Interactions electric multipole

The long-range interactions between a pair of molecules are detemiined by electric multipole moments and polarizabilities of the individual molecules. MuJtipoJe moments are measures that describe the non-sphericity of the charge distribution of a molecule. The zeroth-order moment is the total charge of the molecule Q = Yfi- where q- is the charge of particle and the sum is over all electrons and nuclei in tlie molecule. The first-order moment is the dipole moment vector with Cartesian components given by... 

At distances of a few molecular diameters, the interaction will be dominated by electric multipole interactions for dipolar molecules, such as water, the dominant tenn will be the dipole-dipole interaction ... 

Much of our knowledge of molecules is obtained from experimental studies of the way they interact with electromagnetic radiation, and the recent growth in non-linear spectroscopies and molecular electronics has focused attention on our ability (or otherwise) to predict and rationalize the electric properties of molecules. The idea of an electric multipole is an important one, so let s begin the discussion there. 

Awq, in terms of interactions of the lowest-order electrical multipoles is often... 

FIM can also be used to study properties, such as the surface induced dipole moment and the effective polarizability of some surface atoms, kink site atoms and adsorbed atoms etc. The charge distribution of a surface atom is obviously completely different from that of a free atom because of its interaction with the surface and in addition surface atoms are partially shielded by itinerant charges of the surface. The charge distribution of a surface atom can be described by the magnitudes of the electric multipoles of the atom. 

Consider tlie mutual approach of two noble gas atoms. At infinite separation, there is no interaction between them, and this defines die zero of potential energy. The isolated atoms are spherically symmetric, lacking any electric multipole moments. In a classical world (ignoring the chemically irrelevant gravitational interaction) there is no attractive force between them as they approach one another. When tliere are no dissipative forces, the relationship between force F in a given coordinate direction q and potential energy U is... 

As we have seen in Chapter 11, the energy levels of atoms and ions, depending on the relative role of various intra-atomic interactions, are classified with the quantum numbers of different coupling schemes (11.2)— (11.5) or their combinations. Therefore, when calculating electron transition quantities, the accuracy of the coupling scheme must be accounted for. The latter in some cases may be different for initial and final configurations. Then the selection rules for electronic transitions are also different. That is why in Part 6 we presented expressions for matrix elements of electric multipole (Ek) transitions for various coupling schemes. 

From the work of Casimir, Lifshitz, London and many others [229] we know that the perturbation expression for the dispersion interaction between separated systems can be related to the electric polarizabilities of the interacting species, and also to the correlation of fluctuating electric multipoles on the two systems. In the Present TDDFT context, a useful polarizability form for the second-order dispersion interaction was given by Zaremba and Kohn [231] who derived it directly from second-order perturbation theory ... 

Proceeding as above, the general expression (58) can be used to calculate other particular cases involving higher multipoles, on resorting to Tables 4—7, where the multipole elements are listed. Interactions between magnetic multipoles are also the subject of discussion. The theory of electrostatic interactions for electric multipoles has been dealt with in various approaches by Frenkel, Pople, Jansen, and others, as well as by Gray. The above presentation follows the concise uniform treatment of Kielich. > > ... 

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 ... 

With the general definition of the electric multipole (40) and n-th order field (53), the potential energies of electrostatic interaction between electric multipoles and fields take the general form ... 

Table 2. Distance dependence of the interaction energy between electric multipoles. The exponent to which the distance must be raised is reported.

The vdW bonds which hokl subsystems forming a vdW molecule together are due to pomanait or temporary electric multipole—multipole interactions and not, in contrast to common molecules, to electron pair formation. 

An electric multipole is specified by its value of l as 2l—pole (1=1 dipole, / = 2quadrupole, l = 3 octupole, etc.). Hence, the electrostatic interaction is between l1— 1 poles, the leading term for two dipolar molecules (l = V = 1) being the dipole-dipole interaction. 

Full knowledge of the charge distribution of a molecule requires specification of the charge density at all points. For some purposes the charge density provides excess information thus, the potential outside a sodium ion is independent of the distribution of the electrons, and the interaction of a molecule with a uniform external field is determined by its dipole moment and dipole polarizabilities. The electric multipole moments characterize the charge distribution the first three are defined as follows ... 

Blasse and Bril (15) came to the conclusion that the exchange interaction is active if the S emission band overlaps the 4/- 4/absorption bands of 4, and by electric multipole interaction if the S emission band overlaps allowed absorption bands of A. Their assumption was based mainly on the fact that the exchange interaction depends on the overlap integral only while the multipolar interaction depends on the absorption cross-section in addition to the overlap. 

E. Burgos and H. Bonadeo, Mol. Phys., 44, 1 (1981). Electrical Multipoles and Multipole Interactions Compact Expressions and a Diagrammatic Method. 

General considerations on symmetry [12,13] lead to the result, that an atomic nucleus in a stationary state with spin quantum number / has electric and magnetic multipole moments only of order 2 with 0 < I <21. For electric multipole moments I must be even, while magnetic multipole moments require I to be odd. These rules are strictly obeyed, as long as very tiny parity non-conservation effects, due to weak interaction between nucleons, axe omitted (as is usually done for the nucleus, but see Sect. 6.3, where these effects are briefly discussed for the electronic structure). Thus,... 

In this last section we mention a few cases, where properties other than the energy of a system are considered, which are influenced in particular by the change from the point-like nucleus case (PNC) to the finite nucleus case (FNC) for the nuclear model. Firstly, we consider the electron-nuclear contact term (Darwin term), and turn then to higher quantum electrodynamic effects. In both cases the nuclear charge density distribution p r) is involved. The next item, parity non-conservation due to neutral weak interaction between electrons and nuclei, involves the nuclear proton and neutron density distributions, i.e., the particle density ditributions n r) and n (r). Finally, higher nuclear electric multipole moments, which involve the charge density distribution p r) again, are mentioned briefly. 

The first term is the total charge 2 interacting with the value of the scalar potential at the expansion point, whereas the second term corresponds to higher moments of the charge distribution interacting with spatial derivatives of the electric field at the expansion point. The electric multipoles have the general... 

Note that, contrary to the electric multipole expansion (148), there is no zeroth order term corresponding to a magnetic charge interacting with the value of the vector potential at the expansion point, this reflecting the absence of magnetic monopoles. 

An important class of properties arise from multipolar expansions of the interaction of nuclear moments with the electric and magnetic fields set up by surrounding electrons and nuclei. Restrictions apply to the possible nuclear moments 2 [93]. In general I < 21, where I is the nuclear spin. Furthermore, electric (magnetic) moments are restricted to even(odd) values of /. The lowest nuclear electric multipole is accordingly the electric quadmpole moment... 

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