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Harmonic Vibrational Frequencies: An Evaluation of Hartree-Fock, Møller-Plesset, Quadratic Configuration Interaction, Density Functional Theory, and Semiempirical Scale Factors Anthony P. Scott and Leo Radom* Research School of Chemistry, Australian National UniVersity, Canberra, ACT 0200, Australia ReceiVed: April 2, 1996; In Final Form: July 30, 1996X Scaling factors for obtaining fundamental vibrational frequencies, low-frequency vibrations, zero-point vibrational energies (ZPVE), and thermal contributions to enthalpy and entropy from harmonic frequencies determined at 19 levels of theory have been derived through a least-squares approach. Semiempirical methods (AM1 and PM3), conventional uncorrelated and correlated ab initio molecular orbital procedures [Hartree- Fock (HF), Møller-Plesset (MP2), and quadratic configuration interaction including single and double substitutions (QCISD)], and several variants of density functional theory (DFT: B-LYP, B-P86, B3-LYP, B3-P86, and B3-PW91) have been examined in conjunction with the 3-21G, 6-31G(d), 6-31+G(d), 6-31G- (d,p), 6-311G(d,p), and 6-311G(df,p) basis sets. The scaling factors for the theoretical harmonic vibrational frequencies were determined by a comparison with the corresponding experimental fundamentals utilizing a total of 1066 individual vibrations. Scaling factors suitable for low-frequency vibrations were obtained from least-squares fits of inverse frequencies. ZPVE scaling factors were obtained from a comparison of the computed ZPVEs (derived from theoretically determined harmonic vibrational frequencies) with ZPVEs determined from experimental harmonic frequencies and anharmonicity corrections for a set of 39 molecules. Finally, scaling factors for theoretical frequencies that are applicable for the computation of thermal contributions to enthalpy and entropy have been derived. A complete set of recommended scale factors is presented. The most successful procedures overall are B3-PW91/6-31G(d), B3-LYP/6-31G(d), and HF/6- 31G(d). 1. Introduction The determination of vibrational frequencies by ab initio computational methods is becoming increasingly important in many areas of chemistry. One such area is the identification of experimentally observed reactive intermediates for which the theoretically predicted frequencies can serve as fingerprints. Another important area is the derivation of thermochemical and kinetic information through statistical thermodynamics. Ab initio harmonic vibrational frequencies (ω) are typically larger than the fundamentals (ν˜) observed experimentally.1 A major source of this disagreement is the neglect of anharmonicity effects in the theoretical treatment. Errors also arise because of incomplete incorporation of electron correlation and the use of finite basis sets. Thus, for example, Hartree-Fock (HF) theory tends to overestimate vibrational frequencies because of improper dissociation behavior, a shortcoming that can be partially compensated for by the explicit inclusion of electron correlation. The overestimation of ab initio harmonic vibrational frequen- cies is, however, found to be relatively uniform, and as a result generic frequency scaling factors are often applied. Good overall agreement between the scaled theoretical harmonic frequencies and the anharmonic experimental frequencies can then usually be obtained. The determination of appropriate scale factors for estimating experimental fundamental frequencies from theoretical harmonic frequencies has received considerable attention in the literature.2-11 Semiempirical methods, such as AM112 and PM3,13 are potentially attractive for the computation of vibrational frequen- cies because of their inherent low computational cost. However, there has been little systematic work reported on the performance of such methods for the prediction of vibrational frequencies. The most comprehensive study to date is an AM1 investigation by Healy and Holder2 on 42 common organic molecules in which the computed harmonic frequencies were found to differ from experiment by an average of 10.4%. We are unaware of any such study for the PM3 method. Pople et al.3 found that the harmonic vibrational frequencies calculated at HF/3-21G for a set of 38 molecules (477 frequencies) had a mean ω(3-21G)/ν˜(expt) ratio of 1.123, which suggested that this level of theory overestimates frequencies by about 12%. A scaling factor of 0.89 for theoretical HF/3-21G harmonic frequencies was proposed as being appropriate for predictive purposes. Hehre et al.4 determined from an HF/6- 31G(d) study of 36 molecules a mean percentage deviation of theoretical harmonic frequencies from experimental fundamen- tals of about 13%, similar to the findings of Pople et al.3 for HF/3-21G. An HF/6-31G(d) theoretical frequency scaling factor of 0.8929 has been widely used in theoretical thermochemical studies.14 Hehre et al.4 also determined that the MP2-fu/6-31G(d) method gave mean percentage deviations of theoretical harmonic frequencies from experimental fundamentals of about 7%. Such an error indicates that an appropriate scale factor for MP2-fu/ 6-31G(d) theoretical frequencies would be 0.921. In a later study, DeFrees and McLean5 found somewhat larger scale factors (of 0.96 for first-row molecules and 0.94 for second- row molecules) by determining an average of the experimental/ theoretical frequency ratios for individual modes. Recently, we determined the scale factors for the HF/6-31G- (d) and MP2-fu/6-31G(d) methods using a set of 122 molecules (1066 frequencies) and a least-squares approach.6 We computed an optimum HF/6-31G(d) frequency scale factor of 0.8953, veryX Abstract published in AdVance ACS Abstracts, September 1, 1996. 16502 J. Phys. Chem. 1996, 100, 16502-16513 S0022-3654(96)00976-8 CCC: $12.00 © 1996 American Chemical Society + + + + D ow nl oa de d by B EI JI N G U N IV S CI T EC H L IB o n Se pt em be r 7 , 2 00 9 | ht tp: //p ub s.a cs. org Pu bl ic at io n D at e (W eb ): Oc tob er 10 , 1 99 6 | doi : 1 0.1 021 /jp 960 976 r similar to the previous standard value of 0.8929, and recom- mended that the standard value remain unchanged. A scale factor appropriate for frequencies computed at the MP2-fu/6- 31G(d) level of theory (0.9427) was also determined, which lies between the values proposed by Hehre4 and DeFrees.5 We also found that, even after removing some exceptionally poorly predicted frequencies for O3 and NO2, the overall root-mean- square (rms) error for the MP2-fu/6-31G(d) method was only slightly smaller than the overall rms error for the HF/6-31G(d) level of theory. Some work has been presented in the literature on harmonic vibrational frequencies determined with more sophisticated correlated methods.15-18 These studies have, however, been generally limited to small polyatomic molecules. Procedures such as QCISD, CCSD, and CCSD(T) have been shown by several researchers to provide excellent agreement with experi- mental harmonic frequencies when used in conjunction with a variety of basis sets (double zeta plus polarization and larger).15-18 While the computational cost of such procedures is very expensive relative to that for HF or MP2, recent and continuing improvements in raw computer speed together with more efficient programs make these methods increasingly feasible. The advent of density functional theory (DFT) has provided an alternative means of including electron correlation in the study of the vibrational frequencies of moderately large molecules.19 Of the myriad of DFT functionals that are available today, perhaps the most prominent are B-LYP and B3-LYP. B-LYP uses a combination of the Becke exchange functional20 (B) coupled with the correlational functional of Lee, Yang, and Parr (LYP),21 while the hybrid B3-LYP procedure uses Becke’s three-parameter exchange functional (B3),22 as slightly modified by Stephens et al.,23 in combination with the LYP correlation functional. Pople et al.24 have shown that B-LYP/6-31G(d) harmonic vibrational frequencies reproduce observed fundamentals with surprising accuracy. They found, for example, an average error of only 13 cm-1 for a small set of molecules with up to three heavy atoms. The same set of molecules gave average errors of 243, 138, and 95 cm-1 for the (unscaled) HF/6-31G(d), MP2- fu/6-31G(d), and QCISD/6-31G(d) methods, respectively. How- ever, if the theoretically determined frequencies are compared with experimental harmonic frequencies, the average error for the B-LYP method is increased while that for the conventional ab initio methods decreases.24 Such a finding indicates that the good agreement between B-LYP harmonic frequencies and experimentally observed anharmonic frequencies is partly fortuitous. In another study, Rauhut and Pulay8 developed scaling factors for the B-LYP/6-31G(d) method based on a set of 20 small molecules with a wide range of functional groups. Their overall frequency scaling factor for the B-LYP/6-31G(d) method was determined to be 0.990 with an rms deviation of 26 cm-1. Rauhut and Pulay, in the same study,8 also developed a scaling factor for the B3-LYP/6-31G(d) method (0.963) that resulted in a slightly lower overall rms deviation of 19 cm-1. This result is in accord with the findings of Finley and Stephens.10 Use of the B and B3 exchange functionals with other correlation functionals such as P8625 and PW9126 to compute vibrational frequencies has received less attention in the recent literature. Hertwig and Koch27 systematically studied vibrational frequencies for the main group homonuclear diatomics and found that the B-P86 method was superior to B-LYP [in conjunction with the 6-311G(d) basis]. Finley and Stephens10 found that, as with B3-LYP, the B3-P86 method performed better than B-LYP (with the TZ2P basis) in the prediction of experimental harmonic frequencies. Other less extensive stud- ies28 have suggested that the choice of exchange functional is a more important consideration than the choice of correlation functional in achieving reliable theoretical frequencies from DFT methods. Apart from an interest in vibrational frequencies in their own right, one important use of theoretically determined vibrational frequencies is in the computation of zero-point vibrational energies (ZPVEs). ZPVE values determined from theoretical harmonic frequencies are used widely and, in particular, in very high level composite ab initio procedures such as the G1 and G2 theories of Pople and co-workers14 and the complete basis set methodologies of Petersson et al.29 In the simplest approximation, the ZPVE of a molecule is evaluated theoretically as where ωi is the ith harmonic vibrational frequency expressed in cm-1 (more rigorously called the harmonic vibrational wavenumber) and Nhc is the appropriate energy conversion constant. This expression is not precise since it does not take into account the effects of anharmonicity. The effect on ZPVEs of anharmonicity can be illustrated by considering a simple diatomic molecule for which the vibrational term values are given by where ωe is the harmonic vibrational frequency, ωexe and ωeye are anharmonicity constants, all in cm-1, and V is the vibrational quantum number.30 This series is often truncated at the second term because the subsequent terms are generally quite small, e.g. ωeye , ωexe. The ZPVE is then given by and the fundamental frequency (ν˜) is given by Extension to polyatomic molecules is straightforward.30 Simply using harmonic frequencies (i.e. ignoring anharmo- nicities) to calculate the ZPVE (ZPVEharm) results in an error in ZPVE (errharm) given by On the other hand, use of fundamental frequencies (such as suitably scaled theoretical harmonic frequencies) to calculate ZPVE (ZPVEfund) results in an error in ZPVE (errfund) of As is clear from eqs 5 and 6, and as shown in recent papers by Grev, Janssen, and Schaefer (GJS)31 and by Del Bene, Aue, and Shavitt,32 the calculation of ZPVEs from theoretical harmonic frequencies requires frequency scale factors that are somewhere in between those that relate theoretical harmonic frequencies to observed fundamentals and those that relate theoretical harmonic frequencies to experimental harmonic frequencies. In our recent study,6 we found, from a least-squares study of the same set of 24 molecules as used by GJS,31 HF/6-31G(d) and MP2-fu/6-31G(d) scale factors for ZPVE of 0.9135 and 0.9646, respectively. In accord with the suggestions by GJS, ZPVE/Nhc ) 1/2∑ i ωi (1) G(V) ) ωe(V + 1/2) - ωexe(V + 1/2)2 + ωeye(V + 1/2)3 + . . . (2) ZPVE/Nhc ) G(0) ) 1/2ωe - 1/4ωexe (3) ν˜ ) G(0 f 1) ) ωe - 2ωexe (4) err harm/Nhc ) G(0) - (ZPVEharm/Nhc) ) -1/4ωexe (5) err fund/Nhc ) G(0) - (ZPVEfund/Nhc) ) 3/4ωexe (6) Harmonic Vibration Frequencies J. Phys. Chem., Vol. 100, No. 41, 1996 16503 + + + + D ow nl oa de d by B EI JI N G U N IV S CI T EC H L IB o n Se pt em be r 7 , 2 00 9 | ht tp: //p ub s.a cs. org Pu bl ic at io n D at e (W eb ): Oc tob er 10 , 1 99 6 | doi : 1 0.1 021 /jp 960 976 r these factors are indeed larger than those relating theoretical harmonic vibrational frequencies to experimental fundamentals (0.8953 and 0.9427, respectively). Knowledge of vibrational frequencies also plays a vital role in determining the thermal contributions to enthalpy and entropy [∆Hvib(T) and Svib(T), respectively] which can be expressed33 as where N is Avogadro’s number and µi ) hcν˜i/kT, in which ν˜i is the ith fundamental frequency in cm-1. Inspection of eqs 7 and 8 indicates that small frequencies contribute more to the thermal contributions to enthalpy and entropy than do larger frequencies. This can be readily confirmed by reference to Figure 1 which plots ∆Hvib(T) (left- hand axis) and Svib(T) (right-hand axis) as a function of frequency (ν˜i). We note that as the vibrational frequency tends to zero (see insert in Figure 1), ∆Hvib(T) reaches a limiting value which, within the confines of the harmonic oscillator model, is equal to RT ()2.479 kJ mol-1 at 298.15 K). However, since many very low frequencies are rotational in nature, it is often more appropriate to calculate the thermal component of enthalpy associated with very low frequencies using a free rotor ap- proximation. In such circumstances, there is a contribution of 1/2RT for each such frequency. The “cross-over” frequency at which ∆Hvib(T) equals 1/2RT at 298.15 K is about 260 cm-1. We note also that as the vibrational frequency tends to zero, the value for Svib(T) tends to infinity. The use of scaling factors potentially allows Vibrational frequencies and thermochemical information of useful accuracy to be obtained from procedures of only modest computational cost. Widespread application to molecules of moderate size is then possible. In the present study, we examine the performance of 19 such procedures, with particular emphasis on density functional methods, since these have received relatively little previous attention in the literature. Specifically, we have computed harmonic vibrational frequencies for a large standard suite of test molecules at many of the levels of theory currently in popular use. The methods employed include the semiempirical procedures AM1 and PM3, the conventional ab initio procedures HF/3-21G, HF/6-31G(d), HF/6-31+G(d), HF/6-31G(d,p), HF/ 6-311G(d,p), HF/6-311G(df,p), MP2-fu/6-31G(d), MP2-fc/6- 31G(d), MP2-fc/6-31G(d,p), MP2-fc/6-311G(d,p), and QCISD- fc/6-31G(d), and the DFT procedures B-LYP/6-31G(d), B-LYP/ 6-311G(df,p), B-P86/6-31G(d), B3-LYP/6-31G(d), B3-P86/6- 31G(d), and B3-PW91/6-31G(d). From these theoretical frequencies, we have determined a set of recommended scaling factors that relate theoretical harmonic frequencies to experimental fundamentals and have determined the rms errors for the various theoretical procedures. We have also determined a set of recommended scaling factors for the calculation of ZPVEs. Finally, we also present here, for the first time, a set of frequency scaling factors for calculating low- frequency vibrations and the thermal contributions to enthalpies and entropies. The present work provides the most comprehensive com- pendium of theoretically determined harmonic vibrational frequencies and related scale factors available to date. 2. Theoretical Procedures We define three sets of molecules for our present work. The first, designated F1, is the full set of 122 molecules and a total of 1066 vibrational frequencies (after counting degenerate modes) used in our previous paper.6 This set is made up of the union of the set of polyatomic molecules listed by Shimanou- chi34 that contain no more than 4 heavy atoms of the first or second row, with no more than 10 atoms in total, and the set of 24 molecules listed by GJS.31 The second set, named F2, is a subset of F1 and consists of those molecules that contain only H, C, N, O, and F atoms, no more than 4 heavy atoms, and no more than 10 atoms in total. The F2 set of molecules (37 molecules with a total of 477 vibrational frequencies) was used in earlier work by Pople et al.3 The third set, designated Z1, is comprised of 25 of the diatomic molecules from the G2 atomization list14b (all except Si2) together with 14 additional molecules derived from the GJS set of molecules referred to above. This set is similar to that used by Bauschlicher and Partridge.9 We have, however, elected not to include singlet or triplet CH2 in our present study since, to the best of our knowledge, no experimental harmonic vibrational frequencies are available for these species. We have used the Z1 set to study zero-point vibrational energies. The experimental frequencies for the F1 and F2 sets are obtained directly from the compilation of Shimanouchi.34 The experimental ZPVE values for molecules in the Z1 set have been calculated according to standard formulas (cf. eq 3).30,35 The requisite experimental harmonic frequencies (ωe) and anharmonic constants (ωexe) for the diatomic molecules were obtained from the compilation of Huber and Herzberg.36 Experimental harmonic frequencies and associated anharmonic corrections for HCN, CO2, and C2H2 were obtained from Allen et al.,37 those for H2O, H2S, H2CO, HCO, and C2H4 were obtained from Clabo et al.,38 those for CH4 from Lee et al.,39 and those for CH3Cl from Duncan and Law.40 Most of the calculations performed in this study were carried out with the GAUSSIAN 92/DFT package of ab initio programs.41a Calculations for one of the DFT methods (B3- PW91) were performed with GAUSSIAN 94.41b Standard basis sets were used throughout and, unless otherwise stated (see below), the SG1 grid42,43 was used as the quadrature grid for DFT calculations within GAUSSIAN 92. Møller-Plesset perturbation theory truncated at second order (MP2) was employed both with the core electrons of the heavy atoms held frozen (indicated by MP2-fc) and with the core electrons explicitly included (indicated by MP2-fu). Only the frozen- core approximation was employed for the QCISD calculations. The geometry of each molecule was completely optimized at the appropriate level of theory by analytic gradient techniques. Figure 1. Plot of ∆Hvib(T) (kJ mol-1, left-hand axis) and Svib(T) (J K-1 mol-1, right-hand axis), at 298.15 K, as a function of frequency (cm-1). (Insert) Expansion of 0-100 cm-1 region. ∆Hvib(T) ) Nhc∑ i ν˜i e µi- 1 (7) Svib(T) ) R∑[ µi e µi- 1 - ln(1 - e-µi)] (8) 16504 J. Phys. Chem., Vol. 100, No. 41, 1996 Scott and Radom + + + + D ow nl oa de d by B EI JI N G U N IV S CI T EC H L IB o n Se pt em be r 7 , 2 00 9 | ht tp: //p ub s.a cs. org Pu bl ic at io n D at e (W eb ): Oc tob er 10 , 1 99 6 | doi : 1 0.1 021 /jp 960 976 r Harmonic vibrational frequencies were determined by the analytic evaluation of the second derivative of the ener