Antisymmetric eigenfunctions of the spin

In this section we investigate the connections between the symmetric groups and spin eigenfunctions. We have briefly outlined properties of spin operators in Section 4.1. The reader may wish to review the material there. 

One of the important properties of all of the spin operators is that they are symmetric. The total vector spin operator is a sum of the vector operators for individual electrons 

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Antisymmetric eigenfunctions of the spin one obtains for the matrix system,... 

Methods based upon Slater determinantal functions (SDF). When we take this approach, we are, in effect, applying the antisymmetrization requirement first. Only if the orbitals are all doubly occupied among the spin orbitals is the SDF automatically, at the outset, an eigenfunction of the total spin. In all other cases further manipulations are necessary to obtain an eigenfunction of the spin, and these are written as sums of SDFs. 

Symmetric group methods. When using these we, in effect, first construct n-particle (spin only) eigenfunctions of the spin. From these we determine the functions of spatial orbitals that must be multiplied by the spin eigenfunctions in order for the overall function to be antisymmetric. It may be noted that this is precisely what is done in almost all treatments of two electron problems. Generating spatial functions... 

A is the antisymmetrizer, ensuring that the wavefunction changes sign on interchange of two electrons (and thus the wavefunction obeys the Pauli exclusion principle), and 0(S) is a spin projection operator " that ensures that the wavefunction remains an eigenfunction of the spin-squared operator,... 

The ground-state Hartree-Fock wave function o is the Slater determinant mi 2 m of spin-orbitals. This Slater determinant is an antisymmetrized product of the spin-orbitals [for example, see Eq. (10.36)] and, when expanded, is the sum of n terms, where each term involves a different permutation of the electrqns among the spin-orbitals. Each term in the expansion of s an eigenfunction of the MP H for example, for a four-electron system, application of H to a typical term in the I>o expansion gives... 

In dealing with systems containing only two electrons we have not been troubled with the exclusion principle, but have accepted both symmetric and antisymmetric positional eigenfunctions for by multiplying by a spin eigenfunction of the proper symmetry character an antisymmetric total eigenfunction can always be obtained. In the case of two hydrogen atoms there are three... 

The wave funetion obtained eorresponds to the Unrestricted Hartree-Fock scheme and beeomes equivalent to the RHF ease if the orbitals (t>a and (()p are the same. In this UHF form, the UHF wave funetion obeys the Pauli prineiple but is not an eigenfunction of the total spin operator and is thus a mixture of different spin multiplicities. In the present two-eleetron case, an alternative form of the wave funetion which has the same total energy, which is a pure singlet state, but whieh is no longer antisymmetric as required by thePauli principle, is ... 

Write down the nine nuclear spin functions of D2. Show that the three antisymmetric spin functions are eigenfunctions of the operator for the square of the magnitude of the total nuclear spin with the eigenvalue 2ft2. Find the corresponding eigenvalues for the symmetric spin functions. 

Since the reference system s consists of noninteracting particles, the results of Section 6.2 and the Pauli principle show that the ground-state wave function of the reference system is the antisymmetrized product (Slater determinant) of the lowest-energy Kohn-Sham spin-orbitals up of the reference system, where the spatial part 0P(r,) of each spin-orbital is an eigenfunction of the one-electron operator AP that is. 

The eigenfunctions of the one-electron problem of (1.13) and (1.14) are spin orbitals which can be used to construct the antisymmetric eigenfunctions of i non ... 

In discussing the helium atom (Section 1.2) the antisymmetry requirement on the electronic wavefunction was easily satisfied for with only two electrons the function would be written as a product of space and spin factors, one of which had to be antisymmetric, the other symmetric, lliis is possible even for an exact eigenfunction of the Hamiltonian (1.2.1), as well as for an orbital product. The construction of an antisymmetric many-electron function is less easy. We have seen in Section 1.2 that for a general permutation (involving both space and spin variables) an antisymmetric function has the property... 

In the development of the Slater method (Section 3.1) it was noted that the Pauli principle in the form (1.2.27) could always be satisfied by constructing the electronic wavefunction from determinants (i.e. antisymmetrized products) of spin-orbitals. In an earlier section, however, it was shown that for a two-electron system the antisymmetry principle could also be satisfied by writing the wavefunction as a product of individually symmetric or antisymmetric factors—one for spatial variables and the other for spin variables. Since, in the usual first approximation the Hamiltonian does not contain spin variables, it is natural to enquire whether a corresponding exact N-electron wavefunction might be written as a space-spin product in which the spatial factor is an exact eigenfunction of the spinless Hamiltonian (1.2.1). To investigate this possibility, we need a few basic ideas from group theory (Appendix 3). 

If we want the bond eigenfunction corresponding to a bond between a and 6, but none between c and d, we proceed in an analogous manner. Only 3) 4> 5 can be used, as they are the only functions in which a and 6 have opposite spins. This function must be antisymmetric with respect to exchange of the spins on a and h but symmetric with respect to the exchange of the spins on c and d. We readily find the appropriate bond eigenfunction to be... 

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