Multipole Electrical Moments

Table 8 Natural permanent) multipole electric moments of simple molecules...

The charge distribution of a quantum mechanical system can be expanded in terms of its multipole electric moments. The zeroth order moment is the total charge Z... 

In this case the multipole electrical moments of the rank n for any molecule B can be defined in the form (see, for example, [1-6])... 

Sometimes it is useful to represent the multipole electrical moments in a spherical form. The spherical form of these moments allows us to apply effectively the theory of irreducible spherical tensor formalism. For this aim these 2 -pole moments may be written in terms of the regular spherical harmonics using their both complex Rlm r) and real (Rimc r) and Rims r)) forms defined, for m > 0, as... 

It is evident from (2.3.3)-(2.3.5) taking into account (2.2.5) and (2.2.10) that in the presence of an external field the multipole electric moments and polarizabilities of a molecule can be determined as the correspondent derivatives of the energy... 

Yu.N. Kalugina, V.N. Cherepanov, Multipole electric moments and higher polarizabilities of molecules the methodology and some results of ab initio calculations. Atmos. Oceanic Opt 28(5), 406 14 (2015)... 

S. P6rez-Casas, J. Hem idez-Trujillo, M. Costas, "Experimental and theoretical study of aromatic-aromatic interactions. Association enthalpies and central and distributed multipole electric moments analysis", J. Phys. Chem. B, 2003,107,4167-4174. 

The three moments higher than the quadrupole are the octopole, hexapole and decapoli. Methane is an example of a molecule whose lowest non-zero multipole moment is the octopole. The entire set of electric moments is required to completely and exactly describe the distribution of charge in a molecule. However, the series expansion is often truncated after the dipole or quadrupole as these are often the most significant. 

VVe therefore return to the point-charge model for calculating electrostatic interactions. If sufficient point charges are used then all of the electric moments can be reproduced and the multipole interaction energy. Equation (4.30), is exactly equal to that calculated from the Coulomb summation. Equation (4.19). 

Electrical moments are useful because at long distances from a molecule the total electronic distribution can be increasingly well represented as a truncated multipole expansion, and thus molecular interactions can be approximated as multipole-multipole interactions (charge-charge, charge-dipole, dipole-dipole, etc.), which are computationally particularly... 

The dipole polarizability can be used in place of the dipole moment function, and this will lead to Raman intensities. Likewise, one can compute electrical quadrupole and higher multipole transition moments if these are of interest. 

Figures 5.7 and 5.8 sketch a picture of the first two permanent electric moments (au) for a selection of noncentrosymmetric and centrosymmetric molecules, respectively. The notation is the same as that given in Mag-nasco et al. (1988). It is understood that the point-like multipoles are placed at the centre of mass of the molecule, their sign in relation to the molecular structure of the monomer being of fundamental importance in determining the nature of the electrostatic interaction (attractive or repulsive). The numbers shown in each figure are from SCF calculations and so are little larger than those given in Table 5.2.

The radial deformation of the valence density is accounted for by the expansion-contraction variables (k and k ). The ED parameters P, Pim , k, and k are optimized, along with conventional crystallographic variables (Ra and Ua for each atom), in an LS refinement against a set of measured structure factor amplitudes. The use of individual atomic coordinate systems provides a convenient way to constrain multipole populations according to chemical and local symmetries. Superposition of pseudoatoms (15) yields an efficient and relatively simple analytic representation of the molecular and crystalline ED. Density-related properties, such as electric moments electrostatic potential and energy, can readily be obtained from the pseudoatomic properties [53]. 

The multipole expansion has already been used in certain quantum chemical calculations [59-65]. As localized orbitals are concentrated in certain spatial region, they can also be represented by their multipole moments. In the following we investigate whether the Coulomb integrals in terms of localized orbitals can be substituted by the multipole expansion of electric moments. 

The idea of using the multipole expansion for the evaluation of intermolecular interactions is not new. It was found that for the moments of the entire molecules the convergence is questionable especially at smaller distances. The convergence can be improved, however, if the charge distributions of molecules are divided into small parts, and the multipole expansion takes the electric moments of these parts into consideration separately [66]. 

Some of the most important properties that a quantum mechanical calculation provides are the electric multipole moments of the molecule. The electric multipoles reflect the distribution of charge in a molecule. The simplest electric moment (apart from the total net charge on the molecule) is the dipole. The dipole moment of a distribution of charges located at positions r, is given by ij r/. If there are just two charges +q and -q separated by a distance... 

The rotation of nuclei with electric multipoles gives rise to formation of magnetic multipoles. Nuclei can therefore also be divided according to the magnetic moments in the same way as according to their electrical moments. 

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]... 

These multipoles behave as tensors of increasing rank k) and alternating parity given by (—1) for electric moments and (—1) for magnetic moments. 

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