ST gaps) of organic diradicals and low-lying excited states of some common organic molecules. The results demonstrate that this scheme improves existing translated functionals and gives more accurate results, even with minimal active spaces.

Multiconfiguration Pair-Density Functional Theory for Vertical Excitation Energies in Actinide Molecules

Modeling actinides with electronic structure theories is challenging because these systems present a strong ligand field and metal-ligand covalency. We systematically investigate the effectiveness of pair-density functional theory (PDFT) for the calculation of vertical excitation energies in An(III), [AnIIICl6]3-, and [AnVIO2]2+ (An = U, Np, Pu, and Am). We compare the performance of PDFT, hybrid PDFT, and multistate PDFT with traditional active-space methods followed by perturbation theory, like multistate CASPT2, and with experimental data. Overall, multistate PDFT gives quantitative agreement with multistate CASPT2 at a significantly reduced computational cost.

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

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

The General Atomic and Molecular Electronic Structure System (GAMESS): Novel Methods on Novel Architectures

The primary focus of GAMESS over the last 5 years has been the development of new high-performance codes that are able to take effective and efficient advantage of the most advanced computer architectures, both CPU and accelerators. These efforts include employing density fitting and fragmentation methods to reduce the high scaling of well-correlated (e.g., coupled-cluster) methods as well as developing novel codes that can take optimal advantage of graphical processing units and other modern accelerators. Because accurate wave functions can be very complex, an important new functionality in GAMESS is the quasi-atomic orbital analysis, an unbiased approach to the understanding of covalent bonds embedded in the wave function. Best practices for the maintenance and distribution of GAMESS are also discussed.

A Perspective on Sustainable Computational Chemistry Software Development and Integration

The power of quantum chemistry to predict the ground and excited state properties of complex chemical systems has driven the development of computational quantum chemistry software, integrating advances in theory, applied mathematics, and computer science. The emergence of new computational paradigms associated with exascale technologies also poses significant challenges that require a flexible forward strategy to take full advantage of existing and forthcoming computational resources. In this context, the sustainability and interoperability of computational chemistry software development are among the most pressing issues. In this perspective, we discuss software infrastructure needs and investments with an eye to fully utilize exascale resources and provide unique computational tools for next-generation science problems and scientific discoveries.

Understanding Antiferromagnetic and Ligand Field Effects on Spin Crossover in a Triple-Decker Dimeric Cr(II) Complex

Two possible explanations for the temperature dependence of spin-crossover (SCO) behavior in the dimeric triple-decker Cr(II) complex ([(η5-C5Me5)Cr(μ2:η5-P5)Cr(η5-C5Me5)]+) have been offered. One invokes variations in antiferromagnetic interactions between the two Cr(II) ions, whereas the other posits the development of a strong ligand-field effect favoring the low-spin ground state. We perform multireference electronic structure calculations based on the multiconfiguration pair-density functional theory to resolve these effects. We find quintet, triplet, and singlet electronic ground states, respectively, for the experimental geometries at high, intermediate, and low temperatures. The ground-state transition from quintet to triplet at an intermediate temperature derives from increased antiferromagnetic interactions between the two Cr(II) ions. By contrast, the ground-state transition from triplet to singlet at low temperature can be attributed to increased ligand-field effects, which dominate with continued variations in antiferromagnetic coupling. This study provides quantitative detail for the degree to which these two effects can act in concert for the observed SCO behavior in this complex and others subject to temperature-dependent variations in geometry.

An Accurate Density Coherence Functional for Hybrid Multiconfiguration Density Coherence Functional Theory

We present hybrid multiconfiguration density coherence functional theory (HMC-DCFT), and we optimize a density coherence functional by parametrization against a diverse data set of 59 bond energies and 60 barrier heights. We compare the results to calculations on the same data set by CASSCF, CASPT2, six Kohn-Sham and hybrid Kohn-Sham exchange-correlation functionals, and three on-top functionals for pair-density functional theory (PDFT) and hybrid PDFT. The new functional has better accuracy than all compared methods.

Intruder-free cumulant-truncated driven similarity renormalization group second-order multireference perturbation theory

Accurate multireference electronic structure calculations are important for constructing potential energy surfaces. Still, even in the case of low-scaling methods, their routine use is limited by the steep growth of the computational and storage costs as the active space grows. This is primarily due to the occurrence of three- and higher-body density matrices or, equivalently, their cumulants. This work examines the effect of various cumulant truncation schemes on the accuracy of the driven similarity renormalization group second-order multireference perturbation theory. We test four different levels of three-body reduced density cumulant truncations that set different classes of cumulant elements to zero. Our test cases include the singlet-triplet gap of CH2, the potential energy curves of the XΣg+1 and AΣu+3 states of N2, and the singlet-triplet splittings of oligoacenes. Our results show that both relative and absolute errors introduced by these cumulant truncations can be as small as 0.5 kcal mol-1 or less. At the same time, the amount of memory required is reduced from O(NA6) to O(NA5), where NA is the number of active orbitals. No additional regularization is needed to prevent the intruder state problem in the cumulant-truncated second-order driven similarity renormalization group multireference perturbation theory methods.

Unraveling the Spin-State Energetics of FeN4 Complexes with Ab Initio Methods

A systematic analysis was conducted to explore the spin-state energetics of a series of 19 FeN4 complexes. The performance of a large number of multireference methods was assessed, highlighting the significant challenges associated with accurately describing the spin-state energetics of FeN4 complexes. Most multireference methods were found to be susceptible to errors originating from the reference CASSCF wavefunction, leading to an overstabilization of high-spin states. Nonetheless, a few multireference methods, namely, CASPT2/CC, DSRG-MRPT3, and LDSRG(2), demonstrated promising performance compared to the benchmark CCSD(T) method. Furthermore, our study revealed that FeN4 complexes having a quintet ground state are exceedingly rare. Accordingly, only one specific model (Fe(L2)) and one synthesized complex (Fe(OTBP)) have the quintet ground state among the studied complexes. This scarcity of quintet FeN4 complexes highlights the unique nature of these systems and raises intriguing questions regarding the factors influencing spin states, such as the size of the macrocycle cavity, the introduction of substituents, or the induction of out-of-plane deformation.

Development of Fluorescent Isocoumarin-Fused Oxacyclononyne - 1,2,3-Triazole Pairs

Fluorescent isocoumarin-fused cycloalkynes, which are reactive in SPAAC and give fluorescent triazoles regardless of the azide nature, have been developed. The key structural feature that converts the non-fluorescent cycloalkyne/triazole pair to its fluorescent counterpart is the pi-acceptor group (COOMe, CN) at the C6 position of the isocoumarin ring. The design of the fluorescent cycloalkyne/triazole pairs is based on the theoretical study of the S1 state deactivation mechanism of the non-fluorescent isocoumarin-fused cycloalkyne IC9O using multi-configurational ab initio and DFT methodologies. The calculations revealed that deactivation proceeds through the electrocyclic ring opening of the α-pyrone cycle and is accompanied by a redistribution of electron density in the fused benzene ring. We proposed that the S1 excited state deactivation barrier could be increased by introducing a pi-acceptor group into a position that is in direct conjugation with the formed C=O group and has a reduced electron density in the transition state. As a proof of concept, we designed and synthesized two fluorescent isocoumarin-fused cycloalkynes IC9O-COOMe and IC9O-CN bearing pi-acceptors at the C6 position. The importance of the nature of a pi-acceptor group was shown by the example of much less fluorescent CF3 -substituted cycloalkyne IC9O-CF3 .

The mechanism of visible light-induced C-C cross-coupling by Csp3-H bond activation

Csp3-C cross-coupling by activating Csp3-H bonds is a dream reaction for the chemical community, and visible light-induced transition metal-catalysis under mild reaction conditions is considered a powerful tool to achieve it. Advancement of this research area is still in its infancy because of the chemical and technical complexity of this catalysis. Mechanistic studies illuminating the operative reaction pathways can rationalize the increasing amount of experimental catalysis data and provide the knowledge allowing faster and rational advances in the field. This goal requires complementary experimental and theoretical mechanistic studies, as each of them is unfit to clarify the operative mechanisms alone. In this tutorial review we summarize representative experimental and computational mechanistic studies, highlighting weaknesses, strengths, and synergies between the two approaches.

Assessment of DFT functionals for a minimal nitrogenase [Fe(SH)4H]- model employing state-of-the-art ab initio methods

We have designed a [Fe(SH)4H]- model with the fifth proton binding either to Fe or S. We show that the energy difference between these two isomers (∆E) is hard to estimate with quantum-mechanical (QM) methods. For example, different density functional theory (DFT) methods give ∆E estimates that vary by almost 140 kJ/mol, mainly depending on the amount of exact Hartree-Fock included (0%-54%). The model is so small that it can be treated by many high-level QM methods, including coupled-cluster (CC) and multiconfigurational perturbation theory approaches. With extrapolated CC series (up to fully connected coupled-cluster calculations with singles, doubles, and triples) and semistochastic heat-bath configuration interaction methods, we obtain results that seem to be converged to full configuration interaction results within 5 kJ/mol. Our best result for ∆E is 101 kJ/mol. With this reference, we show that M06 and B3LYP-D3 give the best results among 35 DFT methods tested for this system. Brueckner doubles coupled cluster with perturbaitve triples seems to be the most accurate coupled-cluster approach with approximate triples. CCSD(T) with Kohn-Sham orbitals gives results within 4-11 kJ/mol of the extrapolated CC results, depending on the DFT method. Single-reference CC calculations seem to be reasonably accurate (giving an error of ∼5 kJ/mol compared to multireference methods), even if the D1 diagnostic is quite high (0.25) for one of the two isomers.

Predicting Magnetic Coupling and Spin-Polarization Energy in Triangulene Analogues

Triangulene and its analogue metal-free magnetic systems have garnered increasing attention since their discovery. Predicting the magnetic coupling and spin-polarization energy with quantitative accuracy is beyond the predictive power of today's density functional theory (DFT) due to their intrinsic multireference character. Herein, we create a benchmark dataset of 25 magnetic systems with nonlocal spin densities, including the triangulene monomer, dimer, and their analogues. We calculate the magnetic coupling (J) and spin-polarization energy (ΔE_spin) of these systems using complete active space self-consistent field (CASSCF) and coupled-cluster methods as high-quality reference values. This reference data is then used to benchmark 22 DFT functionals commonly used in material science. Our results show that, while some functionals consistently correctly predict the qualitative character of the ground state, achieving quantitative accuracy with small relative errors is currently not feasible. PBE0, M06-2X, and MN15 are predicting the correct electronic ground state for all systems investigated here and also have the lowest mean absolute error for predicting both ΔE_spin (0.34, 0.32, and 0.31 eV) and J (11.74, 12.66, and 10.64 meV). They may therefore also serve as starting points for higher-level methods such as the GW or the random phase approximation. As other functionals fail for the prediction of the ground state, they cannot be recommended for metal-free magnetic systems.

A Combined Wave Function and Density Functional Approach for K-Edge X-ray Absorption Near-Edge Spectroscopy: A Case Study of Hydrated First-Row Transition Metal Ions

The prediction of X-ray absorption spectra (XAS) of transition metal complexes has important and broad application areas in chemistry and biology. In this letter, we have investigated the predictive ability of multiconfiguration pair-density functional theory (MC-PDFT) for X-ray absorption spectra by calculating the metal K pre-edge features of aquated 3d transition metal ions in common oxidation states. MC-PDFT results were compared with experimentally measured spectra as well as analyzed against results from restricted active-space second-order perturbation theory (RASPT2) and time-dependent density functional theory (TDDFT). As expected, TDDFT performs well for excited states that can be accurately represented by singly excited configurations but fails for excited states where higher order excitations become important. On the other hand, both RASPT2 and MC-PDFT provide quantitatively accurate results for all excited states irrespective of their character. While core-level spectroscopy with RASPT2 is accurate, it is computationally expensive. Our results show that MC-PDFT performs equally well with significantly lower computational cost and is an encouraging alternate approach for X-ray spectroscopies.

Linearized Pair-Density Functional Theory

Multiconfiguration pair-density functional theory (MC-PDFT) is a post-SCF multireference method that has been successful at computing ground- and excited-state energies. However, MC-PDFT is a single-state method in which the final MC-PDFT energies do not come from diagonalization of a model-space Hamiltonian matrix, and this can lead to inaccurate topologies of potential energy surfaces near locally avoided crossings and conical intersections. Therefore, in order to perform physically correct ab initio molecular dynamics with electronically excited states or to treat Jahn-Teller instabilities, it is necessary to develop a PDFT method that recovers the correct topology throughout the entire nuclear configuration space. Here we construct an effective Hamiltonian operator, called the linearized PDFT (L-PDFT) Hamiltonian, by expanding the MC-PDFT energy expression to first order in a Taylor series of the wave function density. Diagonalization of the L-PDFT Hamiltonian gives the correct potential energy surface topology near conical intersections and locally avoided crossings for a variety of challenging cases including phenol, methylamine, and the spiro cation. Furthermore, L-PDFT outperforms MC-PDFT and previous multistate PDFT methods for predicting vertical excitations from a variety of representative organic chromophores.

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.

Dipole Moments and Transition Dipole Moments Calculated by Pair-Density Functional Theory with State Interaction

We develop response-function algorithms for dipole moments and transition dipole moments for compressed multistate pair-density functional theory (CMS-PDFT). We use the method of undetermined Lagrange multipliers to derive analytical expressions and validate them using numerical differentiation. We test the accuracy of the magnitudes of predicted ground-state and excited-state dipole moments, the orientations of these dipole moments, and the orientation of transition dipole moments by comparison to experimental data. We show that CMS-PDFT has good accuracy for these quantities, and we also show that, unlike methods that neglect state interaction, CMS-PDFT yields correct behavior for the dipole moment curves in the vicinity of conical intersections. This work, therefore, opens the door to molecular dynamic simulations in strong electric fields, and we envision that CMS-PDFT can now be used to discover chemical reactions that can be controlled by an oriented external electric field upon photoexcitation of the reactants.

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

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

Reimagining the Wave Function in Density Functional Theory: Exploring Strongly Correlated States in Pancake-Bonded Radical Dimers

Pancake bonding between π-conjugated radicals challenges conventional electronic structure approximations, due to the presence of both dispersion (van der Waals) interactions and "strong" electron correlation. Here we use a reimagined wave function-in-density functional theory (DFT) approach to model pancake bonds. Our generalized self-interaction correction extends DFT's reference system of noninteracting electrons, by introducing electron-electron interactions within an active space. We show that a small variation on our previous derivation recovers a DFT-corrected complete active space method proposed by Pijeau and Hohenstein. Comparison of the two approaches shows that the latter provides reasonable dissociation curves for single bonds and pancake bonds, including excited states inaccessible to conventional linear response time-dependent DFT. The results motivate broader adoption of wavefunction-in-DFT approaches for modeling pancake bonds.

New Gradient Correction Scheme for Electronically Nonadiabatic Dynamics Involving Multiple Spin States

It has been recommended that the best representation to use for trajectory surface hopping (TSH) calculations is the fully adiabatic basis in which the Hamiltonian is diagonal. Simulations of intersystem crossing processes with conventional TSH methods require an explicit computation of nonadiabatic coupling vectors (NACs) in the molecular-Coulomb-Hamiltonian (MCH) basis, also called the spin-orbit-free basis, in order to compute the gradient in the fully adiabatic basis (also called the diagonal representation). This explicit requirement destroys some of the advantages of the overlap-based algorithms and curvature-driven algorithms that can be used for the most efficient TSH calculations. Therefore, although these algorithms allow one to perform NAC-free simulations for internal conversion processes, one still requires NACs for intersystem crossing. Here, we show that how the NAC requirement is circumvented by a new computation scheme called the time-derivative-matrix scheme.