Born equation/ model

The Born equation thus derived is based on very simple assumptions that the ion is a sphere and that the solvents are homogeneous dielectrics. In practice, however, ions have certain chemical characters, and solvents consist of molecules of given sizes, which show various chemical properties. In the simple Born model, such chemical properties of ions as well as solvents are not taken into account. Such defects of the simple Born model have been well known for at least 60 years and some attempts have been made to modify this model. On the other hand, there has been another approach that focuses on short-range interactions of an ion with solvent molecules. 

Strictly speaking, a chiral species cannot correspond to a true stationary state of the time-dependent Schrodinger equation H

time scale for such spontaneous racemization is extremely long. The wavefunction of practical interest to the (finite-lived) laboratory chemist is the non-stationary Born-Oppenheimer model Eq. (1.2), rather than the true T of Eq. (1.1). 

Extraction processes that proceed according to the model of ion-pair extraction are described by a formalism different from that presented in section 16.4.2, and are based on partition of single ions and their association in the organic phase [76] (see also section 2.6). The Born equation has been widely used to describe the transfer of an ion of the charge q and radius r from vacuum to the liquid (water) of the dielectric constant e ... 

In summary, the adiabatic approximation defined within the Born-Oppenheimer model leads to the equation. 

The Born equation is based on the simple model of a spherical ion with a single charge at its centre. Such an ion has no dipole moment and no higher multipole moments, but real molecular ions are of course much more complex. Since the electrical charge is distributed among all the atoms of the... 

The expression for the excess Gibbs energy is built up from the usual NRTL equation normalized by infinite dilution activity coefficients, the Pitzer-Debye-Hiickel expression and the Born equation. The first expression is used to represent the local interactions, whereas the second describes the contribution of the long-range ion-ion interactions. The Bom equation accounts for the Gibbs energy of the transfer of ionic species from the infinite dilution state in a mixed-solvent to a similar state in the aqueous phase [38, 39], In order to become applicable to reactive absorption, the Electrolyte NRTL model must be extended to multicomponent systems. The model parameters include pure component dielectric constants of non-aqueous solvents, Born radii of ionic species and NRTL interaction parameters (molecule-molecule, molecule-electrolyte and electrolyte-electrolyte pairs). 

Influence of the molecular environment on the structure and dynamics of molecular subsystems will be outlined referring to the solvation free energy (Chapter 4). Implicit solvent models based on the Poisson-Boltzmann (PB) equation and the Generalized Born (GB) model is discussed in 5 and 6. The PB or GB models are used for studies of molecular electrostatic properties and allow proper assignments of positions of protons (hydrogen atoms) within the given (bio)molecular structure. 

The nature of the solvent is represented in the Born equation only by the static dielectric permittivity e, which includes all types of polarization. The Born assumption on the structureless nature of the solvent may be only approximately fulfilled in the case when the ion is cdhsiderably larger than the solvent molecules. The problem of ion size and the applicability of the dielectric model has been discussed by Evans et al. [23]. 

A comparison of anion solvation by methanol, a protic solvent, and dimethylformamide, a dipolar aprotic solvent, is instructive. The electrostatic contribution, d/i , to the Gibbs free energy of solvation per mole of an ion is sometimes estimated quite successfully (Stokes, 1964) from the Bom model, in which a charged sphere of radius r is transferred from vacuum to a medium of uniform dielectric constant, c. The Bom equation (17) suggests that an anion should be similarly solvated in methanol and in DMF, because these solvents have effectively the same dielectric constant (33-36). The Born equation makes no allowance for chemical interactions, such as hydrogen-bonding and mutual... 

Among the many approximate models for solvation free energy evaluation, the most frequently used is the generalized Born (GB) model. It evaluates the solvation energy using the following equation ... 

The applicability of donicities to cation-solvent interactions is most convincingly demonstrated by the polarographic reduction of various metal ions in solvents of different donicity. The observed variation of half-wave potentials with solvent donicity can be explained neither in terms of the Born equation nor by simple microscopic electrostatic models in view of the magnitude of the dipole moments of solvent molecules. The concept also provides the basis for an interpretation of complex formation reactions and the behaviour of electrolytes (ion pair equilibria) in a large number of EPD solvents. 

The generalised Born equation has been incorporated into both molecular mechanics calculations (by Still and co-workers [Still et al. 1990 Qiu et al. 1997]) and semi-empirical quantum mechanics calculations (by Cramer and Truhlar, in an ongoing series of models called SMI, SM2, SM3, etc. [Cramer and Truhlar 1992 Chambers et al. 1996]). In these treatments, the two terms in Equation (11.61) are combined into a single expression of the following form ... 

The generalized Born (GB) model has been developed as a fast substitute of fhe PB equation [28-31]. The GB model can be tailored to match PB results for elecfrosfafic solvafion energies obtained by either the MS or the vdW protocol. The errors of GB resulfs in reproducing the PB counterparts are at least of fhe order of fypical mufafional effects on binding affinities. Therefore caution should be exercised when applying the GB model to calculate mutational effects. 

The need for computationally facile models for dynamical applications requires further trade-offs between accuracy and speed. Descending from the PB model down the approximations tree. Figure 7.1, one arrives at the generalized Born (GB) model that has been developed as a computationally efficient approximation to numerical solutions of the PB equation. The analytical GB method is an approximate, relative to the PB model, way to calculate the electrostatic part of the solvation free energy, AGei, see [18] for a review. The methodology has become particularly popular in MD applications [10,19-23], due to its relative simplicity and computational efficiency, compared to the more standard numerical solution of the Poisson-Boltzmann equation. 

Use of the Born equation for the calculation of free energies and enthalpies of solvation is based on a model of a continuous dielectric... 

In order to obtain the matrix equations above, one must decide how to construct, and subsequently discretize, the cavity surface. The most widely used methods take the cavity to be a union of atom-centered spheres [77], as suggested in Fig. 11.1(a). The electrostatic solvation energy is quite sensitive to the radii of these spheres (it varies as in the Born ion model), and highly parameterized constructions that exploit information about the bonding topology [6] or the charge states of the atoms [31] are sometimes employed. The details of these parameterizations are beyond the scope of the present work, especially given that careful reconsideration of these parameters is probably necessary for classical biomolecular electrostatics calculations. 

An entirely different approach to the treatment of electrostatic interactions is to eliminate them entirely in favor of implicit solvation techniques [28, 315] which either solve the Poisson-Boltzmann equation [144, 337] or employ the Generalized Born [291] model of excluded volumes. 

The Born equation is a very simple but surprisingly useful dielectric continuum model for solvation energetics. From the Bom equation... 

The protein matrix can be modeled as a continuum, but generally a much lower dielectric constant (of the order of 2-20) must be chosen than for water (78) to fit experimental data. Since reaction fields of lower dielectric constant disfavor more highly charged complexes, differences in redox potentials for metal site models in polar solvents vs. the active site itself can be significant. Simple estimates can be done by using the Born equation (Equation (9)). For example, the electrode potential of a ML (—1/—2) redox couple with a 400 pm radius is predicted to shift positively by 1.3 V when transferred from water to a medium with dielectric constant 4. 

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