Multipole electron systems

The original FMM has been refined by adjusting the accuracy of the multipole expansion as a function of the distance between boxes, producing the very Fast Multipole Moment (vFMM) method. Both of these have been generalized tc continuous charge distributions, as is required for calculating the Coulomb interactioi between electrons in a quantum description. The use of FMM methods in electronic structure calculations enables the Coulomb part of the electron-electron interaction h be calculated with a computational effort which depends linearly on the number of basi functions, once the system becomes sufficiently large. 

The combination of the neutron structural model with a multipole model of the X-ray data measured at a matching temperature, has enabled us to obtain detailed information about the electron density distribution. This has revealed new information about the bonding in m-enol systems. 

Many molecules contain chemically equivalent atoms, which, though in a different crystal environment, have, to a good approximation, the same electron distribution. Such atoms may be linked, provided equivalent local coordinate systems are used in defining the multipoles. In particular, for the weakly scattering hydrogen atoms, abundant in most organic molecules, this procedure can lead to more precisely determined population parameters. 

These correlated fluctuations themselves ride on a further set of coherent fluctuations taking place at a much lower frequency scale and normally attributed to the phonons, the traditional exchange Bosons associated with superconductivity. Real systems are never devoid of ionic or nuclear motion, and at the very least it is now Hamiltonian (3) (and eventually its extension to alloys) that applies for a full discussion of superconductivity density fluctuations in the nuclear coordinates are omnipresent and of course their effects on electronic ordering have been evident for quite some time. An elementary estimate of the relative importance of (monopole) polarization arising from phonons and the (multipole) equivalents arising from internal fluctuations, primarily of a dipole character, can now be easily given. 

As we are dealing with spherical harmonics, and as we are trying to model the aspherical atomic electron density, the orientation of the local atom centered coordinate system is, in principle, arbitrary, appropriate linear combinations always giving the same result. However, in practice it is helpful to choose a local coordinate system such that the multipoles are oriented in rational directions, and thus the most important multipole populations will lie in directions that would be expected to represent chemical bonds or lone pairs [2,20], e.g. for an sp2 hybridized atom, defining one bond as the x direction, the trigonal plane as the xy plane, and z perpendicular defines three lobes of the 33+... 

The redistribution of the valence electron density due to chemical bonding may be obtained from summing the multipole populations or Fourier transforming appropriately calculated structure factors, having removed the contribution from neutral spherical atoms, to produce a so-called deformation density map [2], This function was introduced by Roux et al. [23] and has been widely used since then. The deformation electron density represents the difference between the electron density of the system, p(r), and the electron... 

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