The Complete Active Space Self-Consistent Field Method and its Applications in Electronic Structure Calculations

Metal-Ligand Bonds in Rare Earth Metal-Biphenyl Complexes

A series of theoretical methods, including density functional theory, multiconfiguration molecular orbital theory, and ab initio valence bond theory, are devoted to understanding the metal-ligand bonds in M-BP (BP = biphenyl; M = Sc, Y, or La) complexes. Different from most transition metal-BP complexes, the most stable metal-biphenyl conformers are not half-sandwich but clamshell. Energy decomposition analysis results reveal that the M-BP bonds in the clamshell conformers possess extra-large orbital relaxation. According to the wave function analysis, 2-fold donations and 2-fold back-donations exist in the clamshell M-BP bonds. The back-donations from M to BP are quite strong, while donations from BP to M are quite weak. Our work improves our understanding of the metal-ligand bonds, which can be considered as the "reversed" Dewar-Chatt-Duncanson model.

Generalized Formulation of the Density Functional Tight Binding-Based Restricted Ensemble Kohn-Sham Method with Onsite Correction to Long-Range Correction

We present a generalized formulation for the combination of the density functional tight binding (DFTB) approach and the state-interaction state-average spin-restricted ensemble-referenced Kohn-Sham (SI-SA-REKS or SSR) method by considering onsite correction (OC) as well as the long-range corrected (LC) functional. The OC contribution provides more accurate energies and analytic gradients for individual microstates, while the multireference character of the SSR provides the correct description for conical intersections. We benchmark the LC-OC-DFTB/SSR method against various DFTB calculation methods for excitation energies and conical intersection structures with π/π* or n/π* characters. Furthermore, we perform excited-state molecular dynamics simulations with a molecular rotary motor with variations of LC-OC-DFTB/SSR approaches. We show that the OC contribution to the LC functional is crucial to obtain the correct geometry of conical intersections.

Calculation of Core-Excited and Core-Ionized States Using Variational Quantum Deflation Method and Applications to Photocatalyst Modeling

The possibility of performing quantum-chemical calculations using quantum computers has attracted much interest. Variational quantum deflation (VQD) is a quantum-classical hybrid algorithm for the calculation of excited states with noisy intermediate-scale quantum devices. Although the validity of this method has been demonstrated, there have been few practical applications, primarily because of the uncertain effect of calculation conditions on the results. In the present study, calculations of the core-excited and core-ionized states for common molecules based on the VQD method were examined using a classical computer, focusing on the effects of the weighting coefficients applied in the penalty terms of the cost function. Adopting a simplified procedure for estimating the weighting coefficients based on molecular orbital levels allowed these core-level states to be successfully calculated. The O 1s core-ionized state for a water molecule was calculated with various weighting coefficients, and the resulting ansatz states were systematically examined. The application of this technique to functional materials was demonstrated by calculating the core-level states for titanium dioxide (TiO₂) and nitrogen-doped TiO₂ models. The results demonstrate that VQD calculations employing an appropriate cost function can be applied to the analysis of functional materials in conjunction with an experimental approach.

Zero-Field Splitting Calculations by Multiconfiguration Pair-Density Functional Theory

Zero-field splitting (ZFS) is a fundamental molecular property that is especially relevant for single-molecule magnets (SMMs), electron paramagnetic resonance spectra, and quantum computing. Developing a method that can accurately predict ZFS parameters can be very powerful for designing new SMMs. One of the challenges is to include external correlation in an inherently multiconfigurational open-shell species for the accurate prediction of magnetic properties. Previously available methods depend on expensive multireference perturbation theory calculations to include external correlation. In this paper, we present spin-orbit-inclusive multiconfiguration and multistate pair-density functional theory (MC-PDFT) calculations of ZFSs; these calculations have a cost comparable to complete-active-space self-consistent field (CASSCF) theory, but they include correlation external to the active space. We found that combining a multistate formulation of MC-PDFT, namely, compressed-state multistate pair-density functional theory, with orbitals optimized by weighted-state-averaged CASSCF, yields reasonably accurate ZFS results.

Origin of the different reactivity of the high-valent coinage-metal complexes [RCuiii Me3 ]- and [RAgiii Me3 ]- (R=allyl)

High-valent tetraalkylcuprates(iii) and -argentates(iii) are key intermediates of copper- and silver-mediated C-C coupling reactions. Here, we investigate the previously reported contrasting reactivity of [RMiii Me3 ]- complexes (M=Cu, Ag and R=allyl) with energy-dependent collision-induced dissociation experiments, advanced quantum-chemical calculations and kinetic computations. The gas-phase fragmentation experiments confirmed the preferred formation of the [RCuMe]- anion upon collisional activation of the cuprate(iii) species, consistent with a homo-coupling reaction, whereas the silver analogue primarily yielded [AgMe2 ]- , consistent with a cross-coupling reaction. For both complexes, density functional theory calculations identified one mechanism for homo coupling and four different ones for cross coupling. Of these pathways, an unprecedented concerted outer-sphere cross coupling is of particular interest, because it can explain the formation of [AgMe2 ]- from the argentate(iii) species. Remarkably, the different C-C coupling propensities of the two [RMiii Me3 ]- complexes become only apparent when properly accounting for the multi-configurational character of the wave function for the key transition state of [RAgMe3 ]- . Backed by the obtained detailed mechanistic insight for the gas-phase reactions, we propose that the previously observed cross-coupling reaction of the silver complex in solution proceeds via the outer-sphere mechanism.

An efficient implementation of the NEVPT2 and CASPT2 methods avoiding higher-order density matrices

A factorization of the matrix elements of the Dyall Hamiltonian in N-electron valence state perturbation theory allowing their evaluation with a computational effort comparable to the one needed for the construction of the third-order reduced density matrix at the most is presented. Thus, the computational bottleneck arising from explicit evaluation of the fourth-order density matrix is avoided. It is also shown that the residual terms arising in the case of an approximate complete active space configuration interaction solution and containing even the fifth-order density matrix for two excitation classes can be evaluated with little additional effort by choosing again a favorable factorization of the corresponding matrix elements. An analogous argument is also provided for avoiding the fourth-order density matrix in complete active space second-order perturbation theory. Practical calculations indicate that such an approach leads to a considerable gain in computational efficiency without any compromise in numerical accuracy or stability.

iCISCF: An Iterative Configuration Interaction-Based Multiconfigurational Self-Consistent Field Theory for Large Active Spaces

An iterative configuration interaction (iCI)-based multiconfigurational self-consistent field (SCF) theory, iCISCF, is proposed to handle systems that require large active spaces. The success of iCISCF stems from three ingredients: (1) efficient selection of individual configuration state functions spanning the active space while maintaining full spin symmetry; (2) the use of Jacobi rotation for optimization of the active orbitals in conjunction with a quasi-Newton algorithm for the core/active-virtual and core-active orbital rotations; (3) a second-order perturbative treatment of the residual space left over by the selection procedure (i.e., iCISCF(2)). Several examples that go beyond the capability of CASSCF are taken as showcases to reveal the efficacy of iCISCF and iCISCF(2), facilitated by iCAS for imposed automatic selection and localization of active orbitals.

Spin-Orbit Natural Transition Orbitals and Spin-Forbidden Transitions

Natural transition orbitals (NTOs) are in widespread use for visualizing and analyzing electronic transitions. The present work introduces the analysis of formally spin-forbidden transitions with the help of complex-valued spin-orbit (SO) NTOs. The analysis specifically focuses on the components in such transitions that cause their intensity to be nonzero because of SO coupling. Transition properties such as transition dipole moments are partitioned into SO-NTO hole-particle pairs, such that contributions to the intensity from specific occupied and unoccupied orbitals are obtained. The method has been implemented within the restricted active space (RAS) self-consistent field wave function theory framework, with SO coupling treated by RAS state interaction. SO-NTOs have a broad range of potential applications, which is illustrated by the T₂-S₁ state mixing in pyrazine, spin-forbidden versus spin-allowed 4f-5d transitions in the Tb³⁺ ion, and the phosphorescence of tris(2-phenylpyridine) iridium [Ir(ppy)₃].

Reactivity of the Bicyclic Amido-Substituted Silicon(I) Ring Compound Si4 {N(SiMe3 )Mes}4 with FLP-Type Character

The bicyclic amido-substituted silicon(I) ring compound Si4 {N(SiMe3 )Mes}4 2 (Mes=Mesityl=2,4,6-Me3 C6 H2 ) features enhanced zwitterionic character and different reactivity from the analogous compound Si4 {N(SiMe3 )Dipp}4 1 (Dipp=2,6-i Pr2 C6 H3 ) due to the smaller mesityl substituents. In a reaction with the N-heterocyclic carbene NHCMe 4 (1,3,4,5-tetramethyl-imidazol-2-ylidene), we observe adduct formation to give Si4 {N(SiMe3 )Mes}4  ⋅ NHCMe 4 (3). This adduct reacts further with the Lewis acid BH3 to yield the Lewis acid-base complex Si4 {N(SiMe3 )Mes}4  ⋅ NHCMe 4  ⋅ BH3 (4). Coordination of AlBr3 to 2 leads to the adduct 5. Calculated proton affinities and fluoride ion affinities reveal highly Lewis basic and very weak Lewis acidic character of the low-valent silicon atoms in 1 and 2. This is confirmed by protonation of 1 and 2 with Brookharts acid yielding 6 and 7. Reaction with diphenylacetylene only occurs at 111 °C with 2 in toluene and is accompanied by fragmentation of 2 to afford the silacyclopropene 8 and the trisilanorbornadiene species 9.

A Density Matrix Renormalization Group Study of the Low-Lying Excited States of a Molybdenum Carbonyl-Nitrosyl Complex

A density matrix renormalization group-self consistent field (DMRG-SCF) study has been carried out to calculate the low-lying excited states of CpMo(CO)2 NO, a molybdenum complex containing NO and CO ligands. In order to automatically select an appropriate active space, a novel procedure employing the maximum single-orbital entropy for several states has been introduced and shown to be efficient and easy-to-implement when several electronic states are simultaneously considered. The analysis of the resulting natural transition orbitals and charge-transfer numbers shows that the lowest five excited electronic states are excitation into metal-NO antibonding orbitals, which offer the possibility for nitric oxide (NO) photorelease after excitation with visible light. Higher excited states are metal-centered excitations with contributions of metal-CO antibonding orbitals, which may serve as a gateway for carbon monoxide (CO) delivery. Time-dependent density functional theory calculations done for comparison, show that the state characters agree remarkably well with those from DMRG-SCF, while excitation energies are 0.4-1.0 eV red-shifted with respect to the DMRG-SCF ones.

Second-Order CASSCF Algorithm with the Cholesky Decomposition of the Two-Electron Integrals

In this contribution, we present the implementation of a second-order complete active space–self-consistent field (CASSCF) algorithm in conjunction with the Cholesky decomposition of the two-electron repulsion integrals. The algorithm, called norm-extended optimization, guarantees convergence of the optimization, but it involves the full Hessian and is therefore computationally expensive. Coupling the second-order procedure with the Cholesky decomposition leads to a significant reduction in the computational cost, reduced memory requirements, and an improved parallel performance. As a result, CASSCF calculations of larger molecular systems become possible as a routine task. The performance of the new implementation is illustrated by means of benchmark calculations on molecules of increasing size, with up to about 3000 basis functions and 14 active orbitals.

Metal-Metal Bonding in Actinide Dimers: U2 and U2. J Am Chem Soc 2021;143:17023-17028. [PMID: 34609860 DOI: 10.1021/jacs.1c06417]

Understanding direct metal-metal bonding between actinide atoms has been an elusive goal in chemistry for years. We report for the first time the anion photoelectron spectrum of U₂^-. The threshold of the lowest electron binding energy (EBE) spectral band occurs at 1.0 eV, which corresponds to the electron affinity (EA) of U₂, whereas the vertical detachment energy of U₂^- is found at EBE ∼ 1.2 eV. Electronic structure calculations on U₂ and U₂^- were carried out with state-of-the-art theoretical methods. The computed values of EA(U₂) and EA(U) and the difference between the computed dissociation energies of U₂ and U₂^- are found to be internally consistent and consistent with experiment. Analysis of the bonds in U₂ and U₂^- shows that while U₂ has a formal quintuple bond, U₂^- has a quadruple bond, even if the effective bond orders differ only by 0.5 unit instead of one unit. The resulting experimental-computational synergy elucidates the nature of metal-metal bonding in U₂ and U₂^-.

Multireference Algebraic Diagrammatic Construction Theory for Excited States: Extended Second-Order Implementation and Benchmark

We present an implementation and benchmark of new approximations in multireference algebraic diagrammatic construction theory for simulations of neutral electronic excitations and UV/vis spectra of strongly correlated molecular systems (MR-ADC). Following our work on the first-order MR-ADC approximation [J. Chem. Phys. 2018, 149, 204113], we report the strict and extended second-order MR-ADC methods (MR-ADC(2) and MR-ADC(2)-X) that combine the description of static and dynamic electron correlation in the ground and excited electronic states without relying on state-averaged reference wave functions. We present an extensive benchmark of the new MR-ADC methods for excited states in several small molecules, including the carbon dimer, ethylene, and butadiene. Our results demonstrate that, for weakly correlated electronic states, the MR-ADC(2) and MR-ADC(2)-X methods outperform the third-order single-reference ADC approximation and are competitive with the results from equation-of-motion coupled cluster theory. For states with multireference character, the performance of the MR-ADC methods is similar to that of an N-electron valence perturbation theory. In contrast to conventional multireference perturbation theories, the MR-ADC methods have many attractive features, such as a straightforward and efficient calculation of excited-state properties and a direct access to excitations outside of the frontier (active) orbitals.

Simulation and Analysis of the Transient Absorption Spectrum of 4-(N,N-Dimethylamino)benzonitrile (DMABN) in Acetonitrile

4-(N,N-Dimethylamino)benzonitrile (DMABN) is a well-known model compound for dual fluorescence-in sufficiently polar solvents, it exhibits two distinct fluorescence emission bands. The interpretation of its transient absorption (TA) spectrum in the visible range is the subject of a long-standing controversy. In the present study, we resolve this issue by calculating the TA spectrum on the basis of nonadiabatic molecular dynamics simulations. An unambiguous assignment of spectral signals to specific excited-state structures is achieved by breaking down the calculated spectrum into contributions from twisted and nontwisted molecular geometries. In particular, the much-discussed excited-state absorption band near 1.7 eV (ca. 700 nm) is attributed to the near-planar locally excited (LE) minimum on the S1 state. On the technical side, our study demonstrates that the second-order approximate coupled cluster singles and doubles (CC2) method can be used successfully to calculate the TA spectra of moderately large organic molecules, provided that the system in question does not approach a crossing between the lowest excited state and the singlet ground state within the time frame of the simulation.

Multireference Perturbation Theory Combined with PCM and RISM Solvation Models: A Benchmark Study for Chemical Energetics

The polarizable continuum model (PCM) has been one of the most widely used approaches to take into account the solvation effect in quantum chemical calculations. In this paper, we performed a series of benchmark calculations to assess the accuracy of the PCM scheme combined with the second-order complete-active-space perturbation theory (CASPT2) for molecular systems in polar solvents. For solute molecules with extensive conjugated π orbitals, exemplified by elongated conjugated arylcarbenes, we have incorporated the ab initio density matrix renormalization group algorithm into the PCM-CASPT2 method. In the previous work, we presented a combination of the DMRG-CASPT2 method with the reference interaction site model (RISM) theory for describing the solvation effect using the radial distribution function and compared its performance to the widely used density-functional approaches (PCM-TD-DFT). The work here allows us to further show a more thorough assessment of the RISM model compared to the PCM with an equal level of the wave function treatment, the (DMRG-)CASPT2 theory, toward a high-accuracy electronic structure calculations for solvated chemical systems. With the exception that the PCM models are not capable of properly describing the hydrogen bondings, accuracy of the PCM-CASPT2 model is in most cases quite comparable to the RISM counterpart.

Photophysical Deactivation Mechanisms of the Pyrimidine Analogue 1-Cyclohexyluracil

The photophysical relaxation mechanisms of 1-cyclohexyluracil, in vacuum and water, were investigated by employing the Multi-State CASPT2 (MS-CASPT2, Multi-State Complete Active-Space Second-Order Perturbation Theory) quantum chemical method and Dunning's cc-pVDZ basis sets. In both environments, our results suggest that the primary photophysical event is the population of the S11(ππ*) bright state. Afterwards, two likely deactivation pathways can take place, which is sustained by linear interpolation in internal coordinates defined via Z-Matrix scans connecting the most important characteristic points. The first one (Route 1) is the same relaxation mechanism observed for uracil, its canonical analogue, i.e., internal conversion to the ground state through an ethylenic-like conical intersection. The other route (Route 2) is the direct population transfer from the S11(ππ*) bright state to the T23(nπ*) triplet state via an intersystem crossing process involving the (S11(ππ*)/T23(nπ*))STCP singlet-triplet crossing point. As the spin-orbit coupling is not too large in either environment, we propose that most of the electronic population initially on the S11(ππ*) state returns to the ground following the same ultrafast deactivation mechanism observed in uracil (Route 1), while a smaller percentage goes to the triplet manifold. The presence of a minimum on the S11(ππ*) potential energy hypersurface in water can help to understand why experimentally it is noticed suppression of the triplet states population in polar protic solvent.

Theoretical Study of Proton Tunneling in the Imidazole-Imidazolium Complex

Proton tunneling in the hydrogen-bonded imidazole–imidazolium complex ion has been studied theoretically. Ab initio CASSCF/6-311++G(d,p) calculations concerning geometry optimization and vibrational frequencies have been carried out for equilibrium and transition state structures of the system. Two-dimensional double-well model potentials were constructed on the basis of ab initio results and used to analyze the proton dynamics in the hydrogen bond and the influence of the excitation of low-frequency hydrogen-bond vibrations on the proton tunneling splittings. The energy of tunneling-split vibrational sublevels of the high-frequency tunneling mode have been calculated for its ground and first excited vibrational state for the series of excitations of the coupled low-frequency intramolecular hydrogen-bond modes. The promoting and suppressing effect of the low-frequency modes on the proton splittings was shown in the ground and first excited vibrational state of the tunneling mode. The vibrational sublevels form the two separate semicontinuous bands between which the absorption transitions may occur. This mechanism explains the experimentally observed splitting and doublet-component broadening of the high-frequency N–H stretching infrared (IR) absorption band.

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.

Free Energy Computation for an Isomerizing Chromophore in a Molecular Cavity via the Average Solvent Electrostatic Configuration Model: Applications in Rhodopsin and Rhodopsin-Mimicking Systems

We present a novel technique for computing the free energy differences between two chromophore "isomers" hosted in a molecular environment (a generalized solvent). Such an environment may range from a relatively rigid protein cavity to a flexible solvent environment. The technique is characterized by the application of the previously reported "average electrostatic solvent configuration" method, and it is based on the idea of using the free energy perturbation theory along with a chromophore annihilation procedure in thermodynamic cycle calculations. The method is benchmarked by computing the ground-state room-temperature relative stabilities between (i) the cis and trans isomers of prototypal animal and microbial rhodopsins and (ii) the analogue isomers of a rhodopsin-like light-driven molecular switch in methanol. Furthermore, we show that the same technology can be used to estimate the activation free energy for the thermal isomerization of systems i-ii by replacing one isomer with a transition state. The results show that the computed relative stability and isomerization barrier magnitudes for the selected systems are in line with the available experimental observation in spite of their widely diverse complexity.

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.

A Simple Effective Δ SCF Method for Computing Optical Gaps in Organic Chromophores

Photoluminescence effects in organic chromophores are of significant importance and requires precise description of low lying excited states. In this communication, we put forward an alternative time-independent DFT scheme for computing lowest single-particle excitation energy, especially for singlet excited state. This adopts a recently developed "virial"-theorem based model of singlet-triplet splitting which requires a DFT calculation on closed shell ground state and a restricted open-shell triplet excited state, followed by a simple 2 e - integral evaluation. This produces vertical excitation energies in small molecules, linear and non-linear polycyclic aromatic hydrocarbon and organic dyes in comparable accuracy to the TDDFT. We also explore the functional dependency of present method with three different functionals (B3LYP, wB97X and CAM-B3LYP) for polyenes and linear acenes. A systematic comparison with literature value illustrates the validity and usefulness of the present scheme in determining optical gap with fair computational cost.

Projector-Based Quantum Embedding for Molecular Systems: An Investigation of Three Partitioning Approaches

Projector-based embedding is a relatively recent addition to the collection of methods that seek to utilize chemical locality to provide improved computational efficiency. This work considers the interactions between the different proposed procedures for this method and their effects on the accuracy of the results. The interplay between the embedded background, projector type, partitioning scheme, and level of atomic orbital (AO) truncation are investigated on a selection of reactions from the literature. The Huzinaga projection approach proves to be more reliable than the level-shift projection when paired with other procedural options. Active subsystem partitioning from the subsystem projected AO decomposition (SPADE) procedure proves slightly better than the combination of Pipek-Mezey localization and Mulliken population screening (PMM). Along with these two options, a new partitioning criteria is proposed based on subsystem von Neumann entropy and the related subsystem orbital occupancy. This new method overlaps with the previous PMM method, but the screening process is computationally simpler. Finally, AO truncation proves to be a robust option for the tested systems when paired with the Huzinaga projection, with satisfactory results being acquired at even the most severe truncation level.

Multiconfiguration Pair-Density Functional Theory for Transition Metal Silicide Bond Dissociation Energies, Bond Lengths, and State Orderings

Transition metal silicides are promising materials for improved electronic devices, and this motivates achieving a better understanding of transition metal bonds to silicon. Here we model the ground and excited state bond dissociations of VSi, NbSi, and TaSi using a complete active space (CAS) wave function and a separated-pair (SP) wave function combined with two post-self-consistent field techniques: complete active space with perturbation theory at second order and multiconfiguration pair-density functional theory. The SP approximation is a multiconfiguration self-consistent field method with a selection of configurations based on generalized valence bond theory without the perfect pairing approximation. For both CAS and SP, the active-space composition corresponds to the nominal correlated-participating-orbital scheme. The ground state and low-lying excited states are explored to predict the state ordering for each molecule, and potential energy curves are calculated for the ground state to compare to experiment. The experimental bond dissociation energies of the three diatomic molecules are predicted with eight on-top pair-density functionals with a typical error of 0.2 eV for a CAS wave function and a typical error of 0.3 eV for the SP approximation. We also provide a survey of the accuracy achieved by the SP and extended separated-pair approximations for a broader set of 25 transition metal–ligand bond dissociation energies.

Ultrafast strong-field dissociation of vinyl bromide: An attosecond transient absorption spectroscopy and non-adiabatic molecular dynamics study

Attosecond extreme ultraviolet (XUV) and soft x-ray sources provide powerful new tools for studying ultrafast molecular dynamics with atomic, state, and charge specificity. In this report, we employ attosecond transient absorption spectroscopy (ATAS) to follow strong-field-initiated dynamics in vinyl bromide. Probing the Br M edge allows one to assess the competing processes in neutral and ionized molecular species. Using ab initio non-adiabatic molecular dynamics, we simulate the neutral and cationic dynamics resulting from the interaction of the molecule with the strong field. Based on the dynamics results, the corresponding time-dependent XUV transient absorption spectra are calculated by applying high-level multi-reference methods. The state-resolved analysis obtained through the simulated dynamics and related spectral contributions enables a detailed and quantitative comparison with the experimental data. The main outcome of the interaction with the strong field is unambiguously the population of the first three cationic states, D ₁, D ₂, and D ₃. The first two show exclusively vibrational dynamics while the D ₃ state is characterized by an ultrafast dissociation of the molecule via C-Br bond rupture within 100 fs in 50% of the analyzed trajectories. The combination of the three simulated ionic transient absorption spectra is in excellent agreement with the experimental results. This work establishes ATAS in combination with high-level multi-reference simulations as a spectroscopic technique capable of resolving coupled non-adiabatic electronic-nuclear dynamics in photoexcited molecules with sub-femtosecond resolution.

Spin-Flip Pair-Density Functional Theory: A Practical Approach To Treat Static and Dynamical Correlations in Large Molecules

We present a practical approach to treat static and dynamical correlation accurately in large multiconfigurational systems. The static correlation is taken into account by using the spin-flip approach, which is well-known for capturing static correlation accurately at low-computational expense. Unlike previous approaches to add dynamical correlation to spin-flip models which use perturbation theory or coupled-cluster theory, we explore the ability to use the on-top pair-density functional theory approaches recently developed by Gagliardi and co-workers (J. Comput. Theor. Chem., 2014, 10, 3669). External relaxations are performed in the spin-flip calculations through a restricted active space framework for which a truncation scheme for the orbitals used in the external excitation is presented. The performance of the approach is demonstrated by computing energy gaps between ground and excited states for diradicals, triradicals, and linear polyacene chains ranging from naphthalene to dodecacene. Accurate results are obtained using the new approach for these challenging open-shell molecular systems.

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

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

A Fock-operator complete active space self-consistent field (CAS-SCF) method combined with frozen-density embedding

We report the implementation of a Fock-operator complete-active space self-consistent field (CAS-SCF) method combined with frozen-density embedding (FDE) into the KOALA quantum-chemistry program. The implementation is based on configuration interaction from an unrestricted reference determinant and is able to treat electronic configurations such as singlet, triplet, or quintet states embedded in a molecular environment. In order to account for possible spin polarization effects, the FDE contribution is extended to the unrestricted case. We assess the convergence obtained with the implementation at the example of a stretched lithium dimer with significant multi-reference character. The efficiency of the implementation enables the orbital optimization for 25 states in a state-average SA[S₀-S₁₀,T₁-T₁₂,Q₁-Q₂]-CAS(10,10)-SCF calculation for the retinal molecule using a def2-TZVP basis. The FDE ansatz leads to orbitals localized by definition on the target system, thus facilitating the orbital selection required for CAS methods in complex environments.

A Single-Point Mutation in d-Arginine Dehydrogenase Unlocks a Transient Conformational State Resulting in Altered Cofactor Reactivity

Proteins are inherently dynamic, and proper enzyme function relies on conformational flexibility. In this study, we demonstrated how an active site residue changes an enzyme's reactivity by modulating fluctuations between conformational states. Replacement of tyrosine 249 (Y249) with phenylalanine in the active site of the flavin-dependent d-arginine dehydrogenase yielded an enzyme with both an active yellow FAD (Y249F-y) and an inactive chemically modified green FAD, identified as 6-OH-FAD (Y249F-g) through various spectroscopic techniques. Structural investigation of Y249F-g and Y249F-y variants by comparison to the wild-type enzyme showed no differences in the overall protein structure and fold. A closer observation of the active site of the Y249F-y enzyme revealed an alternative conformation for some active site residues and the flavin cofactor. Molecular dynamics simulations probed the alternate conformations observed in the Y249F-y enzyme structure and showed that the enzyme variant with FAD samples a metastable conformational state, not available to the wild-type enzyme. Hybrid quantum/molecular mechanical calculations identified differences in flavin electronics between the wild type and the alternate conformation of the Y249F-y enzyme. The computational studies further indicated that the alternate conformation in the Y249F-y enzyme is responsible for the higher spin density at the C6 atom of flavin, which is consistent with the formation of 6-OH-FAD in the variant enzyme. The observations in this study are consistent with an alternate conformational space that results in fine-tuning the microenvironment around a versatile cofactor playing a critical role in enzyme function.

Theoretical investigation of a novel xylene-based light-driven unidirectional molecular motor

In this study, the working mechanism of the first light-driven rotary molecular motors used to control an eight-base-pair DNA hairpin has been investigated. In particular, this linker was reported to have promising photophysical properties under physiological conditions, which motivated our work at the quantum mechanical level. Cis-trans isomerization is triggered by photon absorption at wavelengths ranging 300 nm-400 nm, promoting the rotor to the first excited state, and it is mediated by an energy-accessible conical intersection from which the ground state is reached back. The interconversion between the resulting unstable isomer and its stable form occurs at physiological conditions in the ground state and is thermally activated. Here, we compare three theoretical frameworks, generally used in the quantum description of medium-size chemical systems: Linear-Response Time-Dependent Density Functional Theory (LR-TDDFT), Spin-Flip TDDFT (SF-TDDFT), and multistate complete active space second-order perturbation theory on state-averaged complete active space self consistent field wavefunctions (MS-CASPT2//SA-CASSCF). In particular, we show the importance of resorting to a multireference approach to study the rotational cycle of light-driven molecular motors due to the occurrence of geometries described by several configurations. We also assess the accuracy and computational cost of the SF-TDDFT method when compared to MS-CASPT2 and LR-TDDFT.

Reduction and Rearrangement of a Boron(I) Carbonyl Complex

The one-electron reduction of a cyclic (alkyl)(amino)carbene (CAAC)-stabilized arylborylene carbonyl complex yields a dimeric borylketyl radical anion, resulting from an intramolecular aryl migration to the CO carbon atom. Computational analyses support the existence of a [(CAAC)B(CO)Ar].- radical anion intermediate. Further reduction leads to a highly nucleophilic dianionic (boraneylidene)methanolate.

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.

Artificial Photosynthesis: Is Computation Ready for the Challenge Ahead?

A tremendous effort is currently devoted to the generation of novel hybrid materials with enhanced electronic properties for the creation of artificial photosynthetic systems. This compelling and challenging problem is well-defined from an experimental point of view, as the design of such materials relies on combining organic materials or metals with biological systems like light harvesting and redox-active proteins. Such hybrid systems can be used, e.g., as bio-sensors, bio-fuel cells, biohybrid photoelectrochemical cells, and nanostructured photoelectronic devices. Despite these efforts, the main bottleneck is the formation of efficient interfaces between the biological and the organic/metal counterparts for efficient electron transfer (ET). It is within this aspect that computation can make the difference and improve the current understanding of the mechanisms underneath the interface formation and the charge transfer efficiency. Yet, the systems considered (i.e., light harvesting protein, self-assembly monolayer and surface assembly) are more and more complex, reaching (and often passing) the limit of current computation power. In this review, recent developments in computational methods for studying complex interfaces for artificial photosynthesis will be provided and selected cases discussed, to assess the inherent ability of computation to leave a mark in this field of research.

The Shape of Full Configuration Interaction to Come

We present a Perspective on what the future holds for full configuration interaction (FCI) theory, with an emphasis on conceptual rather than technical details. Upon revisiting the early history of FCI, a number of its key contemporary approximations are compared on as equal a footing as possible, using a recent blind challenge on the benzene molecule as a testbed [Eriksen et al., J. Phys. Chem. Lett., 2020 11, 8922]. In the process, we review the scope of applications for which FCI continues to prove indispensable, and the required traits in terms of robustness, efficacy, and reliability its modern approximations must satisfy are discussed. We close by conveying a number of general observations on the merits offered by the state-of-the-art alongside some of the challenges still faced to this day. While the field has altogether seen immense progress over the years-the past decade, in particular-it remains clear that our community as a whole has a substantial way to go in enhancing the overall applicability of near-exact electronic structure theory for systems of general composition and increasing size.

Reduced scaling formulation of CASPT2 analytical gradients using the supporting subspace method

We present a reduced scaling and exact reformulation of state specific complete active space second-order perturbation (CASPT2) analytical gradients in terms of the MP2 and Fock derivatives using the supporting subspace method. This work follows naturally from the supporting subspace formulation of the CASPT2 energy in terms of the MP2 energy using dressed orbitals and Fock builds. For a given active space configuration, the terms corresponding to the MP2-gradient can be evaluated with O(N⁵) operations, while the rest of the calculations can be computed with O(N³) operations using Fock builds, Fock gradients, and linear algebra. When tensor-hyper-contraction is applied simultaneously, the computational cost can be further reduced to O(N⁴) for a fixed active space size. The new formulation enables efficient implementation of CASPT2 analytical gradients by leveraging the existing graphical processing unit (GPU)-based MP2 and Fock routines. We present benchmark results that demonstrate the accuracy and performance of the new method. Example applications of the new method in ab initio molecular dynamics simulation and constrained geometry optimization are given.

Comparison of Methods for Active Orbital Selection in Multiconfigurational Calculations

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

Compressed-State Multistate Pair-Density Functional Theory

Multiconfiguration pair-density functional theory (MC-PDFT) is a multireference method that can be used to calculate excited states. However, MC-PDFT potential energy surfaces have the wrong topology at conical intersections because the last step of MC-PDFT is not a diagonalization of a model-space Hamiltonian matrix, as done in, for example, multistate second-order perturbation theory (MS-CASPT2). We have previously proposed methods that solve this problem by diagonalizing a model-space effective Hamiltonian matrix, where the diagonal elements are MC-PDFT energies for intermediate states, and the off-diagonal elements are evaluated by wave function theory. One previous method is called variational multistate PDFT (VMS-PDFT), whose intermediate states maximize the trace of the effective Hamiltonian, namely, the sum of the MC-PDFT energies of the model-space states; the VMS-PDFT is very robust but is more computationally expensive than another method, extended multistate PDFT (XMS-PDFT), in which the transformation to intermediate states is accomplished without needing any density functional evaluations. However, although VMS-PDFT was accurate in all cases tested, XMS-PDFT was accurate in only some of them. In the present paper, we propose a new method, called compressed-state multistate PDFT (CMS-PDFT), that is as efficient as XMS-PDFT and as accurate as VMS-PDFT. The new method maximizes the trace of the classical Coulomb energy of the intermediate states such that the electron densities of the intermediate states are compressed. We show that CMS-PDFT performs robustly even where XMS-PDFT fails.

Electronic Spectrum of the Tropylium Cation in the Gas Phase

The structure and properties of the tropylium cation (C₇H₇⁺) have enthralled chemists since the prediction by Hückel in 1931 of the remarkable stability for cyclic, aromatic molecules containing six π-electrons. However, probing and understanding the excited electronic states of the isolated tropylium cation have proved challenging, as the accessible electronic transitions are weak, and there are difficulties in creating appreciable populations of the tropylium cation in the gas phase. Here, we present the first gas-phase S₁ ←S₀ electronic spectrum of the tropylium cation, recorded by resonance-enhanced photodissociation of weakly bound tropylium-Ar complexes. We demonstrate that the intensity of the symmetry-forbidden S₁ ←S₀ transition arises from Herzberg-Teller vibronic coupling between the S₁ and S₂ electronic states mediated by vibrational modes of e₂' and e₃' symmetry. The main geometry change upon excitation involves elongation of the C-C bonds. Multiconfigurational ab initio calculations predict that the S₁ excited state is affected by the dynamical Jahn-Teller effect, which should lead to the appearance of additional weak bands that may be apparent in higher-resolution electronic spectra.

Local Enhancement of Dynamic Correlation in Excited States: Fresh Perspective on Ionicity and Development of Correlation Density Functional Approximation Based on the On-Top Pair Density

We discuss the interplay between the nondynamic and dynamic electron correlation in excited states from the perspective of the suppression of dynamic correlation (SDC) and enhancement of dynamic correlation (EDC) effects. We reveal that there exists a connection between the ionic character of a wave function and EDC. Following this finding we introduce a quantitative measure of ionicity based solely on local functions without referring to valence bond models. The ability to recognize both the SDC and EDC regions underlies the presented method, named CASΠDFT, combining complete active space (CAS) wave function and density functional theory (DFT) via the on-top pair density (Π) function. We extend this approach to excited states by devising an improved representation of the EDC effect in the correlation functional. The generalized CASΠDFT uses different DFT functionals for ground and excited states. Numerical demonstration for singlet π → π* excitations shows that CASΠDFT offers satisfactory accuracy at a fraction of the cost of the ab initio approaches.