Electric field higher-order multipole

Energy of Induced Multipole Interaction.—Interaction between a Multipolar System and External Fields. We calculated above the potential energy of electrostatic interaction between a multipole system and external fields, or thefirst-orderenergyduetothefirstpowerofthefield. Besides that energy, which took account only of the reoriratation of permanent multipoles, we have to take into consideration contributions due to the drcum-stance that an external electric field induces higher-order multipole moments given by the expressions (72) and (79). 

The starting point for nonhnear optics is the constitutive relationship between the polarization induced in a molecule (p) and the electric field components of incident electromagnetic waves ( ). With the electric dipole approximation that ignores magnetic dipoles and higher order multipoles... 

In modem ion trap mass spectrometry, weak higher-order multipole and/or dipole electric fields are always superimposed upon the quadrupole electric field. The weak superimposed higher-order multipole and/or dipole electric fields are found to enhance the mass spectral performance with respect to mass resolution, charge capacity, scan speed, and to chemical mass shift. In fact, all commercial ion trap mass spectrometers are non-linear ion traps [7-9]. In the Finnigan ion trap instrument, the positions of the two end-cap electrodes are stretched out from the theoretical positions to add the higher-order multipoles. In the Bruker ion trap, the profiles of the end-cap electrodes are crafted to create the superimposed higher-order multipoles. In the Varian 4000 ion trap, a dipole and higher-order multipoles are superimposed upon the quadrupole field by a switchable electric circuit [10],... 

The non-linear ion traps with stretched or modified electrode profiles have been discussed previously in detail. In 2004, Varian introduced the 4000 instrument with a new type of non-linear ion trap [9,10], in which a dipole and higher-order multipoles are superimposed upon the quadrnpole field by a switchable electric circuit, shown in Figure 15.2. 

FIGURE 15.2 In the 4000 instrument, a trapping dipole (ca 3%) and other higher-order multipoles are superimposed on the quadrupole field through a series inductor and capacitor. The non-linear field is electrically switchable. 

The condition for this quantity to be non-zero is that the chromophore of interest must have a non-zero magnetic and electric transition dipole moment along the same molecular diiectioa In the absence of pertuibing external fields, this is only trae for molecules that are chiral. Expressions that include higher order multipole contributions to eqs. (8), (9), and (11) can be found in previous theoretical descriptions of CPU theory (Riehl and Richardson, 1976a, 1986). 

If higher-order multipoles in an expansion of the electric field in terms of a discretizated grid of cubic elements are ignored, the local electric field, Eioc (o, r), has the following form ... 

Terms of higher order in the field amplitudes or in the multipole expansion are indicated by. . . The other two tensors in (1) are the electric polarizability ax and the magnetizability The linear response tensors in (1) are molecular properties, amenable to ab initio computations, and the tensor elements are functions of the frequency m of the applied fields. Because of the time derivatives of the fields involved with the mixed electric-magnetic polarizabilities, chiroptical effects vanish as a> goes to zero (however, f has a nonzero static limit). Away from resonances, the OR parameter is given by [32]... 

We will be using this form of the molecule-field interaction repeatedly in this text, however, it should be kept in mind that it is an approximation on several counts. Already Eq. (3.1), an electrostatic energy expression used with a time varying field, is an approximation. Even in this electrostatic limit, Eq. (3.1) is just the first term in an infinite multipole expansion in which the higher-order terms depend on higher spatial derivatives of the electric field. 

The theoretical framework developed above is valid in the electric dipole approximation. In this context, it is assumed that the nonlinear polarization PfL(2 >) is reduced to the electric dipole contribution as given in Eq. (1). This assumption is only valid if the surface susceptibility tensor x (2 > >, a>) is large enough to dwarf the contribution from higher orders of the multipole expansion like the electric quadrupole contribution and is therefore the simplest approximation for the nonlinear polarization. At pure solvent interfaces, this may not be the case, since the nonlinear optical activity of solvent molecules like water, 1,2-dichloroethane (DCE), alcohols, or alkanes is rather low. The magnitude of the molecular hyperpolarizability of water, measured by DC electric field induced second harmonic... 

The second-order induction effects pertain to polarization by the charge distributions corresponding to the isolated molecules. However, the induced multipoles introduce a change in the electric field and in this way contribute to further changes in charge distribution. These effects already belong to the third and higher orders of perturbation theory. 

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