Rotation-inversion

Lightman A., Ben-Reuven A. Line mixing by collisions in the far-infrared spectrum of ammonia, J. Chem. Phys. 50, 351-3 (1969) Cross relaxation in the rotational inversion doublets of ammonia in the far infrared, J. Quant. Spectrosc. Radiat. Transfer 12, 449-54 (1972). 

As an example, the group of rotations about an axis is a connected group. The property of connectedness is not the same as the continuous nature of a group. A continuous group, for instance the rotation-inversion group in three dimensions may be disconnected. The parameter space of a continuous disconnected group consists of two or more disjoint subsets such that each subset is a connected space, but where it is impossible to go continuously from a point in one subset to a point in another without going outside the parameter space. 

The group (E, J) has only two one-dimensional irreducible representations. The representations of 0/(3) can therefore be obtained from those of 0(3) as direct products. The group 0/(3) is called the three-dimensional rotation-inversion group. It is isomorphic with the crystallographic space group Pi. 

The full rotation-inversion group 0/(3) has four parameters which may be taken to be (a, P, 7, d) where a, P, 7 are the parameters of 0(3) and d denotes the determinant of an element and can take values 1. The parameter space of 0/(3) thus consists of two disconnected regions. It therefore is a four-parameter continuous compact group which is, however, not connected. It is also not a Lie group because one of its parameters is discrete. 

Electron pair-electron pair, electron pair-polar bond, or polar bond-polar bond interactions cause a significant increase in rotation-inversion barriers of atoms bearing these substituents. 

Proper rotational operations are represented by the n-fold rotation axes n 1000 (n = 2, 3,4, 6). Rotation-inversion axes such as the 2 axis are improper rotation operations, while screw axes and glide planes are combined rotation-translation operations. 

It might be anticipated that computational models would provide good accounts of conformational energy differences and rotation/inversion... 

As with conformational energy differences, SYBYL and MMFF molecular mechanics show marked differences in performance for rotation/inversion barriers. MMFF provides a good account of singlebond rotation barriers. Except for hydrogen peroxide and hydrogen disulfide, all barriers are well within 1 kcal/mol of their respective experimental values. Inversion barriers are more problematic. While the inversion barrier in ammonia is close to the experimental value, barriers in trimethylamine and in aziridine are much too large, and inversion barriers in phosphine and (presumably) trimethylphosphine are smaller than their respective experimental quantities. Overall,... 

Compilations of experimental data relating to conformational energy differences and rotation/inversion barriers may be found in (a) T. A. Halgren andR.B. Nachbar,/. ComputationalChem., 17, 587 (1996) (b) T.A. Halgren,... 

Thus far we have addressed the symmetry of crystalline arrays only in terms of the proper rotations and the rotation-inversion operations (the latter including simple inversion, as 1, and reflection, as 2) that occur in point symmetries, along with the lattice translation operations. However, for a complete discussion of symmetry in crystalline solids, we require two more types of operation in which translation is combined with either reflection or rotation. These are, respectively, glide-reflections (or, as commonly called, glides) and screw-rotations. 

As for the remaining symbols, many have already been used, namely, those for the rotation axes, 2, 3, 4, and 6 and the various screw axes seen end-on. Symbols not previously used are those Jor the l axis (inversion center) and the other three rotation-inversion axes 3, 4, and 6. Recall that 2 is equivalent to m. Finally, there are the symbols for rotation and screw axes that lie parallel to the page, which are distinguished by use of full and half-arrowheads, respectively. 

Because of intramolecular mobility (rotations, inversions) and intermolecular interactions, chemicals shifts depend on temperature, solvent, and concentration. Coupling constants, however, for the most part do not depend on these conditions. 

Studies of the anions CH2OH and CH2SH- have been reported, particularly the rotation-inversion behaviour. Plots of the potential energy surface were analysed in terms of possible paths between the different conformations.144 A more recent paper146 dealt in detail with the protonation of these molecules and of their isomers MeX- [reaction (7)]. Several basis sets of different sizes were used which contained... 

K. Miscellaneous Five-atom Molecules.—We mention in this section calculations on a variety of molecules which are not readily classified. One such molecule is the as yet undetected sulphilimine, H2SNH. A 4-31G basis set computation of the energy surface for rotation-inversion has been presented.170 The optimum HSN angle was much smaller than the observed angle in known compounds R1R2SNH. 

Figure 8.1. Rotation inversion energy levels and allowed transitions of the free NH3 molecule. The selection rules are A7 =...

The symmetry elements, proper rotation, improper rotation, inversion, and reflection are required for assigning a crystal to one of the 32 crystal systems or crystallographic point groups. Two more symmetry elements involving translation are needed for crystal structures—the screw axis, and the glide plane. The screw axis involves a combination of a proper rotation and a confined translation along the axis of rotation. The glide plane involves a combination of a proper reflection and a confined translation within the mirror plane. For a unit cell... 

Symmetry elements include axes of twofold, threefold, fourfold, and sixfold rotational symmetry and mirror planes. There are also axes of rotational inversion symmetry. With these, there are rotations that cause mirror images. For example, a simple cube has three <100> axes of fourfold symmetry, four axes of <111>... 

Therefore, in compliance with the Law of Rational Indices, only n-axes with n = 1,2,3,4 and 6 are allowed in crystals. The occurrence of the inversion center means that the rotation-inversion axes I, 2(= m), 3, 4 and 6 are also possible. 

Numbering (in parentheses) and location of the operations (identity, twofold rotation, inversion, c glide, translation, and n glide). 

The Hermann- Mauguin notation is generally used by crystallographers to describe the space group. Tables exist to convert this notation to the Schoen-flies notation. The first symbol is a capital letter and indicates whether the lattice is primitive. The next symbol refers to the principal axis, whether it is rotation, inversion, or screw, e.g.,... 

Use arabic numerals or combinations of numerals and the italic letter m to designate the 32 crystallographic point groups (Hermann-Mauguin). The number is the degree of the rotation, and m stands for mirror plane. Use an overbar to indicate rotation inversion. 

The first few rotation-inversion levels of NH3 are presented in Fig. 17. They are specified by the quantum numbers J, the total rotational angular momentum,... 

Fig. 17. Energy levels of the rotation-inversion spectrum of ammonia. The quantum numbers (J,K) are given for each level. The heavy arrows indicate the inversion transitions detected in interstellar space and their frequencies in MHz. Thin arrows indicate the rotation-inversion transitions located in the submillimeter wave region. Dashed arrows indicate some collision induced transitions...

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