Johnson•, which is based on a combination of Heisenberg's matrix theory and Jordan's operator calculus methods. The elements in this transformation matrix then represent the coefficients in the linear expansion of the L-S wavefunctions in terms of Slater functions. This will be achieved if these variables are co-credentally transformed with the type form.
The two functions which are then obtained may be regarded as simple functions and L and S coupled to obtain J, by a repetition of the same process. The resulting function t~J" is thus expressed as a linear combination of the original one-electron functions. These coefficients were selected and fixed as the correct L-S functions with which to proceed in the evaluation of the matrix elements by general integration.
We now turn to the calculation, by the direct 'spinor' method, of the interconfiguration interactions that we need to calculate the transition probabilities for the two electron atoms in intermediate coupling. Therefore, we are now concerned with the non-dlagonal matrix elements of the electrostatic interaction (e2/r~2). The angular momentum of the electrons in configuration I will be denoted by 11, 12 and in configuration II by A.,~ ~.
After introducing the substitution (17) it is possible to split .1l.z: in terms of some invariants that can be constructed from the ).~ spinor components. The integral (16) is somewhat simplified when we introduce the operator (18) since many of the terms resulting after multiplying the integrand turn out to be zero. This is easily seen by an examination of the typical term of which the integral is.
If we now introduce this substitution into the symbolic integrals of J~, the corresponding substitution is in terms of spinor components. We now proceed to expand (18) in the usual way using the binomial theorem. In the case of operators (22) and (23), it is necessary to make one more change before entering them into the integro Because the meaning of the spinor components is ~~in nc_~.
In this expression, the Qs, the numerical coefficients of the radial integrals, are defined by (26), and for a non-vanishing matrix element to exist, 'l:: and Once the standard curves are calculated, it is only a short procedure to evaluate almost any required radial integral of the type considered here. Therefore, the sum of all final states M to obtain the total intensity of the transition J+l in J, and-. The factors CJ(i)0, which result from the normalization integral {39), can be separated into the product of two factors, UJ(i)0 = f(J,L,6) CL(i)~ where f is independent of values of l~,. This is, of course, neglecting, as far as energy is concerned, the spin-orbit term in the Hamiltonian.). However, absolute intensities and multiple relative intensities may be related. is significantly affected, especially if two interacting configurations happen to lie close to each other and thus create small values for some of the denominators occurring in the expression. Hij is independent of J, and hence the fact that expressions arising in applications of group theory in quantum mechanics involving two vector quantities and their sum (eg J = L + S) can be written as the product of two terms one of which contains all the dependence on the sum {J) and is completely independent of the subcomponents of the other two (i.e. 1. If now the diagonal elements of the spin-orbit term in the Hamiltonian are considered the usual system of levels of energy in terms of configurations, L1, S, and J, are obtained, except that energy perturbations due to interconfigurational interactions of the electrostatic term are further included. When considering the "interconfiguration" interaction caused by the off-diagonal matrix elements of the spin-orbit term in the Ham- .. iltonian, together with that binterconfig- .. h '. part of the congratulation of the diagonal elements mentioned above, two new features are introduced. First, if it is assumed that 'these off-diagonal elements are negligible either in absolute magnitude or compared to similar electrostatic interaction terms, then the same form obtains for the perturbed wave functions{37). Therefore, they are multiple distinct central energy levels, and as such '· are still independent of J; so the theorem ·is still valid. Since the matrix elements are diagonal in J but not in L and si3'1~ as were the corresponding electrostatic interaction elements, the additional terms that are introduced include multiples of different types than the original. Before proceeding further in trying to obtain an estimate of the magnitude of these anomalies, we must consider some facts about the spin-orbit interaction between the configurations. However, in the case of intensity anomalies, it is seen that the electrostatic term alone, even when the energies of the involved configurations overlap, cannot have an effect on some of the relative intensities. In the end it was clear that only the introduction of the interconfiguration spin-orbit interaction introduced anomalies, and therefore, since it was the only term present, it could not be neglected. Nevertheless, the above restriction on it, to among the members of the same "configuration" series, makes all its terms small. When the interaction of the central field with the electrons is taken into account, the diagonal and off-diagonal elements within the configuration of the spin-orbit term are introduced. Then the diagonal elements of the electrostatic interaction are taken into account, so that so far each configuration is independent. However, in this case, the E0s that appear in the denominators of the expansion coefficients, Rij, are no longer independent of J, but are just the usual energy levels specified by J, L, and S. Houston, and for thanks for many useful discussions and suggestions during the progress of the work.TRANSITION INTENSITIES
