According to the principles of classical dynamics the funda mental frequencies of libration are given by Correspondence Principle.—It will be noticed that there is a close similarity between (43) and (43a), and, in fact, when J1, J2, etc., are individually very big compared with Al2, etc., the two formulae (43) and (43a), the former giving frequencies of radiation calculated from the classical theory and the latter those given by the quantum theory, become identical. If the quantum theory is to represent the facts this asymptotic ap proach to the classical theory, when large energies are involved, is essential, as has already been pointed out. Now one of the striking facts about spectra is that while the frequency (or wave number) of any spectral line is equal to the difference of two spectral terms (combination principle), the converse is not al ways true. That is to say the difference between a pair of spectral terms does not in every case represent an observed frequency (or wave number). Using the language of the quantum theory we may say that the atom is not at liberty to make any transition it likes from one stationary state to another.
Certain restrictions have to be placed on the values of the integers, Si, s2, etc., in (43a). These restrictions have found their expression in the past in the form of selection rules (Rubinowicz, Sommerfeld and others), mostly deduced from the principle of conservation of angular momentum. They were superseded by the comprehensive correspondence principle of Bohr. According to this principle the integers s2, etc., in (43a) must have the same values as those of the classical formula (43). In simple harmonic motion (the type of motion of the pendulum described above when the amplitude is small), the energy associated with a stationary state is equal to nhw, if w is the frequency of oscillation, and according to Bohr's postulate where p is the frequency of the radiation emitted when a transi tion from a stationary state corresponding to a higher to one of a lower energy level takes place, i.e., where s is an integer. Now according to the classical theory s is restricted to the value 1 and the correspondence principle requires that we must adopt this value in applying the quantum theory. When applied to our simple Bohr atom the correspondence prin ciple requires that (the change of the azimuthal quantum num ber or Bohr's k) during a transition must be + i or — 1 (unless the atom is under the influence of an external field). If we were concerned only with this special type of atom and neglect, as we have done, relativistic refinements, this would not be of the slightest consequence, since the frequency of the emitted radia tion is determined by the change of and is not, there fore, affected (for a given change of n) by the changes in the individual integers.
If, however, we turn to a more complicated sort of atom, e.g., a neutral sodium atom, it becomes of great importance. Accord ing to Bohr this atom consists of a nucleus having a charge of 11 units of positive electricity, with so electrons travelling in orbits arranged relatively closely round it and a more isolated electron travelling in an orbit which we may regard as roughly elliptic. A sodium atom therefore resembles the simple hydrogen atom. In deed, if we were to imagine the so inner electrons to be very close indeed to the nucleus, the outer electron would behave in almost exactly the same way as the electron in the hydrogen atom, since we have an atom consisting of a small but massive inner structure, with a net charge of one positive unit. In consequence, however, of the fact that the inner orbits do extend out from the nucleus, the outer electron may, if or k is small, have its peri helion (or perinucleon) actually within the boundary of the inner system of orbits. If k is large the orbit of the outer electron may be quite outside this boundary. In these circumstances the energy of the outer electron is not determined by the sum but by the individual values of and k. For a given n the energy is least (numerically greatest, if we re member the sign of E) when k = i and increases progressively with k. We shall have a set of spectral terms for which k= 1, another set for which k= 2 and so on. Bohr supposes k= I, 2, 3, 4 • . . for S, P, D, F terms respectively. The circumstance that k changes by + I or — 1, and not by zero or any other number, corresponds to the spectroscopic fact that we have combinations of S and P, P and D, or D and F terms, but not for instance combinations of S and D, or S and F terms.