Dispersion energies

Dispersion forces act between all molecules, although they are absent in the interaction of a proton and an atom. They result from intermolecular correlations in the fluctuations of the electronic coordinates of the molecules, and are a consequence of the quantum-mechanical nature of the electron. 

By writing the frequencies of two dipole oscillators, and using the zero-point energy expression for a harmonic oscillator with the same proper frequency (equation 3.24), London [28] was able to derive the following expression for the dispersion energy between two molecules of polarizability a at a distance R  

This stabilization has to do with zero-point oscillator energies, so dispersion energy is a consequence of the uncertainty principle, like all zero-point energy effects (Section 3.1). 

The reader will have noticed that there is no attempt to relate dispersion energies in the London model with some parts of the quantum mechanical expectation energies. It is not entirely clear, or at least it is not easily explained, if and how London forces 

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Unfortunately none of the various proposed forms of the potential theory satisfy this criterion Equation XVII-78 clearly does not Eq. XVII-79 would, except that / includes the constant A, which contains the dispersion energy Uo, which, in turn, depends on the nature of the adsorbent. Equation XVII-82 fares no better if, according to its derivation, Uo reflects the surface polarity of the adsorbent (note Eq. VI-40). It would seem that after one or at most two layers of coverage, the adsorbate film is effectively insulated from the adsorbent. 

Long-range forces are most conveniently expressed as a power series in Mr, the reciprocal of the intemiolecular distance. This series is called the multipole expansion. It is so connnon to use the multipole expansion that the electrostatic, mduction and dispersion energies are referred to as non-expanded if the expansion is not used. In early work it was noted that the multipole expansion did not converge in a conventional way and doubt was cast upon its use in the description of long-range electrostatic, induction and dispersion interactions. However, it is now established [8, 9, 10, H, 12 and 13] that the series is asymptotic in Poincare s sense. The interaction energy can be written as... 

Some electric properties of molecules are described in section Al.5.2.2 because the coefficients of the powers of Mr turn out to be related to them. The electrostatic, mduction and dispersion energies are considered m turn in section Al.5.2.3, section Al.5.2.4 and section Al.5.2.5, respectively. 

Hence, the same teclmiques used to calculate are also used for Cg. Note that equation (A1.5.28) has a geometrical factor whose sign depends upon the geometry, and that, unlike tlie case of the two-body dispersion interaction, the triple-dipole dispersion energy has no minus sign in front of the positive coefficient Cg. For example, for an equilateral triangle configuration the triple-dipole dispersion is repulsive and varies... 

The details of the second-order energy depend on the fonn of exchange perturbation tiieory used. Most known results are numerical. However, there are some connnon features that can be described qualitatively. The short-range mduction and dispersion energies appear in a non-expanded fonn and the differences between these and their multipole expansion counterparts are called penetration tenns. 

Kumar A, Fairley G R G and Meath W J 1985 Dipole properties, dispersion energy coefficients and integrated oscillator strengths for SFg J. Chem. Phys. 83 70... 

Kumar A and Meath W J 1992 Dipole oscillator strength properties and dispersion energies for acetylene and benzene Mol. Phys. 75 311... 

Meath W J and Kumar A 1990 Reliable isotropic and anisotropic dipole dispersion energies, evaluated using constrained dipole oscillator strength techniques, with application to interactions involving H2, N2 and the rare gases Int. J. Quantum Chem. Symp. 24 501... 

Knowles P J and Meath W J 1986 Non-expanded dispersion energies and damping functions for A 2 and 2 Chem. Phys. Lett. 124 164... 

Wheatley R J and Meath W J 1993 Dispersion energy damping functions, and their relative scale with interatomic separation, for (H,He,Li)-(H,He,Li) interactions Mol. Phys. 80 25... 

The Lennard-Jones potential is characterised by an attractive part that varies as r ° and a repulsive part that varies as These two components are drawn in Figure 4.35. The r ° variation is of course the same power-law relationship foimd for the leading term in theoretical treatments of the dispersion energy such as the Drude model. There are no... 

The dispersion (London) force is a quantum mechanieal phenomenon. At any instant the electronic distribution in molecule 1 may result in an instantaneous dipole moment, even if 1 is a spherieal nonpolar moleeule. This instantaneous dipole induces a moment in 2, which interacts with the moment in 1. For nonpolar spheres the induced dipole-induced dipole dispersion energy function is... 

Recall that regular solution theory deals with nonpolar solvents, for which the dispersion force is expected to be a major contributor to intermolecular interactions. The dispersion energy, from Eq. (8-15), is for 1-2 interactions... 

Nader, L., and Milleron, N. (1979). Dimensions of the People Problem in Energy Research and the Factual Basis of Dispersed Energy Futures. Energy 4(5) 953-967. 

With this basis, one gets for the constant in the London dispersion energy formula E = — j... 

The opposite situation holds for reactions that have negative values for A iiT ° and A S °. These reactions are spontaneous at low temperature because their release of heat disperses energy into the surroundings. The favorable AH ° dominates A G ° as long as T does not become too large, and the reaction is enthalpy-driven. At high temperature, however, the unfavorable A S ° dominates A G °, and the reaction is no longer spontaneous. The effects of temperature on spontaneity are summarized in Table 14-3. 

The free energy changes in metabolic oxidation reactions serve several purposes. Part of this energy appears as heat flows that maintain body temperature and disperse energy to the surroundings. Another portion of the energy... 

The dispersion force between two atoms results in the dispersion energy D, which is approximately proportional to r 6, r being the distance between the atoms ... 

The dispersion energy between two molecules results approximately from the sum of the contributions of atoms of one molecule to atoms of the other molecule. 

This dispersion energy is coupled to an exchange-dispersion component ... 

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