Electrostatic potential real-space

Since the electrostatic potential is closely related to the electronic density, it may be useful to discuss how the information that can be obtained from V(r) differs from that provided by the p(r). Both are real physical properties, related by Eqs. (3.1) and (3.4). An important difference between V(r) and p(r) is that the electrostatic potential explicitly reflects the net effect of all of the nuclei and electrons at each point in space, whereas the electron density directly represents only the concentration of electrons at each point. A molecule s interactions with another chemical system is affected by its total charge distribution, both positive and negative, and thus can be better understood in terms of its electrostatic potential than its electronic density alone. Examples illustrating this point have been discussed elsewhere (Politzer and Daiker 1981 Politzer and Murray 1991). 

Conventional HRTEM operates at ambient temperature in high vacuum and directly images the local structure of a catalyst at the atomic level, in real space. In HRTEM, as-prepared catalyst powders can be used without additional sample preparation. The method does not normally require special treatment of thin catalyst samples. In HRTEM, very thin samples can be treated as WPOs, whereby the image intensity can be correlated with the projected electrostatic potential of the crystal, leading to the atomic structural information characterizing the sample. Furthermore, the detection of electron-stimulated XRE in the EM permits simultaneous determination of the chemical composition of the catalyst. Both the surface and sub-surface regions of catalysts can be investigated. 

When the electrostatic properties are evaluated by AF summation, the effect of the spherical-atom molecule must be evaluated separately. According to electrostatic theory, on the surface of any spherical charge distribution, the distribution acts as if concentrated at its center. Thus, outside the spherical-atom molecule s density, the potential due to this density is zero. At a point inside the distribution the nuclei are incompletely screened, and the potential will be repulsive, that is, positive. Since the spherical atom potential converges rapidly, it can be evaluated in real space, while the deformation potential A(r) is evaluated in reciprocal space. When the promolecule density, rather than the superposition of rc-modified non-neutral spherical-atom densities advocated by Hansen (1993), is evaluated in direct space, the pertinent expressions are given by (Destro et al. 1989)... 

The self-part of the Ewald electrostatic potential given in Eq. (6.14) can be derived in a fashion similar to our derivation of the real-space contribution in Appendix F.1.1.1. Starting from Poisson s formula [see Eq. (6.3)] and inserting Eq. (6.10b) for the charge deasity (r ), we have... 

The most widely used method for Coulomb force is Ewald summation, which was first introduced by Ewald in 1921 [3]. The total electrostatic potential is rewritten as a summation of three terms a short-range term that is calculated in real space, a long-range term that can be... 

The summation of the short-range forces can normally be readily converged directly in real space until the terms become negligible within the desired accuracy. However, other terms may decay slowly with distance, particularly since the number of interactions increases as 47rr Np, where Np is the particle number density. In particular, the electrostatic energy is conditionally convergent since the number of interactions increases more rapidly with distance than the potential (which is proportional to 1/r) decays. Hence, the two classes of energy components will be considered separately. 

Ewald summation is one of the procedures developed to solve the problems just mentioned. While VDW has rapid potential drop across certain interatomic distances due to its 6-12 exponential function, the electrostatic interaction s convergence over the interatomic distance variation is very slow due to its 1/r dependency. The use of a two-step summation (one in real space and one in reciprocal space) for the periodic system will give a more accurate value for the electrostatic interactions (63). One summation is carried out in reciprocal space the other is carried out in real space. Based on Ewald s formulation, the simple lattice sum can be reformulated to give absolutely convergent summations that define the principal value of the electrostatic potential, called the intrinsic potential. Given the periodicity present in both crystal calculations and in dynamics simulations using periodic boundary conditions, the Ewald formulation becomes well suited for the calculation of electrostatic energy and force. 

Za is the charge on nucleus A, located at Ra, and p(r) is the electronic density function, which we compute from the molecular wave function. V(r) is commonly termed the "electrostatic" potential, as molecular systems are most practically treated as static distributions of electronic charge around rigid nuclear frameworks. It is a real physical property which expresses the net electrical effect of the nuclei and electrons at each point in space r, and it has emerged as an effective tool for studying molecular reactive behavior [34-37]. An important feature of V(r) is that it can be determined experimentally by diffraction methods [37-39], as well as computationally. 

Finally, we list some interesting developments related to alternative forms of electrostatics calculations. Pask and Steme pointed out that real-space electrostatic calculations for periodic systems require no information from outside the central box. Rather, we only need the charge density within the box and the appropriate boundary conditions to obtain the electrostatic potential for the infinite system. These ideas were used earlier in an initial MG effort to compute Madelung constants in crystals. ° So long as charge balance exists inside the box, the computed potential is stable and yields an accurate total electrostatic energy. Thus, questions about conditional summation of the 1/r potential to obtain the physical electrostatic energy are unnecessary. This has also been noted in the context of Ewald methods for the simulation of liquids (See Chapter 5 in Ref. 227). 

The electrostatic potential V r) that is created in the space around a molecule by its nuclei and electrons is a well-established guide to chemical reactivity and molecular interactive behavior. " Unlike many of the other quantities used now and earlier as indices of reactivity, e.g., atomic charges, the electrostatic potential is a real physical property, one that can be determined experimentally by diffraction methods as well as computationally. However, V r) is most commonly obtained computationally. With the recent advances in computer technology, it is currently being applied to a variety of significant chemical systems. 

Electrostatics is a 3-D theory, because it describes forces between particles that exist in a 3-D world. Nevertheless, it is often useful to consider versions of the Poisson and PB equations where the field depends on only one or two spatial variables, and the 3-D Laplacian operator is replaced by l-D or 2-D analogs. In the context of electrostatics, these reduced dimensional versions arise in situations with spatial symmetry such that the dependence of the field along one or two Cartesian directions vanishes. Since it is easier to notate the l-D than the 3-D case, we shall often consider the simpler versions when discussing real-space lattice methodology. In order to derive the l-D update equation (13) from the steepest descents principle embodied by equation (12), we should construct a l-D version of the action indicated in equation (8). That is, suppose the charge distribution depends only on x (and is uniform in the y. z directions). Then the electric potential

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In contrast, truncation at a computationally feasible (i.e., small) value of R. produces artifacts when the system is strongly ionic because the potential energy is dominated by the slowly varying 1/r terms.To address this, a popular approach known as Ewald or lattice sum (similar to the one used to calculate the lattice energy of ionic crystals) is used to sum the electrostatic interactions in the simulation box and all of its replicas. This is done by rewriting the sum of the 1/r terms as a sum of a rapidly converged series in real space (so a small cutoff can be used for these terms) and a much more slowly varying smooth function that can be approximated by a few cosine and sine terms in reciprocal (k) space.These are expensive calculations that scale like... 

As is any potential including the electrostatic scalar potential between the poles of an electric dipole and the magnetostatic scalar potential between the poles of a permanent magnet, the A potential is an ongoing set of longitudinal EM energy flows between the time domain (imaginary plane) and real 3-space... 

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