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New Accurate Benchmark Energies for Large Water Clusters: DFT Is Better Than Expected Tony Anacker* and Joachim Friedrich In this work, we use MP2 and coupled-cluster with single, double, and perturbative triple excitations [CCSD(T)] as well as their corresponding explicitly correlated (F12) counterparts to compute the interaction energies of water icosamers. The incremental scheme is used to compute benchmark energies at the CCSD(T)/CBS(45) and CCSD(T)(F12*)/cc-pVQZ-F12 level of theory. The four structures, dodecahedron, edge sharing, face sharing, and fused cubes, are part of the WATER27 test set and therefore, highly accurate interaction energies are required. All methods applied in this work lead to new benchmark energies for these four systems. To obtain these values, we carefully analyze the convergence of the interaction ener-
gies with respect to the basis set. Furthermore, we investigate the influence of the basis set superposition error and the corevalence correlation. The interaction energies are: dodecahedron 2198.6 kcal/mol, edge sharing 2209.7 kcal/mol, face sharing 2208.0 kcal/mol, and fused cubes 2208.0 kcal/mol. For water clusters, we recommend to use the PW6B95 density functional of Truhlar in combination with Grimme’s dispersion correction (D3), as the mean absolute error is 0.9 and the root C 2014 Wiley mean-squared deviation is only 1.4 kcal/mol. V Periodicals, Inc.
Introduction
improve the results is the extrapolation to the CBS limit. The correlation-consistent polarized valence X-tuple f basis sets, cc-pVXZ, and their counterparts with tight, cc-pCVXZ, or diffuse, aug-cc-pCXZ,[13–16] functions are well-suited for such extrapolations.[17–19] Another way to obtain interaction energies at the CBS level is to use explicitly correlated (F12) methods. In these methods, the coulomb cusp, which is responsible for the slow convergence of the correlation energy in Gaussian basis sets, is explicitly parameterized into the wave function. Therefore, the F12 models like MP2-F12[20] or CCSD(T)(F12)[21–23] converge significantly faster to the CBS limit compared to the original models.[24–27] Due to the unfavorable scaling behavior of CCSD(T), this high-level method is only applicable to small and mediumsized molecules. To overcome this problem, many different low-order scaling techniques were proposed during the last decades.[5,28–62] In this work, we use the incremental scheme proposed by Stoll,[28–30,63] which is a generalization of the Bethe–Goldstone expansion as introduced to quantum chemistry by Nesbet.[64,65] In our previous work, we implemented a fully automated version of the incremental scheme for CCSD,[66] CCSD(T),[67–69] and explicitly correlated coupled-cluster.[70–72] The applicability of the incremental scheme was shown for
Recently, Goerigk and Grimme presented a quantum chemistry benchmark database for general main group thermochemistry, kinetics, and noncovalent interactions (GMTKN24) consisting of 24 different, chemically relevant subsets.[1] The purpose of such databases is to evaluate the general accuracy of different density functionals and to analyze which functional is optimal for different application classes.[2,3] The reference values in the GMTKN24 are based on theoretical or experimental reference values. A general accepted benchmark method is coupled-cluster with single, double, and perturbative triple excitations [CCSD(T)] in combination with an appropriate one-particle basis set and an extrapolation to the complete basis set (CBS). For water clusters, it was found that the computationally cheap MP2 method is also suitable to compute sufficiently accurate interaction energies, when using a large basis set. Considering the WATER27 subset, Goerigk and Grimme found very big errors in the interaction energies of water icosamers for recently developed, accurate density functionals. They used the MP2/CBS benchmark values from Xantheas and coworkers,[4] which should be fine for these systems.[5,6] However, in this work, we show that a large part of the errors attributed to density functional theory (DFT) were due to error in the benchmarks. When using a finite basis set in quantum chemical calculations, a main source of error is the basis set superposition error (BSSE). When computing interaction energies of dimers, one can account for the BSSE, with the counterpoise (CP) correction of Boys and Bernadi.[7] For larger clusters, one has to use a generalized version the CP correction.[8–10] In this work, we apply the site-site function counterpoise (SSFC) approach of Wells and Wilson[8] using our automated implementation.[11,12] It is well-known that the correlation energy converges slowly with respect to the one-particle basis set. A way to 634
Journal of Computational Chemistry 2014, 35, 634–643
DOI: 10.1002/jcc.23539
T. Anacker, J. Friedrich Department of Theoretical Chemistry, Chemnitz University of Technology, Straße der Nationen 62, Chemnitz D-09111, Saxony, Germany E-mail: [email protected] Contract grant sponsor: Fonds der Chemischen Industrie (Material Cost Allowances program) C 2014 Wiley Periodicals, Inc. V
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Table 1. Interaction energies Eint (kcal/mol) of the four structures calculated with MP2 and MP2 plus CP correction. Dodecahedron Basis aVDZ’ aVTZ’ aVQZ’ aV5Z’ CBS(23) CBS(34) CBS(45) Old CBS limit [4]
MP2 2206.1 2203.0 2200.9 2199.7 2203.9 2199.9 2198.9
Edge sharing
MP210.5CP MP21CP 2190.9 2195.7 2197.5 2198.0
2175.7 2188.3 2194.0 2196.3 2194.9 2198.7 2198.7
MP2
Face sharing
MP210.5CP MP21CP
2216.1 2213.0 2210.7 2209.4 2214.2 2209.7 2208.7
2200.1
2200.0 2205.3 2207.2 2207.7
2217.9
2183.9 2197.5 2203.6 2206.0 2205.0 2208.5 2208.4
MP2 2214.5 2211.0 2208.8 2207.5 2212.1 2207.9 2206.7
Fused cubes
MP210.5CP MP21CP 2198.2 2203.3 2205.2 2205.8
2181.9 2195.6 2201.7 2204.1 2203.1 2206.6 2206.5
MP2
MP210.5CP MP21CP
2213.3 2210.5 2208.4 2207.0 2212.0 2207.5 2206.3
2215.0
2197.4 2202.9 2204.8 2205.4
2181.5 2195.2 2201.3 2203.7 2202.9 2206.2 2206.1
2212.6
The CBS extrapolations are added at the bottom of the table.
organic[67,71,73] and inorganic compounds[66] and molecular clusters.[71,74–77] To establish new reference values for the interaction energies of four water icosamers, we investigate this interaction energies at MP2 and CCSD(T) level using basis sets up to quintuple-f quality, we analyze the CP contribution, perform a CBS extrapolation, compute the core-valence contributions, and apply explicitly correlated CCSD(T) and MP2. We systematically analyze the convergence of the interaction energy to the CBS limit with all of these methods to show that the actual benchmarks of the four water icosamers from Xantheas and coworkers[4] are not accurate enough. Finally, we analyze the errors in the interaction energies for some recently developed density functionals and show that their accuracy was underestimated, due to inaccuracies in the reference energies.
For the explicitly correlated methods, we reduce the error in the same manner. With this method, we have the possibility to truncate the incremental expansion at second order for water clusters, while keeping the required accuracy.[12,71]
Theory
CBS extrapolation
Incremental scheme
The dominant source of error in most calculations is the truncation of the one-particle basis set. Therefore, linear extrapolations are attempted to achieve the CBS limit. In this work, the CBS limit for the Hartree–Fock (HF) energy is calculated with the extrapolation of Petersson and coworkers[17] in combination with the augmented correlation-consistent polarized valence X-tuple basis sets:
The incremental scheme is used to obtain the CCSD(T) correlation energy Ecorr for the large water clusters studied in this work. Within this framework, the total system is divided into small one-site domains consisting of localized orbitals[66,72,73,78]. Based on these domains, the correlation energy is computed for all domains, domain pairs, and triples of domains. The total correlation energy is computed with the incremental expansion[30,66]: X DeX (1) Ecorr 5 X2PðDÞÙjXj
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New Accurate Benchmark Energies for Large Water Clusters: DFT Is Better Than Expected Tony Anacker* and Joachim Friedrich In this work, we use MP2 and coupled-cluster with single, double, and perturbative triple excitations [CCSD(T)] as well as their corresponding explicitly correlated (F12) counterparts to compute the interaction energies of water icosamers. The incremental scheme is used to compute benchmark energies at the CCSD(T)/CBS(45) and CCSD(T)(F12*)/cc-pVQZ-F12 level of theory. The four structures, dodecahedron, edge sharing, face sharing, and fused cubes, are part of the WATER27 test set and therefore, highly accurate interaction energies are required. All methods applied in this work lead to new benchmark energies for these four systems. To obtain these values, we carefully analyze the convergence of the interaction ener-
gies with respect to the basis set. Furthermore, we investigate the influence of the basis set superposition error and the corevalence correlation. The interaction energies are: dodecahedron 2198.6 kcal/mol, edge sharing 2209.7 kcal/mol, face sharing 2208.0 kcal/mol, and fused cubes 2208.0 kcal/mol. For water clusters, we recommend to use the PW6B95 density functional of Truhlar in combination with Grimme’s dispersion correction (D3), as the mean absolute error is 0.9 and the root C 2014 Wiley mean-squared deviation is only 1.4 kcal/mol. V Periodicals, Inc.
Introduction
improve the results is the extrapolation to the CBS limit. The correlation-consistent polarized valence X-tuple f basis sets, cc-pVXZ, and their counterparts with tight, cc-pCVXZ, or diffuse, aug-cc-pCXZ,[13–16] functions are well-suited for such extrapolations.[17–19] Another way to obtain interaction energies at the CBS level is to use explicitly correlated (F12) methods. In these methods, the coulomb cusp, which is responsible for the slow convergence of the correlation energy in Gaussian basis sets, is explicitly parameterized into the wave function. Therefore, the F12 models like MP2-F12[20] or CCSD(T)(F12)[21–23] converge significantly faster to the CBS limit compared to the original models.[24–27] Due to the unfavorable scaling behavior of CCSD(T), this high-level method is only applicable to small and mediumsized molecules. To overcome this problem, many different low-order scaling techniques were proposed during the last decades.[5,28–62] In this work, we use the incremental scheme proposed by Stoll,[28–30,63] which is a generalization of the Bethe–Goldstone expansion as introduced to quantum chemistry by Nesbet.[64,65] In our previous work, we implemented a fully automated version of the incremental scheme for CCSD,[66] CCSD(T),[67–69] and explicitly correlated coupled-cluster.[70–72] The applicability of the incremental scheme was shown for
Recently, Goerigk and Grimme presented a quantum chemistry benchmark database for general main group thermochemistry, kinetics, and noncovalent interactions (GMTKN24) consisting of 24 different, chemically relevant subsets.[1] The purpose of such databases is to evaluate the general accuracy of different density functionals and to analyze which functional is optimal for different application classes.[2,3] The reference values in the GMTKN24 are based on theoretical or experimental reference values. A general accepted benchmark method is coupled-cluster with single, double, and perturbative triple excitations [CCSD(T)] in combination with an appropriate one-particle basis set and an extrapolation to the complete basis set (CBS). For water clusters, it was found that the computationally cheap MP2 method is also suitable to compute sufficiently accurate interaction energies, when using a large basis set. Considering the WATER27 subset, Goerigk and Grimme found very big errors in the interaction energies of water icosamers for recently developed, accurate density functionals. They used the MP2/CBS benchmark values from Xantheas and coworkers,[4] which should be fine for these systems.[5,6] However, in this work, we show that a large part of the errors attributed to density functional theory (DFT) were due to error in the benchmarks. When using a finite basis set in quantum chemical calculations, a main source of error is the basis set superposition error (BSSE). When computing interaction energies of dimers, one can account for the BSSE, with the counterpoise (CP) correction of Boys and Bernadi.[7] For larger clusters, one has to use a generalized version the CP correction.[8–10] In this work, we apply the site-site function counterpoise (SSFC) approach of Wells and Wilson[8] using our automated implementation.[11,12] It is well-known that the correlation energy converges slowly with respect to the one-particle basis set. A way to 634
Journal of Computational Chemistry 2014, 35, 634–643
DOI: 10.1002/jcc.23539
T. Anacker, J. Friedrich Department of Theoretical Chemistry, Chemnitz University of Technology, Straße der Nationen 62, Chemnitz D-09111, Saxony, Germany E-mail: [email protected] Contract grant sponsor: Fonds der Chemischen Industrie (Material Cost Allowances program) C 2014 Wiley Periodicals, Inc. V
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Table 1. Interaction energies Eint (kcal/mol) of the four structures calculated with MP2 and MP2 plus CP correction. Dodecahedron Basis aVDZ’ aVTZ’ aVQZ’ aV5Z’ CBS(23) CBS(34) CBS(45) Old CBS limit [4]
MP2 2206.1 2203.0 2200.9 2199.7 2203.9 2199.9 2198.9
Edge sharing
MP210.5CP MP21CP 2190.9 2195.7 2197.5 2198.0
2175.7 2188.3 2194.0 2196.3 2194.9 2198.7 2198.7
MP2
Face sharing
MP210.5CP MP21CP
2216.1 2213.0 2210.7 2209.4 2214.2 2209.7 2208.7
2200.1
2200.0 2205.3 2207.2 2207.7
2217.9
2183.9 2197.5 2203.6 2206.0 2205.0 2208.5 2208.4
MP2 2214.5 2211.0 2208.8 2207.5 2212.1 2207.9 2206.7
Fused cubes
MP210.5CP MP21CP 2198.2 2203.3 2205.2 2205.8
2181.9 2195.6 2201.7 2204.1 2203.1 2206.6 2206.5
MP2
MP210.5CP MP21CP
2213.3 2210.5 2208.4 2207.0 2212.0 2207.5 2206.3
2215.0
2197.4 2202.9 2204.8 2205.4
2181.5 2195.2 2201.3 2203.7 2202.9 2206.2 2206.1
2212.6
The CBS extrapolations are added at the bottom of the table.
organic[67,71,73] and inorganic compounds[66] and molecular clusters.[71,74–77] To establish new reference values for the interaction energies of four water icosamers, we investigate this interaction energies at MP2 and CCSD(T) level using basis sets up to quintuple-f quality, we analyze the CP contribution, perform a CBS extrapolation, compute the core-valence contributions, and apply explicitly correlated CCSD(T) and MP2. We systematically analyze the convergence of the interaction energy to the CBS limit with all of these methods to show that the actual benchmarks of the four water icosamers from Xantheas and coworkers[4] are not accurate enough. Finally, we analyze the errors in the interaction energies for some recently developed density functionals and show that their accuracy was underestimated, due to inaccuracies in the reference energies.
For the explicitly correlated methods, we reduce the error in the same manner. With this method, we have the possibility to truncate the incremental expansion at second order for water clusters, while keeping the required accuracy.[12,71]
Theory
CBS extrapolation
Incremental scheme
The dominant source of error in most calculations is the truncation of the one-particle basis set. Therefore, linear extrapolations are attempted to achieve the CBS limit. In this work, the CBS limit for the Hartree–Fock (HF) energy is calculated with the extrapolation of Petersson and coworkers[17] in combination with the augmented correlation-consistent polarized valence X-tuple basis sets:
The incremental scheme is used to obtain the CCSD(T) correlation energy Ecorr for the large water clusters studied in this work. Within this framework, the total system is divided into small one-site domains consisting of localized orbitals[66,72,73,78]. Based on these domains, the correlation energy is computed for all domains, domain pairs, and triples of domains. The total correlation energy is computed with the incremental expansion[30,66]: X DeX (1) Ecorr 5 X2PðDÞÙjXj
