Electrostatic multipole term

Components defined in such way naturally correspond to hierarchy of theoretical models of gradually decreasing complexity MP2, SCF, first order interaction energy E, electrostatic term E i, electrostatic multipole term be defined in the section 2.2). Such consistent division could be very useful in construction of simplified models based on more accurate theories. 

Another problem arises from the presence of higher terms in the multipole expansion of the electrostatic interaction. While theoretical formulas exist for these also, they are even more approximate than those for the dipole-dipole term. Also, there is the uncertainty about the exact form of the repulsive interaction. Quite arbitrarily we shall group the higher multipole terms with the true repulsive interaction and assume that the empirical repulsive term accounts for both. The principal merit of this assumption is simplicity the theoretical and experimental coefficients of the R Q term are compared without adjustment. Since the higher multipole terms are known to be attractive and have been estimated to amount to about 20 per cent of the total attractive potential at the minimum, a rough correction for their possible effect can be made if it is believed that this is a preferable assumption. 

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... 

A factor -2 included in the last term here compensates for the use of Rydberg units and for the omission of the negative electronic charge in potential functions derived from Eq. (7.14). Hence the electrostatic multipole moments of atomic cell r/( are... 

The COSMO method is also interesting as the basis of a very successful COSMO-RS method, which extends the treatment to solvents other than water [27,28]. The COSMO method is very popular in quantum chemical computations of solvation effects. For example, 29 papers using COSMO calculations were published in 2001. However, we are not aware of its use together with MM force fields. Compared with the BE method, COSMO introduces one more simplification, that of Eq. (22). On the other hand, the matrix A in Eq. (21) is positively defined [25], which makes solution of the system of linear equations simpler and faster. Also, because both A and B matrices contain only electrostatic potential terms, their computation in quantum chemistry is easier than calculation of the electric field terms in Eq. (12). Another potential benefit is that the long-range electrostatic potential contribution is easier to expand into multipoles than the electric field needed in BE methods, which may benefit linear-scaling approaches. 

One quickly recognizes the endohedral potential, Eq.(l), in the first term of Eq.(32). Since, as discussed in Section 3.4, the electrostatic potential is almost constant in the vicinity of the cage center, for small guest atoms, ions, or molecules one can neglect the higher-order multipole terms in Eq.(32) and write... 

Since empirical force fields do not accurately estimate the true interatomic forces, it is difficult a priori to say how accurate the fast multipole approximation to the exact Coulomb potential and forces (exact in terms of the sum over partial charges) should be. Probably a good rule is to make sure that at each atom the approximate electrostatic force is within a few percent relative error of the true electrostatic force, obtained by explicitly summing over all atom pairs, i.e., IF — FJ < 0.05 F , for all atoms i, where F is the... 

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). 

The first term in (4) is the band energy contribution, as given by the GPM. The second term represents electrostatic interactions of higher multipoles and is usually neglibly small. 

A common feature of many mesogenic molecules is the presence of polar substituents and aromatic cores [3]. The electrostatic interactions between such groups can be incorporated into a molecular potential with the addition of dipolar and quadrupolar terms, respectively. Rather than represent these permanent electrostatic interactions by using a model in which a charge distribution is scattered over the surface of the molecule, it is very common to use one (or more) point multipoles [2,29]. Thus for an electrostatic Gay-Berne model, the pair potential is given by the sum... 

Consequently, we introduce the second approximation which is to use an approximate electrostatic potential in Eq.(4-21) to determine inter-fragment electronic interaction energies. Thus, the electronic integrals in Eq. (4-21) are expressed as a multipole expansion on molecule J, whose formalisms are not detailed here. If we only use the monopole term, i.e., partial atomic charges, the interaction Hamiltonian is simply given as follows ... 

When the non-electrostatic terms are semiempirical, they also make up in an average way for systematic deficiencies in the treatment of electrostatics, e.g., for the truncation of the distributed multipole representation of the solute charge density at the monopole term on each center. 

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