Classical multipole approximation

While exchange- and dispersion-induced dipole components are of a quantum nature, the multipole-induced dipole components can be modeled by classical relationships, if the quantum effects are small. For many systems of practical interest, multipolar induction generates the dominant dipole components. The classical multipole induction approximation has been very successful, except for the weakly polarizable partners (e.g., He atoms) [193]. It models the dipole induced in the collisional partner by polarization in the molecular multipole fields. 

Besides the linear polarization contribution, the hyperpolarizabilities may be accounted for, as well as the field gradients (and higher derivatives of the multipolar fields) taken at the molecular center if the fields are very non-uniform. The total dipole induced by electric fields in a molecule may be written [89] 

for example, the induced dipole model is truncated at the order R 6 in the separation R between the molecular centers, account may be made of the dipoles induced by multipoles up to order — 4 (hexadecapole). Moreover, dipoles induced by derivatives of the local field at their center 

Collision-induced dipoles manifest themselves mainly in collision-induced spectra, in the spectra and the properties of van der Waals molecules, and in certain virial dielectric properties. Dipole moments of a number of van der Waals complexes have been measured directly by molecular beam deflection and other techniques. Empirical models of induced dipole moments have been obtained from such measurements that are consistent with spectral moments, spectral line shapes, virial coefficients, etc. We will briefly review the methods and results obtained. 

Spectroscopic measurement. Specifically, if the induced dipole moment and interaction potential are known as functions of the intermolecular separation, molecular orientations, vibrational excitations, etc., an absorption spectrum can in principle be computed potential and dipole surface determine the spectra. With some caution, one may also turn this argument around and argue that the knowledge of the spectra and the interaction potential defines an induced dipole function. While direct inversion procedures for the purpose may be possible, none are presently known and the empirical induced dipole models usually assume an analytical function like Eqs. 4.1 and 4.3, or combinations of Eqs. 4.1 through 4.3, with parameters po, J o, 32, etc., to be chosen such that certain measured spectral moments or profiles are reproduced computationally. 

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Specifically, the eleven profiles include the obvious single transitions, i.e., the rotovibrational transitions in just one of the two colliding H2 molecules these are the Si(0), Si(l), and Qi (1) transitions in one of the two interacting molecules. Double transitions in both collisional partners are also taking place, such as the simultaneous transitions gi(l)+So(0) (which occur near the Si(0) transition frequency) and Q (1) + So(l) (near Si(l)), Fig. 3. 32. Intensities of all these lines are known from theory (classical multipole approximation, Chapter 6) their superposition reproduces the measurement closely, Fig. 3.33. 

Fig. 4.2. The principal induced dipole components of H2-H2 ground state vibrational averages are shown in the left-hand plot. Overlap (dotted), multipole-induced terms (solid lines) classical multipole approximation (dashed). Right-hand plot ditto, but vibrational transition elements, v = 0 - v = 1. Molecule 2 undergoes the vibrational transition [281].

Table 3.7 also lists ternary spectral moments for a few systems other than H2-H2-H2. For the H2-He-He system, the pairwise-additive dipole moments are also known from first principles. The measured spectral moments are substantially greater than the ones calculated with the assumption of pairwise additivity - just as this was seen in pure hydrogen. For the other systems listed in the Table, no ab initio dipole surfaces are known and a comparison with theory must therefore be based on the approximate, classical multipole model. 

Asymptotic formulae. For a discussion of induced dipoles in highly polarizable species, it is often sufficient to consider the so-called classical multipole induction approximation in its simplest form (i.e., neglecting field gradients and hyperpolarizabilities). In such a case, one needs to know only the vibrational matrix elements of the multipole moments,... 

The results of energy partitioning in Li+... OH2 obtained with a number of different basis sets are listed in Table 3. Since intermolecular overlap is small in these kind of complexes (Table 1), we expect the electrostatic model to be a good approximation for classical contributions to the total energy of interaction. Indeed, ZlE cou is to a good approximation proportional to the dipole moment of the water molecule calculated with the same basis set. This can be seen even more clearly in Table 4 where zIEcou is compared with ion-dipole and ion-quadrupole energies obtained from the classical expression of the multipole expansion series 45,95-97) ... 

Induced Interaction between Two Multipole Systems. Equation (58) defines in general form the classical electrostatic interaction of two electric systems having permanent multipoles and pj" , in conformity with the classical theory of Keesom. In the classical approach also, as shown by Debye and Falkenhagen, one has to take into consideration energies due to interactions between the permanent multipoles of the one system and electric multipoles induced in the other, and vice versa. Restricting the problem in a first approximation to the energy arising from the mutual interaction of dipoles, we can write ... 

A very useful tool to rationalize the nature of these intermolecular bonds will be the analysis of the dominant components of the interaction energy. Using classical mechanics, when the two molecules are far enough the interaction energy between two molecules A and B fixed in the space can be approximated as the electrostatic interaction energy (E ) between two sets of multipoles [1,24, 25]. The leading terms of the electrostatic interaction energy between two undistorted A and B molecules can be written as a power series ... 

The calculation of the Coulomb excitation can be treated by a semi-classical approximation. If v is the velocity, and Zg and Z are the charges on the target nucleus and the bombarding particle respectively, the quantity ri = 2Z Z e l%v is greater than unity, and the cross sections can be calculated from the classical trajectories of the bombarding particles. In general the total cross section for Coulomb excitation for multipole order Z is ... 

It has become apparent that the preceding discussion of synchrotron radiation gives a false impression that multipoles do not offer a significant reduction in synchrotron losses,when in fact the point I was trying to make was that the improvement factor involves a difficult calculation and cannot be approximated by a simple estimate. Neither the single-particle nor the blackbody formula can give a reasonable answer, and classical diffusion is probably not as relevant as maximum B and 3 in comparing different devices. 

All the methods considered in the preceding section relies, strictu sensu, on analytical expressions of the MEP. This section is dedicated to a specific strategy of elaboration of into simpler analytical expressions via multipole expansions. The use of a multipole expansion may reduce the computation time by several order of magnitude. Multipole expansions are however approximate, and may led to considerable errors. There is no unique way of getting a multipole expansion once F (r) (or r r)) is known, and the different methods vary greatly in reliability, computer time, and other characteristics. The variety of the approaches, and the importance of the subject suggest that we treat this topic in a specific section. The expression of the MEP in terms of the so-called multipole expansion is a classic topic [64-65], described in many textbooks [7, 66-68]. 

ASC = apparent surface charge BEM = boundary-element method CPHF = coupled perturbed Hartree-Fock C/RF = classical reaction field GBA = generalized Bom approximation FDM = finite-difference method FEM = finite-element method MPE = multipole expansion PD = potential derived SOS = sum over states SPT = scaled particle theory. 

The classical problem of electrostatics. The electrostatic interaction energy ms can be calculated in several ways. Historically at least the most important approach is based on the multipole expansion of the interaction integral in equation (4). Other methods are based on the apparent surface charge approach, the image charge approximation as well as finite difference and finite element based techniques. 

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