Electrostatic Contribution Distributed Multipoles

The electrostatic interaction, which is defined as the classical Coulombic interaction between the undistorted charge distributions of the isolated molecules, is the easiest to derive from wavefunctions. When there is no overlap of the charge distributions of the molecules, all that is required is a representation of the molecular charge density. The traditional, and simplest, representation of the molecular charge distribution is in terms of the total multipole moments. The first nonvanishing multipole moment could often be derived from experi- 

Electrostatic Distributed multipoles e.g., DMA, CAMM, Topologically partitioned electric properties Polyatomics, up to polypeptides 

Dispersion Distributed dispersion Linear molecules, e.g., N2, C2H2 

Exchange-repulsion Overlap modeF Diatomics, small organic molecules 

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The potential outside the charge distribution and due to it is simply related to the moments, as is the interaction energy when an external field is applied.14 The multipole moments are thus very useful quantities and have been extensively applied in the theory of intermolecular forces, particularly at long range where the electrostatic contribution to the interaction may be expanded in moments. Their values are related to the symmetry of the system thus, for instance, a plane of symmetry indicates that the component of n perpendicular to it must be zero. Such multipoles are worth calculating in their own right. 

The multipole part can efficiently be estimated from the distributed multipole analysis191. In this way the electrostatic penetration contribution is obtained. One may note that the accuracy of the electrostatic term can be increased by keeping the penetration part from Eq. (1-184), and replacing the Hartree-Fock distributed multipole moments by some correlated, e.g. MP2 moments. Finally, the intramonomer correlation term and the dispersion energy can be evaluated from the expression,... 

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. 

The contribution to the predicted electrostatic potential of the anisotropic atomic multipoles (Q , / > 0), which represent the lone pair and n-electron density, rapidly become less important as the distance between the molecules increases. This not only results from the inverse power of R increasing with I, but also from the cancellation between the contributions from different multipoles and different atoms. For example, there is generally an atomic dipole component along a bond that opposes the polarity implied by the atomic charges, as shown in the results of distributed multipole analyses (DMAs) of the azabenzenes. ° Thus, the accuracy gained by using a distributed multipole model is very dependent on the relative separation and orientation of the molecules, as well as the actual distribution of charge in the molecule. 

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 ... 

Note that a distinction is made between electrostatic and polarization energies. Thus the electrostatic term, Ue e, here refers to an interaction between monomer charge distributions as if they were infinitely separated (i.e., t/°le). A perturbative method is used to obtain polarization as a separate entity. The electrostatic and polarization contributions are expressed in terms of multipole expansions of the classical coulomb and induction energies. Electrostatic interactions are computed using a distributed multipole expansion up to and including octupoles at atom centers and bond midpoints. The polarization term is calculated from analytic dipole polarizability tensors for each localized molecular orbital (LMO) in the valence shell centered at the LMO charge centroid. These terms are derived from quantum calculations on the... 

Stone s DMA method has been applied in several other papers. Our review cannot be exhaustive but we would like to quote two additional papers using this approach because they give additional information on the basic problems of the electrostatic approach. Price, Harrison and Guest [89] examined the DMA description of the MEP of a large molecule, with formula C63H113N11O12, obtained from a 3-21G SCF wavefunction. The description of the electrostatic potential obtained in such a way is comparable to that obtained with potential derived atomic charges (PD-AC) to which we shall refer later on in more detail. The superiority of a distribute multipole description, in describing the anisotropic contributions to the MEP on the van der Waals surface is shown clearly. 

A useful alternative approach is to isolate the components of the perturbation expansion, namely the repulsion, electrostatic interaction, induction, and dispersion terms, and to calculate each of them independently by the most appropriate technique. Thus the electrostatic interaction can be calculated accurately from distributed multipole descriptions of the individual molecules, while the induction and dispersion contributions may be derived from molecular polarizabilities. This approach has the advantage that the properties of the monomers have to be calculated only once, after which the interactions may be evaluated easily and efficiently at as many dimer geometries as required. The repulsion is not so amenable, but it can be fitted by suitable analytic functions much more satisfactorily than the complete potential. The result is a model of the intermolecular potential that is capable of describing properties to a high level of accuracy. 

Other contributions to the interaction energy are usually less important. The induction energy can be calculated using distributed multipoles and polarizabilities for the individual molecules as for the electrostatic energy it is necesary to include contibutions at least to... 

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