Born’s approximation

The quantum-mechanical ionization cross section is derived using one of several approximations—for example, the Born, Ochkur, two-state, or semi-classical approximations—and numerical computations (Mott and Massey, 1965). In some cases, a binary encounter approximation proves useful, which means that scattering between the incident particle and individual electrons is considered classically, followed by averaging over the quantum-mechanical velocity distribution of the electrons in the atom (Gryzinski, 1965a-c). However, Born s approximation is the most widely used one. This is discussed in the following paragraphs. 

As co) increases, the experimental scattering lobe is no longer symmetric to Zcol but turns its symmetry axis essentially into the direction of the momentum-transfer vector K=kout—kjn as indicated by 9 in Fig. 32. In Born s approximation the lobe would be exactly symmetric to K. This reflects the fact that electronic angular momentum of the atom is transferred into linear momentum of the scattered electron. 

This corresponds to excitation of a single molecule in one of its first excited states. In a simple disordered system of N symmetrical molecules with volume V and temperature T 10 K, the scattering process may be described with Born s approximation [7, 8]. In this approximation, the excited electronic level contributes primarily to the sum under the electronic state. 

Despite the fact that the relative velocity is not large with respect to the internal motion, Born s approximation is still fitting if all the matrix elements are relatively small with 17the largest of them. This situation does not usually occur for slow collisions, and therefore we are led to make other slightly better approximations in Eq. (178). 

Both inoleciilar and qiiantnin mechanics in ethods rely on the Born-Oppenheimer approximation. In qnantiinn mechanics, the Schrddmger equation (1) gives the wave function s and energies of a inolecii le. 

Fig. 5.2 Radial distribution curves, Pv

Marcus uses the Born-Oppenheimer approximation to separate electronic and nuclear motions, the only exception being at S in the case of nonadiabatic reactions. Classical equilibrium statistical mechanics is used to calculate the probability of arriving at the activated complex only vibrational quantum effects are treated approximately. The result is... 

The direct access to the electrical-energetic properties of an ion-in-solution which polarography and related electro-analytical techniques seem to offer, has invited many attempts to interpret the results in terms of fundamental energetic quantities, such as ionization potentials and solvation enthalpies. An early and seminal analysis by Case etal., [16] was followed up by an extension of the theory to various aromatic cations by Kothe et al. [17]. They attempted the absolute calculation of the solvation enthalpies of cations, molecules, and anions of the triphenylmethyl series, and our Equations (4) and (6) are derived by implicit arguments closely related to theirs, but we have preferred not to follow their attempts at absolute calculations. Such calculations are inevitably beset by a lack of data (in this instance especially the ionization energies of the radicals) and by the need for approximations of various kinds. For example, Kothe et al., attempted to calculate the electrical contribution to the solvation enthalpy by Born s equation, applicable to an isolated spherical ion, uninhibited by the fact that they then combined it with half-wave potentials obtained for planar ions at high ionic strength. 

According to the Born-Oppenheimer approximation, the motions of electrons are much more rapid than those of the nuclei (i.e. the molecular vibrations). Promotion of an electron to an antibonding molecular orbital upon excitation takes about 10-15 s, which is very quick compared to the characteristic time for molecular vi-... 

H. Koppel, W. Domcke, and L. S. Cederbaum, Multimode molecular dynamics beyond the Born-Oppenheimer approximation, Adv. Chem. Phys. 57, 59-246(1984). 

A detailed discussion of the theoretical evaluation of the adiabatic correction for a molecular system is beyond the scope of this book. The full development involves, among other matters, the investigation of the action of the kinetic energy operators for the nuclei (which involve inverse nuclear masses) on the electronic wave function. Such terms are completely ignored in the Born-Oppenheimer approximation. In order to go beyond the Born-Oppenheimer approximation as a first step one can expand the molecular wave function in terms of a set of Born-Oppenheimer states (designated as lec (S, r ))... 

The important fact that must be remembered is that in the Born-Oppenheimer approximation, Equation 2.8, the potential energy for vibrational motion is Eeiec(S) which is independent of isotopic mass of the atoms. In the adiabatic approximation, the potential energy function is Eeiec(S)+C and this potential will depend on nuclear mass if C depends on nuclear mass. 

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... 

Born-Oppenheimer approximation, 22, 219 Bound state, 209-210 Bid, with alkenes, 260 Brillouin s theorem, 241 Bromide ion (Br ) effect on Ao, 181 trails effect, 181 as X ligand, 176 Bromine (Br2) sigma bond, 77 Bromochloromethane, 13 ll-Bromo- ii/o-9-chloro-7-... 

Born s idea of the dielectric continuum solvation approximation became very popular, and many researchers worked on its further development. Hence a brief overview of the most important development steps will be given, but it is impossible to mention all the different modifications and all workers who have been contributing to this field. Readers who seek a broader overview are referred to some reviews on continuum solvation methods, e.g., by Cramer and Truhlar [22] or by Tomasi and Persico [23]. The goal of the history given here is to enable... 

In the previous chapter, we have seen how Born s simple and successful idea of a dielectric continuum approximation for the description of solvation effects has been developed to a considerable degree of perfection. Almost all workers in this area have been trying to obtain more efficient and more precise methods for the solution of dielectric boundary conditions combined with molecular electrostatics, but the question of the validity of Born s basic assumption has rarely been discussed. This will be done in the following sections, with a surprising result. 

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... 

Chapter 3 describes radiationless transitions in the tunneling electron transfers in multi-electron systems. The following are examined within the framework of electron Green s function approach the dependence on distance, the influence of crystalline media, and the effect of intermediate particles on the tunneling transfer. It is demonstrated that the Born-Oppenheimer approximation for the wave function is invalid for longdistance tunneling. 

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