Additivity of the second-order dispersion energy

The dispersion energy is a second-order correction, eq. (13.12) on p. 695 gives the formula for the interaction of two molecules. For three molecules we obtain the following formula for the dispersion part of the second-order effect (cf. the discussion on the induction energy on p. 736) 

In the first term we can integrate over the coordinates of C. Then the first term displayed in the above formula turns out to be the dispersion interaction of A andB, 

One of the third-order energy terms represents a correction to the dispersion energy. The correction as shown by Axilrod and Teller has a three-body character. The part connected to the interaction of three distant instantaneous dipoles on A, B and C reads as 

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Additivity of the Second-order Dispersion Energy Non-additivity of the Third-order Dispersion Interaction... 

The interaction energy of N molecules is not pairwise additive i.e., it is not the sum of the interactions of all possible pairs of molecules. Among the energy corrections up to the second order, the exchange and, first of all, the induction terms contribute to the non-additivity. The electrostatic and dispersion (in the second order) contributions ate pairwise additive. 

Based on the above-mentioned assumptions, the mass, momentum and energy balance equations for the gas and the dispersed phases in two-dimensional, two-phase flow were developed [14], In order to solve the mass, momentum and energy balance equations, several complimentary equations, definitions and empirical correlations were required. These were presented by [14], In order to obtain the water vapor distribution the gas phase the water vapor diffusion equation was added. During the second drying period, the model assumed that the particle consists of a dry crust surrounding a wet core. Hence, the particle is characterized by two temperatures i.e., the particle s crust and core temperatures. Furthermore, it was assumed that the heat transfer from the particle s cmst to the gas phase is equal to that transferred from the wet core to the gas phase i.e., heat and mass cannot be accumulated in the particle cmst, since all the heat and the mass is transferred by diffusion through the cmst from the wet core to the surrounding gas. Based on this assumption, additional heat balance equation was written. 

For multi-molecular assemblies one has to consider whether the total interaction energy can be written as the sum of pairwise interactions. The first-order electrostatic interaction is exactly pairwise additive, the dispersion only up to second order (in third order a generally small three-body Axilrod-Teller term appears [73]) while the induction is not at all pairwise it is non-linearly additive due to the interference of electric fields from different sources. Moreover, for polar systems the inducing fields are strong enough to change the molecular wave functions significantly. 

The dispersion energy is the universal attractive glue that leads to the formation of condensed phases. It is additive at second order in perturbation theory, and the form of the three-body term that arises at third order (the tripledipole dispersion term) is also well known from perturbation theory. This Axilrod-Teller term " was the only addition to the pair potential for argon that was required to quantitatively account for its solid and liquid state properties. This may be grounds for optimism that other nonadditive dispersion terms are negligible. Whether this can be extended to less symmetrical organic molecules and their typical crystalline and liquid environments has not yet been established however. 

In addition to the above methods utilizing conventional ionization modes, the field ionization technique has appeared [75]. The very intense electric field (about 1 V/A), produced by an electrode, results in the ionization of molecules in the gas phase. This soft ionization technique is often used competitively with Cl, since it does not pollute the source and may yield sufficiently reproducible results. The transit time of ions in the source is on the order of 10 to 10 second. The radical molecular ions (M ) produced are characterized by a low internal energy, and thus can be detected easily. As a result of dispersion within the source, however, sensitivity is about two orders of magnitude lower than that of El. As in the case of El, the fragments produced by FI can furnish interesting structural data on carbohydrates, amino acids, peptides and cardenolides [76],... 

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