Vibronic functions

Vo + V2 and = Vo — 2 (actually, effective operators acting onto functions of p and < )), conesponding to the zeroth-order vibronic functions of the form cos(0 —4>) and sin(0 —(()), respectively. PL-H computed the vibronic spectrum of NH2 by carrying out some additional transformations (they found it to be convenient to take the unperturbed situation to be one in which the bending potential coincided with that of the upper electi onic state, which was supposed to be linear) and simplifications (the potential curve for the lower adiabatic electi onic state was assumed to be of quartic order in p, the vibronic wave functions for the upper electronic state were assumed to be represented by sums and differences of pairs of the basis functions with the same quantum number u and / = A) to keep the problem tiactable by means of simple perturbation... 

The complete wave function of a molecule is called the rovibronic wave function. In the simplest approximation, the rovibronic function is a product of rotational, vibrational, and electronic functions. For certain applications, the rotational motion is first neglected, and the vibrational and electronic motions are treated together. The rotational motion is then taken into account. The wave function for electronic and vibrational motion is called the vibronic wave function. Just as we separately classified the electronic and vibrational wave functions according to their symmetries, we can do the same for the vibronic functions. In the simplest approximation, the vibronic wave function is a product of electronic and vibrational wave functions, and we can thus readily determine its symmetry. For example, if the electronic state is an e2 state and the vibrational state is a state, then the vibronic wave function is... 

We consider now the complete basis set of vibronic functions I f m, n) obtained as the direct product of the electronic functions l/+) and 1/1) by the phonons wave functions Im, n), where the integer numbers m and n (positive or zero) label the occupation numbers of the boson operators b b+ and b b, respectively. The Renner-Teller Hamiltonian (14) when applied to any vibronic function I f m,n) couples it with three states (at most) in fact it holds that ... 

In the Migdal model the retarded vibron function is calculated from the... 

In the more complicated case of a tetrahedral system with an adiabatic potential having four equivalent minima of dipolar type, Hvihl and the operators of Cartesian components of the dipole moment are given by the following matrices of dimension 4x4, provided that the overlap integrals of the vibronic functions in different minima are neglected ... 

The vibronic functions ne) belong to the representation A], A2, and E of the symmetry point group. Therefore in the group Td the matrix elements of the operator of the dipole moment (belonging to the T2 representation) calculated by the functions in (36) are identically equal zero. Hence the dipole transitions between the states (36) are forbidden as pure rotational or vibronic-rotational ones. 

The intensities of the spectral lines and the depolarization coefficients are functions of the reduced matrix elements of the polarizability tensor calculated by vibronic functions. In order to estimate the possibility of observation of the pure rotational Raman spectra under consideration, one has to consider in more detail the polarizability operator. Its components belonging to the line y of representation f can be presented in the form of a power series with respect to the displacement qriri active in the Jahn-Teller effect (the other components can be neglected as not active in the pure rotational Raman spectrum under consideration) ... 

Although it was not necessary for the above development, the CBO and ABO approximations should be briefly reviewed In both approximations the vibronic wave function BO(q, g) is written as a product of an electronic wave function ji(q, g) and a nuclear vibrational wave function x(0- In the CBO), the electronic functions are taken to be independent of g (they are all evaluated at some g0), while in the ABO the ip are evaluated parametrically for each value of g. Thus, the vibronic functions are... 

The ground state vibronic functions for each molecule are products of zero-point vibrational functions and a zeroth-order electronic function 0( , Qo). Thus the inclusion of 10.108 into 10.110 results in the disappearance of all terms involving (r, Qo) due to the orthogonality of and the ground state vibrational function. The result is... 

Generally D R) is computed at various positions along the energy surface, fitted to a convenient functional form (polynomial in R or trigonometric functions if torsional nuclear motion is involved), and evaluated using the vibrational (vibronic) functions Xv In the Franck-Condon approximation, D(R) is assumed to be constant, so that... 

Figure 13 Vibronic functions in MgO the X S" " and a fl potential curves coupled by spin-orbit interaction (top), the levels for the unperturbed vibrational functions in the two potential curves, and some of the vibronic functions (the square thereof) with minimal and maximum E/fl mixing. The dotted lines indicate the unperturbed vibrational species...

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