Eigenfunction spin functions

For axially symmetric complexes, the parameter E is zero, and the spin functions S,ms) are eigenfunctions of the spin Hamiltonian ... 

A. Express the wavefunction (eigenfunction) as the sum of orthogonal, normalized wavefunctions typically the latter would be spin functions denoted by pj... 

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] ... 

We make a small digression and note that the spin-degeneracy problem we have alluded to before is evident in Eq. (5.102). It will be observed that / = 1,..., /x in the index of e s pnp these functions are linearly independent since the efj are. There are, thus, fi linearly independent spin eigenfunctions of eigenvalue S(S + 1). Each of these has a full complement of Ms values, of course. In view of Eq. (5.40) the number of spin functions increases rapidly with the number of electrons. Ultimately, however, the dynamics of a system governs if many or few of these are important. 

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. 

All electrons are characterized by a spin quantum number. The electron spin function is an eigenfunction of the operator and has only two eigenvalues, h/2 the spin eigenfunctions... 

Since our spin functions are eigenfunctions of S2, we can drop the last term in Eq. (70), because it contributes a constant term to all energy levels and hence drops out when we compute energy differences. Many workers prefer to add a term — 5(5+ 1) to the spin Hamiltonian to get a Hamiltonian which transforms readily under a coordinate rotation. We thus have forJt ... 

If we use spin functions quantized with respect to the z axis, these functions can become eigenfunctions of the Zeeman part of the Hamiltonian when... 

In magnetic resonance we are often confronted with the problem of obtaining a solution to a Hamiltonian which has only spin operators. To find the allowed energies and eigenfunctions, we generally start out with a convenient set of spin functions < , which represent the spin system but are not eigenfunctions of the Hamiltonian,. The eigenfunction ifi can, however, be constructed from a linear sum of the s ... 

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. 

Mj — + i and - respectively. (The same symbols were used previously to designate electronic spin functions with ms = the context will indicate whether a and ft mean electronic or nuclear spin functions.) Since the operator (8.42) is Hermitian, its eigenfunctions (8.43) form a complete set of well-behaved functions for the problem of two nuclei with spin Moreover, since there are only a finite number of functions in this complete set, the secular determinant in (2.77) is of finite order and easy to deal with. The complete set is then... 

Each spin orbital is a product of a space function fa and a spin function a. or ft. In the closed-shell case the space function or molecular orbitals each appear twice, combined first with the a. spin function and then with the y spin function. For open-shell cases two approaches are possible. In the restricted Hartree-Fock (RHF) approach, as many electrons as possible are placed in molecular orbitals in the same fashion as in the closed-shell case and the remainder are associated with different molecular orbitals. We thus have both doubly occupied and singly occupied orbitals. The alternative approach, the unrestricted Hartree-Fock (UHF) method, uses different sets of molecular orbitals to combine with a and ft spin functions. The UHF function gives a better description of the wavefunction but is not an eigenfunction of the spin operator S.2 The three cases are illustrated by the examples below. 

The significance of this set of spatial functions will become clear shortly. The spin functions t are also orthonormal, and are eigenfunctions of S2 and Sz with the eigenvalues S and M, respectively, ... 

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