Electrostatic interactions Coulombic

The fourth term on the right-hand side of eq. (11.3) is the electrostatic interaction (Coulomb s law) between pairs of charged atoms i and j, separated by distance r j. Since electrostatic interactions fall off slowly with r (only as r-1) they are referred to as long-range and, for an infinite system such as a periodic solid, special techniques, such as the Ewald method, are required to sum up all the electrostatic interactions (cf. Section 7.1) (see e.g. Leach, Jensen (Further reading)). The... 

Specific electrolyte solutions have been discovered and intensively smdied which can produce a phase with a higher, and a phase with a lower salt concentration. The reasons for phase separation may be different. For aqueous solutions of large organic ions (e.g., tetra-n-pentylammonium bromide), the phenomenon is ascribed to hydrophobic interaction for large ions in low permittivity solvents (e.g., tetra-n-pentylammoniumpicrate in 1-chloropentane), it is due to the long-range electrostatic interactions (coulombic phase separation). 

The viscosity of electrolyte solutions is also of research interest because of the long-range electrostatic interactions (Coulombic forces) between ions (Harrap and Heymaim, 1951 Stokes and Mills, 1965 Home, 1972 Horvath, 1985 Chandra and Bagchi, 2000a,b Esteves etal, 2001 Anderko et al, 2002 Jiang and Sandler, 2003). 

Whereas dispersion forces are straightforward to model, this is not the case with electrostatic interactions. Coulombic interactions act at significantly longer distances, falling off with r instead of r , and require special treatment, with a method called Ewald summation. A discussion of Ewald s method is beyond the scope of this book, and the interested reader is referred to the text by Allen and Tildesley (Further reading). 

Ihi.. same molecule but separated by at least three bonds (i.e. have a 1, h relationship where n > 4). In a simple force field the non-bonded term is usually modelled using a Coulomb piilential term for electrostatic interactions and a Lennard-Jones potential for van der IV.uls interactions. 

VVe therefore return to the point-charge model for calculating electrostatic interactions. If sufficient point charges are used then all of the electric moments can be reproduced and the multipole interaction energy. Equation (4.30), is exactly equal to that calculated from the Coulomb summation. Equation (4.19). 

There is a very convenient way of writing the Hamiltonian operator for atomic and molecular systems. One simply writes a kinetic energy part — for each election and a Coulombic potential Z/r for each interparticle electrostatic interaction. In the Coulombic potential Z is the charge and r is the interparticle distance. The temi Z/r is also an operator signifying multiply by Z r . The sign is - - for repulsion and — for atPaction. 

Electrostatics is the study of interactions between charged objects. Electrostatics alone will not described molecular systems, but it is very important to the understanding of interactions of electrons, which is described by a wave function or electron density. The central pillar of electrostatics is Coulombs law, which is the mathematical description of how like charges repel and unlike charges attract. The Coulombs law equations for energy and the force of interaction between two particles with charges q and q2 at a distance rn are... 

The energy of solvation can be further broken down into terms that are a function of the bulk solvent and terms that are specifically associated with the first solvation shell. The bulk solvent contribution is primarily the result of dielectric shielding of electrostatic charge interactions. In the simplest form, this can be included in electrostatic interactions by including a dielectric constant k, as in the following Coulombic interaction equation ... 

EIectrosta.tlcs. Electrostatic interactions, such as salt bridges, result from the electrostatic attraction that occurs between oppositely charged molecules. These usually involve a single cation, eg, the side chain of Lys or Arg, or the amino terminus, etc, interacting with a single anion, eg, the side chain of Glu or Asp, or the carboxyl terminus, etc. This attractive force is iaversely proportional to the distance between the charges and the dielectric constant of the solvent, as described by Coulomb s law. 

The only problem with the foregoing approach to molecular interactions is that the accurate solution of Schrddinger s equation is possible only for very small systems, due to the limitations in current algorithms and computer power. Eor systems of biological interest, molecular interactions must be approximated by the use of empirical force fields made up of parametrized tenns, most of which bear no recognizable relation to Coulomb s law. Nonetheless the force fields in use today all include tenns describing electrostatic interactions. This is due at least in part to the following facts. 

If classical Coulombic interactions are assumed among point charges for electrostatic interactions between solute and solvent, and the term for the Cl coefficients (C) is omitted, the solvated Eock operator is reduced to Eq. (6). The significance of this definition of the Eock operator from a variational principle is that it enables us to express the analytical first derivative of the free energy with respect to the nuclear coordinate of the solute molecule R ,... 

Calculation of the energies and forces due to the long-range Coulomb interactions between charged atoms is a major problem in simulations of biological molecules (see Chapter 5). In an isolated system the number of these interactions is proportional to N-, where N is the number of charged atoms, and the evaluation of the electrostatic interactions quickly becomes intractable as the system size is increased. Moreover, when periodic... 

In the original, elementary treatment governed by Eq. 4 above, one might initially expect contributions to the barrier from several sources. There is first the Coulomb integral Q, which will contain angle dependent terms from the electrostatic interaction of the electrons and protons ar the two ends of the molecule. In this treatment the only orbitals used are Is on each H atom and tetra-... 

The diazonio group is a somewhat more complex substituent for such evaluations because it is charged, in contrast to the majority of substituents on which the Hammett treatment is based. The electrostatic interaction of the diazonio and other charged groups was calculated by Hoefnagel et al. (1978) and by Exner (1978). The substituent constants they obtained, including the effects of coulomb interactions, are only slightly different from those of Lewis and Johnson (1959). 

Group (1) Cations and anions which are incapable of donor-acceptor interactions. These are the large univalent ions. Bonding is purely by Coulomb and Madelung electrostatic interactions. From the Lewis point of view these are not acids or bases. They have no cement-forming potential. 

The first two terms on the right-hand side of Eq. (83) are usually assumed to be harmonic, as given for example by Eq. (6-74). The third term is often developed in a Fourier series, as given by Eq. (82). The potential function appropriate to the interaction between nonbonded atoms is taken to be of the Lennard-Jones type (Section 6.7.3). In all of these cases the necessary force constants are estimated by comparing the results obtained from a large number of similar molecules. If electrostatic interactions are to be considered, effective atomic charges must be suggested and Coulomb s law applied directly [see Eq. (6-81)]. 

The GEM force field follows exactly the SIBFA energy scheme. However, once computed, the auxiliary coefficients can be directly used to compute integrals. That way, the evaluation of the electrostatic interaction can virtually be exact for an perfect fit of the density as the three terms of the coulomb energy, namely the nucleus-nucleus repulsion, electron-nucleus attraction and electron-electron repulsion, through the use of p [2, 14-16, 58],... 

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