Equivalent Multipoles

As has been mentioned, the electrostatic potential created by the molecule plays an important role in the ionic reactions as well as in molecular interaction phenomena. As the problem has been presented and discussed in detail in an article recently published in this series 34) shall not discuss it here. 

It is quite obvious that the best representation of this potential consists in constructing contour lines maps (Figs. 1, 2). Such pictures are often very effective, but have the limitation that several sections are necessary to represent the potential in space, and they do not lend themselves easily to further calculations. These diagrams, sometimes not easy to interpret, will be discarded by the chemist in favor of a more tangible and familiar description, based on point charges and electric multipoles, especially dipoles. The problem is then to reproduce as well as possible the electrostatic field created by the molecules by means of these point charges and these dipoles. 

General Expression of the Electrostatic Potential Created by a Molecule 

The potential created by a molecule is completely defined by the charge density values in every point of space. This density may be written  

In the following, we shall always assume that the nuclei are fixed. It is well known that the molecule is actually in a perpetual state of vibration, the nuclei oscillating around equilibrium positions corresponding to the minima of energy. We shall suppose that the nuclei are fixed in these positions and we shall define the coordinate system with reference to them. 

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Instead of trying to derive the charges from the various equivalent multipoles, one can try directly to use the wave functions describing the electrons in the molecule. The problem is neither simple nor easy. 

Gaussian also predicts dipole moments and higher multipole moments (through hexadecapole). The dipole moment is the first derivative of the energy with respect to an applied electric field. It is a measure of the asymmetry in the molecular charge distribution, and is given as a vector in three dimensions. For Hartree-Fock calculations, this is equivalent to the expectation value of X, Y, and Z, which are the quantities reported in the output. 

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. 

The problem is also a challenge from both group theoretical [6,7] and experimental [8] point of view. In the following we will use a method which is based on a multipole expansion of the Coulomb interaction between electrons on a same molecule [9,10]. Thereby we systematically include electronic transitions which go beyond the usual Hartree-Fock scheme and hence our approach is equivalent to a full configuration interaction calculation. The details of our technique are given in Ref. [10]. 

One of the advantages of the molecular (one center) and multicenter multipole expansions of the MEP is that, through truncating the series after some terms, one can get an analytical expression. The molecular multipole expansion, in contrast with the multicenter multipole expansion, diverges at distances of chemical interest. The various multicenter multipole expansion [87, 89, 93-96] practically equivalent to each other [97]. 

Further comparison between PCM and MPE was reported in the paper by Rinaldi et al. [26] using several multipole moment distribution types. Again, an excellent agreement was obtained. The authors reported examples for which the computational time with the MPE method represents only 30% of the time required for the equivalent computation with PCM (for details on code versions, see that paper). MPE and PCM results are also found comparable for IR properties [91] as well as for UV spectra [91,92] of substituted coumarins and chromones at the TD-DFT level. 

Energy calculations for ionic lattices show 6a = 1l6. Molten salts can form transient dipoles and multipoles as ion pairs and clusters, but it is unlikely that these contribute to a dielectric constant. For charge transfer in molten salts, the equivalent of an FC process is the change in electroneutrality length as valence changes. 

Fig. 3 A comparison of different coarse grain lipid models. The Shelley model " of DMPC, and Marrink and Essex models of DPPC are compared to their atomistic equivalents (for ease of comparison, hydrogen atoms of the atomistic models are not shown). Solid lines represent harmonic bonds connecting CG particles, and the CG particle types for the Shelley and Marrink models are labelled (the labels are the same as those used in the main text). The point charges (represented by + and —) and point dipoles (represented by arrows) are shown for the Essex model (the charges and dipoles are located at the centre of their associated CG particle). The Shelley and Marrink models use LJ particles (represented by spheres), while the Essex model uses a combination of LJ particles (spheres) and Gay-Berne particles (ellipsoids). Finally, the blob model proposed by Chao et al is also shown for comparison. This model represents groups of atoms as rigid non-spherical blobs that use interaction potentials based on multipole expansions.

One simple but very effective logic procedure developed for DHF keeps track of the derivatives. An integer list is constructed for each particular derivative where there is one integer in the list for each parameter. For first-and second-multipole polarizabilities, there will be nine parameters, three for the first moment and six for the second moment, after ignoring equivalent elements in V. One way of ordering them is V, V, V, Vj,y, Vyy, Vy, and K22, and then nine integers, ordered the same, are associated with each a derivative ... 

Note also that neither CHEEP nor PDQC assigns identical charges to the two hydrogens that are equivalent by symmetry. The differences are small, on the order of 0.001 e, but nonnegligible. This effect, which had been noted previously in fitting multipoles to potentials, was treated by aligning the major symmetry axis of the molecule with one of the Cartesian axes, a measure that obscures, but does not fix, the problem. 

A general equation can be derived that describes the variation in direction of the valence electron density about the nucleus. The distortion from sphericity caused by valence electrons and lone-pair electrons is approximated by this equation, which includes a population parameter, a radial size function, and a spherical harmonic function, equivalent to various lobes (multipoles). In the analysis the core electron density of each atom is assigned a fixed quantity. For example, carbon has 2 core electrons and 4 valence electrons. Hydrogen has no core electrons but 1 valence electron. Experimental X-ray diffraction data are used to deri e the parameters that correspond to this function. The model is now more complicated, but gives a better representation of the true electron density (or so we would like to think). This method is useful for showing lone pair directionalities, and bent bonds in strained molecules. Since a larger number of diffraction data are included, the geometry of the molecular structure is probably better determined. 

Whitehead C, Breneman C, Sukumar N, Ryan M. Transferable atom equivalent multicentered multipole expansion method. J Comput Chem 2003 24 512-29. 

The explicit form of the interaction Hamiltonian // ,(() consists of a series of multipolar terms, but for most purposes the electric-dipole (El) approximation is sufficient. Although the results are calculated within this approximation for each molecular center detailed analysis of the coupling provides results equivalent to the inclusion of higher-order multipole terms for the pair. The same assumption underlies the well-known coupled-chromophore model of optical rotation (Kuhn 1930 Boys 1934 Kirkwood 1937). The Hamiltonian for the system may thus be written as... 

Indeed, this idea lies at the heart of the venerable notion of assigning partial charges to the atoms in a molecule [17]. One may think of this practice as a representation of the true charge distribution of the molecule by a series of distributed multipoles (in this case, limited to monopoles) at various sites, namely, atomic centers. Even if limited to monopoles, the act of spreading them out over the entire molecule is equivalent in some sense to simulation of high orders of molecular center-based multipoles. 

This uses a multipole wiggler and will have operational modes for focussed Laue diffraction work and monochromatic experiments. The small source sizes should allow an equivalently small focal spot from a grazing incidence mirror system. Exposure times in the microsecond range for a macromolecular crystal should be feasible. Depending on the current achieved in single bunch mode it may be possible, at least for smaller unit cell sizes, to record a Laue pattern from one of the single bunches with an intrinsic time resolution therefore of the bunch width. (Feasibility experiments of this kind have been conducted at CHESS but on an undulator (Szebenyi et al 1989).)... 

All successful water models make use of a distribution of point charges rather than of point multipoles. The main reason is that the directional properties of intermolecular hydrogen bonds can be obtained efficiently with oifly 3 or 4 point charges. Furthermore, Goldman and Backx [39] have shown that model molecules with such distributions of point charges are more effective as solvents (for instance in their ability to dissociate ion pairs) than molecules with equivalent point dipoles and quadrupoles. 

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