The Second-Order Perturbation Theory

The perturbations in this case are between a singlet and a triplet state. The perturbation Hamiltonian, H, of the second-order perturbation theory is spin-orbital coupling, which has the effect of mixing singlet and triplet states. 

Lattice vibrations are calculated by applying the second order perturbation theory approach of Varma and Weber , thereby combining first principles short range force constants with the electron-phonon coupling matrix arising from a tight-binding theory. 

Here, V is the electrostatic potential created by the solvent that acts on the charge in the center of the cavity. Recalling (2.30), the second-order perturbation theory yields... 

Indeed, both expressions predict quadratic dependence of AA on the dipole moment of the solute. As in the previous example, it is of interest to test whether this prediction is correct. Such a test was carried out by calculating AA for a series of model solutes immersed in water at different distances from the water-hexane interface [11]. The solutes were constructed by scaling the atomic charges and, consequently, the dipole moment of a nearly spherical molecule, CH3F, by a parameter A, which varied between 0 and 1.2. The results at two positions - deep in the water phase and at the interface - are shown in Fig. 2.3. As can be seen from the linear dependence of A A on p2, the accuracy of the second-order perturbation theory... 

In fact, we have already used a modeling strategy when Po(AU) was approximated as a Gaussian. This led to the second-order perturbation theory, which is only of limited accuracy. A simple extension of this approach is to represent Pq(AU) as a linear combination of n Gaussian functions, p, (AU), with different mean values and variances [40]... 

Ua and Ub. Assume further that the second-order perturbation theory applies. This means that Po(AU) can be represented as a bivariate Gaussian. Then, AA, from (2.30), is given by... 

In a formal sense, a track is generated by correlated energy loss events along the direction of the momentum of the penetrating particle. Figure 3.2 shows the formation of a track according to Mott in the second-order perturbation theory... 

In the second-order perturbation theory in vq, the required retarded GF (25) is equal to... 

In general, all three contributions to the second order perturbation theory stabilize the molecular association AB. zJApol is the analogue to the classical polarization energy and becomes identical with the function (2 C R -n) calculated... 

In this paper, the main features of the two-step method are presented and PNC calculations are discussed, both those without accounting for correlation effects (PbF and HgF) and those in which electron correlations are taken into account by a combined method of the second-order perturbation theory (PT2) and configuration interaction (Cl), or PT2/CI [100] (for BaF and YbF), by the relativistic coupled cluster (RCC) method [101, 102] (for TIF, PbO, and HI+), and by the spin-orbit direct-CI method [103, 104, 105] (for PbO). In the ab initio calculations discussed here, the best accuracy of any current method has been attained for the hyperfine constants and P,T-odd parameters regarding the molecules containing heavy atoms. 

The second order perturbation theory term with two one-loop self-energy operators does not generate any logarithm squared contribution for the state with nonzero angular momentum since the respective nonrelativistic wave function vanishes at the origin. Only the two-loop vertex in Fig. 3.24 produces a logarithm squared term in this case. The respective perturbation potential determined by the second term in the low-momentum expansion of the two-loop Dirac form factor [111] has the form... 

In the region where k > a we may safely neglect the binding energies in the denominators of the second-order perturbation theory and thus simplify the integrand. After integration one obtains... 

Fig. 9. Comparison between the second-order perturbation theory and quasi-degenerate perturbation theory for the D value of ( 20) +.

The differences in the energy levels as they are produced by the second-order perturbation theory for the SH. 

Restricting our discussion to the subspace spanned by the terms 6Aig and 4 Tig, the matrix element of the spin-orbit operator have been evaluated by Weissbluth [59] using the formalism pioneered by Griffith [56] and ending at the eigenvalue problem of the 18 x 18 dimension (which is partly factored— Table 34). Then the second-order perturbation theory yields the energies of the lowest multiplets as... 

Then for a large energy gap Az = e(4A2g) - e(6Aig), and with the help of the substitution D = 2/5Az, the second-order perturbation theory yields the approximate roots of the form listed in Table 35. 

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... 

A rather more justified theoretical basis exists for the estimates of the corrections for the incompleteness of the angular parts of the orbital basis. It has been shown in [19,20] that the correction to the energy of an atom in the second order perturbation theory is... 

In addition to the molecular weight of the free polymer, there axe other variables, such as the nature of the solvent, particle size, temperature, and thickness of adsorbed layer which have a major influence on the amount of polymer required to cause destabilization in mixtures of sterically stabilized dispersions and free polymer in solution. Using the second-order perturbation theory and a simple model for the pair potential, phase diagrams relat mg the compositions of the disordered (dilute) and ordered (concentrated) phases to the concentration of the free polymer in solution have been presented which can be used for dilute as well as concentrated dispersions. Qualitative arguments show that, if the adsorbed and free polymer are chemically different, it is advisable to have a solvent which is good for the adsorbed polymer but is poor for the free polymer, for increased stability of such dispersions. Larger particles, higher temperatures, thinner steric layers and better solvents for the free polymer are shown to lead to decreased stability, i.e. require smaller amounts of free polymer for the onset of phase separation. These trends are in accordance with the experimental observations. 

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