Article pubs.acs.org/JCTC
Frequency Scale Factors for Some Double-Hybrid Density Functional Theory Procedures: Accurate Thermochemical Components for HighLevel Composite Protocols Bun Chan*,†,‡ and Leo Radom*,† †
School of Chemistry and ARC Centre of Excellence for Free Radical Chemistry and Biotechnology, University of Sydney, Sydney, New South Wales 2006, Australia S Supporting Information *
ABSTRACT: In the present study, we have obtained geometries and frequency scale factors for a number of double-hybrid density functional theory (DH-DFT) procedures. We have evaluated their performance for obtaining thermochemical quantities [zero-point vibrational energies (ZPVE) and thermal corrections for 298 K enthalpies (ΔH298) and 298 K entropies (S298)] to be used within high-level composite protocols (using the W2X procedure as a probe). We find that, in comparison with the previously prescribed protocol for optimization and frequency calculations (B3-LYP/cc-pVTZ+d), the use of contemporary DH-DFT methods such as DuT-D3 and DSD-type procedures leads to a slight overall improved performance compared with B3-LYP. A major strength of this approach, however, lies in the better robustness of the DH-DFT methods in that the largest deviations are notably smaller than those for B3-LYP. In general, the specific choices of the DH-DFT procedure and the associated basis set do not drastically change the performance. Nonetheless, we find that the DSD-PBE-P86/aug′-cc-pVTZ+d combination has a very slight edge over the others that we have examined, and we recommend its general use for geometry optimization and vibrational frequency calculations, in particular within high-level composite methods such as the higher-level members of the WnX series of protocols. The scale factors determined for DSD-PBE-P86/aug′-cc-pVTZ+d are 0.9830 (ZPVE), 0.9876 (ΔH298), and 0.9923 (S298).
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INTRODUCTION Advances in computational quantum chemistry have enabled highly accurate first-principles thermochemistry predictions. Notably, the development of the Wn series of composite protocols by Martin and co-workers has provided a set of tools with accuracies ranging from ∼3 kJ mol−1 for the W1 method1 to ∼1 kJ mol−1 for W4.2 Building on the foundations of the Wn procedures, we have in recent years devised the WnX set of composite methods3−5 that are comparable in accuracy to their Wn counterparts but with a considerably reduced demand on computational resources. In our investigations within the WnX series, the highest-level WnX-type procedure that we have developed is W3X-L.5 In that study, we quantified the uncertainties associated with the various components of W3X-L for the G2/97 set6,7 of thermochemical properties. Thus, of the total uncertainty of 1.5 kJ mol−1, about half is determined to be associated with the vibrationless electronic energies provided by the single-point energy components within W3X-L. The remaining half can be attributed to the zero-point vibrational energies (ZPVEs) and thermal corrections for enthalpies (ΔH298) calculated with the B3-LYP/cc-pVTZ+d density functional theory (DFT) procedure. Such an analysis highlights that the electronic structure component of W3X-L has reached a high level of accuracy. With W3X-L using the computationally demanding CCSDT(Q) procedure as its highest-level component, it would seem © XXXX American Chemical Society
that, within the traditional framework of Wn-type composite methods, further significant improvement in this component can only be achieved at a very high computational cost. To this end, a more cost-effective approach to an improved performance would be to attempt to reduce the uncertainties associated with the ZPVEs and ΔH298. Their calculation with B3-LYP is the least demanding component of the complete W3X-L protocol, and there is thus much room for improvement in this part, with potentially minimal impact on the total computational cost. Indeed, in a recent study,8 Martin and co-workers have examined a range of computational chemistry methods for their performance in the calculation of ZPVEs and vibrational frequencies. Their results, together with earlier developmental and assessment studies on double-hybrid (DH) DFT procedures,9−12 suggest that DH-DFT may provide a convenient means for obtaining an improved performance over conventional DFT methods for geometry optimization and the calculation of ZPVEs. In a complementary manner to these studies, the main objective of the present investigation is to determine the effect of using improved geometries and vibrational frequencies on the total energies obtained with a high-level composite procedure. We hope that our results will shed light on the potential limit of the accuracy that may be Received: May 27, 2016
A
DOI: 10.1021/acs.jctc.6b00554 J. Chem. Theory Comput. XXXX, XXX, XXX−XXX
Article
Journal of Chemical Theory and Computation
Table 1. Frequency Scale Factors Determined for B3-LYP/VTZ+d and the Various Double-Hybrid DFT Procedures for Fundamentals (freq), Low Frequencies (low freq, ≤ 1000 cm−1), Zero-Point Vibrational Energies (ZPVE), and Thermal Corrections for 298 K Enthalpies (ΔH298) and Entropies (S298)
a
method
basis seta
freq
low freq
ZPVE
ΔH298
S298
B3-LYP B2-PLYP-D3 DuT-D3 DuT-D3 DuT-D3 DuT-D3 DSD-PBE-P86 DSD-B-P86 DSD-PBE-PBE
VTZ+d VTZ VTZ TZVP mAVTZ A′VTZ+d A′VTZ+d A′VTZ+d A′VTZ+d
0.9681 0.9610 0.9532 0.9541 0.9539 0.9546 0.9605 0.9592 0.9609
1.0028 0.9986 0.9871 0.9838 0.9899 0.9897 0.9954 0.9944 0.9979
0.9886 0.9830 0.9756 0.9766 0.9765 0.9769 0.9830 0.9817 0.9834
0.9926 0.9924 0.9813 0.9781 0.9831 0.9828 0.9876 0.9864 0.9899
0.9970 0.9965 0.9855 0.9815 0.9874 0.9871 0.9923 0.9912 0.9948
VTZ+d = cc-pVTZ+d, VTZ = cc-pVTZ, TZVP = def2-TZVP, mAVTZ = maug-cc-pVTZ, and A′VTZ+d = aug′-cc-pVTZ+d.
constant. Vibrational enthalpy is calculated with the equation
achieved cost-effectively in this component of a high-level composite procedure.
(
ΔHT(vib) = R ∑ Θ
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hν kB
+
1 eΘ /T − 1
)
with R being the gas
constant and T being the temperature. Vibrational entropy is
COMPUTATIONAL DETAILS Standard ab initio molecular orbital theory and density functional theory (DFT) calculations were carried out with Gaussian 0913 and Molpro 2012.14 For geometry optimizations and the computation of harmonic vibrational frequencies, we have examined the use of a number of DH-DFT procedures as well as B3-LYP/cc-pVTZ+d, which has been previously prescribed for the WnX-type protocols.3−5 We focus on DHDFT procedures because their potential for obtaining reliable vibrational frequencies has already been demonstrated,8,15 and we deem this class of methods to be cost-effective as a component in high-accuracy composite procedures. The DH-DFT procedures that we have examined include B2PLYP-D3,16 DuT-D3,17 DSD-PBE-P86,18 DSD-B-P86,18 and DSD-PBE-PBE.18 For DuT-D3, the zero-damping variant19 of the dispersion correction was employed according to ref 13, whereas the D3(BJ) version20 was used for B2-PLYP-D321 and the three DSD-type procedures.18 The B2-PLYP procedure is arguably the earliest example within the class of DH-DFT methods that is widely used today, whereas the DSD-type methods represent some of the latest DH-DFT methods. Due to the inclusion of an MP2 component in DH-DFT, together with the more demanding requirement on the basis set for MP2 when compared with DFT, it is often recommended that DHDFT computations should be conducted in conjunction with a quadruple-ζ basis set.21 In order to avoid the costs associated with such a basis set, the DuT-D3 method has been designed to be used with a smaller triple-ζ basis set. We believe that a good performance when combined with a triple-ζ basis set is a desirable feature in terms of computational efficiency. In the present study, we have investigated the use of a number of triple-ζ basis sets including cc-pVTZ+d (for B3-LYP),22 ccpVTZ,23 def2-TZVP,24 maug-cc-pVTZ,25 and aug′-cc-pVTZ +d.22 These basis sets will be abbreviated, respectively, as VTZ +d, VTZ, TZVP, mAVTZ, and A′VTZ+d. To obtain theoretical zero-point vibrational energies (ZPVEs) and thermal corrections for enthalpies at 298 K (ΔH298), we used standard thermochemical formulas together with scaled vibrational frequencies.26 Specifically, ZPVEs are Θ obtained by ZPVE = ∑ 2 where Θ is the vibrational temperature as defined by Θ =
1 2
obtained by ST(vib) = R ∑
(
Θ/T eΘ /T − 1
)
− ln(1 − e−Θ / T ) .
Scale factors for ZPVE, ΔH298, and S298 were determined using the methodology of Merrick et al.27 Thus, the root-meansquare deviation from benchmark values for contributions from each scaled vibrational mode is minimized. Single-point energy calculations were carried out at the W2X level5 to determine the effect on the energies of the various geometries. In order to examine how the deviations in energies scale with the size of the system, in addition to the data sets employed in ref 21, we have compiled a new test set (LF10) that consists of ten larger molecules. Experimental benchmark data for this set were obtained from the NIST CCCBDB.28 Energies in the text are given in kJ mol−1 and entropies in J mol−1 K−1.
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RESULTS AND DISCUSSION Overall Performance of B3-LYP/VTZ+d. In the present study, we have obtained frequency scale factors for a total of nine theoretical protocols, and the values are shown in Table 1. The scale factors are determined using the Z2 set of 48 ZPVEs and, for properties other than ZPVE, the F1 set of 1066 frequencies from 122 molecules.21 We have determined scale factors suitable for standard fundamentals as well as scale factors suitable for low frequency vibrations, with low frequency defined as
