Born approximation extended

Despite the fact that Bohr s stopping power theory is useful for heavy charged particles such as fission fragments, Rutherford s collision cross section on which it is based is not accurate unless both the incident particle velocity and that of the ejected electron are much greater than that of the atomic electrons. The quantum mechanical theory of Bethe, with energy and momentum transfers as kinematic variables, is based on the first Born approximation and certain other approximations [1,2]. This theory also requires high incident velocity. At relatively moderate velocities certain modifications, shell corrections, can be made to extend the validity of the approximation. Other corrections for relativistic effects and polarization screening (density effects) are easily made. Nevertheless, the Bethe-Born approximation... 

Most other calculations of positronium formation in positron-helium scattering have employed much simpler methods of approximation, but results have usually been obtained over energy ranges extending well beyond the Ore gap. It must therefore be borne in mind that the experimental results include contributions from positronium formation into excited states as well as into the ground state. The Born approximation, used first by Massey and Moussa (1961) and subsequently by Mandal, Ghosh... 

Habashy et al., (1993) and Torres-Verdin and Habashy (1994) developed the extended Born approximation, which replaces the total field in the integral (9.73) not by the background field, like in the Born approximation, but by its projection onto a scattering tensor F (r) ... 

Following ideas of the extended Born approximation outlined above, we use the fact that the Green s tensor Ge (j" 1 r) exhibits either singularity or a peak at the point where r = r. Therefore, one can expect that the dominant contribution to the integral Ge [ActAE "] in equation (9.83) is from some vicinity of the point r = r. Assuming also that A (r) is slowly varying within domain D, one can write... 

We have noticed that the background of the modified Born approximation and the new first order QL approximation is the same. The main difference is that in the case of the Born approximation the starting point (zero order approximation) for the iteration process is the zero anomalous field, while in the QL approach we start with the anomalous field proportional to the background field. In principle we can extend our approach to computing all iterations bj (9.149). In this case we will obtain a complete analog of the Born series. For example, the second order QL approximation is equal to... 

In accord with our interest we restrict our exposition in this section to statistical treatments which contain as an element the quantum mechanical cross-section or transition probability discussed in Section IV. Such statistical approaches which have been applied to chemical reactions may be conveniently divided into three categories those based on the Pauli equation or similar considerations (Section V-A), a modified Boltzmann equation (Section V-B), or a quantum statistical formulation of the Onsager theory (Section V-C). These treatments have not had notable success in comparison with experiment, probably because of the implicit Born approximation or its equivalent. It is therefore of considerable importance to extend this type of treatment to cross-sections other than that derived with the Born approximation. The method presented in Section V-C would seem to offer the best hope in this direction. 

The Born-Oppenheimer approximation takes us a long way in that direction by assuming that the rapid motion of the electrons is separable from the nuclear motion. The Coulomb forces acting on the electrons and nuclei in a molecule are of comparable magnitude, but force is mass times acceleration, and each nucleus has at least 10 times more mass than each electron. Therefore, the forces accelerate the electrons to much higher speeds than the nuclei. The Born-Oppenheimer approximation extends this difference in speed to the limit in which we treat the electrons as traveling around stationary nuclei. This means we don t have to solve for the motions of the electrons and nuclei at the same time the electronic and nuclear coordinates are separable, in a way similar to the separation of angular and radial coordinates for the one-electron atom (Section 3.1). In the one-electron atom, we solve the... 

In filtration, the particle-collector interaction is taken as the sum of the London-van der Waals and double layer interactions, i.e. the Deijagin-Landau-Verwey-Overbeek (DLVO) theory. In most cases, the London-van der Waals force is attractive. The double layer interaction, on the other hand, may be repulsive or attractive depending on whether the surface of the particle and the collector bear like or opposite charges. The range and distance dependence is also different. The DLVO theory was later extended with contributions from the Born repulsion, hydration (structural) forces, hydrophobic interactions and steric hindrance originating from adsorbed macromolecules or polymers. Because no analytical solutions exist for the full convective diffusion equation, a number of approximations were devised (e.g., Smoluchowski-Levich approximation, and the surface force boundary layer approximation) to solve the equations in an approximate way, using analytical methods. 

Born-Oppenheimer approximation, matrix elements, 186-191 coordinate origins, 137-138 extended Born-Oppenheimer equations closed path matrix quantization, 171 — 173... 

The extended Born-Oppenheimer approximation based on the nonadiabatic coupling terms was discussed on several occasions [23,25,26,55,56,133,134] and is also presented here by Adhikari and Billing (see Chapter 3). 

We wanted to extend this approach to include dynamical effects on line shapes. As discussed earlier, for this approach one needs a trajectory co t) for the transition frequency for a single chromophore. One could extract a water cluster around the HOD molecule at every time step in an MD simulation and then perform an ab initio calculation, but this would entail millions of such calculations, which is not feasible. Within the Born Oppenheimer approximation the OH stretch potential is a functional of the nuclear coordinates of all the bath atoms, as is the OH transition frequency. Of course we do not know the functional. Suppose that the transition frequency is (approximately) a function of a one or more collective coordinates of these nuclear positions. A priori we do not know which collective coordinates to choose, or what the function is. We explored several such possibilities, and one collective coordinate that worked reasonably well was simply the electric field from all the bath atoms (assuming the point charges as assigned in the simulation potential) on the H atom of the HOD molecule, in the direction of the OH bond. 

At this stage we are at the very beginning of development, implementation, and application of methods for quantum-mechanical calculations of molecular systems without assuming the Born-Oppenheimer approximation. So far we have done several calculations of ground and excited states of small diatomic molecules, extending them beyond two-electron systems and some preliminary calculations on triatomic systems. In the non-BO works, we have used three different correlated Gaussian basis sets. The simplest one without r,y premultipliers (4)j = exp[—r (A t (8> Is) "]) was used in atomic calculations the basis with premultipliers in the form of powers of rj exp[—r (Aj (8> /sjr])... 

In order to solve die Poisson equation for an arbitrary cavity, recourse to numerical methods is required. An altemative approach that has seen considerable development involves computing die polarization free energy using an approximation to the Poisson equation that can be solved analytically, and diis is the Generalized Bom (GB) approach. As its name implies, the GB method extends the Born Eq. (11.12) to polyatomic molecules. The fundamental equation of the GB method expresses the polarization energy as... 

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. 

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