Distributed multipole expansions

The multipole expansion gives exactly that expression. If the charge distribution shown has an overall charge Q, an electric dipole pe, an electric quadrupole 0g, and so on, then we write... 

The original FMM has been refined by adjusting the accuracy of the multipole expansion as a function of the distance between boxes, producing the very Fast Multipole Moment (vFMM) method. Both of these have been generalized tc continuous charge distributions, as is required for calculating the Coulomb interactioi between electrons in a quantum description. The use of FMM methods in electronic structure calculations enables the Coulomb part of the electron-electron interaction h be calculated with a computational effort which depends linearly on the number of basi functions, once the system becomes sufficiently large. 

The charge distribution of the molecule can be represented either as atom centred charges or as a multipole expansion. For a neutral molecule, the lowest-order approximation considers only the dipole moment. This may be quite a poor approximation, and fails completely for symmetric molecules which do not have a dipole moment. For obtaining converged results it is often necessarily to extend the expansion up to order 6 or more, i.e. including dipole, quadrupole, octupole, etc. moments. 

In connection with electronic strucmre metlrods (i.e. a quantal description of M), the term SCRF is quite generic, and it does not by itself indicate a specific model. Typically, however, the term is used for models where the cavity is either spherical or ellipsoidal, the charge distribution is represented as a multipole expansion, often terminated at quite low orders (for example only including the charge and dipole terms), and the cavity/ dispersion contributions are neglected. Such a treatment can only be used for a qualitative estimate of the solvent effect, although relative values may be reasonably accurate if the molecules are fairly polar (dominance of the dipole electrostatic term) and sufficiently similar in size and shape (cancellation of the cavity/dispersion terms). 

In the literature slightly different definitions of the multipole expansion are found, depending on how the pre-factor (2( — l)/2 is distributed between expansion equation and the definition of die coefficients. Cf. (Ward [251], eq. 5.2)... 

Each coefficient of the multipole expansion is computed by a numerical integration - after aligning and normalizing the found orientation distribution. 

Electrical moments are useful because at long distances from a molecule the total electronic distribution can be increasingly well represented as a truncated multipole expansion, and thus molecular interactions can be approximated as multipole-multipole interactions (charge-charge, charge-dipole, dipole-dipole, etc.), which are computationally particularly... 

One of the simplest orientational-dependent potentials that has been used for polar molecules is the Stockmayer potential.48 It consists of a spherically symmetric Lennard-Jones potential plus a term representing the interaction between two point dipoles. This latter term contains the orientational dependence. Carbon monoxide and nitrogen both have permanent quadrupole moments. Therefore, an obvious generalization of Stockmayer potential is a Lennard-Jones potential plus terms involving quadrupole-quadrupole, dipole-dipole interactions. That is, the orientational part of the potential is derived from a multipole expansion of the electrostatic interaction between the charge distributions on two different molecules and only permanent (not induced) multipoles are considered. Further, the expansion is truncated at the quadrupole-quadrupole term. In all of the simulations discussed here, we have used potentials of this type. The components of the intermolecular potentials we considered are given by ... 

The dielectric constant is a macroscopic property of the material and arises from collective effects where each part of the ensemble contributes. In terms of a set of molecules it is necessary to consider the microscopic properties such as the polarizability and the dipole moment. A single molecule can be modeled as a distribution of charges in space or as the spatial distribution of a polarization field. This polarization field can be expanded in its moments, which results in the multipole expansion with dipolar, quadrupolar, octopolar and so on terms. In most cases the expansion can be truncated to the first term, which is known as the dipole approximation. Since the dipole moment is an observable, it can be described mathematically as an operator. The dipole moment operator can describe transitions between states (as the transition dipole moment operator and, as such, is important in spectroscopy) or within a state where it represents the associated dipole moment. This operator describes the interaction between a molecule and its environment and, as a result, our understanding of energy transfer. 

Although it is possible to determine the complete electron density distribution using the Fourier transform of the observed structure factors, Eq. (1), the errors inherent in the structure factor amplitudes and, in the case of non-centrosymmetric structures, the errors in their phases introduce significant noise and bias into the result. Because of this, it has become normal practice to model the electron density by a series of pseudo-atoms consisting of a frozen, spherical core and an atom centered multipole expansion to represent the valence electron density [2,17]. 

The electrostatic potential at any point, V(r), is the energy required to bring a single positive charge from infinity to that point. As each pseudo atom in the refined model consists of the nucleus and the electron density distribution described by the multipole expansion parameters, the electrostatic potential may be calculated by the evaluation of... 

EMTP is the electrostatic interaction energy calculated as a sum of multipole-multipole interactions using the overlap multipole expansion of the SCF electron density distributions of the host and guest182). 

The equilibrium state is determined by a minimization of the free energy. The total interaction in colloidal systems is mainly determined by electrostatic interactions and can be divided in three components. The first component occurs at interactions between net charged molecules or molecules with asymmetric charged distribution. These charge distributions can often be described by multipole expansions, i.e., a combination of monopole, dipole, quadrupole, etc., and is a fruitful approach if each multipole expansion can be described by one or two terms. The interaction is given by the sum of interactions between the terms, where the first contribution is the interaction between ions and is given by Coulomb s law and is the main contribution in systems with... 

Since the single-center multipole expansion of the interaction energy is divergent, one could use a kind of multicenter expansion. One can hope that the multipole expansion will provide better results if multipole moments and polarizabilities localized at various points of a molecule are used instead of global multipole moments and polarizabilities. This idea forms the basis of the so-called distributed multipole analysis of the electrostatic, induction, and dispersion interactions between molecules187 195. 

Inserting for each pair (a, b) the multipole expansion of r 1 with respect to centers located at sites a and b, cf. Eq. (1-141), and using Eqs. (1-142) and (1-145) one gets the following expression for the multicenter distributed multipole expansion of the electrostatic energy ... 

Inserting the multipole expansions of the operators rfj1 and rfA with respect to the pairs of sites (a, b) and (a1, b ), respectively, and defining the distributed polarizability tensor,... 

As said above, there is no unique way to represent the solute s charge distribution by a multicentric multipole expansion. This point has been discussed by Stone [71] in the general case and by Rinaldi et al. [26] in the context of the MPE method. The... 

The dispersion interaction arises from fluctuations in the charge distribution of <2, leading to transient induction in S and vice versa. By applying the multipole expansion in the dipole approximation in both Q and S, I4uf sp is reduced to... 

Multipoles. One can express [15] the multipole expansion of the energy of a charge distribution in an external field by defining the free energy G of a localized charge distribution p(r), which is placed in an external potential (which has no charge distribution associated with it) ... 

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