Polyatomic molecule equilibrium structure

These arguments can be extended to linear and non-linear polyatomic molecules for which zero-point structure, in terms of bond lengths and angles, is isotope-dependent but for which equilibrium structure is not. 

In this chapter, the diverse coupling constants and MEC components identified in the combined electronic-nuclear approach to equilibrium states in molecules and reactants are explored. The reactivity implications of these derivative descriptors of the interaction between the electronic and geometric aspects of the molecular structure will be commented upon within both the EP and EF perspectives. We begin this analysis with a brief survey of the basic concepts and relations of the generalized compliant description of molecular systems, which simultaneously involves the electronic and nuclear degrees-of-freedom. Illustrative numerical data of these derivative properties for selected polyatomic molecules, taken from the recent computational analysis (Nalewajski et al., 2008), will also be discussed from the point of view of their possible applications as reactivity criteria and interpreted as manifestations of the LeChatelier-Braun principle of thermodynamics (Callen, 1962). 

Difficulty arises when we attempt to specify bond force constants in polyatomic molecules. It was mentioned in Chapter 1 that molecular geometry of itself does not imply the existence of chemical bonds. Now force constants are the measure of the resistance of the molecule to infinitesimal displacements from the equilibrium configuration that is, to infinitesimal alterations in molecular geometry. And just as molecular geometry does not require the structure of the molecule to be stated in terms of bonds, so the force constants which express its resistance to change do not need to be stated in terms of bonds. Indeed, it is often inconvenient to do so. 

The simplest structural procedure is to use ground state (v = 0) effective moments via equations of the form of Eqs. (2), (8) or (10). Such structures are commonly known as r or effective structures since they utilize only Aq. Bq, Cq values. For most polyatomic molecules it is customarily assumed that the interatomic distances are isotopicalfy invariant, even though this is true only for a rigid, non-vibrating or equilibrium molecule. Therefore, data from several isotopic species can be invoked. Thus, for OCS, if Bq is determined for each of the isotopes 16-12-32 and 16-13-32, the two Bq values pamit determination of the C-0 and C-S distances from two equations with the form of Eq. (10). Indeed, various isotopomer combinations are possible, as summarized in Table 2. It is seen that the ro structure parameters vary rather widely (far outside experimental error), depending upon which pair is selected for the calculation. This is a clear example of the deleterious effects of uncorrected zero-point vibration terms (a or s), along with the assumption of isotopic invariance. 

The most extensively used spectroscopic structural method for polyatomic molecules is known as the substitution (rg) method and was proposed by Costain [29] some 40 years ago. In this method, the equations of Kraitchman (Eq. (11)) are used with ground state (Iq) moments of inertia for a parent (or normal) molecule and an appropriate collection of isotopomers. It has been generally supposed that the resulting rg coordinates and structural parameters are better approximations to the equilibrium (r ) parameters than the simpler rg effective parameters. This idea arises fix>m the following simple argument. Consider a diatomic parent molecule with moment Iq and one of the isotopomers with moment Iq. From the diatomic... 

For polyatomic molecules, even the simplest linear ones, there is no proof that the rg structure should be closer to equilibrium, but it has nevertheless been assumed to be the case. Because the Kraitchman equations rely most heavily on AI values upon isotopic substitution, it has seemed reasonable to suppose that even an incomplete cancellation of vibration-rotation terms would lead to a structure that was closer to equilibrium than the ro structures. For this reason, and because rg structures are simply not possible, the rg method has been used extensively to obtain structures for polyatomic molecules. Unfortunately, although the early work of Costain... 

New experimental teehniques have increased the capacity of rotationally resolved spectroscopy tremendously. They are generating experimental data at an unprecedented rate and with excellent precision. As a result of all these developments, we know a lot more with much greater precision and accuracy about many more molecules. This includes equilibrium structures of polyatomic molecules. 

The simplest, most direct, and most precise determination of bond distances and bond angles from rotational constants is from equilibrium values of these constants. Equilibrium parameters have a well-defined interpretation and are virtually invariant to isotopic substitution. Unfortunately, the required spectra in the first excited vibrational states are nearly always very difficult to obtain. In addition, the rotational constants must be free of the effects of perturbations and resonances. As a result, equilibrium structures have been obtained only for diatomic molecules and a few small polyatomic molecules. An example is the structure of S02 obtained by Morino et al.19 (Table 1). Also shown in the table is the approximate re structure called the rm structure by Watson.17... 

The observed vibration frequencies of a molecule depend on two features of the molecular structure the masses and equilibrium geometry of the molecule and the potential eneigy surface, or force field, governing displacements from equilibrium. These are described as kinetic and potential effects, respectively for a polyatomic molecule the form and the frequency of each of the 3N—6 normal vibrations depend on the two effects in a complicated way. The object of a force field calculation is to separate these effects. More specifically, if the kinetic parameters are known and the vibration frequencies are observed spectroscopically, the object is to deduce the potential eneigy surface. A major difficulty in this calculation is that the observed frequencies are often insufficient to determine uniquely the form of the potential energy surface, and it is necessary to use data on the frequency shifts observed in isotopically substituted molecules or data on vibration/rotation interaction constants observed in high resolution spectra in order to obtain a unique solution. 

There are two quite different ways of describing the equilibrium structure of molecular liquids, one based on the rotational invariant expansions and another based on the interaction-site model [19, 40]. It is clearly advantageous to use the latter in developing theories for dynamics of molecular liquids because it is capable of treating the general class of polyatomic fluids without too much difficulties. This feature is in contrast to other theories based on the rotational invariant expansions [41, 42, 43, 44, 45], in which theories become very complicated when there is no symmetry in a molecule. 

In those favorable cases of predominantly direct ionization in which the photoionization cross-section curve consists of a series of steplike thresholds, the situation is particularly simple. Each step corresponds to the production of ions in a vibrationally excited state as shown in Fig. 3 for NO. For polyatomic molecules with many normal vibrational modes, the vibrational structure can get hopelessly complicated yet often only a few normal modes are predominantly excited and these can often be identified from the observed spacing of steps. A particularly simple case is shown in Fig. 10. Photoionization of the pyramidal NH3 molecule produces an NH3 ion which has a planar equilibrium configuration. By the Franck-Condon principle, one expects predominant excitation of the symmetric out-of-plane bending mode of the NH3 ion. This expectation is confirmed by the energy spacing of the observed steps, which corresponds to production of the ion with successively higher vibrational excitation predominantly in this particular normal mode. 

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