Harmonic Models in Cartesian and Internal Coordinates to Simulate the Absorption Spectra of Carotenoids at Finite Temperatures

Abstract

When large structural displacements take place between the ground state (GS) and excited state (ES) minima of polyatomic molecules, the choice of a proper set of coordinates can be crucial for a reliable simulation of the vibrationally resolved absorption spectrum. In this work, we study two carotenoids that undergo structural displacements from GS to ES minima of different magnitude, from small displacements for violaxanthin to rather large ones for β-carotene isomers. Their finite-temperature (77 and 300 K) spectra are simulated at the harmonic level, including Duschinsky effect, by time-dependent (TD) and time-independent (TI) approaches, using (TD)DFT computed potential energy surfaces (PES). We adopted two approaches to construct the harmonic PES, the Adiabatic (AH) and Vertical Hessian (VH) models and, for AH, two reference coordinate frames: Cartesian and valence internal coordinates. Our results show that when large displacements take place, Cartesian coordinates dramatically fail to describe curvilinear displacements and to account for the Duschinsky matrix, preventing a realistic simulation of the spectra within the AH model, where the GS and ES PESs are quadratically expanded around their own equilibrium geometry. In contrast, internal coordinates largely amend such deficiencies and deliver reasonable spectral widths. As expected, both coordinate frames give similar results when small displacements occur. The good agreement between VH and experimental line shapes indicates that VH model, in which GS and ES normal modes are both evaluated at the GS equilibrium geometry, is a good alternative to deal with systems exhibiting large displacements. The use of this model can be, however, problematic when imaginary frequencies arise. The extent of the nonorthogonality of the Dushinsky matrix in internal coordinates and its correlation with the magnitude of the displacement of the GS and ES geometries is analyzed in detail.

Ground and excited states of zinc phthalocyanine, zinc tetrabenzoporphyrin, and azaporphyrin analogs using DFT and TDDFT with Franck-Condon analysis

“…This has been debated in recent years between Cartesian and internal coordinate systems. 52,53 While it has been shown that internal coordinate system yield better results for the vibronic equations due to the fact that there are less coordinates displaced versus the Cartesian system, some of the same researchers have produced significant work and comparisons in Cartesian coordinates. 54,55 In addition to comparisons between the adiabatic and vertical Hessian.…”

Going beyond the vertical approximation with time‐dependent density functional theory

“…This is particularly true for the AH model, since the necessity to project S 0 and S 1 normal modes computed at very different geometries, yields huge broadenings caused by exaggerated Duschinsky mixings, an artifact connected to the inadequacy of normal coordinates to describe large distortions of the molecular structure. Several groups are currently working to extend the capability of available models toward large and flexible compounds, adopting for instance a description of normal modes in internal coordinates and/or separating few large‐amplitude anharmonic motions to be treated at quantum or classical levels of theory . Different schemes for introducing anharmonic corrections on the frequencies and vibrational wavefunctions have been proposed considering either a perturbation expansion of the anharmonic potential, or exploiting prescreening techniques similar to those adopted in TI harmonic approaches .…”

Single-Hessian thawed Gaussian approximation

“…In the context of global harmonic methods, there are two natural possibilities of constructing a final-state harmonic potential using the initial-state Hessian: one can either compute the potential energy and gradient of the final-state potential energy surface at the initial geometry, which results in the vertical gradient model, or optimize the geometry in the final electronic state, which gives the adiabatic shift model. 13,50,52 Both the vertical gradient and adiabatic shift models are examples of displaced harmonic systems, and thus ignore mode distortion and mixing (the Duschinsky effect) between the two electronic states. In the results section, we discuss only the adiabatic shift model and, for consistency with the other methods discussed in this work, refer to it as the initial harmonic model.…”