Analytical approach for the excited-state Hessian in time-dependent density functional theory: Formalism, implementation, and performance

Abstract

The paper presents the formalism, implementation, and performance of the analytical approach for the excited-state Hessian in the time-dependent density functional theory (TDDFT) that extends our previous work [J. Liu and W. Z. Liang, J. Chem. Phys. 135, 014113 (2011)] on the analytical Hessian in TDDFT within Tamm-Dancoff approximation (TDA) to full TDDFT. In contrast to TDA-TDDFT, an appreciable advantage of full TDDFT is that it maintains the oscillator strength sum rule, and therefore yields more precise results for the oscillator strength and other related physical quantities. For the excited-state harmonic vibrational frequency calculation, however, full TDDFT does not seem to be advantageous since the numerical tests demonstrate that the accuracy of TDDFT with and without TDA are comparable to each other. As a common practice, the computed harmonic vibrational frequencies are scaled by a suitable scale factor to yield good agreement with the experimental fundamental frequencies. Here we apply both the optimized ground-state and excited-state scale factors to scale the calculated excited-state harmonic frequencies and find that the scaling decreases the root-mean-square errors. The optimized scale factors derived from the excited-state calculations are slightly smaller than those from the ground-state calculations.

Advances in molecular quantum chemistry contained in the Q-Chem 4 program package

“…A more distinctive capability is the availability of an implementation of TDDFT analytical frequencies (greatly extending an existing analytical configuration interaction with single substitutions [CIS] frequency code [127]), both in the Tamm-Dancoff approximation [128] and for full TDDFT [129]. In addition, analytical TDDFT gradients and frequencies have been extended [130] to include the smooth polarisable continuum models for solvation that are discussed in Section 6.1.…”

Optical signatures of borico dyes: a TD-DFT analysis

“…Amongst all possible first-principle approaches for investigating excited-state properties, Time-Dependent Density Functional Theory (TD-DFT) is certainly the most popular, with a continuously increasing number of published applications [13][14][15][16]. Beyond its widespread access in several quantum-chemical programs, three factors might explain the success of TD-DFT: 1) the possibility to tackle very extended molecules, e.g., possessing 300 atoms or even more if simplified approaches are used [17,18]; 2) the ease to combine TD-DFT with several environmental models allowing to accurately account for solvent and/or biochemical media [19,20]; 3) the availability of analytic first and second derivatives, allowing the optimization of the excited-state structures, as well as the determination of their vibrational patterns at a relatively small computational cost [21][22][23][24][25]. No other excited-state approach, but the much less accurate Configuration Interaction Singles (CIS) scheme, combines these key advantages.…”

A general time-dependent route to Resonance-Raman spectroscopy including Franck-Condon, Herzberg-Teller and Duschinsky effects

“…Furthermore, cheaper methods for the computation of harmonic frequencies of excited-state must be found. In our opinion, this goal can be accomplished by using methods where analytical second derivatives are available 65,66 . Finally, thanks to the implementation of the TD-RR theory inside GAUSSIAN, it is straightforward to extend it to the study of molecules in solution by using the polarizabile continuum model.…”

“…The necessary third and semi-diagonal fourth derivatives of the potential energy are obtained by numerical differentiation of the analytic harmonic force constants along the mass-weighted normal coordinates (Q) with the GAUSSIAN default step, 40,41 . Even if analytic formulae for TD-DFT energy second derivatives have been proposed 65,66 , due to the unavailability of them in our TD-DFT calculations and to reduce the computational times, the anharmonic frequencies of the excited state have been calculated by using the extrapolation procedure presented before.…”