Rotations in spin space

If we use single determinant (38) in Eq. (34) we get the wrong answer because it has equal population of spin up and spin down, just as does 1,0>. This inconsistency is never a practical problem, at least for simple systems. The lack of invariance under rotations in spin space does, however, distinguish Xa-like models from HF (Fukutome, 1981), or to be more descriptive what Sykja and Calais (1982) call generalized Hartree-Fock (GHF). Any first-order density matrix can be decomposed into four mathematically equivalent pieces... 

Bmte-force fast MAS is not the only means by which line narrowing can be achieved in solid-state NMR. A particularly ingenious alternative approach, first presented over 30 years ago by Waugh and co-workers, involves the removal of the dipolar broadening by specific multiple-pulse techniques, where radiofrequency pulses achieve rotations in spin space [51, 52]. These rotations can complement the effect of the physical rotation of the sample combined rotation and multiple-pulse spectroscopy (CRAMPS) [53—55] yields well-resolved H spectra [56]. 

Many molecular hamiltonians commute with the total spin angular momentum operator, a fact that leads to the consideration of transformation properties of electron field operators under rotations in spin space. Basis functions, natural for such studies, are... 

A rotation in spin space about an axis, the magnitude and direction of which can be given by a vector 0 = (Q. Qy 0 ), produces transformed electron field... 

Actually, the components of the bosonized SB spin operator act as generators of rotations in spin space, i.e., Si(pi ) satisfies the spin algebra. Since and commute with the SB t-J Hamiltonian, the constraints in Eq. 5 and Eq. 6 can be ensured by introducing the time-independent Lagrange multipliers and = consequence, the... 

Spatial symmetry may be utilized in a similar way. According to the theorems in Appendix 3 (p. 541), the expansion of any wavefunction W of given symmetry species contains only symmetry functions of the same species. The situation is precisely analogous to that which arises in the case of spin for the eigenvalues (S, M) are in fact the labels that define the different basis functions (Af = S, S - 1,... -S) of a (2S + 1)-dimensional representation Dj of the group of rotations in spin space, and therefore correspond to the species labels (or, i) used in Appendix 3. Functions of pure symmetry species, with respect to spatial symmetry operations, may again be built up by linear combination of the basic determinants for molecules, this is easily accomplished by the methods of Appendix 3, and adequate illustrations appear in later sections. 

Much of the beauty of high-resolution molecular spectroscopy arises from the patterns formed by the fine and hyperfine structure associated with a given transition. All of this structure involves angular momentum in some sense or other and its interpretation depends heavily on the proper description of such motion. Angular momentum theory is very powerful and general. It applies equally to rotations in spin or vibrational coordinate space as to rotations in ordinary three-dimensional space. 

The presented technique, finite pulse r.f. driven recoupling (fpRFDR), restores homonuclear dipolar interactions based on constructive usage of finite pulse-width effects in a phase- and symmetry-cycled 7c-pulse train in which a rotor-synchronous n pulse is applied every rotation period. The restored effective dipolar interaction has the form of a ZQ dipolar Hamiltonian for static solids, whose symmetry in spin space is different from that obtained by conventional r.f. driven recoupling (RFDR) techniques. It has been demonstrated that the efficiency of recoupling by fpRFDR is not strongly dependent on chemical shift differences or resonance offsets in contrast to previous recoupling... 

The product operator can clearly describe the spin behavior, and is the base of many multinuclear multidimensional NMR pulse sequences.14,16,20 The pulse sequence is a technique to visualize the invisible phenomena by rotation in complex space. This technique has much potential. 

In the presence of spin-orbit interaction the transformations of k and the spin rotations are no longer independent. The spins are frozen in the lattice and the operations of the point gronp amonnt to simultaneons rotations in k space and spin space ... 

In spin space, this requires a general rotation in the space of the functions 0[x], 0[y], and 0[z]. This can be accomplished by a rotation in the space of one-electron spin functions and this is equivalent to choosing a particular set of molecular axes in real... 

Note that NAOs for a and P spin have identical spatial forms (as required to ensure rotational invariance against different coordinate choices in spin space). The populations of spin-up and spin-down electrons in NAO i of atom A, therefore, have well-defined meaning in the NPA framework. 

Spin-adapted rotations The spin-free nonrelativistic Hamiltonian commutes with the total and projected spin operators. We are therefore usually interested only in wave functions with well-defined spin quantum numbers. Such functions may be generated from spin tensor operators that are totally symmetric in spin space. For optimizations, we need consider only singlet opeiatois since these are the only ones that conserve the spin of the wave function. Spin poturbations, on the other hand, may mix spin eigenstates and require the inclusion also of triplet rotations. 

In contrast to the spin-orbital rotation operator in (3.3.4), we have in (3.3.19) and (3.3.21) introduced irreducible tensor components in spin space and at the same time made a clear distinction between real and imaginary rotational parameters. This procedure makes it easy to select the components of ic that are necessary for a particular task. For example, in optimizations of real orbitals, we may neglect all complex rotations and rotations that mix spin states as well as the complex phase factors. The orbital-rotation operator then reduces to... 

For a free noninteracting spinning particle, invariance with respect to translations and rotations in three dimensional space, i.e., invariance under the inhomogeneous euclidean group, requires that the momenta pl and the total angular momenta J1 obey the following commutation rules... 

---