By V. Y. Veldre
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Extra info for Atomic Collisions: The Theory of Electron-Atom Collision
6. R. Pauncz, Int. J. Quantum Chem. ll, 369 (1977). 7. J. H van Vleck and A. Sherman, Rev. Mod. Phys. 7,167 (1935). 8. P. O. LOwdin and O. Goscinski, Int. J. Quantum Chem. 38, 533 (1970). 9. J. E. Graebenstetter, T. J. Tseng, and F. Grein, Int. J. Quantum Chem. 10, 143 (1976). 1. Serber-Type Construction of Spin Eigenfunctions In the description of the electronic structure of atoms and molecules, the notion of the electron pair plays an important role. Pairs of electrons associated with inner shells, lone pairs, or chemical bonds occur as basic building blocks in the formation of a many-electron wave function.
Going step-by-step backwards we can trace the complete genealogy of the spin function. Later we shall find useful an alternative notation, (6) closely related to the graphical scheme. Here we start from the origin and to each segment we Sec. 7 Construction from One-Electron Spin Eigenfunctions 21 assign the number 1 if it goes upward, or the number 2 if it goes downward. The resulting sequence of numbers will be called a branching-diagram symbol (Bi): Each bik is either 1 (addition) or 2 (subtraction).
This concludes the proof of the theorem. As an example, for N = 5, S =!. M = !. 5. So far we have expressed the S2 eigenfunctions in terms of the primitive spin functions. This is an orthogonal transformation among the two sets of 2N functions. We can invert this orthogonal transformation by taking its transposed matrix, and we can express the primitive spin functions in terms of the S2 eigenfunctions. (JL - v) are given in the following form: OI(JL, v) = cllX(N, S, S; 1) + ... (}2(JL, v) = C2IX(N, S, S; 1) + c22X(N, S, S; 2) + ...