Molecular structure Born-Oppenheimer approximation

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. 

The problem, as Woolley addressed it, is that quantum mechanical calculations employ the fixed, or "clamped," nucleus approximation (the Born-Oppenheimer approximation) in which nuclei are treated as classical particles confined to "equilibrium" positions. Woolley claims that a quantum mechanical calculation carried out completely from first principles, without such an approximation, yields no recognizable molecular structure and that the maintenance of "molecular structure" must therefore be a product not of an isolated molecule but of the action of the molecule functioning over time in its environment.47... 

Background Philosophy. Within the framework of the Born-Oppenheimer approximation (JJ ), the solutions of the Schroedin-ger equation, Hf = Ef, introduce the concept of molecular structure and, thereby, the total energy hyperspace provided that the electronic wave function varies only slowly with the nuclear coordinates, electronic energies can be calculated for sets of fixed nuclear positions. The total energies i.e. the sums of electronic energy and the energy due to the electrostatic re-... 

Under the Born-Oppenheimer approximation, two major methods exist to determine the electronic structure of molecules The valence bond (VB) and the molecular orbital (MO) methods (Atkins, 1986). In the valence bond method, the chemical bond is assumed to be an electron pair at the onset. Thus, bonds are viewed to be distinct atom-atom interactions, and upon dissociation molecules always lead to neutral species. In contrast, in the MO method the individual electrons are assumed to occupy an orbital that spreads the entire nuclear framework, and upon dissociation, neutral and ionic species form with equal probabilities. Consequently, the charge correlation, or the avoidance of one electron by others based on electrostatic repulsion, is overestimated by the VB method and is underestimated by the MO method (Atkins, 1986). The MO method turned out to be easier to apply to complex systems, and with the advent of computers it became a powerful computational tool in chemistry. Consequently, we shall concentrate on the MO method for the remainder of this section. 

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... 

Using the Born-Oppenheimer approximation, electronic structure calculations are performed at a fixed set of nuclear coordinates, from which the electronic wave functions and energies at that geometry can be obtained. The first and second derivatives of the electronic energies at a series of molecular geometries can be computed and used to find energy minima and to locate TSs on a PES. 

Whereas the quantum-mechanical molecular Hamiltonian is indeed spherically symmetrical, a simplified virial theorem should apply at the molecular level. However, when applied under the Born-Oppenheimer approximation, which assumes a rigid non-spherical nuclear framework, the virial theorem has no validity at all. No amount of correction factors can overcome this problem. All efforts to analyze the stability of classically structured molecules in terms of cleverly modified virial schemes are a waste of time. This stipulation embraces the bulk of modern bonding theories. 

The classical idea of molecular structure gained its entry into quantum theory on the basis of the Born Oppenheimer approximation, albeit not as a non-classical concept. The B-0 assumption makes a clear distinction between the mechanical behaviour of atomic nuclei and electrons, which obeys quantum laws only for the latter. Any attempt to retrieve chemical structure quantum-mechanically must therefore be based on the analysis of electron charge density. This procedure is supported by crystallographic theory and the assumption that X-rays are scattered on electrons. Extended to the scattering of neutrons it can finally be shown that the atomic distribution in crystalline solids is identical with molecular structures defined by X-ray diffraction. 

The Born-Oppenheimer separation19-22 of the electronic and nuclear motions in molecules is probably the most important approximation ever introduced in molecular quantum mechanics, and will implicitly or explicitly be used in all subsequent sections of this chapter. The Born-Oppenheimer approximation is crucial for modern chemistry. It allows to define in a rigorous way, within the quantum mechanics, such useful chemical concepts like the structure and geometry of molecules, the molecular dipole moment, or the interaction potential. In this approximation one assumes that the electronic motions are much faster than the nuclear... 

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... 

According to the Born-Oppenheimer approximation, the potential function of a molecule is not influenced by isotopic substitution. Frequency shifts caused by isotopic substitution therefore provide experimental data in addition to the fundamentals which can yield information about the structure of a species. However, the half-widths of absorptions are too large to be resolved by the experimental techniques which are normally used, which is why these methods cannot reveal small isotopic shifts (some cm ). The half-widths of the bands are reduced drastically by applying the matrix-isolation technique (c.f. Sec. 4.4). The absorptions of many matrix-isolated species can therefore be characterized with the help of isotopic substitution, i.e., the molecular fragment which is involved in the vibration can be identified. The large - Si/" Si shift of the most intense IR absorption of matrix-isolated S=Si=S from 918 cm to 907 cm, for instance, demonstrates that silicon participates considerably in this vibration (Schnoeckel and Koeppe, 1989). The same vibration is shifted by 4 cm if only one atom is substituted by a atom. The band at 918 cm must be assigned to the antisymmetric stretching vibration, since the central A atom in an AB2 molecule with Doo/rsymmetry counts twice as much as the B atoms in the G-matrix (c.f. Wilson et al., 1955). 

The geometrical and electronic structure for molecular systems in general will depend on the balance between the different terms in the Hamiltonian i.e. electron-nucleus, electron-electron and nucleus-nucleus interaction including the valence as well as the core electrons of the constituent atoms. The full Hamiltonian for the molecular system is normally separated into a Hamiltonian Hn for the nuclei and another one Hgi for the electrons with fixed positions for the nuclei according to Born Oppenheimer approximation [31]. 

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. 

In representing molecular structures, we consider the nuclei to be fixed in specific positions while the electrons move rapidly around them (the Born-Oppenheimer approximation). 

For example, the Born-Oppenheimer approximation is ubiquitous. The separation of the electronic and nuclear motion is most often an excellent approximation. However, it is also fundamental to the concept of molecular structure. The model of fixed nuclei surrounded by electrons which accommodate almost instantly any change in the nuclear positions is basic to qualitative and quantitative discussions of molecular structure. 

Section 2 of this chapter notes will be devoted to the framework for separation of the ionic and electronic dynamics through the Born-Oppenheimer approximation. Atomic motion, with forces on the ions at each timestep evaluated through an electronic structure calculation, can then be propagated by Molecular Dynamics simulations, as proposed by first-principle Molecular Dynamics. This allows for a description of the electronic reorganisation following the atomic motion, e.g. bond rearrangements in chemical reactions. 

Woolley, R. G. 1977. "Molecular Structure and the Born-Oppenheimer Approximation." Chemical Physics Letters, 45 393—398. 

Woolley, R.G. 1985. The Molecular Structure Conundrum. Journal of Chemical Education 62(12) 1082. Woolley, R.G. and Sutcliffe, B.T. 1977. Molecular Structure and the Born-Oppenheimer Approximation. Chemical Physics Letters 45(2) 393. 

Perturbations are defined as deviations in the quantum-number variation of any observable from that predicted by a zero-order molecular structural model based on the Born-Oppenheimer approximation. This section is intended as an outline of the ingredients of molecular structural models. 

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