The total interaction

Interaction type Distance dependence of potential energy Typical energy (kJ moT ) Comment 

Dipole-dipole 1/r 2 Between stationary polar molecules 

London (dispersion) l/r 2 Between all types of molecules and ions 

To treat the myriad interactions in bioiogical assemblies quantitatively, we need simple 

One such approximation is to express the short-range repulsive potential energy as inversely proportional to a high power of r. 

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The total interaction between two slabs of infinite extent and depth can be obtained by a summation over all atom-atom interactions if pairwise additivity of forces can be assumed. While definitely not exact for a condensed phase, this conventional approach is quite useful for many purposes [1,3]. This summation, expressed as an integral, has been done by de Boer [8] using the simple dispersion formula, Eq. VI-15, and following the nomenclature in Eq. VI-19 ... 

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]. 

Here we consider the total interaction between two charged particles in suspension, surrounded by tlieir counterions and added electrolyte. This is tire celebrated DLVO tlieory, derived independently by Derjaguin and Landau and by Verwey and Overbeek [44]. By combining tlie van der Waals interaction (equation (02.6.4)) witli tlie repulsion due to the electric double layers (equation (C2.6.lOI), we obtain... 

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... 

Since the term hydration refers to aqueous solutions only, the word solvation was introduced as a general term for the process of forming a solvate in solution. The terms solvation and heat of solvation were introduced at a time when little or nothing was known about polar molecules. We know now that, when an atomic ion is present in a solvent, the molecular dipoles are subject to the ionic field, whose intensity falls off in 1/r2. We cannot draw a sphere round the ion and say that molecules within this sphere react with the ion to form a solvated ion, while molecules outside do not. The only useful meaning that can now be attached to the term solvation is the total interaction between ion and solvent. As already mentioned, this is the sense in which the term is used in this book. 

The dy. p and dp yi orbitals each interact with four point charges in precisely the same way as does the dyi yi orbital. Again the repulsion relates to electron density, so the total interaction of the combination is (4/ /2) + (4/ /2) = 16 of our repulsion units. In other words, the d.2 and dp y2 orbitals are degenerate in octahedral symmetry. 

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. 

However, due to the availability of numerous techniques, it is important to point out here the differences and equivalence between schemes. To summarize, two EDA families can be applied to force field parametrization. The first EDA type of approach is labelled SAPT (Symmetry Adapted Perturbation Theory). It uses non orthogonal orbitals and recomputes the total interaction upon perturbation theory. As computations can be performed up to the Coupled-Cluster Singles Doubles (CCSD) level, SAPT can be seen as a reference method. However, due to the cost of the use of non-orthogonal molecular orbitals, pure SAPT approaches remain limited... 

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. 

Recent analysis has shown that this approximation is, in general, insufficient.6 This is due to the long-range character of the interaction of the electron with the medium polarization. The zeroth-order states determined from Eqs. (8) taking into account the total interaction of the electron with the total inertial polarization of the medium VeP may not describe the states of the electron localized in the donor or in the acceptor sites. Since the polarization varies due to thermal fluctuations, at certain configurations of the... 

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). 

Similar expressions to equations (8) to (10) may be derived for spherical particles. The form of the total interaction (i.e. 

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. 

A rigourous way to evaluate the total interaction potential energy, U(q(N- ), would be the formulation and resolution of the Schrodinger equation for the whole system at each configuration. However, given the size of the samples where the statistical simulations are performed, this method is impracticable. 

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... 

One further theoretical method that merits consideration at this point is the topological theory of molecular structure exemplified by Bader (1985, 1990). In this method a topological description of the total electron density in the molecule is used. A major advantage of this method is that it allows the total interaction between various centres to be probed. Cremer et al. (1983) used the Bader method to examine the homotropylium cation [12] and concluded that it was indeed homoaromatic. 

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. 

In most MC (11,12) and MD (12,13) studies, a small number (N) of particles are placed in a cell of fixed volume (V) and the total interaction potential energy (U ) from all pairwise interaction potentials (U j) between particles i and j is calculated ... 

To determine the movement of molecules, the following algorithm (15) is often used. The force acting on the ith atom in a molecule (Fj) is determined from the spatial derivative of the total interaction potential energy of that particle ... 

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... 

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