Electrostatic moments

Distributed Electrostatic Moments Based on the Electron Localization Function Partition... 

That way, the Distributed Electrostatic Moments based on the ELF Partition (DE-MEP) allows computing of local moments located at non-atomic centres such as lone pairs, a bonds and n systems. Local dipole contributions have been shown to be useful to rationalize inductive polarization effects and typical hydrogen bond interactions. Moreover, bond quadrupole polarization moments being related to a n character enable to discuss bond multiplicities, and to sort families of molecules according to their bond order. 

Pilme J, Piquemal J-P (2008) Advancing beyond charge analysis using the electronic localization function Chemically intuitive distribution of electrostatic moments. J Comput Chem 29 1440... 

The IAM model further assumes the atoms in a crystal to be neutral. This assumption is contradicted by the fact that molecules have dipole and higher electrostatic moments, which can indeed be derived from the X-ray diffraction intensities, as further discussed in chapter 7. The molecular dipole moment results, in part, from the nonspherical distribution of the atomic densities, but a large component is due to charge transfer between atoms of different electronegativity. A population analysis of an extended basis-set SCF wave function of HF, for example, gives a net charge q of +0.4 electron units (e) on the H atom in HF for CH4 the value is +0.12 e (Szabo and Ostlund 1989). 

The electrostatic moments of a distribution p(r) in terms of the clmp are given by... 

Expression (4.31) is widely applied to calculate the error in properties derived from the least-squares variables. We will use it in chapter 7 for the calculation of the standard deviations of the electrostatic moments derived from the parameters of the multipole formalism. 

Once the partitioning has been accomplished, net atomic charges, atomic electrostatic moments, and other physical properties are obtained by straightforward integration using the expectation value expression... 

Nevertheless, multipole partitioning leads to very acceptable molecular electrostatic moments, as fully discussed in chapter 7. 

The moments of a charge distribution provide a concise summary of the nature of that distribution. They are suitable for quantitative comparison of experimental charge densities with theoretical results. As many of the moments can be obtained by spectroscopic and dielectric methods, the comparison between techniques can serve as a calibration of experimental and theoretical charge densities. Conversely, since the full charge density is not accessible by the other experimental methods, the comparison provides an interpretation of the results of the complementary physical techniques. The electrostatic moments are of practical importance, as they occur in the expressions for intermolecular interactions and the lattice energies of crystals. 

The first electrostatic moment from X-rays was obtained by Stewart (1970), who calculated the dipole moment of uracil from the least-squares valence-shell populations of each of the constituent atoms of the molecule. Stewart s value of... 

For a site of point symmetry 1, the electrostatic moment 3...a, of order... 

In a different form, the traceless moment operators can be written as the Cartesian spherical harmonics c,mp multiplied by r, which defines the spherical harmonic electrostatic moments ... 

An element of an electrostatic moment tensor can only be nonzero if the distribution has a component of the same symmetry as the corresponding operator. In other words, the integrand in Eq. (7.1) must have a component that is invariant under the symmetry operations of the distribution, namely, it is totally symmetric with respect to the operations of the point group of the distribution. As an example, for the x component of the dipole moment to be nonzero, p(r)x must have a totally symmetric component, which will be the case if p(r) has a component with the symmetry of x. The symmetry restrictions of the spherical electrostatic moments are those of the spherical harmonics given in appendix section D.4. Restrictions for the other definitions follow directly from those listed in this appendix. 

Atomic Electrostatic Moments in Terms of the Parameters of the Multipole Formalism... 

The atomic electrostatic moments of an atom are obtained by integration over its charge distribution. As the multipole formalism separates the charge distribution into pseudoatoms, the atomic moments are well defined. 

When the electrostatic moments are to be obtained by integration over direct space, it is advantageous to use the deformation density rather than the total... 

As noted above, in the traceless definition the /th-order multipoles are the sole contributors to the /th electrostatic moments. This implies that the traceless moments derived from the total density p(r) and from the deformation density Ap(r) are identical, that is, ,mp(p) = 0,mp(Ap) for / > 2. 

Electrostatic Moments of a Subvolume of Space by Fourier Summation... 

The accuracy of the electrostatic moments based on the multipole parameters is a function of the errors in both the population coefficients Tvai and the atomic parameters Pimp- Let M represent the m x m variance-covariance matrix for these parameters, as in chapter 4. Let D be the derivative matrix with elements... 

Not surprisingly, formalisms with very diffuse density functions tend to yield large electrostatic moments. This appears, in particular, to be true for the Hirshfeld formalism, in which each cos 1 term in the expansion (3.48) includes diffuse spherical harmonic functions with / = n, n — 2, n — 4,... (0, 1) with the radial factor rn. For instance when the refinement includes cos4 terms, monopoles and quadrupoles with radial functions containing a factor r4 are present. For pyridin-ium dicyanomethylide (Fig. 7.3), the dipole moment obtained with the coefficients from the Hirshfeld-type refinement is 62.7-10" 30 Cm (18.8 D), whereas the dipole moments from the spherical harmonic refinement, from integration in direct space, and the solution value (in dioxane), all cluster around 31 10 30 Cm (9.4 D) (Baert et al. 1982). 

The dipole moments are generally affected by intermolecular electrostatic interactions. In a crystal, molecules tend to line up such that opposite charges are proximal, in order to maximize the electrostatic attractions. Induced polarization therefore tends to enhance the electrostatic moments. 

In the derivation of the traceless quadrupole moments from the electrostatic moments, the spherical components are subtracted. Thus, the quadrupole moments can be derived from the second moments, but the opposite is not the case. Spackman (1992) notes that the subtraction introduces an ambiguity in the comparison of quadrupole moments from theory and experiment. The spherical component subtracted is not that of the promolecule, but is based on the distribution itself. It is therefore generally not the same in the two densities being compared. On the other hand, the moments as defined by Eq. (7.1) are based on the total density without the intrusion of a reference state. 

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