Selection rules many-electron atoms

We now consider many-electron atoms. We will assume Russell-Saunders coupling, so that an atomic state can be characterized by total electronic orbital and spin angular-momentum quantum numbers L and S, and total electronic angular-momentum quantum numbers J and Mj. (See Section 1.17.) The electric-dipole selection rules for L, J, and Mj can be shown to be (Bethe and Jackiw, p. 224)... 

Thus a transition between two given electronic states shows many bands, each such band corresponding to a different pair of initial and final vibrational states under high resolution, each band shows many closely spaced lines, each such line corresponding to a different pair of initial and final rotational states. (The electronic spectra of molecules are called band spectra, whereas the electronic spectra of atoms are called line spectra.) Consider the selection rules for electronic transitions. The electric di-... 

For spectroscopic transitions in many-electron atoms, the selection rules under LS-coupling are similar to those for the one-electron atom ... 

In this chapter, we review electronic structure in hydrogenlike atoms and develop the pertinent selection rules for spectroscopic transitions. The theory of spin-orbit coupling is introduced, and the electronic structure and spectroscopy of many-electron atoms is greated. These discussions enable us to explain details of the spectra in Fig. 2.2. Finally, we deal with atomic perturbations in static external magnetic fields, which lead to the normal and anomalous Zeeman effects. The latter furnishes a useful tool for the assignment of atomic spectral lines. 

MANY-ELECTRON ATOMS SELECTION RULES AND SPECTRA... 

Strong El transitions in many-electron atoms are observed only when one electron changes its orbital quantum numbers for this electron, the selection rule A/= +1 must be obeyed (cf. our discussion following Eq. 2.12). To appreciate this, we recall that spatial wavefunctions in many-electron atoms may be expressed (Section 2.3) in terms of products (1, 2,..., p) = i(i)(j>2 2) 4>pip) of one-electron orbitals 0i(l), 2(2),. .., (j>p p Since the pertinent electric dipole operator is p= — eSr, the El transition moment from electronic state 2,. .., p) to state 2,. .., p) = i(l)( 2(2). .. 0p(p)... 

In many-electron atoms, states can have different S quantum numbers. For light elements, where the spin-orbit coupling is weak, the selection rules for atomic spectra are... 

Approximate selection rules. The last three selection rules in Table 7.1 are derived using explicit forms for the atomic wavefunctions. Since these can only be.approximate solutions of Schrodinger s equation in the case of a many-electron atom, rules 4-6 in turn apply only approximately in many cases. Thus rule 4 is obeyed only if each of the states involved can be correctly described by means of a single configuration of electrons. Similarly rules 5 and 6 may be expected to apply only in Russell-Saunders coupling where the spin-orbit interaction is negligible compared with the electrostatic forces between the electrons. 

Atoms have complete spherical synnnetry, and the angidar momentum states can be considered as different synnnetry classes of that spherical symmetry. The nuclear framework of a molecule has a much lower synnnetry. Synnnetry operations for the molecule are transfonnations such as rotations about an axis, reflection in a plane, or inversion tlnough a point at the centre of the molecule, which leave the molecule in an equivalent configuration. Every molecule has one such operation, the identity operation, which just leaves the molecule alone. Many molecules have one or more additional operations. The set of operations for a molecule fonn a mathematical group, and the methods of group theory provide a way to classify electronic and vibrational states according to whatever symmetry does exist. That classification leads to selection rules for transitions between those states. A complete discussion of the methods is beyond the scope of this chapter, but we will consider a few illustrative examples. Additional details will also be found in section A 1.4 on molecular symmetry. 

When you consider the selection rules, which are not particularly restrictive (see Section 7.1.6), governing transitions between these states arising from each configuration, it is not surprising that the electronic spectrum of an atom such as zirconium consists of very many lines. (Remember that the Laporte rule of Equation (7.33) forbids transitions between states arising from the same configuration.)... 

The problem of N bound electrons interacting under the Coulomb attraction of a single nucleus is the basis of the extensive field of atomic spectroscopy. For many years experimental information about the bound eigenstates of an atom or ion was obtained mainly from the photons emitted after random excitations by collisions in a gas. Energy-level differences are measured very accurately. We also have experimental data for the transition rates (oscillator strengths) of the photons from many transitions. Photon spectroscopy has the advantage that the photon interacts relatively weakly with the atom so that the emission mechanism is described very accurately by first-order perturbation theory. One disadvantage is that the accessibility of states to observation is restricted by the dipole selection rule. 

A number of characteristic X-rays can be emitted from an atom if a number of inner shell electrons have been knocked out by high energy particles. The individual characteristic X-rays are marked Ka, Kf5... as mentioned in Chapter 2. It seems that there are many possible ways in which outer shell electrons can fill inner shell vacancies however, the possibilities are limited and such electron transitions in an atom are controlled by the selection rules. 

The majority of atoms in a flame are in the ground state (Eq), therefore, many electronic transitions originate from this state. Such transitions are limited in number, since by quantum-mechanical selection rules some energy levels are not directly accessible from the ground state. 

Since P decay occurs throughout the periodic table, it offers a wider field of study than a decay, which is largely confined to heavy elements. The term P decay encompasses both P (negatron) and P (positron) emission (Curie and JoKot 1934) as well as the process of orbital electron capture discovered in 1937 (Alvarez 1937) which, like P emission, leads to a one-unit decrease in atomic number. In each of the three processes a neutrino is emitted (in P emission it is an antineutrino). The systematic of P decay has been worked out and selection rules for P transitions in terms of spin and parity changes have been established. Many if not most P transitions lead to excited states of the product nuclei. 

For dielectric relaxation to be observed there has to be a change in the polarizability of the media under the influence of an applied field. This selection rule implies that dispersion will only be detected in polar materials. However, in certain cases contributions to the polarizability have been detected in non polar materials and are ascribed to collisional polarization effects. If two non polar molecules collide there is the possibility that distortion of the electron density nd atomic positions may result in the formation of a tiansient dipole or multipole. The induced polarization can be destroyed by further collision with other non activated molecules. The lifetime of these collisionally activated molecules may be many times the collision frequency and it then becomes possible to observe the reorientational motion of the induced dipoles, which act as though they were permanent dipoles. 

An interesting example of a many-electron spectrum is that of He, in which the shown low-energy transitions involve orbital jumps of one of the two electrons. For this case our one-electron atomic selection rules (A/ = 1, Ay = 0, 1) hold for the electron involved in the transition. The He electronic spectrum resembles... 

The octet rule applies to most molecules (and ions) with Period 2 central atoms, but not every one, and not to many with central atoms from Period 3 and higher. Three important exceptions occur in molecules with (1) electron-deficient atoms, (2) odd-electron atoms, and (3) atoms with expanded valence shells. In this discussion, you ll also see that formal charge has limitations for selecting the best resonance form. 

Since nuiny processses demonstrate substantial quantum effects of tunneling, wave packet break-up and interference, and, obviously, discrete energy spectra, symmetry induced selection rules, etc., it is clearly desirable to develop meAods by which more complex dynamical problems can be solved quantum mechanically both accurately and efficiently. There is a reciprocity between the number of particles which can be treated quantum mechanically and die number of states of impcxtance. Thus the ground states of many electron systems can be determined as can the bound state (and continuum) dynamics of diatomic molecules. Our focus in this manuscript will be on nuclear dynamics of few particle systems which are not restricted to small amplitude motion. This can encompass vibrational states and isomerizations of triatomic molecules, photodissociation and exchange reactions of triatomic systems, some atom-surface collisions, etc. 

For a molecule that has a symmetric arrangement of the atomic nuclei, we label all of its electronic and vibrational states according to that symmetry. The labels may help us determine how many electrons are bonding or antibonding, how much electron density lies along the internuclear axis, and what excited states are accessible by electric dipole selection rules. 

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