Harmonic Vibrational Frequencies: Approximate Global Scaling

Article pubs.acs.org/JPCA

Harmonic Vibrational Frequencies: Approximate Global Scaling Factors for TPSS, M06, and M11 Functional Families Using Several Common Basis Sets D. O. Kashinski,*,† G. M. Chase,† R. G. Nelson,† O. E. Di Nallo,† A. N. Scales,† D. L. VanderLey,† and E. F. C. Byrd*,‡ †

Department of Physics and Nuclear Engineering, US Military Academy, West Point, New York 10996, United States Weapons and Materials Research Directorate, Army Research Laboratories, Aberdeen Proving Grounds, Maryland 21005, United States

‡

ABSTRACT: We propose new approximate global multiplicative scaling factors for the DFT calculation of ground state harmonic vibrational frequencies using functionals from the TPSS, M06, and M11 functional families with standard correlation consistent cc-pVxZ and aug-cc-pVxZ (x = D, T, and Q), 6-311G split valence family, Sadlej and Sapporo polarized triple-ζ basis sets. Results for B3LYP, CAM-B3LYP, B3PW91, PBE, and PBE0 functionals with these basis sets are also reported. A total of 99 harmonic frequencies were calculated for 26 gas-phase organic and nonorganic molecules typically found in detonated solid propellant residue. Our proposed approximate multiplicative scaling factors are determined using a least-squares approach comparing the computed harmonic frequencies to experimental counterparts well established in the scientific literature. A comparison of our work to previously published global scaling factors is made to verify method reliability and the applicability of our molecular test set. database, as well as the Sapporo36−38 polarized triple-ζ (SPKTZP) and augmented-TZP (SPK-ATZP) basis sets along with the Sapporo 2012 updates38,39 (SPK-TZP-2012 and SPK-ATZP2012) are employed. Although much work regarding scaling factors has been published and tabulated using these common basis sets,5 to our knowledge, there is no present work in the literature addressing both the scaling factors and associated standard uncertainties for the M06, M11, and TPSS DFT functional families proposed in the current study. In an attempt to support the reliability of our proposed approximate scaling factors we compare a subset of our work to scaling factors and associated standard uncertainties already well established in literature.

1. INTRODUCTION Density functional theory’s (DFT)1 ability to accurately compute ground state observable properties has been remarkable given its ease of use and its ability to effectively balance accuracy with computational speed when compared to traditional highaccuracy ab inito methods. However, one area where DFT does not exhibit as much accuracy is in the calculation of harmonic vibrational frequencies. It is well established that calculated harmonic frequencies are typically an overestimation of their experimental counterparts because of an incomplete description of electron correlation, anharmonicity, and approximations to the Schrödinger equation.2−4 To address this overestimation, the calculation and tabulation of multiplicative linear scaling factors, initially proposed by Pople et al.,2 has become common in the literature as well as in practice. A host of scaling factors for various levels of theory and basis sets can be found on the National Institute of Standards and Technology (NIST) computational chemistry comparison and benchmark database (CCCBDB).5 This article proposes approximate optimized global multiplicative harmonic frequency scaling factors, denoted λ, and their associated absolute uncertainties, denoted δλ, for functionals in the TPSS,6−8 M06,9 and M1110 families as well as the B3LYP,11 CAM-B3LYP,12 B3PW91,11,13−16 PBE,17,18 and PBE017−19 methods. The standard 6-311G series,20−24 correlation-consistent series,25−28 and Sadlej polarized triple-ζ29−33 basis sets, obtained through the Environmental Molecular Sciences Laboratory Basis Set Exchange34,35 (EMSL-BSE) © 2017 American Chemical Society

2. MOLECULES OF INTEREST In this study we are proposing “approximate” scaling factors because the set of test molecules used to generate our results is smaller than the standard molecular sets typically used in ground electronic state harmonic frequency scaling factor studies.5,40−42 The test set, outlined in Table 1, consists of 8 diatomic and 18 polyatomic gas-phase organic and nonorganic molecules. These 26 molecules are a subset of the 122 species included in the F1 test set used by Pople et al.40 and others3,43 as well as the set of 125 species employed by Andersson and Uvdal44 and references Received: December 2, 2016 Revised: February 9, 2017 Published: February 9, 2017 2265

DOI: 10.1021/acs.jpca.6b12147 J. Phys. Chem. A 2017, 121, 2265−2273

Article

The Journal of Physical Chemistry A

Holder,50 establishing the first global harmonic scaling factors for the B3LYP functional using the correlation consistent basis sets. Rahut and Pulay51 employed 20 small species as a molecular training set to establish scaling factors in their study of vibrational force fields. Teixeira et al.47 suggests that harmonic frequency scaling factors are independent of the molecular test set size. All of the studies mentioned above compare the computed results to established experimental counterparts to verify the reliability of reported values. Like the studies above we also verify the applicability of the molecular set in Table 1 and confirm reliability of the approximate factors proposed below by computing scaling factors for functionals and basis sets already in the literature and benchmarking our proposed results to these well established values by direct comparison. This direct comparison of our results to values tabulated in the NIST-CCCDB5 as well as in several other sources is presented in Table 2. Three additional scaling factors were also used as a benchmark, two values computed with M06-2X/aug-cc-pVTZ and one computed with TPSSh/cc-pVTZ (0.956 ± 0.019,5 0.959,48 and 0.968 ± 0.0195 respectively). We report our scaling factors for the M06-2X/augcc-pVTZ and TPSSh/cc-pVTZ combinations in section 3 of this article.

Table 1. Eight Diatomic and 18 Polyatomic Molecules, Commonly Found in the Residue Immediately Following Detonation of Chemical Propellants,45,46 Used To Compute the Scaling Factors in the Current Studya H2 CN CO2 NO2 NH3 HNCO

O2 NH H2O HCO HNO2 CH4

N2 OH NH2 HNO HCHO CH3

CO

NO

HO2 HCN C2H2

CH2 NCO C2H4

a

All harmonic frequencies in the present study were computed for the ground electronic state of these molecules.

within. The 26 molecules used in this study is of interest to our team because these species are commonly found in the residue immediately following detonation of chemical propellants.45,46 Although our test set is smaller than the typical F1 set, its size is not unprecedented. A molecular test set of 82 diatomic molecules is employed by Staroverov et al.49 establishing deviations from experiment for DFT functionals, including TPSS, with the 6-311++G(3df,3pd) basis set. A study by Sinha et al.4 employed a test set of 41 species, derived from the work of Healy and

Table 2. Global Multiplicative Scaling Factors (λ) and Associated Absolute Uncertainties (δλ) Reported as Confidence Intervals (λ ± δλ) for the Indicated Functionals and Basis Sets That Are Used To Benchmark Our Calculationsa basis set

reference

B3LYP

aug-cc-pVDZ

NISTb footnote

0.970 ± 0.026 0.9698 (49)c 0.9747 ± 0.0311f 0.967 ± 0.023 (52) 0.968 ± 0.019 0.959g 0.9676 (37)c 0.965 ± 0.020 (45) 0.969 ± 0.021 0.9705 (34)c 0.965 ± 0.020 (44) 0.970 ± 0.025 0.9709 (45)c 0.9746 ± 0.0294f 0.970 ± 0.025 (57) 0.965 ± 0.070 0.9691 (35)c 0.965 ± 0.020 (44) 0.969 ± 0.021 0.9751 (36)c 0.965 ± 0.020 (44) 0.996 ± 0.023 0.9623 (40)d 0.964 ± 0.023 (52) 0.967 ± 0.021 0.9619 (33)d 0.9729 ± 0.0281f 0.964 ± 0.020 (46) 0.972 ± 0.020 0.9726 (42)e 0.970 ± 0.023 (52)

aug-cc-pVTZ

aug-cc-pVQZ

cc-pVDZ

cc-pVTZ

cc-pVQZ

6-311G(d)

6-311G(d,p)

Sadlej-pVTZ

this work NIST footnote this work NIST footnote this work NIST footnote this work NIST footnote this work NIST footnote this work NIST footnote this work NIST footnote this work NIST footnote this work

B3PW91

PBE

0.965 ± 0.025

0.994 ± 0.031

0.962 ± 0.024 (55) 0.965 ± 0.018

0.993 ± 0.027 (59) 0.992 ± 0.022

0.961 ± 0.021 (47)

0.991 ± 0.025 (49)

0.960 ± 0.021 (47) 0.965 ± 0.026

0.991 ± 0.022 (48) 0.994 ± 0.030

0.963 ± 0.026 (58) 0.962 ± 0.080

0.996 ± 0.031 (67) 0.993 ± 0.025

0.960 ± 0.021 (47)

0.991 ± 0.023 (50)

0.960 ± 0.021 (47) 0.963 ± 0.022

0.991 ± 0.022 (49) 0.990 ± 0.064

0.959 ± 0.024 (54) 0.963 ± 0.021

0.991 ± 0.028 (61) 0.991 ± 0.025

0.959 ± 0.022 (49)

0.990 ± 0.024 (52) 0.995 ± 0.025

0.9674 (43)e 0.963 ± 0.024 (55)

0.996 ± 0.026 (57)

RMS uncertainties, when available, are in standard spectroscopic units of cm−1, in parentheses. bNIST: The NIST Computational Chemistry Comparison and Benchmark Database (NIST-CCCBDB).5 cSee Table 1 in Sinha et al.4 dSee Table 1 in Andersson and Uvdal.44 eSee Table 2 in Halls et al.42 fSee Table 1 in Teixeira et al.47 gSee Web: “Database of Frequency Scale Factors for Electronic Model Chemistries V3 Beta 2” by Zheng et al.48 a

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conclude that use of the G-UF grid has little to no impact on the resulting harmonic frequencies for systems where the larger grid was accessible. We are assuming that this stability holds for the few situations where the G-96 grid does not converge and the G-UF grid must be employed. The basis sets employed in this study, except for the Sapporo sets obtained from Noro,39 were those tabulated on and downloaded from the EMSL-BSE34,35 database. The DFT functionals used are a subset of those included in the standard Gaussian09 E.01 release.52,57 The harmonic vibrational frequencies for each molecule in question were determined from the Hessian computed from the ground state stationary point as part of a geometry optimization calculation. Calculations were completed using the default point group symmetry, selected by Gaussian09, for the particular molecule in question. 3.2. Theoretical Methods Summary. As shown in many other works3−5,40,42−44,47,50,58 the global multiplicative scaling factors, denoted λ, for the harmonic frequencies can be calculated using a least-squares method. By minimizing the residuals, denoted Δ, between the experimental frequencies found in the literature and our calculated frequencies, we can statistically determine the scaling factor given theoretical method.

In an attempt to demonstrate reliability, we apply our resulting scaling factors and associated uncertainties to the formaldehyde (H2CO) and ethylene (C2H4) molecules comparing the experimentally determined ground state fundamental frequencies to the scaled and unscaled calculated ground state harmonic frequencies. Because our test set is smaller than many of those previously published, we expect the root-mean-square (RMS) and absolute uncertainties associated with our calculated scaling factors to be slightly larger than those reported by NIST. A detailed discussion of these comparisons is completed in section 3.

3. METHODS AND THEORY 3.1. Quantum Chemistry Details. All quantum chemistry calculations were completed using the Gaussian09, Revision E.01 suite.52 Calculations utilized one or two compute nodes (two 16-core Intel Xeon E5-2698v353 processors per node) housed in a Cray XC40 high-performance computer (HPC).54 Each geometry optimization and resulting harmonic frequency calculation used semidefault settings. With only a few exceptions, we used keyword Int(Grid = −96032), hereafter denoted G-96, to set the spherical numerical integration grid for all of our DFT calculations. The University of Minnesota Computational Chemistry Web site55 and the Gaussian09 reference Web site56 outlines that this setting uses 96 radial shells around each atom with a spherical product grid of 32 θ points and 64 ϕ points in each shell resulting in 196 608 integration points per atom. This keyword is often cited in benchmarking calculations56 and effectively eliminates small imaginary frequencies.55 In the handful of cases where the G-96 spherical numerical integration grid failed to converge, resulting in a “soft crash” of the Gaussian09 suite, we relaxed the grid criteria to Int(Grid = Ultrafine), hereafter denoted G-UF. The Ultrafine keyword employs a pruned grid with 99 radial shells and 590 angular points resulting in 58 410 integration points per atom.56 In Table 3 we

N

Δ=

C2H4

Using the same notation of the other works referenced above, ωcalc is the ith calculated vibrational frequency, and νexpt is the i i corresponding experimentally determined ith vibrational frequency. The parameter N denotes the total number of harmonic frequencies, in the current study N = 99. The harmonic frequencies are calculated and analyzed in the common spectroscopic units of wavenumbers (cm−1). Equation 1 is then modified to reveal the global multiplicative scaling factor, λ, for a particular theoretical method. λ=

G-96

sym

G-UF

G-96

B2u B2g B3u Au B3g Ag B1u Ag B1u Ag B3g B2u

834.39 979.10 984.97 1060.41 1245.51 1380.52 1479.22 1688.86 3125.83 3139.54 3194.36 3222.64

834.33 979.08 985.92 1060.40 1245.47 1380.51 1479.18 1688.87 3126.07 3139.77 3194.59 3222.87

B2 B1 B2 A1 A1 A1

2939.36 1198.16 1262.89 1530.17 1813.05 2884.65

2939.29 1198.11 1262.87 1530.17 1813.13 2884.60

N

⎡

N

⎤−1

i

⎢⎣

i

⎥⎦

∑ ωicalcνiexpt⎢∑ (ωicalc)2 ⎥

(2)

Like others cited above, to help quantify the level of uncertainty in our proposed scaling factors we compute the global rootmean-square (RMS) of the residuals for each method in question using the scaling factor computed in eq 2. For each vibrational mode in any given molecule we first compute an individual residual, from which the RMS is determined:

H2CO

G-UF

(1)

i

Table 3. Direct Comparison of Ground State Unscaled Harmonic Frequencies Computed at the B3LYP/aug-ccpVTZ Level of Theory, Using the G-96 and the G-UF Integration Grids As Indicated, for the Ethylene (C2H4) and Formaldehyde (H2CO) Moleculesa sym

∑ (λωicalc − νiexpt)2

Δmin = (λωicalc − νiexpt)2 N

(RMS) =

⎛ Δmin ⎞1/2 ⎟ N ⎠

∑ ⎜⎝ i

(3)

(4)

We used eqs 2 and 4 to compute the factors reported in section 3 of this article. The RMS uncertainty computed using eq 4 is used to estimate the absolute uncertainty in our proposed scaling factors as described below. 3.3. Absolute Uncertainty in Scaling Factors (δλ). All measurements, physical or virtual, have an associated uncertainty.59 However, it is common practice throughout literature to report computational chemistry results without any attempt to quantify and bound the uncertainty, resulting from approximations, in the reported values. In most of the studies cited in this manuscript the calculated scaling factors are reported with four figures of precision. Without quantifying the uncertainty in these virtual measurements a statement about the reported

a

Frequencies are reported with two decimal places highlighting the weak dependence on the two numerical methodologies. Frequencies reported in this table are in units of wavenumbers (cm−1).

present harmonic frequencies for the formaldehyde and ethylene molecules calculated at the B3LYP/aug-cc-pVTZ level of theory directly comparing results of the G-96 and G-UF integration grids. The results shown in Table 3 are typical, allowing us to 2267


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scaling factors and associated absolute uncertainty using a confidence interval of the form (λ ± δλ). As explained in Taylor’s59 book, the absolute uncertainty should be rounded to one significant figure (two significant figures if the leading digit is ⩽2) restricting the precision of the stated measured quantity to the decimal place of the absolute uncertainty.60 For example, we compute the M06/aug-cc-pVTZ scaling factor to be (0.963 ± 0.030) which would then be reported as 0.96 ± 0.03. In this article, however, we will always report an extra significant figure (two in total) for all of our proposed absolute uncertainties resulting in a scaling factor that has three significant figures. We adhere to this format to stay consistent with what is reported in Irikura et al.58 as well as what is reported in the NIST-CCCDB.5 Using the relationships presented by Irikura et al. in conjunction with our data analysis, we are able to approximate the absolute uncertainty in our proposed factors: ⎡ (λω calc − ν expt)2 ⎤1/2 ⎡ 1 i i ⎥ ⎢ δλ ≈ ∑ ⎢ ⎢ ⎥⎦ ⎢⎣ N N i ⎣ N

N

∑

⎤−1/2

(ωicalc)2 ⎥

i

⎥⎦ (5)

Using our relationship to calculate the RMS shown in eq 4, we can express eq 5 in a simplified notation: Figure 1. Number of calculated harmonic frequencies that fall into bins that are multiples of 10% difference from experiment. The larger plot is a zoomed in version of the inlet showing that 98% of all our calculations fall within −10% to +20% relative difference from experiment with 90% of the calculated frequencies falling within a ±10% relative difference from experiment. A total of 28 308 frequencies were calculated in this study.

δλ ≈

(RMS) 1 N

N

∑i (ωicalc)2

(6)

The absolute uncertainty (δλ) gives a tangible sense of maximum and minimum ranges of the scaling factor yielding a justifiable argument for the reported precision of the proposed scaling factors. The absolute uncertainty of the scaling factor has the same dimensions as the scaling factor itself (e.g., unitless). This quality makes the task of understanding the uncertainty less obtuse than if one only reports the RMS uncertainty in units of wavenumbers (cm−1).

4. RESULTS AND DISCUSSION 4.1. Analysis of Data. A detailed statistical analysis was completed on the calculated frequencies. As expected, modes associated with H−X bonds exhibit larger relative differences (5−10%) when compared to experimental values. The umbrella modes of CH3 and NH3 as well as the torsion modes of HNCO showed 3−8% relative uncertainties for most of the levels of theory addressed in this article. The vast majority of our calculations (90%) were within a ±10% relative difference from Table 4. Direct Comparison of Experimentally Determined Fundamental Frequencies to Their Scaled and Unscaled Counterparts Computed at the B3LYP/aug-cc-pVTZ Level of Theory in the Harmonic Approximation for the Formaldehyde Molecule (H2CO)a

Figure 2. Confidence intervals for several of our calculated scaling factors are compared to those reported in the NIST-CCCDB for the DFT functionals and basis sets indicated. The numerical values of these confidence intervals are detailed in Table 2. Intervals in solid black are those reported by NIST, and the dashed intervals are those presently computed.

factor’s accuracy, especially to four significant figures, cannot justifiably be made. The first study attempting to establish a rigorous method for reporting uncertainties in global harmonic scaling factors was by Irikura et al.58 We will not summarize the work of Irikura et al. in this article, but we will use their resulting methodology to estimate the uncertainty in our proposed harmonic scaling factors. Using the reporting standards outlined by Taylor,59 Irikura et al.,58 and the NIST-CCCDB,5 we will report our proposed

sym

exp62

unscaled

NIST-scaled

present study

B1 B2 A1 A1 A1 B2

1167 1249 1500 1746 2782 2843

1198 1263 1530 1813 2885 2939

1160 ± 23 1222 ± 24 1481 ± 29 1755 ± 34 2793 ± 55 2845 ± 56

1156 ± 24 1219 ± 25 1477 ± 30 1750 ± 36 2784 ± 57 2836 ± 58

a

Absolute uncertainties are computed using the method of quadrature.61 Frequencies reported in this table are in units of wavenumbers (cm−1). 2268


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cc-CpVTZ basis sets showed a weak performance when employed across all functionals. Direct comparison of the frequencies calculated with these levels of theory to their experimental counterparts resulted in a majority of the results having relative differences greater than ±10%. This poorer performance is reflected in the larger RMS uncertainties (and thus absolute uncertainties) in the scaling factors associated with these basis sets. 4.2. Scaling Factor Results. A subset of the NIST-CCCDB confidence intervals presented in Table 2 is compared on a number line in Figure 2 to associated confidence intervals computed by us, benchmarking the methods of this study. The confidence intervals reported in Table 2 and those compared in Figure 2 suggest that the methods used in this study are reliable and have the ability to yield reasonably accurate results within typical margins of uncertainty. To explicitly demonstrate the reliability of our calculated scaling factors, we consider case studies of formaldehyde and ethylene. Tables 4 and 5 show the direct comparison of experimental frequencies to the scaled and unscaled harmonic frequencies calculated at the B3LYP/aug-cc-pVTZ level of theory. No significant discrepancy is found when both sets of scaled harmonic frequencies are directly compared to each other, or to experiment, as the respective differences are within the estimated absolute uncertainties. However, as commented by Pernot and Cailliez65 on the works of Irikura et al.66 and Teixeira et al.,47 directly computing a specific harmonic frequency’s uncertainty by quadrature from a method-dependent scaling factor may be an inadequate way of approximating the uncertainty in a specific calculated frequency. The scaled frequencies in Tables 4 and 5 are shown in the present work, computed in the same fashion as those hosted on the NIST-CCCBDB,5 to explicitly demonstrate that the objective quantities of the present work (our proposed scaling factors), approach the reliability of those hosted on the NIST-CCCBDB. Like the works of Irikura et al.,58,66,67 we are

Table 5. Direct Comparison of Experimentally Determined Fundamental Frequencies to Their Scaled and Unscaled Counterparts Computed at the B3LYP/aug-cc-pVTZ Level of Theory in the Harmonic Approximation for the Ethylene Molecule (C2H4)a sym

exp63,64

unscaled

NIST-scaled

present study

B2u B2g B3u Au B3g Ag B1u Ag B1u Ag B3g B2u

826 942 949 1025 1220 1343 1444 1623 2989 3024 3084 3104

834 979 985 1060 1245 1381 1479 1689 3126 3140 3195 3223

808 ± 16 948 ± 19 953 ± 19 1026 ± 20 1206 ± 24 1336 ± 26 1432 ± 28 1635 ± 32 3026 ± 59 3039 ± 60 3092 ± 61 3120 ± 61

805 ± 17 945 ± 19 951 ± 20 1023 ± 21 1202 ± 25 1332 ± 27 1428 ± 29 1630 ± 33 3017 ± 62 3030 ± 62 3083 ± 63 3110 ± 64

a

Absolute uncertainties are computed using the method of quadrature.61 Frequencies reported in this table are in units of wavenumbers (cm−1).

their experimental counterparts. Figure 1 shows a bar-graph plot of the number of calculated frequencies that fall into bins that are multiples of 10% relative difference from experiment. The largest anomalies were CH3 M06HF/cc-pVDZ with a −556 cm−1 (−92%) difference from the experimental result of 606.5 cm−1, NH3 M06HF/6-311G with a −615 cm−1 (−65%) difference from the experimental result of 950.0 cm−1, and HNCO M06-2X/6-311G with a −321 cm−1 (−56%) difference from the experimental result of 577.4 cm−1. These three extremes were not included in the calculation of the scaling factors for those levels of theory. In general, the M06HF functional showed a weaker performance than all other functionals assessed. The 6-311G and

Table 6. Proposed Approximate Global Multiplicative Scaling Factors (λ) and Associated Absolute Uncertainties (δλ) Reported as Confidence Intervals (λ ± δλ) for the Indicated Basis Sets and Functionals within the TPSS Functional Familya

a

basis set

TPSS

TPSSh

RevTPSS

aug-cc-pVDZ aug-cc-pVTZ aug-cc-pVQZ aug-cc-CpVTZ aug-cc-CpVQZ cc-pVDZ cc-pVTZ cc-pVQZ cc-CpVTZ cc-CpVQZ Sadlej-TZP SPK-TZP SPK-ATZP SPK-TZP-2012 SPK-ATZP-2012 6-311G G(d,p) ++G(d,p) ++G(2d2p) ++G(2df2pd) ++G(3df3pd)

0.985 ± 0.025 (55) 0.982 ± 0.022 (48) 0.982 ± 0.021 (47) 0.982 ± 0.021 (48) 0.982 ± 0.021 (47) 0.988 ± 0.029 (65) 0.981 ± 0.022 (49) 0.982 ± 0.021 (47) 0.982 ± 0.035 (79) 0.982 ± 0.021 (47) 0.986 ± 0.023 (50) 0.981 ± 0.022 (48) 0.981 ± 0.021 (47) 0.982 ± 0.022 (48) 0.981 ± 0.021 (47) 0.997 ± 0.049 (108) 0.981 ± 0.023 (51) 0.981 ± 0.022 (49) 0.981 ± 0.022 (49) 0.981 ± 0.021 (48) 0.981 ± 0.021 (47)

0.971 ± 0.023 (51) 0.968 ± 0.020 (44) 0.968 ± 0.019 (43) 0.968 ± 0.020 (44) 0.967 ± 0.019 (43) 0.972 ± 0.026 (58) 0.968 ± 0.020 (45) 0.968 ± 0.019 (44) 0.969 ± 0.034 (76) 0.968 ± 0.019 (44) 0.971 ± 0.021 (47) 0.968 ± 0.020 (44) 0.968 ± 0.020 (44) 0.968 ± 0.020 (45) 0.968 ± 0.020 (44) 0.982 ± 0.046 (101) 0.967 ± 0.021 (46) 0.967 ± 0.020 (45) 0.967 ± 0.020 (45) 0.967 ± 0.020 (44) 0.967 ± 0.019 (44)

0.987 ± 0.025 (55) 0.983 ± 0.022 (49) 0.983 ± 0.022 (48) 0.984 ± 0.022 (49) 0.983 ± 0.021 (48) 0.989 ± 0.030 (67) 0.984 ± 0.023 (52) 0.983 ± 0.022 (49) 0.984 ± 0.036 (80) 0.983 ± 0.022 (48) 0.987 ± 0.022 (50) 0.983 ± 0.023 (50) 0.983 ± 0.022 (48) 0.983 ± 0.023 (50) 0.983 ± 0.022 (48) 0.999 ± 0.048 (105) 0.983 ± 0.024 (52) 0.982 ± 0.022 (49) 0.982 ± 0.023 (50) 0.982 ± 0.021 (49) 0.982 ± 0.021 (47)

Our RMS uncertainty, in standard spectroscopic units of cm−1, is given in Parentheses. 2269


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Table 7. Proposed Approximate Global Multiplicative Scaling Factors (λ) and Associated Absolute Uncertainties (δλ) Reported as Confidence Intervals (λ ± δλ) for the Indicated Basis Sets and Functionals within the M06 Functional Family and the M11 Functionala

a

basis set

M06

M06-2x

M06-HF

M06-L

M11


0.962 ± 0.031 (69) 0.963 ± 0.030 (69) 0.957 ± 0.028 (65) 0.963 ± 0.030 (70) 0.957 ± 0.028 (65) 0.962 ± 0.033 (75) 0.960 ± 0.031 (71) 0.957 ± 0.028 (64) 0.963 ± 0.042 (94) 0.957 ± 0.028 (64) 0.968 ± 0.034 (76) 0.961 ± 0.032 (73) 0.961 ± 0.032 (72) 0.961 ± 0.032 (74) 0.962 ± 0.032 (73) 0.970 ± 0.049 (109) 0.960 ± 0.032 (72) 0.961 ± 0.031 (71) 0.961 ± 0.029 (66) 0.959 ± 0.031 (70) 0.961 ± 0.032 (71)

0.950 ± 0.029 (65) 0.949 ± 0.027 (61) 0.948 ± 0.026 (60) 0.949 ± 0.027 (61) 0.948 ± 0.026 (60) 0.951 ± 0.032 (72) 0.948 ± 0.027 (62) 0.948 ± 0.026 (61) 0.949 ± 0.038 (87) 0.948 ± 0.025 (61) 0.952 ± 0.030 (68) 0.948 ± 0.027 (61) 0.948 ± 0.027 (61) 0.948 ± 0.027 (61) 0.948 ± 0.027 (61) 0.959 ± 0.044 (101) 0.946 ± 0.028 (65) 0.947 ± 0.028 (63) 0.947 ± 0.026 (61) 0.947 ± 0.027 (62) 0.947 ± 0.027 (63)

0.937 ± 0.035 (81) 0.932 ± 0.035 (82) 0.931 ± 0.036 (84) 0.931 ± 0.034 (80) 0.931 ± 0.036 (84) 0.938 ± 0.036 (84) 0.931 ± 0.035 (82) 0.931 ± 0.035 (83) 0.931 ± 0.043 (101) 0.931 ± 0.034 (79) 0.934 ± 0.036 (85) 0.930 ± 0.035 (83) 0.930 ± 0.035 (80) 0.930 ± 0.035 (82) 0.930 ± 0.034 (79) 0.950 ± 0.046 (107) 0.931 ± 0.034 (79) 0.930 ± 0.034 (79) 0.929 ± 0.034 (79) 0.930 ± 0.036 (83) 0.927 ± 0.034 (79)

0.963 ± 0.031 (71) 0.965 ± 0.029 (65) 0.959 ± 0.028 (64) 0.966 ± 0.029 (65) 0.959 ± 0.028 (63) 0.964 ± 0.036 (84) 0.963 ± 0.030 (67) 0.959 ± 0.028 (63) 0.965 ± 0.040 (91) 0.959 ± 0.028 (63) 0.968 ± 0.032 (71) 0.963 ± 0.030 (68) 0.964 ± 0.032 (71) 0.964 ± 0.031 (68) 0.965 ± 0.031 (69) 0.975 ± 0.046 (102) 0.963 ± 0.031 (70) 0.963 ± 0.030 (67) 0.963 ± 0.029 (64) 0.963 ± 0.028 (65) 0.965 ± 0.029 (66)

0.961 ± 0.035 (79) 0.954 ± 0.030 (68) 0.956 ± 0.031 (70) 0.955 ± 0.030 (68) 0.956 ± 0.031 (70) 0.961 ± 0.035 (79) 0.953 ± 0.030 (68) 0.955 ± 0.030 (69) 0.954 ± 0.040 (91) 0.955 ± 0.030 (69) 0.960 ± 0.034 (77) 0.954 ± 0.030 (70) 0.954 ± 0.030 (70) 0.953 ± 0.030 (69) 0.954 ± 0.030 (70) 0.970 ± 0.057 (127) 0.958 ± 0.032 (72) 0.957 ± 0.032 (73) 0.954 ± 0.029 (67) 0.953 ± 0.030 (69) 0.954 ± 0.030 (69)

Our RMS uncertainty, in standard spectroscopic units of cm−1, is given in parentheses.

Table 8. Proposed Approximate Global Multiplicative Scaling Factors (λ) and Associated Absolute Uncertainties (δλ) Reported as Confidence Intervals (λ ± δλ) for the Indicated Basis Sets and Functionalsa basis set

B3LYP

B3PW91

CAM-B3LYP

PBE

PBE0


0.967 ± 0.023 (52) 0.965 ± 0.020 (45) 0.965 ± 0.020 (44) 0.966 ± 0.020 (44) 0.965 ± 0.019 (44) 0.970 ± 0.025 (57) 0.965 ± 0.020 (44) 0.965 ± 0.019 (44) 0.966 ± 0.034 (76) 0.964 ± 0.019 (44) 0.970 ± 0.023 (52) 0.965 ± 0.020 (44) 0.964 ± 0.020 (44) 0.964 ± 0.020 (44) 0.964 ± 0.020 (44) 0.978 ± 0.045 (101) 0.964 ± 0.020 (46) 0.963 ± 0.020 (45) 0.963 ± 0.019 (44) 0.963 ± 0.020 (44) 0.964 ± 0.020 (44)

0.962 ± 0.024 (55) 0.961 ± 0.021 (47) 0.960 ± 0.021 (47) 0.961 ± 0.021 (47) 0.960 ± 0.021 (47) 0.963 ± 0.026 (58) 0.960 ± 0.021 (47) 0.950 ± 0.021 (47) 0.961 ± 0.034 (78) 0.960 ± 0.021 (47) 0.963 ± 0.024 (55) 0.960 ± 0.021 (47) 0.960 ± 0.021 (47) 0.960 ± 0.021 (48) 0.960 ± 0.021 (47) 0.972 ± 0.044 (99) 0.959 ± 0.022 (49) 0.959 ± 0.022 (49) 0.958 ± 0.021 (47) 0.958 ± 0.021 (48) 0.959 ± 0.021 (48)

0.956 ± 0.025 (58) 0.954 ± 0.023 (51) 0.954 ± 0.023 (52) 0.955 ± 0.022 (51) 0.954 ± 0.023 (51) 0.958 ± 0.027 (62) 0.954 ± 0.022 (51) 0.954 ± 0.023 (51) 0.955 ± 0.035 (80) 0.953 ± 0.022 (51) 0.959 ± 0.026 (60) 0.954 ± 0.022 (51) 0.954 ± 0.023 (52) 0.954 ± 0.022 (52) 0.954 ± 0.023 (52) 0.965 ± 0.046 (104) 0.953 ± 0.023 (52) 0.953 ± 0.023 (53) 0.952 ± 0.022 (51) 0.952 ± 0.023 (52) 0.953 ± 0.023 (52)

0.993 ± 0.027 (59) 0.992 ± 0.022 (49) 0.991 ± 0.022 (48) 0.991 ± 0.022 (49) 0.991 ± 0.022 (48) 0.996 ± 0.031 (67) 0.991 ± 0.023 (50) 0.991 ± 0.022 (49) 0.992 ± 0.036 (79) 0.990 ± 0.022 (49) 0.996 ± 0.026 (58) 0.991 ± 0.022 (49) 0.990 ± 0.022 (49) 0.991 ± 0.022 (49) 0.991 ± 0.022 (48) 1.005 ± 0.050 (107) 0.990 ± 0.024 (52) 0.990 ± 0.023 (51) 0.990 ± 0.022 (49) 0.989 ± 0.022 (49) 0.989 ± 0.022 (49)

0.956 ± 0.026 (58) 0.955 ± 0.022 (51) 0.955 ± 0.022 (51) 0.956 ± 0.022 (51) 0.955 ± 0.022 (51) 0.956 ± 0.027 (61) 0.954 ± 0.022 (51) 0.955 ± 0.022 (58) 0.955 ± 0.035 (80) 0.954 ± 0.022 (58) 0.957 ± 0.026 (59) 0.954 ± 0.022 (51) 0.954 ± 0.022 (51) 0.955 ± 0.023 (51) 0.955 ± 0.022 (51) 0.965 ± 0.043 (98) 0.953 ± 0.023 (53) 0.953 ± 0.023 (53) 0.953 ± 0.022 (51) 0.953 ± 0.023 (52) 0.953 ± 0.023 (52)

a

If the functional/basis set combination is established in the literature, we compared our result to the literature values in Table 2. Our RMS uncertainty, in standard spectroscopic units of cm−1, is given in parentheses

computed by eq 4, for each scale factor are presented in parentheses. Our results suggest that the scaling factors have a stronger dependence on the DFT functional choice than on the choice of basis set. This weak dependence on the basis set choice was also suggested by Friese et al.68 for correlation consistent basis sets used in conjunction with methods other than DFT. The small discrepancies observed in our proposed scaling factors,

not attempting to compute scaled harmonic frequencieswe are proposing approximate harmonic frequency scaling factors. Tables 6−8 report our proposed approximate multiplicative global harmonic frequency scaling factors computed by eq 2 and the associated absolute uncertainties computed by eq 6 in the form of a confidence interval for the DFT functional methods and basis sets indicated in the tables. The RMS uncertainties, 2270


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when various basis set options are compared for a specific functional are essentially insignificant when one considers the uncertainty associated with these scaling factors. Like others47,68 we conclude that our proposed scaling factors have an accuracy of two significant figures (two decimal places) and, for the current study, are independent of the triple-ζ basis set chosen (with the exception of 6-311G and cc-CpVTZ which performed poorly across all functionals). Because of the insignificant discrepancies seen in the third (and higher) decimal places of our calculated factors when using the cc-pVTZ, 6-311G(d,p), and SPK-TZP (as well as the SPK-TZP-2012 update36−38) basis sets, we can conclude that these sets perform as well as, and are computationally cheaper to employ than, their larger counterparts.

5. SUMMARY In this article we have summarized the methods commonly used to compute global harmonic frequency scaling factors, outlined the method for estimating the absolute uncertainty in the calculated scaling factors, and benchmarked a subset of our results to their counterparts well established in the literature. We conclude that our scaling factors compare very favorably with the established values. Finally, we propose approximate global harmonic scaling factors, using common basis sets, for a set of functionals in the TPSS, M06, and M11 DFT functional families not currently reported in the literature. We also report our results for the B3LYP, CAM-B3LYP, B3PW91, PBE, and PBE0 functionals already present in the literature. The current work also suggests that our proposed scaling factors have a very weak dependence on the basis sets chosen in this study and that significantly increasing the size of the basis sets does not necessarily lead to improved results.

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AUTHOR INFORMATION

Corresponding Authors

*D. O. Kashinski. E-mail: [email protected]. *E. F. C. Byrd. E-mail: [email protected]. ORCID

D. O. Kashinski: 0000-0001-5098-5547 Notes

The authors declare no competing financial interest.

■

ACKNOWLEDGMENTS The quantum chemistry calculations were completed on hardware located at the ARL DoD Supercomputing Resource Center (DSRC) at Aberdeen Proving Grounds (APG) MD. Computing time and other support was provided by DoD High Performance Modernization Program (DoD-HPCMP). D.O.K., G.M.C., R.G.N., A.N.S., and D.L.V. acknowledge support from USMA. E.F.C.B. acknowledges support from ARL-WMRD. D.O.K and E.F.C.B thank DoD-HPCMP for resources and other support. D.O.K, G.M.C, and O.E.D, thank DoD-HPCMP for travel support. D.O.K acknowledges the USMA Faculty Research Fund (FRF) for hardware support. The views expressed herein are those of the authors and do not purport to reflect the position of the United States Military Academy, Army Research Laboratory, Department of the Army, or Department of Defense.

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REFERENCES

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