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Herzberg-Teller coupling in coherent multidimensional spectroscopy: analytical response functions for multilevel systems

Filippo Troiani

TL;DR

The paper develops an analytic framework to describe Herzberg–Teller non-Condon coupling in coherent multidimensional spectroscopy within a displaced harmonic oscillator model for an arbitrary number of electronic states $N$ and response order $M$. By combining a coherent-state vibrational treatment with a FC baseline, it shows HT corrections factorize as complex oscillatory functions of the waiting times that modulate the FC response, yielding replicas of the FC spectra displaced by integer multiples of the vibrational frequency $\omega$ upon Fourier transformation. Closed-form expressions are provided for $M\le 3$ and a recursive scheme for $M>3$, enabling detailed interpretation of non-Condon effects through FC–HT interference across first-, second-, and third-order processes. The framework generalizes to multiple vibrational modes (without Duschinsky rotations) and can accommodate coherent initial states and nonlinear non-Condon terms, offering a versatile tool for analyzing vibronic dynamics in molecular and solid-state systems.

Abstract

Coherent multidimensional spectroscopy enables detailed investigations of vibronic effects in molecular and solid-state systems. We present explicit analytical expressions for multidimensional nonlinear response functions in the presence of Herzberg-Teller (non-Condon) coupling, within the displaced harmonic oscillator model. The formulation applies to electronic systems with an arbitrary number N of electronic states and to response functions of arbitrary order M in the light-matter interaction. We show that Herzberg-Teller coupling introduces additional oscillatory factors in the time-domain response functions, leading, upon Fourier transformation, to replicas of the Franck-Condon multidimensional spectra shifted by integer multiples of the vibrational frequencies. The present results provide a general analytical framework for the interpretation of non-Condon effects in coherent multidimensional spectroscopies.

Herzberg-Teller coupling in coherent multidimensional spectroscopy: analytical response functions for multilevel systems

TL;DR

The paper develops an analytic framework to describe Herzberg–Teller non-Condon coupling in coherent multidimensional spectroscopy within a displaced harmonic oscillator model for an arbitrary number of electronic states and response order . By combining a coherent-state vibrational treatment with a FC baseline, it shows HT corrections factorize as complex oscillatory functions of the waiting times that modulate the FC response, yielding replicas of the FC spectra displaced by integer multiples of the vibrational frequency upon Fourier transformation. Closed-form expressions are provided for and a recursive scheme for , enabling detailed interpretation of non-Condon effects through FC–HT interference across first-, second-, and third-order processes. The framework generalizes to multiple vibrational modes (without Duschinsky rotations) and can accommodate coherent initial states and nonlinear non-Condon terms, offering a versatile tool for analyzing vibronic dynamics in molecular and solid-state systems.

Abstract

Coherent multidimensional spectroscopy enables detailed investigations of vibronic effects in molecular and solid-state systems. We present explicit analytical expressions for multidimensional nonlinear response functions in the presence of Herzberg-Teller (non-Condon) coupling, within the displaced harmonic oscillator model. The formulation applies to electronic systems with an arbitrary number N of electronic states and to response functions of arbitrary order M in the light-matter interaction. We show that Herzberg-Teller coupling introduces additional oscillatory factors in the time-domain response functions, leading, upon Fourier transformation, to replicas of the Franck-Condon multidimensional spectra shifted by integer multiples of the vibrational frequencies. The present results provide a general analytical framework for the interpretation of non-Condon effects in coherent multidimensional spectroscopies.
Paper Structure (26 sections, 82 equations, 2 figures)

This paper contains 26 sections, 82 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Model and method
  3. Results
  4. First-order response functions
  5. Second-order response functions
  6. Third-order response functions
  7. Higher-order response functions
  8. Resulting spectral features
  9. Conclusions
  10. Commutation relations and notation
  11. First-order response function
  12. FC term
  13. First-order HT terms
  14. Second-order HT terms
  15. Second-order response function
  16. ...and 11 more sections

Figures (2)

  • Figure 1: Double-sided Feynman diagrams corresponding to (a) first- and (b,c) second-order processes; $|j\rangle$ and $|k\rangle$ represent the involved electronic states, with $j>k$.
  • Figure 2: Double-sided Feynman diagrams corresponding to third-order processes; $|j\rangle$, $|k\rangle$, and $|l\rangle$ represent the involved electronic states, with $k>j,l$ in (a), $k<j,l$ in (b,c).