Systematic studies on the computation of nuclear magnetic resonance ...
Systematic Studies on the Computation of Nuclear Magnetic Resonance Shielding Constants and Chemical Shifts: The Density Functional Models ANAN WU,1 YING ZHANG,1 XIN XU,1 YIJING YAN2 1
State Key Laboratory for Physical Chemistry of Solid Surface, Center for Theoretical Chemistry, Department of Chemistry, Xiamen University, 361005, China 2 Department of Chemistry, Hong Kong University of Science and Technology, Kowloon, Hong Kong Received 11 August 2006; Revised 24 October 2006; Accepted 1 November 2006 DOI 10.1002/jcc.20641 Published online 22 August 2007 in Wiley InterScience (www.interscience.wiley.com).
Abstract: We present a systematic density functional investigation on the prediction of the 13C, 15N, 17O, and 19F NMR properties of 23 molecules with 21 density functionals. Extensive comparisons are made for both 13C magnetic shieldings and chemical shifts with respect to the gas phase experimental data and the best CCSD(T) results. We find that the OPBE and OPW91 exchange-correlation functionals perform significantly better than some popular functionals such as B3LYP and PBE1PBE, even surpassing, in many cases, the standard wavefunction-based method MP2. Further analysis has been performed to explore the individual role played by various exchange and correlation functionals. We find that the B88 and PBE exchange functionals have a too strong tendency of deshielding, leading to too deshielded magnetic shielding constants; whereas the OPTX exchange functional performs remarkably well. We claim that the main source of error arises from the exchange functional, but correlation functional also makes important contribution. We find that the correlation functionals may be grouped into two classes. class A, such as LYP and B98, leads to deshielded NMR values, deteriorating the overall performance; whereas class B, such as PW91 and PBE, generally increases the absolute shieldings, which complements the exchange functionals, leading to improved results in the calculation of NMR data. q 2007 Wiley Periodicals, Inc.
J Comput Chem 28: 2431–2442, 2007
Key words: density functional theory; nuclear magnetic resonance; magnetic shieldings; chemical shifts; GIAO
Introduction Nuclear magnetic resonance (NMR) spectroscopy has gained an increasing popularity in many chemical applications.1–7 Because of its sensitivity to the molecular composition, geometry, conformation, and environment, NMR spectroscopy leads to valuable information about the electronic environment of a nucleus, and hence has been extensively used in identifying the fragment of a system and in determining the three-dimensional structures of biomolecules.8–10 As a supplementary to experiments, quantum chemical calculation of NMR properties has become a useful tool for the interpretation of the experimental NMR data.1–6 During the last decades, a number of methods have been developed for the calculations of the NMR chemical shifts. It has been generally accepted that, although the Hartree-Fock (HF) method provides chemical shifts with reasonable accuracy, especially for organic molecules, more accurate prediction of these properties requires tak-
ing into account the electron correlation effects.3,11–15 While the post Hartree-Fock methods, such as the second-order MøllerPlesset perturbation approach (MP2), can be quite computationally demanding for a chemically interesting large system, density functional theory (DFT) offers an alternative to cover the electron correlation effect in an unspecific way with a computational cost which is the same order of HF.11 Today DFT has been successfully applied in predicting various molecular properties such This article contains supplementary material available via the Internet at http://www.interscience.wiley.com/jpages/0192-8651/suppmat Correspondence to: X. Xu; e-mail: [email protected] Contract/grant sponsor: National Science Foundation of China; contract/ grant numbers: 20525311, 20533030, 20021002, 20423002 Contract/grant sponsor: Ministry of Science and Technology of China; contract/grant numbers: 2004CB719902 Contract/grant sponsor: Ministry of Education and Xiamen University
q 2007 Wiley Periodicals, Inc.
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as polarizibility, vibrational frequency, etc., often giving comparable or even better results than higher computationally demanding MP2.11 In this respect, DFT methods possess a promising perspective for the prediction of the NMR properties, as opposed to the conventional ab initio methods in quantum chemistry. Using large Gaussian type basis sets and the gauge including atomic orbital (GIAO)16 technique, Cheeseman et al. tested the performance of some popular exchange-correlation functionals such as LSDA,17–19 BPW91,20,21 BLYP,20,22 B3PW91,21,23 and B3LYP22,23 for the prediction of NMR chemical shifts over a set of organic and inorganic molecules.24 They claimed that all DFT functionals, except the local density approach, which is undoubtedly inferior, provided a similar performance and statistically showed an improvement over the HF model.24 For the 13 C chemical shift, the BPW91 and BLYP functionals were rated the best.24 It was also shown that in general for organic molecules the MP2 method significantly outperformed all the DFT approaches examined in their works.24 With the same approach but slightly different basis sets, Rauhut et al. reached a similar conclusion.25 Adamo and Barone investigated the performance of the PBEPBE26 and PBE1PBE27 functionals and compared the absolute magnetic shieldings, instead of the chemical shifts, with the MP2 results. They drew a conclusion that the PBE1PBE functional delivered competitive results for the well behaved systems with respect to the MP2 method.28,29 It seems to us, however, further validation on PBE1PBE is needed to take its performance on the chemical shift prediction into account based on the reference values adequately chosen. With the development of DFT, several new functionals have been proposed recently. By fitting against the HF energies of atoms, Handy et al. presented an optimized exchange functional OPTX.30 When combined with the LYP or PBE correlation functional, OLYP and OPBE were shown to give a substantial improvement over the classical GGA functionals such as BLYP, BPW91, PBEPBE, etc. in calculating the heats of formation, ionization potentials, and electron affinities with the G2 set.31 Also, the corresponding hybrid form O3LYP was claimed to be competitive with the commonly used B3LYP in many aspects.31 Magyarfalvi and Pulay did a careful investigation of DFT performance on the prediction of magnetic shieldings with respect to the CCSD(T) results. They concluded that OLYP gives a marked improvement over BLYP.32 In a recent communication, we showed that OPBE is even better for the prediction of NMR properties.33 Xu et al. introduced a new exchange functional X3 by combining the B88 and PW91 exchange functionals.34,35 In conjunction with LYP, X3LYP exhibited a promising performance, particularly in dealing with weakly interacting systems.34,35 So far, most of these newly developed functionals are just validated against the thermodynamic data. There is a lack of systematic studies on their performance in calculating NMR spectroscopy parameters.32,33 In this contribution, we presented our results for a systematic DFT/GIAO investigation of the 13C, 15 N, 17O, and 19F NMR properties of 23 molcules with 21 density functionals. Extensive comparisons were made for both 13C magnetic shieldings and chemical shifts with the wavefunction based HF and MP2 calculations. In particular, we partitioned the effects of the correlation and exchange functionals to examine the role each plays in the prediction of the NMR spectroscopy.
Table 1. Testing Set Which Contains 22 Comparisons for Comparisons for Hetero Atoms of 15N, 17O, and 19F.
Carbon atoms
Hetero atoms: nitrogen, oxygen, and fluorine
13
C, and 11
CH4, CHCH, CH2CH2, CH3CH3, CH2CCH2, CH2CCH2, CH3CH2CH3, CH3CH2CH3, C6H6, CH3COCH3, CH3OH, CH3NH2, CH3CN, CH3F, CHF3, CF4 HCN, CH3CN, CH3COCH3, H2CO, CO, CO2 NH3, CH3NH2, CH3CN, HCN, N2, HF, CH3F, CHF3, H2O, CO, CO2
Computational Methods
Caution must be taken when comparing the calculated magnetic shieldings (e) or chemical shifts (e) at the equilibrium geometry with the actually measured ones (0, 0) in the experiments. There are several sources of error, such as approximations of the methods, improper geometries, and incomplete basis sets, etc., which will significantly degrade this comparison. It is now well established that in the framework of Kohn-Sham formulism, the usual Hohenberg-Kohn theorems do not hold for a proper description of molecules in the presence of an external magnetic field because the exchange-correlation functionals should not only depend on the electron density, but also on the current density induced by the external magnetic field.11 On the other hand, just as pointed out by Lee et al., the inclusion of the leading term of the current density dependency in a density functional does not seem to increase the accuracy of the computed NMR shieldings and chemical shifts.36 Hence, it is now a commonplace to use the regular current-density-independent functionals for the calculations of NMR properties.36 Along this line, here we have examined 21 different density functionals, including eight exchange and four correlation functionals, such as the O3,30,31 X3,34,35 B3,23 B98,37 B88,20 OPTX,30 PW91,21 and PBE26 exchange functionals; the B98,37 LYP,22 PBE,26 and PW9121 correlation functionals; as well as the B971 exchange-correlation functional.38 For the calculations of the nuclear magnetic shielding tensor, the GIAO technique was employed to circumvent the gauge-origin problem.3,39 Previous investigations showed that GIAO based methods exhibit a good basis set convergence and valence triple- with at least one set of polarization functions is required for a relatively accurate calculation.24 Therefore, we adopted the 6 311 þ G (2d,p) basis set40 recommended by Cheeseman et al.24 in our calculations. It is well known that NMR properties are very sensitive to the structural variation. To distinguish the geometric influence from the effects of density functionals, the latter is the focus of the present investigation, we used equilibrium geometries or near equilibrium geometries from the experiments in all our calculations.41 A set of 23 molcules has been employed as a testing set in the present work (Table1). It contains molecules with single, double, and triple bonds, and systems requiring multiple determinants for proper descriptions. While reliable experimental data are available for this set of molecules,24,41,42 high-level benchmark calculations for most molecules in this set are also available for comparison.43
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Computation of NMR Shielding Constants and Chemical Shifts 13 Table 2. Statistic Data for C Chemical Shieldings at the Hartree-Fock, DFT, and MP2 Levels of Theory. The Best Two Methods are in Bold.
Method HF MP2 B3LYP O3LYP PBE1PBE X3LYP B971 B98B98 B98LYP B98PBE B98PW91 BB98 BLYP BPBE BPW91 OB98 OLYP OPBE OPW91 PBEB98 PBELYP PBEPBE PBEPW91
Minimum errora 0.2 5.2 4.1 0.5 0.1 3.7 2.0 1.3 0.8 0.3 0.1 5.5 5.6 1.3 1.8 0.1 0.8 0.0 0.1 5.1 4.6 1.5 2.0
(C2H2) (CF4) (CH4) (C2H2) (CH4) (CH4) (CH4) (C2H6) (C6H6) (CH3CH2CH3) (CH3CH2CH3) (CO2) (CO2) (CO2) (CO2) (HCN) (C2H2) (C2H4) (CH3CN) (CH4) (CH4) (CO2) (CO2)
Maximum errora 20.6 18.1 18.5 6.4 10.5 18.6 11.3 11.1 11.9 14.4 13.8 19.5 19.2 13.1 13.9 12.8 7.4 7.5 7.0 19.9 19.6 13.5 14.2
(CO) (CO) (CH2CCH2) (CH2CCH2) (CH2CCH2) (CH2CCH2) (CH2CCH2) (CO) (CO) (CO) (CO) (CF4) (CF4) (CF4) (CF4) (CH3COCH3) (CF4) (CO2) (CO2) (CF4) (CF4) (CF4) (CF4)
MADb MADc 7.0 8.9 9.1 3.2 4.2 9.0 5.0 5.0 5.0 3.4 3.3 11.9 11.4 6.9 7.4 3.9 3.6 2.3 2.1 11.9 11.4 6.9 7.4
7.4 4.7 13.6 7.3 9.0 13.5 9.6 7.6 7.2 4.7 4.9 16.1 15.5 11.2 11.7 7.5 6.9 3.4 3.7 16.0 15.4 11.2 11.7
Signed errors are calculated using (theor. expt.). Minimum and maximum errors are chosen based on the absolute deviation from the experiment. b Mean absolute deviation (MAD) with respect to the experimental shieldings.24,42,47–51 c Mean absolute deviation (MAD) with respect to the CCSD(T)/ 13s9p4d3f values.12,43 a
At this point, we would like to point out that all our data, as well as those theoretical results cited here for comparisons,12,43 were obtained in the nonrelativistic frame work. The experimental data24,42 with which we compared and contrasted our calculated results were obtained from the experimentally measured spin-rotation constants, also using a nonrelativistic formulation. Because shielding constants have a substantial contribution from orbital and spin angular momenta in atomic core region of valence orbitals, the relativistic effects can not be omitted in molecules containing heavy elements.44,45 All our calculations were performed with the GAUSSIAN 03 package.46
Results and Discussion Normally, quantum chemical calculations of NMR properties of a molecule provide the absolute magnetic shieldings (e) and chemical shifts (e) at the equilibrium geometry in the gas phase. When comparing with the experimental NMR data (0, 0), one should bear in mind that several effects may have a great influence on the experimental values, in particular the absolute magnetic shieldings. The rovibrational effect, the intermolecular interactions, and the solvent effects can significantly
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change the experimental values of sensitive nuclei, such that up to 40.0 ppm occurs in the case of the 19F magnetic shieldings of F2.12 Therefore, the choice of a proper reference is of great importance for validating the accuracy of a method. For this purpose, the published gas phase experimental data44 with rovibrational corrections are best suited because the influence of the intermolecular interactions and the solvent effects are largely suppressed. However, exact evaluation of rovibratinal contributions is not possible on a routine basis, because it requires the calculations of analytic derivatives of high order. Previous investigations have demonstrated that sufficient accuracy can be achieved with the benchmark CCSD(T)/13s9p4d3f calculations.42 In this work, we take the measured gas phase NMR data as reference, aiming to examine the performance of some modern functionals from a practical point of view. We also use the rovibrationally corrected experimental data, if available, or the CCSD(T) results as reference for the comparison, trying to shed light on the underling physics of an exchange or correlation functional. Magnetic Shieldings
Tables2 and 3 report the statistic data of the absolute shieldings for 13C and hetero atom (15N, 17O, and 19F) with respect to the experimental NMR data and/or CCSD(T) values, respectively. Table 3. Statistic Data for Chemical Shieldings of Hetero Atoms of 15
N, 17O, and 19F at the Hartree-Fock, DFT and MP2 Levels of Theory. The Best Two Methods are in Bold. Method
Minimum errora
Maximum errora
HF MP2 B3LYP O3LYP PBE1PBE X3LYP B971 B98B98 B98LYP B98PBE B98PW91 BB98 BLYP BPBE BPW91 OB98 OLYP OPBE OPW91 PBEB98 PBELYP PBEPBE PBEPW91
0.5 10.8 1.6 0.9 0.1 1.9 1.8 1.8 2.1 1.8 1.5 0.1 0.0 0.9 0.7 0.9 0.8 0.1 0.1 0.8 0.7 0.9 1.2
42.3 31.9 21.1 16.8 18.9 21.5 18.2 44.1 48.4 42.4 42.6 27.2 30.4 25.5 25.7 20.8 24.2 19.5 19.6 30.3 33.6 28.6 28.8
(CO2) (CH3NH2) (HF) (CO) (NH3) (HF) (HF) (HF) (HF) (CO) (CO) (HF) (HF) (HF) (HF) (HF) (HF) (NH3) (N2) (HF) (HF) (NH3) (NH3)
(N2) (N2) (CH3CN) (CHF3) (N2) (CH3CN) (CH3CN) (CHF3) (CHF3) (CHF3) (CHF3) (CHF3) (CHF3) (CHF3) (CHF3) (CHF3) (CHF3) (CHF3) (CHF3) (CHF3) (CHF3) (CHF3) (CHF3)
MADb 14.5 21.0 11.2 7.1 9.1 11.3 9.1 10.8 11.9 9.9 9.9 12.4 11.7 9.8 10.1 7.2 7.4 5.8 5.9 12.5 11.9 9.9 10.2
Signed errors are calculated using (theor. expt.). Minimum and maximum errors are chosen based on the absolute deviation from the experiment. b Mean absolute deviation (MAD) with respect to the experimental shieldings.24,42,47–51 a
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Figure 1. Mean absolute deviation for the calculated experimental value of 0 and the CCSD(T) value of e.
Besides the data calculated by 21 density functionals, the corresponding statistic data of HF, and MP2 are also included for comparison. The calculated and experimental details may be found in the supplementary material. It has been demonstrated that it is a rather challenging task to obtain accurate magnetic shieldings and at least triple excitation is required to achieve quantitative accuracy.12,42 On the other hand, previous investigation has shown that DFT covers a fairly large portion of three-electron correlation.52 Hence we believe that it is not impossible to design an exchange-correlation functional, which can surpass the MP2 method in the prediction of the magnetic shieldings. From the data listed in Table 2, it is apparent that the OPTX exchange functional generally provides good results for the 13C shieldings, leading to a mean absolute deviation (MAD) with respect to the experimental data less than 4 ppm (i.e., O3LYP: 3.2; OB98: 3.9; OLYP: 3.6; OPBE: 2.3; OPW91: 2.1 ppm). Noteworthily, OPBE or OPW91 offers a promising method for the prediction of the gas-phase NMR shieldings (0), giving both the maximum deviation and MAD that are significantly lower than those of other methods. In comparison with the experimental 0, MP2 systematically overestimates the 13C shieldings and leads to a MAD of 8.9 ppm. Considering the rovibrational corrections are always highly systematic and are all negative,42 a better agreement between the MP2 results and the experimental e would be anticipated. Take methane as an example, the discrepancy between the MP2 number and the experimental 0 is 7.1 ppm, which is reduced to 3.8 if comparison is made to the experimental e (see the supplementary material for detail). Using the benchmark CCSD(T) results as reference,
13
C magnetic shieldings with respect to the
MAD of MP2 drops from 8.9 to 4.7 ppm; while that of HF is less affected, changing from 7.0 to 7.4 ppm. Statistic data reported in Table 2 clearly show that switching the reference from the experimental 0 to the CCSD(T) e generally degrades the performance of the DFT methods. While the most popular B3LYP functional leads to MADs of 9.1 and 13.6 ppm with respect to the different reference states, the newly developed X3LYP behaves no better than B3LYP. OPBE and OPW91, on the other hand, distinguish themselves as the best methods with impressive MADs of 3.4 and 3.7 ppm, respectively, with respect to the CCSD(T) e (Table 2). We find that the DFT results exhibit a strong functional dependency (c.f. Fig. 1). For instance, the predicted 13C shieldings of methane with various density functionals range from 188.6 (BB98) to 201.6 ppm (B98PBE). Among all exchange functionals under consideration, the B88 and PBE exchange functionals systematically predict too deshielded 13C, and are not recommended because both MADs (BB98: 11.9, BLYP: 11.4, BPBE: 6.9, BPW91: 7.4; PBEB98: 11.9, PBELYP: 11.4, PBEPBE: 6.9, PBEPW91: 7.4) and the maximum deviations with respect to the experimental 0 are significantly higher than those of the others (Table 2). Including the rovibrational corrections, or taking the CCSD(T) e as reference will further deteriorate the results. The PBE and PW91 correlation functionals give values much closer to the experiments than does LYP or B98. The improvements depend on the exchange functionals. Hence OPBE (MAD ¼ 2.3) is better than OLYP (MAD ¼ 3.6) by 1.3; while BPBE (MAD ¼ 6.9) is better than BLYP (MAD ¼ 11.4) by 4.5 (Table 2). Interestingly, on the contrary to the observation of Cheeseman et al.24 and Rauhut et al.,25 the data reported in this work reveal
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Computation of NMR Shielding Constants and Chemical Shifts
Figure 2. Mean absolute deviation for the calculated mental value of 0 and the CCSD(T) value of e.
that mixture of the HF exchange with the GGA exchange may substantially improve the agreements with experimental shieldings (B3LYP: 9.1, BLYP: 11.2; PBE1PBE: 4.2, PBEPBE: 6.9; O3LYP: 3.2, OLYP: 3.6, Table 2). This is, however, in line with the usual observation in the calculations of other molecular properties, i.e., HF counterbalances GGA, giving an overall improvement. CF4 delivers a typical example (see the supplementary material for details). Although HF overshoots the experimental 13C shielding by 19.3 ppm; different flavors of GGAs undershoot the experiment by 19.2 (BLYP), 13.5 (PBEPBE), and 7.4 ppm (OLYP). The hybrid form improves the GGA description by 7.8 (B3LYP), 9.2 (PBE1PBE), and 2.1 ppm (O3LYP). The PBE1PBE functional was recommended by Adamo and Barone as the best functional currently available.28 Taking the experimental 0 as reference, our calculations lead to a MAD of 4.2 ppm, substantially better than that predicted at the MP2 level of theory (8.9 ppm). This is in agreement with the previous investigation28,29 even though some of the quantitative values differ due to the use of slightly different testing set and the geometries. Comparing with OPW91 (MAD ¼ 2.1 ppm) and OPBE (MAD ¼ 2.3 ppm), however, the PBE1PBE model underestimates all the 13C shieldings, which implies that the situation will get worse if the rovibrational corrections are taken into account. In fact, when the CCSD(T) values are taken as reference, our results clearly show that the PBE1PBE model is inferior to MP2, or even HF. The corresponding MADs for PBE1PBE, MP2, and HF are 9.0, 4.7, and 7.4 ppm, respectively. For the shieldings of hetero atoms (15N, 17O, and 19F), errors are generally larger than those for 13C (c.f. Tables 2 and 3),
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13
C chemical shifts with respect to the experi-
although OPTX exchange functional still shows good performance, being clearly superior to other methods. In particular, OPW91 and OPBE lead to MADs around 5.9 ppm; whereas, MADs obtained with other DFT methods are generally larger than 10 ppm (Table 3). The wavefunction-based HF and MP2 methods also encounter difficulty with the hetero atoms, giving MADs of 14.5 and 21.0, respectively. Indeed, it has been shown that the density functionals provide a significant improvement over the MP2 method in a more stringent test, i.e., in the case of ozone and metal oxo complexes.24,28,29,47 The lack of accurate gas phase experimental 15N, 17O, and 19 F absolute shieldings limits our discussion. Generally speaking, we find that GGAs perform better for 15N, than for 19F; whereas hybrid functionals generally perform better for 19F, than for 15N (see the supplementary material for the detail). For instance, we see that the maximum error occurs at 19F in CHF3 for all GGAs; whereas hybrid functionals (B3LYP, PBE1PBE, X3LYP, and B971) lead to large errors for 15N in the triple bond environments such as in CH3CN, HCN, and N2. It must be mentioned, in this connection, that accurate prediction of the 17 O shieldings is a rather challenging task, because it requires explicit consideration of nonspecific and specific solvation effects. An empirical scheme has been suggested by Wu et al.53 Chemical Shifts
When compared with the absolute shieldings, the chemical shifts are normally less affected by the various effects because the corrections are generally systematic and error cancellations may occur.42 In fact, it is the chemical shift that is actually measured
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13 Table 4. Statistic Data for C Chemical Shifts at the Hartree-Fock, DFT, and MP2 Levels of Theory. The Best Two Methods are in Bold.
Method HF MP2 B3LYP O3LYP PBE1PBE X3LYP B971 B98B98 B98LYP B98PBE B98PW91 BB98 BLYP BPBE BPW91 OB98 OLYP OPBE OPW91 PBEB98 PBELYP PBEPBE PBEPW91
Minimum errora 0.4 0.2 0.2 0.3 0.0 0.2 0.1 0.4 0.2 0.2 0.1 1.0 0.4 0.3 0.4 0.1 0.0 0.1 0.1 0.6 1.3 0.2 0.1
(C2H2) (CH3COCH3) (CH3CN) (CH2CCH2) (CH3CN) (CH3CN) (CO2) (CH2CCH2) (CH2CCH2) (CH2CCH2) (CH2CCH2) (CO2) (CO2) (C6H6) (C6H6) (CH2CCH2) (CH3CN) (CH3CH2CH3) (CH3CH2CH3) (CO2) (CO2) (CO2) (CO2)
Maximum errora 21.3 10.9 14.4 7.1 13.7 14.9 9.8 15.5 14.7 13.9 13.9 14.4 13.5 12.7 12.7 6.3 6.0 6.0 5.9 17.0 16.1 15.3 15.3
(CO) (CO) (CH2CCH2) (H2CO) (H2CO) (CH2CCH2) (H2CO) (H2CO) (H2CO) (H2CO) (H2CO) (H2CO) (H2CO) (H2CO) (H2CO) (H2CO) (CF4) (CO2) (CO2) (H2CO) (H2CO) (H2CO) (H2CO)
MADb
MADc
7.2 2.1 5.2 2.4 4.4 5.5 3.1 7.6 7.7 5.8 5.9 5.8 5.7 4.2 4.3 2.9 3.0 2.0 2.0 7.1 7.1 5.4 5.5
7.1 2.2 6.1 3.0 5.5 6.5 4.2 7.9 7.8 6.4 6.5 6.6 6.4 5.1 5.2 3.4 3.3 2.5 2.4 7.8 7.6 6.3 6.4
Signed errors are calculated using (theor. expt.). Minimum and maximum errors are chosen based on the absolute deviation from the experiment. b Mean absolute deviation (MAD) with respect to the experimental shifts.24,42,47–51 c Mean absolute deviation (MAD) with respect to the CCSD(T)/13s9p4d3f values.12,43 a
with high accuracy in the NMR spectroscopy. Thus the prediction of chemical shifts (e) is a primary goal of the quantum chemical investigation. The calculated chemical shifts (e) for 13C are compared with the experimental values (0) and the best CCSD(T) results (e) (as shown in Fig. 2). The chemical shifts are given relative to the corresponding gaseous values of CH4, NH3, H2O, and HF (see supplementary material). The statistical data are summarized in Tables4 and 5. MP2 is the most widely used nondensity functional method, providing a good agreement with the experimentally measured 0. Our calculations give 13C MADs of 2.1 and 2.2 ppm, when the experimental 0 and the CCSD(T) e are taken as reference, respectively, confirming that MP2 is indeed good for the prediction of chemical shifts. However, it is noteworthy to recall that large MADs for absolute shielding calculations are encountered at the MP2 level (8.9 and 4.7 ppm, Table 2). Hence, the success of MP2 in the prediction of chemical shifts should be attributed to a systematic error cancellation when considering the differences between two shieldings. It is encouraging to see that OPBE and OPW91, being the absolute winners in the prediction of magnetic shieldings (Tables 2–5), again give the best performance in the prediction of 13C chemical shifts (Table 4). Using the experimental 0 as reference, OPBE or OPW91 leads to 13C
MAD of 2.0 ppm, which is slightly increased to 2.5 ppm when using the CCSD(T) e as reference. We note that OPBE also provides good performance in the calculation of various molecular properties and has been recommended to be applied in studying a broad spectrum.54 Comparing data in Tables 2 and 4, it is evident that the deviations from the experimental chemical shifts are generally smaller than the corresponding values for the absolute shieldings. Besides OPBE and OPW91, other methods with OPTX being the exchange functional generally exhibit a remarkable performance in the calculations of chemical shifts. The MADs are 2.4 ppm for O3LYP, 2.9 for OB98, 3.0 for OLYP with respect to the experimental 0. This superiority is retained when taking the benchmark CCSD(T) results as reference. The next best quality shift within the framework of density functional theory is provided by B971 (3.1 ppm, Table 4 and Fig. 2). We note that Keal and Tozer have shown in a recent study55 that the revised version of B971, namely B972, performs quite well in the calculation of 77Se NMR shielding constants. The performance of the B98 exchange functional is degraded when compared with its own performance in the prediction of magnetic shieldings. Other commonly used hybrid functionals such as B3LYP and PBE1PBE, as well as the newly developed X3LYP do not show an impressive performance. Table 5. Statistic Data for Chemical Shifts of Hetero Atoms
of 15N, 17O, and 19F at the Hartree-Fock, DFT, and MP2 Levels of Theory. The Best Two Methods are in Bold. Method
Minimum errora
Maximum errora
HF MP2 B3LYP O3LYP PBE1PBE X3LYP B971 B98B98 B98LYP B98PBE B98PW91 BB98 BLYP BPBE BPW91 OB98 OLYP OPBE OPW91 PBEB98 PBELYP PBEPBE PBEPW91
0.1 0.1 9.9 2.2 6.4 9.3 7.7 0.1 1.6 2.5 2.2 9.1 5.4 6.9 6.8 0.8 0.1 0.6 0.4 10.5 6.8 8.3 8.3
40.9 19.0 17.9 18.4 18.8 18.9 16.4 45.9 50.5 45.1 45.0 27.3 30.4 26.4 26.4 21.7 25.0 21.2 21.2 31.1 34.3 30.1 30.1
(CH3NH2) (CH3F) (CH3NH2) (CO) (CH3NH2) (CH3F) (CH3NH2) (N2) (N2) (N2) (N2) (CO) (CO) (CO) (CO) (CO) (N2) (N2) (N2) (CO) (CO) (CO) (CO)
(N2) (N2) (CH3CN) (CHF3) (N2) (CH3CN) (CH3CN) (CHF3) (CHF3) (CHF3) (CHF3) (CHF3) (CHF3) (CHF3) (CHF3) (CHF3) (CHF3) (CHF3) (CHF3) (CHF3) (CHF3) (CHF3) (CHF3)
MADb 17.9 7.8 13.9 9.3 13.4 14.4 12.0 15.8 16.3 14.3 14.4 14.2 13.6 12.2 12.3 8.7 9.0 7.7 7.7 15.8 15.3 13.8 13.9
Signed errors are calculated using (theor. expt.). Minimum and maximum errors are chosen based on the absolute deviation from the experiment. b Mean absolute deviation (MAD) with respect to the experimental shifts.24,42,47–51 a
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Figure 3. Mean absolute deviation for the calculated 13C, 15N, 17O, ical shifts with respect to the corresponding experimental values.
We notice that even though the exchange functional plays a decisive role in the prediction of the magnetic properties, the correlation functional also makes an important contribution. Just like what we find in the prediction of the absolute shieldings, PW91 and PBE are the best correlation functionals examined here in the prediction of the chemical shifts. Hence OPBE (MAD ¼ 2.0) is better than OLYP (MAD ¼ 3.0) by 1.0; whereas BPBE (MAD ¼ 4.2) is better than BLYP (MAD ¼ 5.7) by 1.3. Naturally, the role of correlation is more significant in the prediction of shiledings than in shifts. Table 5 summarizes the statistic data for chemical shifts of hetero atoms of 15N, 17O, and 19F at the Hartree-Fock, DFT, and MP2 levels of theory. Figure 3 makes a direct comparison between the calculated shifts and calculated shieldings. Unlike the situation seen for 13C, where improvement is clearly evident for the prediction of shifts over shieldings for all methods, this happens only for MP2. For all DFT methods, we see that errors are more significant for shifts than for shieldings. We notice that for many DFT methods, the minimum error for shielding occur at the reference molecule, impairing the function of error cancellation in shift calculation. Although the statistical performance in shift calculation may be improved by chosen proper reference molecules,56 the deficiency for hetero atoms calls for further improvement of the density functionals. Exchange and Correlation Effects
To better understand the effects of various exchange and correlation functionals, further analysis is made in term of the different exchange functionals for the prediction of the 13C magnetic shield-
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F magnetic shieldings and chem-
ings. Figure 4 presents a comparison of the calculated 13C magnetic shieldings obtained by HF and DFT exchange only methods (B98, OPTX, B88 and PBE) with respect to the CCSD(T) e. Figure 4A depicts the differences between HF and CCSD(T). As (eHF eCCSD(T)) is generally negative, leading to a mean deviation of 4 ppm, we see that HF has a tendency of deshielding. This observation indicates that for a DFT exchange only method, a deshielding tendency would be expected when the correlation effect is ignored. Figure 4B depicts the differences between the 13C shieldings obtained with exchange only methods of B88 and PBE and those of CCSD(T). The deshielding tendency for B88 and PBE is obvious, ranging from 5.0 (CO) to 21.0 ppm (CH2CCH2), which leads to a mean deviation of 14 ppm. In fact, the deshielding effect is so much exaggerated that it overshoots that of HF by 10 ppm. We notice that GGA methods using B88 or PBE as exchange functional generally underestimate the absolute magnetic shieldings with respect to the CCSD(T) results. We believe that the main source of error shall be attributed to the deficiency of these exchange functionals. For the B98 and OPTX exchange functionals, on the other hand, the situation is much better. Although there is still a tendency that 13C is too deshielded, the deviations are halved when compared with that of B88 and PBE. Thus the B98 and OPTX exchange functionals lead to a mean deviation of 4 ppm in the absolute shieldings using the CCSD(T) values as reference, being comparable to that of HF. Recall that GGAs using OPTX as exchange functional consistently provides the best performance for both magnetic shieldings and chemical shifts, signifying the importance of the exchange functional.
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Figure 4. The effect of exchange on the calculated 13C magnetic shieldings. (A) eHF eCCSD(T). The average deviation of the HF values from the CCSD(T) values is around 4 ppm. (B) eEx-only eCCSD(T). The average deviation of the B88 or PBE exchange-only values from the CCSD(T) values is around 14 ppm. (C) eEx-only eCCSD(T). The average deviation of the B98 or OPTX exchangeonly values from the CCSD(T) values is around 4 ppm.
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Figure 4. (continued)
Here we emphasize that there are detailed differences in the performance of B98, OPTX from that of HF. For instance, both the B98 and OPTX exchange functionals dramatically overestimate the absolute shieldings for CO, leading to a deviation of 14.5 and 5.1 ppm, respectively, for this molecule with respect to the CCSD(T) number, while HF underestimates this value by 22.6 ppm. CF4 molecule provides the other extreme. The B98 and OPTX exchange functionals lead to negative deviations of 7.8 and 7.0 ppm, whereas the corresponding number for HF is 18.5 ppm. We believe that understanding of these differences will be of great help for the development of new density functional for the accurate prediction of NMR properties. While for the previous NMR calculations within the framework of DFT, the major focus has been putting on the investigation of the performance of the whole exchange-correlation functionals.24,25,28,29,32,33,54–58 Few has been devoted to the investigation of the contribution by an individual exchange and correlation functional. Especially, the effect of correlation functionals on the prediction of the NMR data is more or less overlooked. Figure 5 signifies the correlation effect based on the calculated 13C shieldings with respect to the HF data. Figure 5A compares the values of (eMP2 eHF) with those of (eCCSD(T) eHF). While the former leads to an average number of 7.7 ppm; the latter gives 4.2 ppm. Thus MP2 has a tendency to exaggerate the correlation effect. As the (eMP2 eHF) values are mostly too positive, we correlate this observation with the fact that the MP2 approach tends to overestimate shieldings.
Figures 5B and 5C depict the correlation contributions to the calculated 13C shieldings. These numbers are obtained by subtracting the calculated values using the exchange only method of OPTX from those using the OLYP, OB98, OPBE, and OPW91 functionals. Different exchange functional may be used, which is seen to not affect the general feature of Figures 5B and 5C. Comparison between Figures 5B and 5C clearly demonstrates that the correlation functionals may be divided into two classes, namely class A and class B. While class A correlation functionals tend to lead to more deshielded values (Fig. 5B), being qualitatively different from MP2 and CCSD(T), class B functionals increase the values (Fig. 5C), in qualitative agreement with MP2 and CCSD(T). Therefore, the introduction of class A correlation functionals systematically lowers the absolute shieldings and hence further worsens the agreement between the calculated magnetic shieldings and the reference data because the exchange functionals generally give deshielded numbers. On the contrary, class B correlation functionals systematically increase the absolute shieldings, which complement the exchange functionals and lead to an overall improved agreement between the calculated values and the reference values. From Figure 5, one can see that the B98 and LYP correlation functionals belong to class A; and the PBE and PW91 correlation functionals belong to class B. This is in consistency with the previous conclusions based on the sum-over-state calculations, in which BPW91 is clearly superior to BLYP,47,56 and is also in line with the data shown in Tables 2–5. It is worthy to point out that although correlation
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Figure 5. The effect of correlation on the calculated 13C magnetic shieldings. (A) eCCSD(T) eHF and eMP2 eHF. For CO the corresponding values are 22.6 and 38.7 ppm, which are truncated at 22 ppm in the figure. (B) eOB98 eOPTX and eOLYP eOPTX. (C) eOPW91 eOPTX and eOPBE eOPTX.
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Figure 5. (continued)
effects of PBE and PW91 follow the same direction as do the CCSD(T) method, the magnitudes are much smaller, leaving room for further improvement.
Conclusion In this work, we have systematically examined the performance of some popular and newly developed density functionals in the calculations of NMR magnetic shieldings and chemical shifts. Careful comparisons have been made in terms of shieldings and shifts with respect to the experimental data and the best CCSD(T) results. The following conclusions can be drawn from this work: 1. When the calculated NMR data are compared with the experimental data, caution has to be taken, as there are many sources of errors, such as the rovibrational correction, the intermolecular interactions and solvent effects, etc., which will degrade the comparison. An improper choice of the reference may result in misleading conclusions. 2. In contrast to the systematical overestimation of the magnetic shieldings predicted by MP2, DFT methods generally lead to too deshielded values. The source of errors may be traced back to the deficiency of the exchange functionals. We find that although the B88 and PBE exchange functionals are not recommended for the NMR calculations; the OPTX exchange functional consistently delivers the best absolute shieldings and chemical shifts.
3. The correlation functionals may be grouped into two classes. class A, such as LYP, generally lowers the absolute shieldings and hence leads to an even deshielded value when combined with an exchange functional. On the other hand, class B, such as PBE or PW91, has a positive contribution, which complements the exchange functional, leading to a better result in the prediction of NMR data. 4. We find that the OPBE and OPW91 exchange-correlation functionals perform remarkably well along the whole set of systems, surpassing, in many aspects, the standard wavefunction-based method MP2. We conclude that OPBE or OPW91 is a good DFT functional currently available for the prediction of NMR data. Future work will be on the application of these functionals to the calculations of NMR spectroscopy of large molecules of chemical interest and on the development of new density functionals with higher accuracy.
References 1. Grant, D. M.; Harris, R. K., Eds. Encyclopedia of Nuclear Magnetic Resonance, Vol. 1–8; Wiley: Chichester, UK, 1996. 2. Kutzelnigg, W.; Fleischer, U.; Schindler, M. In NMR Basis Principles and Progress, Vol. 23; Springer: Heidelberg, 1990; p. 165. 3. Helgaker, T.; Jaszunski, M.; Ruud, K. Chem Rev 1999, 99, 293. 4. Takashima, H.; Hada, M.; Nakatsuji, H. J Phys Chem 1995, 99, 7951.
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5. Chestnut, D. B. In Reviews in Computational Chemistry, Vol. 8; Lipkowitz, K. B.; Boyd, D. B., Eds.; VCH: New York, 1996; p. 245. 6. Fukui, H. Prog Nucl Magn Reson Spectrosc 1997, 31, 317. 7. Jameson, C. J. Annu Rev Phys Chem 1996, 47, 135. 8. Kanelis, V.; Donaldson, L.; Muhandiram, D. R.; Rotin, D.; FormanKay, J. D.; Kay, L. E. J Biomol NMR 2000, 16, 253. 9. Wuthrich, K. Biosci Rep 2003, 23, 119. 10. Wuthrich, K. Angew Chem Int Ed 2003, 42, 3340. 11. Koch, W.; Holthausen, M. C. InA Chemist’s Guide to Density Functional Theory; Wiley-VCH: Verlag, 2001; p. 204. 12. Gauss, J.; Stanton, J. F. J Chem Phys 1996, 104, 2574, and references therein. 13. Gauss, J.; Stanton, J. F. J Chem Phys 1995, 102, 251. 14. Gauss, J.; Stanton, J. F. J Chem Phys 1995, 103, 3561. 15. Stanton, J. F.; Gauss, J. J Chem Phys 1995, 103, 8913. 16. London, F. J Phys Radium 1937, 8, 397. 17. Kohn, W.; Sham, L. J. Phys Rev A 1965, 140, 1133. 18. Slater, J. C. Quantum Theory of Molecules and Solids, Vol. 4: The Self-Consistent Field for Molecules and Solids; McGraw-Hill: New York, 1974. 19. Vosko, S. H.; Wilk, L.; Nusair, M. Can J Phys 1980, 58, 1200. 20. Becke, A. D. Phys Rev B 1988, 38, 3098. 21. Perdew, J. P.; Wang, Y. Phys Rev B 1992, 45, 13244. 22. Lee, C.; Yang, W.; Par, R. G. Phys Rev B 1988, 37, 785. 23. Becke, A. D. J Chem Phys 1993, 98, 5648. 24. Cheeseman, J. R.; Trucks, G. W.; Keith, T. A.; Frisch, M. J. J Chem Phys 1996, 104, 5497, and references therein. 25. Rauhut, G.; Puyear, S.; Wolinski, K.; Pulay, P. J Phys Chem 1996, 100, 6310. 26. Perdew, J. P.; Burke, K.; Ernzerhof, M. Phys Rev Lett 1996, 77, 3865. 27. Perdew, J. P.; Burke, K.; Ernzerhof, M. Phys Rev Lett 1996, 78, 1396. 28. Adamo, C.; Barone, V. Chem Phys Lett 1998, 298, 113. 29. Adamo, C.; Barone, V. J Chem Phys 1999, 110, 6158. 30. Handy, N. C.; Cohen, A. J. Mol Phys 2001, 99, 403. 31. Cohen, A. J.; Handy, N. C. Mol Phys 2001, 99, 607. 32. Magyarfalvi, G.; Pulay, P. J Chem Phys 2003, 119, 1350. 33. Zhang, Y.; Wu, A.; Xu, X.; Yan, Y. J. Chem Phys Lett 2006, 421, 383. 34. Xu, X.; Zhang, Q. S.; Muller, R. P.; Goddard, W. A., III. J Chem Phys 2005, 122, 014105. 35. Xu, X.; Goddard, W. A., III. Proc Natl Acad Sci USA 2004, 101, 2673. 36. Lee, C.; Handy, N. C.; Colwell, S. M. J Chem Phys 1995, 103, 10095. 37. Schmider, H. L.; Becke, A. D. J Chem Phys 1998, 108, 9624. 38. Cohen, A. J.; Handy, N. C. Chem Phys Lett 2000, 316, 160.
39. Ditchfield, R. J Chem Phys 1972, 56, 5688. 40. Krishman, R.; Binkley, J. S.; Seeger, R.; Pople, J. A. J Chem Phys 1980, 72, 650. 41. Sychrovsky´, V.; Gra¨fenstein, J.; Cremer, D. J Chem Phys 2000, 113, 3530, and references therein. 42. Jameson, A. K.; Jameson, C. J. Chem Phys Lett 1987, 134, 461. 43. Auer, A. A.; Gauss, J.; Stanton, J. F. J Chem Phys 2003, 118, 10407, and references therein. 44. Vaara, J.; Pyykko¨, P. J Chem Phys 2003, 118, 2973. 45. Hada, M.; Fukuda, R.; Nakatsuji, H. Chem Phys Lett 2000, 321, 452. 46. Frisch, M. J.; Trucks, G. W.; Schlegel, H. B.; Scuseria, G. E.; Robb, M. A.; Cheeseman, J. R.; Montgomery, J. A. Jr.; Vreven, T.; Kudin, K. N.; Burant, J. C.; Millam, J. M.; Iyengar, S. S.; Tomasi, J.; Barone, V.; Mennucci, B.; Cossi, M.; Scalmani, G.; Rega, N.; Petersson, G. A.; Nakatsuji, H.; Hada, M.; Ehara, M.; Toyota, K.; Fukuda, R.; Hasegawa, J.; Ishida, M.; Nakajima, T.; Honda, Y.; Kitao, O.; Nakai, H.; Klene, M.; Li, X.; Knox, J. E.; Hratchian, H. P.; Cross, J. B.; Adamo, C.; Jaramillo, J.; Gomperts, R.; Stratmann, R. E.; Yazyev, O.; Austin, A. J.; Cammi, R.; Pomelli, C.; Ochterski, J. W.; Ayala, P. Y.; Morokuma, K.; Voth, G. A.; Salvador, P.; Dannenberg, J. J.; Zakrzewski, V. G.; Dapprich, S.; Daniels, A. D.; Strain, M. C.; Farkas, O.; Malick, D. K.; Rabuck, A. D.; Raghavachari, K.; Foresman, J. B.; Ortiz, J. V.; Cui, Q.; Baboul, A. G.; Clifford, S.; Cioslowski, J.; Stefanov, B. B.; Liu, G.; Liashenko, A.; Piskorz, P.; Komaromi, I.; Martin, R. L.; Fox, D. J.; Keith, T.; Allaham, M. A.; Peng, C. Y.; Nanayakkara, A.; Challacombe, M.; Gill, M. W.; Johnson, B.; Chen, W.; Wong, M. W.; Gonzalez, C.; People, J. A. Gaussian 03, Gaussian, Inc., Wallingford, CT, 2004. 47. Malkin, V. G.; Malkina, O. L.; Casida, M. E.; Salahub, D. R. J Am Chem Soc 1994, 116, 5898. 48. Jameson, C. J.; Jameson, A. K.; Oppusungu, D.; Wille, S.; Burrell, P. M. J Chem Phys 1981, 74, 81. 49. Wasylishen, R. E.; Bryce, D. L. J Chem Phys 2002, 117, 10061. 50. Makulski, W.; Jackowski, K. Chem Phys Lett 2001, 341, 369. 51. Jameson, C. J.; Jameson, A. K.; Burrell, P. M. J Chem Phys 1980, 73, 6013. 52. He, Y.; Gra¨fenstein, J.; Kraka, E.; Cremer, D. Mol Phys 2000, 98, 1639. 53. Wu, A.; Cremer, D.; Gauss, J. J Phys Chem A 2004, 107, 8737. 54. Swart, M.; Ehlers, A.; Lammertsma, K. Mol Phys 2004, 102, 2467. 55. Keal, T. W.; Tozer, D. J. Mol Phys 2005, 103, 1007. 56. Olsson, L.; Cremer, D. Ph.D. Thesis, Department of Theoretical Chemistry, Goteborg University, Goteborg, 1996. 57. Helgaker, T.; Wilson, P. J.; Amos, D. J.; Handy, N. C. J Chem Phys 2000, 113, 2983. 58. Wilson, P. J.; Tozer, D. J. J Chem Phys 2002, 116, 10139.
Journal of Computational Chemistry
DOI 10.1002/jcc
State Key Laboratory for Physical Chemistry of Solid Surface, Center for Theoretical Chemistry, Department of Chemistry, Xiamen University, 361005, China 2 Department of Chemistry, Hong Kong University of Science and Technology, Kowloon, Hong Kong Received 11 August 2006; Revised 24 October 2006; Accepted 1 November 2006 DOI 10.1002/jcc.20641 Published online 22 August 2007 in Wiley InterScience (www.interscience.wiley.com).
Abstract: We present a systematic density functional investigation on the prediction of the 13C, 15N, 17O, and 19F NMR properties of 23 molecules with 21 density functionals. Extensive comparisons are made for both 13C magnetic shieldings and chemical shifts with respect to the gas phase experimental data and the best CCSD(T) results. We find that the OPBE and OPW91 exchange-correlation functionals perform significantly better than some popular functionals such as B3LYP and PBE1PBE, even surpassing, in many cases, the standard wavefunction-based method MP2. Further analysis has been performed to explore the individual role played by various exchange and correlation functionals. We find that the B88 and PBE exchange functionals have a too strong tendency of deshielding, leading to too deshielded magnetic shielding constants; whereas the OPTX exchange functional performs remarkably well. We claim that the main source of error arises from the exchange functional, but correlation functional also makes important contribution. We find that the correlation functionals may be grouped into two classes. class A, such as LYP and B98, leads to deshielded NMR values, deteriorating the overall performance; whereas class B, such as PW91 and PBE, generally increases the absolute shieldings, which complements the exchange functionals, leading to improved results in the calculation of NMR data. q 2007 Wiley Periodicals, Inc.
J Comput Chem 28: 2431–2442, 2007
Key words: density functional theory; nuclear magnetic resonance; magnetic shieldings; chemical shifts; GIAO
Introduction Nuclear magnetic resonance (NMR) spectroscopy has gained an increasing popularity in many chemical applications.1–7 Because of its sensitivity to the molecular composition, geometry, conformation, and environment, NMR spectroscopy leads to valuable information about the electronic environment of a nucleus, and hence has been extensively used in identifying the fragment of a system and in determining the three-dimensional structures of biomolecules.8–10 As a supplementary to experiments, quantum chemical calculation of NMR properties has become a useful tool for the interpretation of the experimental NMR data.1–6 During the last decades, a number of methods have been developed for the calculations of the NMR chemical shifts. It has been generally accepted that, although the Hartree-Fock (HF) method provides chemical shifts with reasonable accuracy, especially for organic molecules, more accurate prediction of these properties requires tak-
ing into account the electron correlation effects.3,11–15 While the post Hartree-Fock methods, such as the second-order MøllerPlesset perturbation approach (MP2), can be quite computationally demanding for a chemically interesting large system, density functional theory (DFT) offers an alternative to cover the electron correlation effect in an unspecific way with a computational cost which is the same order of HF.11 Today DFT has been successfully applied in predicting various molecular properties such This article contains supplementary material available via the Internet at http://www.interscience.wiley.com/jpages/0192-8651/suppmat Correspondence to: X. Xu; e-mail: [email protected] Contract/grant sponsor: National Science Foundation of China; contract/ grant numbers: 20525311, 20533030, 20021002, 20423002 Contract/grant sponsor: Ministry of Science and Technology of China; contract/grant numbers: 2004CB719902 Contract/grant sponsor: Ministry of Education and Xiamen University
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as polarizibility, vibrational frequency, etc., often giving comparable or even better results than higher computationally demanding MP2.11 In this respect, DFT methods possess a promising perspective for the prediction of the NMR properties, as opposed to the conventional ab initio methods in quantum chemistry. Using large Gaussian type basis sets and the gauge including atomic orbital (GIAO)16 technique, Cheeseman et al. tested the performance of some popular exchange-correlation functionals such as LSDA,17–19 BPW91,20,21 BLYP,20,22 B3PW91,21,23 and B3LYP22,23 for the prediction of NMR chemical shifts over a set of organic and inorganic molecules.24 They claimed that all DFT functionals, except the local density approach, which is undoubtedly inferior, provided a similar performance and statistically showed an improvement over the HF model.24 For the 13 C chemical shift, the BPW91 and BLYP functionals were rated the best.24 It was also shown that in general for organic molecules the MP2 method significantly outperformed all the DFT approaches examined in their works.24 With the same approach but slightly different basis sets, Rauhut et al. reached a similar conclusion.25 Adamo and Barone investigated the performance of the PBEPBE26 and PBE1PBE27 functionals and compared the absolute magnetic shieldings, instead of the chemical shifts, with the MP2 results. They drew a conclusion that the PBE1PBE functional delivered competitive results for the well behaved systems with respect to the MP2 method.28,29 It seems to us, however, further validation on PBE1PBE is needed to take its performance on the chemical shift prediction into account based on the reference values adequately chosen. With the development of DFT, several new functionals have been proposed recently. By fitting against the HF energies of atoms, Handy et al. presented an optimized exchange functional OPTX.30 When combined with the LYP or PBE correlation functional, OLYP and OPBE were shown to give a substantial improvement over the classical GGA functionals such as BLYP, BPW91, PBEPBE, etc. in calculating the heats of formation, ionization potentials, and electron affinities with the G2 set.31 Also, the corresponding hybrid form O3LYP was claimed to be competitive with the commonly used B3LYP in many aspects.31 Magyarfalvi and Pulay did a careful investigation of DFT performance on the prediction of magnetic shieldings with respect to the CCSD(T) results. They concluded that OLYP gives a marked improvement over BLYP.32 In a recent communication, we showed that OPBE is even better for the prediction of NMR properties.33 Xu et al. introduced a new exchange functional X3 by combining the B88 and PW91 exchange functionals.34,35 In conjunction with LYP, X3LYP exhibited a promising performance, particularly in dealing with weakly interacting systems.34,35 So far, most of these newly developed functionals are just validated against the thermodynamic data. There is a lack of systematic studies on their performance in calculating NMR spectroscopy parameters.32,33 In this contribution, we presented our results for a systematic DFT/GIAO investigation of the 13C, 15 N, 17O, and 19F NMR properties of 23 molcules with 21 density functionals. Extensive comparisons were made for both 13C magnetic shieldings and chemical shifts with the wavefunction based HF and MP2 calculations. In particular, we partitioned the effects of the correlation and exchange functionals to examine the role each plays in the prediction of the NMR spectroscopy.
Table 1. Testing Set Which Contains 22 Comparisons for Comparisons for Hetero Atoms of 15N, 17O, and 19F.
Carbon atoms
Hetero atoms: nitrogen, oxygen, and fluorine
13
C, and 11
CH4, CHCH, CH2CH2, CH3CH3, CH2CCH2, CH2CCH2, CH3CH2CH3, CH3CH2CH3, C6H6, CH3COCH3, CH3OH, CH3NH2, CH3CN, CH3F, CHF3, CF4 HCN, CH3CN, CH3COCH3, H2CO, CO, CO2 NH3, CH3NH2, CH3CN, HCN, N2, HF, CH3F, CHF3, H2O, CO, CO2
Computational Methods
Caution must be taken when comparing the calculated magnetic shieldings (e) or chemical shifts (e) at the equilibrium geometry with the actually measured ones (0, 0) in the experiments. There are several sources of error, such as approximations of the methods, improper geometries, and incomplete basis sets, etc., which will significantly degrade this comparison. It is now well established that in the framework of Kohn-Sham formulism, the usual Hohenberg-Kohn theorems do not hold for a proper description of molecules in the presence of an external magnetic field because the exchange-correlation functionals should not only depend on the electron density, but also on the current density induced by the external magnetic field.11 On the other hand, just as pointed out by Lee et al., the inclusion of the leading term of the current density dependency in a density functional does not seem to increase the accuracy of the computed NMR shieldings and chemical shifts.36 Hence, it is now a commonplace to use the regular current-density-independent functionals for the calculations of NMR properties.36 Along this line, here we have examined 21 different density functionals, including eight exchange and four correlation functionals, such as the O3,30,31 X3,34,35 B3,23 B98,37 B88,20 OPTX,30 PW91,21 and PBE26 exchange functionals; the B98,37 LYP,22 PBE,26 and PW9121 correlation functionals; as well as the B971 exchange-correlation functional.38 For the calculations of the nuclear magnetic shielding tensor, the GIAO technique was employed to circumvent the gauge-origin problem.3,39 Previous investigations showed that GIAO based methods exhibit a good basis set convergence and valence triple- with at least one set of polarization functions is required for a relatively accurate calculation.24 Therefore, we adopted the 6 311 þ G (2d,p) basis set40 recommended by Cheeseman et al.24 in our calculations. It is well known that NMR properties are very sensitive to the structural variation. To distinguish the geometric influence from the effects of density functionals, the latter is the focus of the present investigation, we used equilibrium geometries or near equilibrium geometries from the experiments in all our calculations.41 A set of 23 molcules has been employed as a testing set in the present work (Table1). It contains molecules with single, double, and triple bonds, and systems requiring multiple determinants for proper descriptions. While reliable experimental data are available for this set of molecules,24,41,42 high-level benchmark calculations for most molecules in this set are also available for comparison.43
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Computation of NMR Shielding Constants and Chemical Shifts 13 Table 2. Statistic Data for C Chemical Shieldings at the Hartree-Fock, DFT, and MP2 Levels of Theory. The Best Two Methods are in Bold.
Method HF MP2 B3LYP O3LYP PBE1PBE X3LYP B971 B98B98 B98LYP B98PBE B98PW91 BB98 BLYP BPBE BPW91 OB98 OLYP OPBE OPW91 PBEB98 PBELYP PBEPBE PBEPW91
Minimum errora 0.2 5.2 4.1 0.5 0.1 3.7 2.0 1.3 0.8 0.3 0.1 5.5 5.6 1.3 1.8 0.1 0.8 0.0 0.1 5.1 4.6 1.5 2.0
(C2H2) (CF4) (CH4) (C2H2) (CH4) (CH4) (CH4) (C2H6) (C6H6) (CH3CH2CH3) (CH3CH2CH3) (CO2) (CO2) (CO2) (CO2) (HCN) (C2H2) (C2H4) (CH3CN) (CH4) (CH4) (CO2) (CO2)
Maximum errora 20.6 18.1 18.5 6.4 10.5 18.6 11.3 11.1 11.9 14.4 13.8 19.5 19.2 13.1 13.9 12.8 7.4 7.5 7.0 19.9 19.6 13.5 14.2
(CO) (CO) (CH2CCH2) (CH2CCH2) (CH2CCH2) (CH2CCH2) (CH2CCH2) (CO) (CO) (CO) (CO) (CF4) (CF4) (CF4) (CF4) (CH3COCH3) (CF4) (CO2) (CO2) (CF4) (CF4) (CF4) (CF4)
MADb MADc 7.0 8.9 9.1 3.2 4.2 9.0 5.0 5.0 5.0 3.4 3.3 11.9 11.4 6.9 7.4 3.9 3.6 2.3 2.1 11.9 11.4 6.9 7.4
7.4 4.7 13.6 7.3 9.0 13.5 9.6 7.6 7.2 4.7 4.9 16.1 15.5 11.2 11.7 7.5 6.9 3.4 3.7 16.0 15.4 11.2 11.7
Signed errors are calculated using (theor. expt.). Minimum and maximum errors are chosen based on the absolute deviation from the experiment. b Mean absolute deviation (MAD) with respect to the experimental shieldings.24,42,47–51 c Mean absolute deviation (MAD) with respect to the CCSD(T)/ 13s9p4d3f values.12,43 a
At this point, we would like to point out that all our data, as well as those theoretical results cited here for comparisons,12,43 were obtained in the nonrelativistic frame work. The experimental data24,42 with which we compared and contrasted our calculated results were obtained from the experimentally measured spin-rotation constants, also using a nonrelativistic formulation. Because shielding constants have a substantial contribution from orbital and spin angular momenta in atomic core region of valence orbitals, the relativistic effects can not be omitted in molecules containing heavy elements.44,45 All our calculations were performed with the GAUSSIAN 03 package.46
Results and Discussion Normally, quantum chemical calculations of NMR properties of a molecule provide the absolute magnetic shieldings (e) and chemical shifts (e) at the equilibrium geometry in the gas phase. When comparing with the experimental NMR data (0, 0), one should bear in mind that several effects may have a great influence on the experimental values, in particular the absolute magnetic shieldings. The rovibrational effect, the intermolecular interactions, and the solvent effects can significantly
2433
change the experimental values of sensitive nuclei, such that up to 40.0 ppm occurs in the case of the 19F magnetic shieldings of F2.12 Therefore, the choice of a proper reference is of great importance for validating the accuracy of a method. For this purpose, the published gas phase experimental data44 with rovibrational corrections are best suited because the influence of the intermolecular interactions and the solvent effects are largely suppressed. However, exact evaluation of rovibratinal contributions is not possible on a routine basis, because it requires the calculations of analytic derivatives of high order. Previous investigations have demonstrated that sufficient accuracy can be achieved with the benchmark CCSD(T)/13s9p4d3f calculations.42 In this work, we take the measured gas phase NMR data as reference, aiming to examine the performance of some modern functionals from a practical point of view. We also use the rovibrationally corrected experimental data, if available, or the CCSD(T) results as reference for the comparison, trying to shed light on the underling physics of an exchange or correlation functional. Magnetic Shieldings
Tables2 and 3 report the statistic data of the absolute shieldings for 13C and hetero atom (15N, 17O, and 19F) with respect to the experimental NMR data and/or CCSD(T) values, respectively. Table 3. Statistic Data for Chemical Shieldings of Hetero Atoms of 15
N, 17O, and 19F at the Hartree-Fock, DFT and MP2 Levels of Theory. The Best Two Methods are in Bold. Method
Minimum errora
Maximum errora
HF MP2 B3LYP O3LYP PBE1PBE X3LYP B971 B98B98 B98LYP B98PBE B98PW91 BB98 BLYP BPBE BPW91 OB98 OLYP OPBE OPW91 PBEB98 PBELYP PBEPBE PBEPW91
0.5 10.8 1.6 0.9 0.1 1.9 1.8 1.8 2.1 1.8 1.5 0.1 0.0 0.9 0.7 0.9 0.8 0.1 0.1 0.8 0.7 0.9 1.2
42.3 31.9 21.1 16.8 18.9 21.5 18.2 44.1 48.4 42.4 42.6 27.2 30.4 25.5 25.7 20.8 24.2 19.5 19.6 30.3 33.6 28.6 28.8
(CO2) (CH3NH2) (HF) (CO) (NH3) (HF) (HF) (HF) (HF) (CO) (CO) (HF) (HF) (HF) (HF) (HF) (HF) (NH3) (N2) (HF) (HF) (NH3) (NH3)
(N2) (N2) (CH3CN) (CHF3) (N2) (CH3CN) (CH3CN) (CHF3) (CHF3) (CHF3) (CHF3) (CHF3) (CHF3) (CHF3) (CHF3) (CHF3) (CHF3) (CHF3) (CHF3) (CHF3) (CHF3) (CHF3) (CHF3)
MADb 14.5 21.0 11.2 7.1 9.1 11.3 9.1 10.8 11.9 9.9 9.9 12.4 11.7 9.8 10.1 7.2 7.4 5.8 5.9 12.5 11.9 9.9 10.2
Signed errors are calculated using (theor. expt.). Minimum and maximum errors are chosen based on the absolute deviation from the experiment. b Mean absolute deviation (MAD) with respect to the experimental shieldings.24,42,47–51 a
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Figure 1. Mean absolute deviation for the calculated experimental value of 0 and the CCSD(T) value of e.
Besides the data calculated by 21 density functionals, the corresponding statistic data of HF, and MP2 are also included for comparison. The calculated and experimental details may be found in the supplementary material. It has been demonstrated that it is a rather challenging task to obtain accurate magnetic shieldings and at least triple excitation is required to achieve quantitative accuracy.12,42 On the other hand, previous investigation has shown that DFT covers a fairly large portion of three-electron correlation.52 Hence we believe that it is not impossible to design an exchange-correlation functional, which can surpass the MP2 method in the prediction of the magnetic shieldings. From the data listed in Table 2, it is apparent that the OPTX exchange functional generally provides good results for the 13C shieldings, leading to a mean absolute deviation (MAD) with respect to the experimental data less than 4 ppm (i.e., O3LYP: 3.2; OB98: 3.9; OLYP: 3.6; OPBE: 2.3; OPW91: 2.1 ppm). Noteworthily, OPBE or OPW91 offers a promising method for the prediction of the gas-phase NMR shieldings (0), giving both the maximum deviation and MAD that are significantly lower than those of other methods. In comparison with the experimental 0, MP2 systematically overestimates the 13C shieldings and leads to a MAD of 8.9 ppm. Considering the rovibrational corrections are always highly systematic and are all negative,42 a better agreement between the MP2 results and the experimental e would be anticipated. Take methane as an example, the discrepancy between the MP2 number and the experimental 0 is 7.1 ppm, which is reduced to 3.8 if comparison is made to the experimental e (see the supplementary material for detail). Using the benchmark CCSD(T) results as reference,
13
C magnetic shieldings with respect to the
MAD of MP2 drops from 8.9 to 4.7 ppm; while that of HF is less affected, changing from 7.0 to 7.4 ppm. Statistic data reported in Table 2 clearly show that switching the reference from the experimental 0 to the CCSD(T) e generally degrades the performance of the DFT methods. While the most popular B3LYP functional leads to MADs of 9.1 and 13.6 ppm with respect to the different reference states, the newly developed X3LYP behaves no better than B3LYP. OPBE and OPW91, on the other hand, distinguish themselves as the best methods with impressive MADs of 3.4 and 3.7 ppm, respectively, with respect to the CCSD(T) e (Table 2). We find that the DFT results exhibit a strong functional dependency (c.f. Fig. 1). For instance, the predicted 13C shieldings of methane with various density functionals range from 188.6 (BB98) to 201.6 ppm (B98PBE). Among all exchange functionals under consideration, the B88 and PBE exchange functionals systematically predict too deshielded 13C, and are not recommended because both MADs (BB98: 11.9, BLYP: 11.4, BPBE: 6.9, BPW91: 7.4; PBEB98: 11.9, PBELYP: 11.4, PBEPBE: 6.9, PBEPW91: 7.4) and the maximum deviations with respect to the experimental 0 are significantly higher than those of the others (Table 2). Including the rovibrational corrections, or taking the CCSD(T) e as reference will further deteriorate the results. The PBE and PW91 correlation functionals give values much closer to the experiments than does LYP or B98. The improvements depend on the exchange functionals. Hence OPBE (MAD ¼ 2.3) is better than OLYP (MAD ¼ 3.6) by 1.3; while BPBE (MAD ¼ 6.9) is better than BLYP (MAD ¼ 11.4) by 4.5 (Table 2). Interestingly, on the contrary to the observation of Cheeseman et al.24 and Rauhut et al.,25 the data reported in this work reveal
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Figure 2. Mean absolute deviation for the calculated mental value of 0 and the CCSD(T) value of e.
that mixture of the HF exchange with the GGA exchange may substantially improve the agreements with experimental shieldings (B3LYP: 9.1, BLYP: 11.2; PBE1PBE: 4.2, PBEPBE: 6.9; O3LYP: 3.2, OLYP: 3.6, Table 2). This is, however, in line with the usual observation in the calculations of other molecular properties, i.e., HF counterbalances GGA, giving an overall improvement. CF4 delivers a typical example (see the supplementary material for details). Although HF overshoots the experimental 13C shielding by 19.3 ppm; different flavors of GGAs undershoot the experiment by 19.2 (BLYP), 13.5 (PBEPBE), and 7.4 ppm (OLYP). The hybrid form improves the GGA description by 7.8 (B3LYP), 9.2 (PBE1PBE), and 2.1 ppm (O3LYP). The PBE1PBE functional was recommended by Adamo and Barone as the best functional currently available.28 Taking the experimental 0 as reference, our calculations lead to a MAD of 4.2 ppm, substantially better than that predicted at the MP2 level of theory (8.9 ppm). This is in agreement with the previous investigation28,29 even though some of the quantitative values differ due to the use of slightly different testing set and the geometries. Comparing with OPW91 (MAD ¼ 2.1 ppm) and OPBE (MAD ¼ 2.3 ppm), however, the PBE1PBE model underestimates all the 13C shieldings, which implies that the situation will get worse if the rovibrational corrections are taken into account. In fact, when the CCSD(T) values are taken as reference, our results clearly show that the PBE1PBE model is inferior to MP2, or even HF. The corresponding MADs for PBE1PBE, MP2, and HF are 9.0, 4.7, and 7.4 ppm, respectively. For the shieldings of hetero atoms (15N, 17O, and 19F), errors are generally larger than those for 13C (c.f. Tables 2 and 3),
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C chemical shifts with respect to the experi-
although OPTX exchange functional still shows good performance, being clearly superior to other methods. In particular, OPW91 and OPBE lead to MADs around 5.9 ppm; whereas, MADs obtained with other DFT methods are generally larger than 10 ppm (Table 3). The wavefunction-based HF and MP2 methods also encounter difficulty with the hetero atoms, giving MADs of 14.5 and 21.0, respectively. Indeed, it has been shown that the density functionals provide a significant improvement over the MP2 method in a more stringent test, i.e., in the case of ozone and metal oxo complexes.24,28,29,47 The lack of accurate gas phase experimental 15N, 17O, and 19 F absolute shieldings limits our discussion. Generally speaking, we find that GGAs perform better for 15N, than for 19F; whereas hybrid functionals generally perform better for 19F, than for 15N (see the supplementary material for the detail). For instance, we see that the maximum error occurs at 19F in CHF3 for all GGAs; whereas hybrid functionals (B3LYP, PBE1PBE, X3LYP, and B971) lead to large errors for 15N in the triple bond environments such as in CH3CN, HCN, and N2. It must be mentioned, in this connection, that accurate prediction of the 17 O shieldings is a rather challenging task, because it requires explicit consideration of nonspecific and specific solvation effects. An empirical scheme has been suggested by Wu et al.53 Chemical Shifts
When compared with the absolute shieldings, the chemical shifts are normally less affected by the various effects because the corrections are generally systematic and error cancellations may occur.42 In fact, it is the chemical shift that is actually measured
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13 Table 4. Statistic Data for C Chemical Shifts at the Hartree-Fock, DFT, and MP2 Levels of Theory. The Best Two Methods are in Bold.
Method HF MP2 B3LYP O3LYP PBE1PBE X3LYP B971 B98B98 B98LYP B98PBE B98PW91 BB98 BLYP BPBE BPW91 OB98 OLYP OPBE OPW91 PBEB98 PBELYP PBEPBE PBEPW91
Minimum errora 0.4 0.2 0.2 0.3 0.0 0.2 0.1 0.4 0.2 0.2 0.1 1.0 0.4 0.3 0.4 0.1 0.0 0.1 0.1 0.6 1.3 0.2 0.1
(C2H2) (CH3COCH3) (CH3CN) (CH2CCH2) (CH3CN) (CH3CN) (CO2) (CH2CCH2) (CH2CCH2) (CH2CCH2) (CH2CCH2) (CO2) (CO2) (C6H6) (C6H6) (CH2CCH2) (CH3CN) (CH3CH2CH3) (CH3CH2CH3) (CO2) (CO2) (CO2) (CO2)
Maximum errora 21.3 10.9 14.4 7.1 13.7 14.9 9.8 15.5 14.7 13.9 13.9 14.4 13.5 12.7 12.7 6.3 6.0 6.0 5.9 17.0 16.1 15.3 15.3
(CO) (CO) (CH2CCH2) (H2CO) (H2CO) (CH2CCH2) (H2CO) (H2CO) (H2CO) (H2CO) (H2CO) (H2CO) (H2CO) (H2CO) (H2CO) (H2CO) (CF4) (CO2) (CO2) (H2CO) (H2CO) (H2CO) (H2CO)
MADb
MADc
7.2 2.1 5.2 2.4 4.4 5.5 3.1 7.6 7.7 5.8 5.9 5.8 5.7 4.2 4.3 2.9 3.0 2.0 2.0 7.1 7.1 5.4 5.5
7.1 2.2 6.1 3.0 5.5 6.5 4.2 7.9 7.8 6.4 6.5 6.6 6.4 5.1 5.2 3.4 3.3 2.5 2.4 7.8 7.6 6.3 6.4
Signed errors are calculated using (theor. expt.). Minimum and maximum errors are chosen based on the absolute deviation from the experiment. b Mean absolute deviation (MAD) with respect to the experimental shifts.24,42,47–51 c Mean absolute deviation (MAD) with respect to the CCSD(T)/13s9p4d3f values.12,43 a
with high accuracy in the NMR spectroscopy. Thus the prediction of chemical shifts (e) is a primary goal of the quantum chemical investigation. The calculated chemical shifts (e) for 13C are compared with the experimental values (0) and the best CCSD(T) results (e) (as shown in Fig. 2). The chemical shifts are given relative to the corresponding gaseous values of CH4, NH3, H2O, and HF (see supplementary material). The statistical data are summarized in Tables4 and 5. MP2 is the most widely used nondensity functional method, providing a good agreement with the experimentally measured 0. Our calculations give 13C MADs of 2.1 and 2.2 ppm, when the experimental 0 and the CCSD(T) e are taken as reference, respectively, confirming that MP2 is indeed good for the prediction of chemical shifts. However, it is noteworthy to recall that large MADs for absolute shielding calculations are encountered at the MP2 level (8.9 and 4.7 ppm, Table 2). Hence, the success of MP2 in the prediction of chemical shifts should be attributed to a systematic error cancellation when considering the differences between two shieldings. It is encouraging to see that OPBE and OPW91, being the absolute winners in the prediction of magnetic shieldings (Tables 2–5), again give the best performance in the prediction of 13C chemical shifts (Table 4). Using the experimental 0 as reference, OPBE or OPW91 leads to 13C
MAD of 2.0 ppm, which is slightly increased to 2.5 ppm when using the CCSD(T) e as reference. We note that OPBE also provides good performance in the calculation of various molecular properties and has been recommended to be applied in studying a broad spectrum.54 Comparing data in Tables 2 and 4, it is evident that the deviations from the experimental chemical shifts are generally smaller than the corresponding values for the absolute shieldings. Besides OPBE and OPW91, other methods with OPTX being the exchange functional generally exhibit a remarkable performance in the calculations of chemical shifts. The MADs are 2.4 ppm for O3LYP, 2.9 for OB98, 3.0 for OLYP with respect to the experimental 0. This superiority is retained when taking the benchmark CCSD(T) results as reference. The next best quality shift within the framework of density functional theory is provided by B971 (3.1 ppm, Table 4 and Fig. 2). We note that Keal and Tozer have shown in a recent study55 that the revised version of B971, namely B972, performs quite well in the calculation of 77Se NMR shielding constants. The performance of the B98 exchange functional is degraded when compared with its own performance in the prediction of magnetic shieldings. Other commonly used hybrid functionals such as B3LYP and PBE1PBE, as well as the newly developed X3LYP do not show an impressive performance. Table 5. Statistic Data for Chemical Shifts of Hetero Atoms
of 15N, 17O, and 19F at the Hartree-Fock, DFT, and MP2 Levels of Theory. The Best Two Methods are in Bold. Method
Minimum errora
Maximum errora
HF MP2 B3LYP O3LYP PBE1PBE X3LYP B971 B98B98 B98LYP B98PBE B98PW91 BB98 BLYP BPBE BPW91 OB98 OLYP OPBE OPW91 PBEB98 PBELYP PBEPBE PBEPW91
0.1 0.1 9.9 2.2 6.4 9.3 7.7 0.1 1.6 2.5 2.2 9.1 5.4 6.9 6.8 0.8 0.1 0.6 0.4 10.5 6.8 8.3 8.3
40.9 19.0 17.9 18.4 18.8 18.9 16.4 45.9 50.5 45.1 45.0 27.3 30.4 26.4 26.4 21.7 25.0 21.2 21.2 31.1 34.3 30.1 30.1
(CH3NH2) (CH3F) (CH3NH2) (CO) (CH3NH2) (CH3F) (CH3NH2) (N2) (N2) (N2) (N2) (CO) (CO) (CO) (CO) (CO) (N2) (N2) (N2) (CO) (CO) (CO) (CO)
(N2) (N2) (CH3CN) (CHF3) (N2) (CH3CN) (CH3CN) (CHF3) (CHF3) (CHF3) (CHF3) (CHF3) (CHF3) (CHF3) (CHF3) (CHF3) (CHF3) (CHF3) (CHF3) (CHF3) (CHF3) (CHF3) (CHF3)
MADb 17.9 7.8 13.9 9.3 13.4 14.4 12.0 15.8 16.3 14.3 14.4 14.2 13.6 12.2 12.3 8.7 9.0 7.7 7.7 15.8 15.3 13.8 13.9
Signed errors are calculated using (theor. expt.). Minimum and maximum errors are chosen based on the absolute deviation from the experiment. b Mean absolute deviation (MAD) with respect to the experimental shifts.24,42,47–51 a
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Figure 3. Mean absolute deviation for the calculated 13C, 15N, 17O, ical shifts with respect to the corresponding experimental values.
We notice that even though the exchange functional plays a decisive role in the prediction of the magnetic properties, the correlation functional also makes an important contribution. Just like what we find in the prediction of the absolute shieldings, PW91 and PBE are the best correlation functionals examined here in the prediction of the chemical shifts. Hence OPBE (MAD ¼ 2.0) is better than OLYP (MAD ¼ 3.0) by 1.0; whereas BPBE (MAD ¼ 4.2) is better than BLYP (MAD ¼ 5.7) by 1.3. Naturally, the role of correlation is more significant in the prediction of shiledings than in shifts. Table 5 summarizes the statistic data for chemical shifts of hetero atoms of 15N, 17O, and 19F at the Hartree-Fock, DFT, and MP2 levels of theory. Figure 3 makes a direct comparison between the calculated shifts and calculated shieldings. Unlike the situation seen for 13C, where improvement is clearly evident for the prediction of shifts over shieldings for all methods, this happens only for MP2. For all DFT methods, we see that errors are more significant for shifts than for shieldings. We notice that for many DFT methods, the minimum error for shielding occur at the reference molecule, impairing the function of error cancellation in shift calculation. Although the statistical performance in shift calculation may be improved by chosen proper reference molecules,56 the deficiency for hetero atoms calls for further improvement of the density functionals. Exchange and Correlation Effects
To better understand the effects of various exchange and correlation functionals, further analysis is made in term of the different exchange functionals for the prediction of the 13C magnetic shield-
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19
F magnetic shieldings and chem-
ings. Figure 4 presents a comparison of the calculated 13C magnetic shieldings obtained by HF and DFT exchange only methods (B98, OPTX, B88 and PBE) with respect to the CCSD(T) e. Figure 4A depicts the differences between HF and CCSD(T). As (eHF eCCSD(T)) is generally negative, leading to a mean deviation of 4 ppm, we see that HF has a tendency of deshielding. This observation indicates that for a DFT exchange only method, a deshielding tendency would be expected when the correlation effect is ignored. Figure 4B depicts the differences between the 13C shieldings obtained with exchange only methods of B88 and PBE and those of CCSD(T). The deshielding tendency for B88 and PBE is obvious, ranging from 5.0 (CO) to 21.0 ppm (CH2CCH2), which leads to a mean deviation of 14 ppm. In fact, the deshielding effect is so much exaggerated that it overshoots that of HF by 10 ppm. We notice that GGA methods using B88 or PBE as exchange functional generally underestimate the absolute magnetic shieldings with respect to the CCSD(T) results. We believe that the main source of error shall be attributed to the deficiency of these exchange functionals. For the B98 and OPTX exchange functionals, on the other hand, the situation is much better. Although there is still a tendency that 13C is too deshielded, the deviations are halved when compared with that of B88 and PBE. Thus the B98 and OPTX exchange functionals lead to a mean deviation of 4 ppm in the absolute shieldings using the CCSD(T) values as reference, being comparable to that of HF. Recall that GGAs using OPTX as exchange functional consistently provides the best performance for both magnetic shieldings and chemical shifts, signifying the importance of the exchange functional.
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Figure 4. The effect of exchange on the calculated 13C magnetic shieldings. (A) eHF eCCSD(T). The average deviation of the HF values from the CCSD(T) values is around 4 ppm. (B) eEx-only eCCSD(T). The average deviation of the B88 or PBE exchange-only values from the CCSD(T) values is around 14 ppm. (C) eEx-only eCCSD(T). The average deviation of the B98 or OPTX exchangeonly values from the CCSD(T) values is around 4 ppm.
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Figure 4. (continued)
Here we emphasize that there are detailed differences in the performance of B98, OPTX from that of HF. For instance, both the B98 and OPTX exchange functionals dramatically overestimate the absolute shieldings for CO, leading to a deviation of 14.5 and 5.1 ppm, respectively, for this molecule with respect to the CCSD(T) number, while HF underestimates this value by 22.6 ppm. CF4 molecule provides the other extreme. The B98 and OPTX exchange functionals lead to negative deviations of 7.8 and 7.0 ppm, whereas the corresponding number for HF is 18.5 ppm. We believe that understanding of these differences will be of great help for the development of new density functional for the accurate prediction of NMR properties. While for the previous NMR calculations within the framework of DFT, the major focus has been putting on the investigation of the performance of the whole exchange-correlation functionals.24,25,28,29,32,33,54–58 Few has been devoted to the investigation of the contribution by an individual exchange and correlation functional. Especially, the effect of correlation functionals on the prediction of the NMR data is more or less overlooked. Figure 5 signifies the correlation effect based on the calculated 13C shieldings with respect to the HF data. Figure 5A compares the values of (eMP2 eHF) with those of (eCCSD(T) eHF). While the former leads to an average number of 7.7 ppm; the latter gives 4.2 ppm. Thus MP2 has a tendency to exaggerate the correlation effect. As the (eMP2 eHF) values are mostly too positive, we correlate this observation with the fact that the MP2 approach tends to overestimate shieldings.
Figures 5B and 5C depict the correlation contributions to the calculated 13C shieldings. These numbers are obtained by subtracting the calculated values using the exchange only method of OPTX from those using the OLYP, OB98, OPBE, and OPW91 functionals. Different exchange functional may be used, which is seen to not affect the general feature of Figures 5B and 5C. Comparison between Figures 5B and 5C clearly demonstrates that the correlation functionals may be divided into two classes, namely class A and class B. While class A correlation functionals tend to lead to more deshielded values (Fig. 5B), being qualitatively different from MP2 and CCSD(T), class B functionals increase the values (Fig. 5C), in qualitative agreement with MP2 and CCSD(T). Therefore, the introduction of class A correlation functionals systematically lowers the absolute shieldings and hence further worsens the agreement between the calculated magnetic shieldings and the reference data because the exchange functionals generally give deshielded numbers. On the contrary, class B correlation functionals systematically increase the absolute shieldings, which complement the exchange functionals and lead to an overall improved agreement between the calculated values and the reference values. From Figure 5, one can see that the B98 and LYP correlation functionals belong to class A; and the PBE and PW91 correlation functionals belong to class B. This is in consistency with the previous conclusions based on the sum-over-state calculations, in which BPW91 is clearly superior to BLYP,47,56 and is also in line with the data shown in Tables 2–5. It is worthy to point out that although correlation
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Figure 5. The effect of correlation on the calculated 13C magnetic shieldings. (A) eCCSD(T) eHF and eMP2 eHF. For CO the corresponding values are 22.6 and 38.7 ppm, which are truncated at 22 ppm in the figure. (B) eOB98 eOPTX and eOLYP eOPTX. (C) eOPW91 eOPTX and eOPBE eOPTX.
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Figure 5. (continued)
effects of PBE and PW91 follow the same direction as do the CCSD(T) method, the magnitudes are much smaller, leaving room for further improvement.
Conclusion In this work, we have systematically examined the performance of some popular and newly developed density functionals in the calculations of NMR magnetic shieldings and chemical shifts. Careful comparisons have been made in terms of shieldings and shifts with respect to the experimental data and the best CCSD(T) results. The following conclusions can be drawn from this work: 1. When the calculated NMR data are compared with the experimental data, caution has to be taken, as there are many sources of errors, such as the rovibrational correction, the intermolecular interactions and solvent effects, etc., which will degrade the comparison. An improper choice of the reference may result in misleading conclusions. 2. In contrast to the systematical overestimation of the magnetic shieldings predicted by MP2, DFT methods generally lead to too deshielded values. The source of errors may be traced back to the deficiency of the exchange functionals. We find that although the B88 and PBE exchange functionals are not recommended for the NMR calculations; the OPTX exchange functional consistently delivers the best absolute shieldings and chemical shifts.
3. The correlation functionals may be grouped into two classes. class A, such as LYP, generally lowers the absolute shieldings and hence leads to an even deshielded value when combined with an exchange functional. On the other hand, class B, such as PBE or PW91, has a positive contribution, which complements the exchange functional, leading to a better result in the prediction of NMR data. 4. We find that the OPBE and OPW91 exchange-correlation functionals perform remarkably well along the whole set of systems, surpassing, in many aspects, the standard wavefunction-based method MP2. We conclude that OPBE or OPW91 is a good DFT functional currently available for the prediction of NMR data. Future work will be on the application of these functionals to the calculations of NMR spectroscopy of large molecules of chemical interest and on the development of new density functionals with higher accuracy.
References 1. Grant, D. M.; Harris, R. K., Eds. Encyclopedia of Nuclear Magnetic Resonance, Vol. 1–8; Wiley: Chichester, UK, 1996. 2. Kutzelnigg, W.; Fleischer, U.; Schindler, M. In NMR Basis Principles and Progress, Vol. 23; Springer: Heidelberg, 1990; p. 165. 3. Helgaker, T.; Jaszunski, M.; Ruud, K. Chem Rev 1999, 99, 293. 4. Takashima, H.; Hada, M.; Nakatsuji, H. J Phys Chem 1995, 99, 7951.
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Journal of Computational Chemistry
DOI 10.1002/jcc
