Spin eigenstates

Hartke B and Carter E A 1992 Spin eigenstate-dependent Hartree-Fock molecular dynamics Chem. Phys. Lett. 189 358... 

This linear combination is obtained by projection of one of the determinants on the spin eigenstates with S even or odd ... 

Some uni-configurational wave functions consist of only one determinant. This is called a single-determinant wave function. A single-determinant can be a spin-eigenstate wave function only if the eigenfunctions possess the values of... 

It is evident that a 2-RDM that corresponds to a Hamiltonian eigenstate also corresponds to a pure-spin state. However, when one is working with an approximated RDM, it is important that this RDM should correspond to a spin eigenstate. 

The application of quantum-mechanical methods to the prediction of electronic structure has yielded much detailed information about atomic and molecular properties.13 Particularly in the past few years, the availability of high-speed computers with large storage capacities has made it possible to examine both atomic and molecular systems using an ab initio variational approach wherein no empirical parameters are employed.14 Variational calculations for molecules employ a Hamiltonian based on the nonrelativistic electrostatic nuclei-electron interaction and a wave function formed by antisymmetrizing a suitable many-electron function of spatial and spin coordinates. For most applications it is also necessary that the wave function represent a particular spin eigenstate and that it have appropriate geometric symmetry. 

Electron-spin-based phenomena are attracting attention for the possibility of logic devices based on spin rather than on the flow of charge for possible quantum information applications. Retention of a spin lifetime in the material is important for these considerations. The conduction-band states at the zone center are approximately spin eigenstates, and therefore, they have relatively long spin lifetimes because they are made up of s-type orbitals. This is in contrast to valence-band states that are made up of p-type orbitals for which the spin-orbit interaction mixes the orbital part of the p state with the spin part of the electron wave... 

We find the effect of 0 on the spin eigenstates (3.79) by using their representation as vectors in spin space. 

For future reference, we summarize the action of the spin operators on the two spin eigenstates ... 

Figure 2.23. Odd alternant hydrocarbon radicals a) Schematic representation of the frontier orbital energy levels and of the various configurations that are obtained by single excitations from the ground configuration o. (It should be remembered that spin eigenstates cannot be represented correctly in these diagrams.) b) Energies of these configurations and effect of first-order configuration interaction.

It is a fundamental fact of quantum mechanics, that a spin-independent Hamiltonian will have pure spin eigenstates. For approximate wave functions that do not fulfill this criterion, e.g. those obtained with various unrestricted methods, the expectation value of the square of the total spin angular momentum operator, (5 ), has been used as a measure of the degree of spin contamination. is obviously a two-electron operator and the evaluation of its expectation value thus requires knowledge of the two-electron density matrix. 

Taking advantage of the symmetry properties of the Sanibel coefficients, this linear combination (6) was seen to contain only spin eigenstates with even S quantum number, i.e, singlets, quintuplets, etc.. ... 

The integrations in (40) obviously yield unity, since (41) contains two normalized spin density factors in fact (40) is a precise analogue of (25) and (28) in the hydrogen molecule example. In any spin eigenstate the coupling density (41) will be nonzero but the integrations over all space will evidently yield a simple scalar product coupling between the subsystems A and B, however complicated they may be. This result is quite independent of the distance between A and B. 

A third contribution of importance in hyperfine spectroscopy is the nuclear quadrupole moment for nuclei with I > 1/2, e.g., H, N, and They lead to an additional line broadening, arising from the shift of the nuclear spin eigenstates by the quadrupolar coupling ... 

The situation is somewhat different in the solid state [50], where stable radical pairs are usually formed by electron transfer over a distance of more than 1 nm and are then localized rather than diffusing. In this case it becomes significant that the To state is a coherent superposition of spin eigenstates of the pair the same is true for the S state [51]. Evolution of this zero-quantum coherence can be observed indirectly, and information on the coupling between the two spins can be obtained from out-of-phase ESEEM experiments [52]. In photosynthetic reaction centers, this technique was used to measure distances between the radicals up to approximately 3 nm [53]. In a cascade of electron transfers, out-of-phase ESEEM spectra depend on the couplings in both the primary and secondary pair and on the time constant for the secondary electron transfer [54]. 

These six CSFs can be combined to form spin eigenstates three states with local triplet coupling on both magnetic centers and three more with at least one magnetic center in a locally excited (non-Hund) state. 

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