By symmetry

We now turn to electronic selection rules for syimnetrical nonlinear molecules. The procedure here is to examme the structure of a molecule to detennine what synnnetry operations exist which will leave the molecular framework in an equivalent configuration. Then one looks at the various possible point groups to see what group would consist of those particular operations. The character table for that group will then pennit one to classify electronic states by symmetry and to work out the selection rules. Character tables for all relevant groups can be found in many books on spectroscopy or group theory. Ftere we will only pick one very sunple point group called 2 and look at some simple examples to illustrate the method. 

The Ag (100) surface is of special scientific interest, since it reveals an order-disorder phase transition which is predicted to be second order, similar to tire two dimensional Ising model in magnetism [37]. In fact, tire steep intensity increase observed for potentials positive to - 0.76 V against Ag/AgCl for tire (1,0) reflection, which is forbidden by symmetry for tire clean Ag(lOO) surface, can be associated witli tire development of an ordered (V2 x V2)R45°-Br lattice, where tire bromine is located in tire fourfold hollow sites of tire underlying fee (100) surface tills stmcture is depicted in tlie lower right inset in figure C2.10.1 [15]. 

H(I) and H(II). This fact does not provide any information on the nuclear sti ucture of this species at the energy minimum. By symmetry, it is clear that the system has three equivalent minima on the ground-state surface, which were designated as the three diatomic pairs. The nuclear geometry of each of these minima is quite different from that of the other two. 

Figure 25 shows the results of the 2 distortion induced by a degenerate e 2 vibration that removes the Dsa degeneracy (compare Fig. 23). By symmetry, five...

Now we can calculate the ground-state energy of H2. Here, we only use one basis function, the Is atomic orbital of hydrogen. By symmetry consideration, we know that the wave function of the H2 ground state is... 

When Cj symmetry is present, irreducible representations of the double group (see Table II) so that = 0 by symmetry. In this case, there are only... 

In the case of ethylene, because of 2-fold symmetry, odd terms drop out of the series, V3, V5,... = 0. In the case of ethane, because of 3-fold symmeti-y, even temis drop out, V2, V4,... = 0. Terms higher than three, even though permitted by symmetry, are usually quite small and force fields can often be limited to three torsional terms. Like cubic and quaitic terms modifying the basic quadratic approximation for stretching and bending, terms in the Fourier expansion of Ftors (to) beyond n = 3 have limited use in special cases, for example, in problems involving octahedrally bound complexes. In most cases we are left with the simple expression... 

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. 

Enforcing the molecular symmetry will also help orbital-based calculations run more quickly. This is because some of the integrals are equivalent by symmetry and thus need be computed only once and used several times. 

Many molecules, such as carbon monoxide, have unique dipole moments. Molecules with a center of inversion, such as carbon dioxide, will have a dipole moment that is zero by symmetry and a unique quadrupole moment. Molecules of Td symmetry, such as methane, have a zero dipole and quadrupole moment and a unique octupole moment. Likewise, molecules of octahedral symmetry will have a unique hexadecapole moment. 

For systems where the transition structure is not defined by symmetry, it may be necessary to ensure that the starting geometry does not have any symmetry. This helps avoid converging to a solution that is an energy maximum of some other type. 

If the transition state can be defined by symmetry, do a normal geometry optimization calculation with the symmetry constrained. 

In a molecule such as the asymmetric rotor formaldehyde, shown in Figure 5.1(f), the a, b and c inertial axes, of lowest, medium and highest moments of inertia, respectively, are defined by symmetry, the a axis being the C2 axis, the b axis being in the yz plane and the c axis being perpendicular to the yz plane. Vibrational transition moments are confined to the a, b or c axis and the rotational selection mles are characteristic. We call them... 

The most common type of sequence, one with Av = 0 shown in Figure 7.18, is always allowed by symmetry because, whatever the symmetry of the vibration involved. 

The answer, very often, is that they do not obtain any intensity. Many such vibronic transitions, involving non-totally symmetric vibrations but which are allowed by symmetry, can be devised in many electronic band systems but, in practice, few have sufficient intensity to be observed. For those that do have sufficient intensity the explanation first put forward as to how it is derived was due to Herzberg and Teller. 

In the two-photon spectmm in Figure 9.29 the vibrations vand have symmetry species h3 , and b2u, respectively. Show that the ISag, IThg, Ihag and 15q transitions are allowed by symmetry. 

Any symmetric object is built up from smaller pieces that are identical and that are related to each other by symmetry. An icosahedron can therefore be divided into a number of smaller identical pieces called symmetry-related units. Protein subunits are asymmetric objects hence, a symmetry axis cannot pass through them. The minimum number of protein subunits that can form a virus shell with icosahedral symmetry is therefore equal to... 

Figure 18.1 A crystal is built up from many billions of small identical units, or unit cells. These unit cells are packed against each other in three dimensions much as identical boxes are packed and stored in a warehouse. The unit cell may contain one or more than one molecule. Although the number of molecules per unit cell is always the same for all the unit cells of a single crystal, it may vary between different crystal forms of the same protein. The diagram shows in two dimensions several identical unit cells, each containing two objects packed against each other. The two objects within each unit cell are related by twofold symmetry to illustrate that each unit cell in a protein cr) stal can contain several molecules that are related by symmetry to each other. (The pattern is adapted from a Japanese stencil of unknown origin from the nineteenth century.)...

An orbital correlation diagram can be constructed by examining the symmetry of the reactant and product orbitals with respect to this plane. The orbitals are classified by symmetry with respect to this plane in Fig. 11.9. For the reactants ethylene and butadiene, the classifications are the same as for the consideration of electrocyclic reactions on p. 610. An additional feature must be taken into account in the case of cyclohexene. The cyclohexene orbitals tr, t72. < i> and are called symmetry-adapted orbitals. We might be inclined to think of the a and a orbitals as localized between specific pairs of carbon... 

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