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. 

It wiU be recalled that in section 3.10 we developed the nuclear Hamiltonian by calculating the magnetic vector potential at nucleus a arising from the spin and orbital motion of the electrons. Clearly we should now also include the nuclear spin contribution to A, the complete magnetic vector potential being (3.249) plus the additional term from the other nucleus a. ... 

Where Coulomb potential of nucleus At atomic shell electrons... 

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. 

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... 

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... 

These minimums reflect the fact of an acquisition of excess electron charge by the fluorine atoms in the crystal and their existence is a consequence of a specific dependence ESP of negative charged atom on the distance (Fig.8). The reliability of existence of two-dimensional minimums in crystals is confirmed by direct calculations of the ESP by the Hartree-Fock method. An analysis shows the more negative charge of isolated ions (deeper negative minimum of potential). At the same time the minimum position shifts to the nucleus. The K-parameter influences the characteristics of the minima only marginally. 

Results of similar accuracy as relativistic TFDW are found with a simple procedure based on near-nuclear correction which leave space for further improvements. For the reasons mentioned at the end of previous section the direct way to improve the present approach seems to be the refinement of the near nuclear corrections, a problem that we have just tackled with success in the non-relativistic framework [31,32]. The aim was to describe the near-nuclear region accurately by means of using the quantum mechanical exact asymptotic expression up to of the different ns eigenstates of Schodinger equation with a fit of the semiclassical potential at short distancies to the exact asymptotic behaviour (with four terms) of the potential near the nucleus. The result is that the density below Tq becomes very close to Hartree-Fock values and the improvement of the energy values is large (as an example, the energy of Cs+ is improved from the Ashby-Holzman result of-189.5 keV up to -205.6, very close to the HF value of -204.6 keV). This result makes us expect that a similar procedure in the relativistic framework may provide results comparable to Dirac-Fock ones. 

Etor state Torsional potential energy local Hamiltonian Magnetic field strength at nucleus of a... 

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. 

FIG. 8.4 Relation between the change in Is orbital energy for first-row atoms and the potential at the nuclear position due to the valence electrons. Both are expressed relative to the corresponding hydride (i.e., CH4, NH3, H20, and HF for C, N, O, and F, respectively). The label in the brackets specifies the nucleus being considered. Source Data in Schwartz (1970). 

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. 

---