The Nuclear Coordinate Dependence of Matrix Elements

I 5 The Nuclear Coordinate Dependence of Matrix Elements rewrite Equation 5.16 as... 

The acetylene A <- X electronic transition is a bent <- linear transition that would be electronically forbidden ( - ) at the linear structure. The usual approximation is to ignore the possibility that the electronic part of the transition moment depends on nuclear configuration and to calculate the relative strengths of vibrational bands as the square of the vibrational overlap between the initial and final vibrational states (Franck-Condon factor). A slightly more accurate picture would be to express the electronic transition moment as a linear function of Q l (the fra/w-bending normal coordinate on the linear X1 state) in such a treatment, the transition moment would be zero at the linear structure and the vibrational overlap factors would be replaced by matrix elements of Qfl- Nevertheless, as long as one makes use of low vibrational levels of the A state, neglect of the nuclear coordinate dependence of the electronic excitation function is unlikely to affect the predicted dynamics or to complicate any proposed control scheme. 

In the BO representation, the nuclear kinetic energy matrix is not diagonal because of the nuclear coordinate dependence of the wavefunction. The off-diagonal elements of the nuclear kinetic energy are non-adiabatic couplings. In order to discuss the relationship between vibronic coupling and non-adiabatic coupling, we present the Born-Oppenheimer approximation. 

The intensity of a vibronic band is proportioned to the square of the electronic dipole moment matrix element between the initial and final states. These states are well described by Bom-Oppenheimer products of electronic and nuclear functions. For a symmetry allowed transition the nuclei coordinate dependence of the electronic function can be ignored so that the intensity of a vibronic band can be factored into a purely electronic factor that is independent of nuclear coordinate times the square of the overlap of the vibrational functions (Franck-Condon factor). 

The integral in large parentheses is over the electronic coordinates r only, and still depends on the nuclear coordinates R. At this stage we invoke the Condon approximation, which is familial from the theory of electronic spectroscopy. Because of the large nuclear mass the wave-functions Xi and Xf are much more strongly localized than the electronic wavefunctions 4>i and some value R, and it suffices to replace the electronic matrix element by its value M at R. So we write ... 

In an electronic adiabatic representation, however, the electronic Hamiltonian becomes diagonal,i.e. ( a 77e C/3) = da,0Va, where the adiabatic Va potentials for initial (A,B,B ) and final (X) electronic states were described in Ref.[31]. The couplings between different electronic states arises from the matrix elements of the nuclear kinetic operator Tn, giving rise to the so-called non-adiabatic coupling matrix elements (NACME) and are due to the dependence of the electronic functions on the nuclear coordinates. The actual form of these matrix elements depends on the choice of the coordinates. 

In Eqs. 31, 36, and 37 we have employed the Condon approximation [66], factoring Tjf out of the full vibronic matrix element, with the understanding that 7)y is to be evaluated for values of the nuclear coordinates pertinent to the configuration or range of configurations of the system in which the ET process occurs. The validity of the Condon factorization depends on the extent to which 7)y varies with the coordinates Q, a topic to which we return below. The coordinates of interest in this connection include the reaction coordinate [t] in Figure 3), as well as others such as conformational modes of the DBA system. The influence of fluctuations in these coordinates (and hence in the magnitude of 7)y) on the overall kinetics depends in detail on the relationship between the timescale for such fluctuations and the time-scales of the other dynamic processes [97]. 

This is because the electric dipole operators, ftp, act only on the electronic coordinates, the contribution to the transition moment from the nuclear coordinates being negligible. The resulting matrix elements, 9ge(Q), are still parametrically dependent on Q through the electronic states. This dependence is supposed to be a weak function of the internuclear coordinates and is therefore described by a rapidly converging Taylor series expanded about the equilibrium configuration, Q = 0, of the ground electronic state ... 

The effective nuclear motion Hamiltonian, depending only on the q, is expressed in terms of matrix elements of the Hamiltonian just as before, between pairs of functions like (21) but with internal coordinate parts like (24) integrated over the z as well as the angular factors. Doing this yields an equation rather like (23) but with coupling between different electronic states, labelled by p. The term analogous to (23) become... 

The nuclear wave functions are the projections of the total wave functions on the electronic PES involved. They depend on the nuclear coordinate Q and on the time t. is the initial vibrational state = 0 of the ground state. The matrix elements of the Hamiltonian (3.12), describing the neutral molecular states, are given by... 

In this chapter, we shall extend the description given in Ref. [27]. This will include the effect of Duschinsky mixing for some (totally symmetric) modes. We shall also give an extensive discussion of the dependence of the electronic matrix element on the nuclear coordinates. 

Third, as a consequence of the foregoing, the use of IDs is suitable in investigating the dependence of the electronic matrix element for radiationless transition on the nuclear coordinates. This problem can be solved, as has been shown in Chapter 5, by considering the matrix element as one that of an operator that depends upon both electronic and nuclear displacements and by introducing a q-centroid approximation for the electronic factor. The latter is obtained as an average with DSWVO factor. The familiar Condon approximation can be so improved as to write the whole matrix element as a product of a vibrational overlap integral and an electronic factor, the latter being evaluated at some (j-centroid for the nuclear positions. 

Since the vibrational eigenstates of the ground electronic state constitute an orthonomial basis set, tire off-diagonal matrix elements in equation (B 1.3.14) will vanish unless the ground state electronic polarizability depends on nuclear coordinates. (This is the Raman analogue of the requirement in infrared spectroscopy that, to observe a transition, the electronic dipole moment in the ground electronic state must properly vary with nuclear displacements from... 

Here q and Q symbolize the sets of all electronic and nuclear coordinates q (i = 1,. . ., 3 ) and Qt (i = 1,2,.. . , 3AT — 6), respectively. The derivatives are taken at the coordinate values Qf and the summation runs over all nuclear coordinates of independent vibrations. The expansion may be carried out with respect to different types of nuclear coordinates, i.e. symmetry coordinates and normal coordinates of the ground or the excited states. If the Q, s are normal coordinates and the Q s are taken at the potential minimum of an electronic state E the coordinate values are by definition Q = 0 for all i. In this case the matrix elements of the electron dependent part in the second term of Eq. (1) should vanish due to the minimal condition, i.e. 

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