Density functional theory predictions of anharmonicity and spectroscopic constants for diatomic molecules

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1 JOURNAL OF CHEMICAL PHYSICS VOLUME 115, NUMBER 6 8 AUGUST 2001 Density functional theory predictions of anharmonicity and spectroscopic constants for diatomic molecules Mutasem Omar Sinnokrot and C. David Sherrill Center for Computational Molecular Science and Technology, School of Chemistry and Biochemistry, Georgia Institute of Technology, Atlanta, Georgia Received 19 April 2001; accepted 25 May 2001 The reliability of density functional theory and other electronic structure methods is examined for anharmonicities and spectroscopic constants of the ground electronic states of several diatomic molecules. The equilibrium bond length r e, harmonic vibrational frequency e, vibrational anharmonicity e x e, rotational constant B e, centrifugal distortion constant D e, and vibration-rotation interaction constant e have been obtained theoretically for BF, CO, N 2,CH, and H 2. Predictions using Hartree Fock, coupled-cluster singles and doubles CCSD, coupled cluster singles and doubles with perturbative triples CCSD T, and various density functional methods S-VWN, BLYP, and B3LYP have been made using the 6-31G*, aug-cc-pvdz, and aug-cc-pvtz basis sets and compared to experimental values. Density functional theory predictions of the spectroscopic constants are reliable particularly for B3LYP and often perform as well as the more expensive CCSD and CCSD T estimates American Institute of Physics. DOI: / I. INTRODUCTION Predictions of the spectroscopic constants of diatomic molecules e.g., r e, e, e x e, B e, D e, and e are often used to benchmark new theoretical methods or study electron correlation or basis set effects. 1 8 Such predictions are also useful for as yet unknown or poorly characterized electronic states of diatomics. 9,10 While the theoretical prediction of equilibrium geometries and harmonic vibrational frequencies for molecules is routine, predictions of properties related to higher derivatives of the potential energy surface are relatively rare. Nevertheless, theoretical estimates of anharmonicity are critical for high accuracy predictions of experimentally observed fundamental vibrational frequencies; harmonic computations often overestimate fundamental frequencies by a few ( 1 3) percent. Likewise, anharmonic corrections are necessary to transform vibrationally averaged structures to equilibrium structures. Theoretically computed anharmonic force fields are therefore valuable for obtaining equilibrium geometries from, e.g., microwave and electron diffraction techniques. Botschwina and co-workers 11 have investigated the differences between equilibrium and ground state moments of inertia at several levels of theory, and Kochikov et al. 12 have recently demonstrated that an accurate equilibrium structure for SF 6 can be obtained by electron diffraction and empirically scaled ab initio quadratic and cubic force constants. Gauss, Stanton, and co-workers have computed the anharmonicity corrections to equilibrium rotational constants using coupled-cluster with single, double, and perturbative triple substitutions CCSD T Ref. 13 and have used these corrections to obtain highly accurate equilibrium structures from experimentally observed, vibrationally averaged rotational constants for several molecules 14 including cyclopropane 15 and benzene. 16 Electronic structure estimates of vibrational anharmonicity and vibration-rotation interaction have been very carefully studied for the Hartree Fock HF and configuration interaction singles and doubles CISD methods by Schaefer and co-workers 17,18 for a series of asymmetric top and linear molecules. Although harmonic frequency predictions can change significantly between the Hartree Fock and CISD methods, these workers found that Hartree Fock with a polarized double-zeta basis DZP was sufficient to provide good agreement with experiment for the asymmetric top molecules considered. 17 The subsequent study of linear polyatomic molecules 18 did not show as good agreement between DZP HF anharmonic constants and experiment, presumably because the linear molecules all contained multiple bonds, making electron correlation effects more important. Dunning and co-workers 1 3 have studied the accuracy of theoretical vibrational anharmonicities, vibration-rotation interaction constants, and other spectroscopic constants for many diatomic molecules using Hartree Fock, generalized valence bond GVB, GVB CI, complete-active-space selfconsistent-field CASSCF, and CASSCF second-order configuration interaction SOCI methods. These authors found reasonable predictions at all levels of theory considered, but basis sets of at least polarized triple-zeta quality and dynamic electron correlation GVB-CI or CASSCF SOCI were required to obtain high accuracy. Feller and Sordo 6 have recently reported full coupled-cluster singles, doubles, and triples CCSDT spectroscopic constants for first row diatomic hydrides and find excellent agreement with experiment even without correcting for core-valence correlation or relativistic effects; moreover, no significant differences were found between predictions using full iterative triples or perturbative triples according to the CCSD T Ref. 13 prescription. Additionally, full configuration interaction quartic /2001/115(6)/2439/10/$ American Institute of Physics

2 2440 J. Chem. Phys., Vol. 115, No. 6, 8 August 2001 M. O. Sinnokrot and C. D. Sherrill force fields have been reported by Van Huis et al. 19 for NH 2. Although density functional theory DFT Ref. 20 is rapidly replacing more traditional correlated ab initio methods for use in many chemical problems, we are unaware of any systematic study of DFT predictions of anharmonic spectroscopic constants. However, recent work 21,12 on benzene and SF 6 suggests that the hybrid functional B3LYP Refs. 22, 23 can provide accurate anharmonic force fields. Moreover, the study of benzene by Miani et al. 21 indicates that the B3LYP spectroscopic constants are significantly more accurate than the Hartree Fock results. In this study we use several different DFT approaches in conjunction with three different basis sets to determine the spectroscopic constants of BF, CO, N 2,CH, and H 2. For comparison, spectroscopic constants were also predicted using Hartree Fock, coupled-cluster singles and doubles CCSD, 24 and coupledcluster singles and doubles and perturbative triples CCSD T ; 13 the results are compared to the experimental values of Huber and Herzberg. 25 II. THEORETICAL APPROACH The vibrational term values G(v) of a diatomic molecule are G v e v 1 2 e x e v 1 2 2, while the rotational term values F v (J) are given by F v J B v J J 1 D ej 2 J The effective rotational constant B v for vibrational level v depends on the equilibrium rotational constant B e and the vibration-rotation interaction terms e via B v B e e v 1 2. Fundamental frequencies are thus related to harmonic frequencies via e 2 e x e. The term e requires the third derivative of the electronic energy, while e x e requires up to the fourth derivative. 26 In this study we determined the equilibrium bond length r e, harmonic vibrational frequency e, vibrational anharmonicity e x e, rotational constant B e, centrifugal distortion constant D e, and vibration-rotation interaction constant e for the ground states of BF, CO, N 2,CH, and H 2. Spectroscopic constants were determined by a five-point method, in which energies are very tightly converged typically to hartree for five bond lengths uniformly distributed with a spacing of Å around the equilibrium internuclear distance r e. The force constants up to the quintic constant f rrrrr were used to determine these spectroscopic constants as explained elsewhere. 27 Computations were performed using three basis sets of contracted Gaussian functions, namely 6-31G*, 28,29 aug-ccpvdz, and aug-cc-pvtz. 30 The 6-31G* basis set is of double zeta plus polarization quality and has a contraction scheme of (10s4p1d)/ 3s2p1d for first-row elements. The augmented correlation-consistent polarized valence doublezeta basis set, aug-cc-pvdz, is formed from a contraction of (9s4p1d) to 3s2p1d, and is augmented by a diffuse set of 1 3 functions (1s1p1d), resulting in a contraction scheme of (10s5p2d)/ 4s3p2d. The third basis set, the augmented correlation-consistent polarized triple-zeta basis set, aug-ccpvtz, is formed from contracting a set of (10s5p2d1 f ) primitives to 4s3p2d1 f, and is augmented by a diffuse set of functions (1s1p1d1 f ), resulting in a contraction scheme of (11s6p3d2 f )/ 5s4p3d2f. A set of six Cartesian Gaussian functions is used for 6-31G*, whereas the correlationconsistent basis sets employ pure angular momentum sets of five d and seven f functions. The diffuse functions are included primarily to facilitate comparison to forthcoming studies of excited electronic states. Based on the results of Dunning and co-workers, 1 3 we expect little difference between the cc-pvxz and aug-cc-pvxz predictions of the spectroscopic constants of the ground states of the molecules considered here, particularly for the triple- basis sets. Spectroscopic constants were predicted ab initio using Hartree Fock, coupled-cluster with single and double substitutions CCSD, 24 and coupled-cluster with single, double, and perturbative triple substitutions CCSD T. 13 Predictions were also made using the following Kohn Sham density functional theory methods: 20 the local spin density approximation with Slater exchange 31 and the correlation functional of Vosko, Wilk, and Nusair 32 denoted S-VWN ; the generalized gradient approximation GGA methods which pair Becke s 1988 exchange functional 33 with the correlation functionals of Lee, Yang, and Parr 34 BLYP ; and a hybrid 22 gradient-corrected functional which mixes in Hartree Fock exchange, B3LYP. 23,35 Geometries were optimized using analytic gradients for Hartree Fock, 36,37 CCSD, 38 CCSD T, 39,40 and density functional theory. All electrons were correlated in the coupled-cluster procedures. All computations were performed using Q-CHEM 1.2 Ref. 41 except the coupled-cluster results which used the ACES II package. 42 III. RESULTS AND DISCUSSION Total electronic energies and spectroscopic constants are presented in Tables I BF, II CO, III (N 2 ), IV (CH ), and V (H 2 ). Errors versus experimental values from Huber and Herzberg 25 are displayed in Figs. 1 (r e ), 2 (B e ), 3 ( e ), 4(D e), 5 ( e ),and6( e x e ). Table VI summarizes the average absolute relative errors for the spectroscopic constants considered at each level of theory. A. Equilibrium bond lengths and rotational constants The errors in bond lengths presented in Fig. 1 are consistent with the errors expected for these levels of theory based on previous systematic studies of equilibrium geometries Although the best geometry predictions for Hartree Fock are obtained with the aug-cc-pvdz basis, for all other theoretical methods this basis gives much poorer geometries than 6-31G* or aug-cc-pvtz. The coupledcluster methods usually improve the Hartree-Fock geometry predictions substantially; with a 6-31G* basis, bond lengths are slightly overestimated, and they are quite accurately predicted with the aug-cc-pvtz basis. An unusually large error in the BF bond length Å makes our average ab-

3 J. Chem. Phys., Vol. 115, No. 6, 8 August 2001 DFT for diatomic molecules 2441 TABLE I. Spectroscopic constants for the ground electronic state of BF. a Method Energy r e e e x e B e D e e 6-31G* HF e aug-cc-pvdz HF e aug-cc-pvtz HF e G* CCSD e aug-cc-pvdz CCSD e aug-cc-pvtz CCSD e G* CCSD T e aug-cc-pvdz CCSD T e aug-cc-pvtz CCSD T e G* S-VWN e aug-cc-pvdz S-VWN e aug-cc-pvtz S-VWN e G* BLYP e aug-cc-pvdz BLYP e aug-cc-pvtz BLYP e G* B3LYP e aug-cc-pvdz B3LYP e aug-cc-pvtz B3LYP e Experiment e a Energies in hartrees, bond lengths in Å, and other quantities in cm 1. Core electrons correlated. Experimental data from Huber and Herzberg Ref. 25. solute aug-cc-pvdz CCSD bond length error Å noticeably larger than expected from Helgaker s study; 46 this error is even larger for aug-cc-pvdz CCSD T. As one can see from Fig. 1 or Table VI, S-VWN and BLYP DFT predictions do not improve over Hartree-Fock for the bond lengths of these molecules except with the large augcc-pvtz basis. However, the B3LYP DFT bond lengths are very good with the 6-31G* and aug-cc-pvtz basis sets, with average absolute errors even smaller than CCSD or CCSD T. The equilibrium rotational constants B e are simply related to the equilibrium bond lengths via B e h/(8 2 r e 2 ). Hence, we expect the general trends observed for bond length errors to persist for B e errors, although underestimates of bond lengths lead to overestimates of B e and vice versa. Consistent with this view, the B e errors in Fig. 2 are qualitatively like a mirror image through zero error of the bond length errors in Fig. 1. Most errors in B e are within 6%, with Hartree Fock typically overestimating and other methods typically underestimating experiment. The BLYP and B3LYP results are substantially improved over S-VWN and rival or surpass the accuracy of the CCSD and CCSD T B e predictions, depending on basis set. Just as in the bond length predictions, the aug-cc-pvdz basis set gives the least accurate results. Our best results average absolute errors of 0.7% for B3LYP or CCSD T with the large basis compare very favorably with the 0.8% average absolute error found by Peterson et al. using a much more intricate and expensive TABLE II. Spectroscopic constants for the ground electronic state of CO. a Method Energy r e e e x e B e D e e 6-31G* HF e aug-cc-pvdz HF e aug-cc-pvtz HF e G* CCSD e aug-cc-pvdz CCSD e aug-cc-pvtz CCSD e G* CCSD T e aug-cc-pvdz CCSD T e aug-cc-pvtz CCSD T e G* S-VWN e aug-cc-pvdz S-VWN e aug-cc-pvtz S-VWN e G* BLYP e aug-cc-pvdz BLYP e aug-cc-pvtz BLYP e G* B3LYP e aug-cc-pvdz B3LYP e aug-cc-pvtz B3LYP e Experiment e a Energies in hartrees, bond lengths in Å, and other quantities in cm 1. Core electrons correlated. Experimental data from Huber and Herzberg Ref. 25.

4 2442 J. Chem. Phys., Vol. 115, No. 6, 8 August 2001 M. O. Sinnokrot and C. D. Sherrill TABLE III. Spectroscopic constants for the ground electronic state of N 2. a Method Energy r e e e x e B e D e e 6-31G* HF e aug-cc-pvdz HF e aug-cc-pvtz HF e G* CCSD e aug-cc-pvdz CCSD e aug-cc-pvtz CCSD e G* CCSD T e aug-cc-pvdz CCSD T e aug-cc-pvtz CCSD T e G* S-VWN e aug-cc-pvdz S-VWN e aug-cc-pvtz S-VWN e G* BLYP e aug-cc-pvdz BLYP e aug-cc-pvtz BLYP e G* B3LYP e aug-cc-pvdz B3LYP e aug-cc-pvtz B3LYP e Experiment e a Energies in hartrees, bond lengths in Å, and other quantities in cm 1. Core electrons correlated. Experimental data from Huber and Herzberg Ref. 25. multireference configuration interaction MRCI procedure in conjunction with a larger cc-pvqz basis set. 3 B. Harmonic vibrational frequencies Figure 3 displays the relative errors in harmonic vibrational frequencies e. The Hartree Fock predictions show a typical 3% 17% overestimate in most cases, whereas the CCSD frequencies are considerably improved, with most errors within 5%. Errors are further reduced with the CCSD T method. Our aug-cc-pvtz CCSD average absolute error of 3.1% is quite similar to the TZ(2df,2pd) CCSD average error of 3.7% in the systematic study of Thomas et al. 44 The DFT methods are generally as accurate as CCSD for the current set of vibrational frequencies, with errors usually within 5%; depending on the basis set, the DFT predictions are almost as good or better than those of CCSD T. S-VWN and BLYP tend to underestimate, while B3LYP tends to overestimate harmonic frequencies. Basis set effects are generally modest, and predictions are not always improved with the larger basis set. However, we note that the aug-cc-pvdz basis substantially underestimates e with several of the methods considered. C. Centrifugal distortion constants The centrifugal distortion constant D e, which accounts for the lengthening of the bond with higher angular momentum J, is given by D e 4B e 3 / e 2. Errors in theoretical pre- TABLE IV. Spectroscopic constants for the ground electronic state of CH. a Method Energy r e e e x e B e D e e 6-31G* HF e aug-cc-pvdz HF e aug-cc-pvtz HF e G* CCSD e aug-cc-pvdz CCSD e aug-cc-pvtz CCSD e G* CCSD T e aug-cc-pvdz CCSD T e aug-cc-pvtz CCSD T e G* S-VWN e aug-cc-pvdz S-VWN e aug-cc-pvtz S-VWN e G* BLYP e aug-cc-pvdz BLYP e aug-cc-pvtz BLYP e G* B3LYP e aug-cc-pvdz B3LYP e aug-cc-pvtz B3LYP e Experiment e a Energies in hartrees, bond lengths in Å, and other quantities in cm 1. Core electrons correlated. Experimental data from Huber and Herzberg Ref. 25.

5 J. Chem. Phys., Vol. 115, No. 6, 8 August 2001 DFT for diatomic molecules 2443 TABLE V. Spectroscopic constants for the ground electronic state of H 2. a Method Energy r e e e x e B e D e e 6-31G* HF e aug-cc-pvdz HF e aug-cc-pvtz HF e G* CCSD e aug-cc-pvdz CCSD e aug-cc-pvtz CCSD e G* S-VWN e aug-cc-pvdz S-VWN e aug-cc-pvtz S-VWN e G* BLYP e aug-cc-pvdz BLYP e aug-cc-pvtz BLYP e G* B3LYP e aug-cc-pvdz B3LYP e aug-cc-pvtz B3LYP e Experiment e a Energies in hartrees, bond lengths in Å, and other quantities in cm 1. Experimental data from Huber and Herzberg Ref. 25. dictions of D e are presented in Fig. 4, which shows that theory almost always underestimates D e, by up to 20%. Estimates of D e show a much larger improvement than the previously discussed spectroscopic constants when the larger aug-cc-pvtz basis is used, errors being reduced by about a factor of two except with Hartree Fock. All three DFT methods perform very well for D e often as well as CCSD T, with errors within 4% for the aug-cc-pvtz basis. D. Vibration-rotation interaction constants The vibration-rotation interaction constants e depend on the third derivative of the potential energy and relate the effective rotational constant B v for vibrational level v to the equilibrium rotational constant B e via Eq. 3. Figure 5 displays the errors in predicted values. Except for CH, theory usually underestimates e, by up to 23% for Hartree Fock. Most other methods are typically within 15%. Similarly to FIG. 1. Error in theoretically predicted bond lengths.

6 2444 J. Chem. Phys., Vol. 115, No. 6, 8 August 2001 M. O. Sinnokrot and C. D. Sherrill FIG. 2. Relative error in theoretically predicted rotational constants B e. FIG. 3. Relative error in theoretically predicted harmonic vibrational frequencies.

7 J. Chem. Phys., Vol. 115, No. 6, 8 August 2001 DFT for diatomic molecules 2445 FIG. 4. Relative error in theoretically predicted centrifugal distortion constants D e. FIG. 5. Relative error in theoretically predicted vibration rotation interaction constants e.

8 2446 J. Chem. Phys., Vol. 115, No. 6, 8 August 2001 M. O. Sinnokrot and C. D. Sherrill FIG. 6. Relative error in theoretically predicted anharmonic constants e x e. D e, DFT methods are competitive with CCSD T and slightly more reliable than CCSD, and the aug-cc-pvtz predictions are substantially improved over those using the smaller basis sets. With the larger basis set, the CCSD T, S-VWN, BLYP, and B3LYP average absolute errors in e see Table VI are 3.6%, 3.6%, 3.8%, and 4.8%. These results compare well with the 1.5% average absolute error inferred from the much more expensive cc-pvqz MRCI results of Peterson et al. for first-row homonuclear diatomics. 3 The leading term in the difference between the zeropoint vibrationally averaged rotational constant B 0 and the equilibrium rotational constant B e is e /2 see Eq. 3. For the cases considered here, this difference is about 0.5% 2.5% of B e. As noted above, we find average absolute errors of 0.7% for aug-cc-pvtz B3LYP or CCSD T predictions of B e. Thus, the corrections for vibrational motion (B 0 B e ) can be larger than the errors in theoretical estimates of B e. Given the very small errors in e predictions, the difference TABLE VI. Average absolute relative errors percent for spectroscopic constants bond length errors in Å, not relative. a Method r e e e x e B e D e e 6-31G* HF aug-cc-pvdz HF aug-cc-pvtz HF G* CCSD aug-cc-pvdz CCSD aug-cc-pvtz CCSD G* CCSD T aug-cc-pvdz CCSD T aug-cc-pvtz CCSD T G* S-VWN aug-cc-pvdz S-VWN aug-cc-pvtz S-VWN G* BLYP aug-cc-pvdz BLYP aug-cc-pvtz BLYP G* B3LYP aug-cc-pvdz B3LYP aug-cc-pvtz B3LYP a Core electrons were correlated for coupled-cluster. Errors were computed relative to experimental data from Huber and Herzberg Ref. 25.

9 J. Chem. Phys., Vol. 115, No. 6, 8 August 2001 DFT for diatomic molecules 2447 (B 0 B e ) can be estimated with very high accuracy with any of the theoretical methods considered here. Once the e /2 vibrational correction is included, the majority of the error in theoretical estimates of the vibrationally averaged rotational constant B 0 again comes from the errors in the equilibrium constant B e itself. E. Vibrational anharmonic constants Computations of the vibrational anharmonic constants e x e require up to the fourth derivative of the potential energy. Errors in the theoretical predictions for this constant are presented in Fig. 6. This quantity appears to be somewhat more difficult to compute accurately on a relative basis than the other constants in this study; the estimates are usually only good to about 30%. Except for Hartree Fock and S-VWN, the larger aug-cc-pvtz yields much better results than the double- basis sets. The aug-cc-pvtz average absolute errors for CCSD, BLYP, and B3LYP are about 5%, with the Hartree-Fock errors about three times as large and the CCSD T average absolute error being 2.3%. Our best results compare favorably with an average absolute error of 2.5% for much costlier cc-pvqz MRCI predictions of first row homonuclear diatomics 3 and 3.0% for aug-cc-pvtz CCSDT predictions of first row diatomic hydrides. 6 Since the leading anharmonic correction to the harmonic vibrational frequencies is just 2 e x e, the observed relative errors in e x e translate into nearly the same percentage error in the correction from harmonic frequencies e to experimentally observed fundamental frequencies e. Given that the anharmonic correction is usually small anyway on the order of a few percent, and considering the quite small errors in DFT predictions of e, this suggests that density functional theory is capable of providing rather accurate predictions of fundamental frequencies e. IV. CONCLUSIONS Density functional theory provides reliable predictions of the spectroscopic constants of several diatomic molecules, including constants depending on anharmonicity such as vibration-rotation interaction constants e and vibrational anharmonicities e x e. While not as accurate as highly sophisticated multireference CI methods, DFT predictions are frequently as good or better than more expensive CCSD results and are often competitive with CCSD T. S-VWN performs the worst and B3LYP performs the best of the DFT methods considered. By considering anharmonic corrections, DFT appears capable of providing very accurate predictions of vibrationally averaged rotational constants B 0 and fundamental vibrational frequencies. ACKNOWLEDGMENTS This research was supported by the National Science Foundation Grant No. CHE One of the authors C.D.S. acknowledges a Camille and Henry Dreyfus New Faculty Award and an NSF CAREER Award Grant No. CHE The authors thank Alfred Park for assistance with the analysis. The Center for Computational Science and Technology is funded through a Shared University Research SUR grant from IBM and Georgia Tech. 1 D. E. Woon and T. H. Dunning, J. Chem. Phys. 99, K. A. Peterson, R. A. Kendall, and T. H. Dunning, J. Chem. Phys. 99, K. A. Peterson, R. A. Kendall, and T. H. Dunning, J. Chem. Phys. 99, T. D. Crawford and H. F. Schaefer, J. Chem. Phys. 104, C. D. Sherrill, A. I. Krylov, E. F. C. Byrd, and M. Head-Gordon, J. Chem. Phys. 109, D. Feller and J. A. Sordo, J. Chem. Phys. 112, X. Li and J. Paldus, Mol. Phys. 98, H. Meissner and J. Paldus, Quantum Chem. 80, K. Okada and S. Iwata, J. Electron Spectrosc. Relat. Phenom. 108, R. Wesendrup, L. Kloo, and P. Schwerdtfeger, Int. J. Mass. 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