Multipole formalism, application

In the following section we discuss application of the multipole formalism to a series of Fe(II) porphyrins. A next step towards derivation of the electronic wavefunction of a transition metal complex may be based on the LCAO formalism for the molecular orbitals. Test calculations with such a formalism, using a theoretical data set, are described in the final part of this article. 

The poly tensor approach introduced by Applequist [169] is a terrific organization of the problem of electrical interaction for high-level calculation because it can be continued uniformly to any order of multiple moment, any distribution of moments, and any number of interacting species. Furthermore, it can incorporate multipole polarization and hyperpolarization [170]. As such, it provides a scheme that can be coded for computer application in an open-ended fashion while also providing the formal analysis needed to extract functional forms of different electrical interaction pieces. 

It has recently become clear that classical electrostatics is much more useful in the description of intermolecular interactions than was previously thought. The key is the use of distributed multipoles, which provide a compact and accurate picture of the charge distribution but do not suffer from the convergence problems associated with the conventional one-centre multipole expansion. The article describes how the electrostatic interaction can be formulated efficiently and simply, by using the best features of both the Cartesian tensor and the spherical tensor formalisms, without the need for inconvenient transformations between molecular and space-fixed coordinate systems, and how related phenomena such as induction and dispersion interactions can be incorporated within the same framework. The formalism also provides a very simple route for the evaluation of electric fields and field gradients. The article shows how the forces and torques needed for molecular dynamics calculations can be evaluated efficiently. The formulae needed for these applications are tabulated. 

The one-centre expansion of the MEP is not adequate, however, for the large number of chemical applications we mentioned in Sect. 4. In fact the expansion theorem holds for points r lying outside a sphere containing all the elements of the charge distribution. In molecules, this condition is never formally satisfied, because Q r) has an exponential decay. The difference between V (r) and the exact multipole expansion of K (r) is generally called the penetration term ... 

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