Spin eigenvector

The Dirac operators a, . ..04, work on the basis vectors ei,. ..04 just as the Pauli spin operators work on the spin eigenvectors a, / . The only other property to be noted is that the Dirac spin-orbitals have two large components and two small components , their ratio being of the order (2moc)" ... 

The MRs of the spin operators are readily obtained using their known properties (as given above) and the MRs of the spin eigenvectors. 

The orthonormality of the spherical harmonics and of the spin eigenvectors psnts (with s = 1/2 and OTs = 1/2) ensures orthonormality of the Pauli spinors,... 

Note that contributions from the secondary sector of the eigenvectors, Uf, do not appear in the residues, for the summation index, r, pertains to spin-orbitals only. 

Pauli spin matrices, geometric phase theory, eigenvector evolution, 14-17... 

The 1-RDM can be diagonalized by a unitary transformation of the spin orbitals (jf), (x) with the eigenvectors being the natural spin orbitals (NSOs) and the eigenvalues , representing the ONs of the latter. 

The left and right eigenvectors are equal only when the populations at all spin sites are the same. As a result of the invariance of modal matrices to matrix functions, the dynamic matrix L can be written as... 

Within a variational treatment, the relative contributions of the spin-and space-symmetry adapted CSFs are determined by solving a secular problem for the eigenvalues (E)) and eigenvectors (CQ of the matrix representation H of the full many-electron Hamiltonian H within this CSF basis ... 

Because of the spin-spin coupling term, (8.41) is not separable into the sum of Hamiltonians for the individual nuclei, and the corresponding Schrodinger equation is not separable. To deal with the problem, we shall use the method of expanding the unknown wave functions in terms of a known complete set of functions. The eigenvalues and eigenfunctions (eigenvectors) are obtained as the solutions of (2.68) [or (2.77)] and (2.67). 

However, here ara) afi) f5ct) fifi) are not eigenstates of the energy. Nevertheless, they are still orthogonal and can always be expanded in eigenvectors of the energy. For example, in the present case of spin 1/2 we have ... 

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