Multireference Perturbation Theory Research Articles

Investigating the Nonradiative Decay Pathway in the Excited State of Silepin Derivatives: A Study with Second-Order Multireference Perturbation Wavefunction Theory.

The fluorescence quantum yield for fluorescent organic molecules is an important molecular property, and tuning it up is desired for various applications. For the computational estimation of the fluorescence quantum yield, the theoretical prediction of the nonradiative decay rate constant has become an attractive subject of study. The rate constant of thermally activated nonradiative decay is related to the activation energy in the photoreaction; thus, the accuracy and reliability of the excited-state potential energies in the quantum chemical computation are critical. In this study, we employed a second-order multireference perturbation wavefunction theory for studying the thermally activated decay via conical intersection (CI) of 1,1-dimethyldibenzo[b,f]silepin derivatives. The correlation between the computed activation energy to reach the CI geometry in the S1 state and the experimentally determined fluorescence quantum yield implied that silepins nonradiatively decay via the CI triggered by the twisting of the central C-C bond. Geometry optimization of the transition state using multireference perturbation theory drastically reduced the estimated activation energy. Our computation gave reasonable predictions of the activation free energies of photoexcited 1,1-dimethyldibenzo[b,f]silepin. The energy profiles and geometry optimizations using proper quantum chemical methods played a critical role in reliable estimation of the rate constant and fluorescence quantum yield.

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

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

Excited-State Geometry Optimization of Small Molecules with Many-Body Green’s Functions Theory

We present a benchmark study of gas phase geometry optimizations in the excited states of carbon monoxide, acetone, acrolein, and methylenecyclopropene using many-body Green’s functions theory within the GW approximation and the Bethe–Salpeter equation (BSE) employing numerical gradients. We scrutinize the influence of several typical approximations in the GW-BSE framework; we used one-shot G0W0 or eigenvalue self-consistent evGW, employing a fully analytic approach or plasmon-pole model for the frequency dependence of the electron self-energy, or performing the BSE step within the Tamm–Dancoff approximation. The obtained geometries are compared to reference results from multireference perturbation theory (CASPT2), variational Monte Carlo (VMC) method, second-order approximate coupled cluster (CC2) method, and time-dependent density-functional theory (TDDFT). We find overall a good agreement of the structural parameters optimized with the GW-BSE calculations with CASPT2, with an average relative error of around 1% for the G0W0 and 1.5% for the evGW variants based on a PBE0 ground state, respectively, while the other approximations have negligible influence. The relative errors are also smaller than those for CC2 and TDDFT with different functionals and only larger than VMC, indicating that the GW-BSE method does not only yield excitation energies but also geometries in good agreement with established higher-order wave function methods.

Efficient multireference perturbation theory without high-order reduced density matrices.

We present a stochastic approach to perform strongly contracted n-electron valence state perturbation theory (SC-NEVPT), which only requires one- and two-body reduced density matrices, without introducing approximations. We use this method to perform SC-NEVPT2 for complete active space self-consistent field wave functions obtained from selected configuration interaction, although the approach is applicable to a larger class of wave functions, including those from orbital-space variational Monte Carlo. The accuracy of this approach is demonstrated for small test systems, and the scaling is investigated with the number of virtual orbitals and the molecule size. We also find the SC-NEVPT2 energy to be relatively insensitive to the quality of the reference wave function. Finally, the method is applied to the Fe(II)-porphyrin system with a (32e, 29o) active space and to the isomerization of [Cu2O2]2+ in a (28e, 32o) active space.

New benzene dimers: a benchmark theoretical investigation

A wide variety of methods have been used to study the retro-dimerization reactions of one cis-fused (1) and four new trans-fused benzene dimers (2–5). These methods, which included both single-reference and multi-reference approaches, were benchmarked against multi-reference Moller-Plesset perturbation theory of the second order (MRMP2) based on CASSCF(12,12) wavefunctions. Each of the single-reference approaches was found to be overestimating the activation barrier for retro-dimerization process. Interestingly, N-electron valence perturbation theory (NEVPT2) energies also diverged for these reactions from MRMP2 values. Canonical MP2 barriers were in better agreement with MRMP2 results than the NEVPT2 numbers, with mean unsigned deviations (MUD) of 5.4 kcal/mol and 6.6 kcal/mol, respectively. The behavior of each of the methods is discussed and compared with the application of the same methods to the standard pericyclic reaction set proposed by Houk et al. (J Phys Chem A 107:11445, 2003). The comparison between NEVPT2 and MRMP2 reaction barriers, when applied to the cycloadditions in the Houk set and the synchronous symmetry-allowed retro-dimerization of the benzene dimer 1, showed that these two methods give almost identical results for these reaction types (MUD of 0.5 kcal/mol).

Benchmarking an Embedded Adaptive Sampling Configuration Interaction Method for Surface Reactions: H2 Desorption from and CH4 Dissociation on Cu(111).

Embedded (emb-) correlated wavefunction (CW) theory enables accurate assessments of both ground- and excited-state reaction mechanisms involved in heterogeneous catalysis. Embedded multireference second-order perturbation theory (emb-MRPT2) based on reference wavefunctions generated via embedded complete active space self-consistent field (emb-CASSCF) theory is currently state-of-the-art. However, the factorial scaling of CASSCF limits the size of active space and the complexity of systems that can be studied. Here, we assess the efficacy of an alternative CW method, adaptive sampling configuration interaction (ASCI)-which enables large active spaces to be used-for studying surface reactions. We couple ASCI with density functional embedding theory (DFET) and benchmark its performance for two reactions: H2 desorption from and CH4 dissociation on the Cu(111) surface. Unlike embedded complete active space second-order perturbation theory (emb-CASPT2) that accurately reproduces a measured H2 desorption barrier, embedded ASCI, using a very large active space (though one that still comprises a small portion of the full set of orbitals) fails to do so. Adding an extra correlation term from embedded Møller-Plesset second-order perturbation theory (emb-MP2) improves the desorption barrier and endothermicity predictions. Thus, the inaccuracy of embedded ASCI comes from the missing dynamic correlation from the many other electrons and orbitals not included in the active space. For CH4 dissociation, again embedded ASCI overestimates the dissociation barrier compared to emb-CASPT2 predictions. Adding dynamic correlation from emb-MP2 helps correct the barrier. However, this composite approach suffers from double counting of correlation within embedded ASCI followed by emb-MP2 calculations. We therefore conclude that the state-of-the-art emb-MRPT2 based on reference wavefunctions generated via emb-CASSCF remains the method of choice for studying surface reactions. emb-ASCI is useful when large active spaces beyond the limit of emb-CASSCF are essential, such as to study complex surface reactions with significant multiconfigurational character (static correlation) but weak dynamic correlation.

Probing ultrafast reaction mechanisms of photo-excited dithizone through transient absorption spectroscopy and computational CASSCF studies

Dithizone, with its derivatives and complexes, is characterized by seven classes of chromic reactions, which make it a remarkable candidate for diverse sensor applications. We now present dithizone’s chromic response to experimental photoexcitation as studied by ultrafast laser spectroscopy, density functional theory (DFT), and complete active space self-consistent field (CASSCF) calculations. After vertical excitation to the first excited state, within 240 fs the reaction proceeds towards an intersystem crossing (ISC) that is coupled with a conical intercept (CI), while the second path proceeds via a second CI. The ISC leads to excited state tautomerization within 1.6 ps to form a triplet excited state product that takes a longer time to return back to reactant ( ∼ 5 n s ), while the first and second CIs both lead to the ground state within 1.6 ps and 12 ps, respectively. Additionally, dynamics through both CIs allow for efficient repopulation of the reactant ground state. Theoretical calculations using minimum energy path analysis and CASSCF calculation (including n-electron valence state perturbation theory multireference perturbation theory correction) gave additional insight into ground and excited state conformational changes that are observed by means of experimental femtosecond laser pulse excitations of dithizone.

Diboron- and Diaza-Doped Anthracenes and Phenanthrenes: Their Electronic Structures for Being Singlet Fission Chromophores.

We used quantum chemistry methods at the levels of mixed-reference spin-flip time-dependent density functional theory and multireference perturbation theory to study diboron- and diaza-doped anthracenes and phenanthrenes. This class of structures recently surged as potential singlet fission chromophores. We studied electronic structures of their excited states and clarified the reasons why they satisfy or fail to satisfy the energy criteria for singlet fission chromophores. Many studied structures have their S1 states not dominated by HOMO → LUMO excitation, so they cannot be described using the conventional two site model. This is attributed to frontier orbital energy shifts induced by the doping and different charge-transfer energies in different one-electron singlet excitations or, in other words, different polarizations of hole and/or particle orbitals in their S1 and T1 states. There is a mirror relation between the orbital energy shifts induced by diboron- and diaza-dopings, which together with alternant hydrocarbon pairings of occupied and unoccupied orbitals, leads to more mirror relations between the excited states of the two types of doped structures.

Difference projection-after-variation double-hybrid density functional theory applied to the calculation of vertical excitation energies.

The use of projection-after-variation double-hybrid density functional theory is proposed and examined as a difference method for the calculation of excited states. The strengths and weaknesses of the proposed method are discussed with particular reference to connections with linear response coupled-cluster theory. Vertical excitation energies are computed for the 28 molecule benchmark of Schreiber and co-workers in order to compare how the model performs with linear response coupled-cluster theories and multireference perturbation theory. The findings of this study show that the proposed method can achieve standard deviations in the error of computed vertical excitation energies compared to complete active space second-order perturbation theory of similar size to linear response coupled-cluster theories.

Improved stochastic multireference perturbation theory for correlated systems with large active spaces

We identify the dominant computational cost within the recently introduced stochastic and internally contracted FCIQMC-NEVPT2 method for large active space sizes. This arises from the contribution to the four-body intermediates arising from low-excitation level sampled determinant pairs. We develop an effective way to mitigate this cost via an additional stochastic step within the sampling of the required NEVPT2 intermediates. We find this systematically improvable additional sampling can reduce simulation time by 80% without introducing appreciable error. This saving is expected to increase for larger active spaces. We combine this enhanced sampling scheme with full stochastic orbital optimisation for the first time and apply it to find FCIQMC-NEVPT2 energies for spin states of an iron porphyrin system within (24,24) active spaces with relatively meagre computational resources. This active space size can now be considered as routine for NEVPT2 calculations of strongly correlated molecular systems within this improved stochastic methodology.

Multi-reference perturbation theory study on the RaCl molecule promising for the laser cooling

The potential energy curves of the ground and eight low-lying excited atomic and ionic terms of the RaCl molecule are calculated for the first time using the multi-reference perturbation theory method at the CASSCF/XMCQDPT2 level of theory including the spin-orbit coupling. The electronic term energies, equilibrium internuclear distances, dissociation energies, transition dipole moments, sequences of vibrational energies, harmonic vibrational frequencies, Franck–Condon factors, and radiative lifetimes are predicted. The data obtained in this study suggests the possibility of the direct laser cooling of the RaCl molecules.

Reaching Full Correlation through Nonorthogonal Configuration Interaction: A Second-Order Perturbative Approach.

Nonorthogonal multireference methods can predict statically correlated adiabatic energies while providing chemical insight through the combination of diabatic reference states. However, reaching quantitative accuracy using nonorthogonal multireference expansions remains a significant challenge. In this work, we present the first rigorous perturbative correction to nonorthogonal configuration interaction, allowing the remaining dynamic correlation to be reliably computed. Our second-order "NOCI-PT2" theory exploits a zeroth-order generalized Fock Hamiltonian and builds the first-order interacting space using single and double excitations from each reference determinant. This approach therefore defines the rigorous nonorthogonal extension to conventional multireference perturbation theories. We find that NOCI-PT2 can quantitatively predict multireference potential energy surfaces and provides state-specific ground and excited states for adiabatic avoided crossings. Furthermore, we introduce an explicit imaginary-shift formalism requiring shift values that are an order of magnitude smaller than those used in conventional multireference perturbation theories.

Electronic Structure of Neutral and Anionic Iron-Nitrosyl Corrole. A Multiconfigurational and Density Matrix Renormalization Group Investigation.

An elaborate study with multireference second-order perturbation theory has been performed to elucidate the electronic structure and relative energy of three relevant states of FeNO corroles, namely the S = 0 ground state, the lowest S = 1 state, and the anion S = 1/2 state. On the basis of CASSCF and DMRG calculations with an active space including up to 37 orbitals, the electronic structure of the ground state was analyzed, with special emphasis on the diradical nature of the Fe-corrole and Fe-NO bonds. The results essentially confirm an earlier suggestion from B3LYP of a non-innocent corrole•2- bound to an {FeNO}7 unit, although the contribution of diradical character to the iron-corrole bond is found to be limited to 35%. This limited diradical character explains the high relative energy (16.5 kcal/mol) of the corresponding triplet state, where the corrole•2- is ferromagnetically coupled to the S = 1/2 {FeNO}7 unit. Consistent with experimental findings, reduction is found to take place at the corrole ligand, with a calculated electron affinity of 52.5 kcal/mol. The results obtained from the correlated calculations were also compared to DFT with a broad range of functionals.

Improved RISM-CASSCF Optimization via State-Average Treatment and Damping for Characterizing Excited Molecules in Solution with Multireference Perturbation Theory.

Determination of excited states of near-infrared (NIR) bioimaging dyes is a challenging theoretical task because of their energy levels with a small gap and the presence of solvation. In the previous study, we showed that the development of the reference interaction site model coupled with the complete active space second-order perturbation theory, the RISM-CASPT2 method, and its extension with the density matrix renormalization group enabled high accuracy prediction of the photochemical properties of bioimaging-related fluorescent molecules in solution (Shimizu et al., J. Chem. Theory Comput. 2018, 14, 5673-5679). This method, however, has a technical issue in convergence of CASSCF optimization, which was encountered when applying the method to a wider variety of systems; thus, practical applications have been hindered. Here, we present an improved scheme of CASSCF optimization with and without the density matrix renormalization group treatment. Detailed derivations and analysis of the second-order orbital optimization scheme with the inclusion of solvation through RISM revealed the requirement of a correction term to the orbital Hessian matrix. As a practical approach, the state-average RISM-CASPT2 method with damping treatment for solvation is presented for improving the convergence of the calculation under reasonable computational cost. The improved scheme allows for performing accurate and numerically stable theoretical analysis of the bioimaging-related excited state with various types of solvation for a P═O-bridged rhodol derivative, which is recently highlighted as a promising photostable NIR dye molecule.

Taming the excited states of butadiene, hexatriene, and octatetraene using state specific multireference perturbation theory with density functional theory orbitals.

To compute the electronic excitation energies, a state-specific multireference Møller-Plesset perturbation theory (SSMRPT) with a complete active space configuration interaction reference function constructed using the orbitals obtained by the density functional theory (DFT) is presented as an accurate, as well as computationally affordable, and efficient protocol at the level of second order. The global hybrid B3LYP (Becke, 3-parameter, Lee-Yang-Parr) functional has been used to generate orbitals. The present method, called DFT-SSMRPT, uses perturbers that are individual Slater determinants and accounts for the coupling between the nondynamical and dynamical correlation effects. We have applied the new method to compute excitation energies in conjugated systems of π-electrons such as trans-1,3-butadiene, trans,trans-1,3,5-hexatriene, and all-trans-1,3,5,7-octatetraene. The ordering of the excited states is correctly reproduced by the DFT-SSMRPT calculations. The relative ordering of low-lying excited 1Bu and 1Ag states alters when the length of the polyene changes. The results match reasonably well with the literature including experimental and best theoretical findings. The accuracy of the method is sufficient to discern the energy gap between the close low-lying singlet and triplet states. The DFT-SSMRPT appears as an affordable computational ab initio avenue for a qualitatively correct description of excitation energies.

Insights into the Thermal and Photochemical Reaction Mechanisms of Azidoacetonitrile. Spectroscopic and MS-CASPT2 Calculations.

This work studies the photochemical and thermal decompositions of azidoacetonitrile (N3 CH2 CN) from both the experimental and theoretical points of view. The data of the photochemical experiments are taken from the literature, while the thermal decomposition have been carried out by us. In addition, we have performed ab initio calculations of the multiconfigurational type [complete active space self-consistent field (CASSCF) and the multistate multireference perturbation theory (MS-CASPT2)]. It is found that the first step of both type of decompositions is N2 elimination and formation of closed shell singlet nitrene. Afterwards, the nitrene tends to rapidly rearrange into formimidoyl cyanide (HNCHCN). As both reactions progress, the imine isomerizes into formimidoyl isocyanide (HNCHNC). The photoisomerization of the imine takes places thorough a conical intersection, while the same reaction on the ground electronic state occurs via a conventional transition state. The last step of the global reaction is decomposition of the imines into HCN and CNH. In photochemical conditions, the conjunction of the imines and its dissociation products (HCN and CNH) yields adenine.

New benzene dimers: an MRMP2 study

The reaction energies of one cis-fused and four new trans-fused benzene dimers have been studied with state-specific multi-reference Moller-Plesset perturbation theory of the second order (MRMP2) based on reference CASSCF(12,12) wavefunctions. The cis-fused dimer 1, the product of symmetry-allowed [4 + 2] cycloaddition process, has been previously characterized experimentally and theoretically and, in this study, was used as reference. Dimer 1 lies 45 kcal/mol above two isolated benzene molecules, and its retro-dimerization activation energy is ~ 8 kcal/mol. In contrast, the [4 + 2] formation of trans-fused dimer 2 is symmetry-forbidden with an energetic driving force (Erel) of 57.6 kcal/mol. An activation barrier for corresponding retro-dimerization calculated to be 31.0 kcal/mol. The thermal [2 + 2] formation of dimers 3 and 4 are also symmetry-forbidden. They have been calculated to be 79.0 kcal/mol and 105.1 kcal/mol higher in energy than two isolated benzene molecules, respectively. However, the activation energies of 23.2 and 20.5 kcal/mol could allow for kinetic persistence of these compounds. Dimer 5 is the only symmetry-allowed [2 + 2] cycloaddition product. With Erel of 67.5 kcal/mol and Ea for retro-dimerization reaction of 19.2 kcal/mol, it shows high similarity with the symmetry-forbidden dimers. The wave function for each of the dimers at their local minimum geometry contains greater than 80% of the configuration $$\sigma^{4} \pi^{8} \sigma^{*0} \pi^{*0}$$. The transition state for the retro-dimerization of 1 (1TS) also contains more than 80% of the closed-shell configuration $$\sigma^{4} \pi^{8} \sigma^{*0} \pi^{*0}$$. For 2TS–5TS systems, the wavefunctions exhibit significant biradical character (20%, 20%, 14% and 26%, respectively).

Semiclassical Dispersion Corrections Efficiently Improve Multiconfigurational Theory with Short-Range Density-Functional Dynamic Correlation.

Multiconfigurational wave functions are known to describe the electronic structure across a Born-Oppenheimer surface qualitatively correct. However, for quantitative reaction energies, dynamic correlation originating from the many configurations involving excitations out of the restricted orbital space, the active space, must be considered. Standard procedures involve approximations that eventually limit the ultimate accuracy achievable (most prominently, multireference perturbation theory). At the same time, the computational cost increases dramatically due to the necessity to obtain higher-order reduced density matrices. It is this disproportion that leads us here to propose an MC-srDFT-D hybrid approach of semiclassical dispersion (D) corrections to cover long-range dynamic correlation in a multiconfigurational (MC) wave function theory, which includes short-range (sr) dynamic correlation by density functional theory (DFT) without double counting. We demonstrate that the reliability of this approach is very good (at negligible cost), especially when considering that standard second-order multireference perturbation theory usually overestimates dispersion interactions.

Efficient and stochastic multireference perturbation theory for large active spaces within a full configuration interaction quantum Monte Carlo framework.

Full Configuration Interaction Quantum Monte Carlo (FCIQMC) has been effectively applied to very large configuration interaction (CI) problems and was recently adapted for use as an active space solver and combined with orbital optimization. In this work, we detail an approach within FCIQMC to allow for efficient sampling of fully internally contracted multireference perturbation theories within the same stochastic framework. Schemes are described to allow for the close control over the resolution of stochastic sampling of the effective higher-body intermediates within the active space. It is found that while complete active space second-order perturbation theory seems less amenable to a stochastic reformulation, strongly contracted N-Electron Valence second-order Perturbation Theory (NEVPT2) is far more stable, requiring a similar number of walkers to converge the sc-NEVPT2 expectation values as to converge the underlying CI problem. We demonstrate the application of the stochastic approach to the computation of sc-NEVPT2 within a (24, 24) active space in a biologically relevant system and show that small numbers of walkers are sufficient for a faithful sampling of the sc-NEVPT2 energy to chemical accuracy, despite the active space already exceeding the limits of practicality for traditional approaches. This raises prospects of an efficient stochastic solver for multireference chemical problems requiring large active spaces, with an accurate treatment of external orbitals.

The combination of multipartitioning of the Hamiltonian with canonical Van Vleck perturbation theory leads to a Hermitian variant of quasidegenerate N-electron valence perturbation theory.

Many recent developments in the area of multistate multireference perturbation theories focused on methods that use a state-averaged 0th order Hamiltonian. We recently found that the dynamic correlation dressed complete active space method fails in describing ligand field and charge transfer states in a balanced way precisely because it uses a state-averaged 0th order Hamiltonian [L. Lang and F. Neese, J. Chem. Phys. 150, 104104 (2019)]. The multipartitioning idea allows the use of state-specific 0th order Hamiltonians in a multistate framework and could therefore alleviate the mentioned problem. However, the effective Hamiltonian is non-Hermitian in the traditional formulation of multipartitioning, which can lead to unphysical behavior, especially for nearly degenerate states. In order to achieve a more balanced treatment of states with different physical character and at the same time have a Hermitian effective Hamiltonian, we combine in this work multipartitioning with canonical Van Vleck perturbation theory. At the 2nd order, the result is a Hermitian variant of multipartitioning quasidegenerate N-electron valence state perturbation theory. The effect of model space noninvariance of the method is discussed and the benefit of a Hermitian formulation is highlighted with numerical examples. The method is shown to give good results for the calculation of electronic transitions of the [CuCl4]2 -complex and for the calculation of electron paramagnetic resonance parameters, which are two examples where the balance between ligand field and charge transfer configurations is of utmost importance.