Two-electron spin functions

Each of the five Rumer spin eigenfunctions for a six-electron singlet represents a product of three singlet two-electron spin functions ... 

The situation here is completely analogous to that obtained in the restricted open HE theory (ROHF). The states are not eigenfunctions of S, except when all the open-shell electrons have parallel spins (A p = 0 or = 0). This result is a consequence of the expansion (22) used to obtain Eq. (27). Actually, the spin decomposition, Eq. (22), for does not conserve in general the total spin S. However, we can form appropriate linear combinations of two-electron spin functions cri Sj, 8 81,52) that are simultaneously eigenfunctions of and S, and achieve a correct spin decomposition of the 2-RDM [85] ... 

The spatial functions (1.249) and (1.250) must be multiplied by a spin function. For electrons, we have the following four possible two-electron spin functions, three symmetric and one antisymmetric ... 

The spin functions a and (3 (i.e., the components of a spin doublet) belong to the Ej/2 irrep of C2v- But what about singlet and triplet spin functions For this purpose, we look at the action of the symmetry operators on typical two-electron spin functions such as aa and ap. The results, displayed in Table 6, are easily verified. 

Table 6 Action of the Symmetry Operators of Civ on Two-Electron Spin Functions...

For two-electron systems (but not for three or more electrons), the wavefunction can be factored into an orbital function times a spin function. The two-electron spin function... 

The wave function [ ffl(l) f (2) - t ( I ) fa(2)]/V2 is antisymmetric with respect to an interchange of electrons 1 and 2 and needs to be multiplied with one of the three symmetric two-electron spin functions to describe a triplet state T. The other three functions are symmetric, need to be multiplied with the antisymmetric two-electron spin function, and describe three singlet states, So, S, and Sj. The amusing isomorphism of the two-electron ordinary and spin functional space has been discussed elsewhere (Michl, 1991, 1992). 

Now consider the excited states of helium. We found the lowest emted state to have the zeroth-order spatial wave function 2 [ls(l)25(2) - 2s(l)ls(2)] [Eq. (9.105)]. Since this spatial function is antisynunetric, we must multiply it by a symmetric spin function. We can use any one of the three symmetric two-electron spin functions, so instead of the nondegenerate level previously found, we have a triply degenerate level with the three zeroth-order wave functions... 

The functions (13.86) and (13.88) are symmetric with respect to exchange. They therefore go with the antisymmetric two-electron spin factor (11.60), which has 5 = O.Thus (13.86) and (13.88) are the spatial factors in the wave functions for the two states of the doubly degenerate A term. The antisymmetric functions (13.87) and (13.89) must go with the symmetric two-electron spin functions (11.57), (11.58), and (11.59), giving the six states of the A term. These states all have the same energy (if we neglect spin-orbit interaction). 

Now it will be shown that the two-electron spin function a(l)fi(2) — a(2)fi(l) ensures the singlet state. First, let us construct the square of the total spin of the two electrons ... 

The spin functions ap and pa may also form a linear combination ap - pa, which is orthogonal to the previous one, ap + pa. It is obvious that the eigenvalue of is equal to zero for the former spin function. Changing all + signs to - signs in Equation 1.73, we find that ap - Pa is an eigenfunction to with the eigenvalue 0 (equal to S(S + 1) if S = 0). Hence, the two-electron spin function for S = 0 is... 

We now include spin in the He zeroth-order ground-state wave function. The function 15(1)15(2) is symmetric with respect to exchange. The overall electronic wave function including spin must be antisymmetric. Hence we must multiply the symmetric space function 15 (1) 15 (2) by an antisymmetric spin function. There is only one antisymmetric two-electron spin function, so the ground-state zeroth-order wave function for the helium atom including spin is... 

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