Spin operators

An exceiient beginner s introduction to magnetic resonance, spin operators and their manipuiation to predict and anaiyze spectra. 

For a coupled spin system, the matrix of the Liouvillian must be calculated in the basis set for the spin system. Usually this is a simple product basis, often called product operators, since the vectors in Liouville space are spm operators. The matrix elements can be calculated in various ways. The Liouvillian is the conmuitator with the Hamiltonian, so matrix elements can be calculated from the commutation rules of spin operators. Alternatively, the angular momentum properties of Liouville space can be used. In either case, the chemical shift temis are easily calculated, but the coupling temis (since they are products of operators) are more complex. In section B2.4.2.7. the Liouville matrix for the single-quantum transitions for an AB spin system is presented. 

In the usual preparatioii-evohition-detection paradigm, neither the preparation nor the detection depend on the details of the Hamiltonian, except hi special cases. Starthig from equilibrium, a hard pulse gives a density matrix that is just proportional to F. The detector picks up only the unweighted sum of the spin operators,... 

The first theoretical handling of the weak R-T combined with the spin-orbit coupling was carried out by Pople [71]. It represents a generalization of the perturbative approaches by Renner and PL-H. The basis functions are assumed as products of (42) with the eigenfunctions of the spin operator conesponding to values E = 1/2. The spin-orbit contribution to the model Hamiltonian was taken in the phenomenological form (16). It was assumed that both interactions are small compared to the bending vibrational frequency and that both the... 

By applying Eq. (C.13) to the spin operators Si and using Eq. (C.22), one then gets after some matrix multiplications... 

The spin operator S is an irredueible tensor of rank one with the following transformational properties... 

The spin functions a and P which accompany each orbital in lsalsP2sa2sP have been eliminated by carrying out the spin integrations as discussed above. Because H contains no spin operators, this is straightforward and amounts to keeping integrals <( i I f I ( j > only if ( )i and ( )j are of the same spin and integrals... 

To consider the question in more detail, you need to consider spin eigenfunctions. If you have a Hamiltonian X and a many-electron spin operator A, then the wave function T for the system is ideally an eigenfunction of both operators ... 

In the limit of infinite atom separations, or if we switch off the Coulomb repui. sion between two electrons, all four wavefunctions have the same energy. But they correspond to different eigenvalues of the electron spin operator the first combination describes the singlet electronic ground state, and the other three combinations give an approximate description of the components of the first triplet excited state. 

For every electronic wavefunction that is an eigenfunction of the electron spin operator S, the one-electron density function always comprises an spin part... 

The resulting wavefunction is not necessarily an eigenfunction of the spin operator S. This may or may not matter, depending on the application. 

The spinning operation consists of highspeed rotating and suitably shaped tool that creates frictional heat that will permit the stud to conform to the configuration in the tool. Pressure exerted on the tool and... 

The Pauli spin operators, S, encountered in Margenau and Murphy, p. 402, are another example. 

The last term is the intrinsic change in the operator P, which is zero when P does not formally depend on t, as is the case for momentum, angular momentum, and spin operators. In deriving Eq. (7-74), no use was made of the adiabatic property of the w-functions. Therefore, it holds for all time-dependent bases. In the moving representation, w = , D = H by virtue of Eq. (7-49), and (7-74) reverts to Eq. (7-59). 

We then say that the particle has spin and the three components Sl constitute the (pseudovector) spin operator. Note that by virtue of Eq. (9-55) the spin variables are not expressible in terms of the variables q and p. Since the angular momentum variables J are also the infinitesimal generators of rotations we deduce that... 

Hence U commutes with both position and momentum operators, and must, therefore, depend only on the spin operators. If s is a spin operator then since 8 is similar to an angular momentum operator... 

The spin operators may be taken to be the Pauli spin matrices.7... 

Spin operators, taken as Pauli spin matrices, 730... 

The interaction of electron 1 on radical A with nucleus j (spin operator Ij) and of electron 2 on radical B with nucleus k (spin operator I ) is given by equation (20)... 

The existing SCF procedures are of two types in restricted methods, the MO s, except for the hipest (singly) occupied MO, are filled by two electrons with antiparallel spin, while in unrestricted methods, the variation procedure is performed with individual spin orbitals. In the latter, a total wave function is not an eigenvalue of the spin operator S, which is disadvantageous in many applications because of a necessary annihilation of higher multiplets by the projection operator. Since in practical applications the unrestricted methods have not proved to be remarkably superior, we shall call our attention in this review mainly to the restricted methods. 

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