Evaluating vibronic spectra beyond the Condon and Herzberg-Teller approximations with Hagedorn wavepacket dynamics

In most calculations of vibrationally resolved electronic spectra, the transition dipole moment is assumed to be either a constant or a linear function of nuclear coordinates. We describe an efficient method, which goes beyond these so-called Condon and Herzberg-Teller approximations by allowing an arbitrary polynomial dependence of the transition dipole moment on nuclear coordinates. To propagate a wavepacket in the form of a Gaussian multiplied by an arbitrary polynomial, we expand the initial state in Hagedorn’s basis^1-3. In harmonic systems, the coefficients of the Hagedorn functions do not evolve with time, and one only needs to evolve the parameters of the guiding Gaussian and evaluate the (highly nontrivial) autocorrelation function of Hagedorn wavepackets using an exact recursive expression from Ref. 4. To validate the proposed method, we compare its results with exact quantum benchmarks for displaced, distorted, and Duschinsky-rotated harmonic models and for different polynomial dependences of the transition dipole moment. In on-the-fly ab initio applications, the bottleneck is the propagation of the guiding Gaussian, and therefore the proposed method can evaluate approximate spectra beyond the Condon and Herzberg-Teller approximations at a negligible additional cost, similarly as the Hagedorn wavepacket dynamics has been used for evaluating single-vibronic level fluorescence spectra^5,6.

¹ G. A. Hagedorn, “Raising and Lowering Operators for Semiclassical Wave Packets,” Ann. Phys. (NY) 269, 77–104 (1998).

² C. Lubich, From Quantum to Classical Molecular Dynamics: Reduced Models and Numerical Analysis, 12th ed. (European Mathematical Society, Zurich, 2008).

³ C. Lasser and C. Lubich, “Computing quantum dynamics in the semiclassical regime,” Acta Numerica 29, 229–401 (2020).

⁴ J. J. Vanı́ček and Z. T. Zhang, “On Hagedorn wavepackets associated with different Gaussians,” arXiv preprint arXiv:2405.07880 (2024)

⁵ Z. T. Zhang and J. J. L. Vanı́ček, “Single vibronic level fluorescence spectra from Hagedorn wavepacket dynamics,” J. Chem. Phys. 161, 111101 (2024)

⁶ Z. T. Zhang, M. Visegradi, and J. J. Vaníček, “On-the-Fly Ab Initio Hagedorn Wavepacket Dynamics: Single Vibronic Level Fluorescence Spectra of Difluorocarbene,” arXiv:2409.01862, under review in Phys. Rev. A.