Multipole moments complex

The Born equation is based on the simple model of a spherical ion with a single charge at its centre. Such an ion has no dipole moment and no higher multipole moments, but real molecular ions are of course much more complex. Since the electrical charge is distributed among all the atoms of the... 

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]

Equation (1-239) relates the interaction-induced part of the dipole moment of the complex AB to the distortion of the electron density associated with the electrostatic, exchange, induction, and dispersion interactions between the monomers. The polarization contributions to the dipole moment through the second-order of perturbation theory (A/a, A/a, and A/a ) have an appealing, partly classical, partly quantum, physical interpretation. The first-order multipole-expanded polarization contribution (F) is due to the interactions of permanent multipole moments on A with moments induced on B by the external field F, and vice versa. The terms... 

A few practical points remain to be discussed. Acceptance of the counterpoise principle implies that all energy terms contained in the final AE must be interpreted in terms of monomer wavefunctions and properties calculated in the full dimer basis set. As mentioned before, the final representation of may in fact be better than would be expected from an estimate of E in terms of the (/ a(Xa) monomer wavefunctions. On the other hand, the final representation of coui and Ei may contain undesirable artefacts such as an unphysical dipole-dipole contribution to Ecoui of He2. These effects have been termed higher-order BSSE , and they will not be removed by applying the 3 correction. In certain applications it may be desirable to remove these artefacts for example by adding a correction coul(l A(J(A)) polar complexes the multipole energy is corrected by inserting better values of the multipole moments , then the dimer-basis multipole energy is the proper reference. The same applies to the calculation of differential properties of van der Waals molecules. Counterpoise corrections have been applied to electron densities , multipole moments and polarizabilities . 

Large ions normally feature a complex distribution of charges and thus have nonzero electric quadrupole, octupole, and higher multipole moments. However, those produce zero torque in homogeneous fields and thus are not relevant to ion alignment in high-field IMS. 

The spherical harmonics are quite appropriate to express the explicit orientational dependence of the interaction, but in the chemical practice it is customary to introduce a linear transformation of the complex spherical functions Y into real functions expressed over Cartesian coordinates, which are easier to visualize. In Table 8.3 we report the expressions of the multipole moments. 

In spite of the disadvantages of the Cartesian formulation, it is preferred by many workers because the alternative, the spherical tensor formulation, is perceived as mathematically difficult. There is undoubtedly some truth in this view. Moreover the spherical-tensor formulation deals in complex quantities which are more difficult to comprehend than the cartesian-tensor components. However the power and versatility of the spherical tensor approach should not be abandoned lightly, and the main purpose of the present paper is to show that it is possible to combine the best features of the cartesian and spherical-tensor methods. We will show that this hybrid approach leads to very compact expressions for the electrostatic energy and related quantities such as the induction and dispersion energies, and that these can be expressed entirely in terms of real multipole moments referred to molecule-fixed coordinate systems. The transformation between molecule-fixed and space-fixed coordinates can be carried out once and for all, and the analogues in this method of the interaction tensors contain the necessary orientational information. 

We note in passing that it is possible, and much more useful for practical calculations, to use an alternative approach in which the indices t and u refer to real multipole moments, rather than the complex ones defined by (7). The moments are now denoted Qikc and Qiksi defined, for A > 0, by... 

In general case the multipole moments depend on the origin of the coordinate system (except for the charge of a molecule). It follows from the definition (2.1.1) if the vector r is shifted on any vector a. For uncharged molecules the origin dependence appears only for the mullipole moments of the rank n > 2. As a result, because in the book only the uncharged molecules and complexes are considered, their dipole moments and connected with them polarizabilities and first hyperpolarizabilities do not depend on the origin of the coordinate system. 

For further consideration of the dipole moment, polarizability and hyperpolar-izabihty of a complex we need also some expressions for multipole moments (see 2.3.1-2.3.6) ... 

Thus, having necessary properties of free molecules, it is not difficult to calculate the interaction-induced dipole moment using suggested formulas. As the components of the properties (polarizabilities, multipole moments, etc.) are dependent, in general case, on the orientation of molecules (Appendix A) we have a multidimensional surface of the dipole moment for a complex. 

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