Orbits of points

Figure 2.19 The divisions of the vertices of the regular orbit of point symmetry, defining the great rhombicosidodecahedron, into decoration sets about the rotational axes points [C5], row b, [C3], row c and [C2], row d, on the unit sphere.

Wondiatschek (1976) uses the term orbit instead of point configuration. Orbit (more exactly crystallographic orbit of points in space groups) and crystallographic point configuration are synonyma. 

A set of points that is equivalent with respect to a symmetry group is called an orbit. The polyhedra of Fig. 2.17 represent orbits of point groups. The arrows in Fig. 2.29 represent the orbits of plane groups. For the majority of groups, there are several types of orbit that we refer to as general positions and special positions. We will illustrate this important point with the aid of the plane group p2mg (Fig, 2.30). 

The low-lying bonding solution is characterized by the in-phase combination of the atomic orbitals tpi + (f>2), and it is labeled as the rzg molecular orbital of point group Dooh (note that the linear H2+ molecule belongs to this... 

Depending on the application, models of molecular surfaces arc used to express molecular orbitals, clcaronic densities, van dor Waals radii, or other forms of display. An important definition of a molecular surface was laid down by Richards [182] with the solvent-accessible envelope. Normally the representation is a cloud of points, reticules (meshes or chicken-wire), or solid envelopes. The transparency of solid surfaces may also be indicated (Figure 2-116). 

In a second example, the three CH bonds, three CH antibonds, CO bond and antibond, and three 0-atom non-bonding orbitals of the methoxy radical H3C-O also cluster into ai and e orbitals as shown below. In these cases, point group symmetry allows one to identify degeneracies that may not have been apparent from the structure of the orbital interactions alone. 

In summary, the moleeular orbitals of a linear moleeule ean be labeled by their m quantum number, whieh plays the same role as the point group labels did for non-linear polyatomie moleeules, and whieh gives the eigenvalue of the angular momentum of the orbital about the moleeule s symmetry axis. Beeause the kinetie energy part of the... 

To further illustrate these points dealing with orbital symmetry, consider the insertion of CO into H2 along a path which preserves C2v symmetry. As the insertion occurs, the degenerate n bonding orbitals of CO become hi and 62 orbitals. The antibonding n orbitals of CO also become hi and 62. The <5g bonding orbital of H2 becomes ai, and the antibonding H2 orbital becomes 62. The orbitals of the reactant... 

It is assumed that the reader has previously learned, in undergraduate inorganie or physieal ehemistry elasses, how symmetry arises in moleeular shapes and struetures and what symmetry elements are (e.g., planes, axes of rotation, eenters of inversion, ete.). For the reader who feels, after reading this appendix, that additional baekground is needed, the texts by Cotton and EWK, as well as most physieal ehemistry texts ean be eonsulted. We review and teaeh here only that material that is of direet applieation to symmetry analysis of moleeular orbitals and vibrations and rotations of moleeules. We use a speeifie example, the ammonia moleeule, to introduee and illustrate the important aspeets of point group symmetry. 

To illustrate sueh symmetry adaptation, eonsider symmetry adapting the 2s orbital of N and the three Is orbitals of H. We begin by determining how these orbitals transform under the symmetry operations of the C3V point group. The aet of eaeh of the six symmetry operations on the four atomie orbitals ean be denoted as follows ... 

If these rules are applied to the 2px and 2py orbitals of nitrogen within the C3V point group, one obtains... 

We now return to the symmetry analysis of orbital produets. Sueh knowledge is important beeause one is routinely faeed with eonstrueting symmetry-adapted N-eleetron eonfigurations that eonsist of produets of N individual orbitals. A point-group symmetry operator S, when aeting on sueh a produet of orbitals, gives the produet of S aeting on eaeh of the individual orbitals... 

The functions put into the determinant do not need to be individual GTO functions, called Gaussian primitives. They can be a weighted sum of basis functions on the same atom or different atoms. Sums of functions on the same atom are often used to make the calculation run faster, as discussed in Chapter 10. Sums of basis functions on different atoms are used to give the orbital a particular symmetry. For example, a water molecule with symmetry will have orbitals that transform as A, A2, B, B2, which are the irreducible representations of the C2t point group. The resulting orbitals that use functions from multiple atoms are called molecular orbitals. This is done to make the calculation run much faster. Any overlap integral over orbitals of different symmetry does not need to be computed because it is zero by symmetry. 

The Fermi energy is the energy of the highest-energy filled orbital, analogous to a HOMO energy. If the orbital is half-filled, its energy will be found at a collection of points in /c-space, called the Fermi surface. 

The axes of the sp orbitals point toward the corners of a tetrahedron Therefore sp hybridization of carbon is consistent with the tetrahedral structure of methane Each C—H bond is a ct bond m which a half filled Is orbital of hydrogen over laps with a half filled sp orbital of carbon along a line drawn between them... 

In Figure 1.8 the real wave functions for the f, 2p and 3d orbitals are plotted in the form of polar diagrams, the construction of which may be illustrated by the simple case of the 2p orbital. The wave function in Equation (f.43) is independent of 4> and is simply proportional to cos 6. The polar diagram consists of points on a surface obtained by marking off, on lines drawn outwards from the nucleus in all directions, distances proportional to I cos 6 at a constant value of R2i(r). The resulting surface consists of two touching spheres. 

This procedure is based on the observation of the orbital movement of the shaft eenterline. Three signal piekups are employed, of whieh two probes measure the vibration amplitudes of the rotor in two mutually perpendieular direetions. These two signals trace the orbit of the shaft centerline. The third probe is used to register the once-per-revolution reference point and is called the keyphazor. A schematic arrangement of these probes is shown in Figure 17-6. 

One cannot discuss Lewis acid-catalyzed cycloaddition reactions in the present context without trying to understand the reaction course mechanistically, e.g. using a frontier molecular orbital (FMO) point of reasoning, or theoretical calculations of transition state structures. 

---