Multipole components

Wang, Y. Franzen, J. The Non-Linear Ion Trap. Part 3. Multipole Components in Three Types of Practical Ion Trap. Int. J. Mass Spectrom. Ion Proc. 1994, 132, 155-172. 

Molecular multipole components are best described in the molecular (i.e., body-fixed) frame x, y, z. For example, a dipole when aligned with the z-axis is characterized by a single number, the strength p of the dipole. The other two independent components can then simply be expressed in terms of Euler angles or, in this case, of azimuthal and polar angle, q> and 3, between molecular (x, y, z) and laboratory-fixed frame (X, Y, Z). 

Symmetry Properties. Under inversion, for R being replaced by -/ , we have Qfm — (—1 YQem- A dipole is odd under inversion and a quadrupole is even. From the properties of spherical harmonics and the definition of the spherical harmonics, it is easy to see that Q m — (—1 )mQ(-m-If Q = a, P, y designates the Euler angles of the rotation carrying the laboratory frame X, Y, Z, into coincidence with the molecular frame, x, y, z, the body-fixed multipole components Q(m are related to the laboratory-fixed Q(m, according to... 

Fig. 4.1. The four significant induced dipole components of H2-He pairs ground state vibrational average (left panel) and vibrational transition matrix element v = 0 —> v = 1 (center panel) and of the ground state vibrational average of H2-Ar pairs (right panel). Overlap components are dotted multipole-induced components are represented by solid lines dashed lines indicate the associated classical multipole component after [279, 151, 280],...

As an example of these methods, consider the B cyclic theorem for multipole radiation, which can be developed for the multipole expansion of plane-wave radiation to show that the B<3) field is irrotational, divergentless, and fundamental for each multipole component. The magnetic components of the plane wave are defined, using Silver s notation [112] as... 

Using Eqs. (768a) and (769), it is seen that the product is unity if we sum over all multipole components with 1 —> oo in Eq. (768). In all other cases, the B cyclic theorem is... 

Another advantage of the polarization moments is the possibility of describing relaxation processes in the most rational way. Thus, if the relaxation process is isotropic, then the various moments and their components relax independently for details see Section 5.8. All multipole components of certain rank K relax at one and the same rate constant... 

The accuracy of a distributed multipole representation can always be tested by evaluating the effect on the calculated electrostatic energies in the region of interest of increasing the number of interaction sites (e.g., by adding sites at the midpoints of bonds) or the order of multipoles. Similarly, simplifications of the model, such as removing small multipole components, can be evaluated. However, the accuracy of the calculated electrostatic energies is inevitably limited by the quality of the wavefunction and the absence of penetration effects. 

In the days of Mellor s challenge the electrostatic field seemed likely as the mechanism by which chiral information is conveyed. It would be contained in the multipole expansion of expression (3.4). PE Schipper (later in the Department of Theoretical Chemistry, University of Sydney) and I looked at this expansion [131] to select multipole components that were chiral, that is to say components that lacked symmetry to improper rotations. The multipole components (6.1) transform... 

More general results can readily be found. All moments are from now on referred to the same body axis system, the z axis being the common polar axis. The essential basis is in Equation (II.7), which give the transformation properties of the multipole components under the operations of inversion (t), reflection in the xy plane (aft) reflection in a plane containing the z axis (ur), and improper rotation about the z axis by 2nlp, (iCp). 

Its S3unmetry properties are not the same as those of the electric dipole, which transforms under pure rotations as the multipole components (1,0) and (1, 1) and is antisymmetric under inversion in a centre of sjmmetry. The magnetic dipole transforms hi the same way under rotations but is symmetric to inversion. The properties are those of an antisymmetric second-rank tensor, or axial vector, and can be rationalized using the model that a magnetic field has as source a plane current loop. The current direction in the loop is invariant to inversion of the spatial coordinates but reversed by twofold rotation about an axis in the plane. 

The goal of the present section is to take apart the induction mechanism by showing its multipole components. If we insert the multipole representation of V into the induction energy B), then... 

Just think about a multipole component of the form qz" computed with respect to the center of each subsystem. 

Expanding the ejected electron wave function k) into partial waves and summing over the magnetic quantum numbers m, of the ejected electron s spin— which is generally not observed—one can write the electron intensity in terms of multipole components of the density matrix ... 

Figure 4. IM NaCl solution at 30°C and -4e electrode charge. Detail of individual electric multipole component potentials shown in Figure 3 (bottom) for z > 0.8 nm.

Thus the only multipole components which can appear for a homonuclear diatomic are Qqq (the charge or monopole moment), Q2Q (the quadrupole moment), Q q (the hexadecapole moment), and so on. 

A 1, 2 alignment If the vector distributions are prepared with axial symmetry, e.g. with respect to the colliding reagent relative velocity vector, or the polarisation vector of an incident photon, the multipole components with q 0... 

The second term is the creation energy of the induced multipoles. The + superscript indicates a matrix transpose. Vesely [55] has shown for a set of polarizable dipoles that if the induced dipoles are given by Eq. 9, it corresponds to a minimum of the total electrostatic energy, the partial derivative of E with respect to the induced multipole components being zero. 

The first term, Ees> interactions between permanent multipoles, is often called the electrostatic energy and the second one, Ejnd, which results from interactions between permanent and induced multipoles, is the induction energy. From Eq. 9, it can be seen that the induced multipoles can be obtained by inverting an Nn x Nn matrix, or by an iterative process. In molecular dynamics simulations, the iterative process initiated with the induced multipole components of the previous time step is usually used rather than matrix inversion, especially when large matrices are involved. Then the forces on the centres of mass can be obtained by derivation with respect to the centre of mass position ... 

When discussing shielding effectiveness it is necessary to consider two radiation zones the near-field and the far-field (or smooth wave) zones. The distinction between them lies in the distance from the. source of radia tion. If the distance from source to shielding is less thai. one-sixth of the free path wavelength of the radiation to be shielded, the radiation is dominated by the smaller multipole components of the source field and is described as within the near-field zone." Above this zone it is in the far-field zone. A more detailed consideration of the problem can be found in Ref. 98. 

Similar to the distributed-multipole expansion of molecular electrostatic fields, one can derive a distributed-polarizability expansion of the molecular field response. We can start by including the multipole-expansion in the perturbing Hamiltonian term W = Qf(p, where we again use the Einstein sum convention for both superscripts a, referencing an expansion site, and subscripts t, which summarize the multipole components (/, k) in just one index. Using this approximation for the intermolecular electrostatic interaction, the second-order energy correction now reads ... 

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