Multipole moments orientation-dependence

One of the simplest orientational-dependent potentials that has been used for polar molecules is the Stockmayer potential.48 It consists of a spherically symmetric Lennard-Jones potential plus a term representing the interaction between two point dipoles. This latter term contains the orientational dependence. Carbon monoxide and nitrogen both have permanent quadrupole moments. Therefore, an obvious generalization of Stockmayer potential is a Lennard-Jones potential plus terms involving quadrupole-quadrupole, dipole-dipole interactions. That is, the orientational part of the potential is derived from a multipole expansion of the electrostatic interaction between the charge distributions on two different molecules and only permanent (not induced) multipoles are considered. Further, the expansion is truncated at the quadrupole-quadrupole term. In all of the simulations discussed here, we have used potentials of this type. The components of the intermolecular potentials we considered are given by ... 

The spherical form of the multipole expansion is very useful if we are looking for the explicit orientational dependence of the interaction energy. However, in some applications the use the conceptually simpler Cartesian form of the operators V1a 1b may be more convenient. Moreover, unlike the spherical derivation, the Cartesian derivation is very simple, and can be followed by everybody who knows how to differentiate a function of x, y and z 149. To express the operator V,, in terms of Cartesian tensors we have to define the reducible, with respect to SO(3), tensorial components of multipole moments,... 

Although the spherical form of the multipole expansion is definitely superior if the orientational dependence of the electrostatic, induction, or dispersion energies is of interest, the Cartesian form171-174 may be useful. Mutual transformations between the spherical and Cartesian forms of the multipole moment and (hyper)polarizability tensors have been derived by Gray and Lo175. The symmetry-adaptation of the Cartesian tensors of quadrupole, octupole, and hexadecapole moments to all 51 point groups can be found in Ref. (176) while the symmetry-adaptation of the Cartesian tensors of multipole (hyper)polarizabilities to simple point groups has been considered in Refs. (172-175). 

Kusaka et al. (1995b) also present a density functional theory for ion-induced nucle-ation of polarizable multipolar molecules. For a fixed orientation of a molecule, the ion-molecule interaction through the molecular polarizability is independent of the sign of the ion charge, while that through the permanent multipole moments is not. As a result of this asymmetry, the reversible work acquires a dependence on the sign of the ion charge. 

Despite its Httle practical use, some qualitative conclusions can be drawn from Eq. (1). First, there can be large long-range contributions to the interaction, in particular if the adsorbed species possesses low order multipole moments, such as a charge (1a=0) or a dipole moment (1 =1). Asymptotically, the term with the lowest multipole moment will prevail, but at shorter separations other contributions might be more important Second, the electrostatic interaction can be both repulsive or attractive, depending on the sign of the multipole moments and on the orientation of the molecule relative to the surface. 

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. 

The actual values of the atomic multipole moments depend on the coefficients of the hybrid orbitals in the individual bond orbitals (bond polarities), on the orientation of the hybrids and on the degree of hybridization. Since these latter two parameters depend significantly on the geometrical arrangement of the atoms [47], the bond increment method may be an adequate tool for constructing the zeroth order wave function, which nevertheless describes the main trends in the conformational and geometrical dependence of the atomic charge distributions and consequently of the electrostatic potential. 

Interactions become shorter-ranged and weaker as higher multipole moments become involved. When a monopole interacts with a monopole. Coulomb s law says u r) oc r But when a monopole interacts with a distant dipole, coulombic interactions lead to u r) oc r (see Equation (21.26)). Continuing up the multipole series, two permanent dipoles that are far apart interact as u(r) oc r Such interactions can be either attractive or repulsive, depending on the orientations of the dipoles. Table 24.2 gives typical energies of some covalent bonds, and Table 24.3 compares covalent to noncovalent bond strengths. 

Because of orientation-dependent terms in both the moments and the Boltzmann factor values of B are much siore sensitive to molecular anisotropies than the pressure virial coefficient or the gas shear viscosity as a function of temperature. For nonpolar molecules quadrupole moment effects are large in the case of CO2 for example demonstrating the importance of quadrupole moments Q s 4.2 X 10 esitcii)> inferred from B while octopole and even hexadecapole effects can be recognized for more symmetrical molecules e.g. CH and SFg. For polar molecules permanent dipole interactions also come into play and anisotropy of repulsive forces (shape) is also important. The result is a very wide range in magnitudes and sign of B even for relatively simple molecules and comparison of calculated values with experiment is a sensitive test of multipole moments and anisotropies of used in the calculation. All these matters are discussed in detail by Sutter (21). 

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. 

The little difference in the molecular packing makes a huge difference in the HOMO and LUMO levels at the interface, fii a real molecular system it is very unlikely that the dipoles of the DCM molecule will align perfectly, like that in Fig. 2b, but there will be a distribution of molecular orientations and local regions that may have a number of dipoles pointed in the same direction. These types of effects can be created by any multipole moment of a molecule, but as one goes from dipole, to quadrupole, to octupole, and so on, the magnitude of the electric field reduces and becomes more complex. The electric field created by a multipole moment will also depend very much upon the direction of the molecule at the interface (see Fig. 9). 

Starting from a multipole expansion of intramolecular Coulomb interactions, we present an efficient configuration interaction calculation for the electron terms = 2,3, 4, and the hole terms (hf)", n = 2-5. We have studied magnetic moments for the electron and hole terms. The coupling of spin and orbital momenta differs from the Lande g-factor scheme of atoms. The magnetic moments do not depend on the orientation of the molecule with respect to an external magnetic field. 

It should be emphasized again, that the multipole polarizabilities and moments in Eqs. (5.1.10) and (5.1.15) are written there in the laboratory system of coordinates and depend on the mutual orientation of the interacting molecules. As a result, the first dipole hyperpolarizability of two interacting molecules is a function (surface) of several variables Euler angles (rotation of the first and the second molecule), the intermolecular separation R and the internal coordinates when the molecules are considered as nonrigid ones. 

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