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Scaling factors for carbon NMR chemical shifts obtained from DFT B3LYP calculations
Journal of Molecular Structure: THEOCHEM 893 (2009) 1–5
Contents lists available at ScienceDirect
Journal of Molecular Structure: THEOCHEM journal homepage: www.elsevier.com/locate/theochem
Scaling factors for carbon NMR chemical shifts obtained from DFT B3LYP calculations Abil E. Aliev *, Denis Courtier-Murias, Shen Zhou Department of Chemistry, University College London, 20 Gordon Street, London WC1H 0AJ, UK
a r t i c l e
i n f o
Article history: Received 3 September 2008 Received in revised form 17 September 2008 Accepted 17 September 2008 Available online 25 September 2008 Keywords: Chemical shift Scaling factor NMR DFT GIAO
a b s t r a c t A linear scaling of the calculated chemical shifts is used in order to improve the accuracy of the DFT predicted 13C NMR chemical shifts. The widely applied method of GIAO B3LYP/6-311+G(2d,p) using the B3LYP/6-31G(d) optimized geometries is chosen, which allows cost-effective calculations of the 13C chemical shifts in the molecular systems with 100 and more atoms. A set of 27 13C NMR chemical shifts determined experimentally for 22 simple molecules with various functional groups is used in order to determine scaling factors for reproducing experimentally measured values of 13C chemical shifts. The results show that the use of a simple relationship (dscalc = 0.95 dcalc + 0.30, where dcalc and dscalc are the calculated and the linearly scaled values of the 13C chemical shifts, respectively) allows to achieve a three-fold improvement in mean absolute deviations for 27 chemical shifts considered. To test the universal applicability of the scaling factors derived, we have used complex organic molecules such as taxol and a steroid to demonstrate the significantly improved accuracy of the DFT predicted chemical shifts. This approach also outperforms the recently recommended usage of the Hartree-Fock optimized geometries for the GIAO B3LYP/6-311+G(2d,p) calculations of the 13C chemical shifts. Ó 2008 Elsevier B.V. All rights reserved.
1. Introduction Calculations of NMR chemical shifts have become an increasingly popular method amongst chemists interested in the analysis of NMR data and their use for the three-dimensional structural elucidations [1–3]. Various different approaches have been developed and tested, however, the most widely used technique is the GIAO (Gauge Including Atomic Orbital) calculation of NMR chemical shifts at the DFT (Density Functional Theory) B3LYP (Becke-3Lee-Yang-Par) 6-311+G(2d,p) level which is suitable for organic molecules of medium size with a molecular weight usually less than 1000. Relatively accurate values of 1H chemical shifts can be achieved using this technique, but predictions of 13C shifts are rather poor and do not always allow unambiguous solution of stereochemical problems. In their pioneering work focused on the comparison of different models for calculating NMR chemical shifts, Cheeseman et al. [4] recommended the B3LYP/6-311+G(2d,p) level of theory for 13C chemical shift predictions. A further systematic study of the 13C performance of GIAO B3LYP/6-311+G(2d,p) was reported recently by Zhang et al. using a set of 18 molecules (21 chemical shifts) with various functional groups [5]. It was found that the HF, BLYP and B3LYP optimized geometries lead to mean absolute deviations * Corresponding author. Tel: +44 020 7679 4616; fax: +44 020 7679 7463. E-mail address: [email protected] (A.E. Aliev). 0166-1280/$ - see front matter Ó 2008 Elsevier B.V. All rights reserved. doi:10.1016/j.theochem.2008.09.021
(MADs) of 2.36, 5.80 and 4.43 ppm, respectively, from the experimentally measured values. In the case of the best performing HF geometries, a deshielding tendency of GIAO B3LYP/6-311+G(2d,p) was compensated by the HF underestimations of the bond lengths at the geometry optimization step. In our view, such an approach aimed at the cancelation of errors from one type of calculation by the errors from another type of calculation is somewhat illogical despite the significant improvements in MADs achieved. In addition, a large number of B3LYP/6-311+G(2d,p) NMR calculations using B3LYP/6-31G(d) geometries [denoted as B3LYP/6-311+G(2d,p)// B3LYP/6-31G(d)] has already been reported following the initial work by Cheeseman et al. [4]. Thus, a somewhat different approach is needed for their efficient revision. There are also large number of other reports dedicated to the detailed comparison of 10 or more different protocols in order to achieve a better performance of computational techniques [6–9]. From practical point of view, however, there is often a need for a simple approach that would allow to predict 13C chemical shifts in relatively large molecular systems and to account for systematic differences between theory and experiment for a given type of calculation. Here we use such an approach which relies on the most widely used B3LYP/6-311+G(2d,p)//B3LYP/6-31G(d) method and, therefore, does not require recalculations of the already published data. Considering that the computations are carried out for a molecule in considerably different conditions compared to that in a real experiment and that NMR is sensitive to short range structural changes,
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large systematic differences are expected between the calculated and the experimentally measured chemical shifts. These may arise from different sources, such as electron correlation and rovibrational effects [6,10,11]. It is therefore reasonable to derive correlation coefficients for scaling calculated chemical shifts before comparing them to experiment. This approach is somewhat similar to that used by Wong for the determination of the scaling factors for reproducing experimental vibrational frequencies [12]. A detailed justification for the correlation of quantum mechanically calculated 1H and 13C NMR absolute chemical shielding tensors (r, ppm) and empirical 1H and 13C chemical shifts (d, ppm) measured relative to tetramethylsilane (TMS) has also been presented in the past [10,13]. It was shown that at least a linear correlation equation is needed in the case of computational studies of proton NMR chemical shifts referenced to TMS [10]. In the case of the GIAO calculated 13C chemical shifts, the geometries obtained from molecular mechanics methods were used and the use of the empirically scaled DFT B3LYP/3-21G(X,6-31+G*)//MM3 protocol was recommended, although calculations at the B3LYP/6-31G(d)//B3LYP/ 6-31G(d) level were also carried out for comparison [13]. 2. Computational details A set of 23 small molecules providing 28 different 13C chemical shifts determined in the gas phase was used (Table 1) [14–16]. DFT calculations were carried out using Gaussian 03 [17]. Geometry optimizations were performed for a single molecule in a gas phase using B3LYP/6-31G(d) level of theory using tight optimization (Opt = Tight). Two-electron integrals and their derivatives were calculated using an ultrafine grid option (Int = Ultrafine). NMR chemical shifts were computed at the B3LYP/6-311+G(2d,p) level using the GIAO method [18] and are given relative to that of TMS calculated at the same level of theory (B3LYP/6-311+G(2d,p)//B3LYP/6-31G(d)). A similar protocol was adopted in B3LYP/6-311+G(2d,p)//HF/6-31G(d) and OPBE/6-311+G(2d,p)//OPBE/6-31G(d) calculations. In the case of
Table 1 Calculated and experimental [14–16] isotropic
taxol and steroid molecules, simple geometry optimizations followed by the frequency and chemical shift calculations were performed using the same levels of theory as in the case of the set of 23 small molecules. 3. Results and discussion In order to derive universal scaling factors for predicting 13C chemical shifts in organic compounds, we use a set of simple molecules, similar to those considered by Zhang et al. [5]. We note that it is in principle possible to use the experimental and the calculated chemical shifts for a given system to derive individual scaling factors in each case. However, this approach is significantly limited compared to the universal scaling using a basic set of simple compounds and is applicable to systems which predominantly exist in a single known conformation. The isotropic 13C magnetic shieldings (r, ppm) calculated using B3LYP/6-311+G(2d,p)//B3LYP/6-31G(d) together with the corresponding experimental values are included in Table 1. We use this data to determine chemical shifts relative to TMS (Table 2), which is a standard reference compound used for experimental 13C NMR chemical shift measurements. Note that in all the considered cases we use the reference value for TMS determined under the same conditions as the sample value of interest, i.e., in the case of the experimental data we use the value of 188.1 ppm for the isotropic 13 C magnetic shielding in TMS [14], whereas for the calculated data we use the value of 182.49 ppm (Table 1). The conversion of the calculated magnetic shieldings into chemical shifts relative to a standard (TMS for 13C NMR) is known to benefit from a systematic error cancelation [5]. As an example, the MP2 based protocol was shown to lead to a MAD of 10.03 ppm for 13C magnetic shieldings but only to a MAD of 2.80 ppm for 13C chemical shifts [19]. Note that unlike the ‘‘magnetic shielding – chemical shift” correlation used in references [10] and [13], we use ‘‘chemical shift – chemical shift” correlation in this work.
13
C NMR magnetic shieldings (r, in ppm)
Molecule
Experiment
Calculation B3LYP/6-311+G (2d,p)//B3LYP/6-31G(d)
Calculation B3LYP/6-311 +G(2d,p)//HF/6-31G(d)
Calculation OPBE/6-311 +G(2d,p)//OPBE/6-31G(d)
CH4 C2H2 C2H4 C2H6 H2CCCH2 H2CCCH2 CH3CH2CH3 CH3CH2CH3 C6H6 CH3F CHF3 CF4 CO2 CO H2CO CH3OH CH3COCH3 CH3COCH3 HCN CH3CN CH3CN CH3NH2 TMS CH3CHO CH3CHO OCS CS2 H2CN2
195.1 117.2 64.5 180.9 28.9 115.2 169.3 170.9 57.2 116.8 68.4 64.5 58.5 1 1 136.6 13.1 158 82.1 187.7 73.8 158.3 188.1 157.2 6.7 30 8 164.5
189.16 110.43 53.65 173.49 43.87 107.47 160.71 164.49 48.85 109.38 56.07 48.70 49.89 15.93 17.95 128.55 -29.54 151.45 72.75 181.70 63.17 150.14 182.49 148.61 -22.24 19.45 -26.49 160.91
191.41 112.32 56.67 176.00 42.70 110.55 163.22 166.92 51.57 113.31 63.69 57.22 56.50 4.76 11.08 132.30 21.75 153.88 77.88 184.18 68.69 153.46 184.53 151.05 15.06 26.45 20.24 170.05
194.10 119.16 65.09 179.01 23.11 115.17 167.17 171.23 62.98 117.63 65.06 58.20 63.33 0.68 4.71 136.01 11.86 158.53 82.58 185.85 73.01 156.64 187.98 155.60 -6.51 40.45 7.29 166.30
A.E. Aliev et al. / Journal of Molecular Structure: THEOCHEM 893 (2009) 1–5
3
Fig. 1 shows the correlation between the experimental and the calculated 13C chemical shifts. A good linear relationship is observed (r2 = 0.9992) fitted by:
dexp ¼ 0:95 dcalc þ 0:30 where dexp and dcalc are experimental and calculated chemical shifts. The results show that while the GIAO B3LYP/6-311+G(2d,p)// B3LYP/6-31G(d) gives a relatively large MAD of 4.80 ppm (RMS 6.18 ppm) for the set of 27 chemical shifts (Table 2), the subsequent linear scaling leads to a three-fold improvement with the MAD value of 1.57 ppm (RMS 1.89 ppm). Considering that 13C shifts vary over approximately 200 ppm, the RMS error of 1.89 ppm constitutes an error of 1% of the overall 13C chemical shift range. For comparison with other protocols, the MAD value for the GIAO B3LYP/6-311+G(2d,p)//HF/6-31G(d) recommended by Zhang et al. [5] is 2.72 ppm for the same set of 27 chemical shifts. For another functional of OPBE, which was shown to perform relatively better than popular functionals B3LYP and PBE1PBE [20], the corresponding MAD value is 2.85 ppm (RMS 4.46 ppm, from the GIAO OPBE/6-311+G(2d,p)//OPBE/6-31G(d) calculations, Table 1). Interestingly, however, the scatter of the data is significantly greater in the case of OPBE predicted chemical shifts (Fig. 2) compared to that by B3LYP (Fig. 1), with the subsequent scaling of the OPBE calculated 13C chemical shifts leading to only a small improvement with the MAD value of 2.81 ppm (RMS 4.09 ppm). In order to verify the importance of the linear scaling we have used an example of relatively large molecule, taxol (C47H51NO14, Mw = 853, Fig. 3). The known chemical shifts of 41 carbon atoms of this molecule were used by Cheeseman et al. for the GIAO HF/ 6-31G(d)//STO-3G calculations of the 13C chemical shifts [4,20]. The RMS error with respect to the experimental data was found to be 6.4 ppm, whereas the maximum error was 18.7 ppm. These results were reproduced by us prior to further calculations using
Table 2 Calculated [B3LYP/6-311+G(2d,p)//B3LYP/6-31G(d)] and experimental isotropic NMR chemical shifts (d, in ppm)
13
C
Molecule
Experiment
Calculation
Calculation after scaling
CH4 C2H2 C2H4 C2H6 H2CCCH2 H2CCCH2 CH3CH2CH3 CH3CH2CH3 C6H6 CH3F CHF3 CF4 CO2 CO H2CO CH3OH CH3COCH3 CH3COCH3 HCN CH3CN CH3CN CH3NH2 CH3CHO CH3CHO OCS CS2 H2CN2
7 70.9 123.6 7.2 217 72.9 18.8 17.2 130.9 71.3 119.7 123.6 129.6 187.1 189.1 51.5 201.2 30.1 106 0.4 114.3 29.8 30.9 194.8 158.1 196.1 23.6
6.67 72.06 128.84 9.00 226.35 75.02 21.78 18.00 133.64 73.11 126.42 133.79 132.60 198.42 200.44 53.94 212.02 31.04 109.74 0.79 119.32 32.35 33.88 204.72 163.04 208.98 21.58
6.04 68.75 122.70 8.85 215.34 71.57 20.99 17.40 127.26 69.76 120.40 127.40 126.27 188.80 190.72 51.54 201.72 29.78 104.55 1.05 113.65 31.03 32.48 194.79 155.18 198.83 20.80
Fig. 1. Graph showing the linear correlation between the experimentally measured and calculated [GIAO B3LYP/6-311+G(2d,p)//B3LYP/6-31G(d)] 13C NMR chemical shifts. The best fit straight line corresponding to dexp = 0.95 dcalc + 0.30 is also shown (r2 = 0.9992).
GIAO B3LYP/6-311+G(2d,p)//B3LYP/6-31G(d) method. For the calculation of the linearly scaled values of the chemical shifts we used dscalc = 0.95 dcalc + 0.30, where dcalc and dscalc are the calculated and the linearly scaled values of the 13C chemical shifts, respectively. In addition, we have also applied the GIAO B3LYP/6-311+G(2d,p)//HF/ 6-31G(d) method recommended by Zhang et al. [5]. The results are summarized in Table 3. The MAD and RMS errors were found to be the smallest for the linearly scaled GIAO B3LYP/6-311+G(2d,p)// B3LYP/6-31G(d) shifts. In a similar way we have used 17a-(1-aziridinylmethyl)-5aandrostan-17b-ol (Fig. 4, C22H37NO, Mw = 331), for which the 13 C chemical shifts have been assigned unambiguously using 2D INADEQUATE technique [21]. As in the case of taxol, the MAD and RMS errors were found to be the smallest for the linearly scaled GIAO B3LYP/6-311+G(2d,p)//B3LYP/6-31G(d) shifts (Table 3). Finally, we believe the type of linear correlation used above is a better way to proceed than the direct ‘‘theory-experiment” comparison, since the calculation and experimental measurements of NMR chemical shifts are carried out for two different systems: for a single molecule placed in vacuum in calculations and for a multi-molecular system placed under normal conditions in experimental measurements. In addition, temperature dependent small-angle vibrational motions or solvent-solute interactions, which are not accounted for explicitly in the calculations, are likely to affect isotropic chemical shift values, even in the case of conformationally homogeneous molecules. Thus, the equality of the calculated and the experimentally measured chemical shifts is not necessarily desirable. However, as shown previously [10] and confirmed by our results, the existence of the highly linear correlation (with r2 > 0.999) between the calculated and measured values varying over a large range can be critical. On a relevant note of a general nature, the use of scaling factors is not unique to quantum chemical calculations of NMR properties. For example, scaling factors have been derived through a least-squares approach for the calculations of fundamental vibrational frequencies, low-frequency vibrations,
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40.94 (38.77) 26.25 (22.17)
23.50 (20.40) 12.04 (12.22)
41.31 (36.35)
30.51 (26.81) 32.64 (29.03)
50.17 (47.21)
34.17 (32.07)
57.69 (54.91)
14.44 (14.19)
23.88 (26.95)
OH 86.67 (83.00)
49.65 (45.33) 54.72 (51.13)
39.49 (36.20)
26.57 (23.62)
N
31.69 (29.60)
67.58 (67.47) 37.85 (34.59)
34.74 (31.96) 32.34 (28.99)
Fig. 4. Calculated and experimental (in parenthesis) [21] 13C NMR chemical shifts (in ppm) in 17a-(1-aziridinylmethyl)-5a-androstan-17b-ol relative to TMS. Calculated chemical shifts were obtained using linear scaling of the values predicted by the GIAO B3LYP/6-311+G(2d,p)//B3LYP/6-31G(d) method.
4. Conclusions
Fig. 2. Graph showing the linear correlation between the experimentally measured and calculated [GIAO OPBE/6-311+G(2d,p)//OPBE/6-31G(d)] 13C NMR chemical shifts. The best fit straight line corresponding to dexp = 1.02 dcalc 1.05 is also shown (r2 = 0.9983). The scatter of the data around the fitted line is significantly greater than that in Fig. 1.
zero-point vibrational energies, and thermal contributions to enthalpy and entropy from harmonic frequencies determined at various levels of theory [22,23].
A linear scaling of the calculated chemical shifts is used in order to account for the differences in the conditions of the experimental measurements and computational predictions, as well as for possible systematic errors either at the geometry optimization or NMR stages of the calculations. The widely applied method of GIAO B3LYP/6-311+G(2d,p) using the B3LYP/6-31G(d) optimized geometries was chosen, which allows cost-effective calculations of the 13 C NMR chemical shifts in the molecular systems with 100 and more atoms. A set of 27 13C NMR chemical shifts determined experimentally for 22 simple molecules with various functional groups is used in order to determine scaling factors for reproducing
Fig. 3. Calculated and experimental (in parenthesis) [20] 13C NMR chemical shifts (in ppm) in taxol relative to TMS. Calculated chemical shifts were obtained using linear scaling of the values predicted by the GIAO B3LYP/6-311+G(2d,p)//B3LYP/6-31G(d) method.
Table 3 Statistical data illustrating the performance of various methods for the GIAO calculation of the (AzAnd)
13
C chemical shifts in taxol and 17a-(1-aziridinylmethyl)-5a-androstan-17b-ol
Taxol
HF/6-31G(d)//STO-3G (Cheeseman et al.) [4] B3LYP/6-311+G(2d,p)//HF/6-31G(d) B3LYP/6-311+G(2d,p)//B3LYP/6-31G(d) Scaled B3LYP/6-311+G(2d,p)//B3LYP/6-31G(d)
AzAnd
MAD/ppm
RMS/ppm
MAD/ppm
RMS/ppm
4.96 4.74 6.45 2.90
6.43 5.77 7.44 3.71
– 4.19 4.45 2.83
– 4.52 4.79 3.10
A.E. Aliev et al. / Journal of Molecular Structure: THEOCHEM 893 (2009) 1–5
experimentally measured values of 13C chemical shifts in organic compounds. The results show that the use of a simple relationship (dscalc = 0.95 dcalc + 0.30, where dcalc and dscalc are the calculated and the linearly scaled values of the 13C chemical shifts, respectively) allows to achieve a three-fold improvement in mean absolute deviations for the set of 27 chemical shifts considered. To test the universal applicability of the scaling factors derived, we have used complex organic molecules such as taxol and a steroid to demonstrate the significantly improved accuracy of the DFT predicted chemical shifts. This approach also outperforms the recently recommended usage of the Hartree-Fock optimized geometries (HF/ 6-31G(d)) for the GIOA B3LYP/6-311+G(2d,p) calculations of the 13 C NMR chemical shifts.
[5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17]
Acknowledgements We are grateful for the provision of studentship from the EPSRC (D.C.M.) and Univesity College of London for the provision of the computational facilities. References [1] T. Helgaker, M. Jaszunski, K. Ruud, Chem. Rev. 99 (1999) 293. [2] E. Oldfield, Annu. Rev. Phys. Chem. 53 (2002) 349. [3] M. Kaupp, M. Bühl, V.C. Malkin (Eds.), Calculation of NMR and EPR Parameters. Theory and Applications, Wiley-VCH Verlag GmbH & Co., Weinheim, Germany, 2004. [4] J.R. Cheeseman, G.W. Trucks, T.A. Keith, M.J. Frisch, J. Chem. Phys. 104 (1996) 5497.
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Contents lists available at ScienceDirect
Journal of Molecular Structure: THEOCHEM journal homepage: www.elsevier.com/locate/theochem
Scaling factors for carbon NMR chemical shifts obtained from DFT B3LYP calculations Abil E. Aliev *, Denis Courtier-Murias, Shen Zhou Department of Chemistry, University College London, 20 Gordon Street, London WC1H 0AJ, UK
a r t i c l e
i n f o
Article history: Received 3 September 2008 Received in revised form 17 September 2008 Accepted 17 September 2008 Available online 25 September 2008 Keywords: Chemical shift Scaling factor NMR DFT GIAO
a b s t r a c t A linear scaling of the calculated chemical shifts is used in order to improve the accuracy of the DFT predicted 13C NMR chemical shifts. The widely applied method of GIAO B3LYP/6-311+G(2d,p) using the B3LYP/6-31G(d) optimized geometries is chosen, which allows cost-effective calculations of the 13C chemical shifts in the molecular systems with 100 and more atoms. A set of 27 13C NMR chemical shifts determined experimentally for 22 simple molecules with various functional groups is used in order to determine scaling factors for reproducing experimentally measured values of 13C chemical shifts. The results show that the use of a simple relationship (dscalc = 0.95 dcalc + 0.30, where dcalc and dscalc are the calculated and the linearly scaled values of the 13C chemical shifts, respectively) allows to achieve a three-fold improvement in mean absolute deviations for 27 chemical shifts considered. To test the universal applicability of the scaling factors derived, we have used complex organic molecules such as taxol and a steroid to demonstrate the significantly improved accuracy of the DFT predicted chemical shifts. This approach also outperforms the recently recommended usage of the Hartree-Fock optimized geometries for the GIAO B3LYP/6-311+G(2d,p) calculations of the 13C chemical shifts. Ó 2008 Elsevier B.V. All rights reserved.
1. Introduction Calculations of NMR chemical shifts have become an increasingly popular method amongst chemists interested in the analysis of NMR data and their use for the three-dimensional structural elucidations [1–3]. Various different approaches have been developed and tested, however, the most widely used technique is the GIAO (Gauge Including Atomic Orbital) calculation of NMR chemical shifts at the DFT (Density Functional Theory) B3LYP (Becke-3Lee-Yang-Par) 6-311+G(2d,p) level which is suitable for organic molecules of medium size with a molecular weight usually less than 1000. Relatively accurate values of 1H chemical shifts can be achieved using this technique, but predictions of 13C shifts are rather poor and do not always allow unambiguous solution of stereochemical problems. In their pioneering work focused on the comparison of different models for calculating NMR chemical shifts, Cheeseman et al. [4] recommended the B3LYP/6-311+G(2d,p) level of theory for 13C chemical shift predictions. A further systematic study of the 13C performance of GIAO B3LYP/6-311+G(2d,p) was reported recently by Zhang et al. using a set of 18 molecules (21 chemical shifts) with various functional groups [5]. It was found that the HF, BLYP and B3LYP optimized geometries lead to mean absolute deviations * Corresponding author. Tel: +44 020 7679 4616; fax: +44 020 7679 7463. E-mail address: [email protected] (A.E. Aliev). 0166-1280/$ - see front matter Ó 2008 Elsevier B.V. All rights reserved. doi:10.1016/j.theochem.2008.09.021
(MADs) of 2.36, 5.80 and 4.43 ppm, respectively, from the experimentally measured values. In the case of the best performing HF geometries, a deshielding tendency of GIAO B3LYP/6-311+G(2d,p) was compensated by the HF underestimations of the bond lengths at the geometry optimization step. In our view, such an approach aimed at the cancelation of errors from one type of calculation by the errors from another type of calculation is somewhat illogical despite the significant improvements in MADs achieved. In addition, a large number of B3LYP/6-311+G(2d,p) NMR calculations using B3LYP/6-31G(d) geometries [denoted as B3LYP/6-311+G(2d,p)// B3LYP/6-31G(d)] has already been reported following the initial work by Cheeseman et al. [4]. Thus, a somewhat different approach is needed for their efficient revision. There are also large number of other reports dedicated to the detailed comparison of 10 or more different protocols in order to achieve a better performance of computational techniques [6–9]. From practical point of view, however, there is often a need for a simple approach that would allow to predict 13C chemical shifts in relatively large molecular systems and to account for systematic differences between theory and experiment for a given type of calculation. Here we use such an approach which relies on the most widely used B3LYP/6-311+G(2d,p)//B3LYP/6-31G(d) method and, therefore, does not require recalculations of the already published data. Considering that the computations are carried out for a molecule in considerably different conditions compared to that in a real experiment and that NMR is sensitive to short range structural changes,
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A.E. Aliev et al. / Journal of Molecular Structure: THEOCHEM 893 (2009) 1–5
large systematic differences are expected between the calculated and the experimentally measured chemical shifts. These may arise from different sources, such as electron correlation and rovibrational effects [6,10,11]. It is therefore reasonable to derive correlation coefficients for scaling calculated chemical shifts before comparing them to experiment. This approach is somewhat similar to that used by Wong for the determination of the scaling factors for reproducing experimental vibrational frequencies [12]. A detailed justification for the correlation of quantum mechanically calculated 1H and 13C NMR absolute chemical shielding tensors (r, ppm) and empirical 1H and 13C chemical shifts (d, ppm) measured relative to tetramethylsilane (TMS) has also been presented in the past [10,13]. It was shown that at least a linear correlation equation is needed in the case of computational studies of proton NMR chemical shifts referenced to TMS [10]. In the case of the GIAO calculated 13C chemical shifts, the geometries obtained from molecular mechanics methods were used and the use of the empirically scaled DFT B3LYP/3-21G(X,6-31+G*)//MM3 protocol was recommended, although calculations at the B3LYP/6-31G(d)//B3LYP/ 6-31G(d) level were also carried out for comparison [13]. 2. Computational details A set of 23 small molecules providing 28 different 13C chemical shifts determined in the gas phase was used (Table 1) [14–16]. DFT calculations were carried out using Gaussian 03 [17]. Geometry optimizations were performed for a single molecule in a gas phase using B3LYP/6-31G(d) level of theory using tight optimization (Opt = Tight). Two-electron integrals and their derivatives were calculated using an ultrafine grid option (Int = Ultrafine). NMR chemical shifts were computed at the B3LYP/6-311+G(2d,p) level using the GIAO method [18] and are given relative to that of TMS calculated at the same level of theory (B3LYP/6-311+G(2d,p)//B3LYP/6-31G(d)). A similar protocol was adopted in B3LYP/6-311+G(2d,p)//HF/6-31G(d) and OPBE/6-311+G(2d,p)//OPBE/6-31G(d) calculations. In the case of
Table 1 Calculated and experimental [14–16] isotropic
taxol and steroid molecules, simple geometry optimizations followed by the frequency and chemical shift calculations were performed using the same levels of theory as in the case of the set of 23 small molecules. 3. Results and discussion In order to derive universal scaling factors for predicting 13C chemical shifts in organic compounds, we use a set of simple molecules, similar to those considered by Zhang et al. [5]. We note that it is in principle possible to use the experimental and the calculated chemical shifts for a given system to derive individual scaling factors in each case. However, this approach is significantly limited compared to the universal scaling using a basic set of simple compounds and is applicable to systems which predominantly exist in a single known conformation. The isotropic 13C magnetic shieldings (r, ppm) calculated using B3LYP/6-311+G(2d,p)//B3LYP/6-31G(d) together with the corresponding experimental values are included in Table 1. We use this data to determine chemical shifts relative to TMS (Table 2), which is a standard reference compound used for experimental 13C NMR chemical shift measurements. Note that in all the considered cases we use the reference value for TMS determined under the same conditions as the sample value of interest, i.e., in the case of the experimental data we use the value of 188.1 ppm for the isotropic 13 C magnetic shielding in TMS [14], whereas for the calculated data we use the value of 182.49 ppm (Table 1). The conversion of the calculated magnetic shieldings into chemical shifts relative to a standard (TMS for 13C NMR) is known to benefit from a systematic error cancelation [5]. As an example, the MP2 based protocol was shown to lead to a MAD of 10.03 ppm for 13C magnetic shieldings but only to a MAD of 2.80 ppm for 13C chemical shifts [19]. Note that unlike the ‘‘magnetic shielding – chemical shift” correlation used in references [10] and [13], we use ‘‘chemical shift – chemical shift” correlation in this work.
13
C NMR magnetic shieldings (r, in ppm)
Molecule
Experiment
Calculation B3LYP/6-311+G (2d,p)//B3LYP/6-31G(d)
Calculation B3LYP/6-311 +G(2d,p)//HF/6-31G(d)
Calculation OPBE/6-311 +G(2d,p)//OPBE/6-31G(d)
CH4 C2H2 C2H4 C2H6 H2CCCH2 H2CCCH2 CH3CH2CH3 CH3CH2CH3 C6H6 CH3F CHF3 CF4 CO2 CO H2CO CH3OH CH3COCH3 CH3COCH3 HCN CH3CN CH3CN CH3NH2 TMS CH3CHO CH3CHO OCS CS2 H2CN2
195.1 117.2 64.5 180.9 28.9 115.2 169.3 170.9 57.2 116.8 68.4 64.5 58.5 1 1 136.6 13.1 158 82.1 187.7 73.8 158.3 188.1 157.2 6.7 30 8 164.5
189.16 110.43 53.65 173.49 43.87 107.47 160.71 164.49 48.85 109.38 56.07 48.70 49.89 15.93 17.95 128.55 -29.54 151.45 72.75 181.70 63.17 150.14 182.49 148.61 -22.24 19.45 -26.49 160.91
191.41 112.32 56.67 176.00 42.70 110.55 163.22 166.92 51.57 113.31 63.69 57.22 56.50 4.76 11.08 132.30 21.75 153.88 77.88 184.18 68.69 153.46 184.53 151.05 15.06 26.45 20.24 170.05
194.10 119.16 65.09 179.01 23.11 115.17 167.17 171.23 62.98 117.63 65.06 58.20 63.33 0.68 4.71 136.01 11.86 158.53 82.58 185.85 73.01 156.64 187.98 155.60 -6.51 40.45 7.29 166.30
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3
Fig. 1 shows the correlation between the experimental and the calculated 13C chemical shifts. A good linear relationship is observed (r2 = 0.9992) fitted by:
dexp ¼ 0:95 dcalc þ 0:30 where dexp and dcalc are experimental and calculated chemical shifts. The results show that while the GIAO B3LYP/6-311+G(2d,p)// B3LYP/6-31G(d) gives a relatively large MAD of 4.80 ppm (RMS 6.18 ppm) for the set of 27 chemical shifts (Table 2), the subsequent linear scaling leads to a three-fold improvement with the MAD value of 1.57 ppm (RMS 1.89 ppm). Considering that 13C shifts vary over approximately 200 ppm, the RMS error of 1.89 ppm constitutes an error of 1% of the overall 13C chemical shift range. For comparison with other protocols, the MAD value for the GIAO B3LYP/6-311+G(2d,p)//HF/6-31G(d) recommended by Zhang et al. [5] is 2.72 ppm for the same set of 27 chemical shifts. For another functional of OPBE, which was shown to perform relatively better than popular functionals B3LYP and PBE1PBE [20], the corresponding MAD value is 2.85 ppm (RMS 4.46 ppm, from the GIAO OPBE/6-311+G(2d,p)//OPBE/6-31G(d) calculations, Table 1). Interestingly, however, the scatter of the data is significantly greater in the case of OPBE predicted chemical shifts (Fig. 2) compared to that by B3LYP (Fig. 1), with the subsequent scaling of the OPBE calculated 13C chemical shifts leading to only a small improvement with the MAD value of 2.81 ppm (RMS 4.09 ppm). In order to verify the importance of the linear scaling we have used an example of relatively large molecule, taxol (C47H51NO14, Mw = 853, Fig. 3). The known chemical shifts of 41 carbon atoms of this molecule were used by Cheeseman et al. for the GIAO HF/ 6-31G(d)//STO-3G calculations of the 13C chemical shifts [4,20]. The RMS error with respect to the experimental data was found to be 6.4 ppm, whereas the maximum error was 18.7 ppm. These results were reproduced by us prior to further calculations using
Table 2 Calculated [B3LYP/6-311+G(2d,p)//B3LYP/6-31G(d)] and experimental isotropic NMR chemical shifts (d, in ppm)
13
C
Molecule
Experiment
Calculation
Calculation after scaling
CH4 C2H2 C2H4 C2H6 H2CCCH2 H2CCCH2 CH3CH2CH3 CH3CH2CH3 C6H6 CH3F CHF3 CF4 CO2 CO H2CO CH3OH CH3COCH3 CH3COCH3 HCN CH3CN CH3CN CH3NH2 CH3CHO CH3CHO OCS CS2 H2CN2
7 70.9 123.6 7.2 217 72.9 18.8 17.2 130.9 71.3 119.7 123.6 129.6 187.1 189.1 51.5 201.2 30.1 106 0.4 114.3 29.8 30.9 194.8 158.1 196.1 23.6
6.67 72.06 128.84 9.00 226.35 75.02 21.78 18.00 133.64 73.11 126.42 133.79 132.60 198.42 200.44 53.94 212.02 31.04 109.74 0.79 119.32 32.35 33.88 204.72 163.04 208.98 21.58
6.04 68.75 122.70 8.85 215.34 71.57 20.99 17.40 127.26 69.76 120.40 127.40 126.27 188.80 190.72 51.54 201.72 29.78 104.55 1.05 113.65 31.03 32.48 194.79 155.18 198.83 20.80
Fig. 1. Graph showing the linear correlation between the experimentally measured and calculated [GIAO B3LYP/6-311+G(2d,p)//B3LYP/6-31G(d)] 13C NMR chemical shifts. The best fit straight line corresponding to dexp = 0.95 dcalc + 0.30 is also shown (r2 = 0.9992).
GIAO B3LYP/6-311+G(2d,p)//B3LYP/6-31G(d) method. For the calculation of the linearly scaled values of the chemical shifts we used dscalc = 0.95 dcalc + 0.30, where dcalc and dscalc are the calculated and the linearly scaled values of the 13C chemical shifts, respectively. In addition, we have also applied the GIAO B3LYP/6-311+G(2d,p)//HF/ 6-31G(d) method recommended by Zhang et al. [5]. The results are summarized in Table 3. The MAD and RMS errors were found to be the smallest for the linearly scaled GIAO B3LYP/6-311+G(2d,p)// B3LYP/6-31G(d) shifts. In a similar way we have used 17a-(1-aziridinylmethyl)-5aandrostan-17b-ol (Fig. 4, C22H37NO, Mw = 331), for which the 13 C chemical shifts have been assigned unambiguously using 2D INADEQUATE technique [21]. As in the case of taxol, the MAD and RMS errors were found to be the smallest for the linearly scaled GIAO B3LYP/6-311+G(2d,p)//B3LYP/6-31G(d) shifts (Table 3). Finally, we believe the type of linear correlation used above is a better way to proceed than the direct ‘‘theory-experiment” comparison, since the calculation and experimental measurements of NMR chemical shifts are carried out for two different systems: for a single molecule placed in vacuum in calculations and for a multi-molecular system placed under normal conditions in experimental measurements. In addition, temperature dependent small-angle vibrational motions or solvent-solute interactions, which are not accounted for explicitly in the calculations, are likely to affect isotropic chemical shift values, even in the case of conformationally homogeneous molecules. Thus, the equality of the calculated and the experimentally measured chemical shifts is not necessarily desirable. However, as shown previously [10] and confirmed by our results, the existence of the highly linear correlation (with r2 > 0.999) between the calculated and measured values varying over a large range can be critical. On a relevant note of a general nature, the use of scaling factors is not unique to quantum chemical calculations of NMR properties. For example, scaling factors have been derived through a least-squares approach for the calculations of fundamental vibrational frequencies, low-frequency vibrations,
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A.E. Aliev et al. / Journal of Molecular Structure: THEOCHEM 893 (2009) 1–5
40.94 (38.77) 26.25 (22.17)
23.50 (20.40) 12.04 (12.22)
41.31 (36.35)
30.51 (26.81) 32.64 (29.03)
50.17 (47.21)
34.17 (32.07)
57.69 (54.91)
14.44 (14.19)
23.88 (26.95)
OH 86.67 (83.00)
49.65 (45.33) 54.72 (51.13)
39.49 (36.20)
26.57 (23.62)
N
31.69 (29.60)
67.58 (67.47) 37.85 (34.59)
34.74 (31.96) 32.34 (28.99)
Fig. 4. Calculated and experimental (in parenthesis) [21] 13C NMR chemical shifts (in ppm) in 17a-(1-aziridinylmethyl)-5a-androstan-17b-ol relative to TMS. Calculated chemical shifts were obtained using linear scaling of the values predicted by the GIAO B3LYP/6-311+G(2d,p)//B3LYP/6-31G(d) method.
4. Conclusions
Fig. 2. Graph showing the linear correlation between the experimentally measured and calculated [GIAO OPBE/6-311+G(2d,p)//OPBE/6-31G(d)] 13C NMR chemical shifts. The best fit straight line corresponding to dexp = 1.02 dcalc 1.05 is also shown (r2 = 0.9983). The scatter of the data around the fitted line is significantly greater than that in Fig. 1.
zero-point vibrational energies, and thermal contributions to enthalpy and entropy from harmonic frequencies determined at various levels of theory [22,23].
A linear scaling of the calculated chemical shifts is used in order to account for the differences in the conditions of the experimental measurements and computational predictions, as well as for possible systematic errors either at the geometry optimization or NMR stages of the calculations. The widely applied method of GIAO B3LYP/6-311+G(2d,p) using the B3LYP/6-31G(d) optimized geometries was chosen, which allows cost-effective calculations of the 13 C NMR chemical shifts in the molecular systems with 100 and more atoms. A set of 27 13C NMR chemical shifts determined experimentally for 22 simple molecules with various functional groups is used in order to determine scaling factors for reproducing
Fig. 3. Calculated and experimental (in parenthesis) [20] 13C NMR chemical shifts (in ppm) in taxol relative to TMS. Calculated chemical shifts were obtained using linear scaling of the values predicted by the GIAO B3LYP/6-311+G(2d,p)//B3LYP/6-31G(d) method.
Table 3 Statistical data illustrating the performance of various methods for the GIAO calculation of the (AzAnd)
13
C chemical shifts in taxol and 17a-(1-aziridinylmethyl)-5a-androstan-17b-ol
Taxol
HF/6-31G(d)//STO-3G (Cheeseman et al.) [4] B3LYP/6-311+G(2d,p)//HF/6-31G(d) B3LYP/6-311+G(2d,p)//B3LYP/6-31G(d) Scaled B3LYP/6-311+G(2d,p)//B3LYP/6-31G(d)
AzAnd
MAD/ppm
RMS/ppm
MAD/ppm
RMS/ppm
4.96 4.74 6.45 2.90
6.43 5.77 7.44 3.71
– 4.19 4.45 2.83
– 4.52 4.79 3.10
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experimentally measured values of 13C chemical shifts in organic compounds. The results show that the use of a simple relationship (dscalc = 0.95 dcalc + 0.30, where dcalc and dscalc are the calculated and the linearly scaled values of the 13C chemical shifts, respectively) allows to achieve a three-fold improvement in mean absolute deviations for the set of 27 chemical shifts considered. To test the universal applicability of the scaling factors derived, we have used complex organic molecules such as taxol and a steroid to demonstrate the significantly improved accuracy of the DFT predicted chemical shifts. This approach also outperforms the recently recommended usage of the Hartree-Fock optimized geometries (HF/ 6-31G(d)) for the GIOA B3LYP/6-311+G(2d,p) calculations of the 13 C NMR chemical shifts.
[5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17]
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