Dipole and higher moments

Values of the dipole and higher moments have been calculated for a large number of simple molecules. The calculation of fio is somewhat easier than that of a because the former depends only on the ground-state wave function, whereas the latter depends on the excited-state wave functions as well. For simple molecules even relatively crude wave functions yield values of within 20% of experimental values. 

Charge density analyses can provide experimental information on the concentration of electron density around atoms and in intra- and intermolecular bonds, including the location of lone pairs. Transition metal d-orbital populations can be estimated from the asphericity of the charge distribution around such metal centers. A number of physical properties that depend upon the electron density distribution can also be calculated. These include atomic charges, dipole and higher moments, electric field gradients, electrostatic potentials and interaction... 

The polarity of the molecules is usually considered to be measured on a gross scale by the relative permittivity and on a molecular scale by the electrical dipole and higher moments. Molecules lacking a dipole moment (carbon dioxide, for example) may still exert short-range effects due to quadrupole, and so on, moments. Dipolar bonds that are well separated in a molecule may act almost independently on neighboring molecules Hildebrand and Carter (1930) showed that the three isomeric dinitrobenzenes, in their binary solutions in benzene, exhibit nearly identical deviations from Raoult s law, though their dipole moments are different. The part of the electrical influence of a solvent on solute molecules that arises from the polarizability of the solvent molecules may be represented by the refractive index, n, or by functions of n such as the volume polarization, R, given by ... 

It is further possible to compute dipole and higher moments and electrostatic potentials on a grid provided by the user. Population analyses are also supported. 

There are higher multipole polarizabilities tiiat describe higher-order multipole moments induced by non-imifonn fields. For example, the quadnipole polarizability is a fourth-rank tensor C that characterizes the lowest-order quadnipole moment induced by an applied field gradient. There are also mixed polarizabilities such as the third-rank dipole-quadnipole polarizability tensor A that describes the lowest-order response of the dipole moment to a field gradient and of the quadnipole moment to a dipolar field. All polarizabilities of order higher tlian dipole depend on the choice of origin. Experimental values are basically restricted to the dipole polarizability and hyperpolarizability [21, 24 and 21]. Ab initio calculations are an imponant source of both dipole and higher polarizabilities [20] some recent examples include [26, 22] ... 

For polarizable charge distributions, additional classical-type interactions arise from the induced dipole, quadrupole, and higher moments on each monomer, which are proportional to the fields created by the asymmetric charge distribution on the other monomer. The proportionality constants for each multipole field are the monomer polarizabilities aa and ah (a111 for dipole fields, a(Q) for quadrupole fields, etc.). The leading two induction interactions are ... 

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). 

Born s idea was taken up by Kirkwood and Onsager [24,25], who extended the dielectric continuum solvation approach by taking into account electrostatic multipole moments, Mf, i.e., dipole, quadrupole, octupole, and higher moments. Kirkwood derived the general formula ... 

Birefringences are mostly observed in condensed phases, especially pure liquids or solutions, since the strong enhancement of the effects allows for reduced dimensions (much shorter optical paths) of the experimental apparatus. Nowadays measurements of linear birefringences can be carried out on liquid samples with desktop-size instruments. Such measurements may yield information on the molecular properties, molecular multipoles and their polarizabilities. In some instances, for example KE, CME and BE, measurements (in particular of their temperature dependence) have been carried out simultaneously on some systems. From the combination of data, information on electric dipole polarizabilities, dipole and quadrupole moments, magnetizabilities and higher order properties were then obtained. 

Neither atomic charges nor bond dipoles are observables. About the only experimental data for isolated molecules that can be used as parameterization reference are molecular dipoles and higher multipole moments. Substantial effort has also been expended to find electrostatic schemes that can rationalize the behavior of condensed phases (37). However, electrostatic data may be more conveniently obtained from QM calculations. Several schemes exist for partitioning the electron density into atomic charges (38). In general, methods that reproduce the QM-calculated electrostatic field outside the molecular surface are preferred. 

The quadratic terms in the expansions in powers of i/X, , z /X form a quantity Q called the quadrupole moment of the system, and higher powers form higher moments. The rate of emission of radiant energy as a result of the change of quadrupole and higher moments of an atom or molecule is usually negligibly small in comparison with the rate of dipole emission, and in consequence dipole radiation alone is ordinarily discussed. Under some circumstances, however, as when the intensity of dipole radiation is zero and the presence of very weak radiation can be detected, the process of quadrupole emission is important. 

Properties of interest can be static or dynamic (i.e., time or frequency dependent) first, second, or higher order and one- or two-particle type. First-order static one-electron properties (dipole and quadrupole moments, electric... 

We calculate the interaction through the electric field and its derivatives at the site at which a test chiral system is placed. If is the electric potential, and test site the interaction energy with a chiral system specified by electric charge q and dipole, quadrupole and higher moments j>i, Qtj... is given in expression (11.18)... 

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