Properties Related to Nuclear Motion

So far, we have restricted ourselves to the situation that the nuclei are fixed in space, i.e. we have considered molecular properties or contributions to the molecular properties that can be obtained from the electronic Schrodinger equation, Eq. (2.10), alone. In this chapter, and the following chapter we will finally lift this restriction and allow the nuclei to move again. In this chapter, we will look at properties that arise or at least have contributions due to a breakdown of the Born-Oppenheimer approximation. This means that in order to derive quantmn mechanical expressions for these experimentally observable properties we have to take into account the coupling of nuclear and electronic motion, i.e. some of the terms that are neglected in the Born-Oppenheimer approximation. 

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Our task is to find approximate solutions to the time-independent Schrodinger equation (Eq. (2)) subject to the Pauli antisymmetry constraints of many-electron wave functions. Once such an approximate solution has been obtained, we may extract from it information about the electronic system and go on to compute different molecular properties related to experimental observations. Usually, we must explore a range of nuclear configurations in our calculations to determine critical points of the potential energy surface, or to include the effects of vibrational and rotational motions on the calculated properties. For properties related to time-dependent perturbations (e.g., all interactions with radiation), we must determine the time development of the... 

Since the Mossbauer effect is intimately related to any motion of the emitting or absorbing nucleus on either a microscopic or macroscopic scale, Mossbauer spectroscopy provides a potential means by which information on nuclear dynamics, and hence on the dynamics of a system in which the Mossbauer nucleus acts as a probe, can be obtained. Any motion of the Mossbauer nucleus can influence the Mossbauer spectrum in two ways. Firstly, because this motion may be related to the vibrational properties of the system it can influence the recoil-free fraction and hence the absorption intensity of the spectrum itself. Since the absolute absorption intensity is dependent on a large number of other factors, which may be diflicult to determine accurately, any change in recoil-free fraction is most usefully followed as a function of temperature in order to obtain information on the vibrational properties of the system. The second way in which the effects of any motion of the Mossbauer nucleus in the source or absorber are manifested is in the Mossbauer spectroscopic linewidths, as this motion can be thought of as an additional Doppler motion which may partially smear out the resonant absorption. Since the linewidths are also... 

U(qJ is referred to as an adiabatic-to-diabatic transformation (ADT) matrix. Its mathematical sbucture is discussed in detail in Section in.C. If the electronic wave functions in the adiabatic and diabatic representations are chosen to be real, as is normally the case, U(q ) is orthogonal and therefore has n n — l)/2 independent elements (or degrees of freedom). This transformation mabix U(qO can be chosen so as to yield a diabatic electronic basis set with desired properties, which can then be used to derive the diabatic nuclear motion Schrodinger equation. By using Eqs. (27) and (28) and the orthonormality of the diabatic and adiabatic electronic basis sets, we can relate the adiabatic and diabatic nuclear wave functions through the same n-dimensional unitary transformation matrix U(qx) according to... 

NMR and EPR techniques provide unique information on the microscopic properties of solids, such as symmetry of atomic sites, covalent character of bonds, strength of exchange interactions, and rates of atomic and molecular motion. The recent developments of nuclear double resonance, the Overhauser effect, and ENDOR will allow further elucidation of these properties. Since the catalytic characteristics of solids are presumably related to the detailed electronic and geometric structure of solids, a correlation between the results of magnetic resonance studies and cata lytic properties can occur. The limitation of NMR lies in the fact that only certain nuclei are suitable for study in polycrystalline or amorphous solids while EPR is limited in that only paramagnetic species may be observed. These limitations, however, are counter-balanced by the wealth of information that can be obtained when the techniques are applicable. 

The goal of theory and computer simulation is to predict S i) and relate it to solvent and solute properties. In order to accomplish this, it is necessary to determine how the presence of the solvent affects the So —> Si electronic transition energy. The usual assmnption is that the chromophore undergoes a Franck-Condon transition, i.e., that the transition occurs essentially instantaneously on the time scale of nuclear motions. The time-evolution of the fluorescence Stokes shift is then due the solvent effects on the vertical energy gap between the So and Si solute states. In most models for SD, the time-evolution of the solute electronic stracture in response to the changes in solvent environment is not taken into accoimt and one focuses on the portion AE of the energy gap due to nuclear coordinates. 

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