# A different approach for calculating Franck-Condon factors including anharmonicity.

TL;DR: An efficient new procedure for calculating Franck-Condon factors, based on the direct solution of an appropriate set of simultaneous equations, is presented, and both Duschinsky rotations and anharmonicity are included.

Abstract: An efficient new procedure for calculating Franck–Condon factors, based on the direct solution of an appropriate set of simultaneous equations, is presented. Both Duschinsky rotations and anharmonicity are included, the latter by means of second-order perturbation theory. The critical truncation of basis set is accomplished by a build-up procedure that simultaneously removes negligible vibrational states. A successful test is carried out on ClO2 for which there are experimental data and other theoretical calculations.

## Summary (2 min read)

### A different approach for calculating Franck–Condon factors including anharmonicity

- Those phase-space points where the classical Wigner function for the initial state is maximal, subject to a classical energy constraint on the final state, determine propensity rules for the FCF’s.
- The authors have now begun to develop a rigorous theory for vibrational effects in TPA in order to investigate that situation more thoroughly.

### A. General formulation

- The authors denote the vibrational Hamiltonian, wave functions, and energies of the ground electronic state byĤg, ucng g &, and Eng g and their counterparts for an electronic excited state by Ĥe, ucne e &, andEne e .
- Then the respective Schrödinger equations for nuclear motion are given by Ĥgucng g &5Eng g ucng g &, ~1! Sngne. ~6! contains the entire set of Franck–Condon overlaps between the initial vibrational wave function of the ground electronic state and all final vibrational wave functions of the excited electronic state.
- This allows us to solve for the entire set of overlap integrals in which the authors are interested simultaneously.

### C. Mechanical anharmonicity

- Mechanical anharmonicity can be included through a perturbation treatment using the harmonic oscillator Hamiltonian as the zeroth-order approximation.
- Except that all quantities have a superscript~0!.
- Once the solution forSngme (1) has been determined, the first-order corrections to the FCF’s are found as Fngme ~1! 52Sngme ~0! Sngme ~1! .
- Redistribution subject to AIP license or copyright, see http://ojps.aip.org/jcpo/jcpcr.jsp.

### D. Truncation of the vibrational basis set

- It is critical to perform the truncation of the vibrational basis set in a way that is efficient and does not create significant error.
- The authors procedure involves an iterative buildup by increasing the range of vibrational quantum numbers while, simultaneously, removing unimportant states.
- The next step in the cycle is a screening of the states created in this manner which is based on the difference between the quantum number in each mode and the corresponding quantum number for the FC state.
- The latter still increases in size more rapidly than desired.
- It turns out, however, that most of the FC overlaps obtained from Eq.~9! are quite small.

### III. COMPUTATIONAL DETAILS

- Were used to calculate the neutral and cationic force constants, respectively, for the harmonic calculations.
- The first- and second-order corrections to the wave function are given by Eq.~19!.

### IV. RESULTS

- At the harmonic level their theoretical spectrum is essentially the same as that of Moket al.and thus their geometrical parameters for ClO2 1, obtained from the best match between the simulated and experimental spectrum, are also the same as theirs.
- Again, in order to compare the two spectra the intensities of the~0,0,1!.
- In Fig. 3 the authors present the simulated anharmonic spectra calculated by Moket al. and ourselves.

### V. CONCLUSIONS

- In this work a new method to calculate FCF’s taking into account Duschinsky rotations as well as anharmonicity has been developed and implemented.
- The authors harmonic results are in excellent agreement with those of Mok et al. who used a different procedure and both calculations predict the same geometry for ClO2 1.
- This computational efficiency is due in large part to the major truncation of the vibrational basis set.

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