Communication: Extended multi-state complete active space second-order perturbation theory: Energy and nuclear gradients

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

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

Role of the Perfluoro Effect in the Selective Photochemical Isomerization of Hexafluorobenzene

Hexafluorobenzene and many of its derivatives exhibit a chemoselective photochemical isomerization, resulting in highly strained, Dewar-type bicyclohexenes. While the changes in absorption and emission associated with benzene hexafluorination have been attributed to the so-called "perfluoro effect", the resulting electronic structure and photochemical reactivity of hexafluorobenzene is still unclear. We now use a combination of ultrafast time-resolved spectroscopy, multiconfigurational computations, and non-adiabatic dynamics simulations to develop a holistic description of the absorption, emission, and photochemical dynamics of the 4π-electrocyclic ring-closing of hexafluorobenzene and the fluorination effect along the reaction coordinate. Our calculations suggest that the electron-withdrawing fluorine substituents induce a vibronic coupling between the lowest-energy ¹B_2u (ππ*) and ¹E_1g (πσ*) excited states by selectively stabilizing the σ-type states. The vibronic coupling occurs along vibrational modes of e_2u symmetry which distorts the excited-state minimum geometry resulting in the experimentally broad, featureless absorption bands, and a ∼100 nm Stokes shift in fluorescence-in stark contrast to benzene. Finally, the vibronic coupling is shown to simultaneously destabilize the reaction pathway toward hexafluoro-benzvalene and promote molecular vibrations along the 4π ring-closing pathway, resulting in the chemoselectivity for hexafluoro-Dewar-benzene.

First-Principles Nonadiabatic Dynamics Simulation of Azobenzene Photodynamics in Solutions

The photoisomerization of azobenzene is a prototypical reaction of various light-activated processes in material and biomedical sciences. However, its reaction mechanism has been under debate for decades, partly due to the challenges in computational simulations to accurately describe the molecule's photodynamics. A recent study (J. Am. Chem. Soc. 2020, 142 (49), 20,680-20,690) addressed the challenges by combining the hole-hole Tamm-Dancoff Approximated (hh-TDA) density functional theory (DFT) method with the ab initio multiple spawning (AIMS) algorithm. The hh-TDA-DFT/AIMS method was applied to first-principles nonadiabatic dynamics simulation of azobenzene's photodynamics in the vacuum. However, it remains necessary to benchmark this new method in realistic molecular environments against experimental data. In the current work, the hh-TDA-DFT/AIMS method was employed in a quantum mechanics/molecular mechanics setting to characterize the trans azobenzene's photodynamics in explicit methanol and n-hexane solvents, following both the S₁ (nπ*) and S₂ (ππ*) excitations. The simulated absorption and fluorescence spectra following the S₂ excitation quantitatively agree with the experiments. However, the hh-TDA-DFT method overestimates the torsional barrier on the S₁ state, leading to an overestimation of the S₁ state lifetime. The excited-state population decays to the ground state through two competing channels. The reactive channel partially yields the cis azobenzene photoproduct, and the unreactive channel exclusively leads to the reactant. The S₂ excitation increases the decay through the unreactive channel and thus decreases the isomerization quantum yield compared to the S₁ excitation. The solvent slows down the azobenzene's torsional dynamics on the S₁ state, but its polarity minimally affects the reaction kinetics and quantum yields. Interestingly, the dynamics of the central torsion and angles of azobenzene play a critical role in determining the final isomer of the azobenzene. This benchmark study validates the hh-TDA-DFT/AIMS method's accuracy for simulating the azobenzene's photodynamics in realistic molecular environments.

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

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

Ligands enhanced the Ac[triple bond, length as m-dash]Ac triple bond

The multiple bonds between actinide atoms and their derivatives are computationally investigated extensively and compounds with an unsupported actinide-actinide bond, especially in low oxidation states, have attracted great attention. Herein, high level relativistic quantum chemical methods are used to probe the Ac-Ac bonding in compounds with a general formula LAcAcL (L = AsH3, PH3, NH3, H, CO, NO) at both scalar and spin-orbit coupling relativistic levels. H3AsAcAcAsH3, H3PAcAcPH3 and OCAcAcCO compounds show a type of zero valence Ac[triple bond, length as m-dash]Ac triple bond with a 1σ2g1π4u configuration, and H3AsAcAcAsH3 has been found to have the shortest Ac-Ac bond length of 3.012 Å reported so far. The Ac2 unit is very sensitive to the σ donor ligands and can form triple, double and even single bonds when suitable ligands are introduced, up to 3.652 Å with an Ac-Ac single bond in H3NAcAcNH3.

Theoretical Study of the Phenoxy Radical Recombination with the O(3P) Atom, Phenyl plus Molecular Oxygen Revisited

Quantum chemical calculations of the C₆H₅O₂ potential energy surface (PES) were carried out to study the mechanism of the phenoxy + O(³P) and phenyl + O₂ reactions. CASPT2(15e,13o)/CBS//CASSCF(15e,13o)/DZP multireference calculations were utilized to map out the minimum energy path for the entrance channels of the phenoxy + O(³P) reaction. Stationary points on the C₆H₅O₂ PES were explored at the CCSD(T)-F12/cc-pVTZ-f12//B3LYP/6-311++G** level for the species with a single-reference character of the wave function and at the CASPT2(15e,13o)/CBS//B3LYP/6-311++G** level of theory for the species with a multireference character of the wave function. Conventional, variational, and variable reaction coordinate transition-state theories were employed in Rice-Ramsperger-Kassel-Marcus master equation calculations to assess temperature- and pressure-dependent phenomenological rate constants and product branching ratios. The main bimolecular product channels of the phenoxy + O(³P) reaction are concluded to be para/ortho-benzoquinone + H, 2,4-cyclopentadienone + HCO and, at high temperatures, also phenyl + O₂. The main bimolecular product channels of the phenyl + O₂ reaction include 2,4-cyclopentadienone + HCO at lower temperatures and phenoxy + O(³P) at higher temperatures. For both the phenoxy + O(³P) and phenyl + O₂ reactions, the collisional stabilization of peroxybenzene at low temperatures and high pressures competes with the bimolecular product channels.

Role of Triplet States in the Photodynamics of Aniline

The dynamics of excited heteroaromatic molecules is a key to understanding the photoprotective properties of many biologically relevant chromophores that dissipate their excitation energy nonreactively and thereby prevent the detrimental effects of ultraviolet radiation. Despite their structural variability, most substituted aromatic compounds share a common feature of a repulsive ¹πσ* potential energy surface. This surface can lead to photoproducts, and it can also facilitate the population transfer back to the ground electronic state by means of a ¹πσ*/S₀ conical intersection. Here, we explore a hidden relaxation route involving the triplet electronic state of aniline, which has recently been discovered by means of time-selected photofragment translational spectroscopy [J. Chem. Phys. 2019, 151, 141101]. By using the recently available analytical gradients for multiconfiguration pair-density functional theory, it is now possible to locate the minimum-energy crossing points between states of different spin and therefore compute the intersystem crossing rates with a multireference method, rather than with the less reliable single-reference methods. Using such calculations, we demonstrate that the population loss of aniline in the T₁(³ππ*) state is dominated by C₆H₅NH₂ → C₆H₅NH· + H· dissociation, and we explain the long nonradiative lifetimes of the T₁(³ππ*) state at the excitation wavelengths of 294-264 nm.

Wavepacket propagations for the early time dynamics of proton-coupled electron transfer in the charge-transfer state of NH3Cl complex

A charge-transfer (CT) excited state of NH₃Cl, generated by photo-detachment of an electron from the anionic NH₃Cl^- precursor, can be represented as H₂N⁺-H-Cl^- and proceeds to two chemical reactions: one reaction generating NH₂ and HCl resulting from a proton transfer (PT) and the other reaction producing NH₃ and a Cl atom resulting from an electron transfer (ET); both are coupled to form a typical proton-coupled electron transfer (PCET) process. The early time dynamics of this CT were studied using time-dependent wavepacket propagation on three nonadiabatically coupled electronic states in a reduced three-dimensional space. The electronic states were treated using the XMS-CASPT2/aug-cc-pVTZ ab initio methodology. The population dynamics of the three coupled electronic states were analyzed in detail to reveal the initial stage of the PCET process up to ∼100 fs, while the branching ratio, χ = PT/(ET+PT), was determined after wavepacket propagations of up to 2000 fs. Another main result is the dependence of χ on the vibration levels of the initial precursor anion and the isotope substitution of the connecting H atom with deuterium and tritium. Our study reveals the detailed microscopic features of the PCET process embedded in the CT state of the NH₃Cl complex and certain systematic dependences of the branching ratio χ on the above factors.

Multiconfiguration Pair-Density Functional Theory

Kohn-Sham density functional theory with the available exchange-correlation functionals is less accurate for strongly correlated systems, which require a multiconfigurational description as a zero-order function, than for weakly correlated systems, and available functionals of the spin densities do not accurately predict energies for many strongly correlated systems when one uses multiconfigurational wave functions with spin symmetry. Furthermore, adding a correlation functional to a multiconfigurational reference energy can lead to double counting of electron correlation. Multiconfiguration pair-density functional theory (MC-PDFT) overcomes both obstacles, the second by calculating the quantum mechanical part of the electronic energy entirely by a functional, and the first by using a functional of the total density and the on-top pair density rather than the spin densities. This allows one to calculate the energy of strongly correlated systems efficiently with a pair-density functional and a suitable multiconfigurational reference function. This article reviews MC-PDFT and related background information.

Parameterization of a linear vibronic coupling model with multiconfigurational electronic structure methods to study the quantum dynamics of photoexcited pyrene

With this work, we present a protocol for the parameterization of a Linear Vibronic Coupling (LVC) Hamiltonian for quantum dynamics using highly accurate multiconfigurational electronic structure methods such as RASPT2/RASSCF, combined with a maximum-overlap diabatization technique. Our approach is fully portable and can be applied to many medium-size rigid molecules whose excited state dynamics requires a quantum description. We present our model and discuss the details of the electronic structure calculations needed for the parameterization, analyzing critical situations that could arise in the case of strongly interacting excited states. The protocol was applied to the simulation of the excited state dynamics of the pyrene molecule, starting from either the first or the second bright state (S₂ or S₅). The LVC model was benchmarked against state-of-the-art quantum mechanical calculations with optimizations and energy scans and turned out to be very accurate. The dynamics simulations, performed including all active normal coordinates with the multilayer multiconfigurational time-dependent Hartree method, show good agreement with the available experimental data, endorsing prediction of the excited state mechanism, especially for S₅, whose ultrafast deactivation mechanism was not yet clearly understood.

Spin-Orbit Coupling Changes the Identity of the Hyper-Open-Shell Ground State of Ce+, and the Bond Dissociation Energy of CeH+ Proves to Be Challenging for Theory

Cerium (Ce) plays important roles in catalysis. Its position in the sixth period of the periodic table leads to spin-orbit coupling (SOC) and other open-shell effects that make the quantum mechanical calculation of cerium compounds challenging. In this work, we investigated the low-lying spin states of Ce⁺ and the bond energy of CeH⁺, both by multiconfigurational methods, in particular, SA-CASSCF, MC-PDFT, CASPT2, XMS-PDFT, and XMS-CASPT2, and by single-configurational methods, namely, Hartree-Fock theory and unrestricted Kohn-Sham density functional theory with 34 choices of the exchange-correlation functional. We found that only CASPT2, XMS-CASPT2, and SA-CASSCF (among the five multiconfigurational methods) and GAM, HCTH, SOGGA11, and OreLYP (among the 35 single-configuration methods) successfully predict that the SOC-free ground spin state of Ce⁺ is a doublet state, and CASPT2 and GAM give the most accurate multireference and single-reference calculations, respectively, of the excitation energy of the first SOC-free excited state for Ce⁺. We calculated that the ground doublet state of Ce⁺ is an intra-atomic hyper-open-shell state. We calculated the spin-orbit energy (E^SO) of Ce⁺ by the five multiconfigurational methods and found that E^SO calculated by CASPT2 is the closest to the experimental value. Taking advantage of the availability of an experimental D₀ for CeH⁺ as a way to provide a unique test of theory, we showed that all the multiconfigurational methods overestimate D₀ by at least 246 meV (5.7 kcal/mol), and only three functionals, namely, SOGGA, MN15, and GAM, have an error of D₀ that is less than 200 meV (5 kcal/mol).

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

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

A Full-Dimensional Potential Energy Surface and Dynamics of the Multichannel Reaction between H and HO2

In addition to its vital significance in combustion and atmospheric chemistry, the reaction between H' and HO₂ on the ground triplet state represents a prototype with multiple product channels, including H₂ + O₂, OH + OH, O + H₂O, and H + H'O₂. In this work, a full-dimensional accurate potential energy surface (PES) for the title reaction was developed to provide reliable descriptions for all dynamically relevant regions. Using this PES, we adopted the quasi-classical trajectory approach to study the corresponding reaction dynamics, including the reactivity of each product channel and the associated product branching ratio, the product energy distributions, product angular distributions, and associated microscopic mechanisms. For representing distributions of the product energies, such as product translational energy as well as product rotational and vibrational energies, both the traditional histogram and the kernel density estimation (KDE) methods were used and compared. It seems that the features of the resulting distributions in this work are very similar to each other among different methods. The KDE method is suggested for statistics, particularly for those populations with small oscillations in the histogram plot.

Mixed-Reference Spin-Flip Time-Dependent Density Functional Theory (MRSF-TDDFT) as a Simple yet Accurate Method for Diradicals and Diradicaloids

Due to their multiconfigurational nature featuring strong electron correlation, accurate description of diradicals and diradicaloids is a challenge for quantum chemical methods. The recently developed mixed-reference spin-flip (MRSF)-TDDFT method is capable of describing the multiconfigurational electronic states of these systems while avoiding the spin-contamination pitfalls of SF-TDDFT. Here, we apply MRSF-TDDFT to study the adiabatic singlet-triplet (ST) gaps in a series of well-known diradicals and diradicaloids. On average, MRSF displays a very high prediction accuracy of the adiabatic ST gaps with the mean absolute error (MAE) amounting to 0.14 eV. In addition, MRSF is capable of accurately describing the effect of the Jahn-Teller distortion occurring in the trimethylenemethane diradical, the violation of the Hund rule in a series of the didehydrotoluene diradicals, and the potential energy surfaces of the didehydrobenzene (benzyne) diradicals. A convenient criterion for distinguishing diradicals and diradicaloids is suggested on the basis of the easily obtainable quantities. In all of these cases, which are difficult for the conventional methods of density functional theory (DFT), MRSF shows results consistent with the experiment and the high-level ab initio computations. Hence, the present study documents the reliability and accuracy of MRSF and lays out the guidelines for its application to strongly correlated molecular systems.

Ultrafast nonradiative deactivation of photoexcited 8-oxo-hypoxanthine: a nonadiabatic molecular dynamics study

In the scientific endeavor to understand the chemical origins of life, the photochemistry of the smallest life building blocks, nucleobases, has been a constant object of focus and intense research. Here, we report the results of the first theoretical study on the photo-properties of an 8-oxo-hypoxanthine molecule, the chromophore of 8-oxo-inosine, which is relevant to the recently proposed, prebiotically plausible synthetic routes to the formation of purine- and pyrimidine-nucleotides. With ab initio and semi-empirical OM2/MRCI quantum-chemistry calculations, we predict a strong photostability of the 8-oxo-hypoxanthine system and see the origin of this effect in ultrafast nonradiative relaxation through puckering of the 6-membered heterocyclic ring.

Asymmetric Ring Opening in a Tetrazine-Based Ligand Affords a Tetranuclear Opto-Magnetic Ytterbium Complex

We report the formation of a tetranuclear lanthanide cluster, [Yb₄ (bpzch)₂ (fod)₁₀ ] (1), which occurs from a serendipitous ring opening of the functionalised tetrazine bridging ligand, bpztz (3,6-dipyrazin-2-yl-1,2,4,5-tetrazine) upon reacting with Yb(fod)₃ (fod^- =6,6,7,7,8,8,8-heptafluoro-2,2-dimethyl-3,5-octandionate). Compound 1 was structurally elucidated via single-crystal X-ray crystallography and subsequently magnetically and spectroscopically characterised to analyse its magnetisation dynamics and its luminescence behaviour. Computational studies validate the observed M_J energy levels attained by spectroscopy and provides a clearer picture of the slow relaxation of the magnetisation dynamics and relaxation pathways. These studies demonstrate that 1 acts as a single-molecule magnet (SMM) under an applied magnetic field in which the relaxation occurs via a combination of Raman, direct, and quantum tunnelling processes, a behaviour further rationalised analysing the luminescent properties. This marks the first lanthanide-containing molecule that forms by means of an asymmetric tetrazine decomposition.

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.

On the role of symmetry in XDW-CASPT2

Herewith, we propose two new exponents for the recently introduced XDW-CASPT2 method [S. Battaglia and R. Lindh, J. Chem. Theory Comput. 16, 1555-1567 (2020)], which fix one of the largest issues hindering this approach. By using the first-order effective Hamiltonian coupling elements, the weighting scheme implicitly takes into account the symmetry of the states, thereby averaging Fock operators only if the zeroth-order wave functions interact with each other. The use of Hamiltonian couplings also provides a physically sounder approach to quantitate the relative weights; however, it introduces new difficulties when these rapidly die off to zero. The improved XDW-CASPT2 method is critically tested on several systems of photochemical relevance, and it is shown that it succeeds in its original intent of maintaining MS-CASPT2 accuracy for the evaluation of transition energies and at the same time providing smooth potential energy surfaces around near-degenerate points akin to XMS-CASPT2.

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.

Energetics and optimal molecular packing for singlet fission in BN-doped perylenes: electronic adiabatic state basis screening

Singlet fission has the potential to increase the efficiency of photovoltaic devices, but the design of suitable chromophores is notoriously difficult. Both the electronic properties of the monomer and the packing motif in the crystal have a big impact on the singlet fission efficiency. Using perylene as an example, it is shown that doping with boron and nitrogen not only helps to align the energy levels but also shifts the stacking position that is optimal for singlet fission. Among all perylene derivatives doped with one or two BN groups, we identify the most suitable isomer for singlet fission with the help of TD-DFT and CASPT2 calculations. The optimal relative disposition of the two monomer units in a cofacially stacked homodimer is explored using two semiempirical models for the singlet fission rate: The first one is the well-known diabatic frontier orbital model, while the second treats singlet fission as a non-adiabatic transition and approximates the rate as the length squared of the non-adiabatic coupling vector between eigenfunctions of the diabatic Hamiltonian.

Effect of dynamic correlation on the ultrafast relaxation of uracil in the gas phase

High level multi-reference non-adiabatic dynamics simulations reveal that uracil’s photoexcited S₂ state decays very quickly without any significant trapping.

Carbonyl Stretch as a Franck-Condon Active Mode and Driving Force for Excited-State Decay of 8-Methoxy-4-methyl-2H-benzo[g]chromen-2-one from nπ* State

The fluorescence of most organic chromophore is emitted from the ππ* state, whereas the nπ* state, as a dark state, plays an important role in quenching the fluorescence when its energy is close to the ππ* state. Herein, we report a theoretical study on the fluorescence quenching of 8-methoxy-4-methyl-2H-benzo[g]chromen-2-one by the nπ* state and propose a new mechanism for describing the vibronic coupling between the ππ* and nπ* states. By applying extended multistate complete-active-space second-order perturbation theory (XMS-CASPT2) to optimize the geometries, the geometry distortion of the ππ* state along the out-of-plane mode is observed. This geometry distortion causes the stretching vibration of the carbonyl group to be coupled with the C-C bonds of the pyran ring, which become a Franck-Condon active mode upon photoexcitation and provides a driving force for nonradiative decay from the nπ* state, even if it is energetically unfavorable. This mechanism is significantly different from the previously proposed "proximity effect" and cannot be captured by the popularly used time-dependent density functional theory (TDDFT) and complete-active-space self-consistent field (CASSCF) 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.

Intrastrand Photolesion Formation in Thio-Substituted DNA: A Case Study Including Single-Reference and Multireference Methods

The substitution of canonical nucleobases by thiated analogues in natural DNA has been exploited in pharmacology, photochemotherapy, and structural biology. Thionucleobases react with adjacent thymines leading to 6-4 pyrimidine-pyrimidone photoproducts (6-4PPs), which are a major source of DNA photodamage, in particular intrastrand cross-linked photolesions. Here, we study the mechanism responsible for the formation of 6-4PPs in thionucleobases by employing quantum-mechanical calculations. We use multiconfiguration pair-density functional theory, complete active space second-order perturbation theory, and Kohn-Sham density functional theory. Scrutinizing the photochemistry of thionucleobases can elucidate the reaction mechanism of these prodrugs and identify the role that triplet excited states play in the generation of photolesions in the natural biopolymer. Three different possible mechanisms to generate the 6-4PPs are presented, and we conclude that the use of multireference approaches is indispensable to capture important features of the potential energy surface.

Multi-state pair-density functional theory

Multi-configuration pair-density functional theory (MC-PDFT) has previously been applied successfully to carry out ground-state and excited-state calculations. However, because they include no interaction between electronic states, MC-PDFT calculations in which each state's PDFT energy is calculated separately can give an unphysical double crossing of potential energy surfaces (PESs) in a region near a conical intersection. We have recently proposed state-interaction pair-density functional theory (SI-PDFT) to treat nearly degenerate states by creating a set of intermediate states with state interaction; although this method is successful, it is inconvenient because two SCF calculations and two sets of orbitals are required and because it puts the ground state on an unequal footing with the excited states. Here we propose two new methods, called extended-multi-state-PDFT (XMS-PDFT) and variational-multi-state-PDFT (VMS-PDFT), that generate the intermediate states in a balanced way with a single set of orbitals. The former uses the intermediate states proposed by Granovsky for extended multi-configuration quasi-degenerate perturbation theory (XMC-QDPT); the latter obtains the intermediate states by maximizing the sum of the MC-PDFT energies for the intermediate states. We also propose a Fourier series expansion to make the variational optimizations of the VMS-PDFT method convenient, and we implement this method (FMS-PDFT) both for conventional configuration-interaction solvers and for density-matrix-renormalization-group solvers. The new methods are tested for eight systems, exhibiting avoided crossings among two to six states. The FMS-PDFT method is successful for all cases for which it has been tested (all cases in this paper except O3 for which it was not tested), and XMS-PDFT is successful for all eight cases except the mixed-valence case. Since both XMS-PDFT and VMS-PDFT are less expensive than XMS-CASPT2, they will allow well-correlated calculations on much larger systems for which perturbation theory is unaffordable.

A chemical dynamics study on the gas-phase formation of triplet and singlet C5H2 carbenes

Since the postulation of carbenes by Buchner (1903) and Staudinger (1912) as electron-deficient transient species carrying a divalent carbon atom, carbenes have emerged as key reactive intermediates in organic synthesis and in molecular mass growth processes leading eventually to carbonaceous nanostructures in the interstellar medium and in combustion systems. Contemplating the short lifetimes of these transient molecules and their tendency for dimerization, free carbenes represent one of the foremost obscured classes of organic reactive intermediates. Here, we afford an exceptional glance into the fundamentally unknown gas-phase chemistry of preparing two prototype carbenes with distinct multiplicities-triplet pentadiynylidene (HCCCCCH) and singlet ethynylcyclopropenylidene (c-C5H2) carbene-via the elementary reaction of the simplest organic radical-methylidyne (CH)-with diacetylene (HCCCCH) under single-collision conditions. Our combination of crossed molecular beam data with electronic structure calculations and quasi-classical trajectory simulations reveals fundamental reaction mechanisms and facilitates an intimate understanding of bond-breaking processes and isomerization processes of highly reactive hydrocarbon intermediates. The agreement between experimental chemical dynamics studies under single-collision conditions and the outcome of trajectory simulations discloses that molecular beam studies merged with dynamics simulations have advanced to such a level that polyatomic reactions with relevance to extreme astrochemical and combustion chemistry conditions can be elucidated at the molecular level and expanded to higher-order homolog carbenes such as butadiynylcyclopropenylidene and triplet heptatriynylidene, thus offering a versatile strategy to explore the exotic chemistry of novel higher-order carbenes in the gas phase.

Putting Photomechanical Switches to Work: An Ab Initio Multiple Spawning Study of Donor-Acceptor Stenhouse Adducts

Photomechanical switches are light sensitive molecules capable of transducing the energy of a photon into mechanical work via photodynamics. In this Letter, we present the first atomistic investigation of the photodynamics of a novel class of photochromes called donor-acceptor Stenhouse adducts (DASA) using state-of-the-art ab initio multiple spawning interfaced with state-averaged complete active-space self-consistent field theory. Understanding the Z/E photoisomerization mechanism in DASAs at the molecular level is crucial in designing new derivatives with improved photoswitching capabilities. Our dynamics simulations show that the actinic step consists of competing nonradiative relaxation pathways that collectively contribute to DASAs' low (21% in toluene) photoisomerization quantum yield. Furthermore, we highlight the important role the intramolecular hydrogen bond plays in the selectivity of photoisomerization in DASAs, identifying it as a possible structural element to tune DASA properties. Our fully ab initio simulations reveal the key degrees of freedom involved in the actinic step, paving the way for the rational design of new generations of DASAs with improved quantum yield and efficiency.

Analytical First-Order Derivatives of Second-Order Extended Multiconfiguration Quasi-Degenerate Perturbation Theory (XMCQDPT2): Implementation and Application

Analytical gradient theory for the second-order extended multiconfiguration quasi-degenerate perturbation theory (XMCQDPT2), which can be regarded as the multistate version of the multireference second-order Møller-Plesset perturbation theory (MRMP2), is formulated and implemented. The theory is similar to the previous analytical gradient theory for MCQDPT2, but we take into account the intruder state avoidance (ISA) technique and the "extension" of the MCQDPT2 theory by Granovsky. Although the (X)MCQDPT2 theory is not invariant with respect to rotations among the active orbitals, the resulting analytical gradients are accurate. We demonstrate the utility of the current algorithm in optimizing the minimum energy conical intersections (MECIs) of ethylene, butadiene, benzene, the retinal model chromophore PSB3, and the green fluorescent protein model chromophore pHBI. The XMCQDPT2 MECIs are very similar to the XMS-CASPT2 MECIs in terms of molecular conformation and the computed energies. We also discuss possible improvements of the current algorithm.

Excited state dynamics of cis,cis-1,3-cyclooctadiene: UV pump VUV probe time-resolved photoelectron spectroscopy

We present UV pump, vacuum ultraviolet probe time-resolved photoelectron spectroscopy measurements of the excited state dynamics of cis,cis-1,3-cyclooctadiene. A 4.75 eV deep UV pump pulse launches a vibrational wave packet on the first electronically excited state, and the ensuing dynamics are probed via ionization using a 7.92 eV probe pulse. The experimental results indicate that the wave packet undergoes rapid internal conversion to the ground state in under 100 fs. Comparing the measurements with electronic structure and trajectory surface hopping calculations, we are able to interpret the features in the measured photoelectron spectra in terms of ionization to several states of the molecular cation.

Combination Reactions of Propargyl Radical with Hydroxyl Radical and the Isomerization and Dissociation of trans-Propenal

Ab initio investigation for the ground-electronic potential energy surface (PES) of the CH₂CCH + OH combination and the trans-CH₂CHCHO isomerization and decomposition has been performed at the UCCSD(T)/CBS(TQ5)//M06-2X/aug-cc-pVTZ level of theory. Thermal and microcanonical rate constants, as well as branching ratios in the 300-2000 K temperature range have been predicted based on optimized structures and vibrational frequencies of species involved using statistical theoretical VRC-TST and RRKM master equation computations. The calculated results are in good agreement with the prior reported data, particularly as an accurate scaling of the energy barriers was carried out. Based on the view of PES and kinetic-predicted values, the reaction paths leading to C₂H₂ + CO + H₂, CH₃CH + CO, C₂H₄ + CO, C₂H₃ + HCO, and C₃H₃O + H are the prevailing product channels for the C₃H₃ + OH bimolecular reaction under the considered 300-2000 K temperature range. Among those products, CH₃CH + CO is the most dominant one in the low-temperature condition; however, C₂H₂ + CO + H₂ becomes the most favorable product in the high-temperature region. Alternatively, the C₃H₄O dissociation processes leading to C₂H₂ + CO + H₂, C₂H₃ + HCO, C₂H₄ + CO, and CH₂C + CH₂O constitute the major paths, in which, C₂H₂ + CO + H₂ is the most critical one with the ∼62% and ∼59% branching ratios at E = 148 and 182 kcal/mol, respectively. The overall second-order rate constants of the bimolecular reaction C₃H₃ + OH → products obtained at the pressure 760 Torr (Ar) can be illustrated by the modified Arrhenius expression of k(T) = 1.36 × 10^-13T^1.26 exp[(-1.12 ± 0.43 kcal mol^-1)/RT] and/or k(T) = 3.77 × 10¹⁷T^-7.58 exp[(-18.82 ± 0.20 kcal mol^-1)/RT] cm³ molecule^-1 s^-1, covering the temperature range of 300-1300 and/or 1300-2000 K, respectively. The total high-pressure limit rate constant for the C₃H₃ + OH → CH₂CCHOH barrierless processes is in good agreement with the k(T) = 8.30 × 10^-10 T^-0.1 cm³ molecule^-1 s^-1 literature data. Moreover, microcanonical rate constants for the C₃H₄O isomerization and dissociation are in excellent accordance with the previously predicted values given by Chin and Lee. The present study supplies a thorough insight into the mechanisms and kinetics of the C₃H₃ + OH combination as well as the C₃H₄O multistep isomerization/dissociation pathways.

Tracking the ultraviolet-induced photochemistry of thiophenone during and after ultrafast ring opening

Photoinduced isomerization reactions lie at the heart of many chemical processes in nature. The mechanisms of such reactions are determined by a delicate interplay of coupled electronic and nuclear dynamics occurring on the femtosecond scale, followed by the slower redistribution of energy into different vibrational degrees of freedom. Here we apply time-resolved photoelectron spectroscopy with a seeded extreme ultraviolet free-electron laser to trace the ultrafast ring opening of gas-phase thiophenone molecules following ultraviolet photoexcitation. When combined with ab initio electronic structure and molecular dynamics calculations of the excited- and ground-state molecules, the results provide insights into both the electronic and nuclear dynamics of this fundamental class of reactions. The initial ring opening and non-adiabatic coupling to the electronic ground state are shown to be driven by ballistic S-C bond extension and to be complete within 350 fs. Theory and experiment also enable visualization of the rich ground-state dynamics that involve the formation of, and interconversion between, ring-opened isomers and the cyclic structure, as well as fragmentation over much longer timescales.

On the Theoretical Determination of Photolysis Properties for Atmospheric Volatile Organic Compounds

Volatile organic compounds (VOCs) are ubiquitous atmospheric molecules that generate a complex network of chemical reactions in the troposphere, often triggered by the absorption of sunlight. Understanding the VOC composition of the atmosphere relies on our ability to characterize all of their possible reaction pathways. When considering reactions of (transient) VOCs with sunlight, the availability of photolysis rate constants, utilized in general atmospheric models, is often out of experimental reach due to the unstable nature of these molecules. Here, we show how recent advances in computational photochemistry allow us to calculate in silico the different ingredients of a photolysis rate constant, namely, the photoabsorption cross-section and wavelength-dependent quantum yields. The rich photochemistry of tert-butyl hydroperoxide, for which experimental data are available, is employed to test our protocol and highlight the strengths and weaknesses of different levels of electronic structure and nonadiabatic molecular dynamics to study the photochemistry of (transient) VOCs.

Reduced scaling extended multi-state CASPT2 (XMS-CASPT2) using supporting subspaces and tensor hyper-contraction

We present a reduced scaling formulation of the extended multi-state CASPT2 (XMS-CASPT2) method, which is based on our recently developed state-specific CASPT2 (SS-CASPT2) formulation using supporting subspaces and tensor hyper-contraction. By using these two techniques, the off-diagonal elements of the effective Hamiltonian can be computed with only O(N³) operations and O(N²) memory, where N is the number of basis functions. This limits the overall computational scaling to O(N⁴) operations and O(N²) memory. Thus, excited states can now be obtained at the same reduced (relative to previous algorithms) scaling we achieved for SS-CASPT2. In addition, we also investigate how the energy denominators can be factorized with the Laplace quadrature when some of the denominators are negative, which is critical for excited state calculations. An efficient implementation of the method has been developed using graphical processing units while also exploiting spatial sparsity in tensor operations. We benchmark the accuracy of the new method by comparison to non-THC formulated XMS-CASPT2 for the excited states of various molecules. In our tests, the THC approximation introduces negligible errors (≈0.01 eV) compared to the non-THC reference method. Scaling behavior and computational timings are presented to demonstrate performance. The new method is also interfaced with quantum mechanics/molecular mechanics (QM/MM). In an example study of green fluorescent protein, we show how the XMS-CASPT2 potential energy surfaces and excitation energies are affected by increasing the size of the QM region up to 278 QM atoms with more than 2300 basis functions.

Photoelimination of Nitrogen from Diazoalkanes: Involvement of Higher Excited Singlet States in the Carbene Formation

Although diazoalkanes are important carbene precursors in organic synthesis, a comprehensive mechanism of photochemical formation of carbenes from diazoalkanes has not been proposed. Synergies of experiments and computations demonstrate the involvement of higher excited singlet states in the photochemistry of diazoalkanes. In all investigated diazoalkanes, excitation to S₁ results in nonreactive internal conversion to S₀. On the contrary, excitation to higher-lying singlet states (S_n, n > 1) drives the reaction toward a different segment of the S₁/S₀ conical intersection seam and results in nitrogen elimination and formation of carbenes.

Comparison of Spin-Flip TDDFT-Based Conical Intersection Approaches with XMS-CASPT2

Determining conical intersection geometries is of key importance to understanding the photochemical reactivity of molecules. While many small- to medium-sized molecules can be treated accurately using multireference approaches, larger molecules require a less computationally demanding approach. In this work, minimum energy crossing point conical intersection geometries for a series of molecules have been studied using spin-flip TDDFT (SF-TDDFT), within the Tamm-Dancoff Approximation, both with and without explicit calculation of nonadiabatic coupling terms, and compared with both XMS-CASPT2 and CASSCF calculated geometries. The less computationally demanding algorithms, which do not require explicit calculation of the nonadiabatic coupling terms, generally fare well with the XMS-CASPT2 reference structures, while the relative energetics are only reasonably replicated with the MECP structure as calculated with the BHHLYP functional and full nonadiabatic coupling terms. We also demonstrate that, occasionally, CASSCF structures deviate quantitatively from the XMS-CASPT2 structures, showing the importance of including dynamical correlation.

Excited state dynamics of cis,cis-1,3-cyclooctadiene: Non-adiabatic trajectory surface hopping

We have performed trajectory surface hopping dynamics for cis,cis-1,3-cyclooctadiene to investigate the photochemical pathways involved after being excited to the S₁ state. Our calculations reveal ultrafast decay to the ground state, facilitated by conical intersections involving distortions around the double bonds. The main distortions are localized on one double bond, involving twisting and pyramidalization of one of the carbons of that double bond (similar to ethylene), while a limited number of trajectories decay via delocalized (non-local) twisting of both double bonds. The interplay between local and non-local distortions is important in our understanding of photoisomerization in conjugated systems. The calculations show that a broad range of the conical intersection seam space is accessed during the non-adiabatic events. Several products formed on the ground state have also been observed.

The Molpro quantum chemistry package

Molpro is a general purpose quantum chemistry software package with a long development history. It was originally focused on accurate wavefunction calculations for small molecules but now has many additional distinctive capabilities that include, inter alia, local correlation approximations combined with explicit correlation, highly efficient implementations of single-reference correlation methods, robust and efficient multireference methods for large molecules, projection embedding, and anharmonic vibrational spectra. In addition to conventional input-file specification of calculations, Molpro calculations can now be specified and analyzed via a new graphical user interface and through a Python framework.

Combination of a Voronoi-Type Complex Absorbing Potential with the XMS-CASPT2 Method and Pilot Applications

Electronic resonances are metastable (N + 1) electron states, in other words, discrete states embedded in an electronic continuum. While great progress has been made for certain types of resonances-for example, temporary anions created by attaching one excess electron to a closed shell neutral-resonances in general remain a great challenge of quantum chemistry because a successful description of the decay requires a balanced description of the bound and continuum aspect of the resonance. Here, a smoothed Voronoi complex absorbing potential (CAP) is combined with the XMS-CASPT2 method, which enables us to address the balance challenge by appropriate choice of the CAS space. To reduce the computational cost, the method is implemented in the projected scheme. In this pilot application, three temporary anions serve as benchmarks: the π* resonance state of formaldehyde; the π* and σ* resonance states of chloroethene as functions of the C-Cl bond dissociation coordinate; and the ⁴Π_u and ²Π_u resonance states of N₂^-. The convergence of the CAP/XMS-CASPT2 results has been systematically examined with respect to the size of the active space. Resonance parameters predicted by the CAP/XMS-CASPT2 method agree well with CAP/SAC-CI results (deviations of about 0.15 eV); however, as expected, CAP/XMS-CASPT2 has clear advantages in the bond dissociation region. The advantages of CAP/XMS-CASPT2 are further demonstrated in the calculations of ⁴Π_u and ²Π_u resonance states of N₂^- including their ³Σ_u⁺ and ³Δ_u parent states. Three of the involved states (²Π_u, ³Σ_u⁺, and ³Δ_u) possess multireference character, and CAP/XMS-CASPT2 can easily describe these states with a relatively modest active space.

Multireference Electron Correlation Methods: Journeys along Potential Energy Surfaces

Multireference electron correlation methods describe static and dynamical electron correlation in a balanced way and, therefore, can yield accurate and predictive results even when single-reference methods or multiconfigurational self-consistent field theory fails. One of their most prominent applications in quantum chemistry is the exploration of potential energy surfaces. This includes the optimization of molecular geometries, such as equilibrium geometries and conical intersections and on-the-fly photodynamics simulations, both of which depend heavily on the ability of the method to properly explore the potential energy surface. Because such applications require nuclear gradients and derivative couplings, the availability of analytical nuclear gradients greatly enhances the scope of quantum chemical methods. This review focuses on the developments and advances made in the past two decades. A detailed account of the analytical nuclear gradient and derivative coupling theories is presented. Emphasis is given to the software infrastructure that allows one to make use of these methods. Notable applications of multireference electron correlation methods to chemistry, including geometry optimizations and on-the-fly dynamics, are summarized at the end followed by a discussion of future prospects.

Probing competing relaxation pathways in malonaldehyde with transient X-ray absorption spectroscopy

Excited-state intramolecular hydrogen transfer (ESIHT) is a fundamental reaction relevant to chemistry and biology. Malonaldehyde is the simplest example of ESIHT, yet only little is known experimentally about its excited-state dynamics. Several competing relaxation pathways have been proposed, including internal conversion mediated by ESIHT and C[double bond, length as m-dash]C torsional motion as well as intersystem crossing. We perform an in silico transient X-ray absorption spectroscopy (TRXAS) experiment at the oxygen K-edge to investigate its potential to monitor the proposed ultrafast decay pathways in malonaldehyde upon photoexcitation to its bright S₂(ππ*) state. We employ both restricted active space perturbation theory and algebraic-diagrammatic construction for the polarization propagator along interpolated reaction coordinates as well as representative trajectories from ab initio multiple spawning simulations to compute the TRXAS signals from the lowest valence states. Our study suggests that oxygen K-edge TRXAS can distinctly fingerprint the passage through the H-transfer intersection and the concomitant population transfer to the S₁(nπ*) state. Potential intersystem crossing to T₁(ππ*) is detectable from reappearance of the double pre-edge signature and reversed intensities. Moreover, the torsional deactivation pathway induces transient charge redistribution from the enol side towards the central C-atom and manifests itself as substantial shifts of the pre-edge features. Given the continuous advances in X-ray light sources, our study proposes an experimental route to disentangle ultrafast excited-state decay channels in this prototypical ESIHT system and provides a pathway-specific mapping of the TRXAS signal to facilitate the interpretation of future experiments.

A multireference coupled-electron pair approximation combined with complete-active space perturbation theory in local pair-natural orbital framework

The Complete-Active Space Second-order Perturbation Theory (CASPT2) has been one of the most widely-used methods for reliably calculating electronic structures of multireference systems. Because of its lowest level treatment of dynamic correlation, it has a high computational feasibility; however, its accuracy in some cases falls short of needs. Here, as a simple yet higher-order alternative, we introduce a hybrid theory of the CASPT2 and a multireference variant of the Coupled-Electron Pair Approximation (CEPA), which is a class of high level correlation theory. A central feature of our theory (CEPT2) is to use the two underlying theories for describing different divisions of correlation components based on the full internal contraction framework. The external components, which usually give a major contribution to the dynamic correlation, are intensively described using the CEPA Ansatz, while the rests are treated at the CASPT2 level. Furthermore, to drastically reduce the computational demands, we have incorporated the pair-natural orbital (PNO) method into our multireference implementations. This development, thus, requires highly complex derivations and coding, while it has been largely facilitated with an automatic expression and code generation technique. To highlight the accuracy of the CEPT2 approach and to assess the errors caused by the PNO truncation, benchmark calculations are shown on small- to medium-size molecules, illustrating the high accuracy of the present CEPT2 model. By tightening the truncation thresholds, the PNO-CEPT2 energy converges toward the canonical counterpart and is more accurate than that of PNO-CASPT2 as long as the same truncation thresholds are used.