Computation of Deuterium Isotope Perturbation of 13C NMR Chemical

Oct 13, 2010 - A simple model based on the ground (zero point) vibrational level and treating only the C−H(D) degrees of freedom (local mode approac...
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J. Phys. Chem. A 2010, 114, 12283–12290

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Computation of Deuterium Isotope Perturbation of 13C NMR Chemical Shifts of Alkanes: A Local Mode Zero-Point Level Approach Kin S. Yang† and Bruce Hudson* Department of Chemistry, Syracuse UniVersity, Syracuse, New York 13244-4100, United States ReceiVed: June 26, 2010; ReVised Manuscript ReceiVed: September 13, 2010

Replacement of H by D perturbs the 13C NMR chemical shifts of an alkane molecule. This effect is largest for the carbon to which the D is attached, diminishing rapidly with intervening bonds. The effect is sensitive to stereochemistry and is large enough to be measured reliably. A simple model based on the ground (zero point) vibrational level and treating only the C-H(D) degrees of freedom (local mode approach) is presented. The change in CH bond length with H/D substitution as well as the reduction in the range of the zero-point level probability distribution for the stretch and both bend degrees of freedom are computed. The 13C NMR chemical shifts are computed with variation in these three degrees of freedom, and the results are averaged with respect to the H and D distribution functions. The resulting differences in the zero-point averaged chemical shifts are compared with experimental values of the H/D shifts for a series of cycloalkanes, norbornane, adamantane, and protoadamantane. Agreement is generally very good. The remaining differences are discussed. The proton spectrum of cyclohexane- is revisited and updated with improved agreement with experiment. There has been considerable interest in the use of deuterium substitution effects on NMR spectra for establishing structures without significant perturbation. Considerable experimental study has attempted to establish the relationship between experimental H/D shifts and structure. One of the objectives of the present work is to use the available experimental data to establish the reliability of a computational method that can be used to extend this methodology to classes of molecules for which the empirical methods are not easily applied. This includes reactive species and species present in only small amounts. The method is general and can be expanded to other nuclei. Further, the method developed and tested here is applicable to very large molecules. The instantaneous NMR chemical shift of a nucleus depends on the instantaneous position of all of the nuclei of a molecule. This quantity must be averaged over the range of nuclear positions sampled by the molecule. For small molecules with single conformations the distribution may be approximated by a configuration giving the minimum potential energy of the molecule. This is the same for H and for D. The zero-point correction to the chemical shift consists of averaging over the range of nuclear positions spanned by the atomic coordinates in this lowest vibrational level. At finite temperature the inclusion of geometries sampled by excited vibrational states may be required. In any case, the change in chemical shift due to isotopic replacement must be due to a change in this vibrational averaging. We here make the approximations that the zero-point level dominates and that the effect is local to the site of the isotopic replacement. The effect of nuclear motion by all of the atoms may be important to chemical shift of a particular atom, but this will be substantially the same for the unsubstituted and D substituted molecule. This permits a local mode approximation involving only the three degrees of freedom of the single H (and D) atoms. * To whom correspondence should be addressed. E-mail: [email protected]. † Current address: Department of Chemistry, University of Chicago.

A review1 specific to the case of deuterium isotope effects on 13C NMR chemical shifts outlines the theory and prior implementations of this theory aimed at rationalizing or predicting 13C chemical shift changes induced by D substitution.2-4 Most previous work on large molecules of the size discussed here, and all treatments of perturbations over more than one bond, associate the H/D effect with the change in the CH bond length when H is replaced by D. This change in bond-length is due to the well-known and measurable anharmonicity of the CH bond. As noted,1 treatments of small molecules that include the other degrees of freedom show that the bending motions have a significant effect and that this can be opposite in sign to that of the stretching motion. Another approach to computation of this isotope effect is to determine the response of the chemical shifts of interest to displacements along each of the normal modes of the molecule and to then average this response over the mean square amplitude of motion of each normal mode for the H and D cases.5 This method includes all bending degrees of freedom and is particularly easy to implement for perdeuterated species where the symmetry and normal modes of the H and D forms are the same. The treatment used here includes both stretch anharmonicity and bending degrees of freedom in a local mode picture. In any approach, the use of a symmetric potential (harmonic, quartic, etc.) to determine the distribution probability results in an even symmetry zero-point function and thus cancellation of any effects that are linear (or cubic) in displacement. Only quadratic variations in the chemical shift of the sensing 13C atom contribute to the net H/D effect if the potential is symmetric. The present treatment uses a harmonic potential for both bending degrees of freedom and an anharmonic Morse potential for the stretch. The harmonic bending degrees of freedom thus contribute to the net H/D effect in proportion to the second derivative of the chemical shift with respect to bend times the change in the mean square amplitude (“breadth”) due to H/D substitution. The treatment of the stretch degree of freedom includes both the “shift” (change in equilibrium position) of

10.1021/jp105913x  2010 American Chemical Society Published on Web 10/13/2010

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TABLE 1: 13C H/D Chemical Shift for Cyclohexane for a B3LYP 6-311++G(3d,3p) DFT GIAO Treatment and an Initial S6 Symmetry MP2 6-311++G(3d,3p) Geometry n

∆n

row

1

2

3

4

-87.94 -91.03 413.66 174.39 588.05 7.18 9.19 -82.98 -101.38 403.70

-8.70 -6.22 40.94 11.92 52.86 -0.483 -2.73 5.58 30.12 88.56

-3.39 -0.53 15.97 1.02 16.99 -0.474 0.740 5.48 -8.16 14.31

-0.34 -1.70 1.61 3.25 4.86 0.038 0.133 -0.44 -1.47 2.95

-9.94 -5.37 46.55 10.19 56.74 -1.69 -3.42 19.52 38.46 114.72 101.6 103.7

-1.24 -2.06 5.81 3.91 9.72 -0.937 -0.584 10.84 6.56 27.12 20.7 24.9

-0.01 0.00 0.03 0.00 0.03 0.038 -0.272 -0.44 3.05 2.64 2.8

stretch 1st and 2nd (dσ/dR)0 derivatives (d2σ/dR2)0/2 individual stretch δR(dσ/dR)0 contributions (ppb) δ(R2)(d2σ/dR2)0/2 stretch total bend 2nd derivatives in-plane ×1000 (d2σ/dθ2)0 transverse ×1000 bend in-plane contributions (ppb) transverse total axial D shift (ppb)

axial D 1a 2a 3a 4a 5a 6a 7a 8a 9a 10a

stretch (dσ/dR)0 derivatives (d2σ/dR2)0/2 stretch δR(dσ/dR)0 contributions (ppb) δ(R2)(d2σ/dR2)0/2 stretch total bend second derivatives in-plane ×1000 (d2σ/dθ2)0 transverse ×1000 bend in-plane contributions (ppb) transverse total equatorial D shift (ppb) axial/equatorial average (ppb) literature value (ref 8)

equatorial D 1e -80.39 2e -92.29 3e 376.33 4e 175.10 5e 551.43 6e 9.36 7e 7.92 8e -108.22 9e -89.00 10e 354.21 calc 379.0 exp 418

prior treatments and adds the “breadth” change contributions. There are then four contributions to the overall effect. Each contribution and their sum may be positive or negative in sign. Treatments of NMR or other molecular properties using similar thermal averaging methods have been described.6 The present study uses the same model and computational method previously applied to the case of the vicinal 1H resonances of cyclohexane-d.7 In this work the experimental determination of this effect was performed at low temperature so that the axial- and equatorial- species were distinguishable. For a room temperature situation the two deuteroisomers have to be averaged. Cyclohexane is used to illustrate the computational method as applied to the 13C results. The first-order correction for thermal population is introduced empirically. The prior proton chemical shift results for cyclohexane are updated. Computational Method Illustrated with Cyclohexane The 9.4 T (100.61 MHz) 13C NMR spectrum of cyclohexaned1 shows H/D differences in the 13C chemical shift values for the substituted position (1∆) of 418.0 ppb. The neighboring positions, 2∆, and next neighboring, 3∆, positions have shifts relative to H of 103.7 and 24.9 ppb. The experimental errors are reported as (1.4 ppb.8 The 4∆ value is not reported. The sample was at room temperature. Table 1 shows the method of analysis of this spectrum illustrating each component contribution to the deuterium substitution chemical shift changes. The axial deuterium and equatorial deuterium cases are treated separately. All geometry and NMR chemical shift calculations are performed using Gaussian03.9 Average bond lengths and distribution widths are determined from a computed stretch or bend potential surface using the FGH method.10 The initial minimum energy geometry for the molecule is established by optimization using a particular method and basis set, usually MP2/6-311++G(3d,3p). The NMR chemical shifts at that geometry are determined with another method and basis set, usually DFT B3LYP/6311++G(3d,3p). The NMR chemical shifts are then computed for geometries for which one H atom is moved a small amount

away from equilibrium. This is the atom at the position of deuterium substitution. Three or four positions for each sign of displacement from the equilibrium position are used for the stretch and for each bending degree of freedom. The resulting change in chemical shift with displacement is represented as a Taylor series in the displacement. For the stretch this is σ(R) ) σ(R0) + (R - R0)(dσ/dR)0 + (R - R0)2(d2σ/dR2)0/2. The values for each of these derivatives (dσ/dR)0 and (d2σ/dR2)0 are given in rows 1a, 2a, 1e, and 2e of Table 1 where results are given for a B3LYP 6-311++G(3d, 3p) density functional theory treatment of the NMR spectrum using the GIAO method and an S6 symmetry MP2 6-311++G(3d, 3p) geometry. For the stretching deformation, both (dσ/dR)0 and (d2σ/dR2)0 are negative in all cases. The zero-point correction to the 13C or 1H chemical shifts for the parent hydrogen case is the average of σ(R) over the range of values of R and the angular variables that have significant probability in the ground state using the probability distribution for H. The zero-point correction is 〈σ〉H - σ(R0) where the subscript H (or D) means that the averaging is performed over the distribution function for that isotope. If R0 is defined as the average value of R for the H case, 〈R〉H, then since 〈R - R0〉H ) 〈R〉H - R0 ) 0 and this averaging causes the linear contribution to the correction to vanish so only the quadratic term contributes. If this is repeated for D instead of H, and the value of (〈σ〉D - 〈σ〉H) is computed, then the linear term makes a contribution to the shift of 〈(RD - R0)〉(dσ/dR)0 ) (〈R〉D - 〈R〉H) (dσ/dR)0. For the anharmonic potential representing the energy for CH (or CD) bond stretching, 〈RD〉 is less than 〈RH〉. Since both factors are negative this “primary geometric effect” makes a positive contribution to σD - σH. This is the contribution that has been used in previous treatments of the isotope effect for molecules larger than tetrahedral. If, instead, R0 is defined as the position of the minimum of the potential, Rm, then the zero-point correction to the 13C and 1H chemical shifts for the parent hydrogen case, 〈σ〉H - σ(Rm) includes a linear term given by (〈R〉H - Rm) (dσ/dR)m. The corresponding (〈R〉D - Rm) (dσ/dR)m in the calculation of 〈σ〉D

13C

NMR Chemical Shifts of Alkanes

- 〈σ〉H cancels this reference state shift term resulting in the same contribution to 〈σ〉D - 〈σ〉H as when R0 is 〈R〉H. The second derivative term, (R - R0)2(d2σ/dR2)0, when averaged over the H and D distributions, results in a contribution to 〈σ〉D - 〈σ〉H of [〈(R - 〈R〉)2〉D - 〈(R - 〈R〉)2〉H](d2σ/ dR2)0/2 ≡ δ(R2)(d2σ/dR2)/2. This term represents the fact that the H and D atoms span different ranges in their excursions from the equilibrium. The second derivative terms gives the extent to which the chemical shift increases (or decreases) for such excursions in either direction. Since the width of the distribution for the D case is narrower than that for the H case, this term is negative, as is the second derivative term for stretches and so the overall contribution to the effect is positive for this quadratic variation. Values of δR ≡ 〈R〉D - 〈R〉H ) -0.0047 Å and δ(R2) ≡ 〈δR2〉D - 〈δR2〉H ) -0.0019 Å2 are computed for cyclohexane using a Morse fit to the computed stretch potential using FGH.10 These values are found to be the same for axial and equatorial CH bonds. For cyclohexane with an axial D, the multiplication of these 〈RCH〉 - 〈RCD〉 and 〈R2CH〉 - 〈R2CD〉 values by the (dσ/ dR)0 and (d2σ/dR2)0/2 derivative values gives, for the 1∆ case, contributions of 414 and 174 ppb, respectively, for a total of 588 ppb (“stretch total”) in rows 3a-5a of Table 1. The same geometric factors multiply the corresponding derivatives for the n ) 2-4 13C resonances for atoms more distant from the site of D substitution. The same procedure is followed for each of the two bending degrees of freedom. Only the second derivatives of the chemical shift with respect to angle variation are given for the bending degrees of freedom (Table 1, rows 6a, 7a, 6e, 7e) because the potential appears to be harmonic and the linear term cancels when averaged over a symmetric distribution. Computations for the local bending potential give values δ(θ2) ) 〈θ2〉D - 〈θ2〉H ) -11.589°2 (ax.) and -11.569°2 (eq.) for the in-plane bend, whereas the corresponding transverse values are -11.030°2 (ax.) and -11.232°2 (eq.). These geometric effect values multiply the computed second derivatives of the 13C chemical shift of the carbon separated by the number of bonds (n) from the CD substitution to give the bend contributions indicated in Table 1 rows 8a, 9a, 8e, and 9e. For both bends the 1∆ values are all negative, but for n > 1 the values are mostly positive. The total stretch contribution for the axial 1∆ case of 588 ppm (row 5a) is reduced by addition of the negative bend contributions to 404 ppb (row 10a). The axial and equatorial cases of cyclohexane are treated individually and averaged for this room temperature experiment. The 1∆ calculation for the equatorial D case is 354 ppb for the net shift (row 10e). The average of the axial and equatorial values, 379 ppb, is expected at room temperature. The value reported is 418 ppb. It is predicted that a low temperature experiment will yield two values that differ by 404 - 354 ) 50 ppb for the axial and equatorial D species. The values for the 2∆ and 3∆ 13C resonances are in agreement with experiment to almost the stated experimental error. Again, significant differences are expected at low temperature for the axial and equatorial cases. Calculations for the individual conformers are compared with experimental results for cyclohexane at low temperature and for t-butyl cyclohexane in the discussion. The computed value for the 4∆ case, which has not been reported, is small and nearly the same for both conformations. Table 2 shows comparison with experimental values for five computational methods. The third entry corresponds to Table 1. The root means square deviations (rmsd) values are given for all three experimental values followed by that excluding

J. Phys. Chem. A, Vol. 114, No. 46, 2010 12285 TABLE 2: 13C NMR Data for Cyclohexane-d1 Using Various Electronic Structure Methodsa n∆, n: experimental (ref 8): HF 6-311++G(3d,3p) B3LYP 6-31++G(3d,3p) B3LYP 6-311++G(3d,3p) B3PW91 6-311++G(3d,3p) MP2 6-311++G(3d,3p)

1

2

418.0 103.7 319.6 92.9 344.8 91.6 379.0 101.6 362.31 97.66 300.31 93.01

3

4

rmsd

24.9 16.9 -0.5 57/10 17.3 -2.8 43/10 20.7 2.8 23/3 21.18 2.79 32/5 21.06 0.42 68/8

a ppb; axial and equatorial averaged. RMSD is (all values)/(only n ) 2 and 3).

TABLE 3: Low Temperature 1H NMR H/D Shift for Cyclohexane-d1 (ppb) displaced atom vicinal H atom effect

axial D

equatorial D

axial H eq H axial H eq H rmsd

Experimental (ref 7) 6.90 7.50 10.40 7.50 6-311++G (3d,3p) 5.60 6.43 9.94 8.03 difference -1.30 -1.07 -0.46 0.53 0.91 B3LYP 6-311++G (3d,3p) 5.40 5.48 8.73 6.18 difference -1.50 -2.02 -1.67 -1.32 1.85 MP2 6311++G(3d,3p) 6.03 5.89 9.90 7.78 difference -0.87 -1.61 -0.50 0.28 0.96 HF

the 1∆ values. A series of studies with the HF method for increasing basis sets showed that smaller basis sets gave significantly worse results. This same trend is seen in the two cases given for the B3LYP DFT treatment which is overall a significant improvement over the HF case. MP2 does not do as well as DFT. Table 3 shows a comparison with the proton resonance data from ref 7, where use of low temperature permitted separation of the two isotopomers. The results presented here correct an erroneous computation of 〈R2CH〉 - 〈R2CD〉 in ref 7, omission of contributions from the transverse bend, and use here of a larger basis set. The original rmsd of ref 7 was 3.1 ppb; it is 0.91 ppb for the HF calculation of Table 3. The perturbation values are considerably smaller for the H resonances than for the 3∆ 13C values with which they might be compared since the same number of bonds are involved. The MP2 and HF results are similar. The ratio of the computational time is about 15. The computations presented below for other cases follow this pattern except that in most cases the conformational averaging is not required. The initial geometry optimization is with MP2, usually using the 6-311++G(3d,3p) basis set, in some cases with the 6-311++G(2d,2p) basis set. The NMR GIAO calculations were performed with B3LYP 6-311++G(3d,3p) basis in all cases and for some cases with other methods and basis sets as specified. The isotopic bond length shift δR ≡ 〈R〉D - 〈R〉H, shrinkage δ(R2) ≡ 〈δR2〉D - 〈δR2〉H and in-plane and transverse bend shrinkage δ(θ2) ) 〈θ2〉D - 〈θ2〉H parameters are computed for each distinct CH bond for all species investigated except that in the case of adamantane and protoadamantane uniform values are used for all CH bonds. In all cases MP2 computations are used to establish the potential energy variation with bond length or angle deformation for use in determining the mean square amplitudes. The values used for adamantane and protoadamantane are determined from MP2 calculations of the potential energy for planar cyclopentane. The degree of agreement between theory and experiment justifies this approximation as shown below. The assumption that all isotopic species can be treated with the same local bond parameters is analyzed in detail in the Supporting Information. None of the values are in any case adjusted to fit the experimental results.

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Figure 1. Numbering and D substitution for norbornane.

Yang and Hudson

Figure 2. Correlation of observed and calculated 13C isotopic chemical shifts for norbornane. The reported uncertainties for the experimental values are smaller than the size of the circles used.

TABLE 4: Norbornane 13C Chemical Shifts H/D Differences Calculated with B3LYP 6-311++G (3d,3p)a No.

CD 13 C: obs calc 13 C7 C: obs calc 13 C2 exo C: obs calc C2 endo 13C: obs calc

1

C1

2 3 4

a

1 376.4 382.8 7 361.1 365.2 2 364.9 340.2 2 384.0 374.5

2 7 3, 5 4 103.6 91.2 7.3 -36.7 112.4 104.9 12.3 -29.3 1,4 2,3 5,6 88.5 26.2 -6.6 102.2 22.0 -15.6 1 3 4 7 6 87.4 120.2 14.6 12.3 43.7 93.3 108.8 16.4 9.2 29.4 1 3 4 7 6 100.9 89.1 -6.7 -5.2 50.8 94.3 96.5 -5.8 -14.6 70.8

Observed values from ref 11.

Results

Figure 3. Numbering and deuterium substitution for adamantane.

TABLE 5: Adamantane D/H 13C Shiftsa C2D

2

1

4

8

6

7

calc obs

403.56 440.00

94.70 100.00

13.82 13.00

18.40 31.00

1.47 4.00

6.52 3.00

C1D

1

2

3

103δR

calc

466.43

129.36

30.20

-4.94 Å

obs

514.00

127.00

32.00

a

Norbornane. Four monodeutero derivatives of norbornane (Figure 1) have been investigated with respect to deuterium induced changes in 13C chemical shifts.11 The results of the experimental study are compared with the computed results in Table 4 and Figure 2. The rms deviation for all 21 values is 10.6 ppb. This is reduced to 9.7 if the 1∆ values are excluded. The rmsd for the four negative values is 7.4. Figure 2 shows that the correlation line constrained to pass through the origin has a slope very near unity and that the large negative value for C4 in 1 (bridgehead to bridgehead) is correctly predicted. The overall agreement of the computed and observed values for this case is very good and appears to be more uniform than for the protoadamantane case considered below. Adamantane. The isotopic 13C shifts of the two possible monodeutero derivatives of adamantane (Figure 3) have been investigated in two studies.12,13 The results of the more recent and more extensive of these studies is compared to the calculated values in Table 5. The rmsd of the results overall is 20.6 ppb, considerably larger than for norbornane. Most of the deviation for adamantane is due to the two 1∆ values. If these are removed, the rmsd for the 7 pairs of n∆, n > 1 values is only 5.5 ppb. In this treatment of adamantane we have assumed that all of the

103δ(R2) δ(θ2)ip

δ(θ2)tr

-1.87 Å2

-11.307 deg2

-11.769 deg2

Observed values from ref 13a.

CH bonds have the same values of shift δR, radial shrinkage δ(R2) and shrinkage δ(θ2) parameters for both angular deviations. The values used are taken from the case of planar cyclopentane and are given in Table 5. The approximation that the same values apply to both bonds can be tested by plotting the degree of agreement between the computed and observed values of the 13C chemical shifts for the two adamantane isotopomers individually and comparing them. This is shown in the Supporting Information Figure S1. The slopes of the lines constrained to pass through the origin for the C1D and C2D cases are 1.0933 and 1.0894. It is concluded that these are the same and that the assumption that the bonds are the same is compatible with the data. Because of the Td symmetry of adamantane and the large number of sp3 centers, the Gaussian 03 UltraFine grid option was used for the DFT integrals. Even with that it was decided that the symmetry point in the computation of chemical shifts as a function of angular and stretch displacements should be removed in the fit to obtain first and second derivatives. This is attributed to the difficulty with respect to the grid density of

13C

NMR Chemical Shifts of Alkanes

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Figure 4. Structures of protoadamantane-d1 isotopomers.

DFT calculations for symmetric systems with large numbers of tetrahedral atoms. Protoadamantane. The case of the low symmetry ring system protoadamantane provides 53 nonzero values of deuteriuminduced chemical shifts for the eight monodeuterated species shown in Figure 4.14 The results of B3LYP 6-311++G(3d,3p) DFT calculations for these eight species are given in Table 6 where they are compared with the experimental values. The 13C resonances are indicated by the carbon numbering for structure 1a. The degree of agreement is indicated in Figure 5 and in Table S1 of the Supporting Information. The rmsd for all 53 data points is 14 ppb. For the 8 1∆ points the rmsd is 30 ppb; for all others rmsd ) 8 ppb. For n∆, n ) 2, 3, and 4 individually the rmsd values are 8.4, 7.4, and 9.4 respectively. Five negative values of the chemical shift difference are observed; five negative values are calculated and correspond to those observed to be negative. The rms deviation for the five negative values is 11.7. This is considerably larger than for the 8 smallest positive values for which the rmsd is 4.2. The negative numbers appear to have a greater deviation between theory and experiment than do the comparable positive values. The treatment of protoadamantane assumes that all CH bonds have the same structural changes when H replaces D. The consistency of this assumption with the data is discussed in the Supporting Information in Figure S2 and Table S2. Small Cycloalkanes. The small cycloalkanes8 present more of a challenge to these computations. The result of these efforts is shown in Table 7. Cyclopentane is treated as planar. Cyclobutane is treated in both planar and puckered geometry with puckered results being much closer to the experiment. Restricting attention to the B3LYP/6-311G++(3d,3p) case, the rmsd of all (3) values for cyclopentane is 42 ppb but the 2∆

and 3∆ values are quite close to the experiment. Similar results are observed for cyclobutane. For cyclopropane the situation is reversed with the 1∆ case being more accurate than the 2∆ case and the 1∆ case is slightly above the experimental value. For cyclobutane and cyclohexane the computed values for 1∆ is 10% lower than that observed; for cyclopentane it is 20% too low. The average underestimate for the 1∆ cases for protoadamantane, norbornane and adamantane is 6%. Cyclopropane shows by far the largest deviation between the computed and observed 2∆ value. The B3LYP 6311G++(3d,3p) method gives 112.9 ppb versus an observed value of 64.1 ppb. This is despite the fact that the same method and basis gives among the best agreement for the 1∆ value. MP2 gives some improvement for the 2∆ value at the expense of a much worse value for 1∆. Discussion The procedure used here and in previous work7 breaks the problem of computation of the D/H shift into a structural change factor and a chemical shift response factor. The structural change factor refers only to the CH that is substituted by D; the chemical shift response part depends on how far away the perturbed 13C is from the substituted CD and their mutual orientation. The structural change factor depends on the accuracy with which the energy as a function of deformation is computed by the electronic structure method. The chemical shift response factor depends on how well the method used describes the change in electron density at the sensing 13C. Since this is a difference in the D and H values it is less demanding than an absolute chemical shift value. The overall agreement between the computed and observed deuterium induced chemical shifts is quit good especially for the values for n∆ n > 1 which are of greatest interest from a stereochemical or conformational point of view. A linear fit to all 76 n∆, n > 1 values constrained to pass through the origin (Figure S3) gives a slope of 1.013 and R2 ) 0.95. Elimination of two outlier points (one of which is that for cyclopropane just mentioned; the other is the norbornane 2-endo D to 13C6), gives a slope of 1.033 and R2 ) 0.97. The fact that the correlation line for the n∆, n > 1 cases is so close to unity suggests that the simple model used here is approximately complete (or at least that that which has been left out has cancelation of contributions). All values that are observed to be negative are calculated to be negative (and vice versa) but these values appear to have somewhat larger deviations as discussed for protoadamantane

TABLE 6: Protoadamantane: Calculated and Observed 13C Chemical Shift H/D Difference (ppb) D 1a

2

1b 2a

4

2b 3a

5

3b 4a 4b

10

13

C:

cal exp cal exp cal exp cal exp cal exp cal exp cal exp cal exp

1

2

3

86.8 90.0 91.6 90.0

355.6 376.0 359.8 407.0 18.0 31.0 30.5 24.0 -4.6 -12.0 -31.3 -17.0 23.4 29.0 20.4 25.0

93.8 105.0 81.0 83.0 83.6 101.0 86.7 95.0 22.5 20.0 1.1 8.0

84.8 89.0 102.1 104.0

4

44.6 52.0 357.0 374.0 397.0 417.0 97.0 98.0 89.7 95.0 -10.2 -8.0 5.2 5.0

5

-24.0 -6.0 72.9 78.0 107.0 109.0 370.9 392.0 384.1 417.0 21.9 31.0 47.6 50.0

6

7

1.3 5.0

11.3 14.0

20.2 23.0 90.3 90.0 88.8 107.0 86.4 91.0 90.4 104.0

-4.4 -14.0

8

9

10

13.3 20.0 4.5 7.0

9.2 15.0 22.1 27.0

24.8 37.0 6.9 12.0

18.0 31.0 25.6 34.0

63.9 55.0 17.4 15.0 23.0 32.0

393.6 421.0 411.0 453.0

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Yang and Hudson TABLE 7: 13C NMR D/H Shifts for Small Cycloalkanes

HF B3LYP

HF B3LYP B3LYPa B3LYPb B3PW91

HF HF HF B3LYP B3LYP B3LYP B3PW91 B3PW91 PBEPBE PBEPBE MP2 MP2 a

Figure 5. Correlation of calculated and observed 13C D/H chemical shifts for protoadamantane. The upper inset shows the 1∆ values on the same scale. The correlation line is fit to all the data. The error bars shown are twice the reported digitization uncertainty.

and in contrast to norbornane. The negative values occur for 3∆ or 4∆ involving five-member rings. These correspond to rather large interatomic distances. A hypothesis as to the origin of the larger variation between theory and experiment for the negative values is that they are related to excitations of low frequency modes of vibration that move distant components of the molecule relative to each other. This is probably more of a factor for protoadamantane than for norbornane. The origin of the problem with cyclopropane is not clear. It may derive from a deficiency of the basis sets used or some other special consideration associated with the ring strain and

cyclopentane

1

2,5

3,4

experimental (ref 8): 6-311++G(3d,3p) 6-311++G(3d,3p)

373.7 273.5 301.3

101.3 88.0 94.7

-12.1 -27.0 -11.0

cyclobutane

1

2,4

3

experimental (ref 8): 6-311++G(3d,3p) cc-pcqz 6-311++G(3d,3p) 6-311++G(3d,3p) 6-311++G(3d,3p)

363.0 279.2 330.4 265.3 328.4 310.8

146.6 123.2 136.7 131.7 134.0 129.9

27.2 26.4 23.5 5.8 24.3 24.4

cyclopropane

1

2,3

experimental(ref 8): STO-3G 6-311++G(3d,3p) cc-pvqz 6-31G(2d,2p) 6-311++G(3d,3p) cc-pcqz 6-31++G(3d,3p) 6-311++G(3d,3p) 6-31++G(3d,3p) 6-311++G(3d,3p) 6-31G(2d,2p) 6-311++G(3d,3p)

308.7 171.0 266.2 269.5 303.8 320.8 326.5 272.0 303.2 284.2 317.1 216.6 228.8

64.1 58.5 103.9 104.6 102.2 112.9 111.6 98.8 108.2 100.2 111.0 92.9 99.2

Planar; b Puckered. Axial and equatorial averaged.

hybridization of this species. It would be interesting to know if this discrepancy occurs for lower symmetry substituted cyclopropane species. The most likely source of this discrepancy is the need for vibrational averaging that takes into account vibrationally excited levels. This is discussed below. For cyclobutane a planar model using B3LYP 6-311++G(3d,3p) gives a 3∆ value for C3 that is about 6 ppb while the observed value is 27 ppb. Using this same computational method but taking into account the puckered S4 geometry of cyclobutane and averaging axial and equatorial CD values gives a value of 24 ppb. This illustrates the importance of treating the actual geometry of the molecule properly and points to minor errors in the molecular geometry or in conformational averaging as an important contribution to deviations between computed and observed values. This same consideration is likely the origin of the discrepancy between the observed and computed results for cyclopentane. This is, perhaps surprisingly, only significantly out of line for the 1∆ case. Preliminary examination of a nonplanar geometry for cyclopentane shows that large differences are found for the major stretch component of the chemical shift. Dynamical averaging in this case requires treatment of the full pseudorotation coordinate. Returning to the case of cyclohexane, comparison was made in the introductory example with a room temperature experiment. It is also possible to compare the computed results for the individual conformations with experimental results obtained at low temperature or for t-butyl-cyclohexane15 with a single D at position 4 in either of the two locked axial or equatorial positions (Table 8). The low temperature experiment for C6H11D provides only the 1∆ values for which each of the computed values are about 10% too low as is their average. The difference in the two computed values thus agrees with that observed. The same pattern is observed for the 1∆ values for t-butylcyclohexane. The individual 2∆ and 3∆ values for the isomeric t-butyl-cyclohexanes show that the very good agreement with

13C

NMR Chemical Shifts of Alkanes

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TABLE 8: Comparison of Calculated and Observed12 13C Chemical Shift Perturbations for C6H11D at -80 C and t-Butyl-cyclohexane tBu-cyclohexane-d1(4) 1

axial

exp calc equatorial exp calc axial-equatorial exp calc axial-equatorial average exp (RT) exp (ave) calc

∆

444.9 403.7 396.4 354.2 48.5 49.5 418 420.7 379

1

∆

2

∆

3

∆

442.2 98.6 14.7 403.7 88.6 14.3 392.4 106.2 37.7 354.2 114.7 27.1 49.8 -7.6 -23 49.5 -26.2 -12.8 103.7 24.9 417.3 102.4 26.2 379 101.6 20.7

the observed room temperature averaged results is due in part to compensation of errors with opposite signs. As a result, in this case the computed difference between the observed values is not close to the observed value. The vibrational averaging effects that are proposed to be important in respect to the case of the long-range deuterium effects for protoadamantane involve averaging over a low frequency global deformation of the molecule. This thermal excitation population is also present in the case of the reference nondeuterated species. It is only differences in this thermal population that will contribute to the effect observed. Within the context of the present local mode model there is the additional, more tractable and probably more important, issue of thermal population of the two bending degrees of freedom of the CH and CD bonds themselves. The stretching degree of freedom is too high in frequency to be significantly populated at room temperature. However, the bending degree of freedom is around 1000 cm-1 and might have some nonzero-point population at room temperature, especially for the lower frequency CD bending case. Remaining with the case of cyclohexane, it is possible to evaluate the effect of vibrational population of the excited states of the bending degrees of freedom by considering its effect on δ(θ2) ) 〈θ2〉D - 〈θ2〉H. Use of the computed values for the in-plane and out-of-plane values for this change in bending motion width for the axial and equatorial CH and CD bonds results in deviations of computed and experimental values (Table 1) of the 1∆, 2∆, and 3 ∆ values of 36.6, 2.5, and 2.1 ppb. The rms deviation is 22.7 ppb, dominated by the 1∆ contribution. The four computed values of δ(θ2) ) 〈θ2〉D - 〈θ2〉H used in this calculation are nearly the same with an average value of the ) -11.35°2 The effect of thermal population is expected to make this quantity smaller in absolute magnitude because the increase in mean square amplitude due to population of excited states will be greater for the lower frequency CD bond than for the CH bond. When these several values of δ(θ2) ) 〈θ2〉D - 〈θ2〉H are scaled by a multiplicative factor that is optimized to minimize the rms deviation, the scale factor is found to be 0.80 and the resulting average value of δ(θ2) ) 〈θ2〉D - 〈θ2〉H ) -9.12o2 is obtained. With this quantity optimized, the 1∆, 2∆ and 3∆ deviations are 1.7, 8.3, and 2.8 ppb and the rms deviation is reduced to 5.1 ppm. The change in the rms deviation by a factor of 4 and the direction and magnitude of the difference from the original value -11.35°2 to -9.12°2 indicates that the thermal population effect results in improvement in agreement with experiment. It is interesting that this has the effect of bringing the 1∆ value into line with the others experimental results. The same treatment of the vibrational averaging can be applied to the case of cyclopropane where considerable improvement in agreement with experiment is observed because

of the opposite sign of the deviations of the experimental and computed values for the 1∆ and 2∆ 13C frequencies and the opposite signs of the contribution of the bending contributions to the two computed values. A fit to the data results in a scale factor for the bending contribution that is less than 1 as is anticipated. However, this treatment should be performed in a de novo quantitative fashion in order to be convincing. The approach taken here is an extension of previous work in which the entire effect of replacement of H by D is attributed to a shortening of the CX bond length when H is replaced by D. This effect is one of the four components of our calculation where this contraction multiplies the linear variation of the chemical shift with CX length. The correlation between the contribution from the change in bond length and the observed effect is R2 ) 0.53 (intercept )0) or 0.73 (with a variable constant) with a slope of 1.7 or 2.1, respectively. This change in R contribution accounts for about 1/2 of the effect and it is not a very reliable predictor (SI Figure S4). As expected the bond shortening effect is similarly poorly correlated with the total computed effect. However, examination of the data for the 1∆ case shows that the simple displacement model correlates better and with a slope closer to unity than does the full treatment. However, this situation reverses if the 1∆ value for planar cyclopentane is removed from the correlation. This is shown in Supporting Information Figures S5 and S6. There are cases16 in which replacement of H by D results in what appears to be a conformational effect secondary to the change in the CH bond lengths. The case of cyclohexane-d1 has been investigated experimentally in respect to the axial D equatorial D equilibrium constant. It was found that D prefers the equatorial position by about 7%. The case of cyclopropane was investigated here computationally to assess the possibility that replacement of one CH by a CD might result in a change in the geometry of the other bonds in the molecule. This was done by constraining one CH bond to be 4.7 pm shorter than the optimum value and then freely optimizing all other atomic positions subject only to mirror symmetry. The resulting optimized geometry had 1∆ values that were 15 ppb larger than they otherwise would be and 2∆ values that were 5 ppb larger. These are small effects but may have to be included if future computations approach experimental results more closely. It seems likely that the major improvements needed are in the direction of conformational or vibrational averaging and, at least in the case of cyclopropane, some more advanced electronic structure treatment. A possible source of error in the present computation is the implicit assumption that the effect of stretch and bending deformations are additive. This was found to be the case in a computation of 100 values in a 10 × 10 flat grid of H atom positions that spanned a range of R of 10% of the mean value (approximately two standard deviations) and two standard deviations of both bending degrees of freedom. The results could be represented as a sum of stretch and two bend terms to within the numerical accuracy. With the exception of our prior treatment of the 1H resonances of cyclohexane-d1, previous studies of the D/H chemical shift effect for molecules larger than five atom tetrahedral species have been limited to the “primary” term resulting from change in the bond length. This work has added the second derivative terms to the treatment of the D/H isotopic chemical shift for both the stretch and both bend degrees of freedom. These second derivative terms are the leading nonvanishing terms for the bend motions. Further, this work treats only the single H that is replaced by D. This makes the treatment of arbitrarily large

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molecules feasible. The degree of agreement of the computed and observed results supports the argument that this is adequate for most purposes. Lack of conformational or vibrational averaging is believed to be the source of the residual differences in almost all cases. Acknowledgment. This work was supported by NSF CHE0848790. KSY acknowledges support of iREU grant NSF CHE075538. We thank D. G. Allis for ongoing assistance. Supporting Information Available: Figure S1: Correlation of the calculated and observed deuterium induced 13C chemical shifts for the individual deuterium isotopomers of adamantane. Table S1: Calculated and observed deuterium induced 13C chemical shifts for protoadamantane. Figure S2: Correlation of the calculated and observed deuterium induced 13C chemical shifts for the individual deuterium isotopomers of protoadamantane for n∆, n > 1. Figure S3: Correlation of computed and observed deuterium induced chemical shifts for n∆ n > 1 for all isomers and all molecules. Figure S4: Correlation of the observed H/D chemical shifts with the contribution made by the contraction of the CD bond relative to the CH bond for all n ∆ shifts n > 1. Figures S5 and S6: Correlation of observed and calculated values of 1∆ H/D 13C shifts with stretch coefficient only and with the full calculation. This material is available free of charge via the Internet at http://pubs.acs.org. References and Notes (1) Dziembowska, T.; Hansen, P. E.; Rozwadowski, Z. Prog. Nucl. Magn. Reson. Spectrosc. 2004, 45, 1–29. (2) (a) Batiz - Hernandez, H.; Bernheim, R. A. Prog. Nucl. Magn. Reson. Spectrosc. 1967, 3, 63–85. (b) Bernheim, R. A.; Batiz-Hernandez, H. J. Chem. Phys. 1966, 45, 2261–2269. (3) (a) Jameson, C. J. In Isotopes in the Physical and Biomedical Science; Buncel, E., Jones, J. R. Eds.; Elsevier: Amsterdam, 1991; Vol. 2, p 1. (b) Jameson, C. J. In Specialist Periodical Reports Nuclear Magnetic Resonance; Webb, G. A. Ed.; Royal Society of Chemistry: Cambridge, 1994; Vol. 23, p 47. (c) Jameson, C. J. J. Chem. Phys. 1977, 66, 4983–4988. (4) (a) Raynes, W. T.; Fowler, P. W.; Lazeretti, P.; Zanari, R.; Grayson, M. Mol. Phys. 1988, 64, 143–146. (b) W. T. Raynes, W. T.; Nightingale, M. Int. J. Quantum Chem. 1996, 60, 529–534.

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