Monopole potentials

Point-charge potential (monopole potential). By equation (45), the zeroth approximation... 

Exc [jf ] denotes an exchange-current-density functional, which collects all quantum, i.e., exchange and correlation effects. For the sake of simplicity, we assume that no external electromagnetic potentials are present other than the monopole potential of the atomic nuclei. Unfortunately, Eye [/ ] is not known in analytical form, which is the reason why present-day DFT is hampered by more or less accurate approximations to Exc[/ ]-... 

As one can see, the potential at H-bonding distances from the molecule are not too badly represented. Although more sophisticated methods to represent the electrostatic potential surrounding the molecule are clearly feasible, neither STO nor 43IG wave functions reproduce experimental polarity especially well. Thus, it is not clear that it is worth the effort to go beyond the monopole approximation since even our derived monopole potentials begin to take a non-infinitesmal amount of time on the CDC 7600 if we wish a relatively fine grid. 

Equation (2.3) gives the definition of the monopole potential due to a point charge Q. This is a simple definition of an internal potential between two points located at a relatively short distance. Otherwise, — 0 as r — oo. (3n the other hand, when < > — c as r — 0 an electric potential singularity can be established for in the order of r as predicted by eq. (2.3). 

Volumen and Hydratationswarme der lonen. Zeitschrift filr Physik 1 45-48. aan C M and K B Wiberg 1990. Determining Atom-Centred Monopoles from Molecular Electro-itic Potentials. The Need for High Sampling Density in Formamide Conformational Analysis. imal of Computational Chemistry 11 361-373. 

The above potential describes the monopole-monopole interactions of atomic charges Q and Qj a distance Ry apart. Normally these charge interactions are computed only for nonbonded atoms and once again the interactions might be treated differently from the more normal nonbonded interactions (1-5 relationship or more). The dielectric constant 8 used in the calculation is sometimes scaled or made distance-dependent, as described in the next section. 

CM Breneman, KB Wiberg. Determining atom-centered monopoles from molecular electrostatic potentials. The need for high sampling density m formamide conformational analysis. J Comput Chem 11 361-373, 1990. 

These results were analytically reproduced by Taylor (1985), who employed a velocity potential function for a convected monopole. This concept makes it possible to model an elongated vapor cloud explosion by one single volume source which is convected along the main axis at burning velocity, and whose strength varies proportionally to the local cross-sectional cloud area. 

The Coulomb interaction of the (point) nucleus with the potential Vo, which is also part of the monopole interaction, was neglected in (4.5) because it yields only an offset of the total energy. The subscript u in is introduced to distinguish the radius of the uniformly charged sphere from the usual mean square radius which can be obtained from scattering experiments. 

Consequently, we introduce the second approximation which is to use an approximate electrostatic potential in Eq.(4-21) to determine inter-fragment electronic interaction energies. Thus, the electronic integrals in Eq. (4-21) are expressed as a multipole expansion on molecule J, whose formalisms are not detailed here. If we only use the monopole term, i.e., partial atomic charges, the interaction Hamiltonian is simply given as follows ... 

Breneman, C. M. and K. B. Wiberg. 1990. Determining Atom-Centered Monopoles from Molecular Electrostatic Potentials. The Need for High Sampling Density in Formamide Computational Analysis. J. Comp. Chem. 11, 361. 

In a statistical Monte Carlo simulation the pair potentials are introduced by means of analytical functions. In the election of that analytical form for the pair potential, it must be considered that when a Monte Carlo calculation is performed, the more time consuming step is the evaluation of the energy for the different configurations. Given that this calculation must be done millions of times, the chosen analytic functions must be of enough accuracy and flexibility but also they must be fastly computed. In this way it is wise to avoid exponential terms and to minimize the number of interatomic distances to be calculated at each configuration which depends on the quantity of interaction centers chosen for each molecule. A very commonly used function consists of a sum of rn terms, r being the distance between the different interaction centers, usually, situated at the nuclei. In particular, non-bonded interactions are usually represented by an atom-atom centered monopole expression (Coulomb term) plus a Lennard-Jones 6-12 term, as indicated in equation (51). 

In this section, the field equations (31) and (32) are considered in free space and reduced to a form suitable for computation to give the most general solutions for the vector potentials in the vacuum in 0(3) electrodynamics. This procedure shows that Eqs. (86) and (87) are true in general, and are not just particular solutions. On the 0(3) level, therefore, there exist no topological monopoles or magnetic charges. This is consistent with empirical data—no magnetic monopoles of any kind have been observed in nature. 

The absence of magnetic monopole implies the conditions V = 0 and B = -Vx = 0, which are consistent with the relations given above, since B =VAA =0. However, as with the A vector potential, the equality Bx = = Vx / 0 enables us to define a scalar potential x that can be calculated on the basis of Biot-Savart law for a filiform (filament-shaped) circuit... 

Representation of the density n(r) [or, effectively, the electrostatic potential — 0(r)] near any one of the sinks as an expansion in the monopole and dipole contribution only [as in eqn. (230c)] is generally, unsatisfactory. This is precisely the region where the higher multipole moments make their greatest contribution. However, the situation can be improved considerably. Felderhof and Deutch [25] suggested that the physical size of the sinks and dipoles be reduced from R to effectively zero, but that the magnitude of all the monopoles and dipoles, p/, are maintained, by the definition... 

It is analogous to Poisson s equation in electrostatics and the contributions to the potential, 0 = — n, from the point monopoles and dipoles are included on the right-hand side [489],... 

We consider now the Aharonov-Bohm effect as an example of a phenomenon understandable only from topological considerations. Beginning in 1959 Aharonov and Bohm [30] challenged the view that the classical vector potential produces no observable physical effects by proposing two experiments. The one that is most discussed is shown in Fig. 10. A beam of monoenergetic electrons exists from a source at X and is diffracted into two beams by the slits in a wall at Y1 and Y2. The two beams produce an interference pattern at III that is measured. Behind the wall is a solenoid, the B field of which points out of the paper. The absence of a free local magnetic monopole postulate in conventional... 

In practice a molecule with a dipole moment is often mobile. If the dipole is free to rotate and close to a positive charge it tends to rotate until its negative pole points towards the positive charge. On the other hand, thermal fluctuations drive it away from a perfect orientation. On average, a net preferential orientation remains and the dipole is attracted by the monopole. The average potential energy is... 

The atom-centered monopole charges have been determined by fitting the quantum mechanically derived electrostatic potential in the region... 

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