Dipolar interaction energy differ

AGS is the strain-energy difference, AV is the dipolar-interaction energy difference, and D is the dielectric constant. If both R j and Rjj are known, AGS and AV can be determined by plotting the left-hand side of equation 2 against l/D. This should be a straight line with slope AV and intercept AGS. If only R is known, the above plot is repeated for various values of R-j-j. The value that gives the best straight line relationship is chosen for R.jj. 

An interesting comparison can be made between the experimentally determined limits on the interaction energies and those obtained from a simple dipolar model. In this case, the interaction energy difference between two coplanar dipoles is given by ... 

From the dipolar interaction energy, the dipolar shift can be obtained by evaluating from Eq. (2.16) A d,p between two states differing by A A// = 1 and dividing it by the nuclear Zeeman energy hyiBo (Appendix IV) ... 

Eq. (IV.6) is the general formula for the dipolar interaction energy when the principal axis of x is in a generic k direction. The shift is then obtained by calculating the energy difference between two states differing by AM/ = 1 ... 

The electric-tield-induced phase transition in an ER suspension was found to be different from that in general colloidal suspensions. Tao and Martin [55, 56] predicted theoretically tliat the bet structure has an energy lower than that of the fee (face-centered cubic) and other structures, based on dipolar interaction energy calculations. The dipolar interaction energy per particle for various crystal structures is shown in I able 3. The bet crystal structure is shown in Figure 6. 

This expression is of the order of what one would calculate by squaring Equation (15) and summing to get the mean square field. The difference is due to the fact that two nuclear spins can simultaneously flip so as to conserve magnetic energy since they are coupled by the dipolar interaction. This process limits the lifetime of a spin state and gives rise to an additional line breadth. For two nonidentical nuclei such processes do not con-... 

Thus, a comparison of the HOMO-LUMO energies (Table 10) predicts that the 1,3-dipolar cycloaddition should involve the HOMO of the miinchnone (1,3-dipole) interacting with the LUMO of the nitroindole (dipolarophile). These energy differences are lower by 1.7-3.0 eV than the reverse situation involving the LUMO of the miinchnone... 

Analytical expressions were derived for the CW EPR line-shape for spin-polarized radical pairs in the limit where the combined dipolar and exchange interaction is weak relative to the energy differences between the resonances of the two spins.19 The equations were applied to the case of charge-separated sites in Ti02 nanoparticles. This approach simplifies the analysis of the distributions of interspin distances. 

In subunit R2 of ribonucleotide reductase there is a tyrosyl radical (Y ) in close proximity to a di-iron cluster.100 In the protein from E. coli the EPR signal from Y can be observed up to room temperature. However, in the protein from yeast the Y signal broadens above 15 K and is not observable above about 60 K. Saturation recovery measurements at 140 GHz showed that at 60 K the spin-lattice relaxation rates for the Y signal in the yeast protein were about 2 orders of magnitude faster than for the E. coli protein. The temperature dependence of the relaxation enhancement was consistent with the activation energy for the first excited state of the di-iron cluster, so the relaxation enhancement was attributed to interaction with the di-iron cluster. Relaxation enhancements measured at 140 GHz showed little orientation dependence so the enhancement was assigned to isotropic exchange, which is different from the orientation-dependent dipolar interaction observed for the E. coli protein.100... 

Recently, Huisgen and coworkers have reported on the first unequivocal example of a nonconcerted 1,3-dipolar cycloaddition.27 Sustmann s FMO model of concerted cycloadditions envisions two cases in which the stepwise mechanism might compete with the concerted one.21 Two similar HOMO-LUMO interaction energies correspond to a minimum of rate and a diradical mechanism is possible, especially if stabilizing substituents are present. A second case is when the HOMO (l,3-dipole)-LUMO (dipolarophile) is strongly dominant in the transition state. The higher the difference in rr-MO energies of reac-... 

Do the transition states of the 1,3-dipolar cycloadditions with diazomethane benefit from a stabilizing frontier orbital interaction Yes Computations show that the HOMOdia zomethm/LUMOethene interaction (orbital energy difference, -229 kcal/mol) stabilizes the transition state of the 1,3-dipolar cycloaddition to ethene (Figure 15.37) by about 11 kcal/mol. Moreover, computations also show that the HOMOethene/LUMOdjazomethane interaction (orbital energy difference, -273 kcal/mol ) contributes a further stabilization of 7 kcal/mol. 

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