Monopole

Like the curl condition is reminiscent of the Yang-Mills field, the quantization just mentioned is reminiscent of a study by Wu and Yang [76] for the quantization of Dirac s magnetic monopole [77-78]. As will be shown, the present quantization conditions just like the Wu and Yang conditions result from a phase factor, namely, the exponential of a phase and not just from a phase. 

Monopole (T charge) unit negative charge centered on the n iiclen s. 

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. 

Each two-electron integral is the sum of all the terms arising from the charge distribution representative of the first pair of atomic orbitals interacting with the charge distribution representative of the second pair of atomic orbitals. Thus in the simplest case, the (ssiss) interaction is represented by the repulsion of two monopoles, while a (pj pjjlp jjp jj), a much more complicated interaction,... 

Amonopole-monopole interaction, where the integral should converge to... 

This last integral is the interaction of a monopole with the sum of a monopole and a linear quadrupole. This gives rise to a total of four terms. However, since the interaction of the monopole with each of two negative charges of the dipole are the same, only three terms remain. The remaining integrals are developed by similar manipulations. 

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. 

This equation shows that the maximum overpressure, generated by a constant velocity flame fkmt, continually decreases as it propagates. Modeling an explosion of an extended flat vapor cloud by a single monopole located in the cloud s center is not, however, very realistic. 

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. 

Here q is the net charge (monopole), p, is the (electric) dipole moment, Q is the quadrupole moment, and F and F are the field and field gradient d /dr), respectively. The dipole moment and electric field are vectors, and the pF term should be interpreted as the dot product (p F = + EyPy + Ez z)- "I e quadrupole moment and field... 

The interaetion with a magnetic field may similarly be written in term of magnetie dipole, quadrupole ete. moments (there is no magnetie monopole, eorresponding to eleetrie eharge). Sinee the magnetie interaetion is substantially smaller in magnitude than the eleetrie, only the dipole term is normally considered. 

Monooxy-, (usually) monohydroxy- monoxy- monophasisch, a. monophase, single-phase, Monopol, n. monopoly Monopol T.N.). 

Fig. 11. Peptide moiety indicating the monopoles (net charges) on each of the atoms 54 551. This demonstrates that, on the basis of size, charge and accessibility, a polypeptide could more effectively provide selectivity for cations...

The usual multipole expansion is inappropriate for the treatment of London energies associated with the interatomic charge oscillation. However, London23 has given an appropriate method, known as the monopole method. Coulson and Davies7 and Haugh and Hirschfelder16 have applied the monopole method to the inter-... 

We assume that the double bonds in 1,3-butadiene would be the same as in ethylene if they did not interact with one another. Introduction of the known geometry of 1,3-butadiene in the s-trans conformation and the monopole charge of 0.49 e on each carbon yields an interaction energy <5 — 0.48 ev between the two double bonds. Simpson found the empirical value <5 = 1.91 ev from his assumption that only a London interaction was present. Hence it appears that only a small part of the interaction between double bonds in 1,3-butadiene is a London type of second-order electrical effect and the larger part is a conjugation or resonance associated with the structure with a double bond in the central position. 

London23 has treated the case of the attractive force between anisotropic molecules on the dipole-dipole interaction basis as well as on the monopole basis mentioned above. The small anisotropy found for the chlorine atom makes the dipole-dipole formulation appropriate. For the symmetrical orientation in the Cl2 molecule the London formula is... 

Polymers and copolymers are among the most beneficial materials produced by synthetic chemistry. The invention and commercialization of new polymeric materials with radical new properties provides an opportunity to monopolize the market and justify the expense involved in the research and development. The commercialization of new polymers or copolymers always presents scale-up and design challenges. Scientists have recently developed new polymeric materials whose commercial impact has yet to be realized. Examples are semiconductive and conductive polymers and amphiphilic dendritic block copolymers. Other promising materials, such as polymers for (targeted) drug delivery and... 

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