Rotational and spin symmetries

What happens with the interaction between the rotational and spin symmetries once the system is characterized as being defined by at least different spinors Wigner and von Neumann [10] combined both types of symmetries with the permutation aspect [11]. They intuitively reached the idea using atomic spectroscopy that the H operator has to be constructed by two terms H, resulting from the spatial motion of the single electron only (and the electromagnetic interaction with the field of the atomic core), and (//2), which has to visualize the electron spin. For simplicity, we can consider the eigenvalue problem of the spinless wave function i r without the second term as... 

Finally, an example in which all three types of coupling (radial, rotational and spin-orbit) are operative is the charge-transfer reaction Ar ( P)-l- H- Ar( S)-l-H. All three have been calculated as a function of intemuclear separation. The entrance channel correlates with B 2, a Il, A II and b Z states, while the outgoing channel possesses only Z symmetry. (The angular coupling involves the term with a maximum around 0.4 a.u. the radial coupling is only about half of this and finally the spin-orbit operator couples the output channel X Z with the upper state of the ingoing partners. 

The molecules are assumed to be trapped with a separation Az ry = (2nf /y) /, where the dipole-dipole interaction is d /fy = y/2. In this regime the rotation of the molecules is strongly coupled to the spin and the excited states are described by Hunds case (c) states in analogy to the dipole-dipole coupled excited electronic states of two atoms with fine structure. The ground states are essentially spin-independent. In the subspace of one rotational quantum (Hi -b H2 = 1), there are 24 eigenstates of Hm which are linear superpositions of two electron spin states and properly symmetrized rotational states of the two molecules. There are several symmetries that reduce Hin to block diagonal form. First, Hm, conserves the quantum number Y = M / + Ms where Mff = + Mh/2 Ms = Ms + Ms2 are the total rotational and spin projections... 

For all point, axial rotation, and full rotation group symmetries, this observation holds if the orbitals are equivalent, certain space-spin symmetry combinations will vanish due to antisymmetry if the orbitals are not equivalent, all space-spin symmetry combinations consistent with the content of the direct product analysis are possible. In either case, one must proceed through the construction of determinental wavefunctions as outlined above. 

In spin relaxation theory (see, e.g., Zweers and Brom[1977]) this quantity is equal to the correlation time of two-level Zeeman system (r,). The states A and E have total spins of protons f and 2, respectively. The diagram of Zeeman splitting of the lowest tunneling AE octet n = 0 is shown in fig. 51. Since the spin wavefunction belongs to the same symmetry group as that of the hindered rotation, the spin and rotational states are fully correlated, and the transitions observed in the NMR spectra Am = + 1 and Am = 2 include, aside from the Zeeman frequencies, sidebands shifted by A. The special technique of dipole-dipole driven low-field NMR in the time and frequency domain [Weitenkamp et al. 1983 Clough et al. 1985] has allowed one to detect these sidebands directly. 

When spin-orbit coupling is introduced the symmetry states in the double group CJ are found from the direct products of the orbital and spin components. Linear combinations of the C"V eigenfunctions are then taken which transform correctly in C when spin is explicitly included, and the space-spin combinations are formed according to Ballhausen (39) so as to be diagonal under the rotation operation Cf. For an odd-electron system the Kramers doublets transform as e ( /2)a, n =1, 3, 5,... whilst for even electron systems the degenerate levels transform as e na, n = 1, 2, 3,. For d1 systems the first term in H naturally vanishes and the orbital functions are at once invested with spin to construct the C functions. 

Vector spaces which occur in physical applications are often direct products of smaller vector spaces that correspond to different degrees of freedom of the physical system (e.g. translations and rotations of a rigid body, or orbital and spin motion of a particle such as an electron). The characterization of such a situation depends on the relationship between the representations of a symmetry group realized on the product space and those defined on the component spaces. 

The calculated state energies, the transition moments, and the symmetry classification are given in Table 3. The symmetry species of the triplet functions is obtained by taking the direct product of irreducible representation of the space and the spin functions Fx, Fy, Fz, which transform as the rotations Rx, Ry, and Rz-... 

This phenomenon of antiparamagnetic paramagnetic terms clearly needs a name and is called here the Cornwell effect (ideally the Cornwell-Santry effect). Positive contributions to op (which may or may not be positive overall) are expected in heteronuclear diatomics if they have a IT state this excludes, e.g., HF, InF, and TIF. In homonuclear diatomics, the IT -> a excitation is symmetry-forbidden. The possibility has been mentioned for XeF (34), although, from the chemical shift and calculated values of aa, the resultant Op ( F) is negative in XeFg and KrFj (cf. Fig. 7). Another candidate is FC DH, from the evidence of the fluorine chemical shift and spin-rotation interaction (96). According to this interpretation there should be a substantial upheld shift of the... 

Since we have axial symmetry, we can take the axis to be in the plane of H and the z axis, thus making Hy — 0. To diagonalize the energy matrix for the Zeeman energy, we shall rotate our spin-coordinate system such that the new z axis makes an angle a> with the z axis of the molecule. Spin operators in this new coordinate system are related to those in the molecular system by the equations... 

In connection to control in dynamics I would like to take here a general point of view in terms of symmetries (see Scheme 1) We would start with control of some symmetries in an initial state and follow their time dependence. This can be used as a test of fundamental symmetries, such as parity, P, time reversal symmetry, T, CP, and CPT, or else we can use the procedure to discover and analyze certain approximate symmetries of the molecular dynamics such as nuclear spin symmetry species [2], or certain structural vibrational, rotational symmetries [3]. 

Finally, spin-orbit interaction has often been considered as the cause of states of mixed permutational symmetry. There are, however, a variety of other spin interactions which may accomplish such mixing electron spin-electron spin, electron spin-nuclear spin, spin-other-orbit, and spin rotation interactions. That other such spin interactions may enhance spin-forbidden processes in organic molecules is frequently ignored, though they may be of importance.66,136... 

The dynamic state is defined by the values of certain observables associated with orbilal and spin motions of the electrons and with vibration and rotation of [lie nuclei, and also by symmetry properties of the corresponding stationary-state wave functions. Except when heavy nuclei ate present, the total electron spin angular momentum of a molecule is separately conserved with magnitude Sh. and molecular slates are classified as singlet, doublet, triplet., . according to the value of the multiplicity (25 + I). This is shown by a prefix superscript lo the term symbol, as in atoms. 

Symmetry dictates that the representations of the direct product of the factors in the integral (3 /T Hso 1 l/s2) transform under the group operations according to the totally symmetric representation, Aj. The spin part of the Hso spin-orbit operator converts triplet spin to singlet spin wavefunctions and singlet functions to triplet wavefunctions. As such, the spin function does not have a bearing on the symmetry properties of Hso- Rather, the control is embedded in the orbital part. The components of the orbital angular momentum, (Lx, Ly, and Lz) of Hso have symmetry properties of rotations about the x, y, and z symmetry axes, Rx, Ry, and Rz. Thus, from Table 2.1, the possible symmetry... 

Figure 10.96. Hj (0,2)-(18,3) hyperfine, spin-rotation and symmetry-breaking energy level diagram, showing the six AF = AN transitions, (a) denotes the Fermi contact splitting, (b) is the spin-rotation splitting and (c) shows the effect of symmetry breaking.

A fast-mixing nozzle in an FT microwave spectrometer was used to measure rotational constants, centrifugal distortion constants, Cl-nuclear quadrupole- and spin-rotational coupling constants for the isotopomers (CH2)2S- C1F and (CH2)2S- C1F <1996CPL119>. The complex, with symmetry, has an arrangement of the S-CIF nuclei that is about 3.5° off collinearity. The Cl-F axis makes an angle of 95° with thiirane s C-2 axis. 

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