Internuclear

Rotational diffusion coefficient, Dg, internal motion rate parameter, angle between the internal rotation axis and the internuclear axis... 

The Lennard-Jones 12-6 potential contains just two adjustable parameters the collision diameter a (the separation for which the energy is zero) and the well depth s. These parameters are graphically illustrated in Figure 4.34. The Lennard-Jones equation may also be expressed in terms of the separation at which the energy passes through a minimum, (also written f ). At this separation, the first derivative of the energy with respect to the internuclear distance is zero (i.e. dvjdr = 0), from which it can easily be shown that v = 2 / cr. We can thus also write the Lennard-Jones 12-6 potential function as follows ... 

The minimum for the equilibrium internuclear distance in Hj is 2.49 bohrs in this first... 

The amount of computation necessary to try many conformers can be greatly reduced if a portion of the structure is known. One way to determine a portion of the structure experimentally is to obtain some of the internuclear distances from two-dimensional NMR experiments, as predicted by the nuclear Over-hauser effect (NOE). Once a set of distances are determined, they can be used as constraints within a conformation search. This has been particularly effective for predicting protein structure since it is very difficult to obtain crystallographic structures of proteins. It is also possible to define distance constraints based on the average bond lengths and angles, if we assume these are fairly rigid while all conformations are accessible. 

The C-H coupling constants can also be used to calculate interorbital and internuclear angles. For thiazoie, very precise results have been... 

TABLE I1I-7. CALCULATED INTERORBITAL AND INTERNUCLEAR ANGLES IN THIAZOLE (113)... 

FIGURE 2 17 The carbon-carbon double bond in ethylene has a cr component and a tt compo nent The cr component arises from overlap of sp hybridized orbitals along the internuclear axis The tt component results from a side by side overlap of 2p orbitals... 

FIGURE 3 10 Bent bonds in cyclopropane (a) The orbitals involved in carbon-carbon bond formation overlap in a region that is displaced from the internuclear axis (b) The three areas of greatest negative electrostatic potential (red) correspond to those predicted by the bent bond description... 

This difference is shown in the next illustration which presents the qualitative form of a potential curve for a diatomic molecule for both a molecular mechanics method (like AMBER) or a semi-empirical method (like AMI). At large internuclear distances, the differences between the two methods are obvious. With AMI, the molecule properly dissociates into atoms, while the AMBERpoten-tial continues to rise. However, in explorations of the potential curve only around the minimum, results from the two methods might be rather similar. Indeed, it is quite possible that AMBER will give more accurate structural results than AMI. This is due to the closer link between experimental data and computed results of molecular mechanics calculations. 

The collision diameter is at the value of s(r) equal to zero, and die maximum interaction of the molecules is where s(r) is a minimum. The interaction of molecules is thus a balance between a rapidly-varying repulsive interaction at small internuclear distances, and a more slowly varying attractive interaction as a function of r (Figure 3.7). 

Figure 3.7 The Lennard-Jones potential of the interaction of gaseous atoms as a function of the internuclear distance...

Solid state NMR is a relatively recent spectroscopic technique that can be used to uniquely identify and quantitate crystalline phases in bulk materials and at surfaces and interfaces. While NMR resembles X-ray diffraction in this capacity, it has the additional advantage of being element-selective and inherently quantitative. Since the signal observed is a direct reflection of the local environment of the element under smdy, NMR can also provide structural insights on a molecularlevel. Thus, information about coordination numbers, local symmetry, and internuclear bond distances is readily available. This feature is particularly usefrd in the structural analysis of highly disordered, amorphous, and compositionally complex systems, where diffraction techniques and other spectroscopies (IR, Raman, EXAFS) often fail. 

A bond in which the orbitals overlap along a line connecting the atoms (the inter-ntidear axis) is called a sigma (a) bond. The electron distribution in a a bond is cylin-drically symmetric were we to slice through a a bond perpendicular to the internuclear-axis, its cross-section would appear- as a circle. Another way to see the shape of the electron distribution is to view the molecule end-on. 

Each carbon of ethylene uses two of its sp hybrid orbitals to form a bonds to two hydrogen atoms, as illustrated in the first par-f of Figure 2.17. The remaining sp orbitals, one on each carbon, overlap along the internuclear- axis to give a a bond connecting the two carbons. 

Strong sp -sp a bonds are not possible for cyclopropane, because the 60° bond angles of the ring do not permit the orbitals to be properly aligned for effective overlap (Figure 3.10). The less effective overlap that does occur leads to what chemists refer to as bent bonds. The electron density in the carbon-carbon bonds of cyclopropane does not lie along the internuclear- axis but is distr-ibuted along an arc between the two carbon atoms. The r-ing bonds of cyclopropane are weaker than other carbon-carbon a bonds. 

Finally we have to remember to add on the nuclear repulsion and, if we repeat the calculation for a range of values of the internuclear separation, we arrive at the potential energy curves shown in Figure 4.3 for the ground-state (singlet)... 

Now imagine that we rotate the molecule about the internuclear axis. The curved contour will trace out a surface. If we draw a unit outward normal vector to this surface, it will be everywhere perpendicular to the gradient vector (because the gradient vector points along the trajectory). 

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