Electrostatic energies multipoles

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. 

Once the electronic multipole moments have been identified as q, net charge /t, dipole moment etc. the above expression for the electrostatic energy becomes... 

Finally, the electrostatic energy can be written as a sum of interactions between the multipoles of the two molecules... 

The expression for the electrostatic energy given above can be further subdivided to give a sum of interactions between atomic multipoles, which is in turn summed over all possible pairs of interacting atoms. 

The radial part of the electrostatic energy in the multipole approximation is given... 

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 ( )... 

We wish to end this section by saying that the variation-perturbation approach as discussed above, introduces a natural hierarchy of gradually more and more sophisticated models starting from the crude evaluation of the electrostatic energy in the distributed multipole approximation, and ending with the inclusion of the intramolecular and dispersion contributions at the MP2 or even more correlated level. 

It includes the interactions of distributed multipole moments Q (up to a quadrupole) labeled t and u. The T matrix provides the Coulomb energy appropriate for particular multipoles and includes the distance between sites a and b and their relative orientations. The short range (penetration) component of the electrostatic energy, in a manner similar to the Ar-CC>2 case, can be absorbed into the exchange repulsion term. 

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 accuracy of a distributed multipole representation can always be tested by evaluating the effect on the calculated electrostatic energies in the region of interest of increasing the number of interaction sites (e.g., by adding sites at the midpoints of bonds) or the order of multipoles. Similarly, simplifications of the model, such as removing small multipole components, can be evaluated. However, the accuracy of the calculated electrostatic energies is inevitably limited by the quality of the wavefunction and the absence of penetration effects. 

The distributed multipole model incorporates a nearly exact description of the molecular charge distribution into the evaluation of the electrostatic energy. Is the increase in accuracy gained by representing the effects of lone pair and 7i-electron density worth the extra complexity in the potential model Even if there is a significant enhancement, is it worth using such an elaborate model when only crude models, such as the isotropic atom-atom 6-exp potential, are available for the other contributions ... 

There have been many studies that contrast the accuracy of various atomic charge and distributed multipole models. These studies include the extensive tests provided when various distributed multipole methods were first proposed. For example, there are published contour plots of the potential around a water molecule, the amino acid histidine, and variations in the electrostatic energies of nucleic acid bases,which confirm the significance of the atomic anisotropy shown in the color three-dimensional displays of the electrostatic field around uracil and pyrimidine. It is clear that the difference... 

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. ... 

DMA models have also been incorporated into the molecular modeling force field program MOMO, ° which was used to study host-guest complexes, such as benzene in a hexaoxacyclophane host. The difficult problem of accurately modeling the intramolecular electrostatic energy of proteins has also used distributed-multipole-type models. ... 

R. J. Wheatley and J. B. O. Mitchell, /. Comput. Ghent., 15, 1187 (1994). Gaussian Multipoles in Practice—Electrostatic Energies for Intermolecular Potentials. 

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. 

We will be using this form of the molecule-field interaction repeatedly in this text, however, it should be kept in mind that it is an approximation on several counts. Already Eq. (3.1), an electrostatic energy expression used with a time varying field, is an approximation. Even in this electrostatic limit, Eq. (3.1) is just the first term in an infinite multipole expansion in which the higher-order terms depend on higher spatial derivatives of the electric field. 

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