Symmetry axis proper

An example of a molecule with a three-fold rotation axis is the conformation of. vym-1,3,5-triethylcyclohexane shown in Figure B.l. Note that all molecules possess a trivial Ci axis (indeed, an infinite number of them). Note also that if we choose a Cartesian coordinate system where the proper rotation axis is the z axis, and if the rotation axis is two-fold, then for every atom found at position (x,y,z) where x and y are not simultaneously equal to 0 (i.e., not on the z axis itself) there will be an identical atom at position (—x,—y,z). If the rotation axis is four-fold, there will be an identical atom at the three positions (—x,y,z), (x,—y,z), and (—x,—y,z). Note finally that for linear molecules the axis of the molecule is a proper symmetry axis of infinite order, i.e., Cao-... 

One-dimensional irreducible representations are labeled either A or B according to whether the character of a 2irjn (proper or improper) rotation about the symmetry axis of highest order n is +1 or —1, respectively. For the point groups Wl9 and which have no symmetry axis, all one-dimensional representations are labeled A. For... 

Prolate symmetric top, 199, 211 Propane, dipole moment of, 225 Proper axis of symmetry, 53 Proper rotation, 395-396 Proton, 178 Pseudovector, 434 Pulse laser, 137,139 Purcell, E. M., 328, 360 Purely electronic energy, 57 Pure-rotation spectra, 165... 

For a number of SRMs C H, i.e. ( ) isa proper subgroup of This fact has been established by determining the group H directly. In the case where a principle axis coincides with an internal rotation axis for all values of the internal rotation angle, but not being a covering symmetry axis, we have... 

The first structure of a dimeric DmNcd construct (PDB code 2NCD amino acids 281-700 Sablin et al., 1998) turned out to be perfectly symmetric (by contrast to dimers of rat kinesin-1) the two molecules of a dimer are related by a crystallographic twofold axis. The symmetry axis coincides with the axis of the coiled-coil (Fig. 3C). A similar construct (PDB code 1CZ7 amino acids 295-700 Kozielski et al., 1999) crystallized in a different space group with two dimers per asymmetric unit. Although none of the dimers has a proper twofold symmetry, their conformation is not far from that. The deviation from perfect symmetry can be described by 2- and 10-degree torsions, respectively. 

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... 

We say that a body has an n-fold axis of symmetry (also called an n-fold proper axis or an n-fold rotation axis) if rotation about this axis by 360/n degrees (where n is an integer) gives a configuration physically indistinguishable from the original position n is called the order of the axis. For example, BF3 has a threefold axis of symmetry perpendicular to the molecular plane. The symbol for an n-fold rotation eixis is C .The threefold axis in BF3 is a C3 axis. To denote the operation of counterclockwise rotation by (360/ )°, we use the symbol C . The hat distinguishes synunetry operations from symmetry elements. BF3 has three more rotation axes each B—F bond is a twofold symmetry axis (Fig. 12.2). 

This orientation of the molecule reveals that methane possesses three twofold symmetry axes, one each along the x, y, and z axes. Because of this molecular symmetry, the proper molecular orbitals of methane must possess symmetry with respect to these same axes. There are two possibilities the orbital may be unchanged by 180° rotation about the axis (symmetric), or it may be transformed into an orbital of identical shape but opposite sign by the symmetry operation (antisymmetric). The carbon 2s-orbital is symmetric with respect to each axis, but the three 2p-orbitals are each antisymmetric to two of the axes and symmetric with respect to one. The combinations which give rise to molecular orbitals that meet these symmetry requirements are shown in Fig. 1.11. 

A symmetry operation is a transformation of a system that leaves an object in an indistinguishable position. For molecular systems, we need be concerned with only two types of symmetry operations proper rotations (C ) and improper rotations (S ). A C is a rotation around an axis by (360/ )° that has the net effect of leaving the position of the object unchanged. Thus, a C2 is a 180° rotation, a C3 a 120° rotation, and so on. These are termed "proper" rotations, because it is actually physically possible to rotate an object by 180° or 120°. Some examples are shown below, with the atoms labeled only to highlight the operation. 

C is the proper rotation operator. It rotates the system by an angle iTrjn about a particular axis. For example, Q is rotation by Ittjl = tt (or 180°), and Qfx) is rotation by 2-77/4 (or 90°, as shown in Fig. 6.2b). If the rotation is a symmetry element of the molecule, the axis is called a symmetry axis. The symmetry axis with the greatest value of n for a given molecule is assigned to z and is the principal rotation axis (for the rest of this chapter, we shall simply call it the principal axis ). In a few cases, the results are easily expressed in Cartesian coordinates. For example, proper rotation by ir about the z axis transforms a function as follows ... 

A proper rotation operator turns the molecule by a fixed angle about a particular axis, if proper rotation is a symmetry element of a molecule, the axis of that proper rotation is a symmetry axis of the molecule. 

The principal rotation axis is the symmetry axis having the smallest possible proper rotation as a symmetry element. 

These are the transformation equations for a proper rotation of a vector through an angle 0 about a symmetry axis such as C3. The identity operation is a proper rotation through zero degrees. 

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