Coulomb-Born approximation

For 1, de Broglie s wavelength is small enough compared to the classical collision radius b so that a wave packet can be constructed which, approximately, follows the classical Coulomb trajectory [3]. The opposite limit, where the Sommerfeld parameter Zie hv<, denotes the case of weak Coulomb interaction where the Born approximation may be expected to be valid. 

The first (and still the foremost) quantum theory of stopping, attributed to Bethe [19,20], considers the observables energy and momentum transfers as fundamental in the interaction of fast charged particles with atomic electrons. Taking the simplest case of a heavy, fast, yet nonrelativistic incident projectile, the excitation cross-section is developed in the first Born approximation that is, the incident particle is represented as a plane wave and the scattered particle as a slightly perturbed wave. Representing the Coulombic interaction as a Fourier integral over momentum transfer, Bethe derives the differential Born cross-section for excitation to the nth quantum state of the atom as follows. 

Fig. 5.23. Ratio of K-shell ionization cross sections for electrons and positrons scattering from silver and copper at various impact energies -----, Born approximation calculation including a Coulomb correction (see text) -, Born...

A number of popular methods for computing the GB radii were proposed. The most popular one is based on the Coulomb field approximation, in which the generalized Born radii are defined by the equation ... 

Bandrauk (1969) used a distorted wave Born approximation to calculate the inelastic cross sections of atomic alkali halogen collisions. He found a forwardly peaked differential cross section without oscillations. His total cross section decreases with E 1/2 which is the high energy asymptotic behaviour of the LZ cross section (37). The magnitude of the cross section is much larger than the LZ-result, however. His results were not essentially different if instead of a Coulombic Hl2(R) he used a constant or screened Coulombic interaction. 

The first term in the series expansion in Eq. (3) corresponds to the Coulomb field approximation that is used in GB formalisms to calculate GB radii [37]. Born radii calculated solely based on the Coulomb field approximation are therefore not sensitive to the dielectric environment (as they should be). However, recent GB implementations that include an additional higher-order correction term [13,38] allow the e-dependent calculation of GB radii and consequently an accurate reproduction of solvation energies over the entire range of dielectric constants [35]. 

Grycuk,T. [2003]. Deficiency ofthe Coulomb-field approximation in the generalized Born model An improved formula for Born radii evaluation, /. Chem. Phys 119, pp. 4817-4826. 

The validity of Butler s theory has been discussed by a number of authors. Daitch and French, and Gerjuoy have shown that, in effect, Butler s theory is equivalent to the Born approximation. The validity of the latter is not obvious it has been discussed by Austern . As pointed out above the Butler theory has ignored the effect of the Coulomb field and the interaction... 

At high energies the Coulomb scattering is not the simple Rutherford cross-section from a point charge. The finite extension of the charge density of the nucleus must be taken into account. In Born approximation the Coulomb scattering amplitude from a finite sized charged distribution becomes... 

If we set Q2 to zero, i.e. the energy difference factor, we get the on-shell Coulomb amplitude. This leads to precisely the same cross section as the Born approximation except that Q is of course replaced by Qb- Keeping Q, the integral can be performed analytically to give... 

For example, say we wish to discuss the single ionization of helium by a fast proton. In the Born approximation we need a description of the initial and final state. For the initial state clearly the ground state Hartree-Fock field is most appropriate. As we are dealing with a closed shell the Hartree-Fock field is neutral at large distances for continuum states. The final continuum state might leave one electron in the ground state of the helium ion. An appropriate potential for the latter electron is a Coulombic field of charge two. 

Inner shell ionization of electrons to the continuum in ion-atom collisions can occur by two different processes. For low Z (projectile) particles on high Z2 (target) atoms the only available process is Coulomb excitation which is variously treated by plane wave Born approximation (PWBA), the binary encounter approximation (BEA), and the semiclassical approximation (SCA). When Z becomes comparable to 7/1 and the ion velocity v is lower than the velocity of the bound electron in question, v, the electrons adjust adiabati-cally to the approach of the two nuclei and enter molecular orbitals (MO) which in the limit of fused nuclei approach the atomic orbitals of the united atom Z = Z + Z2. This stacking of electrons can lead to a promotion of an innershell electron to the continuum or to a vacant outer orbital by direct curve crossing, rotational coupling, or radial coupling between molecular levels when such channels are available. 

In Chapter IX, Liang et al. present an approach, termed as the crude Bom-Oppenheimer approximation, which is based on the Born-Oppen-heimer approximation but employs the straightforward perturbation method. Within their chapter they develop this approximation to become a practical method for computing potential energy surfaces. They show that to carry out different orders of perturbation, the ability to calculate the matrix elements of the derivatives of the Coulomb interaction with respect to nuclear coordinates is essential. For this purpose, they study a diatomic molecule, and by doing that demonstrate the basic skill to compute the relevant matrix elements for the Gaussian basis sets. Finally, they apply this approach to the H2 molecule and show that the calculated equilibrium position and foree constant fit reasonable well those obtained by other approaches. 

The quaniity, (R). the sum of the electronic energy computed 111 a wave funciion calculation and the nuclear-nuclear coulomb interaciion .(R.R), constitutes a potential energy surface having 15X independent variables (the coordinates R j. The independent variables are the coordinates of the nuclei but having made the Born-Oppenheimer approximation, we can think of them as the coordinates of the atoms in a molecule. 

Vhen calculating the total energy of the system, we should not forget the Coulomb inter-ction between the nuclei this is constant within the Born-Oppenheimer approximation Dr a given spatial arrangement of nuclei. When it is desired to change the nuclear positions,... 

Since HF has a closed-shell electronic structure and no low-lying excited electronic states. HF-HF collisions may be treated quite adequately within the framework of the Born-Oppenheimer electronic adiabatic approximation. In this treatment (4) the electronic and coulombic energies for fixed nuclei provide a potential energy V for internuclear motion, and the collision dynamics is equivalent to a four-body problem. After removal of the center-of-mass coordinates, the Schroedinger equation becomes nine-dimensional. This nine-dimensional partial differential... 

There are several possible ways of introducing the Born-Oppenheimer model " and here the most descriptive way has been chosen. It is worth mentioning, however, that the justification for the validity of the Bom-Oppenheimer approximation, based on the smallness of the ratio of the electronic and nuclear masses used in its original formulation, has been found irrelevant. Actually, Essen started his analysis of the approximate separation of electronic and nuclear motions with the virial theorem for the Coulombic forces among all particles of molecules (nuclei and electrons) treated in the same quantum mechanical way. In general, quantum chemistry is dominated by the Bom-Oppenheimer model of the theoretical description of molecules. However, there is a vivid discussion in the literature which is devoted to problems characterized by, for example, Monkhorst s article of 1987, Chemical Physics without the Bom-Oppenheimer Approximation... ... 

In principle, it should be possible to obtain the electronic energy levels of the molecules as a solution of the Schrodinger equation, if inter-electronic and internuclear cross-coulombic terms are included in the potential energy for the Hamiltonian But the equation can be solved only if it can be broken up into equations which are functions of one variable at a time. A simplifying feature is that because of the much larger mass of the nucleus the motion of the electrons can be treated as independent of that of the nucleus. This is known as the Born-Oppen-heitner approximation. Even with this simplification, the exact solution has been possible for the simplest of molecules, that is, the hydrogen molecule ion, H,+ only, and with some approximations for the H2 molecule. 

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