Symmetric Polyatomic Molecules

Now that we are in possession of a convenient expansion of ry we can go further and establish the interaction law for polyatomic molecules (Balescu [1956]). 

We shall adopt a model conforming entirely to the assumptions of 3. We thus localize the interacting centres on the atoms of the molecules. This is indeed an approximation we could, for example, localize these centres on the valence electrons. In any case our model takes correctly account of the symmetry of the molecule. The chief conclusions of this section will therefore remain valid whatever the refinements of the model adopted as they are based only on the symmetry properties of the molecules. 

In order to perform explicit calculations, we shall assume that the interaction between individual atoms belonging to different molecules is given by a (6-12) Lennard-Jones potential. Then one can write (see 13.3.1) 

Let us stress the fact that 13.5.1) is only valid for symmetric molecules. The equation assumes, in effect, that the parameters s, r are the same for all the peripherjd atoms. If this is the case, the interatomic distances will be equal and the molecule will also be geometrically symmetric. The general formula, including uns5onmetric molecules, would be 

We first have a central term which, being independent of i, j, is simply times the series (13.4.15). This term will thus be the same whatever the shape and the structure of the molecules (see the discussion below). In contrast the structure will afiect the non-central terms in an essential manner. We shall now discuss a few special cases. 

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Only spatially degenerate states exhibit a first-order zero-field splitting. This condition restricts the phenomenon to atoms, diatomics, and highly symmetric polyatomic molecules. For a comparison with experiment, computed matrix elements of one or the other microscopic spin-orbit Hamiltonian have to be equated with those of a phenomenological operator. One has to be aware of the fact, however, that experimentally determined parameters are effective ones and may contain second-order contributions. Second-order SOC may be large, particularly in heavy element compounds. As discussed in the next section, it is not always distinguishable from first-order effects. 

Illustration 6.4 - BOC-MP Calculation of the Heat of Chemisorption for Symmetric Polyatomic Molecules... 

In contrast, the newer UBI-QEP formalism allows the calculation of binding energies for polyatomic molecules without bond-energy partitioning. This capability is especially applicable to symmetric polyatomic molecules, such as ethylene, acetylene and hydrazine, but can also be applied to other polyatomic molecules such as nitrous oxide [20,22]. This approach gives a value of 11.0 kcal mole for the heat of chemisorption of C2H4 on Pt(l 11) [20], which is very close to the value of 12 kcal mole previously calculated. 

For fairly symmetric polyatomic molecules it is possible to obtain an alternative expression for the frequency (or wavenumber) ratio between the normal and the isotopic molecule. For example, for the antisymmetric V3 frequencies of H2O and D2O this ratio is... 

Vibronic-coupling effects are known to be ubiquitous in the electronic spectroscopy of highly symmetric polyatomic molecules as well as crystals, the Jahn-Teller effect being the most widely known example. In this case the electronic... 

The conclusion to be drawn from the above results expresses the fact that non-central terms make only a small contribution to the interaction of symmetric polyatomic molecules, especially at sufficiently low densities where the mean value of djr is small. 

Pack R T 1976 Simple theory of diffuse vibrational structure in continuous UV spectra of polyatomic molecules. I. Collinear photodissociation of symmetric triatomics J. Chem. Phys. 65 4765... 

This Schrodinger equation relates to the rotation of diatomic and linear polyatomic molecules. It also arises when treating the angular motions of electrons in any spherically symmetric potential... 

This is the same as Equation (5.14) for a diatomic or linear polyatomic molecule and, again, the transitions show an equal spacing of 2B. The requirement that the molecule must have a permanent dipole moment applies to symmetric rotors also. 

For a symmetrical diatomic or linear polyatomic molecule with two, or any even... 

Just as for diatomics, for a polyatomic molecule rotational levels are symmetric (5 ) or antisymmetric (a) to nuclear exchange which, when nuclear spins are taken into account, may result in an intensity alternation with J. These labels are given in Figure 6.24. 

As for diatomic molecules (Section 7.2.5.2) fhe vibrational (vibronic) transitions accompanying an electronic transition fall into the general categories of progressions and sequences, as illustrated in Figure 7.18. The main differences in a polyatomic molecule are that there are 3A — 6 (or 3A — 5 for a linear molecule) vibrations - not just one - and that some of these lower the symmetry of the molecule as they are non-totally symmetric. 

As is the case for diatomic molecules, rotational fine structure of electronic spectra of polyatomic molecules is very similar, in principle, to that of their infrared vibrational spectra. For linear, symmetric rotor, spherical rotor and asymmetric rotor molecules the selection mles are the same as those discussed in Sections 6.2.4.1 to 6.2.4.4. The major difference, in practice, is that, as for diatomics, there is likely to be a much larger change of geometry, and therefore of rotational constants, from one electronic state to another than from one vibrational state to another. 

The eigenfunctions of J2, Ja (or Jc) and Jz clearly play important roles in polyatomic molecule rotational motion they are the eigenstates for spherical-top and symmetric-top species, and they can be used as a basis in terms of which to expand the eigenstates of asymmetric-top molecules whose energy levels do not admit an analytical solution. These eigenfunctions IJ,M,K> are given in terms of the set of so-called "rotation matrices" which are denoted Dj m,K ... 

We noted in Section 8.2 that only half the values of j are allowed for homonuclear diatomics or symmetric linear polyatomic molecules (only the even-y states or only the odd- y states, depending on the nuclear symmetries of the atoms). The evaluation of qmt would be the same as above, except that only half of the j s contribute. The result of the integration is exactly half the value in Eq. 8.64. Thus a general formula for the rotational partition function for a linear molecule is... 

The pure-rotational Raman spectrum of a polyatomic molecule provides information on the moments of inertia, hence allowing a structural determination. For a molecule to exhibit a pure-rotational Raman spectrum, the polarizability must be anisotropic that is, the polarizability ellipsoid must not be a sphere. As noted in Section 5.2, a spherical top has a spherical polarizability ellipsoid, and so gives no pure-rotational Raman spectrum. Symmetric and asymmetric tops have asymmetric polarizabilities. The structures of several nonpolar molecules (which cannot be studied by microwave spectroscopy) have been determined from their pure-rotational Raman spectra these include F2, C2H4, and C6H6. 

The Jahn-Teller theorem says nothing about the magnitude of the departure of the equilibrium geometry from a symmetric geometry, and in many cases, the actual departures are less than the magnitudes of zero-point vibrations. For more on the Jahn-Teller theorem, see Herzberg, Electronic Spectra of Polyatomic Molecules, pp. 40-51 F. S. Ham, Int. J. Quantum Chem., 5S, 191 (1971). 

We now consider the rotational fine structure of gas-phase IR bands, beginning with linear molecules. For nearly all known linear polyatomic molecules, the ground electronic state is a 2 state and we do not have to worry about the interaction of rotational and electronic angular momenta. A linear molecule in a 2 electronic state is a symmetric top with 7 =0 the selection rules are [(6.76) and (6.77)]... 

The symmetry of pcl for linear polyatomic molecules follows from the same arguments as for diatomic molecules thus pcl is symmetric with respect to nuclear exchange for 2, 2 ,... functions and antisymmetric for e2, e2, ... functions. 

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