Adiabatic approximation vibrational excitation

Each of the approaches is based on the premise that it makes sense to focus on the Born Oppenheimer potential for the OH stretch for fixed bath variables. Such a potential has vibrational eigenvalues, and for example h times the transition frequency of the fundamental is simply the difference between the first excited and ground state eigenvalues. Thus in essence this is an adiabatic approximation the assumption is that the vibrational chromophore is sufficiently fast compared to the bath coordinates. To the extent that the h times frequency of the chromophore is large compared to kT, and those of the bath are small compared to kT, this separation of time scales exists and so this should be a reasonable approximation. For water, as discussed earlier, some of the bath variables (librations) have frequencies somewhat larger than kT/h, and... 

In the spirit of the adiabatic approximation, the transitions between two vibrational states (belonging to initial and final electronic states) must occur so rapidly that there is no change in the configurational coordinate Q. This is known as the Frank Condon principle and it implies that the transitions between i and / states can be represented by vertical arrows, as shown in Figure 5.12. Let us now assume our system to be at absolute zero temperature (0 K), so that only the phonon level = 0 is populated and all the absorption transitions depart from this phonon ground level to different phonon levels m = 0, 1, 2,... of the excited state. Taking into account Equation (5.25), the absorption probability from the = 0 state to an m state varies as follows ... 

The excitation of a molecule may result in a change of its electron and rotational-vibrational quantum numbers. In the adiabatic approximation," the total wavefunction of a molecule can be presented as a product of the electron wave and the rovibrational wavefunction. In those cases where the former is weakly affected by the changes in the relative position of the nuclei (this is usually the case with lower vibrational levels), we can use the Condon approximation considering the electron wavefunction only at equilibrium configuration of the nuclei. In this case the oscillator strength factorizes into an electron oscillator strength and the so-called Frank-Condon factor, which is the overlap integral of the vibrational wavefunctions of the initial and the final states of the molecule.115,116... 

Vibrational excitation. There are two processes leading to vibrational excitation of a molecule when it collides with an electron the direct excitation and the resonance excitation where the electron is captured by the molecule. The direct vibrational excitation occurs owing to the dependence of the potential of interaction between an electron and a molecule on the internuclear spacings in the molecule. The cross section of direct excitation varies smoothly and is on the order of 10 17-10 16 cm2. The corresponding interaction time is much smaller than the period of nuclear vibrations. For describing this sort of excitation it is sufficient to use the momentum or the adiabatic approximation.105... 

Here Vf(R) is the eigenvalue of the operator V r,R) and Ef n is the total energy of the molecule in the vibronic state (/, n). As follows from (3.168), Vf(R) acts as the potential energy for the nuclei motion. So we obtain that the nuclei equilibrium positions depend on the state / of the electronic subsystem. It can be shown that the adiabatic approximation gives reasonable results when for given values R the distance between surfaces Vf(R) (for simplicity, the state / is considered as non-degenerate) and Vf> R),f f is much larger than the frequencies of intramolecular vibrations. Only in this case can we assume that excitation of small intramolecular vibrations does not induce the transition / —> ... 

It has also been shown by Merrifield (52) that linear terms in appear due to displacements of nuclear equilibrium positions caused by electronic molecular excitation, whereas changes in vibration frequencies give rise to quadratic terms. By derivation of (3.181) the adiabatic approximation for isolated molecules has been applied so that (3.181) can be used for examination of vibronic states by an arbitrary intensity of resonant intermolecular interaction. 

Fig. 14.11. The reaction H2 + OH -> H2O + H (within the vihrational adiabatic approximation). Three sets of the vibrational numbers (roH. t HH) = (0,0), (1, 0), (0, 1) were chosen. Note that the height and position of the barrier depend on the vibrational quantum numbers assumed. An excitation of H2 decreases considerably the barrier height Source T. Dunning, Jr. and E. Krala, from Advances in Molecular Electronic Structure Theory, ed. T. Dunning, Jr., JAI Press, Greenwich, C T (1989), courtesy of the authors.

In this section we will provide the formalism for an extension of the Born-Oppenheimer adiabatic approximation that deals with unstable autoioniz-ing molecules and takes into account the coupling between electronic and nuclear motion. The new formalism will be applied in two cases (i) full collision process of vibrational excitation of H2 molecule by electron impact [8], (ii) half collision process of interatomic Coulombic decay of electronically excited Ne cationic dimer [9,10]. 

We will derive here the expression for the vibrational excitation cross section of electron/molecule collision using the complex adiabatic approximation. Let us consider a scattering event where the electron represented by collides with a molecule which has N internal electrons. We... 

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