Electrostatics classical problem

As other QM continuum models, the PCM model requires the solution of two coupled problems an electrostatic classical problem for the determination of the solvent reaction potential Va induced by the total charge distribution and a quantum mechanical problem for the determination of the wavefunction I of the solute described by the effective QM Hamiltonian (1.1). The two problems are nested and they must be solved simultaneously. 

In conclusion, let us make one more remark. The distribution of charges caused by electrostatic induction is not usually known before-hand, and their determination constitutes one of the classical problems of the theory of electrical fields. 

Swelling of a polyelearolyte chain in salt solutions is one of the classical problems of polymer physics. The first attempt to account for the effect of the electrostatic interactions on conformations of a polyelectrolyte chain was made over 60 years ago by Kuhn, Kunzle, and Katchalsky and by Hermans and Overbeek (K3HO) by implementing a method that later became known as the Flory-like calculations of the chain size. In the framework of this approach, a free energy of polyelectrolyte chain consists of the chain s configurational (elastic) free energy... 

This is identical to the classical problem of the electrostatic potential created by a point charge in a two-dimensional space (y,z). If we define... 

The classical problem of electrostatics. The electrostatic interaction energy ms can be calculated in several ways. Historically at least the most important approach is based on the multipole expansion of the interaction integral in equation (4). Other methods are based on the apparent surface charge approach, the image charge approximation as well as finite difference and finite element based techniques. 

The distribution of charges on an adsorbate is important in several respects It indicates the nature of the adsorption bond, whether it is mainly ionic or covalent, and it affects the dipole potential at the interface. Therefore, a fundamental problem of classical electrochemistry is What does the current associated with an adsorption reaction tell us about the charge distribution in the adsorption bond In this chapter we will elaborate this problem, which we have already touched upon in Chapter 4. However, ultimately the answer is a little disappointing All the quantities that can be measured do not refer to an individual adsorption bond, but involve also the reorientation of solvent molecules and the distribution of the electrostatic potential at the interface. This is not surprising after all, the current is a macroscopic quantity, which is determined by all rearrangement processes at the interface. An interpretation in terms of microscopic quantities can only be based on a specific model. 

E-A. Numerical or analytic solution of the classical electrostatic problem (e.g., Poisson equation) with homogeneous dielectric constant for solvent. 

It follows from the Hellmann-Feynman theorem1,2 that these interactions are essentially Coulombic, and hence can be described accurately in terms of classical electrostatics, given the correct charge distributions. This has been discussed by Hirschfelder et al.3 4 The problem, with regard to solute-solvent interactions, is one of scale, i.e., the large number of molecules of the solvent that must be taken into account in order to realistically represent its effects. Thus, whereas interaction energies of relatively small noncovalently-bound complexes are... 

While several investigators contributed substantially to the resolution of this problem, it was the classic work of Debye and Hiickel ( 1) which provided a simple yet adequate explanation of the effect on thermodynamic properties of the long-range electrostatic forces between ions in solution. The experimental work of that era tended to emphasize dilute solutions at room temperature. 

A through-space electrostatic effect (field effect) due to the charge on X. This model was developed by Kirkwood and Westheimer who applied classical electrostatics to the problem. They showed that this model, the classical field effect (CFE), depended on the distance d between X and Y, the cosine of the angle 6 between d and the X—G bond, the effective dielectric constant and the bond moment of X. 

Continuum models are the most efficient way to include condensed-phase effects into quantum-mechanical calculations, and this is typically accomplished by using the self-consistent reaction field (SCRF) approach for the electrostatic component. Therefore it is very common to replace the quantal problem by a classical one in which the electronic energy plus the coulombic interactions of the nuclei, taken together, are modeled by a classical force field—this approach usually called molecular mechanics (MM) (Cramer and Truhlar, 1996). 

In the preceding section we discussed the problem of the variation of potential with distance from an interface from the highly artificial perspective of a parallel plate capacitor. The variation of potential with distance from a charged surface of arbitrary shape is a classical electrostatic problem. The general problem is described by the Poisson equation,... 

McKean 182> considered the matrix shifts and lattice contributions from a classical electrostatic point of view, using a multipole expansion of the electrostatic energy to represent the vibrating molecule and applied this to the XY4 molecules trapped in noble-gas matrices. Mann and Horrocks 183) discussed the environmental effects on the IR frequencies of polyatomic molecules, using the Buckingham potential 184>, and applied it to HCN in various liquid solvents. Decius, 8S) analyzed the problem of dipolar vibrational coupling in crystals composed of molecules or molecular ions, and applied the derived theory to anisotropic Bravais lattices the case of calcite (which introduces extra complications) is treated separately. Freedman, Shalom and Kimel, 86) discussed the problem of the rotation-translation levels of a tetrahedral molecule in an octahedral cell. 

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