Vibrational wave function models

Fig. 7. Predicted diffusion coefficients for hydrogen (H) and deuterium (D) in niobium, as calculated by Schober and Stoneham (1988) from a model taking account of tunneling between various states of vibrational excitation and comparison with experimental measurements (solid lines). Theoretical curves are shown both for a model using harmonic vibrational wave functions (dashed lines) and for a model with anharmonic corrections (dashed-dotted lines).

The rate of a given reaction depends on the thermal activation conditions of the particle in donor and acceptor, factors which are accounted for in the Marcus model [6,7] or models where the vibrational wave functions are included [8-10], The reaction rate is derived in rather much the same way as for ordinary chemical reactions, using the concept of potential energy surfaces (PES s) [6]. The electronic factor is introduced either as a matrix element H]2 or as an... 

In the simple reflection principle model the vibrational wave function is determined for the lower (ground) electronic state. It is then reflected using the upper electronic potential onto the energy axis as sketched in Figure 7.2. The width of the spectrum is related to the width of the vibrational wave function in the ground state and to the slope of the dissociative potential energy curve. The differences between the model spectrum and the experimental spectrum in Figure 7.2 will be discussed in Section 6.1. 

Substitution of H2O solvent by D2O generally leads to a change in ET kinetics (44, 45, and references therein). In the simplest model of ET coupled to a single solvent vibration involving H, substitution of H by D lowers the frequency of the vibration and increases the localization of the vibrational wave functions. These changes influence the Franck-Condon factor for the ET reaction in a way that depends on the temperature and on reaction energetics. For example, at low temperatures, when the vibrations are in their ground states, the increased localization (for deuteration) of the vibrational wave functions is expected to... 

Another approach has been proposed in Ref. 191. The approach is based on the model of small polaron and makes it possible to extend the range of the model. In particular, it includes the influence of vibration wave functions on the tunnel integral and provides a way of the estimation of diagonal and off-diagonal phonon transfers on the proton polaron mobility. It turns out that the one phonon approximation is able significantly to contribute to the proton mobility. Therefore, we will further deal with matrix elements constructed on... 

For a more complete description of the vibronic levels associated with the Tig(A2 + E) electronic excited state, the wave functions Tj [Eq. f32)] must be augmented to include vibrational wave functions associated with all of the normal modes other than a = 2,5a, and 5e. Here we shall restrict our attention to just those vibronic levels derived from the vibrational modes a = 2,5a, and 5e. Assuming no vibronic couplings involving the ground electronic state of our model system, the rotatory strength of a transition between the lowest vibrational level of the ground electronic state and the j-th vibronic level of the Tjg (t2g + eg) coupled state may be written as... 

The 2D model of the linear B- -H-A fragment, assuming a strong coupling between the proton (AH stretch) and low-frequency (B- -A stretch) coordinates, was introduced by Stepanov [9, 10]. It seems to be the simplest model enabling one to interpret the different specific features of H-bonded systems [11-16]. In terms of this model, the vibrational wave function of the H-bonded system is written as... 

To take into account this feature in the molecular models we can make an introducing correction (broadening) of the potential function dependent on the degree of anharmonic transformation of each vibration mode. The value of the corresponding elements of the displacement vector b would seem to be a natural criterion for the selection of the correction magnitude. However, the position of the area of the maximal overlap of the vibrational wave functions in the case of multidimensional displacement turns out to be a more adequate characteristic. This area is characterized by the point X (the top of the potential barrier, see Fig. 3.2) with coordinates x, (x ) in space of the normalized normal coordinates. 

Hirata et al. used the vibrational self-consistent field (VSCF), vibrational configuration-interaction (VCI), and vibrational second-order Moller-Plesset perturbation (VMP2) methods. The VSCF expressed the vibrational wave functions as products of 52 harmonic oscillator (HO) wave functions. The VCI wave function was a linear combination of the 2000 lowest-energy VSCF model products. The obtained vibrational corrections to V(F, H) and V(F, F) were 18.4Hz and — 42.9 Hz, respectively. The vibrational correction... 

Dracinsky et investigated relative importance of anharmonic corrections to SSCCs for a model set of methane derivatives, differently charged alanine forms, and sugar models. They systematically estimated the importance of the first and second-order property derivatives of SSCCs for vibrational corrections in model compounds. For a vibrational wave function j/ , the vibrationally averaged SSCC was calculated as... 

Single surface calculations with a vector potential in the adiabatic representation and two surface calculations in the diabatic representation with or without shifting the conical intersection from the origin are performed using Cartesian coordinates. As in the asymptotic region the two coordinates of the model represent a translational and a vibrational mode, respectively, the initial wave function for the ground state can be represented as. 

Figure 1.13 Plot of potential energy, V(r), against bond length, r, for the harmonic oscillator model for vibration is the equilibrium bond length. A few energy levels (for v = 0, 1, 2, 3 and 28) and the corresponding wave functions are shown A and B are the classical turning points on the wave function for w = 28...

Hyperfine coupling constants provide a direct experimental measure of the distribution of unpaired spin density in paramagnetic molecules and can serve as a critical benchmark for electronic wave functions [1,2], Conversely, given an accurate theoretical model, one can obtain considerable information on the equilibrium stmcture of a free radical from the computed hyperfine coupling constants and from their dependenee on temperature. In this scenario, proper account of vibrational modulation effects is not less important than the use of a high quality electronic wave function. 

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