Discrete Multipoles

Most discrete MTP implementations are similar in many respects, e.g., limited expansion up to order 2-4, spherical harmonic description, interaction calculation in the atoms local frames. Hence, what distinguishes these force fields and implementations from each other is primarily in how they treat the other interaction terms. Most importantly, static multipoles only consist of a first-order perturbation of the electrostatic operator. Describing second-order effects leads to polarizability—the charge density s ability to respond to an external electric field—a critical aspect of certain systems (e.g., dielectric changes) [62-64]. Here, possible implementations are ordered in terms of increased overall accuracy (and thus computational investment and larger parametrization effort). Given the heavy requirements of such refined force fields, it is important to point out that more is not always better, and each system of interest will call for a fine balance of accuracy and statistical sampling. 

Non-polarizable force fields Static MTP electrostatic descriptions 

Accurate energy-decomposition schemes Other force fields build on both atomic multipoles and polarizability to provide an accurate decomposition of intermolecular energies. The sum of interaction between fragments ab initio (sibfa) [83, 84] decomposes the 

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Current developments of the MPE continuum model focus on the combination of a multicentric multipole moment expansion of the reaction field combined with a discrete charge representation of the solute charge distribution fitting the electrostatic potential. This scheme leads to a simple formulation that parallels generalized-Born (GB) methods, though in the MPE-GB approach, the only parameter that needs to be defined is the cavity surface [76]. 

For any atomic multipole transition, the excited state can be described in terms of the dual representation of corresponding SU(2) algebra, describing the azimuthal quantum phase of the angular momentum. In particular, the exponential of the phase operator and phase states can be constructed. The quantum phase variable has a discrete spectrum with (2j + 1) different eigenvalues. 

The radiation phase of multipole photons has discrete spectrum in the interval (0,2n). In the classical limit of high-intensity coherent field, the eigenvalues of the radiation phase are distributed uniformly over (0, 2ji). 

Despite being quite different from one another, the QM/discrete models are ah. characterized by maintaining the information on the atomic structure of the environment. The most popular formulation of these models is to use the MM force fields to describe the interaction within the ENV part of the system as weU as the nonelectrostatic interactions between the QM subsystem and the ENV. The electrostatic interactions between the two parts are instead kept in the effective QM Hamiltonian as an additional operator which contributes to determine the MS wavefrmction. Such an operator is generahy written in terms of a set of fixed multipoles usuaUy placed on the atoms of the environment molecules. In most cases just the partial charges are considered, but there are instances where multipoles up to the quadrupoles are included. The resulting MS/ENV interaction term thus becomes ... 

Figure 2 A molecular system (perldlnin, a carotenoid) surrounded by a polarizable discrete environment, represented in terms of fixed multipoles and induced dipoles.

Clearly, the considerations that need to be applied to electrolytes are somewhat different. However, Debye-Hiickel screening for all except the most dilute electrolytes helps to limit the range over which discrete sums of Coulomb terms have to be carried out for amorphous systems, and the cell multipole method can be used to compute the longer range contributions (121). Sometimes it is useful to use cutoffs to propagate the MD simulation, and then use Ewald summations to intermittently evaluate the energy to test that the cutoff procedime is sufficiently accurate. 

This balance is achieved by introducing a finite cavity which reflects all waves perfectly. As well as excluding any energy loss, the cavity guarantees normalizability of all waves. Furthermore, it splits up the continuous frequency spectrum of the Hertzian multipoles. It is a discrete set of coupled ingoing and outgoing modes, which interacts with the particles under consideration. 

Koc, S., and Chew, W. C. (2001) Multilevel fast multipole algorithm for the discrete dipole approximation,/ Electromagnet. Wave., 15,1447-1468. 

Fig. 3.18. Normalized differential scattering cross-sections of an oblate cylinder with ksb = 15 and 2ksa = 7.5. The cmves are computed with the TAXSYM routine, discrete somces method (DSM), multiple multipole method (MMP), discrete dipole approximation (DDA) and finite integration technique (CST)...

K.H. Ding, C.E. Mandt, L. Tsang, J.A. Kong, Monte Carlo simulations of pah-distribution functions of dense discrete random media with multipole sizes of particles, J. Electromag. Waves Appl. 6, 1015 (1992)... 

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