Diatomic molecules Bond distances

A Lithium vapour does contain Li, molecules, but the 2s valence electron in the Li atom is efficiently shielded from the effect of the nuclear charge by the inner Is pair of electrons, so the covalent bond in Li, is long (267 pm) and weak (107 kj moL ). With Li a metallic lattice is formed in which the effects of the assembly of the valence electrons combine to give better stability than in the diatomic molecule. The distance between nearest neighbours in the metal is 304 pm, but there are eight such neighbours, and this leads to a stabilization of 161 kJ moL. ... 

If we were to calculate the potential energy V of the diatomic molecule AB as a function of the distance tab between the centers of the atoms, the result would be a curve having a shape like that seen in Fig. 5-1. This is a bond dissociation curve, the path from the minimum (the equilibrium intemuclear distance in the diatomic molecule) to increasing values of tab describing the dissociation of the molecule. It is conventional to take as the zero of energy the infinitely separated species. 

Figure S-1. Form of a potential energy curve for diatomic molecule AB. VfrAa) is the potential energy, Tab is the intemuclear distance, is the equilibrium intemuclear distance, and D is the bond dissociation energy. (The zero point energy is neglected in the figure.)...

The above models consider only one spatial variable which is the bonding distance. It is clear that, for a molecule anything more complex than diatomic, many parameters are needed to define even approximately the potential energy surface. The enormous advances in computational chemistry during the last few years have allowed quantum mechanical calculations on fairly large size molecules. The first attempt to apply quantum mechanics on deformed polymer chains was made... 

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. 

Within the framework of the Bom-Oppenheimer approximation, a diatomic molecule consists of two nuclei that are more-or-less attached by the surrounding electron cloud. Often the specific form of the resulting potential function is not known. However, if a chemical bond is formed between the two nuclei, the potential function displays a minimum at a distance that corresponds to the equilibrium bond length. Furthermore, the energy necessary to break the chemical bond, the dissociation energy, is often evaluated by spectroscopic measurements. It can be concluded, then, that the potential fiinction has the general form shown in Fig. 6. A simple derivation of the Born- Qppenheimer approximation is presented in Section 12.1. 

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... 

The simplest diatomic molecule consists of two nuclei and a single electron. That species, H2+, has properties some of which are well known. For example, in H2+ the internuclear distance is 104 pm and the bond energy is 268kJ/mol. Proceeding as illustrated in the previous section, the wave function for the bonding molecular orbital can be written as... 

The scheme described above, reconforted by the post-HF calculations [57] where the coordinate representing the distance between the nuclei in the diatomic molecule (or any bond in polyatomic molecules), lead to the pervading picture of a diatom connected adiabatically with two non-interacting atoms at infinite distance. From a compuational point of view, this picture is quite useful and widely employed. 

The potential energy of a diatomic molecule depends on only the distance between two bonded atoms. The potential energy of a diatomic molecule can be plotted in two dimensions by plotting PE as a function of the bond length. The curve is known as potential energy curve (Fig. 9.9). 

Figure 2.3 A Morse curve for a diatomic molecule, showing the quantised vibrational energy levels. The minimum on the curve represents the equilibrium bond distance, re...

A schematic curve showing how the dipole moment changes as internuclear distance increases is shown in Fig. 12.2. The figure nicely explains the observation that dipole moment IE s for diatomic molecules can be either normal or inverse depending whether the equilibrium bond length of the diatomic lies to the left (where (9p/9r)e > 0 and consequently Ap > 0) or the right (where (9p/9r)e < 0 and Ap < 0) of the maximum in the plot. The existence of the maximum is readily understandable in qualitative terms. Even though classically the dipole moment... 

Fig. 2. Relationship between the bond lengths of neighboring bonds in sulfur rings indicating strong bond-bond interaction. The distance, d, of a bond between two-coordinated atoms is. a function of the arithmetic mean of the lengths of the two neighboring bonds, V2(di dj). The values were taken from the compounds indicated in the Figure. The curve ends on the right side at d2 = 189 pm, the bond length of the diatomic molecule Sj...

These results can be interpreted successfully in terms of Pauling s valence bond order concept. In the framework of this model, a chemical bond between X and H in diatomic molecule XH or between H and B in a HB molecule can be characterized by empirical valence bond orders Pxh and Phb decreasing exponentially with bond distance ... 

To illustrate this point, consider a composite system composed of two noninteracting subsystems, one with p electrons (subsystem A) and the other with q = N — p electrons (subsystem B). This would be the case, for example, in the limit that a diatomic molecule A—B is stretched to infinite bond distance. Because subsystems A and B are noninteracting, there must exist disjoint sets Ba and Bb of orthonormal spin orbitals, one set associated with each subsystem, such that the composite system s Hamiltonian matrix can be written as a direct sum. 

Table VI lists the equilibrium geometries for the selected closed-shell diatomic molecules. It has long been recognized that the HF method gives reduced bond lengths. From Table VI we see that the correlated bond lengths are mostly longer than the experimental values. Exceptions are the bond distances predicted by CCD and NOF methods for the HCl molecule.

We now turn our attention to the interaction of an atom with a diatomic molecule. We consider three examples (i) Ar-HF, which is a traditional weakly bound system with a binding energy De(Ar-HF) of just 211 cm- (0.6 kcal/mol) (ii) H-CO", which is also weakly bound, Dg(H-CO ) = 8 kcal/mol, but results in large changes in the CO distance and (Hi) H-CO, a molecule whose bond strength is intermediate between the other atom-diatom systems considered and more traditional, strongly bound molecules (the H-CO bond strength is Just 19 kcal/mol). 

A goal of this chapter is to show, for the diatomic molecules under discussion, both the capability of the VB method in providing quantitative estimates of molecular properties and its capability of giving qualitative pictures of the bonding. The quantitative results are illustrated in Table 11.1, where we give values for Rg, the equilibrium bond distance, and determined theoretically with ST03G, 6-3IG,... 

Vibrational energy and transitions As seen in Fig. 3.2a, the bond between the two atoms in a diatomic molecule can be viewed as a vibrating spring in which, as the internuclear distance changes from the equilibrium value rc, the atoms experience a force that tends to restore them to the equilibrium position. The ideal, or harmonic, oscillator is defined as one that obeys Hooke s law that is, the restoring force F on the atoms in a diatomic molecule is proportional to their displacement from the equilibrium position. 

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. 

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