Effective Born radii

A. rather complex procedure is used to determine the Born radii a values of which. calculated for each atom in the molecule that carries a charge or a partial charge. T Born radius of an afom (more correctly considered to be an effective Born radii corresponds to the radius that would return the electrostatic energy of the system accordi to the Bom equation if all other atoms in the molecule were uncharged (i.e. if the other ato only acted to define the dielectric boundary between the solute and the solvent). In Sti force field implementation, atomic radii from the OPLS force field are assigned to ec... 

The bottleneck of GB calculations is the determination of the effective Born radii (a), since their magnitude depends not only on the intrinsic atomic (Bondi, or atomic) radii of the atom (p), but also on the geometry of the rest of the molecule, which modulates the average distance of the atom to the solvent. Original formulations of the method used Equation (4.19) for the computation of Born s radii, but all current version rely on an approximate formalism, such as that developed by Hawkins et al. [34], where a pairwise approximation to atomic overlap is used to simplify Equation (4.19) see Equations (4.20)-(4.22). Using this approach Born s radii can be computed very fast, which makes the method suitable for MD calculations ... 

The GB model is a modification of the Coulomb equation to include the Born radius of the particle or atom which estimates the degree of the particle s burial within the molecule. Equation 5 relates AGdec to the solvent/solute dielectric (e), the separation between the partial atomic charges r, the effective Born radii R(i and /), and the smoothing function fGR. A Debye-Hiickel screening parameter (k) similar to that used in the PB equation is used to account for the monovalent ions. 

Onufriev, A., Case, D., Bashford, D. Effective Born radii in fhe Generalized Bom approximation The imporfance of being perfecf. J. Compuf. Chem. 2002, 23,1297-304. 

Mongan, J., Svrcek-Seiler, W. A., and Onufriev, A. [2007]. Analysis of integrai expressions for effective Born radii, / Chem. Phys. 127, pp. 185101 1-10. 

Onufriev, A, Case, D. A, and Bashford, D. (2002). Effective Born radii in the generalized Born model approximation The importance of being perfect,/. Comput. Chem. 23,14, pp. 1297-1304. 

The quantity in Eq. (11.33) denotes the "perfect" effective Born radius for q, [64], the efficient and accurate computation of which is a major part of the development of GB models. To define let denote the exact polarization energy (obtained by solving Poisson s equation) for the atomic charge q, in a cavity representative of the entire molecule. (That is, we turn off all charges... 

Effective Electrostatic Radius, Born Coefficient, and Solvation Energy... 

The solvation energy described by the Born equation is essentially electrostatic in nature. Born equations 8.116 and 8.120 are in fact similar to the Born-Lande equation (1.67) used to define the electrostatic potential in a crystal (see section 1.12.1). In hght of this analogy, the effective electrostatic radius of an ion in solution r j assumes the same significance as the equilibrium distance in the Born-Lande equation. We may thus expect a close analogy between the crystal radius of an ion and the effective electrostatic radius of the same ion in solution. 

In 1920, Max Born, a Nobel Prize winner, published some work on the free energy of solvation of ions, AGgon [21]. He conceived the idea of approximating the solvent surrounding the ion as a dielectric continuum. Defining a spherical boundary between the ion and the continuum by an effective ion-radius, f lon, he got the simple result... 

One of the diflSculties in applying the Born equation is that the effective radius of the ion is not known further, the calculations assume the dielectric constant of the solvent to be constant in the neighborhood of the ion. The treatment has been modified by Webb who allowed for the variation of dielectric constant and also for the work required to compress the solvent in the vicinity of the ion further, by expressing the effective ionic radius as a function of the partial molal volume of the ion, it was possible to derive values of the free energy of solvation without making any other assumptions concerning the effective ionic radius. 

A number of theoretical difficulties in equating the Born function with this process are believed to be accommodated in the j parameter, which in the Bom model is the ion radius, but in the HKF model is an adjustable parameter called the effective ionic radius. The r j parameters were originally related to crystallographic ionic radii (r ) and ionic charge Zj in a simple linear fashion in the HKF model, and were independent of T and P... 

Thus, with the help of Marcus equation, we can obtain some useful estimates and predictions. The quantitative accuracy of this theory, however, should not be overstated. It was shown above that this theory is based on the same physical model as the Born theory of ion solvation and hence suffers from the same drawbacks. We do not know whether any attempts have been made to take into account the effect of dielectric saturation of the medium in the vicinity of ions in kinetics. An attempt to take into account the spatial dipole correlation while considering the redox reaction MnOi+ /MnO was made by Dolin et al,[237]. As mentioned in section 3.2, the correlation in dipole orientation leads, as it were, to an increase in the effective ionic radius. Consequently, it should somewhat decrease the activation energy. According to estimates in [237], this effect is not strong, but it must increase with decreasing ionic radius. 

In equation (2) Rq is the equivalent capillary radius calculated from the bed hydraulic radius (l7), Rp is the particle radius, and the exponential, fxinction contains, in addition the Boltzman constant and temperature, the total energy of interaction between the particle and capillary wall force fields. The particle streamline velocity Vp(r) contains a correction for the wall effect (l8). A similar expression for results with the exception that for the marker the van der Waals attraction and Born repulsion terms as well as the wall effect are considered to be negligible (3 ). 

The simplest approach to describing the interactions of metal cations dissolved in water with solvent molecules is the Born electrostatic model, which expresses solvation energy as a function of the dielectric constant of the solvent and, through transformation constants, of the ratio between the squared charge of the metal cation and its effective radius. This ratio, which is called the polarizing power of the cation (cf Millero, 1977), defines the strength of the electrostatic interaction in a solvation-hydrolysis process of the type... 

In this case the reason tor the correlation is fairly obvious. The parameter refr is equal to the ionic radius plus a constant, 85 pm, the radius of the oxygen atom in water. Therefore, rt(S is effectively the interatomic distance in the hydrate, and the Born-Lande equation (Eq. 4.13) can be apphed. 

Solvent effects for ions can be described by a similar continuum solvent model the so-called Born model. This model predicts a stabilization proportional to the square of the charge, and inversely proportional to the size (radius) of the ion that is, small and highly charged ions are strongly stabilized in solution. 

For compounds in which the radius ratio is small, another term may be added to (5) to include also the appreciable effect of anion-anion repulsion. Pauling has indeed proposed such a treatment, analogous to Born s, but refined to include radius-ratio effects in doing so, he has been able to predict just how much the interionic distances in each of the alkali halides should depart from strict additivity. In using his modified treatment further for calculation of lattice energies, he has been able to show that the anomalies in melting points and boiling points, mentioned earlier in this chapter, may be correlated, at least semiquantitatively, with the radius ratios. 

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