Recent progress in understanding chemical shifts

SOLID STATE Nuclear Magnetic Resonance

ELSEVIER Solid State Nuclear Magnetic Resonance 6 (1996) 101-125

Review

Recent progress in understanding chemical shifts 1

Angel C. de Dios, Eric Oldfield *

Department of Chemistry, University of Illinois, 600 S. Mathews Avenue, Urbana, IL 61801, USA

Received 18 January 1995; accepted 16 August 1995

Abstract

In the past three or four years computer hardware and software developments have reached the stage where the nuclear magnetic resonance (NMR) spectra of many molecular systems can now be accurately evaluated. Detailed analysis of chemical shifts may soon become a routine part of solid (and liquid) state NMR structure prediction in chemistry and biology, and this Article covers the development of the topic from its earliest beginnings.

Keywords: Chemical shift; Electrostatics; Protein; Shielding; Tensor

1. Introduction

In 1949-1950, Knight observed large differences in nuclear magnetic resonance (NMR) frequency between a metal and its salt [l], and Dickinson 121, Lindstriim [3], Thomas [4], Gutowsky and Hoffman [5] and Proctor and Yu [6] observed large frequency shifts between different chemical compounds, the latter “chemical effects” being denoted chemical shifts. For physi- cists, chemical shifts were a source of both fasci- nation and mild annoyance since as Proctor and Yu stated “Until it is clearly understood, the accuracy of magnetic moments determined under certain chemical conditions remain somewhat in doubt”. For chemists, however, a cornucopia had

* Corresponding author. ’ This work was supported in part by the U.S. National

Institutes of Health (grants HL-19481 and GM-50694). A.C.D. is an American Heart Association Inc. (Illinois Affiliate) Postdoctoral Research Fellow (Grant FW-01).

arrived. In this Article, we trace some of the developments which have led to an understanding of how “chemical conditions” influence NMR resonance shifts. NMR spectroscopy is now one of the most important tools for characterizing the structures of organic, biological, and to a somewhat lesser extent, inorganic chemical systems, but only recently have computational capabilities advanced to the state where the chemical shifts of most molecules can be accurately predicted. If a resonance is detected, its chemical shift can be measured, and at least in principle, useful structural information can be derived. The availability of powerful workstations now greatly facilitates this task, and in what follows we describe theoretical developments, and recent applications to small molecules, as well as macromolecular systems, such as proteins.

In the next few years, it is likely that “turnkey” (hardware and software) packages will become available for routinely predicting the NMR spectra of most organic chemicals and natural prod-

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102 A.C. de Dies, E. Oldfeld /Solid State Nuclear Magnetic Resonance 6 (1996) 101-125

ucts - even systems as large as proteins and nucleic acids. While this might have been a somewhat speculative claim two or three years ago, recent developments support the idea that even protein spectra can now be successfully predicted [7-101, and current progress in the direction of deducing structure from experimental spectra by using quantum chemical methods [ll] offer even more promise in utilizing the chemical shift. Af- ter 45 years, “chemical effects” - at least for first and second row elements - can now be reliably evaluated, and given the amazing improvements in computer hardware price/performance ratios over the same time period (especially when contrasted with the same ratios for NMR spectrometers themselves), it seems likely that computational quantum chemistry will soon become a routine complement to experimental studies of NMR chemical shifts. In what follows, we trace some of the key developments since the first observations of “chemical effects”, and we begin by briefly reviewing work in days when Mflops were unheard of.

2. The early years

As noted early on [l-6], in the presence of an applied magnetic field the observed nuclear magnetic resonance frequency for a particular nucleus is influenced by its chemical environment. Nuclei are normally observed amidst a back- ground of a sea of electrons which are in continuous motion. The applied field induces electrical currents, which in turn induce a secondary local magnetic field at a given nuclear site. Hence, the actual field felt at the nucleus of interest, Z?Oc, becomes slightly different from the applied exter- nal magnetic field, Wxt, giving the familiar relation:

B’G(l-cr)BeX’ (I)

where u is a so-called screening or shielding constant. A positive value of (+ (shielding) thus results in a smaller Bloc, and a lower NMR resonance frequency, while a negative (T (deshielding) results in an increase in resonance frequency. Experimentally, one usually measures a chemical

shift, 6, which has the opposite sign (and a different reference state) to the shielding, u, but the two are simply related.

Now, in a solid, it is easy to see that the screening or shielding - of say a benzene ring or a CO group - will be orientation dependent, and the shielding is in fact more usefully represented as a second rank tensor quantity. Thus, Eq. (1) can be recast as:

BE = (1 - CQ)B;~ (4

where BE is the magnitude of the field felt at the site of the nucleus along the o-direction induced by electronic currents brought about by an applied magnetic field along the P-direction. The shielding is thus composed of nine independent components, although symmetry generally reduces this number. In the liquid and gas phase, or in solution NMR, the free Brownian motion of molecules usually averages the shielding tensor without preference to any particular orientation with respect to the applied field. Thus, what is observed is referred to as the isotropic shielding, which is simply an average of the principal components:

uisO = - :( fill + u22 + 933) (3)

Both the applied magnetic field, B, and the nuclear moment, CL, which acts like a tiny magnet, cause changes in the electronic wavefunctions and energies of the molecule, and these changes in energy due to simultaneous magnetic perturba- tions define shielding from a different, quantum chemical, perspective [12]:

a2E U 4

=- auw B B-O,p=O

(4)

where E is the total molecular eigenenergy. Being electronic in its origin, the history of the

theoretical analysis of the shielding property par- allels advances in evaluating E. Theories of magnetic shielding begin with Ramsey’s formulation [13], which made use of second-order perturbation theory. Upon collecting all terms in energy that are proportional to both p and B obtained after a perturbation treatment, Ramsey catego-

A.C. de Dies, E. Oldfield/Solid State Nuclear Magnetic Resonance 6 (1996) 101-125 103

rized two distinctly different terms which com- prise shielding: a diamagnetic term and paramagnetic term. The diamagnetic contribution, Us, derived from the first-order correction to the energy, involves only the ground state wavefunction, IO>:

Its evaluation is straightforward since Eq. (5) requires only knowledge of the positions of the electrons with respect to the nucleus of interest, rn, and with respect to an origin chosen for the magnetic vector potential, rO. The difficulty in calculating shielding lies however in evaluating the paramagnetic part:

up = - c 4

mainly because it requires both the wavefunctions, (q I, and energies, E,, of the excited states. Once again, since the phenomenon arises from a simultaneous perturbation, L, refers to the angu- lar momentum with respect to the nucleus of interest, while L, corresponds to the origin of the magnetic vector potential. Explicit in both Eqs. (5) and (6) is a gauge origin chosen for the magnetic vector potential. This origin can be placed anywhere, and in Ramsey’s original formulation, it was situated at the nucleus of interest (r. = r,). The shielding, being an observable, is independent of the choice of the gauge origin. However, its parts, the diamagnetic and paramagnetic terms shown in Eqs. (5) and (61, are both gauge-dependent, and much of the history of nuclear magnetic shielding calculations has re- volved around solving this so-called gauge problem.

As mentioned above, the primary challenge in shielding calculations lies in evaluation of the paramagnetic term. The first simplification, introduced by Ramsey, involved use of an average value of (E, - E,,) [14]. However, choosing an appropriate average value implies a knowledge of all of the excited states, which is a very difficult

problem to solve. Thus after Ramsey’s basic formulation, two improved approaches were introduced. In the first, it was suggested that since the paramagnetic term is gauge-dependent, it should be possible to choose a gauge origin in which the contribution from the sum over the excited states is at a minimum. Chan and Das [15] showed that by placing the gauge origin at the center of electronic charge, the excited state contribution is greatly reduced, and in the case of proton shielding in a variety of compounds, their results were very satisfactory. Kern and Lipscomb [16] also employed a gauge transformation in their proton shielding calculations, in which a cancellation of the AE term was again achieved by having the origin in proximity to the centroid of charge distribution.

A distinctly different approach made use of the variational method. Tillieu and Guy [17] represented the new wavefunction as V = pO(l + g . B), where g is a parameter with which the energy is minimized, then later Kolker and Karplus 1181 used a more elaborate function which took into account the nuclear moment: W = p(,,(l +f. B + g. p). The results obtained by using such variational methods were in fairly good agreement with experiment. However, as in the gauge transformation approaches described above, success was limited to proton shieldings only.

Prompted by the limited success of calculating chemical shieldings from the unperturbed wavefunctions, Stevens and coworkers formulated the first coupled Hartree-Fock method of shielding computation [19-231 in which both the magnetic field and nuclear moment were taken into account in the Hamiltonian. This generated wavefunctions correct to first order, and energies correct to second order (excluding configuration interaction and electron correlation arising from more than one determinant function), and the shieldings obtained by their methods for the heavier elements in a variety of diatomics, such as LiH, HF, F2, CO and ClF, were in good agreement with experiment. Large basis sets, however, were .essential in order to adequately describe the first-order wavefunctions, making it very difficult to apply the same methods to larger molecular systems.

104 A.C. de Dies, E. Oldfield/Soiid State Nuclear Magnetic Resonance 6 (1996) 101-125

3. Modem approaches

The coupled Hartree-Fock calculations described above follow exactly the formalisms of Eqs. (5) and (6), that is, contributions from all orbitals in the molecule are evaluated with respect to a single gauge origin, which is usually placed at the site of the nucleus in question. Both diamagnetic and paramagnetic terms are gauge- origin dependent, and a change in the choice of gauge origin leads to additional terms. In principle these additional gauge-dependent terms are exactly equal in magnitude but of opposite sign, rendering the total shielding gauge-invariant. In practice, the desired cancellation does not occur exactly due to the practical difficulties encoun- tered in calculating the paramagnetic term. The paramagnetic term involves electronic excited states, which can only be described adequately with (very large) basis sets which provide an am- ple set of virtual orbitals. In addition, molecular orbitals obtained from self-consistent field (SCF) calculations are normally optimized with respect to an origin within the region of interest, and assigning a common gauge origin is clearly unde- sirable for orbitals far from this origin. For these reasons, common origin calculations only yield accurate values for shielding in the limit of ex- tremely large atomic basis sets. Thus, only the smallest molecules are accessible using these methods.

In order to overcome the unrealistic dependence of the calculated shielding on the choice of origin, several methods have been proposed. The first one can be traced back to the early seminal work of London 1241, and introduces field dependence into the basis functions. These gauge-in- cluding atomic orbitals (GIAOs) differ from the field-independent basis by an exponential gauge factor:

&Jr, B) =&Jr) exp [ -i(B XR,) *r/2c]

(7)

which takes into account the value of the vector potential (B X R,) at the position of the centroid R,, of the atomic orbital &Jr). The incorporation of field dependence into basis functions for

shielding calculations was first demonstrated by Ditchfield 1251. The reduction in the gauge dependence of the calculated shielding values results from the fact that GIAOs closely resemble the first-order wavefunctions in the presence of a magnetic field (see e.g. Ref. [261X GIAOs no longer allow for an arbitrary choice of origin, and more importantly, having each orbital with its own gauge origin leads to a much more accurate computation of what one might refer to as a local paramagnetic term. Although GIAOs were ideal for achieving the desired gauge independence, the gauge factors also led to perturbation of the two-electron part of the Hamiltonian, necessitat- ing additional (time-consuming) two-electron integral calculations. This again restricted utility to small molecules, or big computers [27].

In the 198Os, Kutzelnigg [28] and Schindler and Kutzelnigg [291 proposed an alternative method, individual gauge for localized orbitals (IGLO), in which localized molecular orbitals are made field-dependent. The gauge origin for each localized molecular orbital is placed at its centroid of charge-analogous to previous ideas [15,16]. Immediately after its introduction, shieldings for more than a hundred compounds were calculated [30]. The results were very satisfactory, and since the shieldings were evaluated with localized ?rbitals, individual contributions from var- ious parts of the molecules could be obtained. As in GIAO, however, there was a price to pay for having more than one gauge origin, and in this case, so-called additional “resonance and ex- change” correction terms needed to be evaluated. Moreover, an integral transformation was also necessary to convert from an atomic orbital to a molecular orbital basis. Hansen and Bouman [31] thus proposed another method, which simply left out calculation of the large terms of opposite sign in the diamagnetic and paramagnetic contributions, which should ideally cancel in the limit of an infinite basis. By invoking commutation and closure relations in a localized orbital-local origin (LORG) formalism, the large errors associated with calculating the paramagnetic term in a finite basis set were avoided.

The IGLO and LORG methods are thus very similar, since they both involve localized molecu-

A. C. de Dios, E. Oldfield / Solid State Nuclear Magnetic Resonance 6 (19!?6) 101-125 105

lar orbitals, invoke closure relations, and require integral transformations. GIAO on the other hand, has to evaluate additional two-electron integrals, and does not provide insight regarding bond contributions to shielding. The massive amount of integral calculations in the GIAO method thus made it less attractive. However, by taking advantage of gradient theory [32], Wolinski et al. [33] formulated a much more efficient implementation of the old GIAO, making it about three times faster. In addition, semi-direct ver- sions of both GIAO and IGLO, in which two- electron integrals are computed on an “as- needed” basis, have recently been introduced [34,35], and promise to alleviate basis size con- cerns - albeit at the expense of increased computational time (loss of a factor of = 3).

Going beyond Coupled Hartree-Fock (CHF), while keeping the advantages gained from local origin methods, has also been of recent interest. The first successful results were reported in 1990 by Bouman and Hansen [36], who applied the second-order polarization propagator approxima- tion of Geertsen and Oddershede [37]. The new method, referred to as SOLO (second-order LORG) was shown to satisfactorily reproduce experimental shifts for 31P in small phosphorus- containing compounds [36], as well as 15N in azines, for example [38]. Implementation of Moller-Plesset theory with GIAOs has been made by Gauss [39], and very recently, van Wullen and Kutzelnigg incorporated multi-configuration SCF wavefunctions into the IGLO method [40]. Thus, all three local origin methods now have the option of going beyond Hartree-Fock-again albeit at the expense of an increase in computational time on the order of N, the number of basis functions.

Fortunately, in most cases it is not essential to describe all atoms in a molecule with large basis sets, since this adds tremendously to computational time, which varies between = N4 and (N log Nj2 for CHF calculations [41]. Thus, for accurate shielding calculations, larger basis sets can usually be restricted to the atom or small group of atoms which carry the nucleus of interest, while a significantly smaller set is provided for the other atoms. The amount of computational

time required, and disk storage space for integrals, are thus greatly reduced with such a “locally dense” basis [42], while the results remain satisfactory when compared with experiment, and calculations with a uniform basis. Use of either modern GIAO or IGLO methods on high performance workstations means that shielding in most molecular systems can now be confidently ad- dressed, and we present below numerous illustra- tions on systems of biomolecular interest.

4. Gas phase NMR and absolute shielding scales

Before we move to consider large molecular systems, we need to consider the accuracy of shielding calculations. Here, the proper evaluation of shielding calculations would not have been possible without reference to an extensive gas phase NMR literature ([43], and references cited therein). The reason for this is that shielding calculations are usually performed on an isolated molecule at its equilibrium geometry, and it is therefore essential to have the same conditions when determining the experimental values, if a valid comparison is to be made. With measurements of the spin-rotation coupling constants in molecular beams, and the demonstration of an intimate relation between the spin-rotation constant and the paramagnetic component of shielding [44], absolute shielding values (referenced to the bare nucleus) can be obtained, an essential prerequisite for testing the theoretical methods. Moreover, as soon as the absolute shielding of a particular nucleus in one molecule is known, an absolute shielding scale can be established for the same nucleus in other molecules by determining their chemical shifts with respect to the molecule whose absolute shielding value was obtained from the molecular beam study. There is one caveat, however, which is that the chemical shifts be free of intermolecular effects. Fortunately, gas phase NMR offers the ability to vary the density of a sample, which leads to a direct way of eliminating most intermolecular contributions to shielding.

The dependence of any molecular electronic property, such as shielding, on density in a pure

106 A.C. de Dies, E. Oldfield /Solid State Nuclear Magnetic Resonance 6 (1996) 101-125

0 100 200 300 400 500 600 700 800 900 1000

- bpmi

Fig. 1. Absolute shielding scale for phosphorus-31 (from Ref. 1461).

gas was first formulated by Buckingham and Pople [45] as:

(8)

where the shielding, (T, is expressed as a virial expansion in terms of the density, p. v~(-O(T> is the shielding in the isolated molecule at a given temperature, T, and is derived by an extrapolation to zero density using values obtained from samples of low to intermediate density.

Measurements of a,(T) in a variety of molecules have provided a means of establishing absolute shielding scales. These numbers, although they still contain intramolecular effects, are free of medium effects, and have enabled the establishment of absolute shielding scales for 13C, 15N, 19F, 29Si and 31P (see e.g. Ref. 1461, and references cited therein). An example of an absolute shielding scale (for 31P) is shown in Fig. 1. Shielding calculations for these nuclei have therefore been properly checked by comparison with experimental data obtained under conditions close to what the calculations have assumed. Of course, it is also highly desirable for ab initio methods to reproduce not only the measured absolute shieldings of an isolated molecule frozen in an equilibrium configuration, but also to evaluate the effects that come from dynamics, confor- mational changes, as well as intermolecular interactions, and we now discuss these topics in detail,

since these basic ideas are all of importance for understanding chemical shifts in macromolecules, both in solution and in the solid-state.

5. Intramolecular effects on shielding

The effects of geometrical parameters, such as bond lengths, bond angles and dihedral angles, on nuclear shielding are most readily studied in small molecules [47,48]. The simplest systems to study are diatomics, in which the only parameter that requires consideration is the internuclear separation. Initial theoretical work aimed at in- vestigating how shielding changes with internuclear separation near the equilibrium geometry was carried out by Stevens and coworkers on LiH [20], HF [21], CO [49] and N, [50], using common-origin CHF methods. It was noted that, except for the 7Li in LiH, shielding decreases with increasing internuclear separation, and em- ploying the GIAO method, Ditchfield verified these early results [51]. Chesnut and Foley then observed a systematic variation of the shielding derivative, au/at-, in the hydrides across the Peri- odic Table U.521, and references cited therein). For both first and second row hydrides, the shielding derivative starts out being small and positive for the alkali metals. There is a small increase for the alkaline earth hydrides, then a decrease for boron and aluminum, which continues all the way across the Periodic Table to the halogen hydrides, where the shielding derivatives are (generally) large and negative. The first general explanation for these trends was made by Jameson and de Dios 1531 who proposed a universal shape for the dependence of shielding on internuclear separation (shown in Fig. 21, based on work with 23Na in NaH [48] and 39Ar in Ar, [53]. The explanation for this behavior, which is in accord with the first exact and complete shielding trace for Hz+ by Hegstrom [54], is that from the united atom value, the shielding decreases monotonically until it reaches a minimum, then it continues to rise and approach the value for infinitely separated atoms. The alkali hydrides have equilibrium separations that are longer than the separation where the shielding minimum oc-

A.C. de Dies, E. OldfEld/Sokd State Nuclear Magnetic Resonance 6 (1996) 101-125 107

curs, hence, positive shielding derivatives are observed. On the other hand, halides have bond lengths inside the minimum, where the shielding decreases with increasing internuclear distance.

In systems involving more than two atoms, the number of geometrical factors which can influence shielding increases, and more complicated shielding surfaces, not just traces, may be necessary [55]. Lazzeretti et al. made use of the symmetry coordinates of the molecule in their theoretical study of CH, [56], and the effects on shielding of changing each symmetry coordinate were obtained separately. When the same theoretical strategies were applied to the shielding of i5N in NH, [57] and 31P in PH, [58], the following trends became apparent: (1) the first and second derivatives with respect to stretching coordinates are negative; (2) the experimentally observed secondary deuterium-induced isotope shifts are dominated by stretching contributions, and (3) the experimentally observed temperature dependencies of the shielding come mainly from centrifugal stretching. Representative shielding traces for these molecules are shown in Fig. 3.

In more complex molecules, the addition of heavy atoms often leads to the introduction of a new geometrical factor which requires consideration, the torsion angle. In ethane, Chesnut et al.

h c ‘is

“5 Fig. 2. Universal shape proposed for the dependence of shielding on internuclear separation (from Ref. [53]).

.70

35

0

b

-50 IL -70 -0.2 -0.1 0.0 0.1 0.2 Symmetry Coordinate. Angstroms

Fig. 3. Shielding traces for “‘P in PH, (filled symbols) and “N in NH, (open symbols) as functions of the symmetry displace- ment coordinates: S, (symmetric stretch 0, l ), S, (asymmetric stretch A, A ), S4 (asymmetric bend 0, n ) (from Ref. Dgl).

[59] and Kutzelnigg et al. [60] have concluded that = 90% of the change in 13C shielding observed as a function of the torsion angle change is directly due to the torsion, rather than a change in other geometric factors (i.e. bond length or bond angle changes). For ethane, experimental results are unfortunately not available, however, in more complex hydrocarbons, Barfield [61] and Kurosu et al. [62] have reported work aimed at reproducing the “y-gauche” effect first observed by Paul and Grant 1631, and agreement with experiment has been quite satisfactory. As we discuss below in more detail, torsional effects on shielding are exceptionally important for understanding experimental chemical shift patterns in macromolecules such as proteins, where backbone (4, $1 and sidechain (~1 torsion angles appear to be major contributors to 13C shielding. For example, for alanine residues, C” and Ca shifts are dominated by backbone ~,IJ torsions, while valine C*, Cp shifts are influenced by 4,# and x [8,11], and once the 4,$,x dependencies on shielding are determined, then experimental

108 A. C. de Dios, E. OIdfeM / Solid State Nuclear Magnetic Resonance 6 (I 996) 101-l 25

measurements of say C*, Cp chemical shifts can give important information on what &$, and in some case x values, must be present in order to reproduce the observed shifts.

6. Weak intermolecular interactions and short- range repulsion

For polar molecules, Raynes et al. [64] first suggested a potentially useful categorization of the intermolecular effects on shielding into bulk susceptibility, magnetic anisotropy, electric field and van der Waals contributions:

~1 = apUlk + alanisotropy + a;lectric field

+ @.;an der Waals (9)

Such a categorization is useful since it allows one to formulate computational methods specific for a particular kind of contribution ([65] and references cited therein), although it is a somewhat arbitrary distribution of effects.

The bulk susceptibility correction takes into account the sample shape, orientation and susceptibility factors often of importance when comparing chemical shift results obtained in electro- magnet vs. superconducting magnet geometries, a]anisotropy takes into account the magnetizabilities of adjacent groups, u1 ‘lectric fie’d is an electrostatic field contribution to shielding, discussed in more detail below, and ~“,a” der Was’s is the somewhat enigmatic dispersion contribution, due to the fluctuating instantaneous moments of polarizable molecules [66]. Such a categorization has been of limited use, however, since the electrostatic and van der Waals dispersion contributions are particularly difficult to evaluate, and indeed, for non-polar gases, short-range contributions arising from orbital overlap - not considered in Eq. (9) - appear to be very significant [53]. While shielding in rare gases may appear to be rather removed from our primary interests in biomolecular and solid-state NMR chemical shifts, they are in fact a paradigm for the van der Waals shift, a question of interest for so many years, in proteins.

For example, in recent work a scaling scheme based on the dependence of the chemical shift range of a particular rare gas on the characteris- tic (ao3/r3> of the free atom in its ground state [67], and the scaling of intermolecular potential functions given by Maitland et al. [68], has been proposed to explain observed gas-to-solution shifts of 21Ne, s3Kr and ‘29Xe ([69,70], and references cited therein), as well as the measured second virial coefficient (a,) and its temperature dependence, for ‘29Xe in rare gas mixtures [71,72]. Moreover, the sensitivity of the ‘29Xe chemical shift to its environment has also been utilized to characterize zeolites, polymers and other microp- orous solids [73,74]. For example in zeolites, the ‘29Xe chemical shift is found to be a function of loading, cavity size and temperature, and an understanding of how the chemical shift is influenced by these factors has led to a better understanding of the structure and dynamics of Xe clusters trapped in these cavities, as well as help- ing to clarify the importance of ~\;a” der Was’s on shielding.

Based on earlier ideas, van der Waals dispersion effects might have been thought to be very important in determining rare gas shielding. However, it has been shown that scaling ab initio shielding functions for Ar in Ar, [53], in Ar-NaH, and in Ar-OH, [47,48] (which mimic the main rare gas interactions in the zeolite cage) to those of ‘29Xe has recently permitted the successful interpretation of the trends in 129Xe chemical shifts observed in Xe-zeolite adsorption studies [75,76]. The so-called alpha cage in which the xenon atoms are trapped is illustrated in Fig. 4, which also shows a typical ‘29Xe NMR spectrum. Clearly, there are a large number of resonances due to different cage occupancies. Nevertheless, the experimentally observed chemical shifts and intensity distributions have been successfully computed by using Monte Carlo methods [75] combined with ‘29Xe shielding functions obtained by scaling of Ar functions, and excellent agreement between theory and experiment can be seen in Fig. 5, results which were obtained simply by assuming an additivity of the intermolecular effects. This work also indicated that no specific structures for the Xe, clusters were responsible

A.C. de Dies, E. Oldfield/Solid State Nuclear Magnetic Resonance 6 (1996) 101-12.5 109

/” 250 200 150 100 50

wm

Fig. 4. 129Xe NMR of Xe in zeolite cages. Structure of zeolite A showing the cage in which Xe atoms are trapped together with a typical ‘29Xe NMR spectrum of Xe atoms adsorbed in zeolite NaA (from Ref. [53]).

for the observed shifts. The somewhat larger chemical shift difference which occurs on going from Xe, to Xe, was shown to be due to the cage forcing the Xe atoms together [76], and not to the special configuration originally hypothesized for these large clusters [77,78]. The steep decrease of

250

f & 200

& ‘2 CA Z .g IS0 4J

G

k? R

too

i

c-

150 200 250 300 350 400 450

Temperature, K

Fig. 5. Experimental temperature dependence of the ‘%Xe chemical shifts of Xe-zeolite clusters together with theoretical values obtained by using Grand Canonical Monte Carlo methods, combined with the scaled ab initio shielding functions discussed in the text (from Ref. [75]).

the ‘29Xe shielding function at such close distances is primarily responsible for the greater increments observed as one goes from Xe, to Xe, and Xe, to Xe,, an interpretation made possible only after a quantitative understanding of how the environment influences Xe shielding was achieved. Also of interest is the result that van der Waals dispersion appears unimportant in rare gas shielding.

Although the scaling scheme described above has been successful in interpreting and predicting chemical shift behavior in rare gases, it cannot be expected to be applicable to other types of intermolecular interactions - such as electrostatic interactions and strong hydrogen bonding, where other strategies are needed. These are discussed in the following sections.

7. Electrostatic field effects

Early theoretical work on electric field effects on shielding began with Stephen 1791, Bucking- ham and Lawley [80] and Buckingham [81]. Buck- ingham considered electrostatic field effects as a Taylor series expansion in terms of the field and the field gradient, with the coefficients now being referred to by a number of workers as the shielding polarizabilities and hyperpolarizabilities [82]. These have been obtained semi-empirically [83], by introducing a uniform electric field in the CHF method [84-871, as well as analytically by using derivative Hartree-Fock theory, and values for several nuclei in a large number of small molecules have recently been reported by Augspurger and Dykstra 1881. Following work by Cummins et al. [89] and Jerman and coworkers [90,911 in their studies of electric field effects on quadrupole coupling constants, point charges have also been used to represent perturbing molecules in chemical shielding calculations [92]. Using as a model system fluorobenzene interact- ing with a number of hydrogen fluoride molecules, Fig. 6, it has been shown that point charge repre- sentations, charge field perturbation (CFP), result in almost identical shielding changes to those obtained by using basis sets on the HF molecules, Fig. 7. Based on this work it has been concluded

110 A.C. de Dios, E. Oldfield /Solid State Nuclear Magnetic Resonance 6 (1996) 101-125

Fig. 6. Fluorobenzene-hydrogen fluoride cluster used to model weak electrical interactions on shielding.

that: intermolecular contributions to shielding may often be additive; that this additivity sup- ports the idea that shielding can be expressed as a convergent power series expansion in the potential; that contributions from mutual polariza- tions are negligible, and that shielding polarizability and point-charge models “work” - i.e. give the same results as supermolecule cluster calculations - precisely because of this additiv-

I -2 -1 0 1 i

log (shielding, ppm) full ab lnfflo

Fig. 7. Calculated 19F shieldings of F in fluorobenzene in a series of C,H,F-HF axial dimers calculated using partial atomic charges (CFP-GIAO) plotted vs. those computed using a full ab initio method (from Ref. 1921).

ity. It has also recently been shown that there is good agreement between results obtained by using a full ab initio approach and those derived by using the shielding polarizabilities, when terms up to the hypergradient of the field are considered 1931, although the uniform field and field gradient terms dominate at all but the shortest interaction distances. The speed of the shielding polarizability method for evaluating electrostatic contributions to shielding makes it particularly suitable for use in work when dynamical effects are considered, and an application to a macromolecular system will be discussed later.

8. Shielding tensors solved

The results we have discussed so far relate to predictions of isotropic chemical shifts only. However, successful prediction of a complete shielding tensor is a much more rigorous test of a calculation than prediction of a single, isotropic chemical shift, where there could be (and often have been) “fortuitous” cancellations of error. Thus, the ability to accurately predict shielding tensors and their orientations has been an important goal for some years, as Mehring states in his extensive 1983 compilation of iH, 13C, i5N, i9F, 31P etc. shielding tensors [94].

In the solid state, where most tensors have been determined, intermolecular interactions are naturally of concern, but it seems likely that CFP or shielding polarizability approaches will be able to tackle at least the electrostatic parts of the problem. Of course, with Flygare’s formulation of the relationship between spin-rotation constants and the paramagnetic shielding term, molecular beam measurements ([951, and references cited therein) provide not only absolute shielding values, but the tensor elements as well, and such molecular beam measurements have provided tensor elements which are free of intermolecular effects, thus serving as excellent tools for testing the theoretical methods. Representative early compilations of calculated shielding anisotropies derived from CHF and local origin methods have been given by Schindler and Kutzelnigg [30], Hansen and Bouman [31], Chesnut and Foley

A. C. de Dies, E. Oldfield / Solid State Nuclear Magnetic Resonance 6 (I 996) 101-125 111

[52], Chesnut 1961, and Holler and Lischka [97]. All calculations compare favorably with experiment.

More recently, much larger molecular species have been investigated, e.g. i3C and 15N shielding tensors in the retinal Schiff base of interest in vision 1981, as well as solid saccharides such as methyl+D-glucopyranoside and sucrose [99,100], and Grant et al. have shown how peak assignments can be made, based on ab initio calculation. For example, in the methyl glycoside, per- mutation of C, and C, tensors gives a much better agreement between prediction and experiment for each atom, supporting a reassignment. Clearly, some care needs to be taken when reas- signing an experimental shift based on theory - but the case can be made quite convincing when full tensors are used.

In comparing calculated and observed tensors, one should also give considerable attention to the orientation of the principal components. Direc- tion cosines with respect to a molecular reference frame are normally used, however, direct comparison between calculated and experimental direction cosines is difficult to visualize. In addition to not being able to translate directly errors in di-

3

Fig. 8. Icosahedral axis system used for shielding tensors. Top:

relationship between the six icosahedral axes and the Carte-

sian coordinate system. Bottom: icosahedron showing rela-

tionships between X, y, z and the icosahedral vertices (from

Ref. 1991).

rection cosines to errors in degrees or radians, a fair evaluation involving tensors that are small or almost spherical is difficult. Efforts have been made by Grant and coworkers [99,100] at solving this problem, and they have proposed a novel way of expressing shielding tensors called the icosahedral representation. Here, the shift or shielding tensor is represented by six quantities associated with the shifts measured when the applied field is along the six unique directions of an icosahedron, as shown in Fig. 8. The six icosahedral components are related to the Cartesian components or the principal components via the following rela- tionships:

6, = a2Sxx + b2S,, - 2abS,, = (al, - bm,)2S,,

+ (a& - bm2)2S,2 + (a4 - bm3)2S33

S, = a2Sxx + b2S,, + 2abS,, = (a!, + bm,)2S, 1

+ (al2 + bm2j2b2 + (4 + bmJ2h

6, = a2S,, + b2S,, - 2abS,,=(am, -bn,)2S,,

+ (am2 -bn2)2S,2 + (am, - bnJ2S3,

6, = a2S,, + b2S,, + 2abS,, = (am, + bnl)‘S,,

+ ( am2 + bn2)2S22 + ( am3 + bn,)2S33

S, = a2S,, + b2S,, - 2abS,,= (an, - bf,)2S,,

+ ( an2 - b12)2S22 + ( an3 - bZ3)2S,3

S, = a2SLZ + b2S,, + 2abS,,= (an, + bl,)2S,,

+ (an, + b12)2S22 + ( an3 + bQ2S3,

where S,,, SyY, S,,, SXy, S,, and S,, are the Cartesian components in the laboratory frame. The constants a and b are cos 4 and sin 4 (4 = 31.72”), respectively, Sll, S,, and S,, are the principal components, and li, mi and ni are the direction cosines of the ith principal axis. In this representation, all six components can be compared visually and carry the same weight, thereby enabling a fair comparison between two tensors. Moreover, this representation is equally applicable in any coordinate frame. Thus, as long as the orientation of the shielding tensor is known experimentally, additional evaluations of the calcu-

112 A.C. de Dies, E. Oldfield/Solid State Nuclear Magnetic Resonance 6 (1996) 101-125

lated tensor, primarily the direction of its principal components, can be made.

The shielding tensors for several other large molecular systems have also now been successfully predicted using ab initio methods. For example, there is an excellent correlation between theory and experiment for the fluorobenzenes first studied in the 1950s by Gutowsky and coworkers [.5,1011 whose tensors were represented more than a decade ago as a major challenge for theory [94]. However, in recent work by others [102,103] it has again been suggested that van $r Waals (vdW) dispersion forces influence F shielding. This approach uses a modified Buck- ingham-Stephen description for the electrical contributions to shielding:

S,=AE,+B(P+ (P)) -B(P) (10)

in which the fluctuating field (E2> is derived from the London dispersion formula, such that the overall vdW shift is:

6 vdw = B C (3P,Zi/2r,‘:) (11)

with B taken as 67.7 X low6 A3 eV-’ [103,104]. Unfortunately, this approach is seriously flawed. The principal reason for this is that the entire theory is based on very early semi-empirical work in which numerous parameters (AE, P, I, r, A, E, B) were used in an attempt to fit i9F shifts in fluorobenzenes. However, over the past 30 years

there have been many theoretical developments which have led to a complete revision of these early vdW ideas. In particular, in the rare gases, where vdW dispersion effects were always assumed to dominate, recent work using SOLO and LORG [53] has clearly indicated that dispersion does not affect shielding perceptibly, with ‘29Xe shifts being well accounted for in the gas phase and in zeolite lattices by SCF-level calculations - which do not include vdW dispersion [75]. Also, the values of the A parameter, computed by using derivative Hartree-Fock theory [93,105], are now known to be two orders of magnitude larger than assumed by previous workers. For the fluorobenzenes, GIAO-SCF level calculations show very good correlations between isotropic liquid state chemical shifts and theoretical shielding (23 points, slope = 0.94, R2 = 0.975, 3 ppm r.m.s. deviation over a 63 ppm range) as well as between experimental solid state shifts/shielding tensor elements and computed values (21 points, slope = 0.95, R2 = 0.989, 6.5 ppm r.m.s. deviation over a 237 ppm shift range), as shown in Fig. 9 [9]. These results do not contain any second-order or vdW dispersion contributions, yet accurately predict all shift and shielding tensor elements with no adjustable parameters (the r.m.s. devia- tions are already close to the experimental uncer- tainties in the solid). Of particular importance is the accurate prediction of all pentafluorobenzene

320 - -180 -150 -120 -90

experimental shift (ppm)

-130 -65 0 65 130

experimental shielding (ppm)

Fig. 9. Experimental vs. theoretical 19F chemical shift/shielding for fluorobenzenes. (A) Liquid state isotropic chemical shifts

plotted vs. computed shielding values. (B) Solid-state shielding principal tensor components (in ppm from CLF6) plotted vs

computed shielding tensor elements (from Ref. [9]).

A. C. de Dios, E. Olafield / Solid State Nuclear Magnetic Resonance 6 (1996) IO1 -125 113

chemical shifts, which could not be predicted using the empirical approach [NM]. Since it only takes one counter-example to disprove a hypothesis, these results convincingly lay to rest the idea that vdW dispersion effects contribute to shielding in the fluorobenzenes - and by inference, in larger systems such as proteins. Since there is neither a basis nor need for B, theories relying on it are suspect. The correlations seen in Fig. 9B also explain the long-standing “ortho” effect ([94], and references cited therein), previously posed as a challenge for theory. The “ortho” effect, the shielding of a,, by = 50 ppm on each ortho-F substitution, is clearly seen in Fig. 9B as is the invariance of a,, and gradual changes in gll with substitution, as observed experimentally, and shown graphically in Fig. 10.

Ab initio evaluations of the 13C shielding tensors for each carbon atom in crystalline, zwitterionic L-threonine and L-tyrosine, using the GIAO method have also been obtained recently [8]. Computations involving isolated molecules yield reasonable agreement between experimental and theoretical shift tensor elements, as shown in Fig. 11. However, considerable improvement is achieved by incorporating a point-charge lattice

I

l,3,5C6H3F3 611 ' 622 533

7.3C6H4F2 bl 622 633

%%F 611 [ 622 633

I

',4c6H4F2 611 I 622 633 I 1

12 C6H4F2 611' 622 633

1,2,4,6C6H2Fq 6,,=622 1 ; 633) I I I I

C6% 611=S22 ' 633 I

I I I I I

-100 -50 0 50 100 150 ppm )

Fig. 10. The “ortho effect” as seen in a series of fluoroben-

zene shielding tensor calculations and experiments: a,, is

very sensitive to substitution at the ortho position, (+, , changes

gradually with substitution while flZ2 is not sensitive at all.

Based on date from Ref. [94].

250

‘I

J

experimental shift from TMS (ppm)

Fig. 11. Plot comparing experimental ‘k chemical shift tensor components for L-threonine and L-tyrosine bitterionic forms) with those calculated using the GIAO method (from Ref. [8]).

to represent the local charge field in the crystal, due primarily to better values for (+11 (in the CO sp2 plane and perpendicular to Ca-Co> and uZ2 (perpendicular to the sp2 plane) elements of the carboxyl carbon shielding tensor, shown in Fig. 12. Thus, combined use of charge-field perturbation and the GIAO method permits excellent prediction of 13C shift tensor elements in two zwitterionic, polar amino acids. Moreover, the polarization effects are primarily limited to Co, supporting the idea that for 13C, long-range electrostatic field contributions to shielding will be small, especially for sp3 carbons. The ability to successfully predict 13C shielding tensor elements in these systems provides strong additional support for the adequacy of GIAO/CFP-GIAO methods in predicting 13C chemical shifts in proteins, and other macromolecules as well. Further- more, in a related recent work, Facelli and Grant [106] have noted that theoretical calculations of shielding tensors have already reached a level at which structural information can be obtained which actually surpasses that of high resolution X-ray and neutron diffraction methods. These

114 A. C. de Dies, E. Oldfeld /Solid State Nuclear Magnetic Resonance 6 (I 996) IO1 -125

300 200 100 0

experimental shift from TMS (ppm)

Fig. 12. Experimental and theoretical 13C NMR chemical shift/shielding tensor elements for L-threonine and L-tyrosine, with charge field perturbation (from Ref. 181).

authors note that (especially) solid-state NMR methods should be useful for refining structural data of biomolecules with molecular weights in the lo-20 X lo3 Da range, where diffraction data have large structural errors, since chemical shift data are not sensitive to many of the structural imperfections which degrade diffraction data. While such single crystal studies have not yet been reported, their ideas nevertheless lead us naturally to the question of predicting chemical shifts in macromolecules, such as proteins, where at least in solution, there is already a considerable (= 10’) body of uninterpreted chemical shift data [107], and solid-state data are now beginning to become available. Once chemical shifts in proteins (and DNA, polysaccharides, etc.) become interpretable, then structure prediction, refinement and validation follow naturally.

9. Predicting protein chemical shifts

Protons are the most abundant nuclei in macromolecules such as proteins (or nucleic

acids), and hydrogen is the most sensitive naturally occurring nucleus, so early NMR studies of macromolecules were of course overwhelmingly dominated by ‘H NMR investigations. For most proton resonances, it was found early on that ring currents from aromatic residues dominated ‘H chemical shifts - at least for side-chain groups [108], where r.m.s. differences between ring current prediction and experiment, in, e.g. hen egg white lysozyme and the bovine pancreatic trypsin inhibitor, were found to be small. However, for backbone (Ha and HN) protons, worse agreement was found, and this was suggested to be due to the effects of the anisotropy in the diamagnetic susceptibility of the peptide groups, as well as electrostatic field effects. Such influences were proposed quite early on by Sternlicht and Wilson [1091, who showed an upfield or shielding effect on helix formation from a random coil. However, it was only much later that a large number of groups began to report the general observation that H” are shifted = 0.39 ppm upfield in helices but = 0.37 ppm downfield in sheets (see e.g. Refs. [llO-1141). This soon led to detailed analy- ses of H” shifts in terms of magnetic susceptibility (ring current and peptide group), electrostatic field and random coil shifts [115-1171, with the result that for H”, experimental chemical shifts can now be quite accurately predicted (at least in populous $,I) regions) by using what are basically empirical methods.

For the heavier elements frequently studied in proteins, 13C, r5N and r9F, the situation is more complex. For r3C, good empirical correlations between secondary structure and “secondary” chemical shifts - the shifts from the random coil position - have been reported [118,119], but for 15N, the correlations are much weaker [114,120], and for r9F, only a general picture that buried residues are often deshielded has developed [121]. The reason for the lack of any significant developments until recently in the analysis of shielding of the heavier elements is due primarily to the fact that the problem looks so daunting. The paramagnetic term dominates shielding, and this needs to be evaluated quantum mechanically - in a protein! Moreover, both X-ray and solution NMR structures of proteins have accuracies much

A. C. de Dios, E, Oldfield / Soki State Nuclear Magnetic Resonance 6 (1996) 101-l 25 115

less than typically expected for chemical shift calculations. Fortunately, however, the task can be reduced to manageable proportions by first separating the total shielding, at, into three parts, basically akin to Eq. (9):

a, = 0, + ue + u (12) and using unif:rm bond lengths [122]. a, is the short-range or “electronic” contribution to shielding, and has to be evaluated by quantum chemical means; a, is the electrostatic field contribution, and can be determined either by using the shielding polarizabilities, or by representing atoms outside of a “core fragment” by partial atomic charges - the CFP approach; and a,.,, represents the other, primarily magnetic contributions, such as the long-range peptide and aromatic “ring current” effects noted. If necessary these can be evaluated using the semi-classical approaches used for ‘H NMR.

As with ‘H NMR, there are well defined experimental correlations between secondary structure and chemical shift in 13C NMR, and we show in Fig. 13A the secondary chemical shifts for C* and Cp residues in the data base of Spera and Bax [118]. As can be seen from Fig. 13A, C” resonances are shielded in P-sheet regions but deshielded in cw-helices, while CB shows the opposite trend, both in solution and in the solid-state 11231. How can these trends be investigated in more detail? Clearly, it is not at present feasible

a-heli

k

expt p-sheet

a-hel

A a-helix

? m

theory -sheet

I wm SCp,

Fig. 13. 13C NMR shielding in proteins. (A) Histograms showing separation of a-helical and P-sheet chemical shifts for C” and CD sites in proteins (based on global database of Spera and Bax, Ref. [1181X (B) Histogram showing C” and C@ shifts for model o-helical and p-sheet (alanine only) fragments, based on ab initio shielding surface calculations (from Refs. [118,1221).

to carry out ab initio calculations on a complete protein. However, if it is reasonable to suppose that if 4,$ torsion angle effects dominate shielding - as suggested by the empirical observations on helical and sheet residues - then the shielding changes due to these geometric parameters would propagate through the bonding framework, and would therefore only operate over very short ranges. Thus, only the local geometry might need be considered, i.e. u, in Eq. (lo), which would require only a small number of atoms requiring basis functions. Of course, if more indirect effects were important - such as differential hydrogen bonding patterns - or if longer-range (e.g. helix dipole) electrostatic field effects dominate, then simple 4,3/ correlations might not be readily detected.

The effects of 4, 4 alone on shielding can be determined by using, e.g. the GIAO method, and we first consider as a general example C” and C” shielding trends of alanine residues in proteins. Construction of a simple molecular fragment, N- formyl-L-alanine amide:

permits incorporation of two peptide planes, so that +,I) effects can be studied. Use of a relatively small (6-311G * * ) basis enables a relatively rapid determination of C” and C@ shielding as a function of &$, and results obtained using this fragment are shown in Fig. 13B. There is clearly good general agreement with the experimental trends reported by Spera and Bax [118,122]. Of course, Figs. 13A and 13B show only general trends, and not predictions of specific experimental chemical shifts. This can be achieved by use of a larger basis, and good agreement between theory and experiment has been achieved for specifi- cally assigned C* and Cp resonances of the twelve Ala residues in Staphylococcal nuclease [122] (SNase). The effects of hydrogen bonding can be incorporated by locating formamide molecules on the coordinates of hydrogen bond partners, and long-range electrostatic field effects are included

116 A.C. de Dies, E. Oldfield /Solid State Nuclear Magnetic Resonance 6 (19%) 101-125

by using CFP - although both effects are quite minor. There is obviously good agreement with experiment, as shown previously [122], even though the protein itself has a molecular weight of over 13000 Da.

10. Carbon-13 shielding surfaces

The +,1+5 influences on shielding are not seen in a particularly clear way from the results shown in Fig. 13, and it is desirable to have a more direct way of converting 4, + information into experimental chemical shifts. This would enable spectral assignments to be rapidly verified, and could lead to ways of deducing structure directly from chemical shifts - if 4, I++ effects dominate.

-180 -120 -60 0 60 120 180

Such a nexus between structure and shielding is indeed implicit in the work of Wishart et al. [114] and Spera and Bax [118], and we show in Figs. 14A and 14B results for C”, C? as a function of 4, I,+ [118,124]. The work of Spera and Bax involved use of a “global” data base containing all types of residue, but it nevertheless clearly sug- gests that C*, C? shifts vary within the helical and sheet regions. This behavior is seen much more readily when individual shielding surfaces are computed for particular amino acids, and Figs. 14C and 14D show computed C”, CB shielding surfaces for alanine, again using a locally dense approach [124]. The general trend seen in the empirical surface of the sheet region containing shielded C* but deshielded CB, with the opposite trend in the helical region, is apparent,

ccx

120

60

-60

-180 -120 -60 0 60 120 180

Fig. 14. Global ‘“C 4, $ shielding surfaces for (A) C”, (B) Ca from Spera and Bax [118], together with theoretical alanine ‘“C 4, JI

shielding surfaces for (C) C”: (II) CB, from Ref. 11241.

A. C. & Dies, E. Oldfield /Solid State Nuclear Magnetic Resonance 6 (1994) 101 -I 25 117

130 - 70 60 50 40

experimental shift (ppm from TSP)

Fig. 15. Computed 13Cu shielding for Gly, Ala and Val

residues in SNase and calmodulin; 57 data points, r.m.s. deviation from fitted curve, 1.5 ppm (from Ref. [lo]).

with the added advantage over the empirical sur-

face that the entire region of 4, $ space can in principle be investigated. Figs. 14A-14D also show that there are significant differences between the global and alanine shielding surfaces - due to different alkyl substitution effects.

The effects of different C” substitution patterns are major, and have two results. First, they cause a large range of C” chemical shifts [lo], a difference of about 24 ppm between glycine (most shielded) and valine (most deshielded), Fig. 15, and second, the shapes of the C” shielding surfaces vary considerably from residue to residue. Fig. 16 shows a three-dimensional shielding surface ~(4, $) for glycine, as well as its projection [11,125]. The availability of such shielding surfaces is useful in that it permits an immediate evaluation of a particular C* chemical shift if 4, $ is known, since as noted above, 4, 4 effects dominate C” shielding. Thus, chemical shift assignments can be readily corroborated. Chemical shifts read off from such shielding surfaces are in good agreement with those obtained from individual GIAO calculations [19,122]. Such shielding surfaces offer an entry point into structure prediction and refinement, since if a chemical shift is known, then it is clear that it can only be described by a limited set of 4, $ angles, and 4, I) predictions of = 15” are now possible via use of shielding surfaces. For most amino acids, the effects of side-chain torsions (x1, x2.. . > will also have an effect on shielding, thus, for example for valine, it is necessary to determine (~(4, I), x1> in order to make chemical shift predictions from the known structure. This has recently been achieved

b

w (4 Fig. 16. Theoretical glycine “C 4, I) shielding surface for C” (based on Ref. [l I]).

118 A. C. de Dim, E. Oldfield /Solid State Nuclear Magnetic Resonance 6 (1996) 101-125

[126], and the principal conclusion, based on both shielding surface predictions and spin-spin coupling constant information [127,128], is that x1 torsions for valine can vary considerably between crystal and solution states, at least in the calmodulin system investigated.

11. Electrostatics and dynamics in proteins

We next consider the problems of “N and 19F shielding in proteins. Unlike 13C NMR, there have been very few empirical correlations between structure and experimental chemical shifts, although Glushka et al. reported an “N NMR chemical shift correlation with qQi__r in P-sheets [129,130], and related weak global correlations with 4, rcli_r have also been observed [120]. In addition, as with 13C, both solid-state and solution shifts appear very similar [1311. Unlike 13C, 15N chemical shifts are expected to be difficult to predict, since not only are (b, $, x expected to be important, but strong hydrogen bonding, as well as longer-range electrostatic field effects and sol- vation might also contribute, and for l”F, these latter effects could be dominant. Nevertheless, these problems should all eventually be tractable. As a test case, we consider first “N shielding of valine residues in SNase, and threonine residues in T4 phage lysozyme. These residues have shielding ranges of = 20-30 ppm 1132-1341 -

160

very large values which would hopefully be repro- duced using even simple models for the longer- range interactions.

For valine, the N-formyl-r_-alanyl-L-valyl amide fragment has been used:

;I [ 1 H/c\

NH,

together with a single formamide hydrogen bond partner, while for threonine, alanyl-threonine forms the core fragment. Model compound studies show a = 5 ppm maximal effect due to x’, while hydrogen bonding can cause up to a = 10 ppm effect, although only when NH is directly involved [122]. Fig. 17A shows computed valine 15N shielding in SNase as a function of the experimental chemical shift, without hydrogen bonding or charge field perturbation. The overall scatter decreases on incorporation of hydrogen bond partners, Fig. 17B, and even more with CFP, Fig. 17C, and a basically similar correlation is found with threonine residues in T4 phage lysozyme,

100 110 120 130 140 100 110 120 130 140 100 110 120 130 140 100 106 112 118 124

experimental shift (ppm)

Fig. 17. “N shieldings for valine sites in SNase. (A) Isolated SNase 15N Val fragments; (B) as in (A) but with inclusion of

hydrogen-bond partner; (C) as in (B) but with additional inclusion of charge field perturbation; (D) as in (C) but for threonine sites

in T4-phage lysozyme (from Ref. [122] and our unpublished results).

A.C. de Dios, E. Oldfield/Solid State Nuclear Magnetic Resonance 6 (1996) 101-125 119

Fig. 17D. Since these results are the first predictions of 15N shielding in a protein, they can be regarded as quite promising. In crystalline SNase, Va151 is not observed, possibly due to disorder, but the shifts for the other residues appear very similar in crystals and in solution, and are well predicted from X-ray structures.

The final nucleus we consider is 19F. Fluorine is a non-native probe of structure in proteins, but has been used for many years in NMR studies of protein structure in solution [121,135-1391, and more recently systems as large as the nicotinic acetylcholine receptor have begun to be probed using solid-state techniques 11401. Conventional wisdom has suggested that van der Waals dispersion interactions dominate 19F chemical shifts in proteins, but this now seems unlikely, since the basis for the B term has disappeared [9]. In particular, we now know that 19F chemical shifts and shift tensors can be well described at the CHF level, without vdW dispersion, and we also know that the shielding polarizabilities of fluoro- aromatics are extraordinarily large-about 2000 ppm a.u.-’ [93,105]. Also there is evidence to suggest that electrostatic fields of up to = 0.008 a.u. exist in proteins [141-1431, so electrostatic field induced shifts of = 2000 X 0.008 = 16 ppm are predicted for 19F - about the values found experimentally. It thus seems plausible that a, dominates i9F shielding in proteins, and in order to test this hypothesis, it is necessary to investi- gate a protein having a rigorously assigned i9F

10 0 127 133 103 195 284

NMR spectrum, whose structure is also known. The galactose binding protein (GBP) from Es- chekhiu coli fits the bill since it has five trypto- phan residues, all of which have been labeled with [5F]Trp, and the chemical shifts cover a = 10 ppm range [137].

In an SCF calculation, it is clearly a difficult matter to decide on the correct atomic charges in a system as complex as a protein, and also to decide on whether or not to consider dielectric effects. Since it is known that surface charge field modification does not appreciably perturb the r9F NMR spectrum of hen egg white lysozyme [144], surface charges need not be considered, with only whole electroneutral residues being used to evaluate the electrostatic field contributions to shielding in GBP [122]. Fig. 18A shows the electrostatic field contributions to shielding for four of the five [5F]Trp residues in GBP - results in good accord with experiment 11371. Trpzs4 is exposed to solvent, and here the alternate route of using the shielding polarizabilities is required in order to evaluate solvent effects. Based on the model compound studies discussed above [93], the field and field gradient contributions (dipole and quadrupole shielding polarizabilities) are expected to dominate the electrostatic contributions to shielding, and these have been evaluated by using a molecular dynamics approach. When the field and field gradients are multiplied by the corresponding shielding polarizabilities, the electrostatic contributions to shielding are obtained

-4 -2 0 2 4 6 experiment (Ppm) shielding (ppm)

.4 -2 i i h i experiment (ppm)

Fig. 18. “F shielding results for the five [S-r”F]-Trp residues in the Escherichia coli galactose binding protein. (A) GIAO shielding

results for Trp residues 127, 133, 183 and 195; (B) individual 20 ps shielding trajectories for each of the five [S-F] Trp residues; (C)

graph showing experimental vs. theoretical shielding results from the trajectories in (B) (from Refs. [7,122]).

120 A. C. de Dies, E. Oldfield /Solid State Nuclear Magneiic Resonance 6 (1996) 101-l 25

over, in this case 5 X 20 psec MD trajectories, Fig. 18B. The time-average shielding values obtained from the shielding trajectories are in good accord with the experimental chemical shifts, and the results of both CFP and shielding polarizability approaches are in good accord with experiment (Fig. 18). For the CFP method, the shielding range is slightly overestimated - possibly because of the lack of dielectric contributions. The shielding trajectory method can also of course be combined with, e.g. C”, C? shielding surfaces in order to incorporate the effects of molecular motion, and should be of use in making even more accurate predictions of experimental protein NMR spectra in the future.

The basic physics of the CFP/MSP weak electrical interaction model is quite different from the vdW dispersion approach, also recently proposed as a means of predicting 19F chemical shifts. On the one hand, the SCF-related methods give good accord with experimental shifts as well as shift tensors for all fluorobenzenes, without exception, and with no adjustable parameters, while the vdW dispersion method fails in several cases, and can best be thought of as an empirical fitting procedure, with no physical basis for A,B, etc.

12. Protein structure prediction and refinement

In essentially all cases examined so far, known (generally X-ray) structures have been used in order to predict experimental NMR spectra. In many cases this is an important step in itself, but clearly one of the main goals of our quantum chemical work is also to be able to deduce structure from experimental chemical shifts. Chemists routinely employ empirical methods, essentially look up tables, to deduce structure from shielding in small molecules, and more recently this approach is beginning to be applied to macromolecules such as proteins. We thus consider first the empirical approach to structure prediction, and then show how the method can be extended to incorporation of shielding surfaces obtained by using ab initio methods.

For proteins, there are = 130000 reported

chemical shifts [107], although perhaps only 1 or 2% have been used directly to facilitate structure analysis. The simplest empirical approach is the so-called chemical shift index (CSI) reported by Wishart and coworkers [119,145]. The idea here is simple and is that for CO, H*, C’, and to a lesser extent Cp, there are clear positive or negative secondary chemical shifts from the random coil chemical shift position which are structure (helix, sheet) related - albeit the nature of the relation may be quite different for say C” and C’. The CSI method uses a tri-state method in which, e.g. for C”, a “1” is assigned to a helical shift, a “0” to a random coil shift, or a “ - 1” to a P-sheet shift. When a certain number of consecu- tive residues exhibit, e.g. helical behavior, then the presence of helical secondary structure is inferred. The method is particularly powerful when C*, Ha and C’ indices are combined into a consensus CSI pattern, and the method gives a generally excellent accord with experiment 11191.

The success of the empirical chemical shift index method immediately leads one to ask - why is it successful, and can it be improved? The method is successful because chemical shifts are structure dependent. Since many such chemical shifts can now be predicted using ab initio methods, it is thus logical to see if additional, more detailed, structural information can be obtained, since as shown in Fig. 14, there are considerable chemical shift differences within a particular structure type region. That is, helical and sheet residues do not have just two chemical shifts, rather, a more or less continuous distribution. How then can more detailed structural information be derived from these distribution functions?

One approach is essentially an analog version of the digital CSI method, and makes use of e.g. C”, Cp and H” shift surfaces. The idea is basically to estimate the probability, Z, that a given chemical shift, Pexpt corresponds to a particular 4,J, torsion angle pair. For P =fCc$, $), the probability 2

Z = exp

is given by:

P expt - P(47 $1 * - W (13)

A. C. de Dies, E. Okffiehi /Solid State Nuclear Magnetic Resonance 6 (I 996) 101-125 121

A C 190

90

0

-90

8

4 -‘*O - 180 3

90

0

-90

-iso -W d 9b tad -ho -9b

’

90 180

B 4 (deal D

Fig. 19. Chemical shift surface prediction of peptide backbone 4, I) torsion angles for Ala 60 in SNase; (A) ‘Z6(C”); (B) ‘ZG(Cfi); (C) ‘ZS(C”)S(Cp); (D) 3ZS(C”)S(Cs)S(Hu) (from Ref. [I 11).

where W is a search width related to the experimental error and the error in the surface. Thus, if a given 4,1) yields P = Pexpt, then Z = 1. If 4,$ yields P # Pexpt, then Z < 1. Thus, for a given P a probability or Z-surface can be con- siE:ted. As an example, the Z-surface, Z(r), I)), for C” of Ala6’ in SNase, based on the alanine shielding surface of Fig. 14C and the experimental C” chemical shift, has an “S” shaped contour, typical of Ala-C” residues in helices [ll], but some sheet and turn regions are also possible, based solely on the C” shift, as shown in Fig. 19A. When Cp is used as a constraint, the high-Z or more probable regions are quite different, Fig. 19B. Since the Z-surface represents a probability distribution, the Z” and Za surfaces can be multiplied to yield a ‘2 surface (ZaTp), in which a pronounced helical solution region is apparent [ll]. Fig. 19C. Further refinement with H” yields a final 3Z result, Fig. 19D. For 22 out of 24 $,tj angles for alanine residues in SNase, the Z- surface approach yields an r.m.s. deviation between prediction and (X-ray) experiment of about 10-15”. Two additional 41 errors are = 30” but

this may represent a genuine crystal-solution difference.

As might be expected, additional spectroscopic parameters can be incorporated into the Z- surface approach. For example, in oriented samples, anisotropic chemical shifts and quadrupole splittings can be used as one-dimensional Z- surfaces, since P = f(e) = 3 cos * 8 - 1, where 8 describes the orientation of the group in question in the director axis system. The method also permits combination of chemical shift and J-coupling constant information, since, e.g. 3JH~Ha is a f(4), and several other coupling constants are fC$, $1 [146-1501. By combining chemical shift and J-coupling information from wholly empirical global surfaces, r.m.s. deviation values for 4,$ are about twice as large as with the quantum chemically computed alanine results - but this can be expected to improve considerably when individual chemical shift and J-surfaces are evaluated [151]. Also, since C’ is very well behaved in the CSI approach, it too can be expected to generate additional restraints in the future - although probably not of a direct 4,rl, nature.

From the results presented so far, it appears that the chemical shifts of the lighter heteronu- clei in most biomolecular systems can now be predicted by using quantum chemical methods. In addition, the first steps in the important reverse direction of predicting structure from spectra, also appear promising. Such new methods will have utility in two distinct areas. First, in solution NMR studies of proteins (and potentially, peptides and other macromolecules), there are frequently regions which are poorly restrained by the available NOE and J-coupling data, and chemical shifts could be useful in the refinement process. Second, in the solid or semi-solid state, a number of spectroscopic methods focus on distance restraints based upon dipolar interactions. Here, the addition of torsional restraints based on isotropic chemical shifts can again be used to reduce (and refine) the number of conforma- tional solutions.

When magic-angle sample spinning techniques are mandated by the properties of the system of interest (insoluble, random powder), or when NOE or J-coupling restraints are too weak

122 A.C. de Dies, E. Oldfield/Solid State Nuclear Magnetic Resonance 6 (1996) 101-125

(surface loops, large proteins, small peptides), then chemical shifts offer an important route for structure refinement and prediction, and in recent work, Case and coworkers [152,1531 have begun to demonstrate the utility of ‘H chemical shifts in the refinement of a relatively low resolution solution NMR structure of the heme protein, carbonmonoxymyoglobin. The general approach can be extended to more than a two-dimensional 4,1(1 surface, and recently we have evaluated a three-dimensional shielding hypersurface for valine, a(+, I), x), for use in refinement and prediction [126]. An interesting result of these studies is that chemical shift refined structures have less scatter (in 4, 4, x> than unrestrained structures, and appear very close to X-ray values.

13. Conclusions and prospects

Recent advances in computer performance together with faster algorithms have begun to enable the widespread application of computational chemistry to permit the accurate prediction of both solid-state and liquid-state NMR chemical shifts in most molecular systems - from small organic molecules to macromolecules such as proteins. However, of even more interest in the long term are the development of new methods of directly predicting structure from experimental spectra. In proteins, the empirical chemical shift index and its quantum chemical analog, the chemical shift or Z-surface approach, can facilitate secondary structure prediction and assign- ment verification, and represent the first steps toward structure prediction, refinement and validation in proteins, using the chemical shift parameter. The chemical shifts (and indeed the shielding tensors) of most light (and many heavy) nuclei in most molecular systems can now be accurately predicted via quantum chemistry, using small workstations or workstation clusters. Moreover, the sensitivity of NMR chemical shifts to structural parameters means that chemical shifts and quantum chemistry can be used together to refine structures obtained by using other (e.g. diffraction) methods. For example, typical X-ray bond lengths found in proteins are quite

inconsistent with experimental chemical shifts [lo]. After over forty years of continuous development, most of the “chemical effects” observed in the 1950s can now be predicted. Given the generally “sparse” nature of distance restraints in solid-state systems, the results we have outlined above suggest that chemical shift analysis - both isotropic and anisotropic - will be a welcome addition to solid-state structure determination. At present, chemical shift calculations are still unpleasantly slow. However, recent progress with density functional theory (DFT) [154] rather than Hartree-Fock methods suggest significant speed improvements, together with the incorporation of electron correlation, and even at the Hartree- Fock level, considerable time savings can be expected in favorable cases, e.g. where accurate shielding surface shapes can be obtained using very small basis sets [126]. The evaluation of metal shifts is also now possible using especially DFT methods [154], and it seems likely that use of periodic Hartree-Fock methods [155] may permit analysis of shielding in inorganic crystal lattices in the near future. Given predictable increases in CPU speed over the next few years, the use of ab initio shielding calculations for macromolecular structure prediction, refinement and validation in chemistry and biology is sure to flourish.

Acknowledgments

We are grateful to C.J. Jameson, P. Pulay and C.E. Dykstra for their continuous flow of advice, and P. Pulay, J.F. Hinton, K. Wolinski and W. Kutzelnigg for providing their computer pro- grams.

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Recent progress in understanding chemical shifts

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