As a spectroscopic method, nuclear magnetic resonance (NMR) has seen spectacular growth, both as a technique and in its applications. Today’s applications of NMR span a wide range of scientific disciplines, from physics to biology to medicine. Each volume of Nuclear Magnetic Resonance comprises a combination of annual and biennial reports which together provide comprehensive coverage of the literature on this topic. This Specialist Periodical Report reflects the growing volume of published work involving NMR techniques and applications, in particular NMR of natural macromolecules.

Nuclear Magnetic Resonance Volume 36

A Review of the Literature Published between June 2005 and May 2006

By G.A. Webb

The Royal Society of Chemistry

Copyright © 2007 The Royal Society of Chemistry
All rights reserved.
ISBN: 978-0-85404-362-0

Contents

Preface G. A. Webb, 7,
NMR books and reviews W. Schilf, 22,
Theoretical and physical aspects of nuclear shielding Cynthia J. Jameson and Angel C. de Dios, 50,
Applications of nuclear shielding Shigeki Kuroki, Naoki Asakawa and Hidekazu Yasunaga, 72,
Theoretical aspects of spin–spin couplings Hiroyuki Fukui, 113,
Applications of spin–spin couplings Krystyna Kamienska-Trela and Jacek Wójcik, 131,
Nuclear spin relaxation in liquids and gases R. Ludwig, 170,
Solid-state NMR spectroscopy A. E. Aliev and R. V. Law, 196,
Multiple pulse NMR Daniel Nietlispach, 244,
NMR of proteins and nucleic acids S. J. Matthews, 262,
NMR of carbohydrates, lipids and membranes Elizabeth Hounsell, 285,
Synthetic macromolecules Hiromichi Kurosu and Takeshi Yamanobe, 309,
NMR in living systems Malcolm J. W. Prior, 344,
Nuclear magnetic resonance imaging By Tokuko Watanabe, 363,
NMR of liquid crystals and micellar solutions Maura Monduzzi and Sergio Murgia, 397,

CHAPTER 1

Theoretical and physical aspects of nuclear shielding

Cynthia J. Jameson and Angel C. de Dios

DOI: 10.1039/b618338g

1. Theoretical aspects of nuclear shielding

1.1 General theory

Relativistic effects on molecular magnetic properties, in particular the nuclear magnetic shielding tensor, can be significant in molecules containing heavy atoms. Many approaches have been put forward. One approach is to append the spin-orbit interaction to the non-relativistic theory. Nakatsuji et al. presented an ab initio UHF formalism for calculating the spin-orbit (SO) effect without electron correlationand applied it to many systems,” while Malkin et al. presented a density functional theory (DFT) formalism, and Vaara et al., on the other hand, used multi-configuration self consistent field (MCSCF). While these works showed the importance of including the spin orbit effects, particularly to understanding the observed ‘normal halogen dependence’, this approach clearly is insufficient for the nuclear shielding of heavy nuclei. Several methods have been proposed to reduce the four-component Dirac equation to two-component equations. The most successful two-component relativistic method to date starts with the no-pair formalism and external field projectors, as developed by Douglas and Kroll and Hess. Nakatsuji and co-workers used a two-component quasi-relativistic (QR) theory based on the Douglas-Kroll-Hess transformation, included the change of picture effect which ensures consistency with the Hellmann-Feynman theorem for the QR theory, adopted gauge-including atomic orbitals (GIAO) method for gauge origins, and incorporated electron correlation at the MP2 level in this reporting period. Fukui and Baba developed a two-component method that derived the expression for the nuclear shielding from the Douglas-Kroll-Hess (DKH) transformation of the no-pair equation for a molecule bearing a nuclear moment and is placed in a magnetic field; they used common origin coupled Hartree–Fock to begin with, later introduced GIAOs. In these earlier works, the finite perturbation method was used to compute nuclear shielding values. In this reporting period, Kudo and Fukui derived expressions for the shielding tensor by analytically differentiating the electronic energy of a system based on the Douglas-Kroll-Hess approach.

Earlier, Fukui, Baba and Inomata had derived a two-component formalism using the Breit-Pauli approach; the magnetic vector potential is added to the canonical momentum in the Breit-Pauli Hamiltonian in order to get a gauge-invariant scheme up to order c-4. In the process, a mass correction term in the shielding became apparent for the first time, arising from a second order expression containing the Fermi contact terms and the kinetic energy operators. The same approach was carried out more completely by Manninen et al., avoiding some of the approximations introduced by Fukui et al.

Another alternative two-component approach to NMR properties begins with the zeroth order regular approximation (ZORA) Hamiltonian; an external magnetic field and nuclear moment are introduced. Wolff and Ziegler derived the equations for calculating the nuclear shielding tensor in the ZORA formulation and implemented them within an existing density functional program for which the ZORA part had already been developed by van Lenthe et al. This version of the two-component approach has been applied to many molecular systems involving heavy atoms by Ziegler and others. Fukui et al. have also derived a ZORA version, but without electron correlation.

A third approach which leads to a two-component method of calculating nuclear shielding is the linear response within the elimination of small components (LR-ESC), introduced by Melo et al. They start with a four-component Rayleigh-Schrodinger perturbation theory formalism, and construct a two-component theory by using the elimination of small component scheme, thereby obtaining formal expressions for operators previously neglected. They arrive at expressions similar to Fukui et al. plus some additional terms correcting both the diamagnetic and paramagnetic parts of the shielding. Thus, the formal relation between the Breit-Pauli approach of Fukui et al. and the LR-ESC approach of Melo et al. had been shown.

Yet another formalism is Kutzelnigg’s minimal coupling approach. He developed two-component expressions of magnetic properties starting from the Dirac equation in the presence of a magnetic field and using direct perturbation theory. By carrying out a double perturbation expansion (in c-1 and in the magnetic vector potential) he arrived at formal perturbative expressions based on the Levy-Leblond Hamiltonian. Are these two-component approaches equivalent to each other? It is difficult to ascertain the answer to this question, in part because basis set incompleteness errors do not impact contributions in different approaches uniformly and in part because two-body contributions which are difficult to calculate are sometimes not included.

A full four-component treatment is to adapt Ramsey’s theory to the four-component Dirac equation. In a four-component context all relativistic corrections are included from the beginning. Within the random phase approximation (RPA) method, four-component relativistic calculations of nuclear shielding have been derived by Visscher et al. Four-component calculations have been carried out on small molecular system where it should be noted that refs. 30 and 31 have an error in having neglected positronic excitations in RPA calculations and refs. 28, 32 and 33 have used modest basis sets.

It seems natural to identify the paramagnetic contribution as that originating in operators linear in the magnetic vector potential A, and to identify the diamagnetic contribution as that from operators linear in A. With this definition, various approaches to the calculations of relativistic effects on nuclear magnetic shielding yield different decompositions into para- and diamagnetic components of magnetic properties. There are different definitions of para- and diamagnetic contributions at the four component level yielding the same para- and diamagnetic separation in the non-relativistic limit. In a very useful paper in this reporting period, Zaccari et al. attempts to clear up the situation by analyzing the formal relations connecting the different para- and diamagnetic contributions of three approaches: the Breit-Pauli, the LR-ESC, and the minimal coupling approaches, as they were presented in refs. 19, 26 and 27, respectively. In the process, the equivalence of these three approaches within the elimination of the small component approximation is proven and verified numerically for the HX series of molecules (X = Br, I). Formal relations are presented proving the gauge origin invariance of the full relativistic effect on the nuclear shielding tensor within the LR-ESC approach. Only one-body terms are included in the analysis by Zaccari et al.

Finally, these three equivalent approaches, represented by the Breit-Pauli approach, can be compared to the Douglas-Kroll Hess, and three other two-component methods, and also with the non-relativistic values and the benchmark full 4-component Dirac-Fock with an analytical linear response approach. Table 1 is a comparison of the calculated results for the isotropic nuclear shieldings for noble gas atoms, HX (X = F, Cl, Br, I), and H2X (X = O, S, Se, Te) molecules. Presented in Table 2 are the anisotropies of the shielding in HX (X — F, Cl, Br, I), and H2X (X = O, S, Se, Te) molecules. Kudo and Fukui also developed a second expression based on the method of Barysz-Sadlej-Snijders, in which the off-diagonal block terms in the transformed Dirac Hamiltonian are completely eliminated with respect to purely electrostatic nuclear attraction potential and the magnetic vector potential A. This is an infinite order two-component DKH method (IOTC). They carried out calculations using the same basis sets for the same molecular systems, and these results are included in Tables 1 and 2 with the label IOTC. In addition to these two-component methods already discussed above, Filatov and Cremer recently introduced a normalized elimination of small component (NESC) method which Kudo et al. have extended to include magnetic interactions. Filatov and Cremer have developed the regular approximation to the normalized elimination of the small component (NESC) to the Dirac equation and this was used as a starting point for the development of a second order regular approximation to NESC, with the acronym NESC-SORA. The NESC method corresponds to the projection of the Dirac Hamiltonian onto a set of positive-energy (electronic) states, which guarantees its variational stability. NESC-ZORA and NESC-SORA can easily be implemented in any non-relativistic quantum mechanical program. Both methods have been applied by Kudo et al. to the same set of molecular systems and these results are compared with the other methods in Tables 1 and 2. Incidentally, what is typically called ZORA is the lowest order regular approximation to the equation for the unnormalized elimination of the small component.

We note that of the two-component methods, those which gave the best agreement with the benchmark four-component calculations are the IOTC, NESC-ZORA and NESC-SORA methods. The Breit-Pauli, which has been shown to be formally analytically equivalent to the LR-ESC, gives a poorer result for the full shielding tensor, as shown by the too large anisotropies of I in HI and Te in HTe, for example. The Douglas-Kroll-Hess fails to give a good account of the anisotropy of the shielding of both I and H nuclei in the HI molecule. The last three columns in Tables 1 and 2 seem to indicate that the two component methods can give results which are reasonably close to the four-component calculations. These are all large-basis-set calculations which do not include electron correlation.

Correlated relativistic calculations are, so far, available in the form of DFT-ZORA and the recently introduced DKH-MP2 by Fukuda and Nakatsuji. The latter has been applied to Te shielding calculations, using only modest basis sets, however, and it is not clear whether all relativistic terms have been included. For example, the non-relativistic and DKH isotropic shielding values calculated for Te in H2Te were, respectively 3644 and 5094 ppm which can be compared to the values 3463.5 and 4569.4 ppm in the Table 1. There have been other correlated relativistic calculations such as those using MCSCF functions, but which only included spin orbit contributions. A new two-component method has been developed which solves the Dirac equation at the Kohn-Sham level of theory using a basis for the large component only and avoids some of the approximations used in the standard ZORA method. This is related to the widely used ZORA approach in that it is an unnormalized elimination of small component method, and that it is density functional theory based rather than ab initio. This new approach, labeled DKS2-RI (Dirac-Kohn-Sham with resolution of identity approximation), employs the resolution of identity approach to reformulate the basic Dirac-Kohn-Sham equations before elimination of the small component. No picture-change problems arise in electronic property calculations by this method. The first applications of the method are to calculations of hyperfine constants for atoms. The differences between ZORA and this method are small for Cu and Ag and larger for Au: 6750 vs. 6737; -1909 vs. -1967; and 3134 vs. 2986 all in MHz for ZORA vs. DKS2-RI methods, respectively. The method has also been applied to calculations of hyperfine constants of Hg in HgH, HgF and HgCN. The new DKS2-RI method gives uniformly smaller values than ZORA, and the agreement with experiment is worse for DKS2-RI, except in the case of HgH and Au. Spin–orbit contributions to the hyperfine tensor can be substantial and in the examples calculated to date, it looks like they tend to decrease the magnitude of the isotropic contact hyperfine, while the dipolar part of the hyperfine tensor remains the same. DFT calculations of the 13C hyperfine tensors in metallocenes with and without inclusion of spin-orbit contributions reveal that, depending on the metal atom, the calculated 13C contact hyperfine is 74 to 84% of that obtained without including SO. In a particular nitroxide radical where both signs of contact hyperfine are observed, the magnitudes of the 13C contact hyperfine of the eight inequivalent carbons are all smaller when SO is included in the calculations.

As discussed in vol. 34 of this series, the remarkable experimental values of the shielding tensors measured for the XeF2 molecule provide the clear indication that insights we have derived from mathematical identities which relate one molecular electronic property to another based on non-relativistic Hamiltonians are completely overturned in systems involving very heavy atoms, since the same identities do not hold in the relativistic case. The DFT-ZORA calculations which accompanied the experimental report gave anisotropies of 4469 ppm and 277 ppm for Xe and F nuclei. A recent IOTC-CHF (see above for description of this method) calculation by Kudo et al. gave 4276 and 115 ppm, which can be compared with the experimental values 4245 ppm and 150 ppm, respectively. The isotropic shielding values for Xe and F nuclei are, respectively, 3007 and 339 ppm from DFT-ZORA while they are 3570 and 415 ppm from IOTC-CHF. Differences may be due to neglect of electron correlation in the one case and the real differences between t he unnormalized elimination of small component ZORA method compared to the infinite order two component formalism based on the Barysz-Sadlej-Snijders method.

The search for better approximations to exchange correlation functionals for use in DFT calculations continues in this reporting period. What is the optimum admixture of exact exchange? To find this, hybrids, such as Becke3, have been parametrized on sets of thermochemical data, properties which are directly associated with the total energy. On the other hand, it has been found that relatively large admixtures of exact exchange give better values in shielding calculations. Another approach is to convert non-local and non-multiplicative hybrid exchange–correlation potentials into local and multiplicative Kohn-Sham potentials. To do this, Wilson and Tozer used the iterative approach of Zhao, Morrison and Parr (ZMP), to convert a given electron density to a multiplicative KS potential, which was then applied to the calculation of nuclear shieldings. The advantage of this method is that electron densities derived from any, even non-DFT methods, can be used as the starting point. On the other hand, instead of this ZMP approach, Arbuznikov and Kaupp construct the Kohn–Sham potential by applying the so-called localized Hartree–Fock approximation to the optimized effective potential method. The resulting localized hybrid exchange–correlation potentials provide improved performance at very nearly the same amounts of admixture of exact exchange (close to 0.5) and nearly independent of which standard DFT functional is used (LYP, or PW91, or PBE). These conclusions are based on results for nuclear shielding in 22 light main-group molecules. Furthermore, they find that an adaptable formulation of local hybrid functionals with position-dependent exact exchange admixture enhances flexibility. At every point in space, a local mixing function determines the amount of exact exchange admixture. This has been tested with thermochemical properties but remains to be tried on shielding calculations. Actually, the concept of position-dependent local mixing functions had been proposed first by Becke when he proposed his hybrid functionals. The performance of CAM-B3LYP, a hybrid exchange correlation energy functional based on the Coulomb-attenuating method (CAM), is compared with standard B3LYP.While it gives improvement in long-range properties, for isotropic nuclear shielding the results are comparable to B3LYP.

It has been demonstrated with self-consistent field calculations that the basis-set dependence of magnetic properties can be characterized by a parameter τ, that describes the deviation of the basis set from completeness in the exponent range (which corresponds to distance from the nucleus) relevant for the property of interest. The completeness profile concept introduced by Chong is particularly useful for molecular properties that may be dominated by phenomena occurring close to atomic nuclei, which are then best described by high-exponent basis functions, or for those dominated by the valence regions farther away from nuclei, which are best described by diffuse basis functions. Based on this a scheme for designing Gaussian-type orbital basis sets is proposed. Instead of optimizing the parameters of the basis constructed at the outset so as to be complete for a range of exponents relevant for the specific property. This has been illustrated for nuclear shielding; completeness profiles of existing basis sets and code for generating completeness-optimized basis sets are made available at http://www.chem.helsinki.fi/~Bmanninen/kruununhaka.

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