A computational strategy for geometry optimization of ionic and covalent excited states, applied to butadiene and hexatriene

Reference Energies for Double Excitations: Improvement and Extension

In the realm of photochemistry, the significance of double excitations (also known as doubly excited states), where two electrons are concurrently elevated to higher energy levels, lies in their involvement in key electronic transitions essential in light-induced chemical reactions as well as their challenging nature from the computational theoretical chemistry point of view. Based on state-of-the-art electronic structure methods (such as high-order coupled-cluster, selected configuration interaction, and multiconfigurational methods), we improve and expand our prior set of accurate reference excitation energies for electronic states exhibiting a substantial amount of double excitations [Loos et al. J. Chem. Theory Comput. 2019, 15, 1939]. This extended collection encompasses 47 electronic transitions across 26 molecular systems that we separate into two distinct subsets: (i) 28 "genuine" doubly excited states where the transitions almost exclusively involve doubly excited configurations and (ii) 19 "partial" doubly excited states which exhibit a more balanced character between singly and doubly excited configurations. For each subset, we assess the performance of high-order coupled-cluster (CC3, CCSDT, CC4, and CCSDTQ) and multiconfigurational methods (CASPT2, CASPT3, PC-NEVPT2, and SC-NEVPT2). Using as a probe the percentage of single excitations involved in a given transition (%T1) computed at the CC3 level, we also propose a simple correction that reduces the errors of CC3 by a factor of 3, for both sets of excitations. We hope that this more complete and diverse compilation of double excitations will help future developments of electronic excited-state methodologies.

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.

Quantification of the Ionic Character of Multiconfigurational Wave Functions: The Qat Diagnostic

The complete active space self-consistent field (CASSCF) method is a cornerstone in modern excited-state quantum chemistry providing the starting point for most common multireference computations. However, CASSCF, when used with a minimal active space, can produce significant errors (>2 eV) even for the excitation energies of simple hydrocarbons if the states of interest possess ionic character. After illustrating this problem in some detail, we present a diagnostic for ionic character, denoted as Q at, that is readily computed from the transition density. A set of 11 molecules is considered to study errors in vertical excitation energies. State-averaged CASSCF obtains a mean absolute error (MAE) of 0.87 eV for the 34 singlet states considered. We highlight a strong correlation between the obtained errors and the Q at diagnostic, illustrating its power to predict problematic cases. Conversely, using multireference configuration interaction with single and double excitations and Pople's size extensivity correction (MR-CISD+P), excellent results are obtained with an MAE of 0.11 eV. Furthermore, correlations with the Q at diagnostic disappear. In summary, we hope that the presented diagnostic will facilitate reliable and user-friendly multireference computations on conjugated organic molecules.

Excited states of ortho-nitrobenzaldehyde as a challenging case for single- and multi-reference electronic structure theory

We present a large set of vertical excitation calculations for the ortho-nitrobenzaldehyde (oNBA) molecule, which exhibits a very challenging excited-state electronic structure like other nitroaromatic compounds. The single-reference methods produce mostly consistent results up to about 5.5 eV. By contrast, the CAS second-order perturbation theory (CASPT2) results depend sensitively on the employed parameters. At the CAS self-consistent field level, the energies of the bright ππ * $$ {\pi \pi}^{\ast } $$ states are strongly overestimated while doubly excited states appear too low and mix with these ππ * $$ {\pi \pi}^{\ast } $$ states. This mixing hampers the CASPT2 step, leading to inconsistent results. Only by increasing the number of states in the state-averaging step to about 40-to cover all bright ππ * $$ {\pi \pi}^{\ast } $$ states embedded in the double excitations-and employing extended multistate CASPT2 could CASPT2 results consistent with experiment be obtained. We assign the four bands in the molecule's spectrum: The weakest band at 3.7 eV arises from the n NO 2 π * $$ {n}_{\mathrm{NO}2}{\pi}^{\ast } $$ states, the second one at 4.4 eV from the ππ * $$ {\pi \pi}^{\ast } $$ ( L b $$ {L}_b $$ ) state, the shoulder at 5.2 eV from the ππ * $$ {\pi \pi}^{\ast } $$ ( L a $$ {L}_a $$ ) state, and the maximum at 5.7 eV from the ππ * $$ {\pi \pi}^{\ast } $$ ( B a / B b $$ {B}_a/{B}_b $$ ) states. We also highlight the importance of modern wave function analysis techniques in elucidating the absorption spectrum of challenging molecules.

Benchmarking CASPT3 Vertical Excitation Energies

Based on 280 reference vertical transition energies of various natures (singlet, triplet, valence, Rydberg, n → π∗, π → π∗, and double excitations) extracted from the QUEST database, we assess the accuracy of third-order multireference perturbation theory, CASPT3, in the context of molecular excited states. When one applies the disputable ionization- potential-electron-affinity (IPEA) shift, we show that CASPT3 provides a similar accuracy as its second-order counterpart, CASPT2, with the same mean absolute error of 0.11 eV. However, as already reported, we also observe that the accuracy of CASPT3 is almost insensitive to the IPEA shift, irrespective of the transition type and system size, with a small reduction of the mean absolute error to 0.09 eV when the IPEA shift is switched off.

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.

Dynamical correction to the Bethe-Salpeter equation beyond the plasmon-pole approximation

The Bethe-Salpeter equation (BSE) formalism is a computationally affordable method for the calculation of accurate optical excitation energies in molecular systems. Similar to the ubiquitous adiabatic approximation of time-dependent density-functional theory, the static approximation, which substitutes a dynamical (i.e., frequency-dependent) kernel by its static limit, is usually enforced in most implementations of the BSE formalism. Here, going beyond the static approximation, we compute the dynamical correction of the electron-hole screening for molecular excitation energies, thanks to a renormalized first-order perturbative correction to the static BSE excitation energies. The present dynamical correction goes beyond the plasmon-pole approximation as the dynamical screening of the Coulomb interaction is computed exactly within the random-phase approximation. Our calculations are benchmarked against high-level (coupled-cluster) calculations, allowing one to assess the clear improvement brought by the dynamical correction for both singlet and triplet optical transitions.

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.

Electronic Excited States and UV-Vis Absorption Spectra of the Dihydropyrene/Cyclophanediene Photochromic Couple: a Theoretical Investigation

Dihydropyrene (DHP)/cyclophanediene (CPD) is a fascinating photoswitchable organic system displaying negative photochromism. Upon irradiation in the visible region, the colored DHP can be converted to its open-ring CPD colorless isomer, which can be converted back to DHP by UV light. DHP and CPD thus possess very different absorption spectra whose absorption bands have never been assigned in detail so far. In this work, we characterize the vertical electronic transitions of the first six and seven excited states of DHP and CPD, respectively, aiming for a realistic comparison with experiment. We used state-of-the-art electronic structure methods [e.g., complete active space second-order perturbation theory (CASPT2), n-electron valence-state perturbation theory (NEVPT2), extended multiconfigurational quasi-degenerate perturbation theory (XMCQDPT2), and third-order algebraic diagrammatic construction ADC(3)] capable of describing differential electron correlation. Vertical transition energies were also computed with time-dependent density functional theory (TD-DFT) and compared to these accurate methods. After the reliability of TD-DFT was validated for the main optical transitions, this efficient method was used to simulate the absorption spectra of DHP and CPD in the framework of the Franck-Condon Herzberg-Teller approximation and also using the nuclear ensemble approach. Overall, for both methods, the simulated absorption spectra reproduce nicely the main spectral features of the DHP and CPD isomers, that is, the main four absorption bands of increasing intensity of DHP and the absorption rise below 300 nm for CPD.

A Mountaineering Strategy to Excited States: Highly Accurate Energies and Benchmarks for Medium Sized Molecules

Following our previous work focusing on compounds containing up to 3 non-hydrogen atoms [J. Chem. Theory Comput. 2018, 14, 4360-4379], we present here highly accurate vertical transition energies obtained for 27 molecules encompassing 4, 5, and 6 non-hydrogen atoms: acetone, acrolein, benzene, butadiene, cyanoacetylene, cyanoformaldehyde, cyanogen, cyclopentadiene, cyclopropenone, cyclopropenethione, diacetylene, furan, glyoxal, imidazole, isobutene, methylenecyclopropene, propynal, pyrazine, pyridazine, pyridine, pyrimidine, pyrrole, tetrazine, thioacetone, thiophene, thiopropynal, and triazine. To obtain these energies, we use equation-of-motion/linear-response coupled cluster theory up to the highest technically possible excitation order for these systems (CC3, EOM-CCSDT, and EOM-CCSDTQ) and selected configuration interaction (SCI) calculations (with tens of millions of determinants in the reference space), as well as the multiconfigurational n-electron valence state perturbation theory (NEVPT2) method. All these approaches are applied in combination with diffuse-containing atomic basis sets. For all transitions, we report at least CC3/aug-cc-pVQZ vertical excitation energies as well as CC3/aug-cc-pVTZ oscillator strengths for each dipole-allowed transition. We show that CC3 almost systematically delivers transition energies in agreement with higher-level methods with a typical deviation of ±0.04 eV, except for transitions with a dominant double excitation character where the error is much larger. The present contribution gathers a large, diverse, and accurate set of more than 200 highly accurate transition energies for states of various natures (valence, Rydberg, singlet, triplet, n → π*, π → π*, ...). We use this series of theoretical best estimates to benchmark a series of popular methods for excited state calculations: CIS(D), ADC(2), CC2, STEOM-CCSD, EOM-CCSD, CCSDR(3), CCSDT-3, CC3, and NEVPT2. The results of these benchmarks are compared to the available literature data.

Toward an understanding of electronic excitation energies beyond the molecular orbital picture

Can we gain an intuitive understanding of excitation energies beyond the molecular picture?

Molecular Vertical Excitation Energies Studied with First-Order RASSCF (RAS[1,1]): Balancing Covalent and Ionic Excited States

RASSCF calculations of vertical excitation energies were carried out on a benchmark set of 19 organic molecules studied by Thiel and co-workers [J. Chem. Phys.2008, 128, 13411018397056]. The best results, in comparison with the MS-CASPT2 results of Thiel, were obtained using a RASSCF space that contains at most one hole and one particle in the RAS1 and RAS3 spaces, respectively, which we denote as RAS[1,1]. This subset of configurations recovers mainly the effect of polarization and semi-internal electronic correlation that is only included in CASSCF in an averaged way. Adding all-external correlation by allowing double excitations from RAS1 and RAS2 into RAS3 did not improve the results, and indeed, they were slightly worse. The accuracy of the first-order RASSCF computations is demonstrated to be a function of whether the state of interest can be classified as covalent or ionic in the space of configurations built from orbitals localized onto atomic sites. For covalent states, polarization and semi-internal correlation effects are negligible (RAS[1,1]), while for ionic states, these effects are large (because of inherent diffusiveness of these states compared to the covalent states) and, thus, an acceptable agreement with MS-CASPT2 can be obtained using first-order RASSCF with the extra basis set involving 3p orbitals in most cases. However, for those ionic states that are quasi-degenerate with a Rydberg state or for nonlocal nπ* states, there remains a significant error resulting from all external correlation effects.

Photochemistry of the 1,6-Dideuterio-1,3,5-hexatrienes in Solution: Efficient Terminal Bond Photoisomerization in One-Bond-Twist and Bicycle Pedal Ways

The report that the central bond photoisomerization of the 1,3,5-hexatrienes (Hts) is highly inefficient has encouraged theoreticians to seek conical intersections (CIs) at geometries that can explain rapid nonradiative return to the initially excited isomer. Because they are photochemically silent, torsional relaxations about the terminal double bonds of the Hts have not been evaluated as significant radiationless decay pathways. Study of the photoisomerization of trans,trans,trans- and trans,cis,trans-1,6-dideuterio-1,3,5-hexatrienes ( ttt- and tct-Ht_d2) addresses this issue. Degassed cyclohexane- d₁₂ (C₆D₁₂) and CD₃CN solutions were irradiated at 254 nm in quartz NMR tubes, and the progress of the reactions was followed by ¹H NMR. Photoisomerization rates based on the integration of terminal hydrogen NMR peaks are in reasonable agreement with rates obtained by fitting pure isomer NMR spectra to the phase shift and baseline corrected experimental NMR spectra. The results show that terminal bond isomerization is highly efficient, especially when one considers that central bond isomerization is much more efficient than previously reported and is mainly observed together with terminal bond isomerization. A mechanism involving terminal one-bond-twist (OBT) in competition with a bicycle pedal (BP) process accounts for all terminal and most central bond photoisomerization. OBT central bond isomerization is a minor reaction that is observed primarily in the tct to ttt direction. Most surprising is the prominent role of the BP process in central bond photoisomerization. Proposed initially to account for photoisomerization in free volume constraining media, it is observed here in the absence of medium constraints.

Mechanistic Investigations on the Photoisomerization Reactions of Five-Membered Ring Heterocyclic Molecules Containing Sulfur and Selenium Atoms

The restricted active space self-consistent field method in the 26-electron/27-orbital active space and the 6-311(d) basis set has been used to investigate the mechanisms of the photochemical isomerization reactions concerning the model systems of 1,2,3-thiadiazole and 1,2,3-selenadiazole. The computational works suggest that the preferred reaction paths for both 1,2,3-thiadiazole and 1,2,3-selenadiazole are as follows: reactant → Franck-Condon region → conical intersection → intermediate → transition states → photoproducts. As a result, the structures of the conical intersections, which play a decisive role in these photoisomerization reactions, are obtained. In particular, the present theoretical evidences demonstrate that the potential energy surfaces for the formation of 1,3-diradicals are quite flat. This may explain why their experimental detections are so difficult.

Electronic Structures of Anti-Ferromagnetic Tetraradicals: Ab Initio and Semi-Empirical Studies

The energy relationships and electronic structures of the lowest-lying spin states in several anti-ferromagnetic tetraradical model systems are studied with high-level ab initio and semi-empirical methods. The Full-CI method (FCI), the complete active space second-order perturbation theory (CASPT2), and the n-electron valence state perturbation theory (NEVPT2) are employed to obtain reference results. By comparing the energy relationships predicted from the Heisenberg and Hubbard models with ab initio benchmarks, the accuracy of the widely used Heisenberg model for anti-ferromagnetic spin-coupling in low-spin polyradicals is cautiously tested in this work. It is found that the strength of electron correlation (|U/t|) concerning anti-ferromagnetically coupled radical centers could range widely from strong to moderate correlation regimes and could become another degree of freedom besides the spin multiplicity. Accordingly, the Heisenberg-type model works well in the regime of strong correlation, which reproduces well the energy relationships along with the wave functions of all the spin states. In moderately spin-correlated tetraradicals, the results of the prototype Heisenberg model deviate severely from those of multi-reference electron correlation ab initio methods, while the extended Heisenberg model, containing four-body terms, can introduce reasonable corrections and maintains its accuracy in this condition. In the weak correlation regime, both the prototype Heisenberg model and its extended forms containing higher-order correction terms will encounter difficulties. Meanwhile, the Hubbard model shows balanced accuracy from strong to weak correlation cases and can reproduce qualitatively correct electronic structures, which makes it more suitable for the study of anti-ferromagnetic coupling in polyradical systems.

π-Conjugation in trans-1,3-butadiene: static and dynamical electronic correlations described through quantum Monte Carlo

We investigate the effects of the static and dynamical electronic correlations on the level of conjugation of the trans-1,3-butadiene molecule through Quantum Monte Carlo methods applied to an Antisymmetrized Geminal Power (AGP) wave function, with a Jastrow factor similar to the Gutzwiller ansatz. The degree of conjugation is measured through the convergence of the structural properties of 1,3-butadiene and in particular of the Bond Length Alternation (BLA), that is the difference between the lengths of the single and double carbon bonds. After verifying the different roles of the Fermionic AGP part of our wave function and of the Jastrow factor in recovering electronic correlation, we study the effects of a constrained Active Space AGP (AGPAS), similar to that used in the Complete Active Space (CAS) representation. Through this AGPAS, we are able to identify the effect of the limited active space on the degree of conjugation, showing that in the limit of infinite active space the structural properties converge exactly to those of the atomic AGP, giving a BLA for 1,3-butadiene around 0.1244(5) Å.

Ultrafast Z → E photoisomerisation of structurally modified furylfulgides

Femtosecond transient absorption spectroscopy of Z-fulgides with selected structural motifs revealed fast and direct excited-state dynamics independent of chemical modifications.

Reaction pathways by quantum Monte Carlo: insight on the torsion barrier of 1,3-butadiene, and the conrotatory ring opening of cyclobutene

Quantum Monte Carlo (QMC) methods are used to investigate the intramolecular reaction pathways of 1,3-butadiene. The ground state geometries of the three conformers s-trans, s-cis, and gauche, as well as the cyclobutene structure are fully optimised at the variational Monte Carlo (VMC) level, obtaining an excellent agreement with the experimental results and other quantum chemistry high level calculations. Transition state geometries are also estimated at the VMC level for the s-trans to gauche torsion barrier of 1,3-butadiene and for the conrotatory ring opening of cyclobutene to the gauche-1,3-butadiene conformer. The energies of the conformers and the reaction barriers are calculated at both variational and diffusional Monte Carlo levels providing a precise picture of the potential energy surface of 1,3-butadiene and supporting one of the two model profiles recently obtained by Raman spectroscopy [Boopalachandran et al., J. Phys. Chem. A 115, 8920 (2011)]. Considering the good scaling of QMC techniques with the system's size, our results also demonstrate how variational Monte Carlo calculations can be applied in the future to properly investigate the reaction pathways of large and correlated molecular systems.

Photoreactivity of a push-pull merocyanine in static electric fields: a three-state model of isomerization reactions involving conical intersections

The photochemistry of a prototype push-pull merocyanine is discussed using a simple three-state model. As a derivative of butadiene, the model focuses on two isomerization reactions around the two double bonds of the butadiene backbone. As a molecule substituted by an electron donor and electron acceptor at opposite ends, its structure as well as its photochemistry are expected to be strongly affected by the environment. In polar solvents, a zwitterion transition state for each of the isomerization reactions is stabilized, and its energy is on the same order as that of the biradical one; this leads to the symmetry allowed crossing (S(0)/S(1) conical intersection). It is shown that applying an external electric field or varying the solvent polarity changes the relative energies of the different transition states as well as that of the conical intersection, and thus different photochemical products can be obtained. In particular, the very existence of conical intersections is found to depend on these external parameters. This work provides a theoretical foundation for ideas expressed by Squillacote et al. (J. Am. Chem. Soc. 2004, 126, 1940) concerning the electrostatic control of photochemical reactions.