Monopole interaction

The above potential describes the monopole-monopole interactions of atomic charges Q and Qj a distance Ry apart. Normally these charge interactions are computed only for nonbonded atoms and once again the interactions might be treated differently from the more normal nonbonded interactions (1-5 relationship or more). The dielectric constant 8 used in the calculation is sometimes scaled or made distance-dependent, as described in the next section. 

Amonopole-monopole interaction, where the integral should converge to... 

The Coulomb interaction of the (point) nucleus with the potential Vo, which is also part of the monopole interaction, was neglected in (4.5) because it yields only an offset of the total energy. The subscript u in is introduced to distinguish the radius of the uniformly charged sphere from the usual mean square radius which can be obtained from scattering experiments. 

The electric monopole interaction between a nucleus (with mean square radius k) and its environment is a product of the nuclear charge distribution ZeR and the electronic charge density e il/ 0) at the nucleus, SE = const (4.11). However, nuclei of the same mass and charge but different nuclear states isomers) have different charge distributions ZeR eR ), because the nuclear volume and the mean square radius depend on the state of nuclear excitation R R ). Therefore, the energies of a Mossbauer nucleus in the ground state (g) and in the excited state (e) are shifted by different amounts (5 )e and (5 )g relative to those of a bare nucleus. It was recognized very early that this effect, which is schematically shown in Fig. 4.1, is responsible for the occurrence of the Mossbauer isomer shift [7]. 

Fig. 4.1 The electric monopole interaction between the nuclear charge and the electron density at the nucleus shifts the energy of the nuclear states and gives the Mossbauer isomer shift...

Fig. 4.6 Quadrupole splitting of the excited state of Fe with I = 3/2 and the resulting Mossbauer spectrum. Quadrupole interaction splits the spin quartet into two degenerate sublevels 7, OT/) with energy separation A q = 2 q. The ground state with I = 1/2 remains unsplit. The nuclear states are additionally shifted by electric monopole interaction giving rise to the isomer shift 8...

Fig. 4.9 Magnetic dipole splitting (nuclear Zeeman effect) in pe and resultant Mossbauer spectrum (schematic). The mean energy of the nuclear states is shifted by the electric monopole interaction which gives rise to the isomer shift 5. Afi. g = Sg/tN and A M,e = refer to the...

The monopole interaction 8 , which yields the isomer shift, is easy to treat it is just additive to all transition energies. Thus, the recorded spectrum has uniform shifts of all resonance hues with no change in their relative separations. 

To calculate Mossbauer spectra, which consist of a finite number of discrete lines, the nuclear Hamiltonian, and thus also Hsu, has to be set up and solved independently for the nuclear ground and excited states. The electric monopole interaction, that is, the isomer shift, can be omitted here since it is additive and independent of Mj. It can subsequently be added as an increment 5 to the transition energies of each of the obtained Mossbauer lines. 

From a physical point of view, this new formulation includes exponential terms that are in agreement with the observed ab initio and experimental results. Moreover, it is easy to verify that the new expression converges to the classical one when r increases. That way, at long range, where the multipolar approximation is valid, the exponential part dies whereas, at short distances, the monopole-monopole interaction embodies a part of the penetration energy. Consequently, Emono-mono has the correct dependence at any range. 

Asymmetry in the ligand environment, either geometric or in charge distribution (or both), affect the asymmetry parameter, tp An r = 0 value corresponds to complete axial symmetry, whereas r = 1 corresponds to pure rhombic symmetry. Electric monopole interactions between the nuclear charge distributions and the electrons at the nucleus cause a shift of the nuclear ground and excited states. These interactions are known as the isomer shift, 8. Both the Mossbauer source and the absorber (the sample of interest) experience an isomer shift, and it is customary to quote 8 relative to a standard, usually Fe metal or Na2[Fe(CN)5NO] 2H2O at... 

Mossbauer isomer shift and quadrupole splitting are commonly used to obtain information about the bonding environment around source nuclides. The isomer shift arises from the electric monopole interaction of the nucleus with the electrons and depends on the... 

The electric monopole interaction is a function of the s electron densities at the nucleus. This results in a displacement of the spectrum and is expressed as the velocity of the source (mm s ) necessary to counteract the displacement. This isomer (or chemical) shift, 6, provides information about the coordination number, the valency and spin state of the iron in the compound. 

The total interaction can be determined in two ways, either by a calculation of the contributions from all the terms described above or by making proper approximations. The former way is of course more accurate, but since it leads to unsolvable equations, it is more fruitful to use the second approach. One common assumption in the theory of electrolyte solutions is to replace the solvent by a continuous medium with a dielectric constant grso- This approximation dramatically reduces the number of interacting molecules in the calculations. The only remaining particles are now the ions and the colloids, which they are treated as monopoles interacting according to Coulomb s law reduced by grso- Thus, the expression for the force F is... 

The nuclear ground and excited levels involved in the Mossbauer transition are shifted or split because of the electrostatic interactions between the nuclear charge and the surrounding electric charge (Fig. 2). The first interaction, sometimes called the electric monopole interaction, shifts only the nuclear levels and is related to the perturbation resulting from the electrons inside the nuclear volume. This shift is... 

Mossbauer spectroscopy senses the hyperfine interactions, which are present at the nucleus of the Mossbauer isotope. The electrical monopole interaction causes the isomer shift and the electric quadrupole interaction leads to the quadrupole splitting, which in the case of Fe causes a two-line Mossbauer pattern. The magnetic dipole interaction leads to a magnetically split six-line pattern (Figure 4). In the following text, these interactions and their deduction from Mossbauer spectra will be discussed. 

The electric monopole interaction is proportional to the s-electron density at the iron nnclens [Pg.2821]

Figure 1. Hyperfine interactions for Fe nuclei, showing the nuclear energy level diagram for (a) an unperturbed nucleus (b) electric monopole interaction (isomer shift) (c) electric quadrupole interaction (quadrupole splitting) and (d) magnetic dipole interaction (hyperfine magnetic splitting). Each interaction is shown individually, accompanied by the resulting Mossbauer spectrum.

Dipole interactions are usually weaker than electrostatic monopole interactions but can dominate the intermolecular interactions within a supramolecular assembly. Diederich and coworkers have recently drawn attention onto dipole interactions, and multipolar interactions in general, in such systems based on a statistical analysis of structures [180]. 

The monopole interaction force is equal to the total net charge on the molecule, and the interaction energy of ionic species is dominated by the monopole interactions. 

The method Neglect of Diatomic Differential Overlap (NDDO) was originally developed by Pople and Beveridge [8] and Pople et al. [37]. The ZDO approximation [Eq. (26)] is only applied for orbital pairs centered at different atoms. Consequently, new types of two-center integrals appear compared to the INDO method, (pv pX) and (/t Fb v). This means that not only monopole-monopole interactions are taken into account, but also dipole and quadrupole terms. Thus, in principle, NDDO-based methods should give an improved description of long-range intra- and interm olecular... 

Williams showed that an unrestricted bond dipole model gave erratic results for the bond dipole directions. To pursue the goals of chemical reasonableness and transferability, it seems appropriate to restrict the direction of the bond dipoles. The most natural direction is along the bond. Table 12 shows that restricted bond dipole models represent the electric potential about as well as monopole models. The two models also have nearly the same number of parameters. So the choice between these two models is a matter of convenience. If long distance interaction is considered, as in crystals, use may be made of the fact that dipole-dipole energy converges much faster than monopole-monopole energy. However, as mentioned above, if ions are considered, monopole interactions are still needed. Table 13 summarizes values of restricted bond dipole moments. 

Figure 1.95 Illustration of monopole/monopole interactions where q and q2 are two monopoles ris the distance of separation.

Figure 1.96 Illustration of fixed dipole/monopole interactions where charges qi/-qi separated by a distance / represent the fixed dipole and and q2 is a monopole r is the distance of separation between dipole midpoint and monopole.

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