Geometries and Vertical Excitation Energies in Retinal Analogues Resolved at the CASPT2 Level of Theory: Critical Assessment of the Performance of CASSCF, CC2, and DFT Methods

Internal Conversion between Bright (11Bu+) and Dark (21Ag-) States in s-trans-Butadiene and s-trans-Hexatriene

Internal conversion (IC) between the two lowest singlet excited states, 1¹B_u⁺ and 2¹A_g^-, of s-trans-butadiene and s-trans-hexatriene is investigated using a series of single- and multi- reference wave function and density functional theory (DFT) methodologies. Three independent types of the equation-of-motion coupled-cluster (EOMCC) theory capable of providing an accurate and balanced description of one- as well as two-electron transitions, abbreviated as δ-CR-EOMCC(2,3), DIP-EOMCC(4h2p){N_o}, and DEA-EOMCC(4p2h){N_u} or DEA-EOMCC(3p1h,4p2h){N_u}, consistently predict that the 1¹B_u⁺/2¹A_g^- crossing in both molecules occurs along the bond length alternation coordinate. However, the analogous 1¹B_u⁺ and 2¹A_g^- potentials obtained with some multireference approaches, such as CASSCF and MRCIS(D), as well as with the linear-response formulation of time-dependent DFT (TDDFT), do not cross. Hence, caution needs to be exercised when studying the low-lying singlet excited states of polyenes with conventional multiconfigurational methods and TDDFT. The multistate many-body perturbation theory methods, such as XMCQDPT2, do correctly reproduce the curve crossing. Among the simplest and least expensive computational methodologies, the DFT approaches that incorporate the contributions of doubly excited configurations, abbreviated as MRSF (mixed reference spin-flip) TDDFT and SSR(4,4), accurately reproduce our best EOMCC results. This is highly promising for nonadiabatic molecular dynamics simulations in larger systems.

Describing ground and excited state potential energy surfaces for molecular photoswitches using coupled cluster models

In this article, we use two extensively studied systems, a retinal model system and azobenzene, to explore the use of coupled cluster models for describing ground and singlet excited state potential energy surfaces of photoswitchable systems. While not being suitable for describing nuclear dynamics of photoisomerization, coupled cluster models have useful attributes, such as the inclusion of dynamical correlation, their black box nature, and the systematic improvement offered by truncation level. Results for the studied systems show that when triple excitations (here through the CC3 model) are included, ground and excited state potential energy surfaces for isomerization paths may reliably be generated, also for states of doubly excited character. For ground state equilibrium cis- and trans-azobenzene, the molecular geometry and basis set is seen to significantly impact the vertical excitation energies for the two lowest excited states. Efficient implementations of coupled cluster models can therefore constitute valuable tools for investigating photoswitchable systems and can be used for preliminary black box studies to gather information before more complicated excited state dynamics approaches are pursued.

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.

Initial excited-state relaxation of locked retinal protonated schiff base chromophore. An insight from coupled cluster and multireference perturbation theory calculations

The initial S₁ excited-state relaxation of retinal protonated Schiff base (RPSB) analog with central C11C12 double bond locked by eight-membered ring (locked-11.8) was investigated by means of multireference perturbation theory methods (XMCQDPT2, XMS-CASPT2, MS-CASPT2) as well as single-reference coupled-cluster CC2 method. The analysis of XMCQDPT2-based geometries reveals rather weak coupling between in-plane and out-of-plane structural evolution and minor energetical relaxation of three locked-11.8 conformers. Therefore, a strong coupling between bonds length inversion and backbone out-of-plane deformation resulting in a very steep S₁ energy profile predicted by CASSCF/CASPT2 calculations is in clear contradiction with the reference XMCQDPT2 results. Even though CC2 method predicts good quality ground-state structures, the excited-state structures display more advanced torsional deformation leading to ca. 0.2 eV exaggerated energy relaxation and significantly red shifted (0.4-0.7 eV) emission maxima. According to our findings, the initial photoisomerization process in locked-11.8, and possibly in other RPSB analogs, studied fully (both geometries and energies) by multireference perturbation theory may be somewhat slower than predicted by CASSCF/CASPT2 or CC2 methods. © 2018 Wiley Periodicals, Inc.

Is the choice of a standard zeroth-order hamiltonian in CASPT2 ansatz optimal in calculations of excitation energies in protonated and unprotonated schiff bases of retinal?

To account for systematic error of CASPT2 method empirical modification of the zeroth-order Hamiltonian with Ionization Potential-Electron Affinity (IPEA) shift was introduced. The optimized IPEA value (0.25 a.u.), called standard IPEA (S-IPEA), was recommended but due to its unsatisfactory performance in multiple metallic and organic compounds it has been questioned lately as a general parameter working properly for all molecules under CASPT2 study. As we are interested in Schiff bases of retinal, an important question emerging from this conflict of choice, to use or not to use S-IPEA, is whether the introduction of the modified zeroth-order Hamiltonian into CASPT2 ansatz does really improve their energetics. To achieve this goal, we assessed an impact of the IPEA shift value, in a range of 0-0.35 a.u., on vertical excitation energies to low-lying singlet states of two protonated (RPSBs) and two unprotonated (RSBs) Schiff bases of retinal for which experimental data in gas phase are available. In addition, an effect of geometry, basis set, and active space on computed VEEs is also reported. We find, that for these systems, the choice of S-IPEA significantly overestimates both S₀ →S₁ and S₀ →S₂ energies and the best theoretical estimate, in reference to the experimental data, is provided with either unmodified zeroth-order Hamiltonian or small value of the IPEA shift in a range of 0.05-0.15 a.u., depending on active space and basis set size, equilibrium geometry, and character of the excited state. © 2018 Wiley Periodicals, Inc.

Excited-state minima and emission energies of retinal chromophore analogues: Performance of CASSCF and CC2 methods as compared with CASPT2

This study provides gas-phase S₁ excited-state geometries along with emission and adiabatic energies for methylated/demethylated and ring-locked analogues of protonated Schiff base retinal models comprising system of five conjugated double bonds (PSB5), using second order multiconfiguration perturbation theory (CASPT2). CASPT2 results serve as reference data to assess the performance of CC2 (second-order approximate coupled cluster singles and doubles) and a commonly used CASSCF/CASPT2 protocol, that is, complete active space self-consistent field (CASSCF) geometry optimization followed by CASPT2 energy calculation. We find that the CASSCF methodology fails to locate planar S₁ minimum energy structures for four out of five investigated planar models in contrast to CC2 and CASPT2 methods. However, for those which were found: one planar and two twisted minima, there is an excellent agreement between CASSCF and CASPT2 results in terms of geometrical parameters, one-electron properties, as well as emission and adiabatic energies. CC2 performs well for in-plane S₁ minima and their spectroscopic and electronic properties. However, this picture deteriorates for twisted minima. As expected, the CC2 description of the S₂ electronic state, with strong multireference and significant double excitation character, is very poor, exhibiting errors in transition energies exceeding 1 eV. They may be substantially diminished by recalculating transition energies with CASPT2 method. Our work shows that CASSCF/CASPT2 and CC2 shortcomings may influence gas-phase retinal analogues' excited state description in a dramatic way. © 2017 Wiley Periodicals, Inc.

Why do TD-DFT excitation energies of BODIPY/Aza-BODIPY families largely deviate from experiment? Answers from electron correlated and multireference methods

The vertical excitation energies of 17 boron-dipyrromethene (BODIPY) core structures with a variety of substituents and ring sizes are benchmarked using time-dependent density functional theory (TD-DFT) with nine different functionals combined with the cc-pVTZ basis set. When compared to experimental measurements, all functionals provide mean absolute errors (mean AEs) greater than 0.3 eV, larger than the 0.1-0.3 eV differences typically expected from TD-DFT. Due to the high linear correlation of TD-DFT results with experiment, most functionals can be used to predict excitation energies if corrected empirically. Using the CAM-B3LYP functional, 0-0 transition energies are determined, and while the absolute difference is improved (mean AE = 0.478 eV compared to 0.579 eV), the correlation diminishes substantially (R(2) = 0.961 to 0.862). Two very recently introduced charge transfer (CT) indices, q(CT) and d(CT), and electron density difference (EDD) plots demonstrate that CT does not play a significant role for most of the BODIPYs examined and, thus, cannot be the source of error in TD-DFT. To assess TD-DFT methods, vertical excitation energies are determined utilizing TD-HF, configuration interaction CIS and CIS(D), equation of motion EOM-CCSD, SAC-CI, and Laplace-transform based local coupled-cluster singles and approximate doubles LCC2* methods. Moreover, multireference CASSCF and CASPT2 vertical excitation energies were also obtained for all species (except CASPT2 was not feasible for the four largest systems). The SAC-CI/cc-pVDZ, LCC2*/cc-pVDZ, and CASPT2/cc-pVDZ approaches are shown to have the smallest mean AEs of 0.154, 0.109, and 0.100 eV, respectively; the utility of the LCC2* approach is demonstrated for eight extended BODIPYs and aza-BODIPYs. We found that the problems with TD-DFT arise from difficulties in dealing with the differential electron correlation (as assessed by comparing CCS, CC2, LR-CCSD, CCSDR(T), and CCSDR(3) vertical excitation energies for five compounds) and from contributions of multireference character and double excitations (from analysis of the CASSCF wave functions).

DFT studies on the structural and vibrational properties of polyenes

Detailed density functional theory (DFT) calculations on the structure and harmonic frequencies of model all-trans and all-cis polyenes were undertaken. For the first time, we report on the convergence of selected B3LYP/6-311++G** and BLYP/6-311++G** calculated structural parameters resulting from a systematic increase in polyene size (chains containing 2 to 14 C = C units). The limiting values of the structural parameters for very long chains were estimated using simple three-parameter empirical formulae. BLYP/6-311++G** calculated ν(C = C) and ν(C-C) frequencies for all-trans and all-cis polyenes containing up to 14 carbon-carbon double bonds were used to estimate these values for very long chains. Correction of raw, unscaled vibrational data was performed by comparing theoretical and experimental wavenumbers for polyenes chains containing 3 to 12 conjugated C = C units with both ends substituted by tert-butyl groups. The corrected ν(C = C) and ν(C-C) wavenumbers for all-trans molecules were used to estimate the presence of 9 - 12 C = C units in all-trans polyene pigment in red coral.

Assessment of Approximate Coupled-Cluster and Algebraic-Diagrammatic-Construction Methods for Ground- and Excited-State Reaction Paths and the Conical-Intersection Seam of a Retinal-Chromophore Model

As a minimal model of the chromophore of rhodopsin proteins, the penta-2,4-dieniminium cation (PSB3) poses a challenging test system for the assessment of electronic-structure methods for the exploration of ground- and excited-state potential-energy surfaces, the topography of conical intersections, and the dimensionality (topology) of the branching space. Herein, we report on the performance of the approximate linear-response coupled-cluster method of second order (CC2) and the algebraic-diagrammatic-construction scheme of the polarization propagator of second and third orders (ADC(2) and ADC(3)). For the ADC(2) method, we considered both the strict and extended variants (ADC(2)-s and ADC(2)-x). For both CC2 and ADC methods, we also tested the spin-component-scaled (SCS) and spin-opposite-scaled (SOS) variants. We have explored several ground- and excited-state reaction paths, a circular path centered around the S1/S0 surface crossing, and a 2D scan of the potential-energy surfaces along the branching space. We find that the CC2 and ADC methods yield a different dimensionality of the intersection space. While the ADC methods yield a linear intersection topology, we find a conical intersection topology for the CC2 method. We present computational evidence showing that the linear-response CC2 method yields a surface crossing between the reference state and the first response state featuring characteristics that are expected for a true conical intersection. Finally, we test the performance of these methods for the approximate geometry optimization of the S1/S0 minimum-energy conical intersection and compare the geometries with available data from multireference methods. The present study provides new insight into the performance of linear-response CC2 and polarization-propagator ADC methods for molecular electronic spectroscopy and applications in computational photochemistry.

Impacts of retinal polyene (de)methylation on the photoisomerization mechanism and photon energy storage of rhodopsin

Similar to native rhodopsin, a two-mode space-saving isomerization mechanism drives the photoreaction in (de)methylated rhodopsin analogues.

Pigment violet 19 - a test case to define a simple method to simulate the vibronic structure of absorption spectra of organic pigments and dyes in solution

A typical quinacridone pigment, PV19, has been used to analyze the impact of several computational parameters on the UV/vis absorption band shape in solution, simulated using density functional theory and time-dependent density functional theory levels of approximation. These encompass, (i) the choice of exchange-correlation functional, (ii) the basis set, (iii) the method for non-equilibrium optimization of the excited state geometry, (iv) the approach for evaluating the vibronic band structure, (v) the peak broadening, and (vi) the scaling of the harmonic vibrational frequencies. Among these, the choice of exchange-correlation functional is certainly of the most importance because it can drastically modify the spectral shape. In the case of PV19, the M05-2X and to a lesser extent CAM-B3LYP XC functionals are the most efficient to reproduce the vibronic structure, confirming the important role of exact Hartree-Fock exchange. Still, these functionals are not the most reliable to predict the excitation energies and oscillator strengths, for which M05, a functional with less HF exchange, performs better. For evaluating the vibronic structure, the simple gradient method, where only one step of geometry optimization of the excited state is carried out and the gradients are used to evaluate the Huang-Rhys factors as well as to determine the excited state geometries produces a spectrum that is very similar to the ones obtained with the more involved Duschinsky and geometry methods, opening the way to a fast simulation of the UV/vis absorption spectra of pigments and dyes. Then, the effect of scaling the calculated vibrational frequencies to account for anharmonicity effects as well as for limitation of the method also impacts the shape of the vibronic spectrum and this effect depends on the method used to determine the Huang-Rhys factors. Indeed, scaling the vibrational frequencies by a factor which is typically smaller than 1.0 results in a relative decrease of the 0-1 peak intensity with respect to the 0-0 band when optimizing the geometry of the excited state whereas the effect is opposite and magnified if using the gradient method.