Function coupled spin

The 0. symbol is used to designate the ith combination of spin functions coupling N electrons to give an overall spin of S, and there are/ number of ways of doing this. The... 

The ordinary unrestricted Hartree-Fock (UHF) function is not written like either of these. It is not a pure spin state (doublet) as are these functions. The spin coupled VB (SCVB) function is lower in energy than the UHF in the same basis. 

The important point here is that the ordering of the orbitals in the space part is essentially imposed by the need to couple electrons pairwise. Methods for computing matrix elements of the Hamiltonian over these space functions have been developed by Ruedenberg and co-workers [20], who termed the functions SAAPs spin-adapted antisymmetrized products. 

Their theory, based on the classical Bloch equations, (31) describes the exchange of non-coupled spin systems in terms of their magnetizations. An equivalent description of the phenomena of dynamic NMR has been given by Anderson and by Kubo in terms of a stochastic model of exchange. (32, 33) In the latter approach, the spectrum of a spin system is identified with the Fourier transform of the so-called relaxation function. 

A spin ladder is an array of coupled spin chains. The horizontal chains are called the legs, the vertical ones, rungs. In the case of spin one-half antiferromagnet spin-ladders, these systems show a. remarkable behaviour in function of the number of leg there is a gap in the excitation spectrum of even-leg ladders and, on the contrary, no gap in the excitation spectrum of odd-leg ladders. In terms of correlation lengths, this means that there is short (long) -range spin correlation in even (odd) -legladder (see [24] for a review). 

It is also possible to derive this result by incorporating the condition (2.5) below from the beginning by using the factorization of a two-electron function into a symmetrical spatial function and an antisymmetrical spin function, see equ. (1.16).) The expression in the braces indicates that the two electrons in the final state have opposite spins, i.e., the photoprocess reaches a singlet final state. This can be easily understood, because in LS-coupling spin-orbit effects are absent, and the photon operator does not act on the spin. Therefore, the selection rule... 

The computational model capable of yielding accurate spin-spin coupling constants is the multiconfigurational self-consistent field (MCSCF) model, and before the advent of density functional theory, spin-spin coupling constants in small systems were often... 

In a coupled spin system, the condition that vectors having distinctive frequencies and those can be assigned to each nucleus fails. Further, the coupling inherently means that the frequency of the observed nucleus depends not only on its own environment but also on the state of the neighbouring nuclei as well. Thus the product functions are introduced to take into consideration the perturbation effect of the neighbouring nuclei... 

The vector model of a single spin is the vector representation of the complex number in the individual density matrix of a single nucleus. This density matrix consists of only one complex number thus there is only one vector in the model. In the case of more than one nuclei, the density matrix is larger, there are more single quantum coherences and more vectors belong to one spin set in the model. Moreover, in case of a strongly coupled spin system, the density matrix has different numerical form for different basis sets of the vector space of the simulation (the basis can be one of the ) and

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The calculation method presented here also provides a possible extension of the single spin vector model. This extension is performed in two steps first to weakly coupled spin systems, then to strongly coupled ones. In the first case, the introduction of the well-known product basis functions and their coherences is sufficient while in the latter one the solution is not so trivial. The crucial point is the interpretation of the linear transformation between the basis functions and the eigenfunctions (or coherences) during the detection and exchange processes. These two processes can be described by the population changes of single quantum... 

The simplest molecular constant to understand is the nuclear spin dipolar interaction constant, to, which is found to be, within experimental error, that calculated from the classical interaction of two magnetic moments, i.e. gFgHfrwO /47t oc2)(7 3> =o. On the other hand, calculation of scalar electron-coupled spin-spin interaction constants is notoriously difficult, requiring a molecular electronic wave function of the highest quality. The best available calculation for HF quoted by Muenter and Klemperer is one due to O Reilly [96]. 

When the ionised shell r is one of several open shells in the molecule (as may happen with configurations of e and t% electrons in cubic ligand fields), more complicated formulae must be applied. Suppose that the ground state has two open shells r and s in which the electrons are separately coupled to form functions with spin and symmetry (S3A3) and (54 4 4) respectively. Thus we write this state ... 

This can give rise to difficulties in tightly coupled spin systems when there may be virtually complete mixing of some pairs of wave functions. (40) Most of the problems can be avoided, however, by an appropriate new choice of wave functions (41) and this approach should always be... 

A number of theoretical transfer functions have been reported for specific experiments. However, analytical expressions were derived only for the simplest Hartmann-Hahn experiments. For heteronuclear Hartmann-Hahn transfer based on two CW spin-lock fields on resonance, Maudsley et al. (1977) derived magnetization-transfer functions for two coupled spins 1/2 for matched and mismatched rf fields [see Eq. (30)]. In IS, I2S, and I S systems, all coherence transfer functions were derived for on-resonance irradiation including mismatched rf fields. More general magnetization-transfer functions for off-resonance irradiation and Hartmann-Hahn mismatch were derived for Ij S systems with N < 6 (Muller and Ernst, 1979 Chingas et al., 1981 Levitt et al., 1986). Analytical expressions of heteronuclear Hartmann-Hahn transfer functions under the average Hamiltonian, created by the WALTZ-16, DIPSI-2, and MLEV-16 sequences (see Section XI), have been presented by Ernst et al. (1991) for on-resonant irradiation with matched rf fields. Numerical simulations of heteronuclear polarization-transfer functions for the WALTZ-16 and WALTZ-17 sequence have also been reported for various frequency offsets (Ernst et al., 1991). 

Homonuclear Hartmann-Hahn transfer functions for off-resonant CW irradiation have been derived for two coupled spins 1 /2 (Bazzo and Boyd, 1987 Bothner-By and Shukla, 1988 Elbayed and Canet, 1990) and for the AX 2 spin system (Chandrakumar et al., 1990). In the multitilted frame, Hartmann-Hahn transfer functions under mismatched effective fields are related to polarization- and coherence-transfer functions in strongly coupled spin systems (Kay and McClung, 1988 McClung and Nakashima, 1988 Nakai and McDowell, 1993). Numerical simulations of homonuclear... 

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