Multipole expansion electrostatic energy

Long-range forces are most conveniently expressed as a power series in Mr, the reciprocal of the intemiolecular distance. This series is called the multipole expansion. It is so connnon to use the multipole expansion that the electrostatic, mduction and dispersion energies are referred to as non-expanded if the expansion is not used. In early work it was noted that the multipole expansion did not converge in a conventional way and doubt was cast upon its use in the description of long-range electrostatic, induction and dispersion interactions. However, it is now established [8, 9, 10, H, 12 and 13] that the series is asymptotic in Poincare s sense. The interaction energy can be written as... 

If the long-range mteraction between a pair of molecules is treated by quantum mechanical perturbation theory, then the electrostatic interactions considered in section Al.5.2.3 arise in first order, whereas induction and dispersion effects appear in second order. The multipole expansion of the induction energy in its fill generality [7, 28] is quite complex. Here we consider only explicit expressions for individual temis in the... 

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 ... 

The results of energy partitioning in Li+... OH2 obtained with a number of different basis sets are listed in Table 3. Since intermolecular overlap is small in these kind of complexes (Table 1), we expect the electrostatic model to be a good approximation for classical contributions to the total energy of interaction. Indeed, ZlE cou is to a good approximation proportional to the dipole moment of the water molecule calculated with the same basis set. This can be seen even more clearly in Table 4 where zIEcou is compared with ion-dipole and ion-quadrupole energies obtained from the classical expression of the multipole expansion series 45,95-97) ... 

McKean 182> considered the matrix shifts and lattice contributions from a classical electrostatic point of view, using a multipole expansion of the electrostatic energy to represent the vibrating molecule and applied this to the XY4 molecules trapped in noble-gas matrices. Mann and Horrocks 183) discussed the environmental effects on the IR frequencies of polyatomic molecules, using the Buckingham potential 184>, and applied it to HCN in various liquid solvents. Decius, 8S) analyzed the problem of dipolar vibrational coupling in crystals composed of molecules or molecular ions, and applied the derived theory to anisotropic Bravais lattices the case of calcite (which introduces extra complications) is treated separately. Freedman, Shalom and Kimel, 86) discussed the problem of the rotation-translation levels of a tetrahedral molecule in an octahedral cell. 

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... 

Although the spherical form of the multipole expansion is definitely superior if the orientational dependence of the electrostatic, induction, or dispersion energies is of interest, the Cartesian form171-174 may be useful. Mutual transformations between the spherical and Cartesian forms of the multipole moment and (hyper)polarizability tensors have been derived by Gray and Lo175. The symmetry-adaptation of the Cartesian tensors of quadrupole, octupole, and hexadecapole moments to all 51 point groups can be found in Ref. (176) while the symmetry-adaptation of the Cartesian tensors of multipole (hyper)polarizabilities to simple point groups has been considered in Refs. (172-175). 

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 ... 

It should stressed that unlike in the case of the electrostatic energy, the expressions for the long-range coefficients and the angular function Aj defining the multicenter multipole expansions of the induction and dispersion energies are different. This difference is due to the fact that in the multicenter expansions the products of the D functions, (wj) ) and ( )... 

Interaction BH-BH in the linear configuration, a) Electrostatic interaction energy and the first terms of its multipole expansion, Intermolecular distance, R,and quadrupole and octupole moments refer to the centers of mass, b) SCF interaction energy and its components. 

Flg-Z Ratio of the multipole expanded (eqn. 16) and the imexpand-ed electrostatic energy (eqn. 12) for two parallel ethene molecules (from ref.". Different multipole expansion lengths are shown... 

In this system the long range (multipole) interaction energy has been calculated directly in the form of a spherical expansion (4) electrostatic R, R and R terms, formula (16), dispersion R , R , R ° terms, formula (21) and induction R", R terms, formula (20). The multipole moments used in the electrostatic energy... 

The electrostatic interaction energies are evaluated using the multipole expansion formulas for each intermolecular pair of sites. Explicit expressions for all terms up to R are given in Ref. 117, and for terms up to R in Ref. 118. Stone has provided a general formulation and discussion of the spherical tensor and Cartesian tensor approaches. The program ORIENT incorporates... 

Equations [22]-[24] illustrate why the derivation and programming of the forces and torques, and second derivatives for all the terms up to R in the atom-atom multipole expansion of the electrostatic energy is a nontrivial exercise in classical mechanics. It has been described in detail by Popelier and Stone,and, with the additional derivatives required for modeling molecular crystal structures, by Willock et al. ... 

G. Jansen, C. Hattig, B. A. Hess, and J. G. Angyan, Mol. Phys., 88,69 (1996). Intermolecular Interaction Energies by Topologically Partitioned Electric Properties. 1. Electrostatic and Induction Energies in One-Center and Multicenter Multipole Expansions. 

Due to the non-local character of the Coulomb operator, the decomposition for the electrostatic energy is more complex. In order to distinguish between local and global terms, we need to introduce atom-dependent screening densities, (hard) and (soft), that generate the same multipole expansion as the local density ha — nA A n, where is the nuclear charge of atom A. 

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