Coupling, hyperfine

The nuclear hyperfine coupling is the field at the electron due to the nucleus and its magnitude depends on whether the electron which is interacting with the nucleus is in an s or a p or orbital. Since s orbitals have high electron density at the nucleus, the hyperfine coupling constant will be large and since s orbitals are symmetrical, it will be independent of direction. This interaction is called the isotropic hyperfine coupling (Aijo) or the Fermi contact interaction. In p or d orbitals, where there 

In a strong magnetic field, the electron and nuclear spin vectors, 1 and S are fully decoupled and each have their axis parallel to the applied field. If 6 is the angle between the axis of the dipoles (Fig. 5) and the line joining them, and r is their separation, then the Hamiltonian representing the energy of dipolar interaction is 

When Eq. 13 is applied to the electron wave function by substituting Mj for I and Mg for , the energies of the levels Mj, Mg are given by 

For an electron in an orbital centered on the nucleus in question (Fig. 6), the anisotropic hyperfine coupling follows from Eq. 14, but having in addition a term 3cos a — 1 which represents the average direction of the electron spin vector within the orbital. The anisotropic hyperfine splitting can now be defined as the separation between adjacent energy levels, viz. Mg, Mj and Mg, and equals 

When 0 = 0, the orbital is parallel to the nuclear magnetic moment which itself will be parallel to the external magnetic field if the latter field is much stronger than the field at the nucleus due to the electron (see Fig. 6). This is referred to as the strong-field approximation and it follows that for a p orbital, AanUo ( z ) = f-f -iid when 0 = 90°, Aaniso A x,m) = -jP- The tensor thus has the form - P,- P,- P, the principal value being positive. 

For those familiar with NMR spectroscopy it may be helpful to realize that the ESR g-shift is comparable with the NMR chemical shift. Similarly, electron-nuclear hyperfine coupling can be compared with nuclear-nuclear spin-spin coupling in NMR. (In systems containing more than one unpaired electron per molecule, electron spin-electron spin coupling is, of course, important. For doublet-state radicals, this coupling does not arise it is of great importance in triplet state molecules and in many high-spin transition metal complexes.) 

(— 1 + 1 /2), (— 1 — 1/2) from which both coupling constants can be derived. Since the magnetic moment of the electron is orders of magnitude greater than those of nuclei these splittings are generally small compared with those caused by the electron. 

The link between H coupling in NMR and ESR can be seen by comparing the ESR spectrum for C(CH3)3 radicals and H-C(CH3)3 molecules (looking at the unique proton only) - for the former, the methyl groups rotate rapidly, and the electron sees 9 equivalent protons, giving a 10-line ESR spectrum of binomial relative intensities. Similarly, the unique proton couples to the 9 methyl protons, giving a similar 10-line NMR spectrum. 

The task of interpreting complex isotropic ESR spectra resembles 

This isa3 x 3 x2 = 18 line spectrum and as can be seen (Fig. 3.6) accounts for 1/3 of the spectrum. So lines 1 - 18 are fully accounted for. Measuring from line 1 to line 19 gives a4, and this repeats as a 1 1 1 pattern. Hence it must be due to l4N nuclei (/=1). Each 14N component comprises the same 18 line pattern, giving the total of 54 lines observed. 

The electronic energy levels for the one electron one nucleus system are given quantitatively by the equation 

Coupling of the spin, which is largely localised on the oxygen nucleus, with the nitrogen nucleus (I = 1) results in the characteristic 1 1 1 triplet. Splitting from the protons in the molecule is not resolvable. 

If the nuclei were equivalent protons with aA = aB, then the resultant energy levels would be as shown in Fig. 6. The observed spectrum reveals a triplet of intensities 1 2 1. The extra intensity of the middle line, which arises from the transition (Af = — M, = 0) to (Af, =, Mt = 0), is because the transition is between doubly degenerate energy levels (AfA = +MB = -, MA = —, Mb = +whereas the outer lines represent transitions between non-degenerate nuclear energy levels. Generally, for an electron interacting with n nuclei of spin, the number of ESR transitions observed is equal to 

The examples considered above are relatively simple, but most free radicals contain several magnetic nuclei. The hyperfine coupling from radicals with many magnetic nuclei will produce ESR spectra which are rich in hyperfine lines. The analysis of such spectra can be relatively straightforward and gives useful information regarding the identity of the radical or radicals involved in an electrode reaction. 

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Determination of relative signs of isotropic hyperfine coupling constants J. Chem. Rhys. 63 3515-22... 

The radical cation of 1 (T ) is produced by a photo-induced electron transfer reaction with an excited electron acceptor, chloranil. The major product observed in the CIDNP spectrum is the regenerated electron donor, 1. The parameters for Kaptein s net effect rule in this case are that the RP is from a triplet precursor (p. is +), the recombination product is that which is under consideration (e is +) and Ag is negative. This leaves the sign of the hyperfine coupling constant as the only unknown in the expression for the polarization phase. Roth et aJ [10] used the phase and intensity of each signal to detemiine the relative signs and magnitudes of the... 

The CIDNP spectrum is shown in figure B 1.16.1 from the introduction, top trace, while a dark spectrum is shown for comparison in figure B 1.16.1 bottom trace. Because the sign and magnitude of the hyperfine coupling constant can be a measure of the spin density on a carbon, Roth et aJ [10] were able to use the... 

For example, if the molecular structure of one or both members of the RP is unknown, the hyperfine coupling constants and -factors can be measured from the spectrum and used to characterize them, in a fashion similar to steady-state EPR. Sometimes there is a marked difference in spin relaxation times between two radicals, and this can be measured by collecting the time dependence of the CIDEP signal and fitting it to a kinetic model using modified Bloch equations [64]. 

Force constant (vibrational k Hyperfine coupling constant a, A... 

The microwave spectrum of isothiazole shows that the molecule is planar, and enables rotational constants and NQR hyperfine coupling constants to be determined (67MI41700>. The total dipole moment was estimated to be 2.4 0.2D, which agrees with dielectric measurements. Asymmetry parameters and NQR coupling constants show small differences between the solid and gaseous states (79ZN(A)220>, and the principal dipole moment axis approximately bisects the S—N and C(4)—C(5) bonds. 

The EPR spectra of a number of bridgehead radicals have been measured and the hyperfine couplings measured (see Section 12.2.3). Both the and couplings are sensitive to the pyramidal geometry of the radical." " The reactivity of bridgehead radicals increases with increased pyramidal character." ... 

Analyze the hyperfine coupling in the spectrum of the butadiene radical anion given in Fig. 12.PI I. What is the spin density at each carbon atom according to the McConnell equation ... 

Gaussian computes isotropic hyperfine coupling constants as part of the population analysis, given in the section labeled "Fermi contact analysis the values are in atomic-units. It is necessary to convert these values to other units in order to compare with experiment we will be converting from atomic units to MHz, using the following expressions ri6ltYg ... 

Compute the isotropic hyperfine coupling constant for each of the atoms in HNCN with the HF, MP2, MP4(SDQ) and QCISD methods, using the D95(d,p) basis set Make sure that the population analysis for each job uses the proper electron density by including the Density=Current keyword in the route section. Also, include the 5D keyword in each job s route sectionfas was done in the original study). 

The electron densities for a spin electrons and for spin electrons are always equal in a singlet spin state, but in non-singlet spin states the densities may be different, giving a resultant spin density. If we evaluate the spin density function at the position of certain nuclei, it gives a value proportional to the isotropic hyperfine coupling constant that can be measured from electron spin resonance experiments. 

Table 18.2 Hyperfine coupling constant for a hydrogen atom when the Is orbital is represented as an uncontracted sum of n primitives...

Table 18.3 illustrates some salient points for ROHF calculations. In each case, I optimized the geometry before calculating the hyperfine coupling constants. 

Table 18.3 Hyperfine coupling constants/mT for the vinyl radical. ROHF calculations...

Barone also introduces two new basis sets, EPR-Il and EPR-llI. These are optimized for the calculation of hyperfine coupling constants by density functional methods. EPR-Il is a double zeta basis set with a single set of polarization functions and an enhanced s part. EPR-III is a triple zeta set including diffuse functions, double d polarization functions and a single set off functions. 

In the following, all isotropic hyperfine coupling constants were calculated using the BLYP functional and the EPR-II basis set. A full geometry optimization was done in all cases. 

The effect of vibrational averaging is particularly significant for the carbon hyperfine coupling constant. 

In standard Hiickel n -electron theory, the highest occupied orbital has a node through the position of C3 and so we might expect a zero proton hyperfine coupling constant, even after using McConnell s argument. 

The first derivative is the hyperfine coupling constant g (as measured by ESR), the second derivative with respect to two different nuclear spins is the NMR coupling constant, J (Planck s constant appears owing to the convention of reporting coupling constants in Hertz, and the factor of 1/2 disappears since we implicitly only consider distinct pairs of nuclei). 

The and operators determine the isotropic and anisotropic parts of the hyperfine coupling constant (eq. (10.11)), respectively. The latter contribution averages out for rapidly tumbling molecules (solution or gas phase), and the (isotropic) hyperfine coupling constant is therefore determined by the Fermi-Contact contribution, i.e. the electron density at the nucleus. 

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