Polarizability dipole-quadrupole

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 dipole-dipole ( / /+ ) and dipole-quadrupole dq+ ) coefficients can be derived from free-ion polarizability a and the mean excitation E by applying... 

Electronic parameters include the number of electrons, the number of valence electrons in the outer orbit, the orbital quantum numbers n, the azimuthal quantum number or spdf status, electron radius and energy, polarizability, dipole moment, quadrupole moment, and first ionization energy. 

K. L. C. Hunt. Nonlocal polarizability densities and the effects of short-range interactions on molecular dipoles, quadrupoles and polarizabilities. J. Chem. Phys., 80 393, 1984. 

At this pressure, the polarizability/volume of SF CO2 is a little less than that of n-hexane, which suggests that there are other molecular interactions between CO2 and phenol blue in addition to dispersion and induction. The likely possibilities include electron donor-acceptor forces and dipole-quadrupole interactions. 

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. 

In the region of negligible overlap, the three-body interactions can be calculated, just like the two-body ones, from imaginary frequency polarizabilities. This has been done, for example, by Doran78 for interaction of triplets of noble-gas atoms. He gives the coefficients for dipole-dipole-dipole, dipole-dipole-quadrupole, dipole-quadru-pole-quadrupole, quadrupole-quadrupole-quadrupole, and dipole-dipolc-octupole terms. 

The PPD and shell models are nearly equivalent in this sense, because they model the electrostatic potential via static point charges and polarizable dipoles (of either zero or very small extent). Accuracy can be improved by extending the expansion to include polarizable quadrupoles or higher order terms.The added computational expense and difficulty in parameterizing these higher order methods has prevented them from being used widely. The accuracy of the ESP for dipole-based methods can also be improved by adding off-atom dipolar sites. 

Fig. 5. Differential lieats of sorption in nature cliabazite (4). = N2 = Ar 0 = 02 > = CO = C02. See Table 2 for polarizability, dipole moment, and quadrupole moment values for the gases. Volume adsorbed is expressed as cm 3 of adsorbate as liquid. To convert kj to kcal, divide by 4.184.

The distinguishing feature of dehydrated zeolites as microporous aluminosilicate adsorbents lies in the presence in their voids—i.e., micropores—of cations. These cations compensate excess negative charges of their aluminosilicate skeletons. The cations form, in the zeolite micropores, centers for the adsorption of molecules with a nonuniform distribution of the electron density (dipole, quadrupole, or multiple-bond molecules) or of polarizable molecules. These interactions, which will be called, somewhat conventionally, electrostatic interactions, combine with dispersion interactions and cause a considerable increase in the adsorption energy. As a result, the adsorption isotherms of vapors on zeolites, as a rule, become much steeper in the initial regions of equilibrium pressures as compared with isotherms for active carbons. 

Thus for two polarizable dipole molecules in the gas phase all attractive interactions depend on the inverse of the sixth power of their mutual distance. For two identical molecules, neglecting higher terms (quadrupole interactions etc.). 

That is, the translated polarizability is related to the original quadru-pole-quadrupole polarizability, the dipole-quadrupole polarizability, and the dipole-dipole polarizability. 

The third-rank tensors are mixed dipole-quadrupole polarizabilities ... 

Some of those have long been known a is the electric dipole polarizability (a symmetric polar tensor of dimension P) in length formalism, k (an asymmetric axial tensor of dimension P t) is related to the optical activity, A is the mixed dipole-quadrupole polarizability, x is the magnetic susceptibility (or magnetizability), written as a sum of diamagnetic and paramagnetic components. 

The mixed dipole-quadrupole polarizability satisfies a similar equation involving the electric mixed dipole-quadrupole shieldings... 

Let us illustrate this conclusion by the example of spherical-top molecules. In the degenerate states E, T , T2, and G3/2 there is the E representation in the decomposition of the appropriate symmetric product, and hence in these cases the matrix elements are nonzero for anisotropic components of the tensors of polarizability and quadrupole moment, since E is first met in the decomposition of the spherical representation D2. Moreover, in tetrahedral systems in states of the type T and G3/2 the matrix elements of the dipole moment are nonzero, since [T2] and G3/2 contain the representation T2. 

A rather novel scheme for modeling molecular polarizabilities as distributed dipole polarizabilities has recently been reported [141]. In this approach, the overall quadrupole induced in a molecule by an external field, as calculated with ab initio methods, is decomposed into induced dipoles distributed to atomic sites. In turn, this yields the dipole polarizability values at those sites. In effect, this relates the overall dipole quadrupole polarizability to a distribution of dipole polarizabihties. 

Experimental measurements of Aie BE can also be used to determine quadrupole moments for dipolar systems. The values obtained in this way refer to an origin which is denoted as the effective quadrupole center (EQC) [135]. The EQC is defined as Aie point at which the combinaAon of dipole-quadrupole and dipole-magnetic dipole polarizabilities multiplied with the dipole moment given in Eq. (27)... 

Let us consider the long-range terms first. An accurate description of these requires the monomer moments and polarizabilities to be described accurately. However, the precise role played by the relative importance of the dipole, quadrupole,..., etc., moment depends on the system studied. In polar-polar interactions the dipole-dipole term in Ecoui has a much larger weight in the final A ,otai than Eji.p. Hence for a 5% accuracy in A , ,a, the dipole moment n may have to be accurate to 1-2%, while the error in the polarizability a may be as large as 10%. As the systems get less polar, the relative weight of coui diminishes while that of E isp increases. Hence for a relative accuracy of 5% in A ,<,ai, a may now have to be accurate to 2% or even better. A summary of these requirements is given in Table III. 

The symmetry of linear molecules imposes less constraints on the multipole polarizability tensors than does tetrahedral symmetry, so the forms of the respective tensors for linear molecules are usually more highly complicated than for tetrahedral ones. This certainly complicates interpretation of light scattering results but has much less influence on complications in running our program. By means of the same methods as in the case of tetrahedral molecules we calculate the spherical irreducible components of the dipole-dipole, dipole-quadrupole, and dipole-octopole polarizability tensors as dipole-dipole... 

In addition to the total energy of the system, it is desirable to carry out FCI calibration studies of properties such as dipole moments, polarizabilities, and electrostatic forces. For example, in the O + OH- O + H reaction, the preferred approach of the O atom is determined by the dipole-quadrupole interaction. At long distances, this favors a collinear approach to the H atom, whereas for reaction to occur the O atom must migrate to the O end of OH. An accurate description of weakly interacting sy stems such as van der Waals complexes requires a quantitative description of dipole-induced-dipole or induced-dipole-induced-dipole interactions. Further, the dipole moment and polarizability functions of a molecule determine its infrared and Raman spectral intensities. 

Table 3.4 Properties of the Solvents Used in a Poly(Ethylene-co-Methyl Acrylate)-Hydrocarbon Study. The symbols for polarizability, dipole moment, and quadrupole moment are a, fi, and Q, respectively...

Hass6 has considered five variation functions of this form, shown with their results in Table 47-2. The success of his similar treatment of the polarizability of helium (function 6 of Table 29-3) makes it probable that the value — 1.413e2aj /i28 for W" is not in error by more than a few per cent. Slater and Kirkwood1 obtained values 1.13, 1.78, and 1.59 for the coefficient of — e2al/R6 by the use of variation functions based on their helium atom functions mentioned in Section 29e. An approximate discussion of dipole-quadrupole and quadrupole-quadrupole interactions has been given by Margenau.1... 

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