Axial rotation definition

New solutions can be generated indefinitely by new linear combinations, amounting to an endless number of axial rotations without any progress beyond the definition and rotation of the three-fold degenerate set, defined by (7) and (8). In the solid angle of 4tt the polar axis (Z) can be directed in infinitely many directions, each representing a linear combination of i/ j, ipQ and i/ i of (7) and (8). In a chemical context the bottom line is that each linear combination ( e.g. fui) defines a specific orientation of the polar axis". With this choice of axes there is no second polar axis and no other linear combination (e.g. that solves (4) in the same coordinate system. The tetrahedral carbon atom remains quantum-mechanically undefined. 

The simulation of the EPR-spectrum of N Cgi(COOEt)2 (Fig. 39) taking into account a fine structure interaction is in nice accordance with the experimental spectrum. The simulation was carried out with the hyperfine interaction and g-factor of unmodified N Cgo and the fine structure interaction 0 2=2 G and E = 0.13 G (non-axial term). The shape of the extra lines definitely requires the inclusion of a non-axial term E, indicating that the molecular structure of the adduct induces some non-axiality which is not averaged out by fast axial rotation. The non-axial term was also observed at a measurement at 100°C showing that axial motional averaging does not take place even at this temperature. These results show that like He Cgo the new endohedral compound N Cgo can be used as a probe to monitor exohedral chemical addition reactions. Due to higher sensitivity, EPR requires less material than NMR.3He Cgo and N Cgo are complementary probes since different interactions are measured. 

We designate the length of the ellipsoid along the axis of rotation as 2a and the equatorial diameter as 2b to define the axial ratio a/b which characterizes the ellipticity of the particle. By this definition, a/b > 1 corresponds to prolate ellipsoids, and a/b < 1 to oblate ellipsoids. 

It is noteworthy that dq(e,t) does not satisfy this relation, as equality [J,x, dq] = 2 C q dq+ll (the definition of an irreducible tensor operator) does not hold for it [23]. Integration in (7.18), performed over the spherical angles of vector e, may be completed up to an integral over the full rotational group due to the axial symmetry of the Hamiltonian relative to the field. This, together with (7.19), yields... 

In order to find the character of the representation of rotational motions and librations, the transformation properties of an axial vector (Fig. 2.1-lb) have to be taken into account. This vector is defined not only by its length and orientation, but also by a definite sense of rotation inherent to it. For linear molecules with only two degrees of rotational freedom, the Xr R) values for Cj and a depend on the orientation of the molecular axis to the symmetry elements. All values of XriR) tire also included in Table 2.7-1. 

There is a close connection between symmetry and the constants of the motion (these are properties whose operators commute with the Hamiltonian H). For a system whose Hamiltonian is invariant (that is, doesn t change) under any translation of spatial coordinates, the linear-momentum operator p will commute with H, and p can be assigned a definite value in a stationary state. An example is the free particle. For a system with H invariant under any rotation of coordinates, the operators for the angular-momentum components commute with H, and the toted angular momentum and one of its components are specifiable. An example is an atom. A linear molecule has axial symmetry, rather than the spherical synunetry of an atom here only the axial component of angular momentum can be specified (Chapter 13). 

Atropisomerism is significant because it introduces an element of chirality in the absence of stereogenic atoms. Axial chirality is observed with stereoisomers (or atropisomers) that result from hindered rotation about a single C—C or C—N bond. The barrier of rotation between atropisomers must be high enough to allow for their isolation. A minimum of three or/ho-substituents are generally required for an axially chiral biphenyl to have substantial stability toward racemization at room temperature. For general definitions and descriptions, see references [3-5]. 

For a definition of the various quantum numbers needed to describe rotational bands we refer to Fig. 11, where we show a well-deformed axially symmetric even-even nucleus. The coordinate axes in the lab system will be labeled with x, y, and z. The coordinate axes in the intrinsic system (in which the nucleus is stationary) are labeled with 1, 2, and 3, with the 3-axis pointing along the symmetry axis. 

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