Molecular symmetry defined

We hope that by now the reader has it finnly in mind that the way molecular symmetry is defined and used is based on energy invariance and not on considerations of the geometry of molecular equilibrium structures. Synnnetry defined in this way leads to the idea of consenntion. For example, the total angular momentum of an isolated molecule m field-free space is a conserved quantity (like the total energy) since there are no tenns in the Hamiltonian that can mix states having different values of F. This point is discussed fiirther in section Al.4.3.1 and section Al.4.3.2. 

AMPAC can also be run from a shell or queue system using an ASCII input file. The input file format is easy to use. It consists of a molecular structure defined either with Cartesian coordinates or a Z-matrix and keywords for the type of calculation. The program has a very versatile set of options for including molecular geometry and symmetry constraints. 

A phenomenological description of the differential cross-section for emission of photoelectrons into solid angle O in the lab frame can be written, assuming random molecular orientation and an axis of cylindrical symmetry defined by the photon polarization, as... 

In effect, the division by two is the result of the molecular symmetry, as specified by the character table for the group 0. In general it is useful to define a symmetry number a (= 2 in this case), as shown below. The well-known example of the importance of nuclear spin is that of ortho- arid para-hydrogen (see Section 10.9.5). 

It is also possible that a membrane might have an even lower symmetry than a chiral smectic-C liquid crystal in particular, it might lose the twofold rotational symmetry. This would occur if the molecular tilt defines one orientation in the membrane plane and the direction of one-dimensional chains defines another orientation. In that case, the free energy would take a form similar to Eq. (5) but with additional elastic constants favoring curvature. The argument for tubule formation presented above would still apply, but it would become more mathematically complex because of the extra elastic constants. As an approximation, we can suppose that there is one principal direction of elastic anisotropy, with some slight perturbations about the ideal twofold symmetry. In that approximation, we can use the results presented above, with 4) representing the orientation of the principal elastic anisotropy. 

Crystal lattices can be depicted not only by the lattice translation defined in Eq. (7.2), but also by the performance of various point symmetry operations. A symmetry operation is defined as an operation that moves the system into a new configuration that is equivalent to and indistinguishable from the original one. A symmetry element is a point, line, or plane with respect to which a symmetry operation is performed. The complete ensemble of symmetry operations that define the spatial properties of a molecule or its crystal are referred to as its group. In addition to the fundamental symmetry operations associated with molecular species that define the point group of the molecule, there are additional symmetry operations necessary to define the space group of its crystal. These will only be briefly outlined here, but additional information on molecular symmetry [10] and solid-state symmetry [11] is available. 

There exists no uniformity as regards the relations between localized orbitals and molecular symmetry. Consider for example an atomic system consisting of two electrons in an (s) orbital and two electrons in a (2px) orbital, both of which are self-consistent-field orbitals. Since they belong to irreducible representations of the atomic symmetry group, they are in fact the canonical orbitals of this system. Let these two self-consistent-field orbitals be denoted by Cs) and (2p), and let (ft+) and (ft ) denote the two digonal hybrid orbitals defined by... 

Based on the N-NMR data alone, the corresponding areas in the x-p maps are displayed in Fig. 5C ( N-Val in GS-3/30 and Fig. 5D ( N-Leu in GS-l/lO. The overlap of these regions with the constraints from F-NMR (Fig. 5A,B) thus defines the novel orientation of gramicidin S in membranes at high peptide concentration. The allowed area (common to all four panels) is marked by a dot in Fig. 5 at r = 80° 10° and p = -45° 10. As the value of r is close to a right angle between the molecular symmetry axis (z-axis, see Fig. 1) and the membrane normal, this means that the P-sheet plane (x-y plane) is tilted almost perpendicular with respect to the lipid bilayer plane. The value of p indicates that the P-strands (peptide /-axis) are inclined by about 45° with respect to the membrane normal. 

In this chapter we will familiarize ourselves with basic concepts in molecular symmetry [17]. The presence or absence of symmetry has consequences on the appearance of spectra, the relative reactivity of groups, and many other aspects of chemistry, including the way we will make use of orbitals and their interactions. We will see that the orbitals that make up the primary description of the electronic structure of molecules or groups within a molecule have a definite relationship to the three-dimensional structure of the molecule as defined by the positions of the nuclei. The orientations of the nuclear framework will determine the orientations of the orbitals. The relationships between structural units (groups) of a molecule to each other can often be classified in terms of the symmetry that the molecule as a whole possesses. We will begin by introducing the basic termi-... 

The molecular symmetry number (a) is a measure of the rotational degeneracy of the molecule. It is defined as the number of indistinguishable positions that can be obtained by rigidly rotating the molecule about its center mass. Symmetry numbers for spherical, conical, and cylindrical molecules, shapes with infinite axes of rotation, have o values of approximately 200, 20, and 20, respectively. Chemicals with no axes of symmetry have o = l. 

Complex [Fe6(L8)6] (23) can be described as having idealized D3-molecular symmetry. The iron centers define the apices of a distorted trigonal antiprism in which six tripodal, tris(bidentate) ligands (L8)3 make up the equatorial faces, leaving the top and bottom triangles unoccupied. All six iron(III) ions are... 

The shielding factor is a property of the molecule, but as we see in later examples, the ability of the magnetic field to influence the motion of electrons depends on the orientation of the molecule relative to B0. Hence, O is a second-rank tensor, not a simple scalar quantity. It is always possible to define three mutually orthogonal axes within a molecule such that o may be expressed in terms of three principal components, on, molecular symmetry requires that two of the components of o be equal (and in other instances it is possible to assume approximate equality), so that the components may be expressed relative to the symmetry axis as chemical shielding anisotropy defined (ct — cr,). 

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