Multiple moments

Figure 16.4 Dlustration of the Fast Multiple Moment method...

In just the same way as in the case of fluorescence intensity, the asymptotic equations of motion of polarization moments (5.54), (5.55) and (5.87), (5.88) must coincide with the corresponding equation of motion of classical multiple moments, as introduced by Eq. (2.16). We will show that this is indeed so in the following section. 

Structures correctly estimated, 89, 578-580 distances underestimated (by lO ), Interaction energies overestimated (120-150% or more of experimental energies. Accuracy depends mainly on the estimated multiple moments and polarizabilities of particular molecules and on the counterpoise correction for a given basis set)... 

The poly tensor approach introduced by Applequist [169] is a terrific organization of the problem of electrical interaction for high-level calculation because it can be continued uniformly to any order of multiple moment, any distribution of moments, and any number of interacting species. Furthermore, it can incorporate multipole polarization and hyperpolarization [170]. As such, it provides a scheme that can be coded for computer application in an open-ended fashion while also providing the formal analysis needed to extract functional forms of different electrical interaction pieces. 

Finally, we note that the model itself is not accurate in the ionic crystal phase. The condensed counterions sit on the rods in our model, whereas they really should sit in between the rods. In other words, the structure of the ionic crystal is not captured correctly by the model. Recent calculations argue that multiple moments perpendicular to the rod axis that arise when counterions lie between rods are important to the low-temperature behavior [23], This is probably a source of greater quantitative error at low temperatures than is the Gaussian approximation. 

Tables 6 and 7 summarise the calculations carried out on alkali metal clusters with 2 to 8 valence electrons reported in the papers of Fantucci et al. The tables contain the point group (and spin multiplicity), moments of inertia, L parameter, and, where available, the energy separation (AE) between that structure and the lowest energy structure for that stoichiometry. The structures include those of neutral, cationic and anionic clusters of lithium and sodium.

There have been extensive experimental and theoretical studies devoted to the structural and bonding characterization of weakly bound van der Waals complexes of acetylene. Structures of these complexes can often be determinated experimentally by means of Fourier transform microwave and infrared spectroscopic techniques. On the theoretical side, advanced treatments are required to understand the complex nature of the weak bonding in terms of the relative contributions of polarization and dispersion interactions, interactions of multiple moments, and electrostatic interactions involved in these completes. To determine the interaction energy in a weak complex, it is necessary to use large basis sets with the inclusion of electron correlation interactions. Theoretical calculations have been reported for van der Waals complexes of acetylene with COj [160], CO [161, 162], AICI3 [163], NH3 [164], He [165], Ar [166], H2O [167], HCN [168], HF [169-172], HCl [173, 174], and acetylene itself in the forms of non-covalent dimer [175-180], trimer [175,181], tetramer [175, 182, 183], and pentamer [175]. These calculations are very useful for the determination of multiple isomeric forms of the complex. For example, calculations at the MP2/6-31G level along with IR spectra indicate that the HCN-acetylene complex exists in a linear form in addition to the T-shaped structure observed previously by microwave studies (see Fig. 1-5) [168]. 

The (permanent) multiple moments of the adsorbed molecule can induce electric moments in the solid and interact with them, or vice versa. This interaction between permanent and induced multipole moments is called induction or polarization interaction and is always attractive. Similar to the interaction between the permanent multipole moments it possesses an R " dependence. For small molecules the leading term is the charge-induced dipole moment contribution which is given by [ 14,15]... 

In the presence of an external field, the perturbed Hamiltonian can be written in terms of the multiple moment tensors. For an uncharged molecule... 

Gross and Vrabrec developed the polar terms used for systems consisting of one multipolar molecule with either a dipole or a quadrupole and one nonpolar molecule. Perhaps not surprisingly, therefore the theory was found to fail for systems in which one molecule has a significant dipole and the other a significant quadrupole. Lucas and co-workers use multiple moments determined from quantum mechanical calculations and so the experimental dipole and quadrupole moments are not needed (see Section 8.5.2). In a subsequent paper they compared their PCP-SAFT equation with a new approach in which the PC-SAFT equation is combined with polar terms derived from perturbation theory with a spherical reference (based on a Pade approximation for multipolar species with constants from one-centre LJ MC simulations) and an equation that combines the PC-SAFT equation with the GV term for the dipolar interactions and spherical-reference perturbation theory for all other multipolar terms (dipole-quadrupole and quadrupole-quadrupole). The equation based on the latter approach was found to out-perform both their original PCP-SAFT model and the new polar contribution. 

The interaction of crystalline electric fields (CEF) with the multiple moments of R atom electrons in a site of a crystal lattice of 4/mmm point symmetry is given by the Hamiltonian... 

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