Born-Oppenheimer approximation molecular properties

To be specific we consider electron transfer from a reactant in a solution, such as [Fe(H20)6]2+, to an acceptor, which may be a metal or semiconductor electrode, or another molecule. To obtain wavefunc-tions for the reactant in its reduced and oxidized state, we rely on the Born-Oppenheimer approximation, which is commonly used for the calculation of molecular properties. This approximation is based on the fact that the masses of the nuclei in a molecule are much larger than the electronic mass. Hence the motion of the nuclei is slow, while the electrons are fast and follow the nuclei almost instantaneously. The mathematical consequences will be described in the following. 

It is clear that arbitrary one-particle densities of a molecular system need not have the same topology. In fact, only those belonging to the same structural region will share this property. To make these concepts clearer, consider two nuclear configurations X and Y belonging to the nuclear configuration space in the context of the Born-Oppenheimer approximation. The corresponding one-electrons densities are p r X) and p(r T), respectively. Consider the... 

Initially the properties calculated were energetic in nature and related to IR spectroscopic measurements.64 Bishop s work was the first serious attempt to calculate the rovibronic energies of the hydrogen molecule and molecular ion without using the Born-Oppenheimer approximation (i.e., three- and four-body calculations). Many years later, this work is still cited and the relativistic... 

Ab initio quantum mechanical (QM) calculations represent approximate efforts to solve the Schrodinger equation, which describes the electronic structure of a molecule based on the Born-Oppenheimer approximation (in which the positions of the nuclei are considered fixed). It is typical for most of the calculations to be carried out at the Hartree—Fock self-consistent field (SCF) level. The major assumption behind the Hartree-Fock method is that each electron experiences the average field of all other electrons. Ab initio molecular orbital methods contain few empirical parameters. Introduction of empiricism results in the various semiempirical techniques (MNDO, AMI, PM3, etc.) that are widely used to study the structure and properties of small molecules. 

All aspects of molecular shape and size are fully reflected by the molecular electron density distribution. A molecule is an arrangement of atomic nuclei surrounded by a fuzzy electron density cloud. Within the Born-Oppenheimer approximation, the location of the maxima of the density function, the actual local maximum values, and the shape of the electronic density distribution near these maxima are fully sufficient to deduce the type and relative arrangement of the nuclei within the molecule. Consequently, the electronic density itself contains all information about the molecule. As follows from the fundamental relationships of quantum mechanics, the electronic density and, in a less spectacular way, the nuclear distribution are both subject to the Heisenberg uncertainty relationship. The profound influence of quantum-mechanical uncertainty at the molecular level raises important questions concerning the legitimacy of using macroscopic analogies and concepts for the description of molecular properties. ... 

The molecular mechanics method is used to calculate molecular structures, conformational energies, and other molecular properties using concepts from classical mechanics. Electrons are not explicitly included in the molecular mechanics method, which is justified on the basis of the Born-Oppenheimer approximation stating that the movements of electrons and the nuclei can be separated. Thus, the nuclei may be viewed as moving in an average electronic potential field, and the molecular mechanics method attempts to describe this field by its force field. ... 

Qualitative information about molecular wave functions and properties can often be obtained from the symmetry of the molecule. By the symmetry of a molecule, we mean the symmetry of the framework formed by the nuclei held fixed in their equilibrium positions. (Our starting point for molecular quantum mechanics will be the Born-Oppenheimer approximation, which regards the nuclei as fixed when solving for the electronic wave function see Section 13.1.) The symmetry of a molecule can differ in different electronic states. For example, HCN is linear in its ground electronic state, but nonlinear in certain excited states. Unless otherwise specified, we shall be considering the symmetry of the ground electronic state. 

The Born-Oppenheimer approximation [96], created to simpUfy the electronic calculus for frozen nuclei approximation, breaks down when computing, for instance, the magnetic dipole moment and its derivative with respect to the nuclear velocities or momenta for assessing the molecular properties of surfaces [97]. 

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. 

The molecular partition functions, Q, can be related to molecular properties of reactants and products. The partition function expresses the probability of encountering a molecule, so that the ratio of partition functions for the products versus the reactants of a chemical reaction expresses the relative probability of encountering products versus reactants and, therefore, the equilibrium constant. The partition function can be written as a product of independent factors at the level of various approximations, each of which is related to the molecular mass, the principal moments of inertia, the normal vibration frequency, and the electronic energy levels, respectively. When the ratio of isotopic partition function is calculated, the electronic part of the partition function cancels, at the level of the Born-Oppenheimer approximation, an approximation stating that the motion of nuclei in ordinary molecular vibrations is slow relative to the motions of electrons. 

In a diatomic molecule isotope effects appear in the vibrational, rotational, and electronic spectra. Isotope effects on vibrational spectra can be interpreted within the framework of the Born-Oppenheimer approximation, which is the cornerstone of most theories dealing with the effects of isotopic substitution on molecular properties. In this approximation, the potential energy surface for the vibrational-rotational motions of a molecular system is taken as being independent of the masses of the nuclei. Thus, the nuclei of different masses move on the same potential surface because to good approximation the electronic structure is independent of isotopic substitution. Then, the harmonic vibrational frequencies of the two isotopic variants of a diatomic molecule (prime denotes the lighter molecule) can be given as... 

The purpose of this review article is to present a comprehensive account of what is generally known as the Born-Oppenheimer approximation, its meaning, its implications, its properties, and very importantly also its limitations and how to cure them. This approximation and the underlying idea have been a milestone in the theory of molecules and actually also of electronic matter in general. Still today this approximation is basic to all molecular quantum mechanics and even in those cases where it fails, it remains the reference to which we compare and in terms of which we discuss this failure. 

The Born-Oppenheimer approximation is the cornerstone of theories dealing with the effect of isotopic substitution on molecular properties. This approximation states that electronic and nuclear motion of a molecule can... 

There is no doubt that the Born-Oppenheimer approximation captures the essential feature of the molecular properties of stable molecules. (So many papers have been published on the mathematical and numerical analyses of the Born-Oppenheimer theory, but it is beyond the scope of this book. See Refs. [181, 405, 409] for relevant recent literature.) The validity or the level of accuracy of the Born-Oppenheimer separation is roughly assessed as follows. Let m be the mass of an electron and let M be the mass of the /-th nucleus. The perturbation parameter n is taken as... 

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. 

In this chapter we will therefore discuss the contributions from the nuclear wave-function to the molecular properties derived in the previous chapters. However, in doing so we will still make use of the Born-Oppenheimer approximation. In the following, we will use the static polarizability as example and illustrate how these vibrational corrections can be incorporated (Bishop and Cheung, 1980 Bishop et al., 1980). The expression, which we are going to derive, can then easily be transferred to all linear response properties. A detailed description of vibrational corrections to static and frequency-dependent hyperpolarizabilities can be found in the reviews by Bishop (1990 1998). 

The current standard for quantum calculations are so called electronic structure calculations that focus on the quantum nature of the electrons, largely justified by the so-called Born-Oppenheimer approximation which separates the motion of the light electrons from the heavier nuclei that are assumed to behave essentially as point charges around which the electrons arrange in quantized levels, in other words what in a molecular language are known as molecular orbitals (MOs). This is routinely applied with great success to the study of the structural and electronic properties of a wide range of molecules and materials. ... 

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