Born formula

Kikuchi, O., T. Matsuoka, H. Sawahata, and O. Takahashi. 1994. Ab Initio Molecular Orbital Calculations Including Solvent Effects by Generalized Born Formula. Conformation of Zwitterionic Forms of Glycine, Alanine and Serine in Water. J. Mol. Struct. (Theochem) 305, 79-87. 

It was stated before that the gas-phase lithium affinities of aziridine and oxirane were found to be 47 and 43 kcals/mole, respectively. These values might be somewhat overestimated since the Born formula used to compute the electrostatic attraction between Li and water may underestimate it. 

If we neglect the spatial dispersion of the dielectric permeability, then this formula turns into the well-known Born formula... 

O. Takahashi, H. Sawahata, Y. Ogawa and O. Kikuchi, Incorporation of solvent effects into ab initio molecular orbital calculations by the generalized Born formula. Formulation, parameterization, and applications, J. Mol. Struct. (THEOCHEM), 393 (1997) 141-150. 

This expression may be interpreted as a generalized Born formula for an off-centered charge in a spherical cavity. Accordingly, for a development up to the Zmax order, the percentual error in total energy may easily be shown to be... 

To frame this point, we give simple estimates of temperature and pressure derivatives assuming that the thermodynamic state dependence of the radii may be neglected. We will consider a simple ion and the Born formula (Pettitt, 2000) the interaction contribution to the chemical potential of such a solute is charge on the ion and R is its Born radius see Section4.2. We assume that these radius parameters are independent of the thermodynamic state. Considering the partial molar volume first, we have... 

The polarization, Epoi, induced by a charge on the surrounding lattice can be estimated by means of the classical Born formula [58] ... 

The addition of the ECPs to the cluster gives a better representation of the electrostatic potential hence of the electrostatic contribution to the surface bonding. What is still missing from this simplified approach is the polarization of the host crystal induced by an adsorbed species or by the presence of a defect. This effect can be particularly important for charged adsorbates or defects. The polarization, Epoi, induced by a charge on the surrounding lattice can be estimated by means of the classical Born formula [56] ... 

Analytical solution of the Poisson equation is possible only for a limited number of simple cavities. For example, for a single charge q( in the center of a spherical cavity with radius R, the solvation energy is described by the Born formula ... 

Here qt and qj are point atomic charges and/GB is a distance-dependent function. The summation is over all pairs of atoms in the solute. The fGB function is chosen to satisfy certain boundary conditions for a distance ry = 0,/GB = Rh and for a long distance ry = oo, foB = ry- These conditions turn Eq. (31) into the Born formula when ry = 0 and would recover the Coulomb interactions for ry = oo. Indeed, the total electrostatic energy of interaction between charges at large distance is given by... 

The popular expression for (A) of Marcus can be readily obtained from Eq.(1.130) employing the Born formula (Eq.1.117) for solvation free energy of a redox pair assuming AErmei = 0,... 

If the multipole expansion is truncated at the charge level (i.e., the lowest one) and the center of the expansion is shifted to the different nuclei, a generalized Born approach is achieved. Thus, the Born formula for the free energy of an ion of charge q and radius a in a dielectric of permittivity e, namely. 

The reorganization of the solvent can be expressed through its polarization, which also contains two contributions one from the electronic polarizability of the solvent molecules, and the other from the librational and vibrational motion. Only the latter are slower than the electron exchange as such, and contribute to the solvent reorganization energy Aout- This takes a form that is reminiscent of the Born formula for the energy of solvation ... 

We have multiplied the integral by the Avogadro constant in order to go over from the energy per ion to the energy per mole of the ions. Equation (3.1) is just the Born formula[220] for the solvation energy. 

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