Multipole types

Note that two distinct types of interactions (ion-quadrupole and dipole-dipole) contribute to an overall R 3 dependence, and the number of distinct multipole types having similar R n dependences continues to increase with increasing n. For uncharged systems, the dipole-dipole interaction (5.23d) is expected to dominate, with an angular term that favors parallel alignment of the two dipoles. 

Energies of multipole type. In cases when the external fidd is not uniform, one has to consider, in conformity with the expansion (S2a), interactions between the quadrupole moment and the field gradioit, the octupole and gradient of the field gradirat, and so forth. The potential energy of multipole interaction of ordar 1 being given by equation (54), we can by... 

DMA models have also been incorporated into the molecular modeling force field program MOMO, ° which was used to study host-guest complexes, such as benzene in a hexaoxacyclophane host. The difficult problem of accurately modeling the intramolecular electrostatic energy of proteins has also used distributed-multipole-type models. ... 

Velocity Dependence of the Cross Section. For S-P type interaction, the excitation transfer cross section was proportional to V1 for Case 1, and to tT2/5 for Case 3. For Case 2 the velocity dependence was not as simple. Here the ratio of the angular frequency of the resonant defect [a> = (Ei — Ef/tl) to the relative incident velocity (v)—i.e.> a = to/v is the most important parameter. If the ratio is small compared with the reciprocal of the interaction range a"1, the transfer will approach that of Case 1 (exact resonance). The cross ection will decrease monotonically with t at higher velocities. If a a"1, the cross section will be fairly small compared with that of exact resonance. Further, in the limit of t 0, the cross section would be zero, and would increase with v at low velocity region. Then, it will reach a maximum in between these regions for Case 2. This feature will hold for all inter-multipole types of interaction including the S-P type. However, the detailed and quantitative discussion on the velocity dependence for Case 2 is not this simple. On the other hand, the velocity dependence of the cross section for the resonance type excitation transfer (Cases 1 and 3) can be discussed more straightforwardly, not only for the S-P interaction case but also for other interaction cases (48, 69). 

The raie gas atoms reveal through their deviation from ideal gas behavior that electrostatics alone cannot account for all non-bonded interactions, because all multipole moments are zero. Therefore, no dipole-dipole or dipole-induced dipole interactions are possible. Van der Waals first described the forces that give rise to such deviations from the expected behavior. This type of interaction between two atoms can be formulated by a Lennaid-Jones [12-6] function Eq. (27)). 

Both the Coulomb cmd Lennard-Jones potentials can be considered examples of this type. In the cell multipole method the simulation space is divided into uniform cubic... 

The following types of multipole distributions are used to represent the four types of atomic orbital products. 

A different ion guide is the ion tunnel, which also uses only RF fields to transmit ions. It is not a rod device but consists of a series of concentric circular electrodes. It is perhaps best described as operating like a series of ion traps. This chapter gives details of some of the fundamental characteristics of rod-type transmission guides (multipoles). 

The various sizes and types utilize single-bar, multifield, multipole magnets singly or in factory assembled bar-bundles. They are further designed for easy disassembly and cleaning in systems with existing corrosion build-up or iron content in the water. ... 

N.2 Computational speedup for the direct and reciprocal sums Computational speedups can be obtained for both the direct and reciprocal contributions. In the direct space sum, the issue is the efficient evaluation of the erfc function. One method proposed by Sagui et al. [64] relies on the McMurchie-Davidson [57] recursion to calculate the required erfc and higher derivatives for the multipoles. This same approach has been used by the authors for GEM [15]. This approach has been shown to be applicable not only for the Coulomb operator but to other types of operators such as overlap [15, 62],... 

For polarizable charge distributions, additional classical-type interactions arise from the induced dipole, quadrupole, and higher moments on each monomer, which are proportional to the fields created by the asymmetric charge distribution on the other monomer. The proportionality constants for each multipole field are the monomer polarizabilities aa and ah (a111 for dipole fields, a(Q) for quadrupole fields, etc.). The leading two induction interactions are ... 

Wang, Y. Franzen, J. The Non-Linear Ion Trap. Part 3. Multipole Components in Three Types of Practical Ion Trap. Int. J. Mass Spectrom. Ion Proc. 1994, 132, 155-172. 

The coefficients n, have to obey the condition n, f, imposed by Poisson s electrostatic equation, as pointed out by Stewart (1977). The radial dependence of the multipole density deformation functions may be related to the products of atomic orbitals in the quantum-mechanical electron density formalism of Eq. (3.7). The ss, sp, and pp type orbital products lead, according to the rules of multiplication of spherical harmonic functions (appendix E), to monopolar, dipolar, and quadrupolar functions, as illustrated in Fig. 3.6. The 2s and 2p hydrogenic orbitals contain, as highest power of r, an exponential multiplied by the first power of r, as in Eq. (3.33). This suggests n, = 2 for all three types of product functions of first-row atoms (Hansen and Coppens 1978). 

The multipole formalism described by Stewart (1976) deviates from Eq. (3.35) in several respects. It is a deformation density formalism in which the deformation from the IAM density is described by multipole functions with Slater-type radial dependence, without the K-type expansion and contraction of the valence shell. While Eq. (3.35) is commonly applied using local atomic coordinate systems to facilitate the introduction of chemical constraints (chapter 4), Stewart s formalism has been encoded using a single crystal-coordinate system. 

Permanent electrical multipole term Uq, lattice energies and, 1 176-177 Pemitrous acid, 22 147 Perovskites, ordered, 35 354, 370-371 Perovskite type fluorides, 20 152-166 Peroxides, see also specific compounds fluorinated, 16 109-168 fluoroaUcyl, spectral properties of, 16 154, 155... 

The Hamiltonian of the Coulomb term involves electrostatic potential energy operator for the interaction of all electrons and nuclei of donors with those of acceptors. The electrostatic potential can be expanded into multipole terms of the type... 

In Chapter 4 we will see that two types of induced dipole functions are of a special importance the overlap-induced dipole, Eq. 4.2, an exponential with a range Rq O.lcr, and the multipole-induced dipole, Eq. 4.3, which falls off as R N (N = 4, 5,...). For these, the optical range becomes... 

Fig. 4.3. The y), yS. dependences of the principal induced dipole components of H2-H2 vibrational transition elements v = 0 - v = 1. Overlap (dotted) and multipole-induced terms (solid line type) are shown dashed lines are the classical multipole approximations [281],...

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