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 limit, at a sufficiently large separation that there are no geometric correlations between a chiral solute molecule and a chiral solvent molecule, has been reviewed previously and likewise been shown too small. Of course, there are larger well-known shifts between two molecules of opposite chirality in a chiral solvent due to differing geometry-imposed solute-solvent interactions. There is also the intramolecular counterpart of this. A carbon nucleus influenced by two chiral centers in the same molecule may experience different environments. These effects are not averaged even under unrestricted internal rotation.

Reviews of nuclear shielding calculations have appeared.

B. Ab Initio Calculations. — The nuclear shielding differences between gaseous metal anions and neutral atoms were calculated for the alkali series Li through Cs by combining Hartree-Fock calculations with electron-electron correlation corrections. The latter was obtained by applying the Hellmann-Feynman theorem to the nuclear charge-dependence of the correlation energy,

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For the shielding difference σ(M-) - σ(M),

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where Icorr (Z) is the contribution arising from electron correlation, to the ionization potential for the removal of an electron from the system of nuclear charge Z and isoelectronic with M- to yield a system isoelectronic with M.

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Relativistic contributions to the ionization potential have to be included, expecially for the heavier alkali, and the last term is the leading correction from nuclear motion for nuclear mass mM. The numerical results are shown in Table 1.

These results are for isolated neutral atoms and ions. Comparison with experiments has been made only in solutions where the M- anion is stabilized by various solvents. The observed differences σ(M-)soln - σ(M+)aq are compared with the σ(M)g - σ (M+)aq in order to obtain σ(M-)soln - σ(M)g. The latter are compared with the calculations to show that Na- interacts very weakly with its surroundings [(i.e., σ (M-)soln - σ(M)g ≈ σ(M- )g - σ(M)g], whereas Rb- and Cs- are largely deshielded in solution relative to gaseous M- anions. MAS NMR spectra reveal that the Na+ cation in NaBPh4 crystal is very similar to gaseous Na+ ion, whereas large deshielding is observed in a number of inorganic salts, showing a range of about 60 ppm. The shifts depend on variety of ligand molecules (H2O, ether, or carbonyl) and Na–O interatomic distances, but are independent of the type of counter ion.

31P nuclear shielding calculations in PH3, P4, P2, PN, PF3, PF4+, PF5, PF6-, and PO43- have been reported. A basis set which is triple zeta plus two sets of d polarization functions (66211/6211/11) = [5s, 4p, 2d] appears to be adequate for the series HCl, H2S, PH3, SiH4 when the GIAO-FPT method is used. Comparisons with other theoretical calculations and experiment are shown in Table 2.

The GIAO-FPT calculations using this intermediate-size basis set give very good agreement with experiment, at least as good as the earlier common-origin CHF. [TEXT INCOMPLETE IN ORIGINAL SOURCE] calculations using larger basis sets." The latter method has been applied to P shielding in PF3, PF4+, PF5, PF6-, PO43-, P4, P2, and PN. The results are shown in Tables 3 and 4. The molecules P2 and P4 are at the extremes of the P shielding scale, only the free P atom (961.1 ppm) is known to be more shielded than P4.

Calculations of 29Si shielding in molecules which are typical counterparts of the CH4, C2H6, CH2=CH2 and H2C=O molecules indicate that the trends in σ for the Si compounds are much like that observed for the carbon compounds, as shown in Table 5.

The 29Si shielding tensor components from these conventional CHF calculations using large basis sets, gauge origin at Si, are shown in Table 6.

The sagging pattern observed in the 29Si shielding in the series SiHnF4-n has been reproduced by conventional CHF calculations:

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The basis set used in these calculations is of the same quality as in Refs. 32 and 39. With the Si nucleus as the gauge origin, the diamagnetic contribution increases linearly with F substitution in going from SiH4 to SiF4, but the paramagnetic contribution is a curve with the highest (smallest negative) value at SiH4 dropping markedly with the first F substitution and changing only slightly with additional replacement of H with F atoms. This trend in the paramagnetic contribution has been previously predicted by Radeglia et al., based on semiempirical calculations and analyses of the dependence of 29Si shielding on effective electronegativities or electron densities.

Conventional CHF calculations, using large polarized basis sets, of F shielding in PF4+, PF3, PF5, PF6-, SiF4, and BF3 have been reported. The results are shown in Table 7.

In this review period several papers have appeared which report IGLO calculations of the 13C shielding tensor elements in various compounds, comparisons of these with experiment, and the analyses of the individual MO contributions to the shielding which are standard output of IGLO calculations. By rotating the shielding tensor from the molecular frame in which they are calculated, into a local bond frame in which one component is selected along the bond, the electronic structural basis of C nuclear shielding can be studied. This is a technique which was used earlier for comparing F shielding tensor elements in the CF3X and CHnF4-n series o molecules, where X = H, F, Cl, Br, I. The rotated tensor elements parallel to the C–F bond, in the CFX plane and perpendicular to the CFX plane could then be compared directly in molecules of different symmetry. The 13C shielding analyses went even further By rotating the shielding tensor into a local bond frame with one component along the bond, the explicit structural dependence of the contributions to the shielding are eliminated, thereby exposing detailed information on the dependence of the shielding tensor on the nature of each bond directly attached to the nucleus in question. Comparing "diamagnetic" and "paramagnetic" contributions from individual bonds directly attached to the 13C reveals very useful qualitative trends, some of which would otherwise not have been noticed.

Since multiple local gauge origins are used when gauge factors are associated with each MO in the IGLO method, the partitioning into "diamagnetic" and "paramagnetic" terms, although well-defined in the context of the method, does not have a simple or known relationship with the diamagnetic and the paramagnetic parts of the shielding which are obtained when a unique gauge origin is used in the conventional CHF calculation. The practical advantage of local origin methods such as GIAO, IGLO, and LORG (These methods were discussed in the previous volume of this series.) lies in that they effectively leave out contributions which are of equal or nearly equal magnitude in both the conventional diamagnetic and paramagnetic parts, and in so doing lead to an effective damping of basis set errors resulting from use of a finite basis set. For example, "diamagnetic" terms for C in IGLO calculations in CH4, C2H6, H2C=CH2, HC [equivalent to] CH, CH3OH, CH3NO2, CH3F, and CH3SH, and likewise for the methine carbon in isobutane, bicyclo[l.1.1] pentane, cubane, and tetrahedrane, show contributions from C–C and C–H bonds which are almost spherically symmetric and also nearly constant, so that all of the chemical shift effects, as well as the shielding anisotropy, originate in the "paramagnetic" terms. This is in sharp contrast with the anisotropic diamagnetic terms calculated using a unique origin. For example, the diamagnetic terms for C1 in H2C=CH2, with origin at C1, are 355.7 ppm, 340.6 ppm, 293.1 ppm for the xx, yy, zz components, respectively, with z axis along C=C and X axis perpendicular to the molecular plane.