London formula

London23 has treated the case of the attractive force between anisotropic molecules on the dipole-dipole interaction basis as well as on the monopole basis mentioned above. The small anisotropy found for the chlorine atom makes the dipole-dipole formulation appropriate. For the symmetrical orientation in the Cl2 molecule the London formula is... 

Figure 9.6. Diagrammatic representation of the geometries that correspond to 7 = 0 in the London formula for four electrons in four Is orbitals (e.g., H4).

Let us summarize briefly at this stage. We have seen that the point of degeneracy forms an extended hyperline which we have illnstrated in detail for a four electrons in four Is orbitals model. The geometries that lie on the hyperline are predictable for the 4 orbital 4 electron case using the VB bond energy (Eq. 9.1) and the London formula (Eq. 9.2). This concept can be nsed to provide nseful qualitative information in other problems. Thns we were able to rationalize the conical intersection geometry for a [2+2] photochemical cycloaddition and the di-Jt-methane rearrangement. 

The last important contribution to intermolecular energies that will be mentioned here, the dispersion energy (dEnis). is not accessible in H. F. calculations. In our simplified picture of second-order effects in the perturbation theory (Fig. 2), d mS was obtained by correlated double excitations in both subsystems A and B, for which a variational wave function consisting of a single Slater determinant cannot account. An empirical estimate of the dispersion energy in Li+...OH2 based upon the well-known London formula (see e.g. 107)) gave a... 

The cohesion in crystals of electrically neutral atoms, typified in Figure 5.10 by Ar, is ascribed to van der Waals interaction, generated by zero-point fluctuation of the electron density and its polarization effect on neighbouring atoms. The energy of interaction is given by the London formula, as a function of zero-point vibration frequency, between atoms at a distance d apart, as ... 

As it is obvious, the intensity of the dispersion interaction is dependent on the polarizability of the adsorbate molecule and the adsorbent surface atom thereafter, the constant C can be calculated with the help of the London formula... 

In this way, for each H atom, the calculated dipole pseudospectra a, si] i = 1,2, , N of Table 4.2 can be used to obtain better and better values for the C6 London dispersion coefficient for the H—H interaction a molecular (two-centre) quantity C6 can be evaluated in terms of atomic (one-centre), nonobservable, quantities, a, (a alone is useless). The coupling between the different components of the polarizabilities occurs through the denominator in the London formula (4.19), so that we cannot sum over i or / to get the full, observable,12 aA oraB. 

Using the London formula and the pseudospectra derived previously, we obtain for the leading term of the H-H interaction the results collected in Table 4.3. The table shows that convergence is very rapid for the H-H interaction.13 Unfortunately, the convergence rate for G, (as well as that for a) is not so good for other systems (Magnasco, 2009a). 

Magnasco, V. and Ottonelli, M. (1999) Long-range dispersion coefficients from a generalization of the London formula. Trends Chem. Phys., 7, 215-232. 

In the present work a simple London formula for application of the valence bond (VB) method to the tree-center problem [28] is used in order to stress the importance of triplet excited states of the activating molecule in the catalysis by TM atoms and complexes. Activation of the C-H bond in methane is considered as a conceivable example illustrating essential features of spinuncoupling in catalysis. Theoretically studied reactions of hydrocarbons with the second-row TM atoms [15] are used for illustration of new features. 

See Reference 16, where a derivation of the original London formula is given. 

In any event, whether we derive the modified formula for either the total or the interaction energy, we will need Slater s generalization of the valence-bond method. Before we proceed, however, it may be well to warn the reader that the approximations made in deriving and applying the modified London formula are legion and clearly dubious. It is therefore indeed surprising that even rough relative results are obtained. 

Further details may be gleaned from consulting standard treatises. Suffice it to say here that Eq. (83) with the assumptions discussed above leads, for example, when = 4, to the modified London formula shown below ... 

The IJjjj terms are positive, so that E3 or -f gives less attraction than alone. This applies to the London formula before the introduction of experimental polarizabilities. On this basis, our equations are of the type of the Eqs. (24) and (26) of Reference 2 (p. 65), i.e. 

One also finds that interpair correlation energies are transferable if the two bonds and their relative position e.g. angle) are the same. For large distances between two electron pairs the interpair correlation energies follow asymptotically the London formula for dispersion energies, i.e. they go like R ). In fact the same formalism can be used to calculate interpair correlation energies within a molecule and dispersion energies between different molecules. This dispersion interaction is a correlation effect of the interpair type. 

Equation (46), usually attributed to Hudson and McCoubrey, was also proposed by Srivastava and Madan. It was obtained by equating the attractive portion of the Lennard-Jones potential to the London formula for Cj. The quantities f are the ionization energies of the pure components. A similar derivation based on the Kirkwood-Miiller treatment of dispersion forces leads to equation (47), which requires knowledge of the diamagnetic susceptibilities... 

This is also dipole-dipole interaction but between oscillating, not permanent dipoles. It is a pure quantum-mechanical effect of oscillatory motion of electrons in the ground state. It is desciihed by the London formula (here v is a frequency of a single oscillator considered) ... 

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