Skip to main content
SpringerLink
Log in

Quantitative estimation of uncertainties from wavefunction diagnostics

  • Regular Article
  • Published:
Theoretical Chemistry Accounts Aims and scope Submit manuscript

Abstract

Coupled-cluster calculations with large basis sets are used widely to make predictions of gas-phase thermochemistry. Wavefunction diagnostics are sometimes used to indicate whether or not there is problematic multireference character that may cause errors. Here, we investigate whether existing diagnostics, as well as diagnostics proposed by us, can be used to estimate these errors quantitatively. We calculate the atomization energy of 50 molecules, including known multireference molecules such as CN, C2, O3, ortho-benzyne, formaldehyde oxide, and hydrogen trioxy radical. In addition to the c 20 , T 1, D 1, and %TAE[(T)] diagnostics, we assess the Hartree–Fock HOMO–LUMO gap, maximum occupation number defect, first vertical excitation energy, a direct estimate of multireference effects, and combinations of diagnostics as indicators of errant thermochemistry.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2

Similar content being viewed by others

Notes

  1. Certain commercial materials and equipment are identified in this paper in order to specify procedures completely. In no case does such identification imply recommendation or endorsement by the National Institute of Standards and Technology, nor does it imply that the material or equipment identified is necessarily the best available for the purpose.

References

  1. Raghavachari K, Trucks GW, Pople JA, Head-Gordon M (1989) A 5th-order perturbation comparison of electron correlation theories. Chem Phys Lett 157(6):479–483. doi:10.1016/s0009-2614(89)87395-6

    Article  CAS  Google Scholar 

  2. Watts JD, Bartlett RJ (1993) Triple excitations in coupled-cluster theory: energies and analytical derivatives. Int J Quantum Chem 48(Supplement 27):51–66. doi:10.1002/qua.560480809

    Article  Google Scholar 

  3. Dunning TH Jr (2000) A road map for the calculation of molecular binding energies. J Phys Chem A 104(40):9062–9080. doi:10.1021/jp001507z

    Article  CAS  Google Scholar 

  4. Peterson KA, Feller D, Dixon DA (2012) Chemical accuracy in ab initio thermochemistry and spectroscopy: current strategies and future challenges. Theor Chem Acc 131(1):1–20. doi:10.1007/s00214-011-1079-5

    Article  CAS  Google Scholar 

  5. Feller D, Peterson KA, Dixon DA (2012) Further benchmarks of a composite, convergent, statistically calibrated coupled-cluster-based approach for thermochemical and spectroscopic studies. Mol Phys 110(19–20):2381–2399. doi:10.1080/00268976.2012.684897

    Article  CAS  Google Scholar 

  6. Dixon DA, Feller D, Peterson KA (2012) Chapter one—a practical guide to reliable first principles computational thermochemistry predictions across the periodic table. In: Wheeler RA (ed) Annual reports in computational chemistry, vol 8. Elsevier, pp 1–28. doi:10.1016/B978-0-444-59440-2.00001-6

  7. Hättig C, Klopper W, Köhn A, Tew DP (2011) Explicitly correlated electrons in molecules. Chem Rev 112(1):4–74. doi:10.1021/cr200168z

    Article  Google Scholar 

  8. Peterson KA, Adler TB, Werner H-J (2008) Systematically convergent basis sets for explicitly correlated wavefunctions: the atoms H, He, B–Ne, and Al–Ar. J Chem Phys 128(8):084102. doi:10.1063/1.2831537

    Article  Google Scholar 

  9. Hill JG, Peterson KA, Knizia G, Werner HJ (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(19):194105. doi:10.1063/1.3265857

    Article  Google Scholar 

  10. Johnson RD III (2013) NIST Computational Chemistry Comparison and Benchmark Database, NIST Standard Reference Database Number 101, Release 16a, August 2013. National Institute of Standards and Technology. http://cccbdb.nist.gov

  11. Jiang WY, DeYonker NJ, Wilson AK (2012) Multireference character for 3d transition-metal-containing molecules. J Chem Theory Comput 8(2):460–468. doi:10.1021/ct2006852

    Article  CAS  Google Scholar 

  12. Lee TJ (2003) Comparison of the T 1 and D 1 diagnostics for electronic structure theory: a new definition for the open-shell D 1 diagnostic. Chem Phys Lett 372(3–4):362–367. doi:10.1016/S0009-2614(03)00435-4

    Article  CAS  Google Scholar 

  13. Lee TJ, Taylor PR (1989) A diagnostic for determining the quality of single-reference electron correlation methods. Int J Quantum Chem 36:199–207. doi:10.1002/qua.560360824

    Article  Google Scholar 

  14. Schultz NE, Gherman BF, Cramer CJ, Truhlar DG (2006) Pd n CO (n = 1,2): accurate ab initio bond energies, geometries, and dipole moments and the applicability of density functional theory for fuel cell modeling. J Phys Chem B 110(47):24030–24046. doi:10.1021/jp064467t

    Article  CAS  Google Scholar 

  15. Tishchenko O, Zheng JJ, Truhlar DG (2008) Multireference model chemistries for thermochemical kinetics. J Chem Theory Comput 4(8):1208–1219. doi:10.1021/ct800077r

    Article  CAS  Google Scholar 

  16. Zhao Y, Schultz NE, Truhlar DG (2006) Design of density functionals by combining the method of constraint satisfaction with parametrization for thermochemistry, thermochemical kinetics, and noncovalent interactions. J Chem Theory Comput 2(2):364–382. doi:10.1021/ct0502763

    Article  Google Scholar 

  17. 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(14):144108. doi:10.1063/1.2348881

    Article  Google Scholar 

  18. Gordon MS, Schmidt MW, Chaban GM, Glaesemann KR, Stevens WJ, Gonzalez C (1999) A natural orbital diagnostic for multiconfigurational character in correlated wave functions. J Chem Phys 110(9):4199–4207. doi:10.1063/1.478301

    Article  CAS  Google Scholar 

  19. Fogueri U, Kozuch S, Karton A, Martin JL (2012) A simple DFT-based diagnostic for nondynamical correlation. Theor Chem Acc 132(1):1–9. doi:10.1007/s00214-012-1291-y

    Google Scholar 

  20. Boguslawski K, Tecmer P, Legeza Ö, Reiher M (2012) Entanglement measures for single- and multireference correlation effects. J Phys Chem Lett 3(21):3129–3135. doi:10.1021/jz301319v

    Article  CAS  Google Scholar 

  21. Hanrath M (2008) Wavefunction quality and error estimation of single- and multi-reference coupled-cluster and CI methods: the H(4) model system. Chem Phys Lett 466(4–6):240–246. doi:10.1016/j.cplett.2008.10.046

    Article  CAS  Google Scholar 

  22. Jiang W, DeYonker NJ, Determan JJ, Wilson AK (2011) Toward accurate theoretical thermochemistry of first row transition metal complexes. J Phys Chem A 116(2):870–885. doi:10.1021/jp205710e

    Article  Google Scholar 

  23. Jiang W, Jeffrey CC, Wilson AK (2012) Empirical correction of nondynamical correlation energy for density functionals. J Phys Chem A 116(40):9969–9978. doi:10.1021/jp305341a

    Article  CAS  Google Scholar 

  24. Zhao Y, Tishchenko O, Gour JR, Li W, Lutz JJ, Piecuch P, Truhlar DG (2009) Thermochemical kinetics for multireference systems: addition reactions of ozone. J Phys Chem A 113(19):5786–5799. doi:10.1021/jp811054n

    Article  CAS  Google Scholar 

  25. Nielsen IMB, Janssen CL (1999) Double-substitution-based diagnostics for coupled-cluster and Møller–Plesset perturbation theory. Chem Phys Lett 310(5–6):568–576. doi:10.1016/S0009-2614(99)00770-8

    Article  CAS  Google Scholar 

  26. Cramer CJ (2004) Essentials of computational chemistry: theory and models, 2nd edn. Wiley, New York

    Google Scholar 

  27. Bak KL, Jorgensen P, Olsen J, Helgaker T, Klopper W (2000) Accuracy of atomization energies and reaction enthalpies in standard and extrapolated electronic wave function/basis set calculations. J Chem Phys 112(21):9229–9242. doi:10.1063/1.481544

    Article  CAS  Google Scholar 

  28. Schmidt MW, Gordon MS (1998) The construction and interpretation of MCSCF wavefunctions. Annu Rev Phys Chem 49:233–266. doi:10.1146/annurev.physchem.49.1.233

    Article  CAS  Google Scholar 

  29. Veryazov V, Malmqvist PA, Roos BO (2011) How to select active space for multiconfigurational quantum chemistry? Int J Quantum Chem 111(13):3329–3338. doi:10.1002/qua.23068

    Article  CAS  Google Scholar 

  30. Nguyen MT, Nguyen TL, Ngan VT, Nguyen HMT (2007) Heats of formation of the Criegee formaldehyde oxide and dioxirane. Chem Phys Lett 448(4–6):183–188. doi:10.1016/j.cplett.2007.10.033

    Article  CAS  Google Scholar 

  31. Szabo A, Ostlund NS (1996) Modern quantum chemistry: introduction to advanced electronic structure theory. Dover Publications, New York

    Google Scholar 

  32. Krylov AI (2001) Size-consistent wave functions for bond-breaking: the equation-of-motion spin-flip model. Chem Phys Lett 338(4–6):375–384. doi:10.1016/s0009-2614(01)00287-1

    Article  CAS  Google Scholar 

  33. Irikura KK, Hudgens JW, Johnson RD III (1995) Spectroscopy of the fluoromethylene radicals HCF and DCF by 2 + 1 resonance enhanced multiphoton ionization spectroscopy and by ab initio calculation. J Chem Phys 103:1303–1308. doi:10.1063/1.469807

    Article  CAS  Google Scholar 

  34. Hernandez-Lamoneda R, Bartolomei M, Hernandez MI, Campos-Martinez J, Dayou F (2005) Intermolecular potential of the O2-O2 dimer. An ab initio study and comparison with experiment. J Phys Chem A 109(50):11587–11595. doi:10.1021/jp053728g

    Article  CAS  Google Scholar 

  35. Shiozaki T, Knizia G, Werner HJ (2011) Explicitly correlated multireference configuration interaction: MRCI-F12. J Chem Phys 134(3):034113. doi:10.1063/1.3528720

    Article  Google Scholar 

  36. Langhoff SR, Davidson ER (1974) Configuration interaction calculations on nitrogen molecule. Int J Quantum Chem 8(1):61–72. doi:10.1002/qua.560080106

    Article  CAS  Google Scholar 

  37. Davidson ER, Silver DW (1977) Size consistency in the dilute helium gas electronic structure. Chem Phys Lett 52(3):403–406. doi:10.1016/0009-2614(77)80475-2

    Article  CAS  Google Scholar 

  38. Ivanov VV, Lyakhy DI, Adamowicz L (2005) New indices for describing the multi-configurational nature of the coupled cluster wave function. Mol Phys 103(15–16):2131–2139. doi:10.1080/00268970500083283

    Article  CAS  Google Scholar 

  39. Hehre WJ, Radom L, von Schleyer R P, Pople JA (1986) Ab initio molecular orbital theory. Wiley, New York

    Google Scholar 

  40. Knizia G, Adler TB, Werner H-J (2009) Simplified CCSD(T)-F12 methods: theory and benchmarks. J Chem Phys 130(5):054104–054120. doi:10.1063/1.3054300

    Article  Google Scholar 

  41. Knowles PJ, Hampel C, Werner H-J (1993) Coupled cluster theory for high spin, open shell reference wave functions. J Chem Phys 99(7):5219–5227. doi:10.1063/1.465990

    Article  CAS  Google Scholar 

  42. Knowles PJ, Hampel C, Werner H-J (2000) Erratum: “Coupled cluster theory for high spin, open shell reference wave functions” [J. Chem. Phys. 99, 5219 (1993)]. J Chem Phys 112(6):3106–3107. doi:10.1063/1.480886

    Article  CAS  Google Scholar 

  43. Bross DH, Hill JG, Werner H-J, Peterson KA (2013) Explicitly correlated composite thermochemistry of transition metal species. J Chem Phys 139(9):094302. doi:10.1063/1.4818725

    Article  Google Scholar 

  44. Tew DP, Klopper W, Neiss C, Hättig C (2007) Quintuple-zeta quality coupled-cluster correlation energies with triple-zeta basis sets. PCCP 9(16):1921–1930. doi:10.1039/b617230j

    Article  CAS  Google Scholar 

  45. Hill JG, Mazumder S, Peterson KA (2010) Correlation consistent basis sets for molecular core-valence effects with explicitly correlated wave functions: the atoms B–Ne and Al–Ar. J Chem Phys 132(5):054108. doi:10.1063/1.3308483

    Article  Google Scholar 

  46. Varandas AJC (2008) Extrapolation to the complete-basis-set limit and the implications of avoided crossings: the X 1Σ+ g, B 1Δg, and B’ 1Σ+ g states of C2. J Chem Phys 129(23):234103. doi:10.1063/1.3036115

    Article  CAS  Google Scholar 

  47. Ruscic B (2014) Active Thermochemical Tables (ATcT) values based on version 1.112 of the Thermochemical Network. http://atct.anl.gov/index.html

  48. Werner HJ, Knowles PJ, Knizia G, Manby FR, Schütz M, Celani P, Korona T, Lindh R, Mitrushenkov A, Rauhut G, Shamasundar KR, Adler TB, Amos RD, Bernhardsson A, Berning A, Cooper DL, Deegan MJO, Dobbyn AJ, Eckert F, Goll E, Hampel C, Hesselmann A, Hetzer G, Hrenar T, Jansen G, Köppl C, Liu Y, Lloyd AW, Mata RA, May AJ, McNicholas SJ, Meyer W, Mura ME, Nicklaß A, O’Neill DP, Palmieri P, Peng D, Pflüger K, Pitzer R, Reiher M, Shiozaki T, Stoll H, Stone AJ, Tarroni R, Thorsteinsson T, Wang M (2012) MOLPRO, version 2012.1, a package of ab initio programs

  49. Frisch MJ, Trucks GW, Schlegel HB, Scuseria GE, Robb MA, Cheeseman JR, Scalmani G, Barone V, Mennucci B, Petersson GA, Nakatsuji H, Caricato M, Li X, Hratchian HP, Izmaylov AF, Bloino J, Zheng G, Sonnenberg JL, Hada M, Ehara M, Toyota K, Fukuda R, Hasegawa J, Ishida M, Nakajima T, Honda Y, Kitao O, Nakai H, Vreven T, Jr. JAM, Peralta JE, Ogliaro F, Bearpark M, Heyd JJ, Brothers E, Kudin KN, Staroverov VN, Kobayashi R, Normand J, Raghavachari K, Rendell A, Burant JC, Iyengar SS, Tomasi J, Cossi M, Rega N, Millam JM, Klene M, Knox JE, Cross JB, Bakken V, Adamo C, Jaramillo J, Gomperts R, Stratmann RE, Yazyev O, Austin AJ, Cammi R, Pomelli C, Ochterski JW, Martin RL, Morokuma K, Zakrzewski VG, Voth GA, Salvador P, Dannenberg JJ, Dapprich S, Daniels AD, Farkas Ö, Foresman JB, Ortiz JV, Cioslowski J, Fox DJ (2009) Gaussian 09, Revision C.01. Gaussian, Inc.

  50. Irikura KK (2007) Experimental vibrational zero-point energies: diatomic molecules. J Phys Chem Ref Data 36(2):389–397. doi:10.1063/1.2436891

    Article  CAS  Google Scholar 

  51. Neff M, Rauhut G (2009) Toward large scale vibrational configuration interaction calculations. J Chem Phys 131(12):124129. doi:10.1063/1.3243862

    Article  Google Scholar 

  52. Vosko SH, Wilk L, Nusair M (1980) Accurate spin-dependent electron liquid correlation energies for local spin density calculations: a critical analysis. Can J Phys 58(8):1200–1211. doi:10.1139/p80-159

    Article  CAS  Google Scholar 

  53. Roy TK, Gerber RB (2013) Vibrational self-consistent field calculations for spectroscopy of biological molecules: new algorithmic developments and applications. PCCP 15(24):9468–9492. doi:10.1039/C3CP50739D

    Article  CAS  Google Scholar 

  54. Yagi K, Hirata S, Hirao K (2007) Efficient configuration selection scheme for vibrational second-order perturbation theory. J Chem Phys 127(3):034111. doi:10.1063/1.2748774

    Article  Google Scholar 

  55. Barone V (2005) Anharmonic vibrational properties by a fully automated second-order perturbative approach. J Chem Phys 122(1):014108. doi:10.1063/1.1824881

    Article  Google Scholar 

  56. Booth GH, Cleland D, Thom AJW, Alavi A (2011) Breaking the carbon dimer: the challenges of multiple bond dissociation with full configuration interaction quantum Monte Carlo methods. J Chem Phys 135(8):084104. doi:10.1063/1.3624383

    Article  Google Scholar 

  57. Varandas AJC (2012) Ab initio treatment of bond-breaking reactions: accurate course of HO3 dissociation and revisit to isomerization. J Chem Theory Comput 8(2):428–441. doi:10.1021/ct200773b

    Article  CAS  Google Scholar 

  58. Ivanic J, Schmidt MW, Luke B (2012) High-level theoretical study of the NO dimer and tetramer: has the tetramer been observed? J Chem Phys 137(21):214316. doi:10.1063/1.4769226

    Article  Google Scholar 

  59. Qi B, Su KH, Wang YB, Wen ZY, Tang XY (1998) Gaussian-2 calculations of the thermochemistry of Criegee intermediates in gas phase reactions. Acta Phys Chim Sin 14(11):1033–1039. doi:10.3866/PKU.WHXB19981114

    CAS  Google Scholar 

  60. Cremer D, Gauss J, Kraka E, Stanton JF, Bartlett RJ (1993) A CCSD(T) investigation of carbonyl oxide and dioxirane—equilibrium geometries, dipole-moments, infrared-spectra, heats of formation and isomerization energies. Chem Phys Lett 209(5–6):547–556. doi:10.1016/0009-2614(93)80131-8

    Article  CAS  Google Scholar 

  61. Wenthold PG (2010) Thermochemical Properties of the Benzynes. Aust J Chem 63(7):1091–1098. doi:10.1071/ch10126

    Article  CAS  Google Scholar 

  62. Karton A, Kettner M, Wild DA (2013) Sneaking up on the Criegee intermediate from below: predicted photoelectron spectrum of the CH2OO− anion and W3-F12 electron affinity of CH2OO. Chem Phys Lett 585:15–20. doi:10.1016/j.cplett.2013.08.075

    Article  CAS  Google Scholar 

  63. Welch BL (1947) The generalization of ‘Student’s’ problem when several different population variances are involved. Biometrika 34(1–2):28–35. doi:10.1093/biomet/34.1-2.28

    CAS  Google Scholar 

  64. NIST/SEMATECH e-Handbook of Statistical Methods, Sect. 1.3.5.10—Levene Test for Equality of Variances. http://www.itl.nist.gov/div898/handbook/eda/section3/eda35a.htm. Accessed 7 Sept 2014

  65. Brown MB, Forsythe AB (1974) Robust tests for the equality of variances. J Am Stat Assoc 69(346):364–367. doi:10.1080/01621459.1974.10482955

    Article  Google Scholar 

  66. 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(20):32. doi:10.1063/1.3008061

    Article  Google Scholar 

  67. 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(4–6):165–178. doi:10.1016/j.cplett.2011.05.007

    Article  CAS  Google Scholar 

  68. Pople JA, Head-Gordon M, Fox DJ, Raghavachari K, Curtiss LA (1989) Gaussian-1 theory—a general procedure for prediction of molecular-energies. J Chem Phys 90(10):5622–5629. doi:10.1063/1.456415

    Article  CAS  Google Scholar 

  69. Curtiss LA, Redfern PC, Raghavachari K (2007) Gaussian-4 theory. J Chem Phys 126(8):084108. doi:10.1063/1.2436888

    Article  Google Scholar 

Download references

Acknowledgments

This research was performed while MKS held a National Research Council Research Associateship Award at the National Institute of Standards and Technology. We thank the anonymous reviewers for copious suggestions and patience, and Drs. Russell D. Johnson III, Thomas C. Allison, and Yamil Simon for helpful discussions and comments on the manuscript.

Author information

Authors and Affiliations

  1. Chemical Sciences Division, National Institute of Standards and Technology, Gaithersburg, MD, 20899-8320, USA

    Matthew K. Sprague & Karl K. Irikura

Authors
  1. Matthew K. Sprague
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Karl K. Irikura
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Karl K. Irikura.

Additional information

Dedicated to Professor Thom Dunning and published as part of the special collection of articles celebrating his career upon his retirement.

Electronic supplementary material

Below is the link to the electronic supplementary material.

214_2014_1544_MOESM1_ESM.pdf

Computed electronic energies and applicable corrections (core-valence, spin orbit, and anharmonic ZPE); details of high-spin states used to compute ΔE exc; summary of anharmonic ZPE data; summary of combined diagnostic values; diagnostic performance using raw error in atomization energy for the analysis; plots showing correlation between pairs of diagnostics; plots showing correlation between f and diagnostics; plot showing correlation between raw error and atomization energy; Cartesian coordinates, spectroscopic constants, and vibrational frequencies for all CCSD(T) and MRCI calculations (PDF 520 kb)

214_2014_1544_MOESM2_ESM.xlsx

Summary of combined diagnostic performance; comparison of additional corrections (relativistic, DBOC, higher order) to errors in atomization energies (XLSX 286 kb)

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Sprague, M.K., Irikura, K.K. Quantitative estimation of uncertainties from wavefunction diagnostics. Theor Chem Acc 133, 1544 (2014). https://doi.org/10.1007/s00214-014-1544-z

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s00214-014-1544-z

Keywords

Search

Navigation