Perturbation expressions, first-order

All first-order approximations (pertaining to integral transport theory) considered are equivalent, in accuracy, either to Pid[x where (j) stands for one of the three approximations, < fl> bd> or ( fd> fo the perturbed flux distribution and stands for either ( fl or ( bd-cf> is a better approximation to compared to better approximation to compared to we conclude that all first-order perturbation expressions in integral transport theory formulations considered in this work are equivalent, in accuracy, to some high-order approximation to Pji>. This higher accuracy can be computed, in integral formulations, using the flux and source-importance functions for the unperturbed reactor. 

The optical activity in valence excitations of chiral metal complexes has been effectively treated using the model of an achiral chromophore (metal ion) in a chiral environment (ligands) and this model appears also appropriate for XAS in view of the core nature of the initial orbital state. The zero-order electric and magnetic transition moments arise from different transitions and must be mixed by some chiral environmental potential (V ). Considering the case of a lanthanide ion, and taking tihe electric dipole transitions for the Z.2,3 edge as Ip—Kj), a first-order perturbation expression for the rotational strength looks like ... 

In use of the ellipsoidal model of segment density distribution mentioned above in connection with equation (50), it has been suggested that the first-order perturbation expression equation (186) may in fact be valid for all z with the result ... 

The expressions (4.22)-(4.23) found in chap. 4 for the isomer shift 5 in nonrelativ-istic form may be applied to lighter elements up to iron without causing too much of an error. In heavier elements, however, the wave function j/ is subject to considerable modification by relativistic effects, particularly near the nucleus (remember that the spin-orbit coupling coefficient increases with Z ). Therefore, the electron density at the nucleus l /(o)P will be modified as well and the aforementioned equations for the isomer shift require relativistic correction. This has been considered [1] in a somewhat restricted approach by using Dirac wave functions and first-order perturbation theory in this approximation the relativistic correction simply consists of a dimensionless factor S (Z), which is introduced in the above equations for S,... 

We draw attention to the fact that in the first-order perturbation theory we have = 1 and B %) = 0 in Eq. (4.2.41) for any N, and the expression for the leaving rate of the molecule from the subbarrier states reduces to two well-known special cases. The first of these corresponds to the low-temperature limit - 0 for which166... 

In summary, density functional theory provides a natural framework to discuss solvent effects in the context of RF theory. A general expression giving the insertion energy of an atom or molecule into a polarizable medium was derived. This expression given in Eq (83), when treated within a first order perturbation theory approach (i.e. when the solute self-polarization... 

The cathodic current density for the neutralization of the proton at an electrode was obtained using first-order perturbation theory and was expressed in nonquadratic form as... 

The author has calculated and will publish elsewhere the perturbations of a known system by a force of the form EoF t). As a special case we have the problem of dispersion— namely, an atomic system acted on by a plane wave. In this case we have an electric field o cos 2irv/. If general cylindrical coordinates are chosen with the z direction parallel to the electric field, the expression for an element of the first order perturbation of the 2/ dimensional matrix q is given by ... 

The values of the electron relaxation rates of the coupled metal ion strongly depend both on the relative electron relaxation rates of the isolated ions and on the value of the magnetic coupling constant J. When the absolute value of J (expressed as J /K) is smaller than both electronic relaxation rates, no effect on the electronic relaxation of the pair is expected. When J /H > (electron relaxation rate of the first ion) but smaller than T[ 2) (electron relaxation rate of the second ion), from first order perturbation... 

Integration is over charge density p(r) at an angle between the radius vector r and the symmetry axis. The application of first order perturbation theory (5, 90) then yields the following expression for the energy levels E ,... 

We denote the vibrational wave packet associated with electronic state i by (7,0 and fi2l is the transition dipole moment. Initially the system is in the vibrational ground state on Vt and treating the interaction with the field E[ v (t) within first-order perturbation theory gives the following expression for the nuclear wave packet on Vi-... 

This expression excludes self-interaction. There have been a number of attempts to include into the Hartree-Fock equations the main terms of relativistic and correlation effects, however without great success, because the appropriate equations become much more complex. For a large variety of atoms and ions both these effects are fairly small. Therefore, they can be easily accounted for as corrections in the framework of first-order perturbation theory. Having in mind the constantly growing possibilities of computers, the Hartree-Fock self-consistent field method in various... 

Some qualitative understanding of the CICD can be gained by means of Wentzel-type theory that treats the initial and final states of the decay as single Slater determinants taking electronic repulsion responsible for the transitions as a perturbation. The collective decay of two inner-shell vacancies (see Figure 6.6) is a three-electron transition mediated by two-electron interaction. Thus, the process is forbidden in the first-order perturbation theory, and its rate cannot be calculated by the first-order expressions, such as (1). Going to the second-order perturbation theory, the expression for the collective decay width can be written as... 

Let us make the relation between the potential anisotropy and the final state distribution more quantitative. In first-order perturbation theory Equation (5.4f) can be directly integrated yielding the (approximate) expression... 

If the coupling is zero, the bound states will live forever. However, immediately after we have switched on the coupling they start to decay as a consequence of transitions to the continuum states until they are completely depopulated. Our goal is to derive explicit expressions for the depletion of the bound states l iz) and the filling of the continuum states 2(E,0)). The method we use is time-dependent perturbation theory in the same spirit as outlined in Section 2.1, with one important extension. In Section 2.1 we explicitly assumed that the perturbation is sufficiently weak and also sufficiently short to ensure that the population of the initial state remains practically unity for all times (first-order perturbation theory). In this section we want to describe the decay process until the initial state is completely depleted and therefore we must necessarily go beyond the first-order treatment. The subsequent derivation closely follows the detailed presentation of Cohen-Tannoudji, Diu, and Laloe (1977 ch.XIII). 

The narrow transitions just mentioned are preceeded by two broader bands (they may overlap in tetrahedral complexes) having a4Tig and a T2g as excited levels. They belong predominantly to the sub-shell configuration t2g4 eg though their distance from the first narrow band is a fairly complicated function of A. The second-order perturbation expressions are, assuming C = 4B ... 

---