Expectation values of spin

Energy is not the only property that is so determined by the electron density fragment pc. Since the (non-degenerate, ground state) local electron density pc(r) in any standard domain c fully determines the complete density pit), which in turn fully determines the molecular wavefunction P (up to a phase factor), all molecular properties P which can be expressed as expectation values of spin-free operators defined by the ground state wavefunction P are also determined by the local electron density pc(r) in the standard domain c. Consequently, any such property P is also a unique functional of the local electron density pc(r) within the standard domain c ... 

The most dramatic difference between GVB and UHF is seen if the expectation value of spin along the z axis is computed using Eqs (61) and (62). For the GVB wavefunction the expectation value is zero everywhere whereas for UHF it is... 

Exercise 3.38 Show that expectation values of spin-independent sums... 

The difference compared to equation B 1.13.2 or equation B 1.13.3 is the occurrence of the expectation value of the operator (the two-spin order), characterized by its own decay rate pjg and coupled to the one-spin longitudinal operators by the tenus 8j aud 5. We shall come back to the physical origin of these tenus below. 

In NMR, the magnetization in the xy plane is detected, so it is the expectation value of the operator that is measured. This is just the unweighted sum of all the operators for the individual spins . It may be a fimction of several time variables (multi-dimensional experiments), including the tune during the acquisition. 

As a check for the presence of spin contamination, most ah initio programs will print out the expectation value of the total spin <(A >. If there is no spin contamination, this should equal. v(.v + 1), where s equals times the number of unpaired electrons. One rule of thumb, which was derived from experience with... 

The amount of spin contamination is given by the expectation value of die operator, (S ). The theoretical value for a pure spin state is S S + 1), i.e. 0 for a singlet (Sz = 0), 0.75 for a doublet (S = 1/2), 2.00 for a triplet (S = 1) etc. A UHF singlet wave function will contain some amounts of triplet, quintet etc. states, increasing the (S ) value from its theoretical value of zero for a pure spin state. Similarly, a UHF doublet wave function will contain some amounts of quartet, sextet etc. states. Usually the contribution from the next higher spin state from the desired is... 

The motion of the neutron polarization P(t) - the quantum mechanical expectancy value of neutron spin - is described by the Bloch equation... 

All the spin manipulations are done by a tailored magnetic field that the neutron spin feels during its passage through the instrument. The motion of the expectation value of the neutron spin (s) obeys the Bloch equation ... 

In an alternative formulation of the Redfield theory, one expresses the density operator by expansion in a suitable operator basis set and formulates the equation of motion directly in terms of the expectation values of the operators (18,20,50). Consider a system of two nuclear spins with the spin quantum number of 1/2,1, and N, interacting with each other through the scalar J-coupling and dipolar interaction. In an isotropic liquid, the former interaction gives rise to J-split doublets, while the dipolar interaction acts as a relaxation mechanism. For the discussion of such a system, the appropriate sixteen-dimensional basis set can for example consist of the unit operator, E, the operators corresponding to the Cartesian components of the two spins, Ix, ly, Iz, Nx, Ny, Nz and the products of the components of I and the components of N (49). These sixteen operators span the Liouville space for our two-spin system. If we concentrate on the longitudinal relaxation (the relaxation connected to the distribution of populations), the Redfield theory predicts the relaxation to follow a set of three coupled differential equations ... 

Hyperfine tensors are given in parts B and C of Table II. Although only the total hyperfine interaction is determined directly from the procedure outlined above, we have found it useful to decompose the total into parts in the following approximate fashion a Fermi term is defined as the contribution from -orbitals (which is equivalent to the usual Fermi operator as c -> < ) a spin-dipolar contribution is estimated as in non-relativistic theory from the computed expectation value of 3(S r)(I r)/r and the remainder is ascribed to the "spin-orbit" contribution, i.e. to that arising from unquenched orbital angular momentum. 

The dimensions of the spin spaces for the active electrons in Table 2, cf. Eq. (9)) are certainly not small. It proved difficult to find a spin basis in which very few of the coefficients were large and so we adopted instead a spin correlation scheme cf. Section 4.2). In the present work, we exploited the way in which expectation values of the two-electron spin operator evaluated over the total spin eigenfunction 4, depend on the coupling of the individual spins associated with orbitals ( )/ and j. Negative values indicate singlet character and positive values triplet character. Special cases of the expectation value are ... 

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