Octupoles

Many molecules, such as carbon monoxide, have unique dipole moments. Molecules with a center of inversion, such as carbon dioxide, will have a dipole moment that is zero by symmetry and a unique quadrupole moment. Molecules of Td symmetry, such as methane, have a zero dipole and quadrupole moment and a unique octupole moment. Likewise, molecules of octahedral symmetry will have a unique hexadecapole moment. 

Third-order (octupole) or higher order muitipoie approximations can be empioyed for more accuracy. 

Just as the dipole changes in an external field, so do all the other moments, and we can develop a set of equations for the quadrupole polarizability (and hyperpolarizabilities), the octupole polarizability, and so on. These esoteric quantities are rarely met in chemistry. 

The charge distribution of the molecule can be represented either as atom centred charges or as a multipole expansion. For a neutral molecule, the lowest-order approximation considers only the dipole moment. This may be quite a poor approximation, and fails completely for symmetric molecules which do not have a dipole moment. For obtaining converged results it is often necessarily to extend the expansion up to order 6 or more, i.e. including dipole, quadrupole, octupole, etc. moments. 

In electrostatics the first term represents the contribution to the field due to the net charge, and higher terms represent dipole, quadrupole, octupole, etc. 

The spherical harmonic density functions are referred to as multipoles, since the functions with 1 = 0, 1, 2, 3, 4, etc., correspond to components of the charge distribution p r) which give nonzero contributions to the monopole (/ = 0), dipole (/ = 1), quadrupole (/ = 2), octupole (/ = 3), hexadecapole (/ = 4), etc., moments of the atomic charge distribution. 

Similarly, octupoles and hexadecapoles can be thought of as arising from 2p3d and 3d3d atomic orbital products, which leads to n, = 3 and 4, respectively. The... 

Double deflection dual octupole assembly 1 Einzel V j lens... 

Born s idea was taken up by Kirkwood and Onsager [24,25], who extended the dielectric continuum solvation approach by taking into account electrostatic multipole moments, Mf, i.e., dipole, quadrupole, octupole, and higher moments. Kirkwood derived the general formula ... 

James and Keenan [1959] showed that the octupole-octupole interactions are dominant in the multipole expansion of Vc in (7.57) ... 

---