J. Phys. Chem. B 2005, 109, 3627-3638
3627
Magnetic Effects of Disulfide Bridges: A Density Functional and Semiempirical Study Finton Sirockin† and Annick Dejaegere* UMR7104, Structural Biology and Genomics Laboratory, Institut de Ge´ ne´ tique et de Biologie Mole´ culaire et Cellulaire, CNRS/INSERM/ULP, BP 10142, F-67404 Illkirch Cedex, France ReceiVed: July 29, 2004; In Final Form: NoVember 25, 2004
Density functional chemical shielding calculations are reported for methane and hydrogen disulfide dimers. The calculations show that the contributions of disulfide bridges to the chemical shielding of neighboring protons is sizable at distances that are frequently sampled in protein structures. A semiempirical model of the quantum chemical data is developed. It is shown that magnetic anisotropy effects of disulfide are poorly described by the McConnell equation, both qualitatively and quantitatively. In particular, the ratio of magnetic anisotropy contributions to shielding along and perpendicular to the magnetic anisotropy principal axis do not conform to the predictions of the McConnell equation, and magnetic anisotropy effects are not null along the magic angle axis. A sulfur-based model of the magnetic anisotropy of the disulfide is developed and shown to give much better agreement with the quantum chemical data.
Introduction Advances in NMR instrumentation and methodology have now made it possible to make site-specific chemical shift assignments for a large number of proteins and nucleic acids. Although it has long been known that chemical shifts are sensitive to details of molecular structure and that the chemical shifts in proteins and nucleic acids can give information on secondary and tertiary structure,1-3 their interpretation in terms of three-dimensional structure remains a challenging task. To interpret chemical shift data in terms of structure, theoretical models that follow several lines have been developed. One class of models is purely empirical, or knowledge based; an example of this sort of model is the chemical shift index, which has gained widespread use in the determination of secondary structure.4,5 In its original form, this model used well-known systematic trends in alpha proton chemical shifts as a tool for secondary structure assignment. As more protein structures were solved by NMR and as labeled proteins were more routinely used, the database of shifts was augmented and the model was refined to incorporate 13C data.6 The increase in the number of experimental NMR structures led to the development of more sophisticated empirical models, for example, chemical shift analysis that incorporates many of the ideas originally developed in the context of comparative 3D protein structure prediction.7,8 The advantage of these methods is that the calculations are fast. The disadvantge is that physical insights into the factors that give rise to particular chemical shifts are absent. Quantum chemical calculations of chemical shifts have also progressed significantly in the past few years.14-16 Calculations of 13C shielding in model peptides and small molecules in the gas phase have been used to trace the dependence of carbon peptide shielding on φ and ψ backbone angles.17,18 These calculations provide the most detailed physical description of chemical shifts. However, because of the large amount of computation that is required, these calculations are limited to relatively small systems. † Present address: Novartis Pharma AG, Protease Platform, Postfach CH-4002 Basel, Switzerland.
More empirical approaches aimed at a physical interpretation of observed chemical shifts have also been pursued. Semiempirical theories for chemical shift dispersion that allow the calculation of chemical shifts in proteins have been developed.11,12 Most of these efforts have been aimed at calculating proton chemical shifts, which are generally dominated by environmental effects rather than local structure. These approaches account for ring current, magnetic anisotropy, and electrostatic effects on chemical shifts via the use of classical models. By exploiting the large number of NMR structures and accompanying chemical shift data available in the public domain, like BioMagResBank,9 parameters for these semiempirical models have been developed to reproduce experimental chemical shift data.11-13 Chemical shifts of small molecule systems calculated from quantum mechanics are another important source of data used to develop a consistent set of parameters for these models. In this approach, small molecules in simple geometries and combinations are studied so that different contributions to the shift and their behavior with conformational change can be isolated and quantified.19-24,55,56 These semiempirical methods are routinely used to validate and refine experimental structures. In a recent NMR study of the omega-conotoxin MVIIa by Atkinson et al.,25 a significant breakdown of the semiempirical model for proton chemical shifts for protons in the vicinity of disulfide bridges was observed. This suggests that S-S bridges have specific magnetic effects that have not yet been characterized and therefore not included in the semiempirical models. This highlights a significant limitation of the semiempirical model. Disulfide bonds are ubiquitous in proteins, and small proteins rich in disulfide bridges, like the conotoxins, are an important class of molecules with potentially a very high therapeutic value. Given that these proteins are often relatively small, they are most often studied by NMR spectroscopy26-29 and therefore it is important to understand the specific effects that S-S bridges have on the local chemical shifts and to be able to include these effects in semiempirical models of chemical shifts.
10.1021/jp0466136 CCC: $30.25 © 2005 American Chemical Society Published on Web 02/08/2005
3628 J. Phys. Chem. B, Vol. 109, No. 8, 2005
Sirockin and Dejaegere
In this work, quantum mechanical calculations were used to characterize the effect of disulfide bridges on proton chemical shifts. We show that, for protons close to disulfide bridges, yet at distances from the bridge that are frequently sampled in protein structures, the magnetic effect of disulfide bridges is sizable (above 0.3 ppm on average). The data generated from the quantum mechanical calculations is then used to develop a new physical model of the shielding effects that includes the magnetic anisotropy and electrostratic contributions of the disulfide. In the course of this work, we show that the McConnell equation40 is not appropriate for describing the magnetic anisotropy effects of the disulfide bridge. An alternative equation using a two-center model based on the sulfur atoms is proposed to describe the magnetic anisotropy contribution, and we show that, by using this equation in place of the McConnell equation, much better agreement with the quantum chemical data is obtained. Methods Structures Examined. To investigate secondary chemical shifts caused by the presence of a disulfide bridge a model system consisting of a methane “probe” molecule placed successively at various positions in the vicinity of the disulfide bridge, modeled by the molecule S2H2, was constructed. In such a system, the secondary shift is computed as the isotropic shielding in the methane-disulfide dimer minus that for the isolated methane (both computed using the same method and basis set).
∆σ ) σ - σref
(1)
∆σ has an opposite sign with respect to the chemical shift (δ)
δ ) σref - σ
(2)
A three-dimensional grid was built, centered on the center of the S-S bond and extending in the three directions up to 6 Å by steps of 0.5 Å. One of the methane hydrogen atoms was placed on the grid point, and the corresponding hydrogencarbon bond was oriented along the axis pointing toward the grid origin. This procedure led to 253 ) 15 625 methane-S2H2 dimers that were used to compute secondary shifts (of those, 1901 structures have a distance between the grid point and the center of the S-S bond shorter than 3.0 Å, 520 have a distance between 3.0 and 3.3 Å, and 13 204 have a distance larger than 3.3 Å). The structures of the methane and S2H2 molecules were obtained by geometry optimizations with the DFT approach and B3LYP correlation functional with the 6-311++G** basis set. In principle, each dimer can give four secondary shifts, one for each proton. In practice, only one secondary shift was considered, corresponding to the shift of the methane proton placed on the grid (which we refer to as the probe proton). Similar approaches have been used to estimate the structural shifts in peptides, nucleic acids, and organic molecules.20,22,24,55,56 We also examined the magnetic effect of disulfide bridges at points around the S2H2 molecule. These nucleus-independent chemical shifts (NICS) allow magnetic anisotropy contributions to secondary shifts to be isolated.31 NICS have been used previously as probes for aromaticity,32 and to estimate magnetic anisotropy contribution to chemical shifts.24 For the NICS calculations, the same grid was used as previously. Dummy points corresponding to the positions of the “probe” protons (see above) were generated automatically by a perl script.33 Besides the grid points, NICS were calculated in specific directions to assess the qualitative behavior of the magnetic
Figure 1. Cones with points situated at the magic angle from the molecular principal axis for which NICS have been computed. Angles shown on the top part of the figure are in degrees. By convention, points along the direction 0 are defined as points in plane xy (cf. Fig 3) with an angle of 54.74° to the magnetic anisotropy principal axis. The other directions are generated from direction 0 by rotating around the molecular principal axis by steps of 45°.
anisotropy contribution along particular axes, namely, the direction of the molecular principal axis, directions perpendicular to the principal axis, and directions that correspond to the socalled magic angle with respect to the principal axis (see below). These latter points are situated on two symmetrical cones around the molecular principal axis (see Figure 1). Quantum Chemistry Calculations Proton and nucleus-independent shielding tensors were computed using the Gaussian 98 program34 with density functional theory. The correlation functional used was B3LYP.35 The calculations were performed using a 6-31G** and 6-311++G** basis set.36 The two basis sets gave similar results, and only the 6-311++G** data will be presented here. The GIAO method was used for gauge invariance.37 DFT has provided a straightforward approach to incorporate some
Magnetic Effects of Disulfide Bridges
J. Phys. Chem. B, Vol. 109, No. 8, 2005 3629
important effects beyond the Hartree-Fock approximation efficiently and has been shown to give a good description of chemical shifts.15,38 The GIAO method is efficient in terms of the convergence of chemical shift values. It allows reliable results with relatively small basis sets.39 Usual wave functions do not guarantee gauge invariance: the results may depend on the position of the molecule in the Cartesian frame. In the GIAO method, each atomic orbital has its own local gauge origin placed on its center.37 Nucleus-independent chemical shifts (NICS) were computed with Gaussian 98 with the same methods and basis sets as described above for proton shielding. NICS are computed by specifying points in space with which no atom will be associated but where any desired property can be computed, in this case chemical shielding. The quantum mechanical contribution to the secondary shifts not arising from magnetic anisotropy effects has been estimated as the difference between the secondary shifts calculated using the methane probe molecule and the nucleus-independent chemical shifts. The proton shielding and the NICS used in the subtraction are of course estimated at the same level of theory and at identical positions with respect to the S2H2 molecule. If we call σCH4+S2H2 the shielding of the probe proton in the presence of the disulfide molecule, σCH4 the shielding of the probe proton in the isolated methane molecule (31.73 ppm, computed with Gaussian 98, the DFT method, the B3LYP correlation functional, and the 6-311++G** basis set), and σNICS the nucleus-independent shielding, the contribution to shielding not arising from magnetic anisotropy effects is
σnot-anis ) σCH4+S2H2 - σCH4 - σNICS
(3)
Note that eq 3 is only approximate, in that as intermolecular interactions between the S2H2 and methane can modify the S2H2 wave function, they can also affect its magnetic susceptibility and therefore modify the magnetic anisotropy contribution with respect to the NICS value. However, we expect this effect to be small in the distance range we are interested in and eq 3 should be a good approximation to contributions to shielding other than those due to magnetic anisotropy. We checked for possible basis set artifacts in the computed nucleus-independent chemical shift data by using different basis sets (TZVP57 and aug-cc-pV5Z58) for a selected number of points. We also performed basis set superposition error estimates using counterpoise corrections at the 6-311++G** level for a subset of points around the disulfide (both for NICS and for methanedisulfide dimers). These data are presented in the Supporting Information. They confirmed that the 6-311++G** basis set does not give large BSSE errors and is a good compromise between speed and accuracy for the system considered. Empirical Shift Calculations In this section, we describe the different components of the semiempirical model for chemical shift calculations; the development of parameters for these components are detailed below for the magnetic anisotropy, electrostatic, and close-contact empirical shielding computational models. Magnetic Anisotropy. An asymmetry in the magnetic susceptibility of a chemical group leads, in the presence of an external magnetic field, to electrostatic currents in the group. These currents are the source of additional magnetic fields, and thus chemical shifts, at atoms surrounding the chemical group. These effects are especially important for aromatic groups, but single bonds can also have sizable magnetic anisotropies. In
TABLE 1: Coordinates of the Disulfide Atoms and Magnetic Susceptibility Tensor Data for the H2S2 Molecule Obtained with the DFT Method, B3LYP Correlation Functional, and 6-311++G** Basis Seta coordinates (Å) atom
X
Y
Z
S S H H
0.0 0.0 -0.956 0.956
1.057 -1.057 1.250 -1.250
-0.055 -0.055 0.882 0.882
magnetic susceptibility anisotropy (×10-6 cm3/mol) -8.11 molecular magnetic susceptibility (cgs) eigenvalue
X
Y
Z
-44.6 -28.8 -28.6
-0.204 0.0 0.979
0.979 0.0 0.204
0.0 1.0 0.0
a The first eigenvalue, which is -44.6 × 106 cm3/mol, corresponds to a nearly y-axial eigenvector (-0.204, 0.979, 0.000) oriented roughly from one hydrogen to the other. The deviation from the y axis is 0.296 rad or 17.0°.
this section, we describe the models we used for the magnetic anisotropy of the sulfur-sulfur bond. Axial Symmetry Model. Magnetic anisotropy effects of single bonds are most often described by the McConnell equation:40
σanis )
1 ∆χ(1 - 3 cos2 θ) 3R3
(4)
where ∆χ is the molecular magnetic susceptibility anisotropy expressed in Å3/molecule, R is the distance between the origin of the magnetic susceptibility tensor and the proton for which chemical shielding is computed, and θ is the angle between the vector joining the proton and the center of magnetic anisotropy and the anisotropy principal axis. Equation 4 assumes axial symmetry of the magnetic susceptibility tensor. In the present work, the value of the molecular magnetic susceptibility anisotropy (∆χ) for the S2H2 molecule was obtained from quantum calculations using DFT, the 6-311++G** basis set, the B3LYP correlation functional, and the CSGT method implemented in Gaussian (cf. Table 1 and the Results section). To check the sensitivity of the computed tensor to basis set effects, it was also computed using the TZVP and aug-cc-pV5Z basis sets (cf. Table 2 in the Supporting Information). Two-Center Model. An alternative approach to the bond anisotropy model of McConnell is to describe the magnetic anisotropy contributions to shielding using atom-centered magnetic susceptibility tensors. In this case, the total magnetic anisotropy is the sum of atomic contributions41,42 and is obtained by the following equation: natoms
σanis )
∑ j)1
1
∑
3R3j i)xx,yy,zz
χii(1 - 3 cos2 θii)
(5)
The geometrical parameters in eq 5 are described in Figure 2. In this equation, the total magnetic anisotropy contribution to shielding at a point a is a sum over all atoms j in the molecule. The contribution of each atom depends on the distance Rj between the atom j and the point a. For each atom, the sum over the eigenvalues of the atomic susceptibility tensor (χii) is
3630 J. Phys. Chem. B, Vol. 109, No. 8, 2005
Sirockin and Dejaegere shifts are caused by magnetic field originating in neighboring molecules, interactions with neighboring molecules that deform electron clouds also modify chemical shifts. The contributions of these “electronic deformation” effects to chemical shielding have long been recognized. With the exception of electrostatic interactions, which are first order and thus occur without modification of electronic wave functions, all interactions that contribute to intermolecular energies have the potential to affect chemical shielding. An important contribution to proton chemical shift can arise from distant polar groups, which can polarize the electron cloud surrounding the nucleus, thereby increasing or decreasing the shielding.44 For a proton, the main term is proportional to the projection of the local electric field onto the X-H (here C-H) bond vector.11
Figure 2. Schematic representation of the two-center magnetic anisotropy model. Hydrogens are omitted for clarity. The distance between the point where the magnetic anisotropy contribution to shielding is evaluated and the sulfur atom is Rj. The vector psa1 is one of the molecule’s principal susceptibility axes (the other axes have been omitted for clarity). The vectors xx, yy, and zz are the eigenvectors of the atomic susceptibility tensor. We give as an example two angles that enter eqs 5 and 6: θxx is the angle between Rj and xx (cf. eq 5) and φ1y is the angle between the principal axis and yy (cf. eq 6).
taken. The angles (θii) are defined in Figure 2. The atomic susceptibility tensors are related to the molecular susceptibility tensor by the following equations:43
{
atoms
χmol 11 )
∑
χxx cos2 φ1x + χyy cos2 φ1y + χzz cos2 φ1z
atoms
χmol 22 )
∑ χxx cos2 φ2x + χyy cos2 φ2y + χzz cos2 φ2z
(6)
atoms
χmol 33 )
∑ χxx cos2 φ3x + χyy cos2 φ3y + χzz cos2 φ3z
mol mol χmol 11 , χ22 , and χ33 are the principal values of the molecular susceptibility tensor (cf. Table 1). χxx, χyy, and χzz are the atomic susceptibility tensor terms, and the φ angles are the angles between the principal susceptibility axis of the molecule and the atomic axis. To estimate the values and principal directions of the atomic susceptibility tensor (χii), we use the following approximation:13
{
χxx ) cσxx χyy ) cσyy χzz ) cσzz
(7)
where σii are the atomic shielding tensor principal components. The value of the constant c in eq 7 was determined from eq 6 using the molecular susceptibility tensor and atomic shielding tensors of the sulfur atoms obtained at the B3LYP/6-311++G** level. The atomic susceptibilities were centered on the sulfur atoms, and the hydrogen atoms were not considered, and thus, by symmetry only one value of c is obtained. Equation 7 is overdetermined (three equations for one variable), so that the value taken for c is an average over the three values determined from eq 7. Once the atomic susceptibility tensor was determined, eq 5 was used to compute the magnetic anisotropy contribution to chemical shielding. Polarization Contribution to the Chemical Shift. Apart from magnetic anisotropy effects, where changes in chemical
σpol ) AE(X - H)
(8)
Equation 8 is the first term in a Taylor expansion, but contributions from higher order terms are expected to be very small.22 It was suggested in 1960 by Buckingham that an appropriate value for A for a C-H bond would be -2 × 10-12 esu-1.44 More recent quantum calculations suggest slightly larger values for A ranging from -2.69 × 10-12 to -4.0 × 10-12 esu-1.45,20 To isolate the polarization contribution to chemical shielding, we used a data set where the distance between the probe proton and any disulfide atom is larger than 3.3 Å. For this data set, the contribution of close-contact interactions is expected to be small22 and the quantum chemical change in shielding should be dominated by the polarization contribution. The value of 3.3 Å was used as a cutoff for neglecting close-contact interactions in studying secondary shift effects for protons close to an argon atom22 and should thus be an appropriate value for sulfur which is very similar in size (the van der Waals radii are 1.88 and 1.80 Å for argon and sulfur, respectively46). This set of DFT shifts comprises 13 204 points. Close-Contact Contribution to the Proton Chemical Shift. Dispersion and repulsion interactions have been described as contributing a so-called “close-contact” term to shielding.47 The notion that dispersion contributes to chemical shift was first proposed by Stephen48 and Buckingham.49 Studies of solvent effect on chemical shifts, particularly on shifts of nonpolar probes such as rare gases, have shown that changes in shielding upon solvation are correlated to dispersive interaction energies.50 At close proximity, London forces due to correlations of fluctuating dipoles can induce a buildup of electron density between atoms. The resultant loss of electron density near the nuclei is expected to decrease chemical shielding. It has been shown however that this description of close-contact contributions does not agree quantitatively with changes in shielding calculated by quantum chemical DFT methods.22 Indeed, deshielding at close contact is observed from DFT computations of chemical shifts even though these methods do not adequately describe dispersive forces, and it is observed in a range of distance where exchange-repulsion overcome dispersive interactions. Moreover, dispersive effects on shielding, as estimated for example from the Drude formula, are smaller than those observed in quantum calculations.22 It is thus not entirely clear whether exchange-repulsion or dispersive forces are responsible for the close-contact contribution to shielding. Nevertheless, a rough dependence of shielding on the inverse sixth power of interatomic distances was observed to hold.22 We introduced a close-contact contribution to shielding for a
Magnetic Effects of Disulfide Bridges
J. Phys. Chem. B, Vol. 109, No. 8, 2005 3631
Figure 3. Views of the disulfide molecule with the magnetic anisotropy principal axis (see text) shown as a green line: (left) view of the Oxy plane; (right) view of the Oyz plane. It should be noted that the principal axis passes through the origin and lies in the Oxy plane. The two sulfur atoms are situated 0.055 Å below the Oxy plane.
data set of points that are situated between 3.0 and 3.3 Å from the center of the disulfide bond. To model the close-contact contribution, we used the simple equation sulfur atoms
σclco )
∑
B R6
(9)
where R is the distance between the probe proton and a sulfur atom and the sum is taken over both sulfurs. The constant B has units of ppm‚Å6 and is determined by a least-squares fit procedure (see below). Parameter Fitting Model for Magnetic Anisotropy Contribution. We considered two models for the magnetic anisotropy contribution to chemical shielding, namely, the fully axial and two-center models. To assess which model gives the best description of the quantum chemical NICS data, we compared the NICS calculations to the chemical shifts modeled by eqs 4 and 5, respectively. The molecular and atomic magnetic anisotropy tensors were determined independently from quantum chemical calculations. The comparison of semiempirical magnetic anisotropy contribution and quantum NICS data therefore did not involve parameter fitting. Fit of Magnetic Anisotropy, Polarization, and CloseContact Contribution. The total empirical change in shielding is given by
∆σempirical ) σanis + σpol + σclco
(10)
where the anisotropy (σanis), polarization (σpol), and close-contact (σclco) contributions are computed via the models and equations described above. To compare the total change in shielding obtained from the quantum chemical calculations to the predictions of the semiempirical model, a nonlinear optimization procedure was used.11,22 This procedure yields values for the magnetic anisotropy (c, see eq 7), electrostatic (A, see eq 8), and close-constact (B, see eq 9) coefficients. An estimate of the uncertainty of the fitted parameters is obtained by a jackknife procedure.22 It should be noted that, in this global fitting procedure, the coefficient c for the magnetic anisotropy contribution is used as a variable, contrary to what is done when assessing the magnetic contribution alone (see above). A comparison of the fitted and quantum chemical c values is presented in the Results section. Results Chemical Shift Anisotropy. Computed Magnetic Susceptibility. We computed the magnetic susceptibility tensor and
magnetic anisotropy of the S2H2 molecule, as these would give us a first indication of the magnitude of magnetic anisotropy effects of sulfur-sulfur single bonds.51 The orientation of the magnetic susceptibility tensor is shown in Figure 3. The ab initio results gave a magnetic anisotropy of -12.9 × 10-6 Å3/molecule (-8.1 × 106 cm3/mol; see Table 1), which can be compared to the value calculated at the same level for formamide: -8.5 × 10-6 Å3/molecule (-5.1 × 106 cm3/mol; similar values were obtained for all basis sets tested, cf. Table 1 and Table 2 in the Supporting Information). The ab initio value for the peptide bond of formamide is comparable to the value used in semiempirical models of chemical shifts which is -7.9 × 106 cm3/mol22 and very close to that obtained from Zeeman effects and bulk susceptibility measurements of formamide (-5.51 × 106 cm3/ mol).52,53 Thus, the computed magnetic anisotropy of the S2H2 molecule is larger than that of formamide, which supports the idea that disulfide bridges could have a significant influence on the shifts of neighboring protons. The magnetic anisotropy of the peptide bond is at the origin of the systematic differences between HR chemical shifts in β-sheets and R-helices, which is exploited in the chemical shift index method.5 This value also compares well to that reported for carboncarbon single bonds (7.7 × 106 cm3/mol).43 However, it must be noted that the sign of the magnetic anisotropy of disulfide bonds is opposite to that of carbon-carbon bonds, which means that protons situated along the bond S-S axis will be shielded. Various models have been proposed to describe the effect of magnetic anisotropy centers.51 In what follows, we fit the quantum chemical estimates of magnetic anisotropy effects obtained using the nucleus-independent chemical shift (NICS) to various models and discuss which models give the best agreement with the quantum chemistry data. Axial Symmetry Model. The values of the nucleus-independent chemical shift (NICS) are used as reference data for the magnetic anisotropy contribution to secondary chemical shifts around the disulfide molecule. The computed NICS ranged from -0.19 to +0.35 ppm for points situated 3.0 Å or more from any S2H2 molecule’s atoms. This range of value is consistent with the magnitude of secondary chemical shifts expected for magnetic anisotropy contributions of single bonds. The ab initio results (cf. Table 1) suggested that an axially symmetric model could give a good description of the disulfide bridge magnetic anisotropy. We therefore compared the quantum chemical NICS data to those computed using eq 4, using a value for ∆χ of -12.9 × 10-6 Å3/molecule. The results of the correlation between the two sets of data is presented in Figure 4. It can be seen from Figure 4 that there is a correlation between the two sets of data but that there are many outliers, particularly for negative shielding values. The McConnell equation does
3632 J. Phys. Chem. B, Vol. 109, No. 8, 2005
Sirockin and Dejaegere
Figure 4. Correlation between the Gaussian B3LYP/6-311++G** nucleus-independent chemical shielding (Bq σ (ppm)) and the magnetic anisotropy contribution calculated using the McConnell equation (σanis (ppm)). See text for details. (b) points situated 3.3 Å and more from the S2H2 molecule; (O) points situated between 3.0 and 3.3 Å from the S2H2 molecule. The regression for the whole data set (b and O) has a slope of 0.36, an intercept of -8.6 × 10-4 ppm, and a correlation coefficient (R2) of 0.44.
not account quantitatively for the quantum chemical NICS values. The slope of the regression line is 0.36, which shows that, to obtain quantitative agreement, one should use a much larger value of the magnetic anisotropy of the disulfide bridge than was evaluated by the quantum mechanical calculations of the susceptibility tensor. It is also readily apparent that the ab initio data do not conform qualitatively to the McConnell equation. According to eq 4, the magnetic anisotropy contribution to the shielding along the principal axis and in directions perpendicular to the principal magnetic anisotropy axis should be in the ratio 2:-1 (with, in the case of the S2H2 molecule, increased shielding along the principal axis). NICS data for points along and perpendicular to the principal axis are presented in Figure 5. It can be seen from Figure 5 that the quantum chemical data indeed have the expected parallel/perpendicular ratio for points situated far from the center of the disulfide bridge, but significant deviations from the prediction of the McConnell equation are already apparent 8 Å from the center of the bridge (where the ratio is equal to -2.7) and the discrepancy gets significatively higher as the distance to the bridge gets smaller (cf. Figure 5; similar data are obtained with the TZVP and aug-cc-pV5Z basis sets, cf. the Supporting Information). The McConnell equation also predicts that magnetic anisotropy effects vanish for points along directions at an angle of 54.74° (the so-called “magic angle”) to the magnetic anisotropy principal axis. NICS were computed along four axes at the magic angle with respect to the magnetic anisotropy principal axis, as shown in Figure 1. It can be seen that the NICS differ significantly from zero as the distance to the disulfide bridge decreases; see Figure 6. Moreover, the values of the shielding differs for different axis along the cone, while the McConnell equation predicts that all axis should have identical (null) shielding on the magic angle cone. Since the McConnell equation is a Taylor series expansion limited to the dipole term,24 higher order terms are expected to contribute as the distance to
the magnetic anisotropy center decreases. This situation is similar to the well-documented convergence problems of Taylor series expansion of molecular electric multipole for the computation of electric properties of molecules.54 However, a statistical analysis of NMR structures deposited in the Protein Data Bank10 shows that nearly two protons are found on average at a distance of 3.5 Å from the center of disulfide bridges in deposited protein structures. The probability of finding a proton becomes very low only at distances
