Continuous, charge distribution

When we have to deal with charge distributions rather than point charges, the definitions have to be generalized. What we do is to divide continuous charge distributions into differential charge elements /o(r)dr, and then apply the basic formula for the electrostatic field, and so on. Flere, dr is a differential volume element. Finally, we would have to integrate over the coordinates of the charge... 

The definitions need to be generalized when dealing with continuous charge distributions. Sums such as... 

The original FMM has been refined by adjusting the accuracy of the multipole expansion as a function of the distance between boxes, producing the very Fast Multipole Moment (vFMM) method. Both of these have been generalized tc continuous charge distributions, as is required for calculating the Coulomb interactioi between electrons in a quantum description. The use of FMM methods in electronic structure calculations enables the Coulomb part of the electron-electron interaction h be calculated with a computational effort which depends linearly on the number of basi functions, once the system becomes sufficiently large. 

However, unlike point charges, the continuous charge distributions that occur in quantum chemistry have varying extents and the applicability of the multipole approximation is not only limited by the distance but also by the extent or diffuseness of the charge distribution. This additional complexity makes a transfer of the concepts of the fast multipole method to applications in quantum chemistry less straightforward. Therefore it should come as no surprise that several adaptations to extend the applicability of the FMM to the Coulomb problem with continuous charge distributions have been suggested. These lead to... 

If <72 is smeared out into a continuous charge distribution over volume v2 at charge density p2 and at an average distance r 2 from qi, then... 

The discussion above is a description of problem that requires answers to the following (1) the determination of the distribution of ions around a reference ion, and (2) the determination of the thickness (radius) of the ionic atmosphere. Obviously this is a complex problem. To solve this problem Debye and Huckel used a rather general approach they suggested an oversimplified model in order to obtain an approximate solutions. The Debye-Huckel model has two basic assumptions. The first is continuous dielectric assumption. In this assumption water (or the solvent) is a continuous dielectric and is not considered to be composed of molecular species. The second, is a continuous charge distribution in the ionic atmosphere. Put differently, charges of the ions in the ionic surrounding atmosphere are smoothened out (continuously distributed). 

We will use an example as illustration. The dipole moment vector for formamide has been determined both by diffraction and microwave spectroscopy. As the diffraction experiment measures a continuous charge distribution, the moments derived are defined in terms of the method used for space partitioning, and are not necessarily equal. Nevertheless, the results from different techniques (Fig. 7.1) agree quite well. 

For a continuous charge distribution, the potential is obtained by integration over the space containing the distribution. At a point defined by r, the potential is given by... 

Fourier Series for the Total Electrostatic Energy The Coulombic electronic energy of a continuous charge distribution is defined as... 

This relation enables one to estimate the charge transfer from measured values of A0 and values of d determined for example by LEED. (However, the picture of dipoles consisting of two point charges a distance d apart is a drastic simplification of the actual continuous charge distribution at a surface). This use of work function measurements is applied in Sect. VI to the case of atomic adsorbates. 

In analogy to the classical expression for a continuous charge distribution po(r),... 

Both, the Gouy-Chapman and Debye-Hiickel are continuum theories. They treat the solvent as a continuous medium with a certain dielectric constant, but they ignore the molecular nature of the liquid. Also the ions are not treated as individual point charges, but as a continuous charge distribution. For many applications this is sufficient and the predictions of continuum theory agree with experimental results. At the end of this chapter we discuss the limitations and problems of the continuum model. 

The CDS parameters, on the other hand, are expected neither to be solvent-independent nor to be clearly related to any particular solvent bulk observable, especially insofar as they correct for errors in the NDDO wavefunc-tion and its impact on the ENP terms. The CDS parameters also make up empirically for the errors that inevitably occur when a continuous charge distribution is modeled by a set of atom-centered nuclear charges and for the approximate nature of the generalized Born approach to solving the Poisson equation. Hence, the CDS parameters must be parameterized separately against available experimental data for every solvent. This requirement presents an initial barrier to developing new solvent parameter sets, and at present, published SMx models are available for water only (although a hexadecane parameter seH will be available soon). 

In principle, the ASC method gives an exact solution of the electrostatic problem for this model. In practice, the fact that the source of Vei is confined to a close surface makes the numerical solution of the problem easier. The continuous charge distribution cr(s) is replaced by a set of point charges cavity surface (called tessera) having an area A Sk ... 

If the ions are not single nuclei as we have seen, in the calculation of the electrostatic potential created by them, the continuous charge distributions can be replaced by point charges located on their nuclei. In the case of symmetric ions of the type AB , such as NOi, CO, SOl",. .., the net charges carried by the B nuclei can be expressed in terms of the charge carried by the central atom ... 

The potential energy of interaction between two continuous charge distributions is given by... 

The charge distributions c )a(l)i and < )j< )b may be replaced by a number of point charges Qi, Qj, Qs,..., on each molecule, sufficieutly deusely disposed to represent the total, continuous charge distribution of the electrons. We first calculate the potential in the point r at some distance from the molecule. 

For a continuous charge distribution, we have instead qi Jp(a)daxdayda ... 

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