# 1. Multidimensional Franck-Condon Integrals and their importance

Franck-Condon integrals are often used as handwaving arguments but rearly evaluated quantitatively. However, they can be calculated efficiently and be used qualitatively for

- Assignment of high resolution spectra, e.g. REMPI, ZEKE
- Determination of initial conditions in quantum dynamics
- Information on the excitation induced geometry change
- Refractive index
- Electron transfer rates
- Singlet to triplet transition rates
- and many more

# 2. Normal and local modes

There are two equivalent ways of defining/describing vibrations.

A **normal mode** of an oscillating system is a pattern of motion in which all parts of
the system move sinusoidally with the same frequency. The frequencies of the normal modes of a system
are known as its natural frequencies or resonant frequencies. A physical object, such as a building,
bridge or molecule, has a set of normal modes (and corresponding frequencies) that depend on its
structure and composition.

Local modes are being increasingly used in the description of highly excited vibrational states [1-3].
Such modes are not strict eigenstates of the vibrational Hamiltonian, and the coupling between them has
been a subject of much theoretical interest [3-7].

(1) Henry, B. R. Acc. Chem. Res. 1977, 10, 204.

(2) Sage, M. L.; Jortner, J. Adu. Chem. Phys. 1981, 47, 293.

(3) Child, M. S.; Halonen, L. Adu. Chem. Phys. 1984, 57, 1.

(4) Jaffe, C.; Brumer, P. J. Chem. Phys. 1980, 73, 5646.

(5) Mortensen, 0. S.; Henry, B. R.; Mohammadi, M. A. J. Chem. Phys. 1981, 75,4800.

(6) Sibert, 111, E. L.; Reinhardt, W. P.; Hynes, J. T. J. Chem. Phys. 1982, 77, 3583.

(7) van Roosmalen, 0. S.; Benjamin, I.; Levine, R. D. J. Chem. Phys. 1984, 81, 5986.

The two descriptions are linked by a unitary transformation

http://pubs.acs.org/doi/abs/10.1021/j100407a036

http://www.ingentaconnect.com/content/tandf/tmph/1998/00000093/00000005/art00013

# 3. The Duschinsky effect and the Duschinsky matrix

The normal coordinates of different electronic states differ as they have different force constants.
Thus electronic transitions can cause the normal modes of one electronic state to be rotated or mixed in the normal mode basis of the other electronic state,
a phenomenon first considered by Duschinsky [F. Duschinsky, *Acta Physicochim. URSS*, **7**, 441, 1937]
when extending the Franck-Condon principle from diatomics to polyatomic molecules.
It complicates the evaluation of multidimensional Franck-Condon problem as it prevents these integrals from being reduced to simple products of one-dimensional integrals.
Duschinsky proposed that the two sets of normal coordinates are related to each other by a linear transformation (the *Duschinsky transformation*) involving a matrix
and a displacement vector. The vector gives the displacement between the equilibrium structures of the two states in terms of the initial state normal coordinates and
the matrix can be viewed as an overlap matrix between the normal modes of the two electronic states. The closer the squares of the diagonal elements are to unity
the more the normal modes are similar to each other in form and energy. Large off-diagonal elements indicate a change in the energy ordering or mixing between different normal modes.
This mixing is the mathematical expression of a Duschinsky rotation.

It should be noted, however, that the Duschinsky transformation is not a general treatment, as the relationship between the normal coordinates of different electronic states of a polyatomic molecule is generally neither linear nor orthogonal. However, the linear/orthogonal transformation proposed by Duschinsky is widely accepted.

# 4. Axis switching

The Franck-Condon integrals are strongly dependent on the values of the Duschinsky matrix and the vector and it is therefore crucial to obtain correct quantities for these two mathematical objects. For this purpose the initial and final electronic state geometries must be oriented so that the Eckard conditions are fulfilled. These are generally fulfilled when the atomic coordinates are in standard orientation, the orientation in which the normal mode analysis is conventionally performed. However when the standard orientation of the two electronic states to be overlapped is largely different (due to large geometry differences) the Duschninsky transformation and consequently the FC-Integrals get contaminated by the axes roations. Accounting for these large "axis switching" effects requires a more general transformation that links the normal modes of the two electronic states. This problem has been discussed extensively in the literature.... To be continued