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Pople et al.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. 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 Pulay 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,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. Use of the B and B3 exchange functionals with other correlation functionals such as P86and PW91 to compute vibrational frequencies has received less attention in the recent literature. Hertwig and Koch 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 studies 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-workers and the complete basis set methodologies of Petersson et al. In the simplest approximation, the ZPVE of a molecule is evaluated theoretically as 公式1 where öi 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 公式2 where öe is the harmonic vibrational frequency, öexe and öeye are anharmonicity constants, all in cm-1, and V is the vibrational quantum number. This series is often truncated at the second term because the subsequent terms are generally quite small.公式3，4,5,6 As is clear from eqs 5 and 6, and as shown in recent papers by Grev, Janssen, and Schaefer (GJS) and by Del Bene, Aue, and Shavitt, 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,we found, from a least-squares study of the same set of 24 molecules as used by GJS, 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,图表1 these factors are indeed larger than those relating theoretical harmonic vibrational frequencies to experimental fundamentals (0.8953 and 0.9427, respectively). 公式7,8 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 (îÄi). 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 across-overo 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. |
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