Electrostatic potential at nucleus

However, if we consider a model of ion solvation that uses the electrostatic potential at nucleus ... 

We shall develop here a simple methodology for the computation of AEins, in terms of the electrostatic potential at nucleus V0. Within this frame, we will show that the first term of Eq (18) represents, in the context of the reaction field theory, the ion-solvent interaction energy. 

Theoretical considerations leading to a density functional theory (DFT) formulation of the reaction field (RF) approach to solvent effects are discussed. The first model is based upon isolelectronic processes that take place at the nucleus of the host system. The energy variations are derived from the nuclear transition state (ZTS) model. The solvation energy is expressed in terms of the electrostatic potential at the nucleus of a pseudo atom having a fractional nuclear charge. This procedure avoids the introduction of arbitrary ionic radii in the calculation of insertion energy, since all integrations involved are performed over [O.ooJ The quality of the approximations made are discussed within the frame of the Kohn-Sham formulation of density functional theory. 

Physically, Q represents the amount of charge inside the sphere 5(0, r ) of radius rg centered at the nucleus. We may then associate to this net charge Q, the electrostatic potential Og(rg) = Q/rp which is the electrostatic potential at any point r > rg, that is created by the nucleus and the electronic... 

Stewart s conclusion underscores the need for short-wavelength, low-temperature studies, if very high accuracy electrostatic properties are to be evaluated by Fourier summation. But, as pointed out by Hansen (1993), the convergence can be improved if the spherical atoms subtracted out are modified by the k values obtained with the multipole model. Failure to do this causes pronounced oscillations in the deformation density near the nuclei. For the binuclear manganese complex ( -dioxo)Mn(III)Mn(IV)(2,2 -bipyridyl)4, convergence of the electrostatic potential at the Mn nucleus is reached at 0.7 A" as checked by the inclusion of higher-order data (Frost-Jensen et al. 1995). 

The electron density centered at M is the only central contributor at the nuclear position M, as in this case the nucleus coincides with the field point P, which is excluded from the integrals. For transition metal atoms, the central contributions are the largest contributors to the properties at the nuclear position, which can be compared directly with results from other experimental methods. The electric field gradient at the nucleus, for instance, can be measured very accurately for certain nuclei with nuclear quadrupole resonance and/or Mdssbauer spectroscopic methods, while the electrostatic potential at the nucleus is related to the inner-shell ionization energies of atoms, which are accessible by photoelectron and X-ray spectroscopic methods. 

Now, the most direct interpretation of Eq. (11.5) follows from the observation, suggested by Eqs. (11.1) and (11.2), that/is essentially a relaxation term. In fact, Vk — represents the difference between the electrostatic potential at the h nucleus in the given molecule and the potential that the same nucleus would feel if the atomic orbitals and the equilibrium distances remained the same as in the reference molecule in spite of the change in electron populations. 

The electrostatic potential at any point, V(r), is the energy required to bring a single positive charge from infinity to that point. As each pseudo atom in the refined model consists of the nucleus and the electron density distribution described by the multipole expansion parameters, the electrostatic potential may be calculated by the evaluation of... 

Et is the /-component of the electric field at the site of the nucleus and V is the electrostatic potential at this site. The Laplace equation gives the relation... 

V(r) > 0.06 hartree) and red denoting electron-rich regions (V(r) < -0.06 hartrees). It has recently been proven by Politzer, Murray et al. and extensively used by Rice et al. (see Ch. 8) [39-44] that the patterns of the computed electrostatic potential on the surface of molecules can generally be related to the sensitivity of the bulk material. The electrostatic potential at any point r is given by the following equation in which ZA is the charge on nucleus A, located at RA. 

If the origin of the coordinate system is taken at the nucleus of the rare earth ion, an expression for the electrostatic potential at a point (r, 6, i/>) near the origin due to the surrounding k ions may be written as... 

The atomic charges as defined by Eq. [50] thus uniquely account for the force on nucleus A due to an external field in the z direction also exactly reproduce the molecular dipole moment, so that the electrostatic potential at large distances is also exactly reproduced by these atomic charges. (In fact, it is usually found that these charges reproduce the molecular electrostatic potential better than the total molecular dipole moment. ) Thus, for planar molecules it is possible to define a consistent set of atomic charges that account for both the molecular electrostatic potential and the intermolecular forces. Atomic charges as defined by other methods may reproduce the molecular electrostatic potential very well, but in general will not reproduce the perpendicular forces. 

Appendix 1 Hardness and Chemical Potential in Atomic Ions Related via Electrostatic Potential at the Nucleus... 

The electrostatic potential at arbitrary point (r) near the solute can be divided into two different contributions, namely, the potential from the nucleus and that from the electron clouds. 

The electrostatic potential at the nucleus due to a point charge at a distance r is given by F = q/(4nSor), where f is the permittivity of the vacuum. The interaction of the nuclear quadrupole moment with the electronic environment is expressed by the Hamiltonian... 

The electric field gradient (EFG) is a ground state property of solids that sensitively depends on the asymmetry of the electronic charge density near the probe nucleus. The EFG is defined as the second derivative of the electrostatic potential at the nucleus position written as a traceless tensor. A nucleus with a nuclear spin number / > 1 has a nuclear quadrupole moment (Q) that interacts with the EFG which originates from the nonspherical charge distribution surrounding this nucleus. This interaction... 

The atomic scattering factor for electrons is somewhat more complicated. It is again a Fourier transfonn of a density of scattering matter, but, because the electron is a charged particle, it interacts with the nucleus as well as with the electron cloud. Thus p(r) in equation (B1.8.2h) is replaced by (p(r), the electrostatic potential of an electron situated at radius r from the nucleus. Under a range of conditions the electron scattering factor, y (0, can be represented in temis... 

The Lewis dot formalism shows any halogen in a molecule surrounded by three electron lone pairs. An unfortunate consequence of this perspective is that it is natural to assume that these electrons are equivalent and symmetrically distributed (i.e., that the iodine is sp3 hybridized). Even simple quantum mechanical calculations, however, show that this is not the case [148]. Consider the diiodine molecule in the gas phase (Fig. 3). There is a region directly opposite the I-I sigma bond where the nucleus is poorly shielded by the atoms electron cloud. Allen described this as polar flattening , where the effective atomic radius is shorter at this point than it is perpendicular to the I-I bond [149]. Politzer and coworkers simply call it a sigma hole [150,151]. This area of positive electrostatic potential also coincides with the LUMO of the molecule (Fig. 4). 

For r —> 0 the leading term of the electrostatic potential must be due to the nucleus, so that the boundary condition at r = 0 becomes... 

The article is organized as follows in Section 2, a general discussion concerning the definition of electrostatic potentials in the frame of DFT is presented. In Section 3, the solvation energy is reformulated from a model based on isoelectronic processes at nucleus. The variational formulation of the insertion energy naturally leads to an energy functional, which is expressed in terms of the variation of the electron density with respect to... 

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