Parities, mass and electrons

Mass and Electron Parities, Closed-Shell Ions and Open-Shell Ions 6.6.1 Electron Parity... 

McLafferty rearrangement through a six-atom ring intermediate. Figure 6.11 shows another example and details the rules linking the mass and electron parities. 

The nitrogen rule requires that the molecular mass is always even when the number of nitrogen atoms is even or zero. This results from the fact that nitrogen has a different mass parity and valence electrons parity mass 14 u, five peripheral electrons. Both of these parities are identical in the case of any other atom. It should be noted that this holds only if we consider the mass of the predominant isotope. Thus, the chemical mass of bromine is 80 u, an even number, but its predominant isotope is that of mass 79 u, an odd mass. In the same way, isotopically labelled compounds do not always obey this rule. 

An odd number of nitrogen atoms brings about an odd molecular mass in daltons such as is defined in mass spectrometry NH3 17, CH3NH2 31, and so on. Thus, in the case of an odd number of nitrogens, the earlier rule must be inverted for the ion, the mass parity is the same as the electron parity. 

Notice that odd n (i.e. even parity V (r)) and odd j (i.e. even centre-of-mass wavefunction) implies and even n (i.e. odd parity V (r)) and odd j implies as the reflection operator reflects both the centre-of-mass and relative coordinates, and hence exchanges the electron and hole. 

However, other non-electromagnetic effects also influence the electronic structure. Examples are isotope shifts or electro-weak interactions, that lead to parity nonconservation. The latter result from the motion of a nucleus (in an atom relative to the center of mass) and yield a dependence of energy levels on the nuclear mass (and on the finite size of the nucleus, as we have already discussed in chapters 6 and 9). The energy levels are thus shifted, while the electromagnetic perturbations result in a splitting of these levels. 

Now consider fa. We set up the space-fixed and molecule-fixed coordinate systems with a common origin on the internuclear axis, midway between the nuclei, as in Fig. 4.11. (Previously in this chapter, we put the origin at the center of mass, but the difference is of no consequence.) The electronic wave function depends on the electronic spatial and spin coordinates and parametrically on R. The parity operator does not affect spin coordinates, and we shall only be considering transformations of spatial coordinates in this section. 

Let us now consider the second mechanism, namely, the appearance of the electronic contribution gj due to the interaction with the paramagnetic electronic states. In particular, the singlet terms 1II and of one parity (either u u or g - g) interact because of the non-zero matrix elements of the electron-rotation operator [—l/(2/iro)](J+L- + J L+), where // is the reduced mass, ro is the internuclear distance (in atomic units) and the cyclic components of the vectors are defined in the same way as in [267] = Lx iLy, = Jx iJy connecting the x and y... 

Symbol, mass number nuclear spin I, nuclear parity -Half-life (s = seconds, ps = microseconds, d = days, m = months, or y = years) -Decay mode (emission), and energy (MeV( if to ground state)), separated by / if several modes if in parentheses, mode produces a shortlived daughter, or occurs <10% Emissions a = alpha = 2He4++ P"= electron ... 

There is an important connection between particle-hole symmetry and the relative parity of the particle-hole pair. Consider a basis state created by the removal of an electron from a valence band Wannier orbital on the repeat unit at i — r/2 and the creation of an electron on a conduction band Wannier orbital at R + r/2. This is illustrated in Fig. 6.1. This particle-hole pair has a centre-of-mass coordinate, R, and a relative coordinate, r ... 

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