Dynamic electron correlation effect on conical intersections in photochemical ring-opening reaction of cyclohexadiene: MS-CASPT2 study

Ultrafast Photoinduced Dynamics in 1,3-Cyclohexadiene: A Comparison of Trajectory Surface Hopping Schemes†

Photoinduced nonadiabatic processes play a crucial role in a wide range of disciplines, from fundamental steps in biology to modern applications in advanced materials science. A theoretical understanding of these processes is highly desirable, and trajectory surface hopping (TSH) has proven to be a well-suited framework for a wide range of systems. In this work, we present a comprehensive comparison between two TSH algorithms, the conventional Tully's fewest switches surface hopping (FSSH) scheme and the Landau-Zener surface hopping (LZSH), to study the photoinduced ring-opening of 1,3-cyclohexadiene (CHD) to 1,3,5-hexatriene at the spin-flip time-dependent density functional theory (SF-TDDFT) level of theory. Additionally, we compare our results with a literature study at the extended multistate complete active space second-order perturbation theory method (XMS-CASPT2) level of theory. Our results show that the average population and lifetimes estimated with LZSH using SF-TDDFT are closer to the literature (using multireference methods) than those estimated with FSSH using SF-TDDFT. The latter speaks in favor of applying LZSH in combination with the SF-TDDFT method to study larger and more complex systems such as molecular photoswitches where the CHD molecule acts as a backbone. In addition, we present an implementation of Tully's FSSH algorithm as an extension to the PySurf software package.

Prediction Challenge: Simulating Rydberg photoexcited cyclobutanone with surface hopping dynamics based on different electronic structure methods

This research examines the nonadiabatic dynamics of cyclobutanone after excitation into the n → 3s Rydberg S2 state. It stems from our contribution to the Special Topic of the Journal of Chemical Physics to test the predictive capability of computational chemistry against unseen experimental data. Decoherence-corrected fewest-switches surface hopping was used to simulate nonadiabatic dynamics with full and approximated nonadiabatic couplings. Several simulation sets were computed with different electronic structure methods, including a multiconfigurational wavefunction [multiconfigurational self-consistent field (MCSCF)] specially built to describe dissociative channels, multireference semiempirical approach, time-dependent density functional theory, algebraic diagrammatic construction, and coupled cluster. MCSCF dynamics predicts a slow deactivation of the S2 state (10 ps), followed by an ultrafast population transfer from S1 to S0 (<100 fs). CO elimination (C3 channel) dominates over C2H4 formation (C2 channel). These findings radically differ from the other methods, which predicted S2 lifetimes 10-250 times shorter and C2 channel predominance. These results suggest that routine electronic structure methods may hold low predictive power for the outcome of nonadiabatic dynamics.

Deciphering the Internal Conversion Processes Involved in the Photochemical Ring-Opening of 1,3-Cyclohexadiene by Symmetric sp2-Carbon Substitutions

Chemical substituents hold the potential to markedly influence the photochemical behavior in molecular systems and assist in gaining a comprehensive understanding of nonadiabatic phenomena. In this study, we have conducted a comparative analysis of the influence of chemical substituents on the photochemical ring-opening of 1,3-cyclohexadiene (CHD), considering four systems: CHD, 2,3-dimethylcyclohexadiene (CHD-Me2-1), 1,4-dimethylcyclohexadiene (CHD-Me2-2), and 1,2,3,4-tetramethylcyclohexadiene (CHD-Me4), using electronic structure theory calculations and nonadiabatic molecular dynamics simulations. Employing extended multistate complete active space second-order perturbation (XMS-CASPT2) theory, we optimized reactants, S1 states, conical intersections (CIns), and products, revealing structural and energetic variations consistent with prior research. Nonadiabatic molecular dynamics simulation was used to gain insights into photochemical dynamics at state-averaged complete active space self-consistent field (SA-CASSCF) theory. CHD-Me4 exhibited reduced carbon-carbon single bond rupture rates, responsible for ring-opening, due to substituent proximity. Further, CHD-Me2-2 and CHD-Me4 displayed prolonged excited-state relaxation times, highlighting notable substituents' impact. Analysis of kinetic energy profiles of specific carbon atoms also revealed restrained atomic displacements, particularly in CHD-Me2-2 and CHD-Me4. These findings advance our understanding of how substituents modulate photochemical reactions in cyclohexadiene derivatives, guiding new molecular design and future research.

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

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

Analytic first-order derivatives of CASPT2 with IPEA shift

Complete active space second-order perturbation theory (CASPT2) is useful for accurately predicting properties of complex electronic structures, but it is well known that it systematically underestimates excitation energies. The underestimation can be corrected using the ionization potential-electron affinity (IPEA) shift. In this study, analytic first-order derivatives of CASPT2 with the IPEA shift are developed. CASPT2-IPEA is not invariant with respect to rotations among active molecular orbitals, and two additional constraint conditions are necessary in the CASPT2 Lagrangian to formulate analytic derivatives. The method developed here is applied to methylpyrimidine derivatives and cytosine, and minimum energy structures and conical intersections are located. By comparing energies relative to the closed-shell ground state, we find that the agreement with experiments and high-level calculations is indeed improved by the inclusion of the IPEA shift. The agreement of geometrical parameters with high-level calculations may also be improved in some cases.

Intermolecular-Type Conical Intersections in Benzene Dimer

The equilibrium and conical intersection geometries of the benzene dimer were computed in the framework of the conventional, linear-response time-dependent and spin-flipped time-dependent density functional theories (known as DFT, TDDFT and SF-TDDFT) as well as using the multiconfigurational complete active space self-consistent field (CASSCF) method considering the minimally augmented def2-TZVPP and the 6-31G(d,p) basis sets. It was found that the stacking distance between the benzene monomers decreases by about 0.5 Å in the first electronic excited state, due to the stronger intermolecular interaction energy, bringing the two monomers closer together. Intermolecular-type conical intersection (CI) geometries can be formed between the two benzene molecules, when (i) both monomer rings show planar deformation and (ii) weaker (approximately 1.6-1.8 Å long) C-C bonds are formed between the two monomers, with parallel and antiparallel orientation with respect to the monomer. These intermolecular-type CIs look energetically more favorable than dimeric CIs containing only one deformed monomer. The validity of the dimer-type CI geometries obtained by SF-TDDFT was confirmed by the CASSCF method. The nudged elastic band method used for finding the optimal relaxation path has confirmed both the accessibility of these intermolecular-type CIs and the possibility of the radiationless deactivation of the electronic excited states through these CI geometries. Although not as energetically favorable as the previous two CI geometries, there are other CI geometries characterized by the relative rotation of monomers at different angles around a vertical C-C axis.

Nonadiabatic Dynamics of 1,3-Cyclohexadiene by Curvature-Driven Coherent Switching with Decay of Mixing

The photoinduced ring-opening reaction of 1,3-cyclohexadiene to produce 1,3,5-hexatriene is a classic electrocyclic reaction and is also a prototype for many reactions of biological and synthetic importance. Here, we simulate the ultrafast nonadiabatic dynamics of the reaction in the manifold of the three lowest valence electronic states by using extended multistate complete-active-space second-order perturbation theory (XMS-CASPT2) combined with the curvature-driven coherent switching with decay of mixing (κCSDM) dynamical method. We obtain an excited-state lifetime of 79 fs, and a product quantum yield of 40% from the 500 trajectories initiated in the S₁ excited state. The obtained lifetime and quantum yield values are very close to previously reported experimental and computed values, showing the capability of performing a reasonable nonadiabatic ring-opening dynamics with the κCSDM method that does not require nonadiabatic coupling vectors, time derivatives, or diabatization. In addition, we study the ring-opening reaction by initiating the trajectories in the dark state S₂. We also optimize the S₀/S₁ and S₁/S₂ minimum-energy conical intersections (MECIs) by XMS-CASPT2; for S₁/S₂, we optimized both an inner and an outer local-minimum-energy conical intersections (LMECIs). We provide the potential energy profile along the ring-opening coordinate by joining selected critical points via linear synchronous transit paths. We find the inner S₁/S₂ LMECI to be more crucial than the outer one.

Analytic First-Order Derivatives of (X)MS, XDW, and RMS Variants of the CASPT2 and RASPT2 Methods

Crossings between states involve complex electronic structures, making the accurate characterization of the crossing point difficult. In this study, the analytic derivatives of three complete active space second-order perturbation theory (CASPT2) variants as well as an extension of the restricted active space (RASPT2) are developed. These variants are applied to locating minimum energy conical intersections. Our results demonstrate that the three CASPT2 variants predict qualitatively similar results, but a recently developed variant, the rotated multistate CASPT2 (RMS-CASPT2), is least sensitive to the number of states considered in the calculation. We demonstrate that CASPT2 and the reference self-consistent field calculations predict qualitatively different energetics and bond lengths.

Providing theoretical insight into the role of symmetry in the photoisomerization mechanism of a non-symmetric dithienylethene photoswitch

Dithienylethene (DTE) molecular photoswitches have shown to be excellent candidates in the design of efficient optoelectronic devices, due to their high photoisomerization quantum yield (QY), for which symmetry is suggested to play a crucial role. Here, we present a theoretical study on the photochemistry of a non-symmetric dithienylethene photoswitch, with a special emphasis on the effect of asymmetric substitution on the photocyclization and photoreversion mechanisms. We used the Spin-Flip Time Dependent Density Functional Theory (SF-TDDFT) method to locate and characterize the main structures (conical intersections and minima) of the ground state and the first two excited states, S₁ and S₂, along the ring-opening/closure reaction coordinate of the photocyclization and photoreversion processes, and to identify the important coordinates governing the radiationless decay pathways. Our results suggest that while the main features that characterize the photoisomerization of symmetric DTEs are also present for the photoisomerization of the non-symmetric DTE, the lower energy barrier on S₁ along the cycloreversion reaction speaks in favor of a more efficient and therefore a higher cycloreversion QY for the non-symmetric DTEs, making them a better candidate for molecular optoelectronic devices than their symmetric counterparts.

Analytical Gradient Theory for Spin-Free State-Averaged Second-Order Driven Similarity Renormalization Group Perturbation Theory (SA-DSRG-MRPT2) and Its Applications for Conical Intersection Optimizations

Second-order multireference-driven similarity renormalization group perturbation theory (DSRG-MRPT2) provides an efficient means of correcting the dynamical correlation with the multiconfiguration reference function. The state-averaged DSRG-MRPT2 (SA-DSRG-MRPT2) method is the simplest means of treating the excited states with DSRG-MRPT2. In this method, the Hamiltonian dressed with dynamical correlation is diagonalized in the CASCI state subspace (SA-DSRG-MRPT2c) or the configuration subspace (SA-DSRG-MRPT2). This work develops analytical gradient theory for spin-free SA-DSRG-MRPT2(c) with the density-fitting approximation. We check the accuracy of the analytical gradients against the numerical gradients. We present applications for optimizing minimum energy conical intersections (MECI) of ethylene and retinal model chromophores (PSB3 and RPSB6). We investigate the dependence of the optimized geometries and energies on the flow parameters and reference relaxations. The smoothness of the SA-DSRG-MRPT2(c) potential energy surfaces near the reference (complete active space self-consistent field) MECI is comparable to the XMCQDPT2 one. These results render SA-DSRG-MRPT2(c) theory a promising approach for studies of conical intersections.

On Electron Pair Rearrangements in Photochemical Reactions: 1,3-Cyclohexadiene Ring Opening

1,3-Cyclohexadiene ring opening has been studied within the bonding evolution theory (BET) framework. We have focused on describing for the first time the electron pair rearrangements leading to the cis-1,3,5-hexatriene (HT) product from CHD. The nature of bonding in this process begins with the weakening of the double bonds in the Franck-Condon region. Along the 1¹B surface, the C-C sigma bond weakens. Meanwhile, its density redistributes toward the whole CHD ring, mainly over double bonds. Breaking of this bond occurs on the 2¹A surface due to the symmetrical splitting of pair density from this region. This density redistributes toward the reaction center once the pericyclic minimum is reached. The formation of the double bonds that characterize HT occurs gradually in the ground state. However, near the 2¹A/1¹A intersection, these bonds are partially established.

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.

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.

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.

Ab initio surface hopping excited-state molecular dynamics approach on the basis of spin-orbit coupled states: An application to the A-band photodissociation of CH3 I

Ab initio molecular dynamics approach has been extended to multi-state dynamics on the basis of the spin-orbit coupled electronic states that are obtained through diagonalization of the spin-orbit coupling matrix with the multi-state second-order multireference perturbation theory energies in diagonal elements and the spin-orbit coupling terms at the state-averaged complete active space self-consistent field level in off-diagonal elements. Nonadiabatic transitions over the spin-orbit coupled states were taken into account explicitly by a surface hopping scheme with utilizing the nonadiabatic coupling terms calculated by numerical differentiation of the spin-orbit coupled wavefunctions and analytical nonadiabatic coupling terms. The present method was applied to the A-band photodissociation of methyl iodide, CH₃ I + hv → CH₃ + I (² P_3/2 )/I* (² P_1/2 ), for which a pioneering theoretical work was reported by Amatatsu, Yabushita, and Morokuma. The present results reproduced well the experimental branching ratio and energy distributions in the dissociative products. © 2018 Wiley Periodicals, Inc.

On the applicability of a wavefunction-free, energy-based procedure for generating first-order non-adiabatic couplings around conical intersections

The primal definition of first-order non-adiabatic couplings among electronic states relies on the knowledge of how electronic wavefunctions vary with nuclear coordinates. However, the non-adiabatic coupling between two electronic states can be obtained in the vicinity of a conical intersection from energies only, as this vector spans the branching plane along which degeneracy is lifted to first order. The gradient difference and derivative coupling are responsible of the two-dimensional cusp of a conical intersection between both potential-energy surfaces and can be identified to the non-trivial eigenvectors of the second derivative of the square energy difference, as first pointed out in Köppel and Schubert [Mol. Phys. 104(5-7), 1069 (2006)]. Such quantities can always be computed in principle for the cost of two numerical Hessians in the worst-case scenario. Analytic-derivative techniques may help in terms of accuracy and efficiency but also raise potential traps due to singularities and ill-defined derivatives at degeneracies. We compare here two approaches, one fully numerical, the other semianalytic, where analytic gradients are available but Hessians are not, and investigate their respective conditions of applicability. Benzene and 3-hydroxychromone are used as illustrative application cases. It is shown that non-adiabatic couplings can thus be estimated with decent accuracy in regions of significant size around conical intersections. This procedure is robust and could be useful in the context of on-the-fly non-adiabatic dynamics or be used for producing model representations of intersecting potential energy surfaces with complete obviation of the electronic wavefunctions.

Effects of probe energy and competing pathways on time-resolved photoelectron spectroscopy: the ring-opening of 1,3-cyclohexadiene

Photoelectron spectra for the ring-opening dynamics of 1,3-cyclohexadiene are studied using a model based on quantum molecular dynamics and the Dyson orbital approach.

Estimating the Derivative Coupling Vector Using Gradients

The difference gradient and derivative coupling vectors span the branching planes of conical intersections between electronic states. While gradients are commonly available in many electronic structure methods, the derivative coupling vectors are not always implemented and ready for use in characterizing conical intersections. This Letter shows how the derivative coupling vectors can be computed to high accuracy (direction and magnitude) using energy and gradient information. The new method is based on the combination of a linear-coupling two-state Hamiltonian and a finite-difference Davidson approach for computing the branching plane. Benchmark cases are provided showing these vectors can be efficiently computed near conical intersections.

Photochemical Dynamics of Ethylene Cation C2H4(.)

We present a theoretical study of the nonadiabatic effects in ethylene cation C2H4(+), the simplest π radical cation, after photoexcitation to its three lowest doublet excited states. Two families of conical intersections are found, with minimum energy structures characterized by planar and twisted geometries. Ab initio multiple spawning dynamical calculations suggest that the competition between these relaxation pathways depends strongly on the initial excited state, with excited state lifetimes in the 30-60 fs range. Ultrafast decay via planar geometries deposits the molecule near a bridged minimum on the ground state, allowing prompt H migration events. The alternative pathway mediated by torsional motion induces important backspawned population transfer promoted by hindered rotations. The results allow us to revisit earlier vibrationally-mediated photodissociation experiments and shed light on the electronic relaxation dynamics of a prototypical radical cation subject to strong vibronic interactions.

Hexamethylcyclopentadiene: time-resolved photoelectron spectroscopy and ab initio multiple spawning simulations

Time-resolved photoelectron spectroscopy and ab initio multiple spawning dynamical simulations of hexamethylcyclopentadiene reveal wavepacket evolution in a distinct degree of freedom.

QUANTIZATION OF CHEMICAL REACTION: THE DYNAMIC CORRELATION DIAGRAM METHOD FREE FROM NONCROSSING RULE

Both Fukui's frontier orbital (FO) theory and Woodward–Hoffmann's orbital symmetry conservation (W–H) theory are based on the molecular orbital theory. However, there are some obvious inconsistencies between the two theories in explaining the electron movement. The process of chemical reactions has been explained by the potential surface analysis based on the time-independent Schrödinger equation. However, this approach is not always appropriate for describing unsteady states, because the variable of reaction coordinate should be time t by its very nature. When considering the time-scale of chemical reactions for molecules, there is inherent uncertainty in the energy levels for the midway state of chemical reactions owing to the Heisenberg uncertainty principle. The states which can be accurately described by quantum mechanics exist discontinuously in chemical reactions. Such quantization of chemical reactions solves all noncrossing problems. We also show that such an essential fact leads to new concepts and theories in chemical reactions such as stable molecule, elementary reaction and minimum deformation of orbital phases. Finally, taking the regularity in organic reactions as an example, we demonstrate that FO theory and W–H theory can be unified consistently by the universally applicable "dynamic correlation diagram method."

Photoisomerization of stilbene: a spin-flip density functional theory approach

The photoisomerization process of 1,2-diphenylethylene (stilbene) is investigated using the spin-flip density functional theory (SFDFT), which has recently been shown to be a promising approach for locating conical intersection (CI) points (Minezawa, N.; Gordon, M. S. J. Phys. Chem. A2009, 113, 12749). The SFDFT method gives valuable insight into twisted stilbene to which the linear response time-dependent DFT approach cannot be applied. In contrast to the previous SFDFT study of ethylene, a distinct twisted minimum is found for stilbene. The optimized structure has a sizable pyramidalization angle and strong ionic character, indicating that a purely twisted geometry is not a true minimum. In addition, the SFDFT approach can successfully locate two CI points: the twisted-pyramidalized CI that is similar to the ethylene counterpart and another CI that possibly lies on the cyclization pathway of cis-stilbene. The mechanisms of the cis--trans isomerization reaction are discussed on the basis of the two-dimensional potential energy surface along the twisting and pyramidalization angles.

The Ultrafast Pathway of Photon-Induced Electrocyclic Ring-Opening Reactions: The Case of 1,3-Cyclohexadiene

The photochemically induced electrocyclic ring-opening reaction of 1,3-cyclohexadiene to 1,3,5-hexatriene serves as a prototype for many important reactions in chemistry and in biological systems. Based on experimental and computational studies, a detailed picture of the reaction has now emerged: Excitation to the Franck-Condon region places the molecule on a steeply repulsive part of the 1B potential energy surface, which propels the molecule in exactly the conrotatory direction that conforms to the Woodward-Hoffmann rules of orbital symmetry. Bypassing a cusp in a symmetry-breaking direction, the wave packet enters the 2A state within 55 fs. It continues to move directly toward the 2A/1A conical intersection, where it crosses in approximately 80 fs to the ground state. This article summarizes the published experimental and theoretical work to describe the current understanding of the reaction while pointing to important questions that remain to be addressed.

Optimizing conical intersections by spin-flip density functional theory: application to ethylene

Conical intersections (CIs) of ethylene have been successfully determined using spin-flip density functional theory (SFDFT) combined with a penalty-constrained optimization method. We present in detail three structures, twisted-pyramidalized, hydrogen-migrated, and ethylidene CIs. In contrast to the linear response time-dependent density functional theory, which predicts a purely twisted geometry without pyramidalization as the S(1) global minimum, SFDFT gives a pyramidalized structure. Therefore, this is the first correct optimization of CI points of twisted ethylene by the DFT method. The calculated energies and geometries are in good agreement with those obtained by the multireference configuration interaction (MR-CI) method and the multistate formulation of second-order multireference perturbation theory (MS-CASPT2).