Generation of Many Electron Spin Functions

Generation of Many Electron Spin Functions case 4 Triplet incorporation (black dashed lines) S ... 

The problem is with the Slater determinant. When we build the matrix, we are sometimes forced to make an arbitrary choice which spin function to associate with which spatial function. In the example above, the Is electron is always the a electron. But in the same way that the Is electron has an equal chance of being electron 1 or electron 2, the Is electron also has an equal chance of being the a or jS spin electron. The Slater determinant above is an artificial mixture of the singlet and triplet states, and does not correctly predict the energy of the excited state. However, the Slater determinants for the ground state and the aa and /3/3 spin excited states are accurate. Most of the time, this method for generating antisymmetric many-electron wavefunctions works well. 

How are the additional determinants beyond the HF constructed With N electrons and M basis functions, solution of the Roothaan-Hall equations for the RHF case will yield N/2 occupied MOs and M — N/2 unoccupied (virtual) MOs. Except for a minimum basis, there will always be more virtual than occupied MOs. A Slater detemfinant is determined by N/2 spatial MOs multiplied by two spin functions to yield N spinorbitals. By replacing MOs which are occupied in the HF determinant by MOs which are unoccupied, a whole series of determinants may be generated. These can be denoted according to how many occupied HF MOs have been replaced by unoccupied MOs, i.e. Slater determinants which are singly, doubly, triply, quadruply etc. excited relative to the HF determinant, up to a maximum of N excited electrons. These... 

In a many-electron system, one must combine the spin functions of the individual electrons to generate eigenfunctions of the total Sz =Li Sz(i) ( expressions for Sx = j Sx(i) and Sy = j Sy(i) also follow from the fact that the total angular momentum of a collection of particles is the sum of the angular momenta, component-by-component, of the individual angular momenta) and total S2 operators because only these operators commute with the full Hamiltonian, H, and with the permutation operators Pjj. No longer are the individual S2(i) and Sz(i) good quantum numbers these operators do not commute with Pjj. 

We found the contraction of the f-wave functions to an atomic like character. This has an important consequense, because we see the onset of a cooperative effect, the orbital magnetic moment to spin conduction band electrons interaction and other correlations will generate an effective mass for the f-electrons which, in turn, will, by contraction of the f-wave function, enhance the atomic like character and then enhance the full process, hence the process will not be smooth or show linear behaviour the process will tend to present a sudden change of the f-electrons from a band character to an almost atomic like behaviour. This many-body effect is responsible for further scattering between the conduction electrons and the rare earth or actinide f electron states,... 

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