Born-Oppenheimer approximations vibrational transitions

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.

Below we will use Eq. (16), which, in certain models in the Born-Oppenheimer approximation, enables us to take into account both the dependence of the proton tunneling between fixed vibrational states on the coordinates of other nuclei and the contribution to the transition probability arising from the excited vibrational states of the proton. Taking into account that the proton is the easiest nucleus and that proton transfer reactions occur often between heavy donor and acceptor molecules we will not consider here the effects of the inertia, nonadiabaticity, and mixing of the normal coordinates. These effects will be considered in Section V in the discussion of the processes of the transfer of heavier atoms. 

First, we shall consider the model where the intermolecular vibrations A—B and intramolecular vibrations of the proton in the molecules AHZ,+1 and BHZ2+1 may be described in the harmonic approximation.48 In this case, using the Born-Oppenheimer approximation to separate the motion of the proton from the motion of the other atoms for the symmetric transition, Eq. (16) may be... 

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. 

It is important to point out here, in an early chapter, that the Born-Oppenheimer approximation leads to several of the major applications of isotope effect theory. For example the measurement of isotope effects on vapor pressures of isotopomers leads to an understanding of the differences in the isotope independent force fields of liquids (or solids) and the corresponding vapor molecules with which they are in equilibrium through use of statistical mechanical theories which involve vibrational motions on isotope independent potential functions. Similarly, when one goes on to the consideration of isotope effects on rate constants, one can obtain information about the isotope independent force constants which characterize the transition state, and how they compare with those of the reactants. 

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... 

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... 

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... 

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... 

In the crude Born-Oppenheimer approximations, the oscillator strength of the 0-n vibronic transition is proportional to (FJ)2. Furthermore, the Franck-Condon factor is analytically calculated in the harmonic approximation. From the hamiltonian (2.15), it is clear that the exciton coupling to the field of vibrations finds its origin in the fact that we use the same vibration operators in the ground and the excited electronic states. By a new definition of the operators, it becomes possible to eliminate the terms B B(b + b ), BfB(b + hf)2. For that, we apply to the operators the following canonical transformation ... 

So far, this discussion of selection rules has considered only the electronic component of the transition. For molecular species, vibrational and rotational structure is possible in the spectrum, although for complex molecules, especially in condensed phases where collisional line broadening is important, the rotational lines, and sometimes the vibrational bands, may be too close to be resolved. Where the structure exists, however, certain transitions may be allowed or forbidden by vibrational or rotational selection rules. Such rules once again use the Born-Oppenheimer approximation, and assume that the wavefunctions for the individual modes may be separated. Quite apart from the symmetry-related selection rules, there is one further very important factor that determines the intensity of individual vibrational bands in electronic transitions, and that is the geometries of the two electronic states concerned. Relative intensities of different vibrational components of an electronic transition are of importance in connection with both absorption and emission processes. The populations of the vibrational levels obviously affect the relative intensities. In addition, electronic transitions between given vibrational levels in upper and lower states have a specific probability, determined in part... 

The electronic-transition dipole moment for the G E transition is defined by Mge = ( g A/ ge1 e> where the are the state wave functions and A/ ge is the dilference in dipole moment of the ground and excited states [22]. The intensity of the transition is proportional to Mge - The broad absorption bands usually observed in transition metal systems are composed of progressions in the vibrational modes that correlate with the differences in nuclear coordinates between the vibrationally equilibrated ground and excited state. Since the energy difference between the donor and acceptor is generally solvent-dependent, the distribution of solvent environments that is characteristic of solutions may also contribute to the bandwidth (see further discussion of this point in the sections below). If the validity of the Born Oppenheimer approximation is assumed, the intensity of each of these vibronic components is given by Eq. 11,... 

Radiationless transitions (IC and ISC) represent a conversion of electronic energy of an initial, excited state to vibrational energy in a lower-energy electronic state (cf. Figure 2.1). Within the Born Oppenheimer approximation (Section 1.3), radiationless... 

Vibrational sideline structure retains the polarization of the purely electronic excitation. Within the crude Born-Oppenheimer approximation, only totally symmetric vibrational modes are coupled to an electronic transition. Consequently, all vibrational sidelines have the same polarization as the purely electronic component (electronic origin or zero phonon line). There are two circumstances... 

The material in this chapter is largely organized around the molecular properties that contribute to electron transfer processes in simple transition metal complexes. To some degree these molecular properties can be classified as functions of either (i) the nuclear coordinates (i.e., properties that depend on the spatial orientation and separation, and the vibrational characteristics) of the electron transfer system or (ii) the electronic coordinates of the system (orbital and spin properties). This partitioning of the physical parameters of the system into nuclear and electronic contributions, based on the Born-Oppenheimer approximation, is not rigorous and even in this approximation the electronic coordinates are a function of the nuclear coordinates. The types of systems that conform to expectation at the weak coupling limit will be discussed after some necessary preliminaries and discussion of formalisms. Applications to more complex, extended systems are mentioned at the end of the chapter. 

We can further simplify the equation by applying the Born-Oppenheimer approximation to separate the electronic from the vibrational wave functions, as shown in Equation (9.23), where f, v, and / are the vibrational levels of the final, intermediate, and initial states and Mp(Q) is the pure electronic transition moment. M(Q) is a function of the vibrational or nuclear coordinates and can be expanded as the Taylor series in Equation (9.24), where the summation occurs over all of the normal coordinates ... 

The transition electric dipole moment in eqn [57] can be developed by invoking the Born-Oppenheimer approximation to express the total molecular wave function as a product of electronic and vibrational parts. (Rotational wave functions do not have to be included here since eqn [57] refers to an isotropic system. That is, the equation is a result of a rotational average which is equivalent to a summation over all the rotational states involved in the transition.) A general molecular state can now be expressed as the product of vibrational and electronic parts. Assuming that the initial and final electronic states are the ground state jcg). 

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