More Complex Polyatomic Molecules

In Section 7.4 we considered the molecules formed by second-row elements with H. In eaeh ease there was only one heavy atom situated at the junction of all the symmetry elements, i.e. at the point of the point group. This allows the symmetry labels for the p-orbitals of this atom to be taken direetly from the right-hand column of the standard eharaeter table in Appendix 12. To form the MO diagram, we then considered the SALCs for the H(ls) orbitals and matched symmetry labels to identify the MOs that will give bonding-antibonding interactions. 

In this section we introduce more eomplex molecules in which there are multiple heavy atoms. We now need to identify SALCs for the sets of atoms that are related by symmetry operations. The MO diagram is then eonstructed by matching irreducible representations for the SALCs. 

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More complex polyatomic molecules AB are tackled in essentially the same way as H20. In a qualitative treatment, the most difficult part is working out the group orbitals. In the simple case of HzO, this was done by inspection. Group-theoretical techniques are available in cases where this would not be practicable. The resulting MO energy diagram, after feeding in the appropriate number of electrons, will usually confirm or... 

A key question about the use of any molecular theory or computer simulation is whether the intermolecular potential model is sufficiently accurate for the particular application of interest. For such simple fluids as argon or methane, we have accurate pair potentials with which we can calculate a wide variety of physical properties with good accuracy. For more complex polyatomic molecules, two approaches exist. The first is a full ab initio molecular orbital calculation based on a solution to the Schrddinger equation, and the second is the semiempirical method, in which a combination of approximate quantum mechanical results and experimental data (second virial coefficients, scattering, transport coefficients, solid properties, etc.) is used to arrive at an approximate and simple expression. 

In the case of polyatomic molecules, such as CO2, H2O, O3, etc., the principles discussed above still apply, but the spectra become more complex. Polyatomic molecules do not rotate only about one single axis, but about three mutually perpendicular axes. In addition, the number of vibrational degrees of freedom is also increased. 

Flowever, in order to deliver on its promise and maximize its impact on the broader field of chemistry, the methodology of reaction dynamics must be extended toward more complex reactions involving polyatomic molecules and radicals for which even the primary products may not be known. There certainly have been examples of this notably the crossed molecular beams work by Lee [59] on the reactions of O atoms with a series of hydrocarbons. In such cases the spectroscopy of the products is often too complicated to investigate using laser-based techniques, but the recent marriage of intense syncluotron radiation light sources with state-of-the-art scattering instruments holds considerable promise for the elucidation of the bimolecular and photodissociation dynamics of these more complex species. 

Sensitivity levels more typical of kinetic studies are of the order of lO molecules cm . A schematic diagram of an apparatus for kinetic LIF measurements is shown in figure C3.I.8. A limitation of this approach is that only relative concentrations are easily measured, in contrast to absorjDtion measurements, which yield absolute concentrations. Another important limitation is that not all molecules have measurable fluorescence, as radiationless transitions can be the dominant decay route for electronic excitation in polyatomic molecules. However, the latter situation can also be an advantage in complex molecules, such as proteins, where a lack of background fluorescence allow s the selective introduction of fluorescent chromophores as probes for kinetic studies. (Tryptophan is the only strongly fluorescent amino acid naturally present in proteins, for instance.)... 

Understanding VER in condensed phases has proven difficult. The experiments are hard. The stmcturally simple systems (diatomic molecules) involve complicated relaxation mechanisms. The stmctures of polyatomic molecules are obviously more complex, but polyatomic systems are tractable because the VER mechanisms are somewhat simpler. 

Polyatomic molecules cover such a wide range of different types that it is not possible here to discuss the MOs and electron configurations of more than a very few. The molecules that we shall discuss are those of the general type AFI2, where A is a first-row element, formaldehyde (FI2CO), benzene and some regular octahedral transition metal complexes. 

The bonding in molecules containing more than two atoms can also be described in terms of molecular orbitals. We will not attempt to do this the energy level structure is considerably more complex than the one we considered. However, one point is worth mentioning. In polyatomic species, a pi molecular orbital can be spread over die entire molecule rather than being concentrated between two atoms. 

For polyatomic molecules the situation is somewhat more complex but essentially the same. The effect of intramolecular motion upon the scattering of fast electrons by molecular gases was first described by Debye3 for the particular case of a molecular ensemble at thermal equilibrium. The corresponding average molecular intensity function can be expressed in the following way ... 

The dipole moment, //, for a diatomic molecule (the situation for polyatomic molecules that have several bonds is more complex) can be expressed as... 

Proceeding in the spirit above it seems reasonable to inquire why s is equal to the number of equivalent rotations, rather than to the total number of symmetry operations for the molecule of interest. Rotational partition functions of the diatomic molecule were discussed immediately above. It was pointed out that symmetry requirements mandate that homonuclear diatomics occupy rotational states with either even or odd values of the rotational quantum number J depending on the nuclear spin quantum number I. Heteronuclear diatomics populate both even and odd J states. Similar behaviors are expected for polyatomic molecules but the analysis of polyatomic rotational wave functions is far more complex than it is for diatomics. Moreover the spacing between polyatomic rotational energy levels is small compared to kT and classical analysis is appropriate. These factors appreciated there is little motivation to study the quantum rules applying to individual rotational states of polyatomic molecules. 

The principles discussed for diatomic molecules generally apply to polyatomic molecules, but their spectra are much more complex. For example, instead of considering rotation only about an axis perpendicular to the internuclear axis and passing through the center of mass, for nonlinear molecules, one must think of rotation about three mutually perpendicular axes as shown in Fig. 3-lb Hence we have three rotational constants A, B, and C with respect to these three principal axes. 

Equation (9.38), if restricted to two particles, is identical in form to the radial component of the electronic Schrodinger equation for the hydrogen atom expressed in polar coordinates about the system s center of mass. In the case of the hydrogen atom, solution of the equation is facilitated by the simplicity of the two-particle system. In rotational spectroscopy of polyatomic molecules, the kinetic energy operator is considerably more complex in its construction. For purposes of discussion, we will confine ourselves to two examples that are relatively simple, presented without derivation, and then offer some generalizations therefrom. More advanced treatises on rotational spectroscopy are available to readers hungering for more. 

The representation as a two-dimensional potential energy diagram is simple for diatomic molecules. But for polyatomic molecules, vibrational motion is more complex. If the vibrations are assumed to be simple harmonic, the net vibrational motion of TV-atomic molecule can be resolved into 3TV-6 components termed normal modes of ibrations (3TV-5 for... 

The more complex rotations and vibrations of polyatomic molecules are subject to the same principles, and distribution laws of the same kind apply, as will be shown in the following section. 

It is, of course, not correct to treat the wave function of a polyatomic molecule as localized in the chromophoric group considered responsible for the optical absorption. The carbonyl group in aldehydes and ketones gives rise to absorption which extends from about 3430 A to about 2200 A (as well as to absorption at shorter wavelengths). Nevertheless the carbon-oxygen bond is never broken by absorption at these wavelengths. Frequently an adjacent bond is broken but often more complex processes occur. It is sometimes possible to describe these processes in terms of quantum mechanics but some of them should not be treated as direct dissociations. 

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