The Point-Dipole Approximation

The interaction between the spin magnetic moments of two unpaired electrons, or of an electron and a nucleus gives rise to an anisotropic coupling. The point-dipole approximation applies when the distance R between the electron spins or the electron and nuclear spins is large compared to the extension of the orbital(s) of the 

Type Measurement method Range (A) Measured quantity Theory 

Electron-nucleus ESR, ENDOR, ESEEM 1-2 Electron-nucleus Quantum average 

Electron-nucleus ENDOR, ESEEM 2-4 dipolar coupling Electron-nucleus Point dipole 

Electron-electron ESR 5-15 dipolar coupling Electron-electron Point dipole 

---

For two stacks having the same molecular orientations, the numerical factor 3 (2) in formula (3.3.15) should be substituted by 2 as a result, with

parameters indicated above, the frequency shift yielded in the point-dipole approximation is 2 = - 445 cm"1. On the other hand, the computer simulation of two parallel stacks, each consisting of ten BTCC molecules, gave the frequency... 

The intrachain dipole-dipole interactions of BTCC molecules are responsible for the frequency shift contributing to the first of Eqs. (3.3.8) as (3)//(l -3cos2sign changing at

Coulomb interactions show this boundary angle to be equal to 30°.132 Thus, as far as intrastack interactions of dye molecules are concerned, the point-dipole approximation introduces large errors into the treatment. 

Summing the first and second order contributions (5.9), (5.5) and (5.12a), the total hf matrix for a proton (ligand atom N) in the point-dipole approximation is given by... 

In this section, we discuss the work inquiring into the meaning of r/s, the distance between the two dipoles in Eqs. (12) and (13). The simplest approximation is to assume that r/s is equal to the internuclear distance between the nucleus in the ligand, the relaxation of which is being studied, and the metal ion. This amounts to the point-dipole approximation for both the nuclear and the electron spins. While such an approximation is perfectly... 

The effective distances obtained by Nordenskiold et al. (40) are compared with the internuclear distances in Table I. Clearly, the point dipole approximation is reasonable for the hydrogen nuclei in these complexes, while substantial deviations are observed for the oxygen nuclei. The findings of these early quantum chemical studies were confirmed by Sahoo and Das (41-43). Wilkens et al. have reported DFT calculations using Eq. (16) for a 104 atom model for high-spin Fe(III) rubredoxin (44). Large discrepancies between the effective distances and the input distances for the calculations were found for the hyperfine-shifted nitrogen-15 resonances, as well as for proton and carbon-13 nuclei in cysteines bound to the iron center. 

Kowalewski analyzed the validity of the point dipole approximation in a series of complexes and found that the effective distance from the ligand nucleus to the unpaired spin agreed well with the internuclear distance for XH nuclei. Large deviations from the point dipole approximation have been found for the ligand nuclei directly bound to the metal. 57 J 59... 

As anticipated in Sections 2.2.2 and 3.1, the unpaired electrons should not be considered as point-dipoles centered on the metal ion. They are at the least delocalized over the atomic orbitals of the metal ion itself. The effect of the deviation from the point-dipole approximation under these conditions is estimated to be negligible for nuclei already 3-4 A away [31]. Electron delocalization onto the ligands, however, may heavily affect the overall relaxation phenomena. In this case the experimental Rm may be higher than expected, and the ratios between the Rim values of different nuclei does not follow the sixth power of the ratios between metal to nucleus distances. In the case of hexaaqua metal complexes the point-dipole approximation provides shorter distances than observed in the solid state (Table 3.2) for both H and 170. This implies spin density delocalization on the oxygen atom. Ab initio calculations of R m have been performed for both H and 170 nuclei in a series of hexaaqua complexes (Table 3.2). The calculated metal nucleus distances in the assumption of a purely metal-centered dipolar relaxation mechanism are sizably smaller than the crystallographic values for 170, and the difference dramatically increases from 3d5 to 3d9 metal ions [32]. The differences for protons are quite smaller [32]. 

The TDC method takes into account the shape of the molecules in detail, and its accuracy in the calculation of the Coulomb coupling depends only on the size of the volume elements used in the grid (the cube ). In this way, the TDC method has proven extremely useful in understanding the limitations of the point dipole approximation (PDA) in a variety of systems. In particular, how the PDA fails to describe the coupling when the interchromophoric center-to-center distance is comparable to the molecular dimensions, a situation found, for example, in many of the relevant interactions present in natural light-harvesting antennas. [50]... 

Riplinger, C., Kao, J. P. Y., Rosen, G. M., Kathirvelu, V., Eaton, G. R., Eaton, S. S., Kutateladze, A., and Neese, F. (2009). Interaction of radical pairs through-bond and through-space Scope and limitations of the point-dipole approximation in electron paramagnetic resonance spectroscopy. J. Am. Chem. Soc. 131, 10092—10106. Schiemann, O., and Prisner, T. F. (2007). Applications of electron paramagnetic resonance to distance measurements in biomolecules. Q. Rev. Biophys. 40, 1—53. 

At large distances compared with the size of the molecules, the dipolar transitions may be treated with point dipoles. This approximation is not valid for intermolecular distances met with in condensed phases. However, the point-dipole approximation allows one to discuss the various levels of interaction and proves very useful for the discussion of the general case, as illustrated in Section II. Historically, the point-dipole approximation was the first to be applied to molecular-crystal excitations.17-20... 

Figure 1.2. Excitonic dispersion curves for the first singlet state of the anthracene crystal. These curves are calculated in the point-dipole approximation, the transitions to upper states being accounted for by a constant dielectric permittivity.13,37...

When the point-dipole approximation is no longer valid, the exact distribution of transition charges on the molecule is introduced. The difference between this distribution and that of the point dipoles is important only in short-range interactions and modifies only the analytic part of the dispersion. In particular, the retarded interactions (and the associated properties) are not modified. 

The forces one must include in such a simulation include electrostatic, hydrodynamic, and steric forces. For small particles, Brownian forces might also be present, but since these break up particle structures, it is desirable to use particles big enough (> 1 /xm) to suppress Brownian motion. Ordinarily, particle inertia can be neglected. Simulations can be greatly simplified by making drastic approximations, including the point-dipole approximation, and the Stokes -drag approximation. Both of these approximations are only really valid for widely separated particles. 

In the point-dipole approximation, one assumes that the dipole moment of a particle is not affected by the surrounding particles, and that the ratio Sp/S is near unity. The dipole... 

The simplest form for the hydrodynamic drag force F, namely Stokes drag, is, like the point-dipole approximation, strictly valid only when the particles are widely spaced. In the Stokes -drag approximation, for a shearing flow at shear rate y, we have... 

One can go beyond the point-dipole approximation by generalizing the expression (8-4) for the electrostatic force to include multipole effects ... 

For higher 0, ay levels off, or even has a maximum, evidently because chains are more efficient at momentum transfer when they are a single particle wide than when they are clumped into columns, as they tend to be when 0 is large (Klingenberg et al. 1991b). Note, also, however, that the magnitude of Oy from the simulations is much lower, by a factor or 50 or so, than the experimental values. At least part of this discrepancy can readily be attributed to the point-dipole approximation, which is only accurate when particles are widely spaced and when 1, conditions that aren t met. The magnitude of the errors made in the... 

The anisotropy of the cyclopropane ring was calculated by Forsen and Norin, using the point dipole approximation. In one of several examples they noted that H4 of nortricyclene, 2, absorbed 0.3 ppm to higher field than the corresponding proton of... 

The dinitroxide (20) is one of several nitroxide biradicals used to test the point-dipole approximation for electron-electron dipolar interactions/ ... 

By using simple scaling laws for the polarizability and first hyperpolarizability as a function of the molecular size, Hurst and Munn [93] addressed, within the point dipole approximation, the relationship between nd molecular elongation and found that very large can be obtained for compact molecules, provided that the ratio between the polarizability and the molecular volume is large. 

---