Theoretical and Physical Aspects of Nuclear Shielding

BY CYNTHIA J. JAMESON

A. General Theory — The general concept of nuclear shielding in the presence of electromagnetic radiation is introduced by defining dynamic electromagnetic shielding tensors which describe the linear response in an external spatially uniform periodic electromagnetic field. The diamagnetic terms do not depend on the angular frequency, ω, of the electromagnetic radiation. The dynamic paramagnetic shielding tensor is a generalization of Ramsey's definition for the static property.

The magnetic field induced by the electrons at nucleus I is expressed by Lazzeretti and Zanasi as follows:

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The external magnetic field B is the real part of B0 exp(iωt). [??] is the partial derivative with respect to time. In the first term, σP is the dynamic paramagnetic shielding tensor,

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Zn which L and MI involve the usual angular momentum operators [??]i (RI) centered on I and [??]i at the gauge origin.

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and |a> and |j> are the time-independent perturbed states which are functions of B. o is the same as Ramsey's, and the [??]I(ω) term is called magnetoelectric shielding. Its physical meaning is shown in the equation; by taking the scalar diadic product with the time derivative of external electric field, one obtains the magnetic field induced at the nucleus. The terms in [??]P and λI, give the magnetic fields induced at the nucleus by a time-dependent magnetic field and an external electric field, respectively. λI(ω) is the analogous definition to σP(ω), except that L is replaced by -2mcR, R being [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. [??]P(ω) takes the imaginary part, whereas σP has the real part of the complex integrals MI|j>L|a> and [??]I takes the imaginary part, while λI has the real part of the complex integrals MI|j>R|a>. The generalized treatment shows fundamental relationships among the electromagnetic properties of molecules and the sum rules obeyed by them.

The dispersion (dependence on ω) of the paramagnetic nuclear shielding in H2O has been calculated and found to be significant. For example, for 17O in H2O, in ppm, with the gauge origin at the center of mass,

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a downfield shift of 184 ppm! This theory predicts a frequency dependence of the dimensionless chemical shifts in NMR spectroscopy, analogous to the frequency dependence (dispersion) of the electric dipole polarizabilities.

Is there a possibility of measuring this dispersion of nuclear shielding by conventional NMR spectroscopy? The formalism developed by Lazzeretti and Zanasi is for a magnetic field B, which is the real part of B0 exp(iωt). In the NMR experiment the resultant applied magnetic field has a steady component B0 (static uniform field) along the laboratory z axis and a component of amplitude B1 rotating in the xy plane with angular velocity ω:

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The radiation's magnetic field amplitude is related to the radiation intensity (related to rf transmitter power). Since |B1| is usually orders of magnitude smaller than B0, the resultant B is very nearly equal to B0 and very nearly along the z axis. Therefore, we could write the relevant part of Binduced as

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or

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NMR spectrometers are now routinely available for observing 17O at 67.8 MHz (500 MHz proton), ω = 1 a.u. corresponds to an energy of 1.0 hartree or ω/2π = 6.58 × 1015 Hz. For ω = 0.3 a.u. or 1.97 × 1015 Hz, the calculated change is [σP (ω= 0.3 a.u.) - σP (ω = 0)] = 184 × 10-6. For the time being let us assume that the dispersion is linear (linear dependence of σP on ω). Then the calculated change [σP(ω) - σP(0)] corresponds to a shielding change of 184 × 10-6 (67.8 × 106/1.97 × 1015) (|B1|B0). This is far too small to detect for 17O in H2O. More favorable examples might be molecules having low-lying magnetic-dipole-allowed transitions from the ground state, that is, molecules with very large temperature-independent paramagnetism, such that ωja is small and closer to radiofrequencies. The frequency dispersion of the nuclear shielding could be observed if ωres is of the order of 1013 - 1014 Hz, that is, using B0 fields a factor of about 104 stronger. At such high fields, however, the higher order terms in

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should be taken into consideration, as proposed by Ramsey, i.e., the experiment would no longer be in the first-order Zeeman regime (see this series, Vol. 9, Chapter 1).

Another interesting development in fundamental theory of nuclear shielding has to do with the parity non-conservation (PNC). The PNC contribution to the nuclear magnetic shielding tensor has been derived, based upon a transposition of the Ramsey theory using a molecular hamiltonian including PNC terms. There is no first-order contribution to a from VPNC. The only second-order PNC contribution comes from the molecular hamiltonian Larmor frequency term in B0 • L and VPNC. This contribution is a nine-component second-rank tensor, just as the parity-conserving shielding tensor derived by Ramsey. The magnitude of the PNC contribution increases roughly as Z2. What this means is that a high Z nucleus in the right- and the left-handed optical isomers of an optically active molecule will have different intrinsic nuclear shielding. The NMR splitting which could be observed in two molecules of opposite chirality, owing to parity non-conservation, has been calculated for T1 in three compounds. The calculated splitting is 0.3 × 10-3 to 1.1 × 10-3Hz at 288.5 MHz 205Tl (i.e., 500 MHz proton). This is too small to detect. The chirality-dependent shielding of a sensor nucleus by a chiral perturbing group in the long-range...