Even and odd states

As a starting point let us see what happens, if we consider not only the ground state, but also the first excited state of the tunneling pairs. Both states are connected via thermal excitation with rates k j, and kjj. Since the transition between even and odd states is both symmetry and spin forbidden, these rates connect only states of the same symmetry and the spin is conserved in the transition. 

Even and odd states also occur for molecules, and the selection rule is also valid here. A further discussion of this point will be given in Section 48. 

Here, even and odd refer to the arithmetic sum over all the electrons and this selection mle is called the Laporte mle. An important result of this is that transitions are forbidden between states arising from the same configuration. For example, of the terms given in Equation (7.18) arising from the configuration of the carbon atom,... 

Of course, the distinction between reactive- and bound-state wave functions becomes blurred when one considers very long-lived reactive resonances, of the sort considered in Section IV.B, which contain Feynman paths that loop many times around the CL Such a resonance, which will have a very narrow energy width, will behave almost like a bound-state wave function when mapped onto the double space, since e will be almost equal to Fo - The effect of the GP boundary condition would be therefore simply to shift the energies and permitted nodal structures of the resonances, as in a bound-state function. For short-lived resonances, however, Te and To will differ, since they will describe the different decay dynamics produced by the even and odd n Feynman paths separating them will therefore reveal how this dynamics is changed by the GP. The same is true for resonances which are long lived, but which are trapped in a region of space that does not encircle the Cl, so that the decay dynamics involves just a few Feynman loops around the CL... 

Hence, the GP has a much milder effect on reactive systems than on bound-state systems. This difference has been overlooked in the past, but becomes apparent on noting that an encircling bound-state function contains Feynman paths that loop an infinite number of times around the Cl [28]. Consequently, the encirclement of a bound-state wave function is much stronger than that of a reactive wave function the bound wave function cannot be unwound from around the Cl, whereas the reactive wave function can. One consequence of this is that the separation into even- and odd-looping paths yields no information about the dynamics of a bound state system, in which these two contributions are necessarily equal and opposite [28]. 

Proceeding in the spirit above it seems reasonable to inquire why s is equal to the number of equivalent rotations, rather than to the total number of symmetry operations for the molecule of interest. Rotational partition functions of the diatomic molecule were discussed immediately above. It was pointed out that symmetry requirements mandate that homonuclear diatomics occupy rotational states with either even or odd values of the rotational quantum number J depending on the nuclear spin quantum number I. Heteronuclear diatomics populate both even and odd J states. Similar behaviors are expected for polyatomic molecules but the analysis of polyatomic rotational wave functions is far more complex than it is for diatomics. Moreover the spacing between polyatomic rotational energy levels is small compared to kT and classical analysis is appropriate. These factors appreciated there is little motivation to study the quantum rules applying to individual rotational states of polyatomic molecules. 

The energy is again given by E = cosh f. The equations for even and odd 91 states are easily derived, but we do not need them if we assume that (P states are bonding. 

When a polaron can move between two sites of equal energy, the state of the system (electron+phonons) will split into two states of even and odd parity, separated by an energy AE fc27t2/mpa2, which, when mp is large, may be quite small. The polarizability of this system will lead to the formation of a moment in a field F of order... 

It is easy to show by group-theoretical arguments that the mixing of the even and odd ionic states occurs only if the crystal field lacks a center of symmetry, that is, is not holohedral. The selection rules appropriate for ions in a static crystal field have been given by Hellwege (26). However,... 

Figure 3. In (a) the potential curve is unsymmetric with respect to the equilibrium position 0 of the nucleus. The crystal field in this case causes mixing of the even and odd parity states. In (b) there is symmetry with respect to the nucleus when it is at 0, but vibration carries the ion to the unsymmetric point P [from Ref. (25)].

The one-body theory of a decay applies strictiy to even-even a emitters only. The odd-nucleon a emitters, especially in ground-state transitions, decay at a slower rate than that suggested by the simple one-body formulation as applied to even-even nuclei. Consider the data in Figure 7.9 that shows the a-decay half-lives of the even-even and odd A uranium isotopes. The odd A nuclei have substantially longer half-lives than their even-even neighbors do. 

To observe a 7s — 9 transition requires that there be a 9p admixture in the 9 state. For odd this admixture is provided by the diamagnetic interaction alone, which couples states of and 2, as described in Chapter 9. For even states the diamagnetic coupling spreads the 9p state to all the odd 9( states and the motional Stark effect mixes states of even and odd (. Due to the random velocities of the He atoms, the motional Stark effect and the Doppler effect also broaden the transitions. Together these two effects produce asymmetric lines for the transitions to the odd 9t states, and double peaked lines for the transitions to even 9( states. The difference between the lineshapes of transitions to the even and odd 9i states comes from the fact that the motional Stark shift enters the transitions to the odd 9( states once, in the frequency shift. However, it enters the transitions to the even 9( states twice, once in the frequency shift and once in the transition matrix element. Although peculiar, the line shapes of the observed transitions can be analyzed well enough to determine the energies of the 9( states of >2 quite accurately.25... 

The displacement law states that the spark spectrum (radiation from the ionized atoms) of any element, resembles in structure the arc spectrum (radiation from the neutral atoms) of the preceding element. The modem alternation law applies to both arc and spark spectra it states that even and odd structures characterize the arc spectra of alternate chemical elements which occupy columns I to VIII of the periodic system, while conversely odd and even structures characterize the first spark spectra of the same elements. The experimental verification of these laws has come only recently with the discovery of regularities in the complex spectra which characterize many of the chemical elements. 

Calculated even- and odd-parity levels are compared with experiment in Becker, et al. [BEC84] An adequate description of states with J > 10 requires contributions from Models II and III. Technically, Model II and ill calculations require a shift in main frame computers and are not complete. 

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