A new method to calculate Franck–Condon factors of multidimensional harmonic oscillators including the Duschinsky effect

2008 ◽  
Vol 128 (17) ◽  
pp. 174111 ◽  
Author(s):  
Jia-Lin Chang
Keyword(s):  
Harmonic Oscillators ◽  
New Method ◽  
Condon Factors ◽  
Franck Condon

Franck-Condon factors for multidimensional harmonic oscillators

1998 ◽  
Vol 228 (1-3) ◽  
pp. 227-240 ◽  
Author(s):  
Per-Åke Malmqvist ◽  
Niclas Forsberg
Keyword(s):  
Harmonic Oscillators ◽  
Condon Factors ◽  
Franck Condon

scholarly journals Theoretical Investigations of Photoionization Efficiency of Naphthalene

2019 ◽  
Author(s):  
Chih-Hao Chin ◽  
Tong Zhu ◽  
John ZH Zhang
Keyword(s):  
Electronic Transition ◽  
Density Functional ◽  
Molecular Beam ◽  
Density Functional Theory Calculations ◽  
Harmonic Oscillators ◽  
Equilibrium Geometry ◽  
Complete Active Space ◽  
Adiabatic Representation ◽  
Condon Factors ◽  
Franck Condon

The equilibrium geometry and 48 vibrational normal-mode frequencies of the neutral and cationic ground state and the cationic first excited states of naphthalene isomers were calculated and characterized in the adiabatic representation by using the complete active space self-consistent field (CASSCF) and second order perturbation theory (CASPT2). Photoionization-efficiency (PIE) spectrum of molecular beam conditions in energy range 8 - 11 eV were determined by Kaiser et al. and they were analyzed using time-dependent density functional theory calculations (TDDFT). CASSCF calculations and PIE spectra simulations by one-photon excitation equations were used to optimize the cationic excited (D1) and neutral ground (S0) state structures of naphthalene isomers. The photoionization-efficiency curve was attributed to the S0  D1 electronic transition in naphthalene, and a curve origin was used at 8.14 eV. The ionization-induced geometry changes of the bases are consistent with the shapes of the corresponding molecular orbitals. The displaced harmonic oscillator approximation and Franck-Condon approximation were used to simulate the PIE curve of the D1  S0 transition of naphthalene, and the main vibronic transitions were assigned for the ππ* state. It shows that the vibronic structures were dominated by one of the xxx active totally symmetric modes, with v8 being the most crucial. This indicates that the electronic transition of the D1 state calculated in the adiabatic representation effectively includes a contribution from the adiabatic vibronic coupling through Franck-Condon factors perturbed by harmonic oscillators. The present method can adequately reproduce experimental PIE curve in the molecular beam condition.

scholarly journals Theoretical Investigations of Photoionization Efficiency of Naphthalene

2019 ◽  
Author(s):  
Chih-Hao Chin ◽  
Tong Zhu ◽  
John ZH Zhang
Keyword(s):  
Electronic Transition ◽  
Density Functional ◽  
Molecular Beam ◽  
Density Functional Theory Calculations ◽  
Harmonic Oscillators ◽  
Equilibrium Geometry ◽  
Complete Active Space ◽  
Adiabatic Representation ◽  
Condon Factors ◽  
Franck Condon

The equilibrium geometry and 48 vibrational normal-mode frequencies of the neutral and cationic ground state and the cationic first excited states of naphthalene isomers were calculated and characterized in the adiabatic representation by using the complete active space self-consistent field (CASSCF) and second order perturbation theory (CASPT2). Photoionization-efficiency (PIE) spectrum of molecular beam conditions in energy range 8 - 11 eV were determined by Kaiser et al. and they were analyzed using time-dependent density functional theory calculations (TDDFT). CASSCF calculations and PIE spectra simulations by one-photon excitation equations were used to optimize the cationic excited (D1) and neutral ground (S0) state structures of naphthalene isomers. The photoionization-efficiency curve was attributed to the S0  D1 electronic transition in naphthalene, and a curve origin was used at 8.14 eV. The ionization-induced geometry changes of the bases are consistent with the shapes of the corresponding molecular orbitals. The displaced harmonic oscillator approximation and Franck-Condon approximation were used to simulate the PIE curve of the D1  S0 transition of naphthalene, and the main vibronic transitions were assigned for the ππ* state. It shows that the vibronic structures were dominated by one of the xxx active totally symmetric modes, with v8 being the most crucial. This indicates that the electronic transition of the D1 state calculated in the adiabatic representation effectively includes a contribution from the adiabatic vibronic coupling through Franck-Condon factors perturbed by harmonic oscillators. The present method can adequately reproduce experimental PIE curve in the molecular beam condition.

scholarly journals A general formula for calculation of the two-dimensional Franck–Condon factors

2017 ◽  
Vol 95 (4) ◽  
pp. 340-345 ◽  
Author(s):  
H. Koç ◽  
B.A. Mamedov ◽  
E. Eser
Keyword(s):  
Hermite Polynomials ◽  
Analytical Formula ◽  
Harmonic Oscillators ◽  
Two Dimensional ◽  
Expansion Theorem ◽  
Binomial Expansion ◽  
Condon Factors ◽  
Astrophysical Molecules ◽  
Franck Condon ◽  
Kinetics Of

Knowledge of the Franck–Condon factors (FCFs) and related quantities is essential to understand and to estimate many important aspects of the astrophysical molecules, such as kinetics of the energy transfer, radiative lifetimes, band intensity, and vibrational temperatures. In this view, we propose a new analytical formula of the Franck–Condon integral for two-dimensional harmonic oscillators taking into account the Duschinsky effect. This method is based on the use of the binomial expansion theorem and the Hermite polynomials. With the formula obtained, the FCF of any transition can be computed independently. In this study, the method for FCF calculations was applied to the NO2molecule.

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