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How Large is the Elephant in the Density Functional Theory Room? Frank Jensen* Department of Chemistry, Aarhus University, Langelandsgade 140, DK-8000 Aarhus, Denmark S Supporting Information *
ABSTRACT: A recent paper reported highly accurate density functional theory results for atomization energies and dipole moments using a multiwavelet-based method and compared the results with those obtained by standard Gaussian basis sets of the aug-cc-pVXZ type. Typical errors with the large aug-cc-pV5Z basis set were in the 0.2 kcal/mol range with outliers displaying errors of ∼2 kcal/mol, and these results could be taken as an indication that Gaussian basis sets in general are unsuitable for achieving high accuracy. We show that by choosing Gaussian basis sets optimized for density functional theory, basis set methods are capable of achieving accuracy comparable to that from the multiwavelet approach.
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INTRODUCTION In a recent paper, Jensen et al. reported multiwavelet results using the MRChem program for density functional theory (DFT) total and atomization energies (AE) as well as dipole moments for a benchmark set of data with 211 molecules.1 Their method is capable of achieving micro-Hartree accuracy which allows a calibration of results obtained by numerical and Gaussian-type orbital methods. The results using the aug-ccpVXZ family of basis sets were somewhat discouraging, as maximum atomization errors with the aug-cc-pVQZ and augcc-pV5Z basis sets were found to be ∼10 and ∼2 kcal/mol, respectively, indicating that even these large basis sets are incapable of reducing the basis set error below the commonly quoted chemical accuracy threshold of 1 kcal/mol. Because the aug-cc-pVXZ basis sets in general are considered as being of very high quality, this has implications for DFT calculations using Gaussian basis sets in general. The purpose of the present work is to illustrate that by using Gaussian basis sets optimized for DFT methods instead of basis sets optimized for correlated wave function methods, it is possible to achieve accuracies of ∼0.01 kcal/mol for atomization energies.
error, the corresponding uncontracted pc-n basis sets, denoted upc-n, can be employed. The cc-pVXZ and pcseg-n basis sets differ in three aspects:8−10 (1) exponents for the primitive basis functions (the cc-pVXZ employs a combinations of HF and CISD optimization, while the upc-n and pcseg-n are DFT optimized); (2) composition of the basis sets in terms of basis functions with different angular momentum (s-, p-, d-, etc.), where cc-pVXZ is based on CISD energy analyses, while pc-n is based on DFT energy analyses; and (3) contraction (cc-pVXZ is general contracted using HF orbital coefficients, while pcsegn is segmented contracted based on an orthogonalization procedure7 of the general contracted pc-n using DFT orbital coefficients). A more detailed comparison of basis sets can be found in refs 8−10. Jensen et al. also considered the pc-2 and pc-3 basis sets and noted that they performed better than similar sized basis sets but did not include results at the pc-4 level or from augmented versions of these basis sets. Figure 1 shows the absolute energy error for the spherical atoms H, He, Li, Be, N, Ne, Na, Mg, P, and Ar with the LDASVWN5 functional, where accurate reference energies are available.11 There is for atoms no need to employ basis sets augmented with diffuse functions, and the cc-pVXZ, pcseg-n, and upc-n basis sets have been used in their standard forms. The pc-2,3,4 results were extrapolated by a square-root exponential functional form to provide the results shown in Figure 1.12 The pcseg-xpol results have lower and more consistent errors compared to those in the cc-pV5Z basis set, and the upc-xpol results show that the large majority of the error is due to the basis set contraction. Basis set contraction improves the computational efficiency, and because the
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RESULTS AND DISCUSSION Jensen et al. used the aug-cc-pVXZ basis sets2 for probing the performance of Gaussian-type basis sets for the PBE and PBE0 functionals.3,4 These basis sets, however, have been developed for correlated wave function methods and are not optimum for DFT methods, displaying a relatively slow basis set convergence behavior.5 The pc-n basis sets,6 on the other hand, have been designed explicitly for DFT methods, and the most recent versions, denoted pcseg-n, have been defined for all atoms up to Kr and up to quintuple ζ-quality.7 These basis sets employ a segmented contraction scheme to improve the computational efficiency, but for assessing the contraction © 2017 American Chemical Society
Received: May 18, 2017 Revised: July 19, 2017 Published: July 19, 2017 6104
DOI: 10.1021/acs.jpca.7b04760 J. Phys. Chem. A 2017, 121, 6104−6107
Article
The Journal of Physical Chemistry A
step up in basis set quality. The difference between the pcseg-4 and upc-4 results shows that the contraction error dominates the performance at the quintuple ζ-level. Employing a squareroot exponential extrapolation12 of the upc-2,3,4 results provides the results denoted with Xpol, where the MAD relative to the MRChem results is ∼0.1 milli-Hartree. There is very little difference in the performance for the two employed functionals, which suggests that these levels of accuracies should be valid for DFT methods in general. The availability of the MRChem results allows a direct evaluation of the accuracy of the basis set results, where previously the basis set limiting energies have been estimated from the observed convergence behavior of the upc-n results with increasing n.13 The total energies are only with the upc-4 basis set of chemical accuracy, but relative energies such as atomization energies benefit from substantial error cancellation because core orbitals to a large extent are independent of the molecular environment. Table 2 shows the MAD and MaxAD for atomization energies. At the double-ζ level (cc-pVDZ, pcseg-1), there is little difference in the performance of the different basis sets. At the triple-ζ level, the pcseg-2 basis set performs better than cc-pVTZ, and this difference accelerates at the quadruple-ζ level (cc-pVQZ and pcseg-3), especially for the MaxAD values (7.7 and 0.6 kcal/mol, respectively). Analogous to the total energies in Table 1, the contraction error is the limiting factor for the pcseg-4 basis set. Extrapolation12 of the upc-2,3,4 results provides MAD relative to the MRChem results of ∼0.01 kcal/ mol, with MaxAD being ∼0.08 kcal/mol. Extrapolation of the pcseg-n results provides no improvement due to the contraction errors, but extrapolation of the upc-n results provide a small but consistent improvement. Augmenting the basis sets with diffuse functions provides a small improvement, which diminishes as the underlying basis set becomes more complete. The performance is analogous to the results in Table 1, only slightly dependent on the employed functional. Jensen et al. also examined the dipole moment as a property that is not directly related to the energy and showed that the aug-cc-pV5Z basis set displayed maximum errors in the ∼0.1 D range.1 Table 3 shows a comparison between the MRChem and basis set results, analogous to the energetic results in Table 2. It is well-known that the basis set convergence for electric properties is greatly accelerated by employing basis sets
Figure 1. Absolute deviations in LDA-SVWN5 total energies (Hartrees).
contraction error is predominantly located in the core orbitals, it leads to a substantial cancellation of errors when comparing relative energies for different systems. The magnitude of the contraction error is shown by the difference between the pcsegxpol and upc-xpol results, where the errors with the latter in all cases are below 10−5 Hartrees. We note that the reference energies are only accurate to ∼10−6 Hartrees, and deviations below this limit have no significance. Table 1 shows the mean and maximum absolute deviations (MAD and MaxAD) of total energies over the full benchmark data set (all geometries and spin state information has been taken from ref 1) relative to the reported MRChem results. The errors are reduced by roughly an order of magnitude for each
Table 1. MAD and MaxAD of Total Energies (Hartree) Relative to the MRChem Results over the Benchmark Set of Data DFT MAD
PBE
PBE0
basis/levela
D
T
Q
5
cc-pVXZ aug-cc-pVXZ pcseg-n aug-pcseg-n upc-n aug-upc-n
0.07269 0.05803 0.10495 0.09655 0.09068 0.08189
0.02195 0.01869 0.01326 0.01188 0.00900 0.00747
0.00856 0.00728 0.00116 0.00100 0.00055 0.00045
0.00246
cc-pVXZ aug-cc-pVXZ pcseg-n aug-pcseg-n upc-n aug-upc-n
0.40826 0.32814 0.55702 0.51450 0.52288 0.48030
0.11983 0.10577 0.06690 0.06016 0.04953 0.04214
0.04520 0.04037 0.00360 0.00327 0.00252 0.00230
0.01034
0.00056 0.00053 0.00013 0.00012
Xpol
D
T
Q
5
0.01883 0.01617 0.01404 0.01263 0.00897 0.00750
0.00639 0.00544 0.00164 0.00149 0.00051 0.00042
0.00167
0.00051 0.00050 0.00010 0.00009
0.06641 0.05392 0.10674 0.09840 0.09066 0.08227
0.11068 0.09834 0.06868 0.06163 0.04983 0.04189
0.03910 0.03504 0.00661 0.00630 0.00244 0.00221
0.00771
0.00220 0.00219 0.00120 0.00119
0.39318 0.31821 0.57081 0.52449 0.52626 0.48368
0.00096 0.00093 0.00011 0.00010
Xpol
0.00090 0.00089 0.00007 0.00007
MaxAD
0.00228 0.00226 0.00126 0.00125
0.00495 0.00492 0.00041 0.00038
0.00482 0.00479 0.00027 0.00030
a
The D/T/Q/5 notation for the cc-pVXZ basis sets corresponds to the pc-1,2,3,4 notation, respectively, in terms of basis set quality. Xpol indicates a square-root exponential extrapolation of the pc-2,3,4 results.12 6105
DOI: 10.1021/acs.jpca.7b04760 J. Phys. Chem. A 2017, 121, 6104−6107
Article
The Journal of Physical Chemistry A
Table 2. MAD and MaxAD of Atomization Energies (kcal/mol) Relative to the MRChem Results over the Benchmark Set of Data DFT MAD
PBE
PBE0
basis/levela
D
T
Q
5
cc-pVXZ aug-cc-pVXZ pcseg-n aug-pcseg-n upc-n aug-upc-n
8.141 8.211 8.119 5.735 5.877 4.115
1.508 1.838 1.103 0.551 1.208 0.576
0.871 0.625 0.158 0.107 0.082 0.056
0.179 0.107 0.097 0.020 0.016
Xpol
D
T
Q
5
1.807 1.831 1.073 0.461 1.245 0.565
0.691 0.582 0.096 0.073 0.073 0.048
0.167
0.106 0.100 0.016 0.013
9.426 8.248 8.234 4.854 6.548 3.846
0.058 0.053 0.015 0.011
Xpol
0.067 0.063 0.011 0.009
MaxAD cc-pVXZ aug-cc-pVXZ pcseg-n aug-pcseg-n upc-n aug-upc-n
48.72 54.75 58.40 51.76 50.90 47.92
13.96 20.94 9.79 7.04 9.90 6.71
7.74 10.78 0.61 0.39 0.54 0.34
1.79 0.46 0.44 0.09 0.09
0.47 0.45 0.08 0.08
60.44 55.13 67.01 50.37 57.69 47.28
16.67 19.82 10.07 6.45 10.47 6.25
8.97 10.18 0.50 0.37 0.50 0.37
2.07 0.23 0.23 0.09 0.10
0.25 0.23 0.06 0.11
a
The D/T/Q/5 notation for the cc-pVXZ basis sets corresponds to the pc-1,2,3,4 notation, respectively, in terms of basis set quality. Xpol indicates a square-root exponential extrapolation of the pc-2,3,4 results.12
Table 3. MAD and MaxAD of Molecular Dipole Moments (Debye) Relative to the MRChem Results over the Benchmark Set of Data DFT basis/level MAD
PBE a
PBE0
D
T
Q
5
D
T
Q
5
cc-pVXZ aug-cc-pVXZ pcseg-n aug-pcseg-n upc-n aug-upc-n
0.1355 0.0190 0.1123 0.0189 0.1162 0.0161
0.0624 0.0084 0.0467 0.0060 0.0525 0.0054
0.0272 0.0046 0.0101 0.0008 0.0102 0.0005
0.0126
0.1098 0.0206 0.1163 0.0213 0.1183 0.0186
0.0495 0.0089 0.0468 0.0066 0.0519 0.0061
0.0213 0.0051 0.0086 0.0009 0.0089 0.0005
0.0095
cc-pVXZ aug-cc-pVXZ pcseg-n aug-pcseg-n upc-n aug-upc-n
1.212 0.218 0.870 0.155 0.894 0.141
0.648 0.119 0.293 0.072 0.326 0.069
0.326 0.091 0.069 0.011 0.057 0.007
0.156
1.163 0.234 1.016 0.171 0.951 0.153
0.619 0.123 0.269 0.077 0.345 0.076
0.309 0.095 0.062 0.010 0.064 0.008
0.135
0.0027 0.0004 0.0021 0.0007
0.0019 0.0005 0.0015 0.0002
MaxAD
a
0.038 0.002 0.020 0.009
0.021 0.005 0.016 0.003
The D/T/Q/5 notation for the cc-pVXZ basis sets corresponds to the pc-1,2,3,4 notation, respectively, in terms of basis set quality.
augmented with diffuse functions,14 and this is clearly displayed by the results in Table 3. The aug-pcseg-n basis sets display a convergence better and more uniform than that of the aug-ccpVXZ-type basis sets. Basis set contraction is, as expected, not important for this property, as it primarily depends on an accurate representation of the wave function tail region, and there is thus little difference between the aug-pcseg-n and augupc-n results. We note that several of the dipole moments which must be identical to zero due to the molecular symmetry are reported to have nonzero MRChem values with the largest deviation being BO2 with a value of 0.0015 D.1 These deviations are a consequence of the MRChem calculations being run without symmetry and are comparable to the differences between the MRChem and aug-upc-4 results in Table 3.
reduced well below chemical accuracy. There are no fundamental limitations in designing higher level pc-n basis sets, and we have in special cases employed pc-5- and pc-6-type basis sets for even higher accuracy.15,16 Table 2 shows that basis set errors with the pcseg-2 typically are below a few kcal/mol for atomization energies, where errors in the employed DFT method usually will dominate, and it will rarely be necessary to employ basis sets larger than pcseg-3. If accuracies below ∼1 milli-Hartree in total energy or below ∼0.1 kcal/mol in atomization energies are required, then the uncontracted versions of the pc-basis sets should be employed. At a given ζ-level, previous work has shown that the pcseg-n basis sets for DFT methods have basis set errors lower than those of other alternatives.7 The chemistry and physics communities have traditionally employed different methods and technologies for calculating the electronic structure of molecules and periodic systems. The ability to calculate the same quantities by different methodologies allows an independent validation of the results. The paper by Jensen et al. is an important contribution to this field,1 while the present work shows that the use of basis sets optimized for the employed Hamiltonian is essential for
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CONCLUSION The present results show that the use of Gaussian basis sets optimized for DFT methods is essential to obtain a high accuracy and for achieving a consistent accuracy over a range of molecular systems without high-error outliers. The results with the (aug-)pc-n basis sets suggest that the basis set error can be 6106
DOI: 10.1021/acs.jpca.7b04760 J. Phys. Chem. A 2017, 121, 6104−6107
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The Journal of Physical Chemistry A achieving the full accuracy of the methodology.8,9 The necessity of using basis sets optimized for the particular Hamiltonian, and in some cases also the desired property,17,18 is a drawback of Gaussian basis sets, and the multiresolution approach provides a general framework for achieving a uniform high accuracy. Achieving an ultrahigh accuracy in the orbital representation, however, may in many practical applications not be important, as method errors will eventually provide the limiting accuracy, and the interesting point is thus the time-to-solution for a given target accuracy.
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(13) Jensen, F. Polarization Consistent Basis Sets. Ii. Estimating the Kohn-Sham Basis Set Limit. J. Chem. Phys. 2002, 116, 7372−7379. (14) Jensen, F. Polarization Consistent Basis Sets. Iii. The Importance of Diffuse Functions. J. Chem. Phys. 2002, 117, 9234− 9240. (15) Madsen, L. B.; Jensen, F.; Tolstikhin, O. I.; Morishita, T. Structure Factors for Tunneling Ionization Rates of Molecules. Phys. Rev. A: At., Mol., Opt. Phys. 2013, 87, 1. (16) Madsen, L. B.; Jensen, F.; Tolstikhin, O. I.; Morishita, T. Application of the Weak-Field Asymptotic Theory to Tunneling Ionization of H2o. Phys. Rev. A: At., Mol., Opt. Phys. 2014, 89, 1. (17) Jensen, F. Segmented Contracted Basis Sets Optimized for Nuclear Magnetic Shielding. J. Chem. Theory Comput. 2015, 11, 132− 138. (18) Jensen, F. The Optimum Contraction of Basis Sets for Calculating Spin-Spin Coupling Constants. Theor. Chem. Acc. 2010, 126, 371−382.
ASSOCIATED CONTENT
S Supporting Information *
The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acs.jpca.7b04760. Total and atomization energies and dipole moments for atoms and molecules with all of the basis sets (XLSX)
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AUTHOR INFORMATION
Corresponding Author
*E-mail:
[email protected]. ORCID
Frank Jensen: 0000-0002-4576-5838 Notes
The author declares no competing financial interest.
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ACKNOWLEDGMENTS This work was supported in part by the Danish Natural Science Research Council by Grant 4181-00030B.
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REFERENCES
(1) Jensen, S. R.; Saha, S.; Flores-Livas, J. A.; Huhn, W.; Blum, V.; Goedecker, S.; Frediani, L. The Elephant in the Room of Density Functional Theory Calculations. J. Phys. Chem. Lett. 2017, 8, 1449− 1457. (2) Kendall, R. A.; Dunning, T. H.; Harrison, R. J. Electron-Affinities of the 1st-Row Atoms Revisited - Systematic Basis-Sets and WaveFunctions. J. Chem. Phys. 1992, 96, 6796−6806. (3) Perdew, J. P.; Burke, K.; Ernzerhof, M. Generalized Gradient Approximation Made Simple. Phys. Rev. Lett. 1996, 77, 3865−3868. (4) Adamo, C.; Barone, V. Toward Reliable Density Functional Methods without Adjustable Parameters: The Pbe0Model. J. Chem. Phys. 1999, 110, 6158−6170. (5) Witte, J.; Neaton, J. B.; Head-Gordon, M. Push It to the Limit: Characterizing the Convergence of Common Sequences of Basis Sets for Intermolecular Interactions as Described by Density Functional Theory. J. Chem. Phys. 2016, 144, 194306. (6) Jensen, F. Polarization Consistent Basis Sets: Principles. J. Chem. Phys. 2001, 115, 9113−9125. (7) Jensen, F. Unifying General and Segmented Contracted Basis Sets. Segmented Polarization Consistent Basis Sets. J. Chem. Theory Comput. 2014, 10, 1074−1085. (8) Jensen, F. Atomic Orbital Basis Sets. Wiley Interdisciplinary Reviews-Computational Molecular Science 2013, 3, 273−295. (9) Hill, J. G. Gaussian Basis Sets for Molecular Applications. Int. J. Quantum Chem. 2013, 113, 21−34. (10) Nagy, B.; Jensen, F. Reviews in Computational Chemistry 2017, 30, 93. (11) Kotochigova, S.; Levine, Z. H.; Shirley, E. L.; Stiles, M. D.; Clark, C. W. Atomic Reference Data for Electronic Structure Calculations. Phys. Rev. A: At., Mol., Opt. Phys. 1997, 55, 191. (12) Jensen, F. Estimating the Hartree-Fock Limit from Finite Basis Set Calculations. Theor. Chem. Acc. 2005, 113, 267−273. 6107
DOI: 10.1021/acs.jpca.7b04760 J. Phys. Chem. A 2017, 121, 6104−6107
