Electronic transitions Born-Oppenheimer approximation

The measurements are predicted computationally with orbital-based techniques that can compute transition dipole moments (and thus intensities) for transitions between electronic states. VCD is particularly difficult to predict due to the fact that the Born-Oppenheimer approximation is not valid for this property. Thus, there is a choice between using the wave functions computed with the Born-Oppenheimer approximation giving limited accuracy, or very computationally intensive exact computations. Further technical difficulties are encountered due to the gauge dependence of many techniques (dependence on the coordinate system origin). 

In the light of the accumulated evidence, it appears quite likely that the scattering of nitric oxide from metals does induce electronic transitions which represents a fundamental breakdown of the Born-Oppenheimer approximation. Clearly this falls in the category of electronically nonadi-abatic phenomena that we set out to understand. But there is a broader question. Is the Born-Oppenheimer breakdown significant within a broader chemical context ... 

Fig. 4. Accumulating evidence is starting to show that molecules which undergo large amplitude vibration can interact strongly with metallic electrons in collisions and reactions at metal surfaces. This suggests that the Born-Oppenheimer approximation may be suspect near transition states of reactions at metal surfaces.

The transition from (1) and (2) to (5) is reversible each implies the other if the variations 5l> admitted are completely arbitrary. More important from the point of view of approximation methods, Eq. (1) and (2) remain valid when the variations 6 in a trial function are constrained in some systematic way whereas the solution of (5) subject to model or numerical approximations is technically much more difficult to handle. By model approximation we shall mean an approximation to the form of as opposed to numerical approximations which are made at a lower level once a model approximation has been made. That is, we assume that H, the molecular Hamiltonian is fixed (non-relativistic, Born-Oppenheimer approximation which itself is a model in a wider sense) and we make models of the large scale electronic structure by choice of the form of and then compute the detailed charge distributions, energetics etc. within that model. 

Hence, according to the symmetry selection rule, n —> n transitions are allowed but n —> ti transitions are forbidden. However, in practice the n —> it transition is weakly allowed due to coupling of vibrational and electronic motions in the molecule (vibronic coupling). Vibronic coupling is a result of the breakdown of the Born-Oppenheimer approximation. 

As already noted, in the Born-Oppenheimer approximation, the nuclear motion of the system is subject to a potential which expresses the isotope independent electronic energy as a function of the distortion of the coordinates from the position of the transition state. An analysis of the motions of the N-atom transition state leads to three translations, three rotations (two for a linear molecule), and 3N - 6 (3N- 5 for a linear transition state) vibrations, one which is an imaginary frequency (e.g. v = 400icm 1 where i = V—T), and the others are real vibrational frequencies. The imaginary frequency corresponds to motion along the so-called reaction... 

Before discussing tunneling in VTST where the discussion will focus on multidimensional tunneling, it is appropriate to consider the potential energy surface for a simple three center reaction with a linear transition state in more detail. The reaction considered is that of Equation 6.3. The collinear geometry considered here is shown in Fig. 6.1a it is in fact true that for many three center reactions the transition state can be shown to be linear. The considerations which follow apply to a onedimensional world where the three atoms (or rather the three nuclei) are fixed to a line. We now consider this one-dimensional world in more detail. The Born-Oppenheimer approximation applies as in Chapter 2 so that the electronic energy of... 

In the case of direct vibrational excitation, the vibrational transition probability is given by p, where are the intermediate and ground vibrational states, respectively, and is the vibrational transition moment. The electronic transition probability out of the intermediate state is < n < e ng e > n>, where are the excited and ground electronic states, respectively, and is the electronic dipole moment operator and vibrational state in the upper electronic state. Applying the Born-Oppenheimer approximation, where the nuclear electronic motion are separated, S can be presented as... 

To deduce whether a transition is allowed between two stationary states, we investigate the matrix element of the electric dipole-moment operator between those states (Section 3.2). We will use the Born-Oppenheimer approximation of writing the stationary-state molecular wave functions as products of electronic and nuclear wave functions ... 

Let us assume the availability of a useful body of quantitative data for rates of decay of excited states to give new species. How do we generalize this information in terms of chemical structure so as to gain some predictive insight For reasons explained earlier, I prefer to look to the theory of radiationless transitions, rather than to the theory of thermal rate processes, for inspiration. Radiationless decay has been discussed recently by a number of authors.16-22 In this volume, Jortner, Rice, and Hochstrasser 23 have presented a detailed theoretical analysis of the problem, with special attention to the consequences of the failure of the Born-Oppenheimer approximation. They arrive at a number of conclusions with which I concur. Perhaps the most important is, "... the theory of photochemical processes outlined is at a preliminary stage of development. Extension of that theory should be of both conceptual and practical value. The term electronic relaxation has been applied to the process of radiationless decay. 

Breakdown of the Born-Oppenheimer Approximation. The B-O approximation is based on the independence of the motions of nuclei and electrons. This is generally a reasonable assumption, except at the crossing point of two electronic states where a minor nuclear displacement is linked to the transition between two electronic states (Figure 3.31). 

Here p is the density of vibrational levels of states Sj and Sf at the energy of the electronic transition E. The overlap of the electronic wavefunctions 0i5 0f and of the vibrational wavefunctions (0i 0f) are factorized according to the Born-Oppenheimer approximation just as in the case of radiative transitions. The density of vibrational levels is greater for the lower (final) state Sf... 

By condition 3 we want to ensure that the Born-Oppenheimer approximation can be applied to the description of the simple systems, allowing definition of adiabatic potential-energy curves for the different electronic states of the systems. Since the initial-state potential curve K (f ) (dissociating to A + B) lies in the continuum of the potential curve K+(/ ) (dissociation to A + B + ), spontaneous transitions K ( )->K+(f ) + e" will generally occur. Within the Born-Oppenheimer approximation the corresponding transition rate W(R)—or energy width T( ) = hW(R) of V (R)... 

This is so because no coupling between electronic and nuclear motion is assumed within the Born-Oppenheimer approximation, which in the classical limit leads to separate conservation of the instantaneous heavy-particle motion. Denoting by EA(R,) and (/ ,) the instantaneous kinetic energy at the moment of transition in the upper- and lower potential curve,... 

Studies of the Pgl electron spectra have shown that, in spite of all the mentioned complications, many Pgl systems with molecular targets can still be well described within the theory of simple Pgl, if only the possibility of vibrational transitions is incorporated into the function r(fl). Within the Born-Oppenheimer approximation for both the projectile-target motion and the intramolecular motion, this is done in the following way. We denote by r,(rt) the width belonging to a certain final electronic state, defined as in (11.85). Then r,( ) can, at any distance R, be decomposed as... 

After having defined the partial dissociation wavefunctions l>(R,r E,n) as basis in the continuum, the derivation of the absorption rates and absorption cross sections proceeds in the same way as outlined for bound-bound transitions in Sections 2.1 and 2.2. In analogy to (2.9), the total time-dependent molecular wavefunction T(t) including electronic (q) and nuclear [Q = (R, r)] degrees of freedom is expanded within the Born-Oppenheimer approximation as... 

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