Skip to main content
SpringerLink
Log in

Numerical integration of atomic electron density with double exponential formula for density functional calculation

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

Abstract

In the preceding study, we reported an application of the double exponential formula to the radial quadrature grid for numerical integration of the radial electron distribution function. Three-type new radial grids with the double exponential transformation were introduced. The performance of radial grids was compared between the double exponential grids and the grids proposed in earlier studies by applying to the electron-counting integrals of noble gas atoms and diatomic molecules including alkali metals, halogens, and transition metals. It was confirmed that the change in accuracy of the quadrature approximation depending on atomic or molecular species is not significant for the double exponential integration schemes rather than the other integration schemes. In the present study, we further investigate the accuracy of the double exponential formula for the electron-counting integrals of all the atoms from H to Kr in the periodic table to elucidate the stable performance of the double exponential radial grids. The electron densities of the atoms are calculated with the Gauss-type orbital basis functions at the B3LYP level. The quadrature accuracy and convergence behavior of numerical integration are compared among the double exponential formula and the formulas proposed by Treutler et al. and by Mura et al. The results reveal that the double exponential radial grids remarkably improve the convergence rate toward high accuracy compared with the previous radial grids, particularly for heavy elements in the 4th period, without fine tuning of the radial grids for each atom.

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
Fig. 3
Fig. 4

Similar content being viewed by others

References

  1. Becke AD (1988) J Chem Phys 88:2547–2553

    Article  CAS  Google Scholar 

  2. Lebedev VI (1975) Comp Math Math Phys 15:44–51

    Article  Google Scholar 

  3. Lebedev VI (1976) Comp Math Math Phys 16:10–24

    Article  Google Scholar 

  4. Lebedev VI (1977) Sib Math J 18:99–107

    Article  Google Scholar 

  5. Lebedev VI, Skorokhodov AL (1992) Russ Acad Sci Dokl Math 45:587–592

    Google Scholar 

  6. Lebedev VI (1995) Russ Acad Sci Dokl Math 50:283–286

    Google Scholar 

  7. Lebedev VI, Laikov DN (1999) Dokl Math 59:477–481

    Google Scholar 

  8. Murray CW, Handy NC, Laming GJ (1993) Mol Phys 78:997–1014

    Article  CAS  Google Scholar 

  9. Gill PMW, Johnson BG, Pople JA (1993) Chem Phys Lett 209:506–512

    Article  CAS  Google Scholar 

  10. Pérez-Jordá JM, Becke AD, San-Fabián E (1994) J Chem Phys 100:6520–6534

    Article  Google Scholar 

  11. Treutler O, Ahlrichs R (1995) J Chem Phys 102:346–354

    Article  CAS  Google Scholar 

  12. Mura ME, Knowles PJ (1996) J Chem Phys 104:9848–9858

    Article  CAS  Google Scholar 

  13. Krack M, Köster AM (1998) J Chem Phys 108:3226–3234

    Article  CAS  Google Scholar 

  14. Lin Z, Jaffe JE, Hess AC (1999) J Phys Chem A 103:2117–2127

    Article  CAS  Google Scholar 

  15. Ishikawa H, Yamamoto K, Fujima K, Iwasawa M (1999) Int J Quantum Chem 72:509–523

    Article  CAS  Google Scholar 

  16. Lindh R, Malmqvist P-Å, Gagliardi L (2001) Theor Chem Acc 106:178–187

    Article  CAS  Google Scholar 

  17. Gill PMW, Chien S-H (2003) J Comput Chem 24:732–740

    Article  CAS  Google Scholar 

  18. Köster AM, Flores-Moreno R, Reveles JU (2004) J Chem Phys 121:681–690

    Article  Google Scholar 

  19. Weber V, Daul C, Baltensperger R (2004) Comput Phys Comm 163:133–142

    Article  CAS  Google Scholar 

  20. Kakhiani K, Tsereteli K, Tsereteli P (2009) Comput Phys Comm 180:256–268

    Article  CAS  Google Scholar 

  21. El-Sherbiny A, Poirier RA (2004) J Comput Chem 25:1378–1384

    Article  CAS  Google Scholar 

  22. Chien S-H, Gill PMW (2006) J Comput Chem 27:730–739

    Article  CAS  Google Scholar 

  23. Martin JML, Bauschlicher CW Jr, Ricca A (2001) Comput Phys Comm 133:189–201

    Article  CAS  Google Scholar 

  24. Johnson ER, Wolkow RA, DiLabio GA (2004) Chem Phys Lett 394:334–338

    Article  CAS  Google Scholar 

  25. Papas BN, Schaefer HF III (2006) J Mol Struct Theochem 768:175–181

    Article  CAS  Google Scholar 

  26. Takahashi H, Mori M (1974) Publ RIMS Kyoto Univ 9: 721–741 Journal@rchive, Japan Science and Technology Agency (JST). http://www.journalarchive.jst.go.jp/english/jnltop_en.php?cdjournal=kyotoms1969. Accessed 2 Sept 2011

  27. Mori M (1985) J Comput Appl Math 12, 13: 119–130

    Google Scholar 

  28. Mori M, Sugihara M (2001) J Comput Appl Math 127:287–296

    Article  Google Scholar 

  29. Muhammad M, Mori M (2003) J Comput Appl Math 161:431–448

    Article  Google Scholar 

  30. Mori M (2005) Publ RIMS Kyoto Univ 41: 897–935 Publications of the Research Institute for Mathematical Sciences, Research Institute for Mathematical Sciences, Kyoto University. http://www.kurims.kyoto-u.ac.jp/~prims/list.html. Accessed 2 Sept 2011

  31. Tanaka K, Sugihara M, Murota K, Mori M (2009) Numer Math 111:631–655

    Article  Google Scholar 

  32. Mitani M (2011) Theor Chem Acc 130:645–669

    Article  CAS  Google Scholar 

  33. Bunge CF, Barrientos JA, Bunge AV (1993) At Data Nucl Data Tables 53:113–162

    Article  CAS  Google Scholar 

  34. Becke AD (1993) J Chem Phys 98:5648–5652

    Article  CAS  Google Scholar 

  35. Lee C, Yang W, Parr RG (1988) Phys Rev B 37:785–789

    Article  CAS  Google Scholar 

  36. Ditchfield R, Hehre WJ, Pople JA (1971) J Chem Phys 54:724–728

    Article  CAS  Google Scholar 

  37. Hehre WJ, Ditchfield R, Pople JA (1972) J Chem Phys 56:2257–2261

    Article  CAS  Google Scholar 

  38. Hariharan PC, Pople JA (1973) Theor Chim Acta 28:213–222

    Article  CAS  Google Scholar 

  39. Hariharan PC, Pople JA (1974) Mol Phys 27:209–214

    Article  CAS  Google Scholar 

  40. Gordon MS (1980) Chem Phys Lett 76:163–168

    Article  CAS  Google Scholar 

  41. Francl MM, Petro WJ, Hehre WJ, Binkley JS, Gordon MS, DeFrees DJ, Pople JA (1982) J Chem Phys 77:3654–3665

    Article  CAS  Google Scholar 

  42. Binning RC Jr, Curtiss LA (1990) J Comput Chem 11:1206–1216

    Article  CAS  Google Scholar 

  43. Blaudeau J-P, McGrath MP, Curtiss LA, Radom L (1997) J Chem Phys 107:5016–5021

    Article  CAS  Google Scholar 

  44. Rassolov VA, Pople JA, Ratner MA, Windus TL (1998) J Chem Phys 109:1223–1229

    Article  CAS  Google Scholar 

  45. Rassolov VA, Ratner MA, Pople JA, Redfern PC, Curtiss LA (2001) J Comput Chem 22:976–984

    Article  CAS  Google Scholar 

  46. Schäfer A, Horn H, Ahlrichs R (1992) J Chem Phys 97:2571–2577

    Article  Google Scholar 

  47. Frisch MJ, Trucks GW, Schlegel HB, Scuseria GE, Robb MA, Cheeseman JR, Montgomery JA Jr, Vreven T, Kudin KN, Burant JC, Millam JM, Iyengar SS, Tomasi J, Barone V, Mennucci B, Cossi M, Scalmani G, Rega N, Petersson GA, Nakatsuji H, Hada M, Ehara M, Toyota K, Fukuda R, Hasegawa J, Ishida M, Nakajima T, Honda Y, Kitao O, Nakai H, Klene M, Li X, Knox JE, Hratchian HP, Cross JB, Adamo C, Jaramillo J, Gomperts R, Stratmann RE, Yazyev O, Austin AJ, Cammi R, Pomelli C, Ochterski JW, Ayala PY, Morokuma K, Voth GA, Salvador P, Dannenberg JJ, Zakrzewski VG, Dapprich S, Daniels AD, Strain MC, Farkas O, Malick DK, Rabuck AD, Raghavachari K, Foresman JB, Ortiz JV, Cui Q, Baboul AG, Clifford S, Cioslowski J, Stefanov BB, Liu G, Liashenko A, Piskorz P, Komaromi I, Martin RL, Fox DJ, Keith T, Al-Laham MA, Peng CY, Nanayakkara A, Challacombe M, Gill PMW, Johnson B, Chen W, Wong MW, Gonzalez C, Pople JA (2003) Gaussian 03, Revision B.05, Gaussian, Inc., Pittsburgh, PA

Download references

Author information

Authors and Affiliations

  1. Department of Chemistry for Materials, Graduate School of Engineering, Mie University, 1577 Kurima-machiya, Tsu, Mie, 514-8507, Japan

    Masaki Mitani & Yasunori Yoshioka

Authors
  1. Masaki Mitani
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Yasunori Yoshioka
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Masaki Mitani.

Electronic supplementary material

Rights and permissions

Reprints and permissions

About this article

Cite this article

Mitani, M., Yoshioka, Y. Numerical integration of atomic electron density with double exponential formula for density functional calculation. Theor Chem Acc 131, 1169 (2012). https://doi.org/10.1007/s00214-012-1169-z

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s00214-012-1169-z

Keywords

Search

Navigation