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Basis Set Convergence and Extrapolation of Connected Triple Excitation Contributions (T) in Computational Thermochemistry: The W4-17 Benchmark with Up to k Functions

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Abstract

The total atomization energy of a molecule is the thermochemical cognate of the heat of formation in the gas phase, its most fundamental thermochemical property. We decompose it into different components and provide a survey of them. It emerges that the connected triple excitations contribution is the third most important one, about an order of magnitude less important than the “big two” contributions (mean-field Hartree–Fock and valence CCSD correlation), but 1–2 orders of magnitude more important than the remainder. For the 200 total atomization energies of small molecules in the W4-17 benchmark, we have investigated the basis set convergence of the connected triple excitations contribution (T). Achieving basis set convergence for the valence triple excitations energy is much easier than for the valence singles and doubles correlation energy. Using reference data obtained from spdfghi and spdfghik basis sets, we show that extrapolation from quintuple-zeta and sextuple-zeta yields values within about 0.004 kcal/mol RMS. Convergence to within about 0.01 kcal/mol is achievable with quadruple- and quintuple-zeta basis sets, and to within about 0.05 kcal/mol with triple- and quadruple-zeta basis sets. It appears that radial flexibility in the basis set is more important here than adding angular momenta L: apparently, replacing nZaPa basis sets with truncations of 7ZaPa at L = n gains about one angular momentum for small values of n. We end the article with a brief outlook for the future of accurate electronic structure calculations.

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Notes

  1. 1.

    The term originates in quantitative analytical chemistry and is used here by analogy.

  2. 2.

    We also note in passing that as one goes further down the periodic table, relativistic effects will eventually come to rival the major contributors [64].

  3. 3.

    He recommends eschewing nonlinear 3-point formulas, as they are not size-consistent. We note in passing that Schwenke also presents separate extrapolation coefficients for the Klopper-style [119] singlet-coupled and triplet-coupled CCSD correlation energy components.

  4. 4.

    As a by-product, we can obtain the extrapolation exponents for MP2 and CCSD, which for the def2-{T,Q}ZVPP pair are αMP2 = 2.612 and αCCSD = 3.017, the latter nearly identical to 2.970 from Neese and Valeev [94], and both functionally equivalent to the simple L–3 extrapolation.

  5. 5.

    Unlike the partitioning of the CCSD correlation energy in singlet-coupled and triplet-coupled pairs, which is not uniquely defined for open-shell cases.

References

  1. Hartree DR (1928) The wave mechanics of an atom with a non-coulomb central field. Part II. Some results and discussion. Math Proc Cambridge Philos Soc 24:111–132. https://doi.org/10.1017/S0305004100011920

    ADS  MATH  Google Scholar 

  2. Slater JC (1928) The self consistent field and the structure of atoms. Phys Rev 32:339–348. https://doi.org/10.1103/PhysRev.32.339

    ADS  MATH  Google Scholar 

  3. Gaunt JA (1928) A theory of Hartree’s atomic fields. Math Proc Cambridge Philos Soc 24:328–342. https://doi.org/10.1017/S0305004100015851

    ADS  MATH  Google Scholar 

  4. Fock V (1930) Näherungsmethode zur Lösung des quantenmechanischen Mehrkörperproblems. Zeitschrift für Phys 61:126–148. https://doi.org/10.1007/BF01340294

    ADS  MATH  Google Scholar 

  5. Fock V (1930) “Selfconsistent field” mit Austausch für Natrium. Zeitschrift für Phys 62:795–805. https://doi.org/10.1007/BF01330439

    ADS  MATH  Google Scholar 

  6. Slater JC (1930) Note on Hartree’s method. Phys Rev 35:210–211. https://doi.org/10.1103/PhysRev.35.210.2

    ADS  Google Scholar 

  7. Löwdin PO (1955) Quantum theory of many-particle systems. III. Extension of the Hartree-Fock scheme to include degenerate systems and correlation effects. Phys Rev 97:1509–1520. https://doi.org/10.1103/PhysRev.97.1509

    ADS  MathSciNet  MATH  Google Scholar 

  8. Wigner E (1934) On the interaction of electrons in metals. Phys Rev 46:1002–1011. https://doi.org/10.1103/PhysRev.46.1002

    ADS  MATH  Google Scholar 

  9. Burke K, Cancio A, Gould T, Pittalis S (2016) Locality of correlation in density functional theory. J Chem Phys 145:054112. https://doi.org/10.1063/1.4959126

    ADS  Google Scholar 

  10. Schwinger J (1980) Thomas-Fermi model: The leading correction. Phys Rev A 22:1827–1832. https://doi.org/10.1103/PhysRevA.22.1827

    ADS  MathSciNet  Google Scholar 

  11. Schwinger J (1981) Thomas-Fermi model: The second correction. Phys Rev A 24:2353–2361. https://doi.org/10.1103/PhysRevA.24.2353

    ADS  MathSciNet  Google Scholar 

  12. Scott JMC (1952) The binding energy of the Thomas-Fermi Atom. Philos Mag Ser 7(43):859–867. https://doi.org/10.1080/14786440808520234

    Google Scholar 

  13. Elliott P, Burke K (2009) Non-empirical derivation of the parameter in the B88 exchange functional. Can J Chem 87:1485–1491. https://doi.org/10.1139/V09-095

    Google Scholar 

  14. Karton A, Sylvetsky N, Martin JML (2017) W4–17: A diverse and high-confidence dataset of atomization energies for benchmarking high-level electronic structure methods. J Comput Chem 38:2063–2075. https://doi.org/10.1002/jcc.24854

    Google Scholar 

  15. Chachiyo T (2016) Communication: Simple and accurate uniform electron gas correlation energy for the full range of densities. J Chem Phys 145:021101. https://doi.org/10.1063/1.4958669

    ADS  Google Scholar 

  16. Martin JM (2022) Electron correlation: nature’s weird and wonderful chemical glue. Israel J Chem 62(1–2). https://doi.org/10.1002/ijch.202100111

  17. Shavitt I, Bartlett RJ (2009) Many – Body methods in chemistry and physics. Cambridge University Press, Cambridge

    Google Scholar 

  18. Brueckner KA (1955) Many-body problem for strongly interacting particles II. Linked cluster expansion. Phys Rev 100:36–45. https://doi.org/10.1103/PhysRev.100.36

    ADS  MATH  Google Scholar 

  19. Goldstone J (1957) Derivation of the Brueckner many-body theory. Proc R Soc London Ser A Math Phys Sci 239:267–279. https://doi.org/10.1098/rspa.1957.0037

    ADS  MathSciNet  MATH  Google Scholar 

  20. Langhoff PW, Hernandez AJ (1976) On the brueckner and goldstone forms of the linked-cluster theorem. Int J Quantum Chem 10:337–351. https://doi.org/10.1002/qua.560100838

    Google Scholar 

  21. Purvis GD, Bartlett RJ (1982) A full coupled-cluster singles and doubles model: the inclusion of disconnected triples. J Chem Phys 76:1910–1918. https://doi.org/10.1063/1.443164

    ADS  Google Scholar 

  22. Janesko BG (2021) Replacing hybrid density functional theory: motivation and recent advances. Chem Soc Rev in press. https://doi.org/10.1039/D0CS01074J

    Google Scholar 

  23. Karton A, Taylor PR, Martin JML (2007) Basis set convergence of post-CCSD contributions to molecular atomization energies. J Chem Phys 127:064104. https://doi.org/10.1063/1.2755751

    ADS  Google Scholar 

  24. Scuseria GE, Schaefer HF (1988) A new implementation of the full CCSDT model for molecular electronic structure. Chem Phys Lett 152:382–386. https://doi.org/10.1016/0009-2614(88)80110-6

    ADS  Google Scholar 

  25. Watts JD, Bartlett RJ (1990) The coupled-cluster single, double, and triple excitation model for open-shell single reference functions. J Chem Phys 93:6104–6105. https://doi.org/10.1063/1.459002

    ADS  Google Scholar 

  26. Kucharski SA, Bartlett RJ (1992) The coupled-cluster single, double, triple, and quadruple excitation method. J Chem Phys 97:4282–4288. https://doi.org/10.1063/1.463930

    ADS  Google Scholar 

  27. Trucks GW, Pople JA, Head-Gordon M (1989) A fifth-order perturbation comparison of electron correlation theoriesA1—Raghavachari, K. Chem Phys Lett 157:479–483

    ADS  Google Scholar 

  28. Watts JD, Gauss J, Bartlett RJ (1993) Coupled-cluster methods with noniterative triple excitations for restricted open-shell Hartree-Fock and other general single determinant reference functions. Energies and analytical gradients. J Chem Phys 98:8718–8733. https://doi.org/10.1063/1.464480

    ADS  Google Scholar 

  29. Karton A, Daon S, Martin JML (2011) W4–11: a high-confidence benchmark dataset for computational thermochemistry derived from first-principles W4 data. Chem Phys Lett 510:165–178. https://doi.org/10.1016/j.cplett.2011.05.007

    ADS  Google Scholar 

  30. Kállay M, Surján PR (2001) Higher excitations in coupled-cluster theory. J Chem Phys 115:2945–2954. https://doi.org/10.1063/1.1383290

    ADS  Google Scholar 

  31. Kállay M, Gauss J (2005) Approximate treatment of higher excitations in coupled-cluster theory. J Chem Phys 123. https://doi.org/10.1063/1.2121589

  32. Kállay M, Nagy PR, Mester D et al (2020) The MRCC program system: accurate quantum chemistry from water to proteins. J Chem Phys 152:074107. https://doi.org/10.1063/1.5142048

    ADS  Google Scholar 

  33. Stanton JF (1997) Why CCSD(T) works: a different perspective. Chem Phys Lett 281:130–134. https://doi.org/10.1016/S0009-2614(97)01144-5

    ADS  Google Scholar 

  34. Löwdin PO (1962) Studies in perturbation theory. IV. Solution of eigenvalue problem by projection operator formalism. J Math Phys 3:969–982. https://doi.org/10.1063/1.1724312

    ADS  MathSciNet  MATH  Google Scholar 

  35. Dunning TH (2000) A road map for the calculation of molecular binding energies. J Phys Chem A 104:9062–9080. https://doi.org/10.1021/jp001507z

    Google Scholar 

  36. Riplinger C, Neese F (2013) An efficient and near linear scaling pair natural orbital based local coupled cluster method. J Chem Phys 138:034106. https://doi.org/10.1063/1.4773581

    ADS  Google Scholar 

  37. Riplinger C, Sandhoefer B, Hansen A, Neese F (2013) Natural triple excitations in local coupled cluster calculations with pair natural orbitals. J Chem Phys 139:134101. https://doi.org/10.1063/1.4821834

    ADS  Google Scholar 

  38. Ma Q, Werner H-J (2018) Explicitly correlated local coupled-cluster methods using pair natural orbitals. Wiley Interdiscip Rev Comput Mol Sci 8:e1371. https://doi.org/10.1002/wcms.1371

    Google Scholar 

  39. Nagy PR, Kállay M (2019) Approaching the basis set limit of CCSD(T) energies for large molecules with local natural orbital coupled-cluster methods. J Chem Theory Comput 15:5275–5298. https://doi.org/10.1021/acs.jctc.9b00511

    Google Scholar 

  40. Ramakrishnan R, Dral PO, Rupp M, von Lilienfeld OA (2015) Big data meets quantum chemistry approximations: the Δ-machine learning approach. J Chem Theory Comput 11:2087–2096. https://doi.org/10.1021/acs.jctc.5b00099

    Google Scholar 

  41. Ruscic B, Pinzon RE, Morton ML et al (2004) Introduction to active thermochemical tables: several “Key” enthalpies of formation revisited. J Phys Chem A 108:9979–9997. https://doi.org/10.1021/jp047912y

    Google Scholar 

  42. Ruscic B, Pinzon RE, von Laszewski G et al (2005) Active thermochemical tables: thermochemistry for the 21st century. J Phys Conf Ser 16:561–570. https://doi.org/10.1088/1742-6596/16/1/078

    ADS  Google Scholar 

  43. Ruscic B, Bross DH (2020) Active Thermochemical Tables (ATcT) values based on ver. 1.122p of the Thermochemical Network. http://atct.anl.gov

  44. Sylvetsky N, Peterson KA, Karton A, Martin JML (2016) Toward a W4–F12 approach: can explicitly correlated and orbital-based ab initio CCSD(T) limits be reconciled? J Chem Phys 144:214101. https://doi.org/10.1063/1.4952410

    ADS  Google Scholar 

  45. Harding ME, Vázquez J, Ruscic B, et al (2008) High-accuracy extrapolated ab initio thermochemistry. III. Additional improvements and overview. J Chem Phys 128:114111. https://doi.org/10.1063/1.2835612

  46. Raghavachari K, Trucks GW, Pople JA, Head-Gordon M (1989) A fifth-order perturbation comparison of electron correlation theories. Chem Phys Lett 157:479–483. https://doi.org/10.1016/S0009-2614(89)87395-6

    ADS  Google Scholar 

  47. Karton A (2019) Basis set convergence of high-order coupled cluster methods up to CCSDTQ567 for a highly multireference molecule. Chem Phys Lett 737:136810. https://doi.org/10.1016/j.cplett.2019.136810

    Google Scholar 

  48. Sylvetsky N, Martin JML (2019) Probing the basis set limit for thermochemical contributions of inner-shell correlation: balance of core-core and core-valence contributions. Mol Phys 117:1078–1087. https://doi.org/10.1080/00268976.2018.1478140

    ADS  Google Scholar 

  49. Martin JML, Sylvetsky N (2019) A simple model for scalar relativistic corrections to molecular total atomisation energies. Mol Phys 117:2225–2232. https://doi.org/10.1080/00268976.2018.1509147

    ADS  Google Scholar 

  50. Kramida A, Ralchenko Y, Reader J, ASD Team N (2018) NIST Atomic Spectra Database (version 5.5.6). https://physics.nist.gov/asd. Accessed 1 Aug 2018

  51. Gauss J, Tajti A, Kállay M et al (2006) Analytic calculation of the diagonal Born-Oppenheimer correction within configuration-interaction and coupled-cluster theory. J Chem Phys 125:144111. https://doi.org/10.1063/1.2356465

    ADS  Google Scholar 

  52. Karton A, Rabinovich E, Martin JML, Ruscic B (2006) W4 theory for computational thermochemistry: in pursuit of confident sub-kJ/mol predictions. J Chem Phys 125:144108. https://doi.org/10.1063/1.2348881

    ADS  Google Scholar 

  53. Tajti A, Szalay PG, Császár AG et al (2004) HEAT: high accuracy extrapolated ab initio thermochemistry. J Chem Phys 121:11599–11613. https://doi.org/10.1063/1.1811608

    ADS  Google Scholar 

  54. Bomble YJ, Vázquez J, Kállay M, et al (2006) High-accuracy extrapolated ab initio thermochemistry. II. Minor improvements to the protocol and a vital simplification. J Chem Phys 125:064108. https://doi.org/10.1063/1.2206789

  55. Thorpe JH, Lopez CA, Nguyen TL, et al (2019) High-accuracy extrapolated ab initio thermochemistry. IV. A modified recipe for computational efficiency. J Chem Phys 150:224102. https://doi.org/10.1063/1.5095937

  56. Feller D, Peterson KA, Dixon DA (2008) A survey of factors contributing to accurate theoretical predictions of atomization energies and molecular structures. J Chem Phys 129:204105. https://doi.org/10.1063/1.3008061

    ADS  Google Scholar 

  57. Li S, Hennigan JM, Dixon DA, Peterson KA (2009) Accurate thermochemistry for transition metal oxide clusters. J Phys Chem A 113:7861–7877. https://doi.org/10.1021/jp810182a

    Google Scholar 

  58. Bross DH, Hill JG, Werner H-J, Peterson KA (2013) Explicitly correlated composite thermochemistry of transition metal species. J Chem Phys 139:094302. https://doi.org/10.1063/1.4818725

    ADS  Google Scholar 

  59. Dixon D, Feller D, Peterson K (2012) A Practical guide to reliable first principles computational thermochemistry predictions across the periodic table. Annu Rep Comput Chem 8:1–28. https://doi.org/10.1016/B978-0-444-59440-2.00001-6

    Google Scholar 

  60. Feller D (2013) Benchmarks of improved complete basis set extrapolation schemes designed for standard CCSD(T) atomization energies. J Chem Phys 138:074103. https://doi.org/10.1063/1.4791560

    ADS  Google Scholar 

  61. Feller D, Peterson KA, Dixon DA (2016) The Impact of larger basis sets and explicitly correlated coupled cluster theory on the feller–peterson–dixon composite method. Annu Rep Comput Chem 12:47–48. https://doi.org/10.1016/bs.arcc.2016.02.001

    Google Scholar 

  62. Feller D (2016) Application of a convergent, composite coupled cluster approach to bound state, adiabatic electron affinities in atoms and small molecules. J Chem Phys 144:014105. https://doi.org/10.1063/1.4939184

    ADS  Google Scholar 

  63. Fogueri UR, Kozuch S, Karton A, Martin JML (2012) A simple DFT-based diagnostic for nondynamical correlation. Theor Chem Acc 132:1291. https://doi.org/10.1007/s00214-012-1291-y

    Google Scholar 

  64. Schwerdtfeger P, Smits OR, Pyykkö P (2020) The periodic table and the physics that drives it. Nat Rev Chem 4:359–380. https://doi.org/10.1038/s41570-020-0195-y

    Google Scholar 

  65. Shiozaki T, Hirata S (2007) Grid-based numerical Hartree-Fock solutions of polyatomic molecules. Phys Rev A 76:040503. https://doi.org/10.1103/PhysRevA.76.040503

    ADS  Google Scholar 

  66. Lehtola S (2019) A review on non-relativistic, fully numerical electronic structure calculations on atoms and diatomic molecules. Int J Quantum Chem 119:1–31. https://doi.org/10.1002/qua.25968

    Google Scholar 

  67. Jensen F (2017) How large is the elephant in the density functional theory room? J Phys Chem A 121:6104–6107. https://doi.org/10.1021/acs.jpca.7b04760

    Google Scholar 

  68. Yanai T, Fann GI, Gan Z et al (2004) Multiresolution quantum chemistry in multiwavelet bases: Hartree-Fock exchange. J Chem Phys 121:6680–6688. https://doi.org/10.1063/1.1790931

    ADS  Google Scholar 

  69. Jensen SR, Saha S, Flores-Livas JA et al (2017) The elephant in the room of density functional theory calculations. J Phys Chem Lett 8:1449–1457. https://doi.org/10.1021/acs.jpclett.7b00255

    Google Scholar 

  70. Jensen F (2005) Estimating the Hartree—Fock limit from finite basis set calculations. Theor Chem Acc 113:267–273. https://doi.org/10.1007/s00214-005-0635-2

    Google Scholar 

  71. Karton A, Martin JML (2005) Comment on: “Estimating the Hartree-Fock limit from finite basis set calculations” [Jensen F (2005) Theor Chem Acc 113:267]. Theor Chem Acc 115:330–333. https://doi.org/10.1007/s00214-005-0028-6

    Google Scholar 

  72. Ranasinghe DS, Petersson GA (2013) CCSD(T)/CBS atomic and molecular benchmarks for H through Ar. J Chem Phys 138:144104. https://doi.org/10.1063/1.4798707

    ADS  Google Scholar 

  73. Schwenke DW (2005) The extrapolation of one-electron basis sets in electronic structure calculations: how it should work and how it can be made to work. J Chem Phys 122:014107. https://doi.org/10.1063/1.1824880

    ADS  Google Scholar 

  74. Halkier A, Helgaker T, Jørgensen P et al (1998) Basis-set convergence in correlated calculations on Ne, N2, and H2O. Chem Phys Lett 286:243–252. https://doi.org/10.1016/S0009-2614(98)00111-0

    ADS  Google Scholar 

  75. Martin JML, de Oliveira G (1999) Towards standard methods for benchmark quality ab initio thermochemistry—W1 and W2 theory. J Chem Phys 111:1843–1856. https://doi.org/10.1063/1.479454

    ADS  Google Scholar 

  76. Varandas AJC (2018) Straightening the hierarchical staircase for basis set extrapolations: a low-cost approach to high-accuracy computational chemistry. Annu Rev Phys Chem 69:177–203. https://doi.org/10.1146/annurev-physchem-050317-021148

    ADS  Google Scholar 

  77. Hättig C, Klopper W, Köhn A, Tew DP (2012) Explicitly correlated electrons in molecules. Chem Rev 112:4–74. https://doi.org/10.1021/cr200168z

    Google Scholar 

  78. Kong L, Bischoff FA, Valeev EF (2012) Explicitly correlated R12/F12 methods for electronic structure. Chem Rev 112:75–107. https://doi.org/10.1021/cr200204r

    Google Scholar 

  79. Kesharwani MK, Sylvetsky N, Köhn A et al (2018) Do CCSD and approximate CCSD-F12 variants converge to the same basis set limits? The case of atomization energies. J Chem Phys 149:154109. https://doi.org/10.1063/1.5048665

    ADS  Google Scholar 

  80. Zhang IY, Ren X, Rinke P et al (2013) Numeric atom-centered-orbital basis sets with valence-correlation consistency from H to Ar. New J Phys 15:123033. https://doi.org/10.1088/1367-2630/15/12/123033

    Google Scholar 

  81. Blum V, Gehrke R, Hanke F et al (2009) Ab initio molecular simulations with numeric atom-centered orbitals. Comput Phys Commun 180:2175–2196. https://doi.org/10.1016/j.cpc.2009.06.022

    ADS  MATH  Google Scholar 

  82. te Velde G, Bickelhaupt FM, Baerends EJ et al (2001) Chemistry with ADF. J Comput Chem 22:931–967. https://doi.org/10.1002/jcc.1056

    Google Scholar 

  83. Förster A, Visscher L (2020) Double hybrid DFT calculations with Slater type orbitals. J Comput Chem 41:1660–1684. https://doi.org/10.1002/jcc.26209

    Google Scholar 

  84. Davidson ER, Feller D (1986) Basis set selection for molecular calculations. Chem Rev 86:681–696. https://doi.org/10.1021/cr00074a002

    Google Scholar 

  85. Shavitt I (1993) The history and evolution of Gaussian basis sets. Isr J Chem 33:357–367. https://doi.org/10.1002/ijch.199300044

    Google Scholar 

  86. Peterson KA (2007) Chapter 11 Gaussian basis sets exhibiting systematic convergence to the complete basis set limit. Annu Rep Comput Chem 3:195–206. https://doi.org/10.1016/S1574-1400(07)03011-3

  87. Hill JG (2013) Gaussian basis sets for molecular applications. Int J Quantum Chem 113:21–34. https://doi.org/10.1002/qua.24355

    Google Scholar 

  88. Jensen F (2013) Atomic orbital basis sets. Wiley Interdiscip Rev Comput Mol Sci 3:273–295. https://doi.org/10.1002/wcms.1123

    Google Scholar 

  89. Nagy B, Jensen F (2017) Basis Sets in Quantum Chemistry. In: Parrill AL, Lipkowitz KB (eds) Reviews in Computational Chemistry, vol 30. Wiley, pp 93–149

    Google Scholar 

  90. Dunning TH (1989) Gaussian basis sets for use in correlated molecular calculations. I. The atoms boron through neon and hydrogen. J Chem Phys 90:1007–1023. https://doi.org/10.1063/1.456153

    ADS  Google Scholar 

  91. Almlöf J, Taylor PR (1987) General contraction of Gaussian basis sets. I. Atomic natural orbitals for first- and second-row atoms. J Chem Phys 86:4070–4077. https://doi.org/10.1063/1.451917

    ADS  Google Scholar 

  92. Almlöf J, Taylor PR (1990) General contraction of Gaussian basis sets. II. Atomic natural orbitals and the calculation of atomic and molecular properties. J Chem Phys 92:551–560. https://doi.org/10.1063/1.458458

    ADS  Google Scholar 

  93. Widmark PO, Malmqvist PÅ, Roos BO (1990) Density matrix averaged atomic natural orbital (ANO) basis sets for correlated molecular wave functions—I First row atoms. Theor Chim Acta 77:291–306. https://doi.org/10.1007/BF01120130

    Google Scholar 

  94. Neese F, Valeev EF (2011) Revisiting the atomic natural orbital approach for basis sets: robust systematic basis sets for explicitly correlated and conventional correlated ab initio methods? J Chem Theory Comput 7:33–43. https://doi.org/10.1021/ct100396y

    Google Scholar 

  95. Zobel JP, Widmark P, Veryazov V (2019) The ANO-R Basis Set. J Chem Theory Comput acs.jctc.9b00873. https://doi.org/10.1021/acs.jctc.9b00873

  96. Jensen F (2001) Polarization consistent basis sets: principles. J Chem Phys 115:9113–9125. https://doi.org/10.1063/1.1413524

    ADS  Google Scholar 

  97. Jensen F (2013) Polarization consistent basis sets. VIII. The transition metals Sc-Zn. J Chem Phys 138:014107. https://doi.org/10.1063/1.4773017

  98. Weigend F, Ahlrichs R (2005) Balanced basis sets of split valence, triple zeta valence and quadruple zeta valence quality for H to Rn: Design and assessment of accuracy. Phys Chem Chem Phys 7:3297–3305. https://doi.org/10.1039/b508541a

    Google Scholar 

  99. Zhong S, Barnes EC, Petersson GA (2008) Uniformly convergent n-tuple-ζ augmented polarized (nZaP) basis sets for complete basis set extrapolations. I. Self-consistent field energies. J Chem Phys 129:184116. https://doi.org/10.1063/1.3009651

  100. Ten-no S (2004) Initiation of explicitly correlated Slater-type geminal theory. Chem Phys Lett 398:56–61. https://doi.org/10.1016/j.cplett.2004.09.041

    ADS  Google Scholar 

  101. Köhn A (2009) Explicitly correlated connected triple excitations in coupled-cluster theory. J Chem Phys 130:131101. https://doi.org/10.1063/1.3116792

    ADS  Google Scholar 

  102. Köhn A (2010) Explicitly correlated coupled-cluster theory using cusp conditions. II. Treatment of connected triple excitations. J Chem Phys 133:174118. https://doi.org/10.1063/1.3496373

  103. Møller C, Plesset MS (1934) Note on an approximation treatment for many-electron systems. Phys Rev 46:618–622. https://doi.org/10.1103/PhysRev.46.618

    ADS  MATH  Google Scholar 

  104. Schwartz C (1962) Ground state of the helium atom. Phys Rev 128:1146–1148. https://doi.org/10.1103/PhysRev.128.1146

    ADS  Google Scholar 

  105. Schwartz C (1962) Importance of angular correlations between atomic electrons. Phys Rev 126:1015–1019. https://doi.org/10.1103/PhysRev.126.1015

    ADS  Google Scholar 

  106. Hill RN (1985) Rates of convergence and error estimation formulas for the Rayleigh-Ritz variational method. J Chem Phys 83:1173–1196. https://doi.org/10.1063/1.449481

    ADS  Google Scholar 

  107. Kutzelnigg W, Morgan JD (1992) Rates of convergence of the partial-wave expansions of atomic correlation energies. J Chem Phys 96:4484–4508. https://doi.org/10.1063/1.462811

    ADS  Google Scholar 

  108. Weisstein EW MathWorld—A Wolfram Web Resource. In: Wolfram MathWorld. http://mathworld.wolfram.com/PolygammaFunction.html. Accessed 23 Jul 2018

  109. Helgaker T, Klopper W, Tew D (2008) Quantitative quantum chemistry. Mol Phys 106:2107–2143. https://doi.org/10.1080/00268970802258591

    ADS  Google Scholar 

  110. Bunge CF (1970) Electronic wave functions for atoms. Theor Chim Acta 16:126–144. https://doi.org/10.1007/BF00572782

    Google Scholar 

  111. Carroll DP, Silverstone HJ, Metzger RM (1979) Piecewise polynomial configuration interaction natural orbital study of 1 s2 helium. J Chem Phys 71:4142. https://doi.org/10.1063/1.438187

    ADS  Google Scholar 

  112. Nyden MR, Petersson GA (1981) Complete basis set correlation energies. I. The asymptotic convergence of pair natural orbital expansions. J Chem Phys 75:1843. https://doi.org/10.1063/1.442208

  113. Martin JML (1996) Ab initio total atomization energies of small molecules—towards the basis set limit. Chem Phys Lett 259:669–678. https://doi.org/10.1016/0009-2614(96)00898-6

    ADS  Google Scholar 

  114. Martin JML, Taylor PR (1997) Benchmark quality total atomization energies of small polyatomic molecules. J Chem Phys 106:8620–8623. https://doi.org/10.1063/1.473918

    ADS  Google Scholar 

  115. Barnes EC, Petersson GA, Feller D, Peterson KA (2008) The CCSD(T) complete basis set limit for Ne revisited. J Chem Phys 129:194115. https://doi.org/10.1063/1.3013140

    ADS  Google Scholar 

  116. Martin JML (2018) A simple ‘range extender’ for basis set extrapolation methods for MP2 and coupled cluster correlation energies. AIP Conf Proc 2040:020008. https://doi.org/10.1063/1.5079050

    Google Scholar 

  117. Feller D (1992) Application of systematic sequences of wave functions to the water dimer. J Chem Phys 96:6104. https://doi.org/10.1063/1.462652

    ADS  Google Scholar 

  118. Klopper W, Noga J, Koch H, Helgaker T (1997) Multiple basis sets in calculations of triples corrections in coupled-cluster theory. Theor Chem Acc Theory Comput Model (Theoretica Chim Acta) 97:164–176. https://doi.org/10.1007/s002140050250

    Google Scholar 

  119. Klopper W (2001) Highly accurate coupled-cluster singlet and triplet pair energies from explicitly correlated calculations in comparison with extrapolation techniques. Mol Phys 99:481–507. https://doi.org/10.1080/00268970010017315

    ADS  Google Scholar 

  120. Martin JML (2018) A simple, “range extender” for basis set extrapolation methods for MP2 and coupled cluster correlation energies. AIP Conf Proc 2040:020008. https://doi.org/10.1063/1.5079050

    Google Scholar 

  121. Werner H-J, Knowles PJ, Manby FR et al (2020) The Molpro quantum chemistry package. J Chem Phys 152:144107. https://doi.org/10.1063/5.0005081

    ADS  Google Scholar 

  122. Frisch MJ, Trucks GW, Schlegel HB, et al (2016) Gaussian 16 Revision C.01. Gaussian, Inc., Wallingford, CT

    Google Scholar 

  123. Pritchard BP, Altarawy D, Didier B et al (2019) New basis set exchange: an open, up-to-date resource for the molecular sciences community. J Chem Inf Model 59:4814–4820. https://doi.org/10.1021/acs.jcim.9b00725

    Google Scholar 

  124. Wilson AK, van Mourik T, Dunning TH (1996) Gaussian basis sets for use in correlated molecular calculations. VI. Sextuple zeta correlation consistent basis sets for boron through neon. J Mol Struct Theochem 388:339–349. https://doi.org/10.1016/S0166-1280(96)80048-0

    Google Scholar 

  125. Kendall RA, Dunning TH, Harrison RJ (1992) Electron affinities of the first-row atoms revisited. Systematic basis sets and wave functions. J Chem Phys 96:6796–6806. https://doi.org/10.1063/1.462569

    ADS  Google Scholar 

  126. Woon DE, Dunning TH (1993) Gaussian-basis sets for use in correlated molecular calculations. 3. The atoms aluminum through argon. J Chem Phys 98:1358–1371. https://doi.org/10.1063/1.464303

    ADS  Google Scholar 

  127. Dunning TH, Peterson KA, Wilson AK (2001) Gaussian basis sets for use in correlated molecular calculations. X. The atoms aluminum through argon revisited. J Chem Phys 114:9244–9253. https://doi.org/10.1063/1.1367373

    ADS  Google Scholar 

  128. Bauschlicher CW, Partridge H (1995) The sensitivity of B3LYP atomization energies to the basis set and a comparison of basis set requirements for CCSD(T) and B3LYP. Chem Phys Lett 240:533–540. https://doi.org/10.1016/0009-2614(95)91855-R

    ADS  Google Scholar 

  129. Martin JML (1998) Basis set convergence study of the atomization energy, geometry, and an harmonic force field of SO2: The importance of inner polarization functions. J Chem Phys 108:2791–2800. https://doi.org/10.1063/1.475670

    ADS  Google Scholar 

  130. Martin JML (2006) Heats of formation of perchloric acid, HClO4, and perchloric anhydride, Cl2O7. Probing the limits of W1 and W2 theory. J Mol Struct Theochem 771:19–26. https://doi.org/10.1016/j.theochem.2006.03.035

    Google Scholar 

  131. Woon DE, Dunning TH (1995) Gaussian basis sets for use in correlated molecular calculations. V. Core-valence basis sets for boron through neon. J Chem Phys 103:4572–4585. https://doi.org/10.1063/1.470645

    ADS  Google Scholar 

  132. Peterson KA, Dunning TH (2002) Accurate correlation consistent basis sets for molecular core–valence correlation effects: The second row atoms Al–Ar, and the first row atoms B-Ne revisited. J Chem Phys 117:10548–10560. https://doi.org/10.1063/1.1520138

    ADS  Google Scholar 

  133. Peterson KA, Adler TB, Werner H-J (2008) Systematically convergent basis sets for explicitly correlated wave functions: the atoms H, He, B-Ne, and Al-Ar. J Chem Phys 128:084102. https://doi.org/10.1063/1.2831537

    ADS  Google Scholar 

  134. Peterson KA, Kesharwani MK, Martin JML (2015) The cc-pV5Z-F12 basis set: reaching the basis set limit in explicitly correlated calculations. Mol Phys 113:1551–1558. https://doi.org/10.1080/00268976.2014.985755

    ADS  Google Scholar 

  135. Sylvetsky N, Kesharwani MK, Martin JML (2017) The aug-cc-pVnZ-F12 basis set family: Correlation consistent basis sets for explicitly correlated benchmark calculations on anions and noncovalent complexes. J Chem Phys 147:134106. https://doi.org/10.1063/1.4998332

    ADS  Google Scholar 

  136. Knizia G, Adler TB, Werner H-J (2009) Simplified CCSD(T)-F12 methods: theory and benchmarks. J Chem Phys 130:054104. https://doi.org/10.1063/1.3054300

    ADS  Google Scholar 

  137. Weigend F (2002) A fully direct RI-HF algorithm: Implementation, optimised auxiliary basis sets, demonstration of accuracy and efficiency. Phys Chem Chem Phys 4:4285–4291. https://doi.org/10.1039/b204199p

    Google Scholar 

  138. Weigend F, Köhn A, Hättig C (2002) Efficient use of the correlation consistent basis sets in resolution of the identity MP2 calculations. J Chem Phys 116:3175–3183. https://doi.org/10.1063/1.1445115

    ADS  Google Scholar 

  139. Yousaf KE, Peterson KA (2009) Optimized complementary auxiliary basis sets for explicitly correlated methods: aug-cc-pVnZ orbital basis sets. Chem Phys Lett 476:303–307. https://doi.org/10.1016/j.cplett.2009.06.003

    ADS  Google Scholar 

  140. Feller D, Peterson KA, Crawford TD (2006) Sources of error in electronic structure calculations on small chemical systems. J Chem Phys 124:054107. https://doi.org/10.1063/1.2137323

    ADS  Google Scholar 

  141. Karton A, Tarnopolsky A, Lamère J-F et al (2008) Highly accurate first-principles benchmark data sets for the parametrization and validation of density functional and other approximate methods. derivation of a robust, generally applicable, double-hybrid functional for thermochemistry and thermochemical. J Phys Chem A 112:12868–12886. https://doi.org/10.1021/jp801805p

    Google Scholar 

  142. Marchetti O, Werner HJ (2009) Accurate calculations of intermolecular interaction energies using explicitly correlated coupled cluster wave functions and a dispersion-weighted MP2 method. J Phys Chem A 113:11580–11585. https://doi.org/10.1021/jp9059467

    Google Scholar 

  143. Almlöf J, Taylor PR (1991) Atomic natural orbital (ANO) basis sets for quantum chemical calculations. Adv Quantum Chem 22:301–373. https://doi.org/10.1016/S0065-3276(08)60366-4

    ADS  Google Scholar 

  144. Martin JML, Santra G (2020) Empirical double-hybrid density functional theory: a ‘third way’ in between WFT and DFT. Isr J Chem 60:787–804. https://doi.org/10.1002/ijch.201900114

    Google Scholar 

  145. Kendall RA, Früchtl HA (1997) The impact of the resolution of the identity approximate integral method on modern ab initio algorithm development. Theor Chem Acc 97:158–163. https://doi.org/10.1007/s002140050249

    Google Scholar 

  146. Weigend F, Häser M, Patzelt H, Ahlrichs R (1998) RI-MP2: optimized auxiliary basis sets and demonstration of efficiency. Chem Phys Lett 294:143–152. https://doi.org/10.1016/S0009-2614(98)00862-8

    ADS  Google Scholar 

  147. Stein F, Hutter J, Rybkin VV (2020) Double-Hybrid DFT functionals for the condensed phase: Gaussian and plane waves implementation and evaluation. Molecules 25:5174. https://doi.org/10.3390/molecules25215174

    Google Scholar 

  148. Janesko BG, Henderson TM, Scuseria GE (2009) Screened hybrid density functionals for solid-state chemistry and physics. Phys Chem Chem Phys 11:443–454. https://doi.org/10.1039/b812838c

    Google Scholar 

  149. Körzdörfer T, Brédas JL (2014) Organic electronic materials: recent advances in the dft description of the ground and excited states using tuned range-separated hybrid functionals. Acc Chem Res 47:3284–3291. https://doi.org/10.1021/ar500021t

    Google Scholar 

  150. Baer R, Livshits E, Salzner U (2010) Tuned range-separated hybrids in density functional theory. Annu Rev Phys Chem 61:85–109. https://doi.org/10.1146/annurev.physchem.012809.103321

    Google Scholar 

  151. Refaely-Abramson S, Jain M, Sharifzadeh S et al (2015) Solid-state optical absorption from optimally tuned time-dependent range-separated hybrid density functional theory. Phys Rev B—Condens Matter Mater Phys 92:1–6. https://doi.org/10.1103/PhysRevB.92.081204

    Google Scholar 

  152. Ma Q, Werner H-J (2019) Accurate intermolecular interaction energies using explicitly correlated local coupled cluster methods [PNO-LCCSD(T)-F12]. J Chem Theory Comput 15:1044–1052. https://doi.org/10.1021/acs.jctc.8b01098

    Google Scholar 

  153. Ma Q, Werner H (2020) Scalable electron correlation methods. 7. Local open-shell coupled-cluster methods using pair natural orbitals: PNO-RCCSD and PNO-UCCSD. J Chem Theory Comput 16:3135-3151. https://doi.org/10.1021/acs.jctc.0c00192

  154. Ma Q, Werner H-J (2021) Scalable electron correlation methods. 8. Explicitly correlated open-shell coupled-cluster with pair natural orbitals PNO-RCCSD(T)-F12 and PNO-UCCSD(T)-F12. J Chem Theory Comput 17:902-906. https://doi.org/10.1021/acs.jctc.0c01129

  155. Sylvetsky N, Banerjee A, Alonso M, Martin JML (2020) Performance of localized coupled cluster methods in a moderately strong correlation regime: Hückel-Möbius interconversions in expanded porphyrins. J Chem Theory Comput 16:3641–3653. https://doi.org/10.1021/acs.jctc.0c00297

    Google Scholar 

  156. Hättig C, Tew DP, Köhn A (2010) Communications: Accurate and efficient approximations to explicitly correlated coupled-cluster singles and doubles, CCSD-F12. J Chem Phys 132:231102. https://doi.org/10.1063/1.3442368

    ADS  Google Scholar 

  157. Yang J, Hu W, Usvyat D, et al (2014) Ab initio determination of the crystalline benzene lattice energy to sub-kilojoule/mole accuracy. Science (80)345:640–643. https://doi.org/10.1126/science.1254419

  158. Bomble YJ, Stanton JF, Kállay M, Gauss J (2005) Coupled-cluster methods including noniterative corrections for quadruple excitations. J Chem Phys 123:054101. https://doi.org/10.1063/1.1950567

    ADS  Google Scholar 

  159. Matthews DA, Cheng L, Harding ME et al (2020) Coupled-cluster techniques for computational chemistry: The CFOUR program package. J Chem Phys 152:214108. https://doi.org/10.1063/5.0004837

    ADS  Google Scholar 

  160. Goerigk L, Hansen A, Bauer C et al (2017) A look at the density functional theory zoo with the advanced GMTKN55 database for general main group thermochemistry, kinetics and noncovalent interactions. Phys Chem Chem Phys 19:32184–32215. https://doi.org/10.1039/C7CP04913G

    Google Scholar 

  161. Mardirossian N, Head-Gordon M (2018) Survival of the most transferable at the top of Jacob’s ladder: Defining and testing the _ωB97M(2) double hybrid density functional. J Chem Phys 148:241736. https://doi.org/10.1063/1.5025226

    ADS  Google Scholar 

  162. Dral PO (2020) Quantum chemistry in the age of machine learning. J Phys Chem Lett 11:2336–2347. https://doi.org/10.1021/acs.jpclett.9b03664

    Google Scholar 

  163. Rackers JA, Wang Z, Lu C et al (2018) Tinker 8: software tools for molecular design. J Chem Theory Comput. https://doi.org/10.1021/acs.jctc.8b00529

    Google Scholar 

  164. Senftle TP, Hong S, Islam MM, et al (2016) The ReaxFF reactive force-field: development, applications and future directions. npj Comput Mater 2:15011. https://doi.org/10.1038/npjcompumats.2015.11

  165. Hill G, Peterson KA, Knizia G, Werner H (2009) Extrapolating MP2 and CCSD explicitly correlated correlation energies to the complete basis set limit with first and second row correlation consistent basis sets. J Chem Phys 131:194105

    ADS  Google Scholar 

  166. Brauer B, Kesharwani MK, Kozuch S, Martin JML (2016) The S66x8 benchmark for noncovalent interactions revisited: explicitly correlated ab initio methods and density functional theory. Phys Chem Chem Phys 18:20905–20925. https://doi.org/10.1039/C6CP00688D

    Google Scholar 

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Acknowledgements

This research was funded in part by the Israel Science Foundation (Grant 1969/20) and by the Minerva Foundation (grant 2020/05).

Supporting Information Microsoft Excel workbook with the relevant energetics is available at https://doi.org/10.34933/wis.000243.

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  1. Department of Molecular Chemistry and Materials Science, Weizmann Institute of Science, 7610001, Reḥovot, Israel

    Jan M. L. Martin

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    Taku Onishi

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Martin, J.M.L. (2022). Basis Set Convergence and Extrapolation of Connected Triple Excitation Contributions (T) in Computational Thermochemistry: The W4-17 Benchmark with Up to k Functions. In: Onishi, T. (eds) Quantum Science. Springer, Singapore. https://doi.org/10.1007/978-981-19-4421-5_8

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