Models for electrostatic interactions

For molecules possessing charges, due to electron deficiency or electron excess in their constituent atoms, or for molecules possessing dipole, 

SOUECE Appendix, Table I-A, J.O. Hirschfelder, C.F. Curtiss, and R.B. Bird, Molecular Theory of Gases and Liquids, Wiley, New York, 1960. Table 1-A contains an extensive listing. Note that k is Boltzmann s constant, k = 1.380 x 10 Joule molecule K , and lA = 10 m. 

The simplest electrostatic interaction is the Coulomb potential between two point charges q and Q2, located a distance r apart  

Interlude 1.1 Mathematical Essentials The Euler Angles A description of 

Fortunately for us, the problem of assigning rotational angles in unambiguous ways has been mathematically resolved. There are several different sets of rotational angle representations, and we will consider only one of these here—the so-called Euler angles. Other representations can be foimd in the further list Reading Section at the end of this chapter. 

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Dudley, L. M. and B. L. McNeal. 1987. A model for electrostatic interactions among charged sites of water-soluble, organic polyions. 1. Description and sensitivity. Soil Sci. 143 329. 

The electrostatic free energy change in this equation differs somewhat from that used by Caselli et Indeed, Leodidis and Hatton introduce the excess free energy of the bulk water whereas Caselli et consider only the water pool contribution. Moreover, the main difference is the model for electrostatic interactions. Leodidis and Hatton eliminated the Poisson-... 

Wisz, M.S. and Hellinga, H.W. (2003) An empirical model for electrostatic interactions in proteins incorporating multiple geometry-dependent dielectric constants. Proteins, 51, 360-377. 

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. 

In this model of electrostatic interactions, two atoms (i and j) have point charges q and qj. The magnitude of the electrostatic energy (Veel) varies inversely with the distance between the atoms, Ry. The effective dielectric constant is 8. For in vacuo simulations or simulations with explicit water molecules, the denominator equals eRij. In some force fields, a distance-dependent dielectric, where the denominator is eRy Rjj, represents solvent implicitly. 

A very simple version of this approach was used in early applications. An alchemical charging calculation was done using a distance-based cutoff for electrostatic interactions, either with a finite or a periodic model. Then a cut-off correction equal to the Born free energy, Eq. (38), was added, with the spherical radius taken to be = R. This is a convenient but ill-defined approximation, because the system with a cutoff is not equivalent to a spherical charge of radius R. A more rigorous cutoff correction was derived recently that is applicable to sufficiently homogeneous systems [54] but appears to be impractical for macromolecules in solution. 

SH Bryant, CE Lawrence. The frequency of lon-pair substructures m proteins is quantitatively related to electrostatic potential A statistical model for nonbonded interactions. Proteins 9 108-119, 1991. 

Metal binding by a hydrous oxide from a 10 7 M solution (SOH + Me2+ OMe+ + H+) for a set of equilibrium constants (see Eqs. (i) - (iii) from Example 2.3) and concentration conditions (see text). Corrected for electrostatic interactions by the diffuse double layer model (Gouy Chapman) for 1 = 01 The hydrolysis of Me2+ is neglected. 

The Gordon-Kim interaction functions may be compared with empirical potential functions derived by energy- or net-force minimization methods using known crystal structures. The O—O Gordon-Kim potentials are more repulsive, as illustrated in Fig. 9.2. Spackman points out that the empirical potentials likely contain a significant attractive component because of the inadequate allowance for electrostatic interactions in their derivation. This attractive component is included in the electrostatic interaction in the density functional model. 

Of course, concerns about periodicity only relate to systems that are not periodic. The discussion above pertains primarily to the simulations of liquids, or solutes in liquid solutions, where PBCs are a useful approximation that helps to model solvation phenomena more realistically than would be the case for a small cluster. If the system truly is periodic, e.g., a zeolite crystal, tlien PBCs are integral to the model. Moreover, imposing PBCs can provide certain advantages in a simulation. For instance, Ewald summation, which accounts for electrostatic interactions to infinite length as discussed in Chapter 2, can only be carried out within the context of PBCs. 

Figure 7. A simple model to account for electrostatic interactions in the ES-complex of chymosin and K-casein...

Within the dielectric continuum model, the electrostatic interactions between a probe and the surrounding molecules are described in terms of the interaction between the charges contained in the molecular cavity, and the electrostatic potential these changes experience, as a result of the polarization of the environment (the so-called reaction field). A simple expression is obtained for the case of an electric dipole, /a0, homogeneously distributed within a spherical cavity of radius a embedded in an anisotropic medium [10-12], by generalizing the Onsager model [13]. For the dipole parallel (perpendicular) to the director, the reaction field is parallel (perpendicular) to the dipole, and can be calculated as [10] ... 

For example, suppose one can choose a rigid three-point-charge model of water with an internal geometry of 109.47° and 100 pm for the HOH angle and OH distance, respectively. The interaction energy involves a Lennard-Jones 6-12 potential for electrostatic interactions between water-water and ion-water pairs, (/pair a nonadditive polarization energy, C/pg, and a term that includes exchange repulsion for ion-water and water-water pairs,... 

Generalized Bom (GB) approach. The most common implicit models used for small molecules are the Conductor-Like Screening Model (COSMO) [77,78], the DPCM [79], the Conductor-Like Modification to the Polarized Continuum Model (CPCM) [80,81], the Integral Equation Formalism Implementation of PCM (IEF-PCM) [82] PB models, and the GB SMx models of Cramer and Truhlar [23,83-86]. The newest Minnesota solvation models are the SMD universal Solvation Model based on solute electron density [26] and the SMLVE method, which combines the surface and volume polarization for electrostatic interactions model (SVPE) [87-89] with semiempirical terms that account for local electrostatics [90]. Further details on these methods can be found in Chapter 11 of Reference [23]. 

The MSA is linear in the pair potentials and has been applied to a variety of QA models with electrostatic interactions [305-307]. However, concerning the blocked correlations, the MSA has the same drawbacks as the PY closure (to whicli the MSA, in fact, reduces for pure hard-core models). As a final example, we present the HNC closure defined by... 

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