Interatomic distances diatomic molecules

One of the first predictions made on the basis of steric effects was that the ease of chemisorption of diatomic molecules should strongly depend on the lattice dimensions of the metallic catalysts. The reasoning was that for large interatomic distances, diatomic molecules would have to dissociate to be completely chemisorbed, while for closely packed lattices, repulsion effects would hinder chemisorption. This is exemplified by our first example, the dehydrogenation of cyclohexane. 

This qualitative description of the interactions in the metal is compatible with quantum mechanical treatments which have been given the problem,6 and it leads to an understanding of such properties as the ratio of about 1.5 of crystal energy of alkali metals to bond energy of their diatomic molecules (the increase being the contribution of the resonance energy), and the increase in interatomic distance by about 15 percent from the diatomic molecule to the crystal. 

Thus, two values can be evaluated, W+ and W, according to the signs in Eq. (137). If the appropriate integrals are known, these quantities can be calculated for a given interatomic distance in the diatomic molecule. As indicated above the successive substitution of the two values W+ and W- yields... 

Because the dipole moment of a diatomic molecule is qd, it would appear that if we knew the interatomic distance (bond length) d, we should be able to calculate the atomic charges q. For example, the bond length of the HC1 molecule is 127 pm and the dipole moment is 3.44 X 10 3n C-m, so we have... 

A single particle of (reduced) mass p in an orbit of radius r = rq + r2 (= interatomic distance) therefore has the same moment of inertia as the diatomic molecule. The classical energy for such a particle is E = p2/2m and the angular momentum L = pr. In terms of the moment of inertia I = mr2, it follows that L2 = 2mEr2 = 2EI. The length of arc that corresponds to particle motion is s = rep, where ip is the angle of rotation. The Schrodinger equation is1... 

Potential energy surface (PES) can be understood by making a plot of energy as a function of various interatomic distances in the complex that is formed during the reaction. For simplicity, let us consider a simplet chemical reaction between an atom A and a diatomic molecule BC to yield another atom C and a diatomic molecule AB as... 

The vibrational frequency of a diatomic molecule (a one-dimensional system) is proportional to the square root of force constant (the second derivative of the energy with respect to the interatomic distance) divided by the reduced mass (which depends on the masses of the two atoms). 

Covalent radii are calculated from half the interatomic distance between two singly bonded like atoms. For diatomic molecules such as F2, this is no problem, but for other elements, such as carbon, which do not have a diatomic molecule, an average value is calculated from a range of compounds that contain a C-C single bond. 

Information about the structure of gas molecules haB been obtained by several methods. Spectroscopic studies in the infrared, visible, and ultraviolet regions have provided much information about the simplest molecules, especially diatomic molecules, and a few polyatomic molecules. Microwave spectroscopy and molecular-beam studies have yielded very accurate interatomic distances and other structural information about many molecules, including some of moderate complexity. Molecular properties determined by spectroscopic methods are given in the two books by G. Herzberg, Spectra of Diatomic Molecules, 1950. and Infrared and Raman Spectra, 1945, Van Nostrand Co., New York. The information obtained about molecules by microwave spectroscopy is summarised by C. H. Townes and A. L. Schawlow in their book Microwave Spectroscopy of Gases, McGraw-Hill Book Co., New York, 1955. 

Figure 8.11 Potential energy diagram of the dissociation of a diatomic molecule AB. Horizontal axis, interatomic distance r. A , k2, absorption wavelengths of the excited molecule and its dissociation products...

The discovery of confinement resonances in the photoelectron angular distribution parameters from encaged atoms may shed light [36] on the origin of anomalously high values of the nondipole asymmetry parameters observed in diatomic molecules [62]. Following [36], consider photoionization of an inner subshell of the atom A in a diatomic molecule AB in the gas phase, i.e., with random orientation of the molecular axis relative to the polarization vector of the radiation. The atom B remains neutral in this process and is arbitrarily located on the sphere with its center at the nucleus of the atom A with radius equal to the interatomic distance in this molecule. To the lowest order, the effect of the atom B on the photoionization parameters can be approximated by the introduction of a spherically symmetric potential that represents the atom B smeared over... 

This requires the existence of at least two independently variable nuclear coordinates. Since in a diatomic molecule there is only one variable coordinate—the interatomic distance—so the noncrossing rule can be stated as follows ... 

Fig. 2. Illustration to the JT and PJT effect in diatomic molecule formation. In the homonuclear case (a) the two atomic states at large interatomic distances <2m = l/R = 0 form a double degenerate term which at larger QM (smaller R) splits due to bonding, thus reducing the energy and symmetry quite similar to any other JT E bi problem (cf. Fig. la at Q > 0) for heteronuclear diatomics (b) the bonding picture is that of the pseudo JT effect (cf. Fig. lc at Q > 0).

In the same way that electronegativities determine the polarity of diatomic interactions, ionization radii should define the effective electronic charge clouds that interpenetrate to form diatomic molecules, as shown schematically in Figure 5.3. The overlap of two such spheres defines a lens of focal lengths fixed by the ionization radii, r and r2, at an interatomic distance d = x i + x2-... 

In a diatomic molecule it is of benefit to minimize the steric influence of those valence electrons not directly involved in mediating the interatomic interaction. The competing factors in this case are atomic size, interatomic distance, distribution of valence-electron density and the Pauli principle. 

Starting from ionization radii, r o, and using experimentally measured values of dissociation energy and interatomic distance for homonuclear diatomic molecules, a self-consistent set of characteristic radii, suitable for the point-charge calculation of bonding parameters, of both homonuclear and... 

In order to test the point-charge method experimentally measured dissociation energy and interatomic distance are required for each chemical bond. Dissociation energies for most homonuclear diatomic molecules have been measured spectroscopically and/or thermochemically. Interatomic distances for a large number of these are also known. However, for a large number of, especially metallic diatomic molecules, equilibrium interatomic distances have not been measured spectroscopically. In order to include these elements in the sample it is noted that for those metals with measured re, it is found to be related, on average, to 5, the distance of closest approach in the metal, by re = 0.78(5. On this assumption reference values of interatomic distance (d) become available for virtually all elements, as shown in the data appendix. In some special cases well-characterized dimetal bond lengths have also been taken into account for final assessment of interatomic distance. 

Covalent interaction in diatomic molecules depends on the golden mean t, the interatomic distance d and the radius ratio x r /r2 of the constituent atoms, as summarized in Figure 5.6. The golden mean is a universal constant that matches the geometry and topology of space-time, the radius ratio is a known function of atomic number and dl relates to the optimal wave-mechanical distribution of valence-electron density in the diatomic system. 

Interatomic distance is calculated by mathematical modelling of the electron exchange that constitutes a covalent bond. Such a calculation was first performed by Heitler and London using Is atomic wave functions to simulate the bonding in H2. To model the more general case of homonuclear diatomic molecules the interacting atoms in their valence states are described by monopositive atomic cores and two valence electrons with constant wave functions (3.36). 

The formulation of spatially separated a and 7r interactions between a pair of atoms is grossly misleading. Critical point compressibility studies show [71] that N2 has essentially the same spherical shape as Xe. A total wave-mechanical model of a diatomic molecule, in which both nuclei and electrons are treated non-classically, is thought to be consistent with this observation. Clamped-nucleus calculations, to derive interatomic distance, should therefore be interpreted as a one-dimensional section through a spherical whole. Like electrons, wave-mechanical nuclei are not point particles. A wave equation defines a diatomic molecule as a spherical distribution of nuclear and electronic density, with a common quantum potential, and pivoted on a central hub, which contains a pith of valence electrons. This valence density is limited simultaneously by the exclusion principle and the golden ratio. 

The Vibration of Diatomic Molecules.—In addition to their rotation, we have seen that diatomic molecules can vibrate with simple harmonic motion if the amplitude is small enough. We shall use only this approximation of small amplitude, and our first stop will be to calculate the frequency of vibration. To do this, we must first find the linear restoring force when the interatomic distance is displaced slightly from its equilibrium value / ,. We can get this from Eq. (1.2) by expanding the force in Taylor s series in (r — rt). We have... 

See Sponer, Molekiilspektren und ihre Anwendungen auf chemische Probleme, Vol. I, Springer, 1935, for interatomic distances of diatomic and polyatomic molecules in this chapter. 

The diatomic molecules which show hindered rotation in the solid generally have quite complicated molecular crystals. This is true, for instance, of the halogens. CI2 forms a crystal composed of molecules, each of interatomic distance 1.82 A (compared to 1.98 A in the gas), arranged in a complicated way which we shall not describe. Iodine I2 forms a layer lattice. In Fig. XXIV-3 we show one of the layers, showing... 

Spectral studies of rotational energy levels have proved most profitable for linear molecules having dipole moments, particularly diatomic molecules (for example, CO, NO, and the hydrogen halides). The moment of inertia of a linear molecule may be readily obtained from its rotation spectrum and for diatomic molecules, interatomic distances may he calculated directly from moments of inertia (Exercise 14d). For a mole-... 

The vibration of a diatomic molecule may be of only one kind, an alternate expansion and contraction of the interatomic distance. The simplest mathematical treatment (useful, but approximate) of such a vibration assumes the molecule to be a harmonic oscillator, roughly analogous to a... 

---