Subject multipole

Conceptually, the self-consistent reaction field (SCRF) model is the simplest method for inclusion of environment implicitly in the semi-empirical Hamiltonian24, and has been the subject of several detailed reviews24,25,66. In SCRF calculations, the QM system of interest (solute) is placed into a cavity within a polarizable medium of dielectric constant e (Fig. 2.2). For ease of computation, the cavity is assumed to be spherical and have a radius ro, although expressions similar to those outlined below have been developed for ellipsoidal cavities67. Using ideas from classical electrostatics, we can show that the interaction potential can be expressed as a function of the charge and multipole moments of the solute. For ease... 

When observed structure factors are used, the thermally averaged deformation density, often labeled the dynamic deformation density, is obtained. An attractive alternative is to replace the observed structure factors in Eq. (5.8) by those calculated with the multipole model. The resulting dynamic model deformation map is model dependent, but any noise not fitted by the muitipole functions will be eliminated. It is also possible to plot the model density directly using the model functions and the experimental charge density parameters. In that case, thermal motion can be eliminated (subject to the approximations of the thermal motion formalism ), and an image of the static model deformation density is obtained, as discussed further in section 5.2.4. 

On the U(l) level, the plane wave is subjected to a multipole expansion in terms of the vector spherical harmonics, in which only two physically significant values of M in Eq. (761) are assumed to exist, corresponding to M = +1 and — 1, which translates into our notation as follows ... 

In the parametrization of equ. (4.68) the terms associated with the Legendre polynomials Pk(cos ab) represent that part of the angular correlation which is independent of the light beam, while the terms associated with the bipolar harmonics are due to the multipole expansion of the interactions of the electrons with the electric field vector. The link between geometrical angular functions and dynamical parameters is made by the summation indices ku k2 and k. These quantities are related to the orbital angular momenta of the two individual emitted electrons, and they are subject to the following conditions ... 

If the molecule, from the moment of absorption up to the moment of emission, is not subjected to external influence, e.g. to disorienting collisions or external fields, then the multipole moments bPQ entering into... 

Proceeding as above, the general expression (58) can be used to calculate other particular cases involving higher multipoles, on resorting to Tables 4—7, where the multipole elements are listed. Interactions between magnetic multipoles are also the subject of discussion. The theory of electrostatic interactions for electric multipoles has been dealt with in various approaches by Frenkel, Pople, Jansen, and others, as well as by Gray. The above presentation follows the concise uniform treatment of Kielich. > > ... 

Figure 10 The accumulation multipole setup for PEA. Ions of interest are selected by thequadrupole (a), accumulated in the accumulation octupole (b), isolated by SWIFT (c), and subjected to PEA by IRMPD (d,e) in the analyzer cell.

The potential function V still must be made explicit in order to complete the description of the system. A general multipole expansion in terms of first, second rank, etcetera interactions depending only on the relative orientation between each pair of bodies can be taken, as well as a multipole-field term (e.g., a dipole-field) for the pairwise interaction between each body and field. Finally each stochastic field is subjected to a harmonic potential, to parametrize in the most economical way the amplitude of the stochastic fluctuations. The complete potential is then... 

The multipole electromagnetic held can be quantized in much the same way as plane waves [2]. We have to subject the complex held amplitudes in the expansion (17) to the Weyl-Heisenberg commutation relations of the form... 

In this chapter we have reviewed some results concerning the quantum multipole radiation. Although the representation of quantum electromagnetic radiation in terms of spherical waves of photons known since the first edition in 1936 of the famous book by Heitler on quantum theory of radiation [2], where this subject is discussed in the Appendix, this representation is not a widespread one. The spherical waves of photons are considered in very few advanced monographs on quantum optics [26]. The brilliant encyclopedic monographs [14,15] just touch on the subject. 

Perturbation theory approach appears to be the most natural tool for theoretical investigations of weak intermolecular interactions (1). It provides the basis for present understanding of interactions between atoms and molecules, and defines the asymptotic constraints (via the multipole expansion (2, 3)) on the interaction potential. It is not surprising, then, that since the early 1970 s the convergence properties of various perturbation expansions for the intermolecular interaction energies are subject of extensive theoretical studies (4)-(23). 

Monatomic entities M+z consisting of one nucleus (carrying Z times the electric charge e of a proton) surrounded by K = (Z — z) electrons have been one of the major subjects for quantum-mechanical treatment. If the nucleus is treated as a geometrical point, and no attention is paid to its electric multipole moments, nor to its magnetic moments, the energy levels can be characterized by even or odd parity and by a quantum number J of total angular momentum. If the coordinates (— x, — y,... 

This subject could have been tackled before, when dipoles or multipoles were treated, because the possibility for these systems to work whether alone or in conjunction with other systems also existed. However, this was not really necessary because the subject could be treated in an implicit manner by giving literal symbols to all nodes without specifying whether or not they corresponded to variables with zero values. 

As far as our understanding of the stability is concerned, minimum-average-B is fairly well established. The calculations, even for finite p, or the critical P for multipoles have been done. The value of/3 turns out to be rather low, and this may also be one of its disadvantages. Secondary instabilities are trapped particle instabilities or instabilities associated with atoroidal field added toa multipole. I would say that this subject has not been sufficiently studied, although the primary stability has been well explored. 

All the methods considered in the preceding section relies, strictu sensu, on analytical expressions of the MEP. This section is dedicated to a specific strategy of elaboration of into simpler analytical expressions via multipole expansions. The use of a multipole expansion may reduce the computation time by several order of magnitude. Multipole expansions are however approximate, and may led to considerable errors. There is no unique way of getting a multipole expansion once F (r) (or r r)) is known, and the different methods vary greatly in reliability, computer time, and other characteristics. The variety of the approaches, and the importance of the subject suggest that we treat this topic in a specific section. The expression of the MEP in terms of the so-called multipole expansion is a classic topic [64-65], described in many textbooks [7, 66-68]. 

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