Internuclear distances

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. 

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...

Figure 1.9 A plot of energy versus internuclear distance for two hydrogen atoms. The distance between nuclei at the minimum energy point is the bond length.

Atomic radii. The radii are determined by assuming that atoms in closest contact in an element touch one another. The atomic radius is taken to be one half of the closest internuclear distance, (a) Arrangement of copper atoms in metallic copper, giving an atomic radius of 0.128 nm for copper, (b) Chlorine atoms in a chlorine (Cl2) molecule, giving an atomic radius of 0.099 nm for chlorine. 

For example, Figure 19-3 contrasts the dimensions assigned to the halogens in the elementary state. One-half the measured internuclear distance is called the covalent radius. This distance indicates how close a halogen atom can approach... 

For the sake of simplicity, we will consider a diatomic molecule with the internuclear distance R, but the result is directly general-izable to a system with several internuclear distances Rv R2,. In addition to the trial function = q>(rlt r2,. . ., rN, R), we will now also consider the scaled function ... 

Our derivation of Eq. 11.33 based on the use of the variation principle is different from Slater s original treatment, but so far follows Hirschfelder and Kincaid. Here we will now show that it also permits such a scaling of an arbitrary trial function [Pg.222]

By multiplying Eq. 11.32 by 77 and by using Eq. 11.30 and Eq. 11.31, it is then easily checked that the virial theorem (Eq. 11.33) is satisfied for the scaled function reverse problem of finding the properly scaled func-... 

The calculation of the derivatives T p and Vp means usually a great deal of additional computations, and it is therefore important to observe that, if we are interested only in determining the energy E0 and the internuclear distance R0 for the equilibrium situation, we can use the simpler relations, Eq. 11.25 and Eq. 11.27. In such a case, E0 is the minimum of the quantity... 

Fig. 2. Energy as a function of internuclear distance the full line refers...

The internuclear distance at which the best bond orbitals are tetrahedral orbitals is in each case somewhat larger than the equilibrium distance given by the minimum of the energy curve. It is possible that in actual molecules the repul-... 

Bromine and Chlorine.—Accurate values of the internuclear distances in the molecules Bra and Cl2 are known from band spectral studies, namely, Br-Br = 2.281 A. and Cl-Cl = 1.988 A. The visual method led to results (2.289 A. and 2.009 A., respectively) in satisfactory agreement with these.8 Radial distribution curves for these substances are shown in Fig. 3, the data used being given in Tables II and III. For bromine, with seven rings, three different estimates of intensities lead to the same Br-Br distance, 2.270 A., less... 

Schrodinger s ideas lead to a simple explanation of the forces between atoms> in particular of the previously difficultly understandable repulsive force.f As an illustration we shall calculate the internuclear distances for the hydrogen halides. 

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