Multipole moments phase

The expansion coefficients Pq are called polarization moments or multipole moments. The expansion (2.14) may also be carried out by slightly alternative methods which are presented in Appendix D and differ from the above one by the normalization and by the phase of the complex coefficients Pq. The normalization used in (2.14) agrees with [19]. Considering the formula (B.2) from Appendix B of the complex conjugation for the spherical function Ykq(0, [Pg.30]

Here Modpg is the module or magnitude of the polarization moment, but ipo is its initial phase at the moment of excitation. As can be seen, the magnetic field itself does not alter the absolute value of the multipole moment, changing only its phase... 

The diagonal elements of the density matrix contain the populations of each of the BO states, whereas off-diagonal elements contain the relative phases of the BO states. The components of the density matrix with a = a describe the vibrational and rotational dynamics in the electronic state a, while the rotational dynamics within a vibronic state are described by the density matrix elements with a = a and va = v ,. The density matrix components with na = n a, describe the angular momentum polarization of the state Ja, often referred to as angular momentum orientation and alignment [40, 87-89]. The density matrix may be expanded in terms of multipole moments as ... 

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. 

Electrostatic interactions cannot account for all of the non-bonded interactions in a system. The rare gas atoms are an obvious example all of the multipole moments of a rare gas atom are zero and so there can be no dipole-dipole or dipole-induced dipole interactions. But there clearly must be interactions between the atoms, how else could rare gases have liquid and solid phases or show deviations from ideal gas behaviour Deviations from ideal gas behaviour were famously quanfatated by van der Waais, thus the forces that give rise to such deviations are often referred to as van der Waais forces. 

The multipole moments of the classical models and QM results [55] are compared in Table 9.1. In examining the QM results, the moments increase from 10 to 30% from the gas phase to the liquid phase. The classical models have moments that are generally... 

It is has been known that the atomic multipole moments for atoms in AMOEBA model can be calculated through quantum mechanics method and Stone s distributed multipole analysis [61]. Thus, it is straightforward to obtain the parameters of electric multipole potentials based on the distributed multipole analysis after the EMP sites of Gay-Berne particles are decided or directly from AMOEBA force field. However, the EMP parameters of Gay-Berne particles need to be optimized because they are derived based on the gas-phase ab initio quantum mechanics. One possible solution would be to match GBEMP and AMOEBA results for the electrostatic energies between CG particles and water molecules, or between CG particle dimers, at various separations and/or in different orientations. 

The aerosol interaction forces discussed in the preceding sections of this chapter have involved only electrically charged particles or neutral particles with multipole moments. The importance of charge on all phases of aerosol dynamics is generally recognized and is exploited in numerous ways. It was argued above (Sect. 

Batista, Xantheas and Jonsson have calculated the molecular multipole moments for water molecular clusters and in ice (Ih) using ab initio methods. Taking the computed charge distribution for the whole cluster they have attempted to partition it in the manner of Bader and hence obtain multipole moments for the individual water molecules. Different partitioning schemes lead to widely different values. In all schemes the magnitude of the dipole increases with size of cluster, monotonically, from its value in the gas phase to its value in ice. 

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. 

Once the multipole analysis of the X-ray data is done, it provides an analytical description of the electron density that can be used to calculate electrostatic properties (static model density, topology of the density, dipole moments, electrostatic potential, net charges, d orbital populations, etc.). It also allows the calculation of accurate structure factors phases which enables the calculation of experimental dynamic deformation density maps [16] ... 

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