Coulomb Potential Functions

For later convenience we study the short-range series expansion 

In practice, Eq. (9.88) may be treated only over a finite range of the radial variable r e [0,rmax] fmax oo) provided that r ax is a sufficiently large number that causes Pk rmax) 0 and Qk fmax) 0. With these constraints a tolerable error will be introduced due to the exponentially decaying behavior of the two radial functions. The potential functions Yku, r) require more attention, since a finite r ax leads to the modified upper boundary condition 

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A 3-D structure of an object viewed through an electron microscope is described in terms of the 3-D Coulomb potential function within the object. The image recorded in an electron microscope is a convolution of the projected potential function of the object with the contrast transfer... 

Moreover, -> molecular interaction fields are calculated for each molecule in terms of similarity indices instead of the usual interaction potential functions, such as Len-nard-Jones and Coulomb potential functions. Similarity fields are calculated representing the similarity between molecules and different probe atoms. In particular, the similarity values at the intersections of the regularly spaced grid (1.1 and 2.0 A) relative to the yth physico-chemical property between the ith compound and a probe atom is calculated as ... 

These indices replace the distance functions used in the standard Lennard-Jones and Coulomb potential functions which generate unrealistically extreme values as the surface of the considered molecule is approached. 

The exact mathematical form of each energy term is given in Table I. EV(jw is the standard 6-12 Lennord-Jones potential fmotion. E is the classical coulombic potential function. The... 

Coulomb potential functions and the standard Tripos CoMFA probes (the Csp probe was used for calculation of steric interactions and the probe for calculation of elec-bostatic interactions, respectively). A PCA (factor analysis without axes rotation) was done on the descriptor matrix and a classification of the heteroaromatic substiments into families was performed using the Sybyl hierarchical clustering procedure of the obtained principal component... 

Here, W, as a Coulomb potential function of point charges, separates into three parts in an additive manner, which each depend on the particles of two molecules and their distances... 

This result appears here as a trivial consequence of the additivity of the Coulomb potential functions. But throughout it is not so obvious, which can be seen from the fact that for the short range forces the additivity is not valid even in first order, even though the Coulomb function is the point of departure. There we have a rather complicated superposition mechanism, which even expresses the saturation of the chemical binding, namely the fact that very different expressions of force take place between the atoms, depending on whether one of them has entered in a chemical force involvement with a third atom. In this respect the molecular forces are quite distinct from the homo-polar valence forces. In first order it can be shown that the forces between atomic systems are not susceptible to the presence of a third [atom] only in the exceptional case where no free valencies are present. We see that in this case the theorem is abo valid for the long range forces of second order. In third order we have no additivity in any case. 

By this way, Aizermann successfully converted a linearly nonseparable problem to a very simple linearly separable problem to take the difference of electric fields at every point. It is easy to understand that potential functions other than Coulomb potential function are also applicable in this method. Aizermann also suggested the following function for the evaluation of the field strength around every sample point instead of Coulomb potential function ... 

We first used a promolecular description of the ED distribution function of the various molecules, as reported before [20]. We also apply the formalism obtained for the CD calculated from smoothed electrostatic potential functions through the Poisson equation. Such a CD distribution function was previously considered to design, through its topological properties, reduced point charge models for proteins [33]. This new aspect is considered in comparison with the method described by Good et al. [34] to superpose Coulomb potential functions and implemented by us in combination with a smoothing approach. Finally, we also considered a smoothed version of the APFs developed by Totrov [15]. 

The magnitude and shape of such a mean-field potential is shown below [21] in figure B3.1.4 for the two 1 s electrons of a beryllium atom. The Be nucleus is at the origin, and one electron is held fixed 0.13 A from the nucleus, the maximum of the Is orbital s radial probability density. The Coulomb potential experienced by the second electron is then a function of the second electron s position along the v-axis (coimecting the Be nucleus and the first electron) and its distance perpendicular to the v-axis. For simplicity, this second electron... 

In modem quantum chemistry packages, one can obtain moleculai basis set at the optimized geometry, in which the wave functions of the molecular basis are expanded in terms of a set of orthogonal Gaussian basis set. Therefore, we need to derive efficient fomiulas for calculating the above-mentioned matrix elements, between Gaussian functions of the first and second derivatives of the Coulomb potential ternis, especially the second derivative term that is not available in quantum chemistry packages. Section TV is devoted to the evaluation of these matrix elements. 

Next, we shall consider four kinds of integrals. The first is the expectation value of the Coulomb potential by one nucleus for one of the primitive basis function centered at that nucleus. The second is the expectation value of the Coulomb potential by one nucleus for one of the primitive basis function centered at a different point (usually another nucleus). Then, we will consider the matrix element of a Coulomb term between two primitive basis functions at different centers. The third case is when one basis function is centered at the nucleus considered. The fourth case is when both basis functions are not centered at that nucleus. By that we mean, for two Gaussian basis functions defined in Eqs. (73) and (74), we are calculating... 

Her workers to fit the exchange-correlation potential and the charge density (in the Coulomb potential) to a linear combination of Gaussian-typc functions. 

The correlation functions of the partly quenched system satisfy a set of replica Ornstein-Zernike equations (21)-(23). Each of them is a 2 x 2 matrix equation for the model in question. As in previous studies of ionic systems (see, e.g.. Refs. 69, 70), we denote the long-range terms of the pair correlation functions in ROZ equations by qij. Here we apply a linearized theory and assume that the long-range terms of the direct correlation functions are equal to the Coulomb potentials which are given by Eqs. (53)-(55). This assumption represents the mean spherical approximation for the model in question. Most importantly, (r) = 0 as mentioned before, the particles from different replicas do not interact. However, q]f r) 7 0 these functions describe screening effects of the ion-ion interactions between ions from different replicas mediated by the presence of charged obstacles, i.e., via the matrix. The functions q j (r) need to be obtained to apply them for proper renormalization of the ROZ equations for systems made of nonpoint ions. 

A complete set of intermolecular potential functions has been developed for use in computer simulations of proteins in their native environment. Parameters have been reported for 25 peptide residues as well as the common neutral and charged terminal groups. The potential functions have the simple Coulomb plus Lennard-Jones form and are compatible with the widely used models for water, TIP4P, TIP3P and SPC. The parameters were obtained and tested primarily in conjunction with Monte Carlo statistical mechanics simulations of 36 pure organic liquids and numerous aqueous solutions of organic ions representative of subunits in the side chains and backbones of proteins... 

This is only valid when — V 2mc, however, all atoms have a region close to the nucleus where this is not fulfilled (sinee V -oo for r —> 0). Inserting (8.22) in (8.15), and assuming a Coulomb potential —Z/r (i.e. V is the attraction to a nucleus), gives after renormalization of the (large component) wave function and some rearrangement the following terms... 

It can be seen from Fig. 7 that V is a linear function of the qf This qV relation was pointed out and discussed at some length in the papers in ref. 6. It is not simple electrostatics in that it would not exist for an arbitrary set of charges on the sites, even if the potentials are calculated exactly. The charges must be the result of a self-consistent LDA calculation. The linearity of the relation and fie closeness of the points to the line is demonstrated by doing a least squares fit to the points. The sums that define the potentials V do not converge at all rapidly, as can be seen by calculating the Coulomb potential from the standard formula for one nn-shell after another. The qV relation leads to a special form for the interatomic Coulomb energy of the alloy... 

The integral (1.7), which is the starting point for the expansion of a hydrogen-like lx function in a Gaussian basis, is rather complicated. There is a much simpler counterpart of (1.7) which is relevant for the expansion of the Coulomb potential 1/r in a Gaussian basis, namely... 

The first term is the familiar one-electron operator, the second term represents the Coulomb potential, and the third term is called exchange-correlation potential. HF and DFT differ only in this last term. In HF theory there is only a nonlocal exchange term, while in DFT the term is local and supposed to cover both exchange and correlation. It arises as a functional derivative with respect to the density ... 

The first density correction to the rate constant depends on the square root of the volume fraction and arises from the fact that the diffusion Green s function acts like a screened Coulomb potential coupling the diffusion fields around the catalytic spheres. 

The first two tenns in Eq. (2) represent the kinetic energy of the nuclei and the electrons, respectively. The remaining three terms specify the potential energy as a function of the interaction between the particles. Equation (3) expresses the potential function for the interaction of each pair of nuclei. In general, this sum is composed of terms that are given by Coulomb s law for the repulsion between particles of like charge. Similarly, Eq. (4) corresponds to the electron-electron repulsion. Finally, Eq. (5) is the potential function for the attraction between a given electron (<) and a nucleus (j). 

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