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An Evaluation of Scaling Factors for Multiparameter Scaling Procedures Based on DFT Force Fields Piotr Borowski* Faculty of Chemistry, Maria Curie-Skłodowska University, pl. Marii Curie-Skłodowskiej 3, 20-031 Lublin, Poland S Supporting Information *

ABSTRACT: An extended database of scaling factors for calculating fundamental frequencies within multiparameter scaled quantum mechanical (SQM) force field, and effective scaling frequency factor (ESFF) methods, based on various DFT force fields is reported. Twenty-six density functionals have been examined in conjunction with various Pople’s and Dunning’s basis sets of VDZ and VTZ quality. The calculations were based on a standard training set of 30 molecules proposed by Baker et al., for which 660 vibrational modes were assigned. Six functionals turned out to be particularly well-suited to the calculations oriented toward determination of scaled frequencies. They are B3LYP, B3PW91, B97, B97-1, B97-2, and O3LYP; they are all capable of providing reasonable scaled frequencies even for the small, 631G* basis set (rms 15 cm−1). Note that the reported rms values refer to frequencies obtained using factors optimized to match the observed fundamentals as faithfully as possible; in the case of other molecules, using these factors should result in higher deviations between scaled and fundamental frequencies. These functionals either contain no correlation term (HFS, HFB, and OPTX) or include Slater local exchange42 in conjunction with the VWN local correlation functional.43,44 Evidently, although scaling procedures are expected to (partially) correct shortcomings of the applied methodology, large errors in the calculated harmonic force fields/harmonic frequencies for various molecules of random rather than systematic character result in substantial increase of the overall rms value. The quality of VWN CF-based functionals with regard to the scaled frequencies improves if Slater exchange is combined either with Becke’s45 or Handy and Cohen52 nonlocal exchange (BVWN and BVWN5 as well as OVWN and OVWN5 functionals, respectively). Finally, it should be noted that the order of density functionals is very similar for both SQM and ESFF scaling proceduresminor changes take place only when the rms values obtained with both functionals are very close. 4.3. Comparison of the SQM and ESFF Scaling Procedures. Considerations presented in this section are based on data included in Tables Quality of the method A/B on the sheet statistics of the Supporting Information. Observe again that for every test parameter, i.e., rms, ARPE, and rms mid., 740 values for each scaling procedure are available (cf. item 3 of the jp212201f_si_001.pdf file). Table 2 extracts the overall number of counts nX the scaling procedure X is better as compared with Y, assuming a specified threshold value. As can be seen, ESFF provides, in general, scaled frequencies that are superior to SQM, though in most cases |rmsESFF − rmsSQM| < 0.2 cm−1. The differences larger than 0.2 cm−1 in favor of ESFF were found in 258 cases, and in favor of SQM, in 107 cases. However, there are more counts in favor of SQM for the threshold value as large as 0.5 cm−1. The reason is that SQM is capable of predicting better frequencies for the XH stretching vibrationslarger deviations for frequencies close to 3000 cm−1 for the ESFF-scaled frequencies increase an overall rms value, only slightly affecting the relative errors. Indeed, for the

Table 2. Comparison of the Performance of SQM and ESFF Scaling Procedures Based on Three Tests: rms, ARPE, and rms mid.; See Texta all functionals ESFF

SQM

ESFF

SQM

ESFF

SQM

better rms 0.20 cm−1 0.50 cm−1 0.00 cm−1 524 207 258 107 34 54 better ARPE 0.00% 0.02% 0.05% 602 92 511 72 193 26 better rms mid. 0.00 cm−1 0.20 cm−1 0.50 cm−1 527 208 363 143 130 91 only B3LYP, B3PW91, B97, B97-1, B97-2, and O3LYP functionals ESFF

SQM

0.00 cm−1 112 52 0.00% 143

6

0.00 cm−1 115 50

ESFF

SQM

ESFF

better rms 0.20 cm−1 27 10 better ARPE 0.02% 91 3 better rms mid. 0.20 cm−1 59 17

SQM

0.50 cm−1 0

0 0.05%

5

0 0.50 cm−1

8

3

a

A number of counts for a given procedure that exhibit rms, ARPE, and rms mid. values lower than the specified threshold (lower or equal to; lower in the case of zero) in favor of this procedure as compared with the other one is given.

largest threshold value for ARPE, i.e., 0.05%, the number of counts reads 193÷26 in favor of ESFF (21÷0 in the case of 0.1%, which is not reported in the table). Larger discrepancies between SQM and ESFF fits are observed for poorly working functionals with respect to the force fieldsthe data included in Table 2 clearly indicate that, for the six highest-quality functionals we found (cf. section 4.2), both scaling procedures give the overall rms values that do not differ by more than 0.5 cm−1. 4.4. Comparison of the 11- and 9-Parameter Scaling Frames. The cyan columns of Tables Quality of the method A/B on the sheet statistics provide a comparison of 11- and 9parameter calculations. This statistics is based on comparing the “rms mid.” values; the number of counts nX the scaling frame X is better as compared with Y assuming a specified threshold value (cf. items 2 and 3 of the jp212201f_si_001.pdf file) is given. The results are also summarized in Table 3. In general, the 9-parameter scaling procedure gives higher rms values in the entire range of a vibrational spectrum. This is not a surprisethe fewer parameters included in the least-squares procedure, the more the quality of a fit deteriorates. However, higher flexibility of the 9-parameter set in the description of the XX stretching vibrations36 as compared with the original, 11parameter set21 should imply its better performance in the middle range of a spectrum. As can be seen, 9-parameter scaling (regardless of the method, SQM or ESFF) seems to be only slightly advantageous when considering all functionals (and MP2, if applicable): a number of counts ratio reads 380÷347 in its favor (i.e., n9‑par./n11‑par. ≈ 1). Adopting a higher threshold value of 0.5 cm−1 (the “X ≫ Y” test), a better ratio was found (96÷7). It should be noted, however, that nearly half (47) of the reported 96 counts follows from the six highest-quality 3870

dx.doi.org/10.1021/jp212201f | J. Phys. Chem. A 2012, 116, 3866−3880

The Journal of Physical Chemistry A

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simply shifted toward higher rms values (here: in the sequence B3LYP → BLYP → HFB), and the distance between SQM and ESFF points covers a narrow range of wavenumbers. There is a minor difference in the shape of the diagrams obtained for 11and 9-parameter calculations. Namely, an increase in the rms value was obtained in most cases for 11-parameter scaling in the step from ...+G* to ...G** basis sets (where “...” denotes 6-311 or 6-31). On the other hand, a slight decrease is frequently observed for 9-parameter scaling, indicating that polarization functions on hydrogen atoms are more important than diffuse functions on heavy atoms with respect to frequency calculations for most functionals. The extension of Pople’s basis sets beyond 6-311++G** was also tested. We used a large, 6-311+ +G(3df,3pd) basis set and B3LYP functional only. Despite promoting the calculations to the highest-quality group (cf. item 4 of the jp212201f_si_001.pdf file), the results are significantly worse than those already discussed for the 6311+G** basis sets: an increase of the overall rms value by 0.7−0.8 cm−1 was obtained for 11-parameter SQM and ESFF calculations. Lower discrepancies (0.3−0.4 cm−1 in favor of the 6-311+G** basis set) are observed for 9-parameter calculations. This is to be contrasted with a single parameter frequency scaling procedure,6 in which the rms calculated for the B3LYP functional, after a rather irregular pattern for small basis sets, “converges” to the minimum value along with an increase of the basis set size already for the 6-311+G(3df,p) basis set. No changes were observed upon further extension. The B3LYP/6311++G(3df,3pd) SQM and ESFF scaling factors, reported in Table 4, do not appear in the Supporting Information. It is worthwhile to mention that the total CPU time was 5−7 times longer with this extended basis set than with 6-311+G**. The Dunning aug-cc-pVTZ basis set gives the lowest rms values in almost all cases (for almost all density functionals within both frames of SQM and ESFF scaling procedures). Apparently, an addition of f functions on heavy atoms is essential in accurate prediction of vibrational spectra for this type of basis setin all 104 cases (2[SQM and ESFF] × 2[11‑ and 9‑par. frames] × 26[functionals]), the aug-cc-pVTZ basis set is superior to aug-cc-pVDZ. Note that the differences between the rms values computed with these two basis sets sometimes exceed 1 cm−1 for the highest-quality functionals and 2 cm−1 for functionals that give the lowest-quality fits. Not augmented cctype basis sets perform, on average, worse; notwithstanding, only in 15 out of 104 cases, the aug-cc-pVDZ basis set is somewhat superior to cc-pVTZ. However, the best Pople’s basis set we found for the DFT frequency calculations, i.e., 6311+G**, is superior to cc-pVTZ in almost all cases: in all for the 11-parameter calculations and in all but 4 (functionals) in the case of 9-parameter calculations. All quantum chemistry calculations should be based on a compromise between the quality of the results and the computational effort. The latter strongly depends on the size of the basis set used in the calculations. Table 5 reports the total CPU time needed for both the geometry optimization and the frequency (Hessian) calculation with the specified basis sets, CPU time only for the Hessian calculation, as well as ranking of all basis sets based on the overall rms value. Note that only the largest considered basis sets are included in the table. The CPU times are reported for m-dichlorobenzene (which includes H, as well as second- and third-row atoms) and the O3LYP functional (for which the calculations were carried out consistently on the same processor with all basis sets); however, for other systems and with other functionals, the

Table 3. Comparison of the Performance of 9- and 11Parameter Scaling Frames in the Middle Range of the Vibrational Spectrum (500−2500 cm−1) Based on rms mid. Test; See Texta all functionals SQM

ESFF

total

SQM

ESFF

total

9-par. > 11-par. 9-par. < 11-par. 202 178 380 163 184 347 9-par. ≫ 11-par. 9-par. ≪ 11-par. 58 38 96 0 7 7 only B3LYP, B3PW91, B97, B97-1, B97-2, and O3LYP functionals SQM 61 22 a

ESFF 9-par. > 11-par. 67 9-par. ≫ 11-par. 25

total 128 47

A number of counts for which rmsX |rmsX − rmsY| < 0.5 cm−1 in favor of denote 11- or 9-parameter frames) individually for SQM and ESFF, as methods.

SQM 23 0

ESFF 9-par. < 11-par. 17 9-par. ≪ 11-par. 1

total 40 1

< rmsY (columns X > Y) and X (columns X ≫ Y; X and Y are given. Statistics is made well as collectively for both

functionals, described in section 4.2. Only in one case (ESFF scaling with O3LYP/6-311+G* force fields), 11-parameter scaling turned out to be significantly better. In addition, the ratio n9‑par./n11‑par. ≈ 3 (rather than 1, i.e., 128/40) is obtained in the case of the “X > Y” test. Thus, as long as high-quality force fields are used in the frequency calculations, the 9parameter scaling is preferable in the interpretation of vibrational spectra in the range 500−2500 cm−1. This is, in fact, what we expected when designing the new set of scaling factors.36 Again, random errors in the scaled frequencies for various molecules computed from poor-quality force fields may increase the rms in the middle range of a vibrational spectrum. 4.5. Quality of Scaled Frequencies vs Basis Set Quality. Diagrams presenting the quality of scaled frequencies (defined via an overall rms value) obtained within both 11- and 9-parameter scaling frames with each basis set considered in the present work are shown in Figure 1. The black, red, and green points refer to Pople’s VTZ-, VDZ-, and Dunning’s cc-type basis sets, respectively. Three functionals were chosen: (i) B3LYP, which is the highest-quality functional with respect to frequency calculations (cf. section 4.2), (ii) BLYP, which works reasonably well, and (iii) HFB, the least accurate one. These three functionals are representative for the purpose of an overall discussionit has been confirmed that the remaining functionals behave similarly, in that the diagrams were merely shifted toward lower/higher rms values (which can be easily verified by the reader by performing a quick scan over the remaining data in the sheet statistics; the diagrams presented in Figure 1 were left there, on top of Table rms values B). Minor deviations from general trends are not detrimental. Out of all Pople’s basis sets considered in the present work, 6-311+G** turned out to be the most accurate one. In particular, it provides the best scaled frequencies within the 11parameter scaling frame, and shares the superiority with the 631+G** basis set in the case of 9-parameter scaling. Its extension by adding diffuse functions on hydrogen atoms, as well as removal of some basis functions, usually leads to deterioration of the quality of results. The pattern of basis set quality is nearly the same for all functionals, regardless of the scaling procedure; as can be seen from Figure 1, points are 3871



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Figure 1. Relationships between the rms values for the SQM- and ESFF-scaled frequencies obtained from B3LYP, BLYP, and HFB force fields, and the basis set used in the calculations.

timing pattern should be similar. Since in all calculations seven cycles were needed in the geometry optimization step, total CPU times are good indicators for the comparison. As can be seen, the cc-type basis sets are not preferable in the frequency calculations. The obtained rms values are either the largest of all

(pVDZ-quality basis sets) or impractical from the computational point of view (pVTZ-quality basis sets). Note that the lowering of the rms values offered by Dunning’s pVTZ basis sets, though noticeable (sometimes even a few tenths of cm−1), is associated with a substantial (up to 20 times) increase in the 3872



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calculations, in which Hessian, obtained as a byproduct of some other calculations (e.g., verification of the type of stationary point found in the optimization procedure), is available. 4.6. Quality of Scaled Frequencies vs Quality of Equilibrium Geometries. Differences between the scaled frequencies and the observed fundamentals follow not only from the adopted harmonic approximation but also from an incomplete incorporation of correlation effects and incompleteness of the basis set used in the calculations. The latter two items have a direct effect on the computed equilibrium geometries, at which molecular force field is to be calculated. The DFT approach based on functionals considered in the present work predicts molecular geometries with variable accuracy. This may have a direct effect on the calculated frequencies. The overall rms value between the scaled and the observed frequencies for all molecules is a good measure of the quality of predicted vibrational spectra. Similar measures can be proposed for geometric parameters.5 However, internal coordinates include both bond lengths and angles, and therefore, the average relative percentage errors (ARPE) are preferable. The relevant data are included in sheets geom. expt. and geom. mp2 of the Supporting Information (cf. items 5 and 6 of the jp212201f_si_001.pdf file for further details). Correlation diagrams: the rms values between the SQMscaled and the observed fundamental frequencies vs the ARPE values between theoretical equilibrium and experimental geometric parameters are presented in Figure 2. Note that ARPE values were computed for 26 rather than 30 molecules, and a limited number of internal coordinates. The original, 11parameter scaling frame was chosen. Diagrams based on ESFF scaled frequencies are similar. Results obtained with the largest (6-311++G**, Figure 2a) and the smallest (6-31G*, Figure 2b) Pople’s basis sets considered in the present work are shown. The following discussion is based on results obtained with the 6-311++G** basis set. As can be seen, a very approximate trend reads as follows: the higher the ARPE value for the geometric parameters, the higher the rms value for all scaled frequencies. However, the relation is not linear. A steep increase of rms is initially observed: in the range of ARPE from 0.58 to 0.83%, the rms raises by as much as ca. 6 cm−1. In this range, all but the three functionals HFS, OPTX, and HFB (i.e., functionals lacking the correlation term) can be found. For the lowestquality HFB functional, ARPE was found to be close to 1.5%, which is associated with further increase of rms by ca. 2.5 cm−1. The six highest-quality functionals for the determination of vibrational spectra (cf. section 4.2) are in the lower left corner

Table 4. SQM and ESFF Scaling Factors for 11- and 9Parameter Calculations Obtained for 660 Frequencies of Molecules of Baker’s Training21 Set Using B3LYP/6-311+ +G(3df,3pd) Force Fieldsa 11 parameters

SQM

ESFF

1. XX 2. CCl 3. CH 4. NH 5. OH 6. XXX 7. XXH 8. HCH 9. HNH 10. linear def. 11. torsions rms (in cm−1) ARPE (in %) rms mid. (in cm−1) 9 parameters

0.9393 1.0201 0.9264 0.9156 0.9117 1.0420 0.9655 0.9504 0.9487 0.8800 0.9467 11.29 0.99 10.16 SQM

0.9754 1.0049 0.9625 0.9569 0.9548 1.0027 0.9820 0.9742 0.9729 0.9538 0.9746 11.13 0.94 9.93 ESFF

1. XX(s,c) 2. XX(d,t) + CN(a) 3. CCl 4. CH + OH + NH 5. XXX 6. XXH 7. HXH 8. linear def. 9. torsions rms (in cm−1) ARPE (in %) rms mid. (in cm−1)

0.9575 0.9316 1.0198 0.9243 1.0285 0.9615 0.9525 0.8817 0.9459 11.27 0.95 9.48

0.9826 0.9672 1.0050 0.9617 0.9994 0.9800 0.9751 0.9544 0.9743 11.08 0.91 9.15

The root-mean-square deviations (rms, in cm−1), the average relative percentage errors (ARPE, in %), and the root-mean-square deviations in the range 500−2500 cm−1 (rms mid., in cm−1) are also given. a

computational effort as compared with the 6-311+G** basis set. The total time of the calculations with the 6-311+ +G(3df,3dp) basis set is confined within those of the cc-pVTZ and aug-cc-pVTZ basis sets, and the overall rms value is significantly higher. Thus, as long as the DFT calculations of vibrational spectra (geometry and Hessian) in conjunction with the multiparameter scaling procedures are to be carried out, typical Pople’s basis sets are definitely preferable. Scaling factors for Dunning basis sets may also be useful in the case of other

Table 5. Number of Primitive/Contracted Basis Functions (Column “size”), Total CPU Time (CPU(tot), min.), and CPU Time Needed for Hessian Calculations (CPU(hess), min.) for m-Dichlorobenzene, as Well as an Overall rms Values/Ranking for Each Basis Set (Columns “ranking”)a ranking basis set

size

CPU(tot)

CPU(hess)

11-p. SQM

11-p. ESFF

9-p. SQM

9-p. ESFF

6-311+G** 6-311++G** 6-311++G(3df,3dp) cc-pVDZ (H − p, X − d) cc-pVTZ (H − 2pd, X − 2df) aug-cc-pVDZ (H − 2p, X − 2d) aug-cc-pVTZ (H − 3p2d, X − 3d2f)

346/216 350/220 530/400 266/140 434/304 488/228 740/468

111.2 128.8 763.6 55.2 392.1 266.6 2227.0

71.9 86.3 404.3 31.9 208.4 148.8 1108.7

10.52/1 10.86/3 11.29/5 12.53/7 10.93/4 11.43/6 10.56/2

10.38/1 10.71/3 11.13/5 12.70/7 10.81/4 11.45/6 10.40/2

10.91/3 11.25/4 11.27/5 12.54/7 10.77/2 11.38/6 10.50/1

10.77/3 11.10/5 11.08/4 12.59/7 10.65/2 11.33/6 10.33/1

a

Timing refers to the O3LYP calculations (which were carried out consistently on the same processor for all basis sets), and rms values to B3LYP calculations for all molecules. Data for only the biggest basis sets are reported. 3873



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Figure 2. Correlation diagrams presenting the rms values between the SQM-scaled and the observed fundamental frequencies vs the ARPE values between DFT equilibrium and experimental (rg or rs) geometric parameters (see text) obtained for 26 density functionals considered in this work with (a) 6-311++G** and (b) 6-31G* basis sets.

fully conclusive, unless strengthened by additional considerations. The experimental equilibrium parameters are sometimes available in the literaturethey are typically determined for small molecules from microwave and infrared gas-phase data using extrapolation techniques. However, they are not available for molecules of Baker’s training set. Very accurate equilibrium geometries can be obtained from highly correlated CC-type calculations employing extended basis sets. CCSD is already quite accurate. CCSD(T) calculations employing a large, say, cc-pVTZ, basis set are nearly perfect with respect to the determination of equilibrium bond lengths,5 probably due to fortuitous cancellation of errors arising from truncations in oneand N-electron bases. Further extension of the basis set to that of pVQZ-quality does not provide significant improvement of the final results. MP2 was shown to be somewhat inferior; the mean absolute error for 27 bond lengths of 19 small molecules predicted using the cc-pVTZ basis set was 0.56 pm, which is to be compared with 0.20 pm obtained in the analogous CCSD(T) calculations. CCSD(T) geometric parameters, calculated consistently (i.e., with the same basis set) for all molecules from Baker’s training set, are not available in the literature. Their determination may become extremely timeconsuming for the largest molecules considered in this work. It seems, however, that we have equilibrium parameters at our disposal that are (probably) accurate enough, which were found consistently for all molecules; they were calculated at the MP2 level with large basis sets. Thus, we compared the accuracy of the above-mentioned, MP2/cc-pVTZ equilibrium bond lengths

of the diagram; i.e., they predict the best geometries and the best spectra. However, a number of additional functionals can be found in the narrow range of the (very lowest) ARPE values of 0.58−0.65% found for these functionals. Thus, in this range, basically no correlation between the quality of the predicted geometry and frequencies is observed. This is shown in Figure 2a, in which functionals from this range (and a little outside, toward higher ARPE values) are inside the yellow ellipse. One could conclude that the only factor responsible for an increase of the rms values for these functionals is the predicted curvature of the PES with respect to various geometric parameters. On the other hand, a green ellipse surrounds functionals, for which the relation of rms vs ARPE is roughly linear, indicating a strong correlation between the quality of the calculated force field and the quality of the predicted geometry. It starts from a group of S+B88 and S+HC XF-based density functionals typically with contributions from CFs other than VWN(5) (apart from OVWN(5) functionals), which predict accurate geometries and medium-quality frequencies. Then, there are S and S+B88 XF-based density functionals with VWN(5) correlation terms, which are significantly inferior in both geometry and frequency calculations. Finally, there are the three XF(only)-based density functionals that are mentioned above. Note that in the discussion above the ARPE values we were referring to were computed for equilibrium theoretical parameters, and the corresponding experimental rg or rs values.67 Thus, the discussion presented here may not be 3874



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Figure 3. Correlation diagrams presenting the rms values between the SQM-scaled and the observed fundamental frequencies vs the ARPE values between DFT and MP2/aug-cc-pVTZ equilibrium geometric parameters (see text) obtained for 26 density functionals considered in this work with (a) 6-311++G** and (b) 6-31G* basis sets.

of 19 molecules5 with these predicted by the B3LYP density functional (also known to be accurate). Our B3LYP/cc-pVTZ calculations gave the value of mean absolute error of 0.87 pm, which is significantly larger than the reported 0.56 pm obtained at the MP2/cc-pVTZ level (Table 15 of ref 5; our MP2 value turned out to be 0.54 pm, probably due to truncation errors). We decided to use MP2/aug-cc-pVTZ geometries as our reference in the considerations analogous to those carried out in the previous paragraph; the presence of additional diffuse functions is frequently needed in the case of systems containing third-row elements. Thus, we computed the ARPE values between geometric parameters (all bond lengths and valence angles, as well as torsional angles significantly deviating from zero), which were calculated with the aid of various density functionals and at the MP2/aug-cc-pVTZ level, and correlated them with rms values in the scaled frequencies. The results are presented in Figure 3, again for the largest (6-311++G**) and the smallest (6-31G*) Pople’s basis sets. As can be seen, the overall result is slightly different from that obtained using experimental geometries. As before, the six highest-quality functionals are located in the lower left corner of the diagram; i.e., they predict the lowest errors in the calculated geometries and in the scaled frequencies. The remaining functionals exhibit a roughly linear correlation, though the spread around the trend line reaches the value of ca. 3 cm−1 at some ARPE values. This time, however, the correlationalthough not perfectly linear is observed for most of the considered 26 functionals: most of the functionals exhibiting higher rms values in the scaled

frequencies also exhibit larger ARPE values in equilibrium geometric parameters relative to the MP2 results. This is to be contrasted with the previous finding based on experimental geometries, where a number of functionals were found to exhibit a vertical pattern on such a correlation diagram. Thus, we conclude that errors in the prediction of vibrational spectra in the various types of DFT calculations are to a great extent due to errors in the predicted geometry. The quality of the theoretical vibrational spectra is also dependent on the curvature of the PES when atoms are displaced from their equilibrium positions, as predicted by a given method. Comparison of parts a and b of Figure 3 shows that when the quality of basis set is decreased from 6 to 311+ +G** to 6-31G*, only a minor increase in geometry errors is observed, but errors in calculated frequencies increase by ca. 2 cm−1 (the entire diagram is shifted toward higher rms values). Indeed, a comparison of individual geometric parameters reveals that the differences |ri(6‑311++G**) − ri(6‑31G*)| in bond lengths predicted by B3LYP calculations with both basis sets are of the order of a few thousands of Å (specifically, an average difference was found to be 0.0026 Å), and in valence angles |θi(6‑311++G**) − θi(6‑31G*)|typically up to a few tenths of a degree (only 10 out of 447 valence angles found for all molecules exhibit differences larger than 1°). In addition, most bonds (259 out of 278) are elongated when going from the 6311++G** to the 6-31G* basis set; no trend was observed in the case of angles. The observations for the lowest-quality HFB functional are basically the same (the corresponding quantities 3875



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corresponding fundamentals). This has a direct effect on the calculated CCl scaling factorall factors determined for the MP2 force fields are lower than unity. Note that similar behavior was found in the case of a few density functionals, including B3PW91 and B97-2. Density functionals are capable of predicting molecular geometries with variable accuracy, which also depends on the basis set used in the calculations. In the case of CCl bond lengths, strong dependence on the quality of a functional was found; the quality of the basis set is not so essential. Indeed, the differences between maximum and minimum average CCl bond lengths predicted by different functionals are typically about 0.09 Å (for all basis sets, cf. Table bond lengths on the sheet statistics). These differences calculated for a given functional typically do not exceed 0.01 Å when various basis sets are considered. This is also shown in Figure 4, which

in the above order read: 0.0030 Å, 11 out of 447, and 248 out of 278, respectively). Thus, upon deterioration of the basis set, the curvature of the potential energy surface at minimum becomes significantly affected, while the location of the minimum itself is affectedon averageto a much lesser extent. 4.7. CCl Stretching Vibration. In standard, 11-parameter SQM calculations,21 a separate scaling factor was attributed to the CCl stretch group. It was found to be somewhat greater than the unity for the B3LYP force fields. The 9-parameter SQM calculations, 36 as well as both frames of ESFF calculations,34,36 are in accord with this findingthe calculated harmonic frequencies acquiring significant contribution from CCl stretching (more than 80% for small systems, like 1,1dichloroethylene; in the case of larger systems, some delocalization of this vibration is observed, but contributions greater than 50% are frequently found) calculated from the B3LYP force fields are lower than the observed fundamentals. It was concluded36 that “...The associated factor greater than 1 indicates the necessity of empirical correction on the computed geometry that results in a too long CCl bond and, as a consequence, too low harmonic frequency...” Thus, we carried out MP2 calculations (both molecular geometries and vibrational frequencies) using all basis sets considered in the present work. Note that MP2 scaling factors are reported for the selected basis sets for the reasons given in the next section. The results are summarized in Table 6, which includes average CCl Table 6. Average CCl Bond Lengths Calculated for Four Chlorine-Containing Molecules of Baker’s Training Set, as Well as SQM and ESFF Scaling Factors for the CCl Stretch Group Obtained at the B3LYP and MP2 Levels Using Basis Sets Considered in the Present Work average CCl bond lengths

11-par. SQM scaling factor

11-par. ESFF scaling factor

basis set

B3LYP

MP2

B3LYP

MP2

B3LYP

MP2

6-311++G** 6-311+G** 6-311+G* 6-311G** 6-311G* 6-31++G** 6-31+G** 6-31+G* 6-31G** 6-31G* cc-pVTZ cc-pVDZ aug-cc-pVTZ aug-cc-pVDZ

1.750 1.750 1.750 1.751 1.750 1.750 1.750 1.750 1.750 1.750 1.743 1.751 1.743 1.753

1.729 1.729 1.730 1.729 1.730 1.732 1.731 1.732 1.732 1.733 1.727 1.739 1.727 1.744

1.0345 1.0334 1.0338 1.0431 1.0504 1.0217 1.0212 1.0209 1.0263 1.0257 1.0342 1.0255 1.0374 1.0175

0.9555

1.0084 1.0097 1.0087 1.0154 1.0135 1.0034 1.0033 1.0025 1.0019 1.0012 1.0082 1.0061 1.0093 1.0081

0.9708

0.8532

0.9289 0.9240 0.9443 0.9534

Figure 4. Relationships between the average CCl bond length calculated for four chlorine-containing molecules of Baker’s training set, and the basis set used in B3LYP, BLYP, HFB, and MP2 calculations.

0.9265

presents average CCl bond lengths optimized with the previously chosen B3LYP, BLYP, and HFB functionals, as well as with the MP2 method, using different basis sets. Evidently, only minor changes in the average CCl bond lengths are observed for all the methods with various Pople’s basis sets. Somewhat larger changes are found when Dunning’s pVTZquality basis sets are used, in which case a shorter CCl bond (slightly better, i.e., approaching the MP2 value) is predicted. Figure 5 shows the correlation diagram between the average CCl bond length calculated using all 26 functionals considered in the present work and the CCl scaling factor. Three basis sets, the largest and the smallest Pople’s (6-311++G** and 6-31G*) and the largest Dunning’s basis set, were chosen. As can be seen, for both the SQM and ESFF approaches, the correlation diagram is nearly linear: the longer the CCl bond length (predicted by a given functional), the larger the value of the scaling factor. Note that SQM and ESFF diagrams cross at rCCl close to 1.73 Å, in which case the values of both factors are

0.9644 0.9629 0.9714 0.9780

bond lengths for four chlorine-containing molecules of a training set, as well as SQM and ESFF scaling factors attributed to the CCl stretch group, calculated at both B3LYP and MP2 levels. The remaining data can be found in the sheet statistics of the Supporting Information, in tables named CCl factors and bond lengths. As can be seen, CCl bond lengths optimized using the MP2 method are, on average, shorter by 0.01−0.02 Å than those found using the B3LYP functional. In addition, harmonic frequencies of the CCl stretching vibrations predicted from MP2 force fields were found to be higher than those obtained at the B3LYP level (as well as higher than 3876



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hand, MP2 force fields are obtained as a byproduct of other calculations and for this reason the calculated vibrational spectra may still be useful (if properly interpreted). In fact, we have the relevant data for calculating the MP2 scaling factors. Table 7 contains the SQM and ESFF rms values for scaled frequencies calculated using MP2 force fields, extracted from Table 7. Root-Mean-Square Deviations (in cm−1) between the Observed Fundamentals and SQM- and ESFF-Scaled Theoretical Frequencies Obtained Using MP2 Force Fields with a Variety of Basis Sets 11-parameter

9-parameter

basis set

SQM

ESFF

SQM

ESFF

6-311++G** 6-31++G** cc-pVTZ cc-pVDZ aug-cc-pVTZ aug-cc-pVDZ

29.04 31.00 18.89 21.75 21.80 22.26

27.82 29.08 18.19 20.96 21.01 21.52

28.16 29.88 17.76 21.17 20.63 21.95

27.13 28.24 17.56 20.82 20.29 21.44

the Supporting Information. As can be seen, the MP2 results obtained with the largest Pople’s basis sets are very inaccurate; they are even inferior to those obtained with the lowest-quality density functional in conjunction with the smallest basis set considered in the present work (the MP2 rms values are frequently close to 30 cm−1). These results were obtained in spite of the fact that the MP2 geometries are relatively accurate; ARPE between the MP2/6-311++G** and probably the most accurate MP2/aug-cc-pVTZ equilibrium geometric parameters (cf. section 4.6) is 0.339% (cf. sheet geom. mp2, Table 6-311+ +G**). This value is lower than that obtained with the highestquality density functionals, in which case ARPEs in geometry are greater than 0.4%. Similar observations were made before6the MP2 frequencies scaled with a single frequency factor turned out to be much worse than those calculated with any density functional. Even worse results were obtained with smaller Pople’s basis sets, and for this reason, the scaling factors are reported only for the largest, ...++G**-type basis sets. Nevertheless, the obtained MP2 scaled frequencies cannot be recommended for routine interpretation of vibrational spectra. As expected, significantly better MP2 results were obtained with Dunning’s basis sets. However, the obtained rms values relegate these calculations to the lowest-quality group. 4.9. Statistical Uncertainties of Scaling Factors. In the last section, we would like to comment upon the uncertainties of the reported scaling factors. In the case of the ESFF procedure, they can be easily estimated36 by calculating standard deviations (SD) of the coefficients of linear regression (SQM standard deviations should be similar). A statistical REGLINP routine of the Microsoft Excel program can be used for this purpose. Table 8 reports standard deviations of ESFF scaling factors obtained within the 11-parameter frame with the selected density functionals. As before, B3LYP, BLYP, and HFB in conjunction with the 6-311+G** basis set were chosen. In addition, the basis set effect on the uncertainties of factors obtained for a given functional (here: B3LYP) was checked by performing calculations at the B3LYP/6-31G* level. As can be seen, an increase in the statistical uncertainties is observed along with the deterioration of the quality of the force field: the uncertainties calculated for the HFB scaling factors are, on average, almost 2 times larger compared to B3LYP. This effect is probably due to a significantly larger spread in the calculated

Figure 5. Correlation diagrams between the SQM and ESFF scaling factor attributed to the CCl stretch group and the average CCl bond length calculated for four chlorine-containing molecules of Baker’s training set. Each point refers to one of 26 density functionals considered in the present work.

approximately equal to unity (this should be obvious in relation to ν0 ∼ f 1/2, which means that scaling factors s for localized group vibrations should satisfy s(SQM) ≈ (s(ESFF))2). 4.8. MP2 Scaling Factors. The present work deals with the evaluation of DFT scaling factors. DFT methods are well suited to calculations on large systemsthey are capable of predicting quite accurate geometries of covalently bonded systems, and force fields of reasonable quality, at rather low computational cost. Very accurate CC-type calculations are not yet practical for large systems, but calculations at the MP2 level are nowadays frequently used to determine various molecular properties of medium-size molecules. However, vibrational frequencies obtained with the aid of the MP2 method are not impressiveusing single parameter frequency scaling, the quality of results is poor, comparable to HF.6 On the other 3877



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to the recommended values. Thus, the presented results are fully consistent with those available in the literature. Second, the ranking of various functionals with respect to the quality of the predicted spectra was made. In particular, six functionalsB3LYP, B3PW91, B97, B97-1, B97-2, and O3LYPare shown to be well suited to the calculations oriented toward interpretation of vibrational spectra based on the scaled frequencies. They all provide the lowest rms values (typically lower than 12 cm−1) between the scaled theoretical and the observed fundamental frequencies, regardless of whether medium-quality basis sets are used. Reasonable spectra can be predicted even with small basis sets (say, 6-31G*) using these functionals, which makes them particularly attractive in the calculations on large systems. There is quite a large group of functionals which can be useful, though we do not recommend them when planning calculations of vibrational spectra (having better functionals at our disposal, they should not be used in the first instance). They are capable of giving satisfactory scaled frequencies (12 cm−1 < rms < 15 cm−1) for all but the smallest basis sets. However, there are functionals, in particular those lacking correlation terms (HFB, HFS, and OPTX), which cannot be recommended for frequency calculations at all. Third, various statistical considerations were carried out. In particular, the choice of basis set in conjunction with density functionals was considered. Pople’s medium-size basis sets turned out to be preferable with respect to the force field calculations for the subsequent use in multiparameter scaling. In fact, ...+G**-type basis sets in conjunction with a number of functionals usually give the lowest rms values, probably due to the fortuitous cancellation of various errors. Better results can be obtained using Dunning’s aug-cc-pVTZ basis set (aug-ccpVDZ basis sets do not perform well, indicating that f-type functions are important in the determination of the molecular force fields for these basis sets). However, the gain in the accuracy in the scaled frequencies is associated with a significant (by a factor of 15−20) increase in the computational time as compared with the 6-311+G** basis set. The cc-pVTZ basis set occasionally leads to slight improvement of the scaled frequencies as compared with 6-311+G**, but on average, it performs worse and is also associated with an increase of the CPU time (by a factor of 3−4). Moreover, correlation between the quality of scaled frequencies and the quality of the geometrical parameters predicted by various density functionals was found. It is also shown that reducing the size of the basis set (from 6 to 311+G** to 6-31G*) in DFT calculations has quite a significant effect on the quality of the calculated force fields; the effect on geometric parameters is not so decisive, at least with regard to the well-behaved molecules of Baker’s training set. In addition, MP2 force fields were shown to be inappropriate for the subsequent use in obtaining multiparameter scaled frequencies (as they are in the case of singleparameter harmonic frequency scaling). The rms values obtained with Pople’s basis sets are much higher than with the lowest-quality density functionals. Better results were obtained with Dunning’s basis sets (which is in accord with the general observation that the MP2 method has much higher requirements with respect to the basis set quality in comparison to DFT). Nevertheless, even the lowest-quality density functionals would be preferable with these basis sets. Finally, statistical considerations reveal that 9-parameter scaling is superior to 11-parameter scaling in the middle range of the vibrational spectrum when high-quality force fields are used. Moreover, the performance of the simple ESFF scaling

Table 8. Standard Deviations of the ESFF Scaling Factors Calculated within the 11-Parameter Frame at the Selected Levels of Theory scaling factor

B3LYP/6311+G**

B3LYP/631G*

BLYP/6311+G**

HFB/6311+G**

1. XX 2. CCl 3. CH 4. NH 5. OH 6. XXX 7. XXH 8. HCH 9. HNH 10. linear def. 11. torsions

0.0009 0.0074 0.0003 0.0010 0.0014 0.0038 0.0012 0.0018 0.0098 0.0140 0.0011

0.0011 0.0088 0.0004 0.0013 0.0017 0.0046 0.0014 0.0022 0.0114 0.0169 0.0013

0.0013 0.0100 0.0004 0.0014 0.0019 0.0053 0.0016 0.0025 0.0120 0.0192 0.0015

0.0018 0.0143 0.0006 0.0019 0.0026 0.0074 0.0021 0.0033 0.0156 0.0261 0.0020

harmonic frequencies for a given type of vibrational modes predicted by HFB. On the other hand, no significant increase of uncertainties of B3LYP factors is observed upon deterioration of the basis set quality. In passing, we note that the ratio of statistical uncertainties computed at different theoretical levels remains fairly constant for all factors, and is slightly higher than the ratio of the overall rms values for these levels. For example, when comparing the BLYP/6-311+G** and B3LYP/6311+G** levels, we obtain SDsBLYP/SDsB3LYP ≈ 1.4 for all factors s, which is very close to rmsBLYP/rmsB3LYP ≈ 1.32. Finally, we admit that by a mistake in data handling in one of our previous works36 the uncertainties reported for natural coordinates based scaling factors were overvalued by an order of magnitude. In fact, all SDs were accidentally scaled by the same factor (close to 12), so the conclusions based on their relative values (like those based on data included in Figure 1 of the above reference) are correct. The correct values, specified in the order of factors entering Table 2 of ref 36, are 0.0016, 0.0016, 0.0098, 0.0003, 0.0027, 0.0012, 0.0013, 0.0202, and 0.0013.

5. CONCLUSIONS The purpose of the present work has been to determine scaling factors for obtaining theoretical fundamental vibrational frequencies based on various DFT force fields within multiparameter scaling methods: scaled quantum mechanical (SQM) force field and effective scaling frequency factor (ESFF) methods. Thus, we extended their applicability for these methods, as well as supplemented the existing database of scaling factors for simple, single-parameter harmonic frequency scaling. We have achieved this goal by following the procedure based on Baker’s training set of 30 molecules, for which 660 vibrational modes were identified. The scaling factors were computed using 26 common density functionals and 14 various Pople’s and Dunning’s basis sets of VDZ- and VTZ-quality. In addition, the MP2 scaling factors are also reported for some of the basis sets considered in this work. Thus, an overall number of 370 theoretical levels was considered. We have made an extended set of tables with SQM and ESFF scaling factors for both 11- and 9-parameter scaling frames, and organized them in a Microsoft Excel file. First, it was shown that single-parameter harmonic frequency scaling factors obtained at the selected theoretical levels (HF, MP2, B3LYP, and PBE) using Baker’s training set are very close 3878



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(9) Krasnoshchekov, S. V.; Stepanov, N. F. Russ. J. Phys. Chem. A 2008, 82, 592−602; Translated from: Zh. Fiz. Khim. 2008, 82, 690− 701. (10) Botschwina, P. Chem. Phys. Lett. 1974, 29, 98−101. (11) Bleicher, W.; Botschwina, P. Mol. Phys. 1975, 30, 1029−1036. (12) Botschwina, P.; Meyer, W.; Semkow, A. M. Chem. Phys. 1976, 15, 25−34. (13) Blom, C. E.; Slingerland, P. J.; Altona, C. Mol. Phys. 1976, 31, 1359−1376. (14) Blom, C. E.; Altona, C. Mol. Phys. 1976, 31, 1377−1391. (15) Blom, C. E.; Otto, L. P.; Altona, C. Mol. Phys. 1976, 32, 1137− 1149. (16) Blom, C. E.; Altona, C. Mol. Phys. 1977, 33, 875−885. (17) Blom, C. E.; Altona, C. Mol. Phys. 1977, 34, 177−192. (18) Blom, C. E.; Altona, C.; Oskam, A. Mol. Phys. 1977, 34, 557− 571. (19) Pulay, P.; Fogarasi, G.; Pongor, G.; Boggs, J. E.; Vargha, A. J. Am. Chem. Soc. 1983, 105, 7037−7047. (20) Rauhut, G.; Pulay, P. J. Phys. Chem. 1995, 99, 3093−3100. (21) Baker, J.; Jarzecki, A. A.; Pulay, P. J. Phys. Chem. A 1998, 102, 1412−1424. (22) Kalincsák, F.; Pongor, G. Spectrochim. Acta, Part A 2002, 58, 999−1011. (23) Hedberg, L.; Mills, I. M. J. Mol. Spectrosc. 2000, 203, 82−95. (24) Sundius, T. Vib. Spectosc. 2002, 29, 89−95. (25) SQM version 1.0, Scaled Quantum Mechanical Force Field, 2013 Green Acres Road, Fayetteville, Arkansas 72703. (26) Pople, J. A.; Schlegel, H. B.; Krishnan, R.; DeFrees, D. J.; Binkley, J. S.; Frisch, M. J.; Whiteside, R. A.; Hout, R. F.; Hehre, W. J. Int. J. Quantum Chem. 1981, 20, 269−278. (27) Scott, A. P.; Radom, L. J. Phys. Chem. 1996, 100, 16502−16513. (28) Sinha, P.; Boesch, S. E.; Gu, C.; Wheeler, R. A.; Wilson, A. K. J. Phys. Chem. A 2004, 108, 9213−9217. (29) Yoshida, H.; Ehara, A.; Matsuura, H. Chem. Phys. Lett. 2000, 325, 477−483. (30) Yoshida, H.; Takeda, K.; Okamura, J.; Ehara, A.; Matsuura, H. J. Phys. Chem. A 2002, 106, 3580−3586. (31) Berezin, K. V.; Nechaev, V. V.; Krivokhizhina, T. V. Opt. Spectrosc. 2003, 94, 357−360; Translated from: Opt. Spektrosk. 2003, 94, 398−401. (32) Borowski, P.; Fernández-Gómez, M.; Fernández-Liencres, M. P.; Peña Ruiz, T. Chem. Phys. Lett. 2007, 446, 191−198. (33) Morino, Y.; Kuchitsu, K. J. Chem. Phys. 1952, 20, 1809−1810. (34) Borowski, P.; Drzewiecka, A.; Fernán dez-Góm ez, M.; Fernández-Liencres, M. P.; Peña Ruiz, T. Chem. Phys. Lett. 2008, 465, 290−294. (35) Borowski, P.; Fernández-Gómez, M.; Fernández-Liencres, M. P.; Peña Ruiz, T.; Quesada Rincón, M. J. Mol. Struct. 2009, 924−926, 493−503. (36) Borowski, P.; Drzewiecka, A.; Fernán dez-Góm ez, M.; Fernández-Liencres, M. P.; Peña Ruiz, T. Vib. Spectosc. 2010, 52, 16−21. (37) Borowski, P.; Pilorz, K.; Pitucha, M. Spectrochim. Acta, Part A 2010, 75, 1470−1475. (38) Borowski, P. J. Mol. Spectrosc. 2010, 264, 66−74. (39) Fábri, C.; Szidarovszky, T.; Magyarfalvi, G.; Tarczay, G. J. Phys. Chem. A 2011, 115, 4640−4649. (40) Baker, J.; Woliński, K.; Malagoli, M.; Kinghorn, D.; Woliński, P.; Magyarfalvi, G.; Saebo, S.; Janowski, T.; Pulay, P. J. Comput. Chem. 2009, 30, 317−335. (41) PQS version 3.3, Parallel Quantum Solutions, 2013 Green Acres Road, Fayetteville, Arkansas 72703. (42) Slater, J. C. Quantum Theory of Molecules and Solids; McGrawHill: New York, 1974; Vol. 4. (43) Vosko, S. H.; Wilk, L.; Nusair, M. Can. J. Phys. 1980, 58, 1200− 1211. (44) Ceperley, D. M.; Alder, B. J. Phys. Rev. Lett. 1980, 45, 566−569. (45) Becke, A. D. Phys. Rev. A 1988, 38, 3098−3100. (46) Perdew, J. P.; Zunger, A. Phys. Rev. B 1981, 23, 5048−5079.

procedure with respect to scaled frequencies seems to be somewhat higher as compared with SQM, in that somewhat lower rms values are, on average, found. On the other hand, the consistency of the final force fields and the scaled frequencies is an obvious advantage of the SQM force field scaling approach, in particular, if further analysis on the force constants matrix is desired. In addition, the SQM eigenmodes are also consistent with the improved force constants; thus, they may provide more reliable IR/Raman intensities when used in the transformation of the dipole moment/polarizability derivatives to normal coordinate representation.

■

ASSOCIATED CONTENT

S Supporting Information *

The Microsoft Excel file (located within jp212201f_si_002.zip) containing scaling factors for use with SQM and ESFF scaling procedures (sheets adopting the names of the basis set, and tables, the names of the density functional used in the calculations), as well as various statistical compilations (the remaining sheets) as described in detail in the jp212201f_si_001.pdf file. The pdf file containing tables only w i t h t h e sc a l in g f a c t o r s i s al s o a v a i l a b l e ( t h e jp212201f_si_003.pdf file; tables adopt the names of the density functional, at which the calculations were carried out; the header of each page contains information about the basis set). This material is available free of charge via the Internet at http://pubs.acs.org.

■

AUTHOR INFORMATION

Corresponding Author

*Phone: +48 81 537 56 14. Fax: +48 81 533 33 48. E-mail: [email protected]. Notes

The authors declare no competing financial interest.

■

ACKNOWLEDGMENTS The author would like to thank Dr. Karol Pilorz, his officemate, for valuable discussions, and his assistance in the calculations of statistical uncertainties. Moreover, Jakub Stefaniak and Alison Firth are kindly acknowledged for their help with the preparation of this manuscript.

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REFERENCES

(1) Wilson, E. B., Jr.; Decius, J. C.; Cross, P. C. Molecular Vibrations. The Theory of Infrared and Raman Vibrational Spectra; Dover Publications, Inc.: New York, 1955. (2) Califano, S. Vibrational States; John Wiley & Sons, Ltd.: London, 1976. (3) Hehre, W. J.; Radom, L.; Schleyer, P. v. R.; Pople, J. A. Ab initio Molecular Orbital Theory; Wiley: New York, 1986 and references therein. (4) Fogarasi, G.; Pulay, P. Ab initio Calculation of Force Fields and Vibrational Spectra. In Vibrational Spectra and Structure; Durig, J. R., Ed.; Elsevier: Amsterdam, The Netherlands, 1985; Vol. 14, pp 125− 219. (5) Helgaker, T.; Ruden, T. A.; Jørgensen, P.; Olsen, J.; Klopper, W. J. Phys. Org. Chem. 2004, 17, 913−933 and references therein. (6) Merrick, J. P.; Moran, D.; Radom, L. J. Phys. Chem. A 2007, 111, 11683−11700. (7) Boese, A. D.; Martin, J. M. L. J. Phys. Chem. A 2004, 108, 3085− 3096. (8) Boese, A. D.; Klopper, W.; Martin, J. M. L. Mol. Phys. 2005, 103, 863−876. 3879



Article

(47) Perdew, J. P. Phys. Rev. B 1986, 33, 8822−8824. (48) Perdew, J. P. Unified Theory of Exchange and Correlation Beyond the Local Density Approximation. In Electronic structure of solids; Ziesche, P., Eschrig, H., Eds.; Akademie Verlag: Berlin, 1991; pp 11−20. (49) Lee, C.; Yang, W.; Parr, R. G. Phys. Rev. B 1988, 37, 785−789. (50) Becke, A. D. J. Chem. Phys. 1993, 98, 5648−5652. (51) Wilson, P. J.; Amos, R. D.; Handy, N. C. Chem. Phys. Lett. 1999, 312, 475−484. (52) Handy, N. C.; Cohen, A. J. Mol. Phys. 2001, 99, 403−412. (53) Hoe, W-M; Cohen, A. J.; Handy, N. C. Chem. Phys. Lett. 2001, 341, 319−328. (54) Perdew, J. P.; Burke, K.; Ernzerhof, M. Phys. Rev. Lett. 1996, 77, 3865−3868. (55) Boese, A. D.; Handy, N. C. J. Chem. Phys. 2001, 114, 5497− 5503. (56) Becke, A. D. J. Chem. Phys. 1997, 107, 8554−8560. (57) Hamprecht, F. A.; Cohen, A. J.; Tozer, D. J.; Handy, N. C. J. Chem. Phys. 1998, 109, 6264−6271. (58) Wilson, P. J.; Bradley, T. J.; Tozer, D. J. J. Chem. Phys. 2001, 115, 9233−9242. (59) Hariharan, P. C.; Pople, J. A. Theor. Chim. Acta 1973, 28, 213− 222. (60) Krishnan, R.; Binkley, J. S.; Seeger, R.; Pople, J. A. J. Chem. Phys. 1980, 72, 650−654. (61) Frisch, M. J.; Pople, J. A.; Binkley, J. S. J. Chem. Phys. 1984, 80, 3265−3269. (62) Dunning, T. H. Jr. J. Chem. Phys. 1989, 90, 1007−1023. (63) Woon, D. E.; Dunning, T. H. Jr. J. Chem. Phys. 1993, 98, 1358− 1371. (64) Woon, D. E.; Dunning, T. H. Jr. J. Chem. Phys. 1995, 103, 4572−4585. (65) Borowski, P.; Peña Ruiz, T.; Barczak, M.; Pilorz, K.; PasiecznaPatkowska, S. Spectrochim. Acta, Part A 2012, 86, 571−585. (66) Irikura, K. K.; Johnson, R. D. III; Kacker, R. N. J. Phys. Chem. A 2005, 109, 8430−8437. (67) CRC Handbook of Chemistry and Physics: 1913−1995; Lide, D. R., Ed.; CRC Press, Inc.: Boca Raton, FL, 1995.

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