# Signatures of Duschinsky Rotation in Femtosecond Coherence Spectra

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## Abstract

**:**

## 1. Introduction

## 2. Theoretical Background

## 3. Results and Discussion

## 4. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 1.**(

**left**) One-dimensional displaced harmonic-oscillator (1D DHO) model characterized by the vibrational frequency ${\omega}_{0}$ and displacement $\Delta $. (center) Two-dimensional displaced harmonic-oscillator (2D DHO) model with the Duschinsky rotation angle, $\theta $. (

**right**) Femtosecond coherence spectroscopy (FCS) is the analysis of the excited-state vibrational wavepacket oscillations in transient–absorption spectroscopy. Fourier transformation of the intensity oscillations leads to an amplitude profile as a function of detection frequency, $A\left(\omega \right)$, which typically contains two main peaks separated by a sharp node.

**Figure 2.**FCS simulated using $\gamma /{\omega}_{{0}_{1}}=0.001$, representing gas-phase measurements, for ${\omega}_{eg}=400$. A nonzero rotation angle leads to peak multiplicities that are not present in the $\theta ={0}^{\circ}$ spectra.

**Figure 3.**Simulated FCS created using $\gamma /{\omega}_{{0}_{1}}=4$, representing condensed-phase measurements, for ${\omega}_{eg}=400$. A nonzero rotation angle leads to changes in relative peak amplitudes, node depths, and nodal frequencies relative to those in the $\theta ={0}^{\circ}$ simulations.

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**MDPI and ACS Style**

Arpin, P.C.; Popa, M.; Turner, D.B.
Signatures of Duschinsky Rotation in Femtosecond Coherence Spectra. *AppliedMath* **2022**, *2*, 675-686.
https://doi.org/10.3390/appliedmath2040039

**AMA Style**

Arpin PC, Popa M, Turner DB.
Signatures of Duschinsky Rotation in Femtosecond Coherence Spectra. *AppliedMath*. 2022; 2(4):675-686.
https://doi.org/10.3390/appliedmath2040039

**Chicago/Turabian Style**

Arpin, Paul C., Mihail Popa, and Daniel B. Turner.
2022. "Signatures of Duschinsky Rotation in Femtosecond Coherence Spectra" *AppliedMath* 2, no. 4: 675-686.
https://doi.org/10.3390/appliedmath2040039