Born approximation modified

Humberston and Wallace, 1972), is shown in Figure 6.4. Also shown there is the distribution function obtained using the Born approximation, in which neither the positron nor the atomic wave function is modified by the interaction. This latter curve therefore represents the momentum distribution of the electron in the undistorted hydrogen atom. The distribution function for the accurate wave function is narrower than that for the undistorted case because the positron attracts the electron towards itself and away from the nucleus, thereby enhancing the probability of low values of the momentum of the pair. 

Let us analyze more carefully the modified Born approximation introduced by formula (9.143). Simple calculations show that... 

From equation (9.146) we can see that the modified Born approximation is equal to a conventional Born approximation outside the inhomogeneous domain D ... 

Taking into account formula (9.145), we can obtain the following expression for the modified A-th order Born approximation as the sum of N terms of the Born (or Neumann) series, which helps to understand better the internal structure of the new series ... 

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... 

From the last formula we can see that higher order QL approximations are more accurate than the modified Born approximation of the higher order because they take into account an additional term in the series. 

Comparison of the inequalities (9.147) and (9.169) clearly demonstrates that the accuracy of the Born approximation depends only on the order N, while the accuracy of QL approximation of the same order can be increased by a proper selection of A. This circumstance makes the QL approximation a more efficient tool for EM modeling than a conventional or modified Born series. 

Note that outside inhomogeneity D, the modified Born approximation coincides with the conventional Born approximation ... 

The generalized oscillator strength formulation of the plane wave Born approximation to the calculation of stopping power is modified by introducing radial Green s functions in place of the infinite sums over bound excited states and integrations over the continuum. Some properties of the resulting expressions are examined. 

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. 

A disadvantage of this technique is that isotopic labeling can cause unwanted perturbations to the competition between pathways through kinetic isotope effects. Whereas the Born-Oppenheimer potential energy surfaces are not affected by isotopic substitution, rotational and vibrational levels become more closely spaced with substitution of heavier isotopes. Consequently, the rate of reaction in competing pathways will be modified somewhat compared to the unlabeled reaction. This effect scales approximately as the square root of the ratio of the isotopic masses, and will be most pronounced for deuterium or... 

The Born equation, proposed in 1920, has been modified in various ways in order to get a single equation that can express the experimental ionic solvation energies. In recent years, the so-called mean spherical approximation (MSA) has often been used in treating ion solvation. In the MSA treatment, the Gibbs energy of ion solvation is expressed by... 

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. 

In this approach, the external potential displacements that are responsible for a transition from stage (i) to stage (ii) create conditions for the subsequent CT effects, in the spirit of the Born-Oppenheimer approximation. Clearly, the consistent second-order Taylor expansion at M°(co) does not involve the coupling hardness t A B and the off-diagonal response quantities of Eqs. (168) and (170), which vanish identically for infinitely separated reactants. However, since the interaction at Q modifies both the chemical potential difference and the... 

The field gradient is measured at a fixed point within the molecule, the translational part of the wave-function is thus of no consequence for (qap)-The effect of molecular rotation does, however, modify (qap) but the relationship between the rotating and stationary (qap) s has already been treated in the chapter dealing with microwave spectroscopy. In the present context, we are interested in the field gradients in a vibrating molecule in a fixed coordinate system. The Born-Oppenheimer approximation for molecular wave-functions enables us to separate the nuclear and electronic motions, the electronic wave functions being calculated for the nuclei in various fixed positions. The observed (qap) s will then be average values over the vibrational motion. 

In this chapter we describe the various stages of the factorisation process. Following the separation of translational motion by reference of the particles coordinates to the molecular centre of mass, we separate off the rotational motion by referring coordinates to an axis system which rotates with the molecule (the so-called molecule-fixed axis system). Finally, we separate off the electronic motion to the best of our ability by invoking the Born-Oppenheimer approximation when the electronic wave function is obtained on the assumption that the nuclei are at a fixed separation R. Some empirical discussion of the involvement of electron spin, in either Hund s case (a) or (b), is also included. In conclusion we consider how the effects of external electric or magnetic fields are modified by the various transformations. 

Now we can start iterations by the modified Born series with a quasi-analytical approximation for the anomalous field ... 

We now introduce an excess electron into the bubble, which is located in the center of the helium cluster at a fixed nuclear configuration of the helium balloon. The electronic energy of the excess electron will be calculated within the Born-Oppenheimer separability approximation. We modified the nonlocal effective potential developed by us for surface excess electron states on helium clusters [178-180] for the case of an excess electron in a bubble of radius Rb... 

From the difference in mass of the electrons m and of the nuclei and from equipartition of energy it is evident that the electrons move rapidly compared to the nuclei and therefore will exist in separable, approximate stationary states that are smoothly modified by the motion of the nuclei. This is, of course, the well-known adiabatic approximation of Born and Oppenheimer. In order to solve for these electronic stationary states we consider the nuclei fixed in the following Schrodinger equation ... 

In the case of MCD spectroscopy, for a fully allowed electronic transition based on the rigid shift, Born-Oppenheimer, and Franck-Condon approximations, the intensity equation, according to the modified conventions recommended by Stephens, Piepho and Schatz,is... 

---