It is shown that the exclusion principle merely requires the electrons to be divided into two distinct groups, then, according to our assumption of the distinguishability of electrons, its mathematical representation can be the spatial double antisymmehization or the spatial-spin antisymmetrization; in the latter case an auxiliary spin double symmetrization is also required in order to distinguish the two types of electrons. The second requirement is used to recognize the fact that, although all electrons can be assumed to be indistinguishable under certain condition, their natural characteristic of two distinct groups must still be preserved. It is further shown that the double antisymmetrization (or double symmetrization) determines the spin quantum number M , whereas the Liiwdin spin projector for nonorthogonal eigenfunctions and the Wigner matric basis for orthogonal eigenfunctions will decide the spin quantum number S. Both the spatial double antisymmether and the spin double symmetrizer will reject all the unnecessary eigenfunctions and project out only one unique set of linearly independent eigenfunctions.
AbstractsIn this paper we show that with the equivalent transformation PT = ( -l)p(P")-l the spin function dependent methods such as Slater's method without group theory or Goddard's method with group theory differ only in different antisymmetric requirements from the present Waller-Hartree spin function free method. There exists a one-to-one correspondence between Slater's determinantal wave function and the Waller-Hartree double determinantal wave function. Explicit expressions for the S2 operator, Lowdin's spin projector, matric basis and several different forms of spin-projected functions are given for the Waller-Hartree formalism. The results are compared with other methods including those of Slater, Matsen, Gallup, Goddard and Segal. The differences are quite significant. New spin operators are worked out using creation-destruction operators. A knowledge of group theory is not required in this Waller-Hartree method. We have also shown that the Waller-Hartree method is more convenient than Slater's method with spin
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