Electric Multipole Expansion

The multipole expansion gives exactly that expression. If the charge distribution shown has an overall charge Q, an electric dipole pe, an electric quadrupole 0g, and so on, then we write... 

Since the electric field (F = —dV/dr) normally is fairly uniform at the molecular level, it is useful to write as a multipole expansion. 

In the usual texts a multipole expansion involving spherical Bessel functions and spherical vector harmonics is also introduced [16,23,23,26]. The fields from electric and magnetic dipoles correspond to the lowest-order terms ( =1) in the expansion. If we define dipole by this expansion then our toroidal antenna is an electric dipole. In any event, the fields away from the source are the same. This is perhaps a matter of consistency in definitions. 

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... 

In the parametrization of equ. (4.68) the terms associated with the Legendre polynomials Pk(cos ab) represent that part of the angular correlation which is independent of the light beam, while the terms associated with the bipolar harmonics are due to the multipole expansion of the interactions of the electrons with the electric field vector. The link between geometrical angular functions and dynamical parameters is made by the summation indices ku k2 and k. These quantities are related to the orbital angular momenta of the two individual emitted electrons, and they are subject to the following conditions ... 

The quadmpoie operator transforms as WY V and makes the 1 s —> 3d transition electric quadmpoie allowed when the dY v2 orbital (in molecular coordinates) bisects the k and E vectors (which define the laboratory coordinates). Quadmpoie intensity is usually very low however, at —9000 eV the wavelength of light is — 1.4 A and in this case the long-wave approximation no longer holds and higher terms in the multipole expansion in Equation 1.2 become important. 

In a multipole expansion of the interaction of a molecule with a radiation field, the contribution of the magnetic dipole is in general much smaller than that of the electric dipole. The prefactor for a magnetic dipole transition probability differs from the one for an electric dipole by a2/4 1.3 x 1 () 5. Magnetic dipoles may play an important role, however, when electric dipole transitions are symmetry-forbidden as, e.g., in homonuclear diatomics. 

We will return to the quadrupole interaction in following chapters, but we now re-examine the general expansion of the electrostatic interaction and, in particular, the possibility of other nuclear electrostatic multipole moments. Because our multipole expansion is performed in a coordinate system with origin at the centre of charge of the protons p in the nucleus, the nuclear electric dipole moment is zero. However, this result arises only from our choice of origin and we now show that there are much... 

Terms of higher order in the field amplitudes or in the multipole expansion are indicated by. . . The other two tensors in (1) are the electric polarizability ax and the magnetizability The linear response tensors in (1) are molecular properties, amenable to ab initio computations, and the tensor elements are functions of the frequency m of the applied fields. Because of the time derivatives of the fields involved with the mixed electric-magnetic polarizabilities, chiroptical effects vanish as a> goes to zero (however, f has a nonzero static limit). Away from resonances, the OR parameter is given by [32]... 

The moments and polarizabilities of molecules can be determined by indirect means. In collision experiments, the nature of the interaction is governed by the potential energy surface, itself a function of the molecular properties of the colliding partners. Usually the potential energy is written in a multipole expansion whereby the electrical properties are displayed in the long-range terms [38]. The potential that is generated must satisfy simultaneously... 

Some molecular tensors (electric dipole polarizability, electric and magnetoelectric shielding) are origin independent, as can be immediately found by inspection of definitions (87)-(112). Other tensors depend on the origin assumed for the multipole expansion. For instance, in a change of origin... 

To some purposes, one can define new molecular tensors that are independent of the origin [40]. At any rate, it can be easily proven that the induced moments (69)-(72) and fields (73)-(74) are, order by order, independent of the origin chosen for the multipole expansion, provided that all the terms of the same order of magnitude are retained. Thus, within the quadrupole approximation, both the magnetic field and the electric field gradient must be taken... 

G. Jansen, C. Hattig, B. A. Hess, and J. G. Angyan, Mol. Phys., 88,69 (1996). Intermolecular Interaction Energies by Topologically Partitioned Electric Properties. 1. Electrostatic and Induction Energies in One-Center and Multicenter Multipole Expansions. 

The point multipole expansion of the Coulombic matrix element governing the first-order electric dipole moment in Eq. (3) gives the selection rules of the ligand polarization model through Eq. (4),... 

We will be using this form of the molecule-field interaction repeatedly in this text, however, it should be kept in mind that it is an approximation on several counts. Already Eq. (3.1), an electrostatic energy expression used with a time varying field, is an approximation. Even in this electrostatic limit, Eq. (3.1) is just the first term in an infinite multipole expansion in which the higher-order terms depend on higher spatial derivatives of the electric field. 

The electromagnetic interaction between the sensitizer and activator is responsible for the energy transfer. Transfer via electric dipole-dipole interaction was first described by Forster ) and later Dexter ) expanded the treatment to include higher order electromagnetic and exchange interactions. The electrostatic interaction can be expressed as a multipole expansion using a Taylor s series about the sensitizer-activator separation Rja,... 

Here /x is the electric field at core A, generated by all other cores and nuclei as well as all valence electrons. Since the validity of the underlying multipole expansion breaks down for small distances from the core X, the field has to be multiplied by a cut-off function ... 

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]. 

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