Higher-order fine structure terms

Thus far, we have investigated the various contributions to the effective Hamiltonian for a diatomic molecule in a particular electronic state which arise from the spin-orbit and rotational kinetic energy terms treated up to second order in degenerate perturbation theory. Higher-order effects of such mixing will also contribute and we now consider some of their characteristics. 

Perturbation form Parameter Order of perturbation Rank of spin operator 

In order to appreciate this point more clearly, we confine our attention to the contributions to 3Qff produced by perturbations from the spin-orbit coupling 3Q0 and the electronic Coriolis mixing 30-ot- If we represent an off-diagonal matrix element of the former by (L S) and the latter by (N L), we can describe some examples of these higher order terms, as shown in table 7.1. The third-rank terms appear only in states of quartet or higher multiplicity and the fourth-rank terms in states of quintet (or higher) multiplicity. With the important exception of transition metal compounds, the vast majority of electronic states encountered in practice have triplet multiplicity or lower. 

This expression for the higher-order spin-rotation interaction probably seems at best daunting and at worse cumbersome. However the great merit of the formulation in equation (7.165) is that it is very straightforward to work out its matrix elements, as we shall see later. 

The experimental evidence for such a contribution to the spin-rotation interaction in the effective Hamiltonian was somewhat elusive in the early days although there are now well documented cases of its involvement, for example for CH in its 4E state [21]. Equation (7.166) suggests one reason why this parameter is not as important in practice as might be expected. The last factor on the right-hand side of (7.166) is just the difference of the rotational constant operators for the upper and lower states. This causes a considerable degree of cancellation in a typical situation because the B value is not expected to vary markedly between the electronic states. 

The first of these reasons can be easily appreciated by reference to equation (7.43) again. It can be seen that the nth order terms have the general form / Eq — 

The parameter ys, first introduced by Brown and Milton [20], is related to C R) in equation (7.165) by 

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DCB is correct to second order in the fine-structure constant a, and is expected to be highly accurate for all neutral and weakly-ionized atoms [8]. Higher quantum electrodynamic (QED) terms are required for strongly-ionized species these are outside the scope of this chapter. A comprehensive discussion of higher QED effects and other aspects of relativistic atomic physics may be found in the proceedings of the 1988 Santa Barbara program [9]. 

Higher-order spin terms are then added when the spacing of the fine structure is found to be a function of the magnetic field. In what follows we shall characterize each material by the value for D and E, and indicate by D that higher order terms were required. The analysis of spectra for the quantitative values of small terms is difficult, particularly when some expected lines have not been observed, and the associated errors hard to determine, so it is best to consult the original papers for terms beyond D and E. 

In Chapter 1, the first order contributions to the annihilation rates from the dominant modes of decay of the S-states of both ortho- and para-positronium (for arbitrary principal quantum number nPs) were given as equations (1.5) and (1.6). These contributions are included in the following equations for the rates for the two ground states, which also contain terms of higher order in the fine structure constant, a ... 

In the visible region of the spectrum water vapour is transparent and all further absorptions of interest occur in the infrared or at even longer wavelengths. These are associated with transitions between vibrational levels of the molecule, the fundamental modes for which are shown in fig. 1.4, and have a fine structure dependent upon the rotational levels involved. Since each of the three normal modes has a direct effect upon the dipole moment of the molecule, they aU lead to absorption bands. Because the interatomic potentials have appreciable anharmonic components from terms of cubic or higher order in the displacements, the relation between... 

This chapter begins with a classical treatment of vibrational motion, because most of the important concepts that are specific to vibrations in polyatomics carry over naturally from the classical to the quantum mechanical description. In molecules with harmonic potential energy functions, vibrational motion occurs in normal modes that are mutually uncoupled. Coupling between vibrational modes inevitably occurs in the presence of anharmonic potentials (potentials exhibiting cubic and/or higher order terms in the nuclear coordinates). In molecules with sufficient symmetry, the use of group theory simplifies the procedure of obtaining the normal mode frequencies and coordinates. We obtain El selection rules for vibrational transitions in polyatomics, and consider the rotational fine structure of vibrational bands. We finally treat breakdown of the normal mode approximation in real molecules, and discuss the local mode formulation of vibrational motion in polyatomics. 

These are of several kinds. First, there are higher-order spin-orbit interactions which appear in a more exact treatment, involving the mutual magnetic interaction of pairs of electrons spin-spin (the spin of one electron with the spin of the other) spin-other orbit (the spin of one with the orbital motion of the other) orbit-orbit. While the magnitude of the effects is small, including them as additional parameters with appropriate coefficients leads to an appreciable improvement in fitting the fine-structure levels, especially for the low terms which are more or less isolated. 

The velocity fields obtained by RANS or LES also illustrate the basic differences between the two approaches (Fig. 7.7). While the field of axial velocity obtained with RANS is very smooth (Fig. 7.7a), the LES field exhibits much more small structures in terms of physics, the LES captures more turbulent activity in terms of numerical resolution, the grid required for RANS can be very coarse because gradients are small. On the other hand, for LES, the grid must be fine and the numerical method non-dissipative in order to capture the small motions evidenced in Fig. 7.7b. Considering also that RANS codes compute only one state (the converged flow) while LES codes must resolve the flow in time, the cost of a t3rpical LES computation is often 100 to 1000 times higher than a RANS computation. However, despite its cost, LES is well adapted for many combustion studies ... 

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