Molecules and Symmetry

Unless otherwise noted, all art on this page is Cengage Learning 2014. 

This axis is the principal axis. If there are more than 1 highest-n-fold axes, choose one to be principal axis. 

FIGURE 13.14 Flowchart for determining the point group of a molecule. It does not include O, T, or Rh(3T Rh( ) is the point group of any single atom or ion. Source Adapted from P. W. Atkins, Physical Chemistry, 5th ed., Freeman, New York, 1994. 

FIGURE 13.15 What are the point groups of these five molecules See Example 13.3. 

The applicability of symmetry to molecules is deeper than just the shape of the molecule. Mathematical equations also have symmetry properties. We have already discussed the concept of odd and even functions. This is a symmetry property. An even function implies that a plane of symmetry exists, typically a plane that intersects the y-axis. You can verify this by looking at plots of cosine, an even function, and sine, an odd function. 

---

Z-matrix was that it mirrors the way chemists think. Molecular construction using the Z-matrix is not particularly difficult for a small molecule, and symmetry can be readily imposed, as in my ethene example above. 

Molecules and symmetry of frame Si-N= bonds in A Si-N=V angles (Y=C, N) in degrees Method References... 

For linear molecules and symmetrie top molecules with a nuclear quadrapole located on the symmetry axis, the principal axes of the field gradient tensor and the inertia tensor coincide. Since two components of the field gradient tensor are equal because of symmetry, only one independent component of the quadrapole couphng tensor remains to be determined in an analysis of the quadrapole hfs ... 

For many-electron systems such as atoms and molecules, it is obviously important that approximate wavefiinctions obey the same boundary conditions and symmetry properties as the exact solutions. Therefore, they should be antisynnnetric with respect to interchange of each pair of electrons. Such states can always be constmcted as linear combinations of products such as... 

We collect syimnetry operations into various syimnetry groups , and this chapter is about the definition and use of such syimnetry operations and symmetry groups. Symmetry groups are used to label molecular states and this labelling makes the states, and their possible interactions, much easier to understand. One important syimnetry group that we describe is called the molecular symmetry group and the syimnetry operations it contains are pemuitations of identical nuclei with and without the inversion of the molecule at its centre of mass. One fascinating outcome is that indeed for... 

The aluminium ion, charge -I- 3. ionic radius 0.045 nm, found in aluminium trifluoride, undergoes a similar reaction when a soluble aluminium salt is placed in water at room temperature. Initially the aluminium ion is surrounded by six water molecules and the complex ion has the predicted octahedral symmetry (see Table 2.5 ) ... 

HMO theory is named after its developer, Erich Huckel (1896-1980), who published his theory in 1930 [9] partly in order to explain the unusual stability of benzene and other aromatic compounds. Given that digital computers had not yet been invented and that all Hiickel s calculations had to be done by hand, HMO theory necessarily includes many approximations. The first is that only the jr-molecular orbitals of the molecule are considered. This implies that the entire molecular structure is planar (because then a plane of symmetry separates the r-orbitals, which are antisymmetric with respect to this plane, from all others). It also means that only one atomic orbital must be considered for each atom in the r-system (the p-orbital that is antisymmetric with respect to the plane of the molecule) and none at all for atoms (such as hydrogen) that are not involved in the r-system. Huckel then used the technique known as linear combination of atomic orbitals (LCAO) to build these atomic orbitals up into molecular orbitals. This is illustrated in Figure 7-18 for ethylene. 

Linear molecules belong to the axial rotation group. Their symmetry is intermediate in complexity between nonlinear molecules and atoms. 

Atoms, linear molecules, and non-linear molecules have orbitals which can be labeled either according to the symmetry appropriate for that isolated species or for the species in an environment which produces lower symmetry. These orbitals should be viewed as regions of space in which electrons can move, with, of course, at most two electrons (of opposite spin) in each orbital. Specification of a particular occupancy of the set of orbitals available to the system gives an electronic configuration. For example,... 

Recall that the symmetry labels e and o refer to the symmetries of the orbitals under reflection through the one Cy plane that is preserved throughout the proposed disrotatory closing. Low-energy configurations (assuming one is interested in the thermal or low-lying photochemically excited-state reactivity of this system) for the reactant molecule and their overall space and spin symmetry are as follows ... 

For symmetric top species, Pave hes along the symmetry axis of the molecule, so the orientation of Pave can again be described in terms of 0 and (j), the angles used to locate the orientation of the molecule s symmetry axis relative to the lab-fixed coordinate system. As a result, the El integral again can be decomposed into three pieces ... 

Molecular orbitals are not unique. The same exact wave function could be expressed an infinite number of ways with different, but equivalent orbitals. Two commonly used sets of orbitals are localized orbitals and symmetry-adapted orbitals (also called canonical orbitals). Localized orbitals are sometimes used because they look very much like a chemist s qualitative models of molecular bonds, lone-pair electrons, core electrons, and the like. Symmetry-adapted orbitals are more commonly used because they allow the calculation to be executed much more quickly for high-symmetry molecules. Localized orbitals can give the fastest calculations for very large molecules without symmetry due to many long-distance interactions becoming negligible. 

MOPAC runs in batch mode using an ASCII input hie. The input hie format is easy to use. It consists of a molecular structure dehned 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. Version 6 and older have limits on the size of molecule that can be computed due to the use of hxed array sizes, which can be changed by recompiling the source code. This input format allows MOPAC to be run in conjunction with a batch job-queueing system. 

Figure 10 12 shows the interaction between the HOMO of one ethylene molecule and the LUMO of another In particular notice that two of the carbons that are to become ct bonded to each other m the product experience an antibondmg interaction during the cycloaddition process This raises the activation energy for cycloaddition and leads the reaction to be classified as a symmetry forbidden reaction Reaction were it to occur would take place slowly and by a mechanism m which the two new ct bonds are formed m separate steps rather than by way of a concerted process involving a sm gle transition state... 

The theory of molecular symmetry provides a satisfying and unifying thread which extends throughout spectroscopy and valence theory. Although it is possible to understand atoms and diatomic molecules without this theory, when it comes to understanding, say, spectroscopic selection rules in polyatomic molecules, molecular symmetry presents a small barrier which must be surmounted. However, for those not needing to progress so far this chapter may be bypassed without too much hindrance. 

---