Theory of atomic spectra

If a chemical equilibrium is established between two different chemical species, then Beer s law ordinarily will not be obeyed, since the concentration of the absorbing species will not usually be directly proportional to the apparent total concentration. For example, consider an acid-base indicator, HX, and assume that HX is the absorbing species. We have the equilibrium, 

The total concentration of HX and X is c = Chx + Cx-. In the simplest case = Cx- = c — Chx then K = (c — Chx) / hx5 which shows that Chx is not simply proportional to c. Therefore Beer s law will not be obeyed. The equilibrium constant can be determined by measuring the absorbance as a function of the concentration. This assumes that only one species absorbs significantly at the wavelength in question. 

The simplest spectra are those obtained from excited atoms. Since all atoms except the hydrogen atom have more than one electron, we need a quantum-mechanical description for multielectron atoms. 

The Schrodinger equation for the hydrogen atom can be solved exactly as we did in Chapter 22. If we attempt to solve the Schrodinger equation for the helium atom, we must deal with the mechanics of three bodies (the nucleus and two electrons), which is not solvable in closed form either in classical mechanics or in quantum mechanics. Therefore we are forced to use approximate methods. 

Suppose that the nucleus has a charge + Ze and is separated from electron 1 by a distance and from electron 2 by r2 the distance between electrons 1 and 2 is r 2 (Fig. 24.7). Since the nucleus is very massive compared to the electrons, we will regard the nucleus as being fixed at the center of mass of the system. The Hamiltonian for the system can then be written as though the system consisted of only the two electrons that are moving in the field of the nucleus and each other. If we write the energy as multiples of E, the hartree, and distances as multiples of ao, the Hamiltonian becomes 

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The Theory of Atomic Spectra, E. U. Condon and G. H. Shortley, Cambridge Univ. Press, Cambridge, England (1963)- Condon and Shortley. 

During the first twenty years or so of this century, an incredibly detailed understanding of atomic line spectra was built up with the application of the, then new, quantum theory. Indeed, the development of quantum theory came about in part by the need to understand these spectral properties. We shall have to review some basic features of the theory of atomic spectra for our present purposes, but we shall leave it for the moment. 

Figure 1.1 shows that the stability sequence revealed by chemical reactions and chemical synthesis corresponds to thermodynamic stabilities. An explanation requires a theory that will explain both. To get it we apply the theory of atomic spectra [9]. The energy of the 4f electrons in an ion with the configuration [Xe]4P, F(4P), can be written [nU+E Q-p(4f )] where U, a negative quantity, is the energy of each 4f electron in the field of the positively charged xenon core, and Frep(4P) represents the repulsion between the n 4f electrons. In Table 1.1, rep(4P) is expressed as a function of the Racah parameters and E. The... 

Table 1.1 The inter-electronic repulsion energies, rep(f"), and the ionization energies, l[f), off configurations according to the theory of atomic spectra.

Condon EU, Shortley GH (1951) The theory of atomic spectra. Cambridge University Press, Cambridge, UK... 

Condon, E. U., Shortley, G. H. Theory of atomic spectra. Chaps. 7 and 8. Cambridge University Press 1935. 

LI. Sobelman, Introduction to the Theory of Atomic Spectra, Pergamon Press, Oxford (1972). 

Condon, E. V. and Shortley, G. H., The Theory of Atomic Spectra, Cambridge University Press London, New York (1957). 

A brief review is given here of the spectroscopic vector model of an atom or ion. In crystal-field theory, the wave function of the isolated ion is taken as the unperturbed state, and the perturbing effect of the electric and magnetic fields is computed. Thus crystal field theory uses the language, nomenclature, and methods employed in the theory of atomic spectra. A complete discussion of these methods can be found in books by Condon and Shortley (5) and by Griffith (/). 

Moreover, for atoms with open shells, the difficulties in calculating the angular parts of the PT expansion grow very rapidly with the increase in the order of expansion terms. Even the methods of their calculations are not developed sufficiently, unlike the usual mathematical apparatus of the theory of atomic spectra. Therefore, in order to successfully apply the PT to complex atoms, further development of the many-body theory... 

B. Judd. Problems of the theory of atomic spectra. In B. Judd and B. Wybourne. Theory of Complex Atomic Spectra, Mir, Moscow, 1973 (in Russian). 

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