Electric Multipole Moments

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... 

Typical properties of the charge distribution are summarized by its various electric multipole moments. The electric dipole moment p. induced in the system by the external field is obviously... 

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... 

Kielich S, Zawodny R (1971) Tensor elements of the molecular electric multipole moments for all point group symmetries. Chem Phys Lett 12 20-24... 

An electric multipole moment of order n in the form (40) has quite generally (2 + 1) independent components, and their number undergoes a further reduction for various molecular symmetries (Table 2). Buckingham, by methods of group theory, calculated the number of independent tensor components of multipoles (40) from n = 1 up to = 6 for 35 point-group symmetries. In Table 3, we give their numbers for the first four moments as well as the number of non-zero components. 

In analogy to the definition of electric dipole moment, electric multipole moments are also defined. In particular, the quadrupole moment Q and the octupole moment U are defined as ... 

A. Salam, A General Formula for the Rate of Resonant Transfer of Energy Between Two Electric Multipole Moments of Arbitrary Order Using Molecular Quantum Electrodynamics. J. Chem. Phys. 122 (2005) 044112. 

We now recall that only the value of the first non-zero electric multipole moment is independent of the origin. Consider a molecular quadrupole moment along the internuclear z axis in a linear molecule at two different origins along the z axis separated by Z. According to Eq. (12)... 

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 ... 

Monatomic entities M+z consisting of one nucleus (carrying Z times the electric charge e of a proton) surrounded by K = (Z — z) electrons have been one of the major subjects for quantum-mechanical treatment. If the nucleus is treated as a geometrical point, and no attention is paid to its electric multipole moments, nor to its magnetic moments, the energy levels can be characterized by even or odd parity and by a quantum number J of total angular momentum. If the coordinates (— x, — y,... 

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. 

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