Electric Multipole Potential

In GBEMP model, point multipoles are included inside Gay-Berne particles for providing a reasonable approximation to the charge density of corresponding all-atom model, and they are usually placed at the mass centers of the Gay-Berne particles and/or specific locations inside the Gay-Berne particles. In some cases, non-interaction EMP sites, which do not involve non-bonded interactions, are used to connect two different Gay-Berne particles. Thus, the interaction energy between two electric multipole sites (/ and j) can be computed as 

Ty is the interaction matrix [58]. In Eq. 13.22, the notations q, d and Q represent charge, dipole and quadrupole moments, respectively. The number of EMP sites included in Gay-Berne particles would determine how accurately the electrostatics would be described as well as how fast the model is able to achieve. Thus, it is critical to decide the number of EMP sites for each Gay-Berne particle through optimizing the balance between accuracy and efficiency. 

The point multipoles can give the accurate description of the electrostatic interactions between Gay-Berne particles separated with a certain distance (independent of the particle sizes). As the two particles are getting too close to each other, the point multipoles are not able to accurately describe the overlap of their charge density, causing the so-called penetration error [59]. An effective solution to avoid the penetration error would be to seek a proper damping function [60]. In current GBEMP model, we employed the damping function defined as 

It is has been known that the atomic multipole moments for atoms in AMOEBA model can be calculated through quantum mechanics method and Stone s distributed multipole analysis [61]. Thus, it is straightforward to obtain the parameters of electric multipole potentials based on the distributed multipole analysis after the EMP sites of Gay-Berne particles are decided or directly from AMOEBA force field. However, the EMP parameters of Gay-Berne particles need to be optimized because they are derived based on the gas-phase ab initio quantum mechanics. One possible solution would be to match GBEMP and AMOEBA results for the electrostatic energies between CG particles and water molecules, or between CG particle dimers, at various separations and/or in different orientations. 

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The electric multipole potential being given generally by equations (44) and (45), we have for the electric field strength ... 

Fig. 24. 1 m NaCl solution at 303 K and surface charge density ao = —4.6 pC cm . Top Atom and ion density profiles. Bottom. Total electrostatic potential and component (monopole, dipole, combined monopole plus dipole, quadrupole and octopole) electric multipole potentials. The total potential was calculated from the total electric charge distribution. Note that the sum of monopole and dipole components is smaller than the quadrupole component. (Reproduced with permission from Ref. 103.)... 

Shen, H., Li, Y, Ren, R, Zhang, D., and Li, G. (2014). An anisotropic coarsegrained modei for proteins based on Gay-Berne and electric multipole potentials,/. Chem. Theory Comput (in press)... 

Let us now turn to spherical particles and let us ask for their reaction to the external magnetic or electric multipole potential... 

Consider tlie mutual approach of two noble gas atoms. At infinite separation, there is no interaction between them, and this defines die zero of potential energy. The isolated atoms are spherically symmetric, lacking any electric multipole moments. In a classical world (ignoring the chemically irrelevant gravitational interaction) there is no attractive force between them as they approach one another. When tliere are no dissipative forces, the relationship between force F in a given coordinate direction q and potential energy U is... 

Such expressions can be easily generalized to cover the case of the electric multipole transition operator with an unspecified value of the gauge condition K of electromagnetic field potential (4.10) or (4.11). 

The permanent electric multipoles (m = 0) are defined by equation (40), the electric multipoles of first order w = 1 by equation (72), and those of the second order w = 2 by equation (79). Similarly, magnetic potential energy of order m — I can be defined, on replacing in equation (83) E by B and p by m. From the general expression [equation (83)] one immediately and quite easily derives all the energies dealt with in the theory of non-linear molecular processes. ... 

With the general definition of the electric multipole (40) and n-th order field (53), the potential energies of electrostatic interaction between electric multipoles and fields take the general form ... 

It is quite obvious that the best representation of this potential consists in constructing contour lines maps (Figs. 1, 2). Such pictures are often very effective, but have the limitation that several sections are necessary to represent the potential in space, and they do not lend themselves easily to further calculations. These diagrams, sometimes not easy to interpret, will be discarded by the chemist in favor of a more tangible and familiar description, based on point charges and electric multipoles, especially dipoles. The problem is then to reproduce as well as possible the electrostatic field created by the molecules by means of these point charges and these dipoles. 

Full knowledge of the charge distribution of a molecule requires specification of the charge density at all points. For some purposes the charge density provides excess information thus, the potential outside a sodium ion is independent of the distribution of the electrons, and the interaction of a molecule with a uniform external field is determined by its dipole moment and dipole polarizabilities. The electric multipole moments characterize the charge distribution the first three are defined as follows ... 

An electric multipole expansion is obtained by inserting the scalar potential in the form suggested by Bloch (141) into of Eq.(120). We then obtain the expression... 

The first term is the total charge 2 interacting with the value of the scalar potential at the expansion point, whereas the second term corresponds to higher moments of the charge distribution interacting with spatial derivatives of the electric field at the expansion point. The electric multipoles have the general... 

Note that, contrary to the electric multipole expansion (148), there is no zeroth order term corresponding to a magnetic charge interacting with the value of the vector potential at the expansion point, this reflecting the absence of magnetic monopoles. 

The representation of hyperfine interactions in the form of multipoles follows from expansion of the potentials of the electric and magnetic fields of a nucleus, conditioned by the distribution of nuclear charges and currents, in a series of the corresponding multipole momenta. It follows from the properties of the operators obtained with respect to the inversion operation that the nucleus can possess non-zero electric multipole momenta of the order k = 0,2,4,..., and magnetic ones with... 

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