Energy total interaction

If the repulsive energy due to the electrostatic interactions is VgR and the steric stabilization is VgR then the total interaction energy is given by 

The interaction energy will affect the rate of particle coagulation or flocculation. A weak electrostatic interaction will not be suffident to overcome a strong van der Waals interaction and result in roapilatipn 

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Fig. XrV-6. (a) The total interaction energy determined from DLVO theory for n-hexadecane drops for a constant ionic strength - 5.0 nm) at various emulsion pH (b) enlargement of the secondary minimum region of (a). (From Ref. 39.)...

The total interaction energy of the nucleus may be expressed as a sum of the individual Hamiltonians given in equation B1.12.1, (listed in table B1.12.1) and are discussed in detail in several excellent books [1, 2, 3 and 4]. 

The combined effect of van der Waals and electrostatic forces acting together was considered by Derjaguin and Landau (5) and independently by Vervey and Overbeek (6), and is therefore called DLVO theory. It predicts that the total interaction energy per unit area, also known as the effective interface potential, is given by V(f) = ( ) + dl ( )- absence of externally imposed forces, the equiHbrium thickness of the Hquid film... 

In a recent paper. Mo and Gao [5] used a sophisticated computational method [block-localized wave function energy decomposition (BLW-ED)] to decompose the total interaction energy between two prototypical ionic systems, acetate and meth-ylammonium ions, and water into permanent electrostatic (including Pauli exclusion), electronic polarization and charge-transfer contributions. Furthermore, the use of quantum mechanics also enabled them to account for the charge flow between the species involved in the interaction. Their calculations (Table 12.2) demonstrated that the permanent electrostatic interaction energy dominates solute-solvent interactions, as expected in the presence of ion species (76.1 and 84.6% for acetate and methylammonium ions, respectively) and showed the active involvement of solvent molecules in the interaction, even with a small but evident flow of electrons (Eig. 12.3). Evidently, by changing the solvent, different results could be obtained. 

Table 6-1. Contribution to the total interaction energy from different energy decomposition schemes...

As the SIBFA approach relies on the use of distributed multipoles and on approximation derived form localized MOs, it is possible to generalize the philosophy to a direct use of electron density. That way, the Gaussian electrostatic model (GEM) [2, 14-16] relies on ab initio-derived fragment electron densities to compute the components of the total interaction energy. It offers the possibility of a continuous electrostatic model going from distributed multipoles to densities and allows a direct inclusion of short-range quantum effects such as overlap and penetration effects in the molecular mechanics energies. 

Saunders, S.R., Anand, M., You, S.S. and Roberts, C.B. (2010) Total interaction energy model to predict nanoparticle dispersibility in C02-expanded solvents, in Computer Aided Chemical Engineering, vol. 28, Elsevier, Amsterdam,... 

Interactions between two unlike ends of the dipoles are negative and, therefore, attractive, while those between two like ends are positive, and thus are repulsive. The total interaction energy is a summation over all ten dipoles, and if we assume that the calculation can be simplified by including only interactions between neighboring dipoles, the total energy can be calculated from equation (5.2). 

In a perturbation theory treatment of the total (not just electrostatic) interaction between the molecule and the point charge, QV(r) is the first-order term in the expression for the total interaction energy (which would include polarization and other effects). 

The series inside the parentheses converges to a sum that is 2 ln2 or 1.38629. This value is the Madelung constant for a hypothetical chain consisting of Na+ and Cl- ions. Thus, the total interaction energy for the chain of ions is —1.38629N0e2/r, and the chain is more stable than ion pairs by a factor of 1.38629, the Madelung constant. Of course NaCl does not exist in a chain, so there must be an even more stable way of arranging the ions. 

Note that, due to their infinite-range character, pure Coulombic potentials can actually lead to significant bond non-additivity for any proposed separation into bonded and nonbonded units. This reflects the fact that classical electrostatics is oblivious to any perceived separation into chemical units, because all Coulombic pairings (whether in the same or separate units) make long-range contributions to the total interaction energy. 

Most of statistical-mechanical computer simulations are based upon the assumption of pairwise additivity for the total interaction energy, what means to truncate the right side of equation (48) up to the two-body term. The remaining terms of the series, which are neglected in this approach, are often known as the nonadditive corrections. 

Let us consider two systems A and B, which interact through their atoms. If the interaction between the systems occur through the atom x of A with the atom k of the molecular system B, one can express the total interaction energy from the local point... 

The presence of polymers or polyelectrolytes have important effects on the Van der Waal interaction and on the electrostatic interaction. Bacterial adhesion, as discussed in Chapter 7.9 may be interpreted in terms of DLVO theory. Since the interaction in bacterial adhesion occurs at larger distances, this interaction may be looked at as occurring in the secondary minimum of the net interaction energy (Fig. 7.4). Particle Size. The DLVO theory predicts an increase of the total interaction energy with an increase in particle size. This effect cannot be verified in coagulation studies. 

This acceptance criteria is often made even more stringent by requiring that as few as 50% of the moves satisfying Equation 3 are actually selected. Note that these procedures always favor moves which lead to reduced total interaction energies. 

It is supposed that this displacement can occur only in the direction normal to the rupture surface, not parallel to the latter. When all the above distances are systematically varied until the minimum of the total interaction energy is reached, then the most probable position of all 6 centers of force is found. It appears that the distance between the Li nuclei in the outermost and the H nuclei in the second layer is smaller (by 0.00032 angstrom) than in the bulk of the crystal, and the distance between the H nuclei in the external and the Li nuclei in the penultimate layer is... 

This equation ignores interactions between the ions. The simplest way to treat the interactions is just to add them to this equation, and assume that the ions remain randomly arranged. Suppose U is the total interaction energy that a given ion would feel if all the other sites were full. When only a fraction x of the sites is occupied, it costs an extra energy Ux to add another ion to the lattice, so fi becomes... 

Total interaction energy is represented by a coulombic term (Vc) calculated between all charges (both electrons and nuclei), plus an electron gas term (1(g) related to local density in the overlap region ... 

TABLE 2.4. Decomposition of Total Interaction Energies (kcal/mol) for Nonconventional Hydrogen-Bonded Complexes ... 

Polanyi (27) as early as 1921 suggested that the activated state consists of free atoms which are adsorbed or bound by large affinities (heat of adsorption or diminution of the homogeneous activation energy) to the catalyst. A more quantitative treatment became possible after the theory of wave mechanics presented equations for the mutual interaction of different covalent bonds. London (28) showed that the total interaction energy of three atoms X, Y, and Z is... 

Using this data for the layer thickness, the total energy of interaction was calculated by summing the electrostatic and steric contributions, the latter of which was calculated according to the method of Ottewill. Total interaction energies at three values of interparticle separation are shown in Figure 11 as a function of polymer dose. 

Figure 11 Calculated total interaction energies at fixed interparticle separation as a function of polymer dose...

There are useful two- and many-electron analogues of the functions discussed above, but when the Hamiltonian contains only one- and two-body operators it is sufficient to consider the pair functions thus the analogue of p(x x ) is the pair density matrix 7t(xi,X2 x i,x ) while that of which follows on identifying and integrating over spin variables as in (4), is H(ri,r2 r i,r2)- When the electron-electron interaction is purely coulombic, only the diagonal element H(ri,r2) is required and the expression for the total interaction energy becomes... 

The A scp term is calculated using the standard CP-method. At the correlated MP2 level, we have shown for several systems [7-10], that the AE terms are usually and systematically smaller than the dominant ( )+ Ecj) terms. The sum of these two terms provides a good approximation to the total interaction energy at the correlated level. It is important to emphasize that the AE values were obtained by making the difference with the values of the CP-corrected subsystems i.e. taking into consideration the "benefit effect" of the superposition of the basis set [3, 6]. As the charge-transfer components are of importance in the two-body interaction, (see a discussion in ref. 10), we will also investigate them separately for the three-body terms in the studied systems. 

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