Potential energy London force

The dispersion force is due to induced dipole interaction between atoms or molecules through electron density fluctuations. According to London, the potential energy of interaction London is given by... 

In addition to the static induction effects included in I/scf, the hot Drude oscillators give rise to a 1/r6, temperature-dependent, attractive term. This jkg Ta2/r6 term is the classical thermodynamic equivalent of the London quantum dispersive attraction IEa2/r6. It corresponds to a small perturbation to the London forces, because k T is at least two orders of magnitude smaller than the typical ionization energy IE. The smaller the temperature of the Drude motion, the closer the effective potential is to the SCF potential, making Eq. (9-57) independent of mo, the mass of the oscillators. 

The pair potential of colloidal particles, i.e. the potential energy of interaction between a pair of colloidal particles as a function of separation distance, is calculated from the linear superposition of the individual energy curves. When this was done using the attractive potential calculated from London dispersion forces, Fa, and electrostatic repulsion, Ve, the theory was called the DLVO Theory (from Derjaguin, Landau, Verwey and Overbeek). Here we will use the term to include other potentials, such as those arising from depletion interactions, Kd, and steric repulsion, Vs, and so we may write the total potential energy of interaction as... 

Lohman, electrode potential, 1457 London forces. 896 Long range interactions, 936 Lorenz and Salie, and partial charge transfer, 922 Lorenz, 1313, 1497 Louis de Broglie postulate. 788 Low energy electron diffraction (LEED), 788, 787, 790... 

One of the more profound manifestations of quantum mechanics is that this curve does not accurately describe reality. Instead, because the motions of electrons are correlated (more properly, the electronic wave functions are correlated), the two atoms simultaneously develop electrical moments that are oriented so as to be mutually attractive. The force associated with tills interaction is referred to variously as dispersion , the London force, or the attractive van der Waals force. In the absence of a permanent charge, the strongest such interaction is a dipole-dipole interaction, usually referred to as an induced dipole-induced dipole interaction, since the moments in question are not permanent. Such an interaction has an inverse sixtli power dependence on the distance between the two atoms. Thus, the potential energy becomes increasingly negative as the two noble gas atoms approach one another from infinity. 

Another explanation must therefore be found. Now we know that besides forces of an electrical character there are others which act between atoms. Even the noble gases attract one another, although they are non-polar and have spherically symmetrical electronic structures. These so-called van der Waals forces cannot be explained on the basis of classical mechanics and London was the first to find an explanation of them with the help of wave mechanics. He reached the conclusion that two particles at a distance r have a potential energy which is inversely proportional to the sixth power of the distance, and directly proportional to the square of the polarizability, and to a quantity

excitation energies of the atom, so that... 

Typical potential energy curves for the interaction of two atoms are illustrated in Figure 11.3. There is characteristically a very steeply rising repulsive potential at short interatomic distances as the two atoms approach so closely that there is interpenetration of their electron clouds. This potential approximates to an inverse twelfth-power law. Superimposed upon this is an attractive potential due mainly to the London dispersion forces. This follows an inverse sixth-power law. The total potential energy is given by... 

Keesom, Debye, and London contributed much to our understanding of forces between molecules [111-113]. For this reason the three dipole interactions are named after them. The van der Waals4 force is the Keesom plus the Debye plus the London dispersion interaction, thus, all the terms which consider dipole-dipole interactions Ctotai = Corient+Cind- -Cdisp. All three terms contain the same distance dependency the potential energy decreases with l/D6. Usually the London dispersion term is dominating. Please note that polar molecules not only interact via the Debye and Keesom force, but dispersion forces are also present. In Table 6.1 the contributions of the individual terms for some gases are listed. 

Figure S.3 Potential energies of interaction between two colloidal particles as a function of their distance of separation, for electrical double layers due to surface charge (VolK London-van der Waals dispersion forces (V ), and the total interaction (VT). From Schramm [426], Copyright 2003, Wiley.

If the London force is expressed as the gradient of the potential energy of interaction 4>> the total flux may be written as the sum of the London and diffusive fluxes ... 

The rate of deposition of particles onto a surface, in the presence of London, double-layer, and gravitational forces, is calculated in terms of the energy of interaction between cell and surface by assuming that Brownian motion over a potential energy barrier is the rate-determining step of the... 

Van der Waals interactions are noncovalent and nonelectrostatic forces that result from three separate phenomena permanent dipole-dipole (orientation) interactions, dipole-induced dipole (induction) interactions, and induced dipole-induced dipole (dispersion) interactions [46]. The dispersive interactions are universal, occurring between individual atoms and predominant in clay-water systems [23]. The dispersive van der Waals interactions between individual molecules were extended to macroscopic bodies by Hamaker [46]. Hamaker s work showed that the dispersive (or London) van der Waals forces were significant over larger separation distances for macroscopic bodies than they were for singled molecules. Through a pairwise summation of interacting molecules it can be shown that the potential energy of interaction between flat plates is [7, 23]... 

The ftrst two terms within brackets define the van der Waals repulsions, which vary as l/r. and the London dispersion attractions, which vary as l/r . The con.stantiT.., is related to the size of the atom pair being considered. r,j is the distance between the atom pairs, and e,j refers to the depth of the potential energy well. It i.s based on the Lcnnard-Jones 6-12 potential. Many force fields u.se functions of this type to describe steric interactions (Fig. 28-9). Only atoms with a 1.4 nonbonded relation.ship to one another (i.e.. with three chemical bonds. separating them) are included in these calculations. The bending and stretching terms include I..1 nonbonded attractive and repulsion terms implicitly. 

The potential energy describing this situation is that of long-range attraction with the form - (l/r "), where r is the distance between a pairofgas phase molecules. This type of attraction primarily arises from dispersion or London forces (see Chapter 5). 

The potential energy function, u(h), where h is the distance of separation, consists of London-van der Waals attractive forces, u and double-layer repulsion forces, u as given below ... 

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