Electrical multipoles measurement

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... 

Whereas the hfs can serve as a local probe for intramolecular fields and field gradients, information about the total charge-density distribution is given by the electric multipole moments, especially the dipole moment of the molecule. Both molecular properties complement each other in giving a more complete picture of the electronic structure and chemical bonding. In section IV we show a way to determine electric dipole moments from high precision Stark effect measurements. The techniques described in sections II-IV will be applicable to many other free radicals. 

Gaussian also predicts dipole moments and higher multipole moments (through hexadecapole). The dipole moment is the first derivative of the energy with respect to an applied electric field. It is a measure of the asymmetry in the molecular charge distribution, and is given as a vector in three dimensions. For Hartree-Fock calculations, this is equivalent to the expectation value of X, Y, and Z, which are the quantities reported in the output. 

So our choices of the two antennas is not unique for separately emphasizing the Lorenz vector and scalar potentials. All that is required is for the two to have the same exterior fields (say, electric dipole fields, or more general multipole fields) with different potentials (related by the gauge condition). In a classical electromagnetic sense, these antennas cannot be distinguished by exterior measurements. This is a classical nonuniqueness of sources. In a QED sense, the same is the case due to gauge invariance in its currently accepted form. 

Birefringences are mostly observed in condensed phases, especially pure liquids or solutions, since the strong enhancement of the effects allows for reduced dimensions (much shorter optical paths) of the experimental apparatus. Nowadays measurements of linear birefringences can be carried out on liquid samples with desktop-size instruments. Such measurements may yield information on the molecular properties, molecular multipoles and their polarizabilities. In some instances, for example KE, CME and BE, measurements (in particular of their temperature dependence) have been carried out simultaneously on some systems. From the combination of data, information on electric dipole polarizabilities, dipole and quadrupole moments, magnetizabilities and higher order properties were then obtained. 

Heck and Williams (1987) observed quantum beats in the decay of the n = 2 states of atomic hydrogen. In the presence of an external electric field the 2s and 2p states can be mixed and their correlation measured. With an applied field of 250 Vcm the modulation periods should be 0.1 ns and 0.6 ns. They were able to observe the second beat period corresponding to interference between the states which reduce to 2si/2 and 2pi/2 in the field-free limit. Williams and Heck (1988) were able to use the technique to determine many of the n = 2 state multipoles (section 8.2.4). 

Another important class of forces, induction or polarization forces, involves permanent moments that induces multipoles in a polarizable species. Polarizability, a, measures the ability of an atomic or molecular species to develop an induced dipole moment, as a response to an applied electric field E. Within the limits of linear response theory, the induced dipole moment is given by the product of polarizability tensor times the electric field E. 

The radial deformation of the valence density is accounted for by the expansion-contraction variables (k and k ). The ED parameters P, Pim , k, and k are optimized, along with conventional crystallographic variables (Ra and Ua for each atom), in an LS refinement against a set of measured structure factor amplitudes. The use of individual atomic coordinate systems provides a convenient way to constrain multipole populations according to chemical and local symmetries. Superposition of pseudoatoms (15) yields an efficient and relatively simple analytic representation of the molecular and crystalline ED. Density-related properties, such as electric moments electrostatic potential and energy, can readily be obtained from the pseudoatomic properties [53]. 

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... 

Since the lowest levels of even-even nuclei almost always have the same parity as the ground state and spin 2, it is generally assumed that Coulomb excitation in heavy elements is mainly an electric quadrupole process and that the cross section observed experimentally measures the E 2 transition probability. Similar assumptions are made for odd-A nuclei. In the majority of nuclei this assumption is justified. However, as Bjerregaard and Huus have shown, the multipole order can be determined directly by measurements with bombarding particles with different specific charge, e.g., protons and deuterons (or a-particles). They have verified that the multipole order is indeed two for the excitation of the lowest excited states in the even-even wolfram nuclei. 

The most basic electrical model of the heart is a bound vector with the variable vector moment m = iLcc see Eq. 6.10. Plonsey (1966) showed that a model with more than one dipole is of no use because it will not be possible from surface measurements to determine the contribution from each source. The only refinement is to let the single bound dipole be extended to a multipole of higher terms (e.g., with a quadrupole). 

An example of a more recent experiment is the smdy of hyperfine structure of highly excited levels of neutral copper atoms, which yielded the magnetic-dipole and electric-quadmpole interaction constants. The experimental results allowed a comparison with theoretical calculations based on multiconfiguration Hartree-Fock methods [858]. Measurements of lifetimes [859] and of state multipoles using level-crossing techniques can be found in [860]. 

Endt (1981) evaluated 1,340 y-ray transitions in A = 91-150 nuclek The transition rates were classified according to electric or magnetic multipole character and were expressed in Weisskopf units. Transitions, which can be mixed in principle, are only listed if the mixing ratio has been measured. The strengths S = Ti Ti) usually scatter appreciably, nevertheless, upper limits could be established fi-om the data. These are 0.01,300,100, and 30 W. u. for El, E2, E3, and E4 and 1, 1, 10, and 30 W. u. for Ml, M2, M3, and M4 radiations, respectively. For upper limits in other regions see (Firestone et al. 1996). The strengths are usually enhanced for E2, and E3, while the El, M2 transitions are usually hindered. 

The Tables 5.4, 5.5, 5.6 show the calculation results of some electrical characteristics for H2,02, N2, CO2, CO, CN, HCl, HCN, NaCl, OH, NaH"", CH4, and H2O molecules, which are important for astrophysical and atmospheric problems. In the work [88] the calculations were carried out using the finite-field method at the (R) CCSD(T) level of theory with different aVXZ basis sets (X = Q, 5). For these cases, the amplitudes of the applied fields have been chosen as follows F = 0.0025 a.u., = 0.0001 a.u., FajSy = 0.00,001 a.u. and Fg,pys = 0.000001 a.u. Multipole moments up to 4th order are presented in Table 5.4. For comparison, in Table 5.4 the other literature data are also given. Table 5.5 presents the calculated and measured values (we have chosen the more reliable ones) of multipole polarizabilities. 

---