Skip to main content

Advertisement

SpringerLink
Log in

Highly accurate O(N) method for delocalized systems

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

Abstract

The elongation method, developed in our groups, is an ab initio method approaching order O(N) type scalability with high efficiency and high accuracy (error <10−8 au/atom in total energy compared to the conventional calculation) that can be applied to any one-dimensional (polymer), two-dimensional (surface) or three-dimensional (solid material) systems. For strongly delocalized systems, however, the accuracy of the original elongation method for the targeted entire systems declines by approximately two orders of magnitude in the total energy as compared to the value obtained by the earlier implemented version of the elongation method for nondelocalized systems. The relatively small differences (10−6–10−8 au) between the elongation method and conventional method total energies have caused more serious errors in the second hyperpolarizability, γ, especially in nano-scale systems which have accompanying strong delocalization. In order to solve this problem, we have incorporated a simple correction technique based on an additional “orbital basis” to the “region basis” in our original elongation method procedures. Some not so-well-localized orbitals are incorporated into the interaction with the attacking molecule. This treatment has been applied to some model nano- and bio-systems that previously have shown strong delocalization, and the high accuracy in the energy obtained for nonstrongly delocalized systems was retained even for the strongly delocalized systems, both for the energies and for the second hyperpolarizabilities. This is a major breakthrough and now expands the systems for which the elongation method can be used to calculate and predict second-order nonlinear optical properties for delocalized systems.

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

Similar content being viewed by others

Abbreviations

RLMO:

Region localized molecular orbital

QFMM:

Quantum fast multipole method

SW-BN/CNT:

Single-wall boron nitride/carbon nanotube

SW-BNNT:

Single-wall boron nitride nanotubes

OTE:

Oligo(2,5-thienylene-ethynylene)

H2TPP:

Free base tetraphenylporphyrin

NLO:

Nonlinear optics

CMO:

Canonical molecular orbital

References

  1. Imamura A, Aoki Y, Maekawa K (1991) A theoretical synthesis of polymers by using uniform localization of molecular orbitals: Proposal of an elongation method. J Chem Phys 95:5419–5431

    Article  CAS  Google Scholar 

  2. Aoki Y, Imamura A (1992) Local density of states of aperiodic polymers using the localized orbitals from an ab initio elongation method. J Chem Phys 97:8432–8440

    Article  CAS  Google Scholar 

  3. Aoki Y, Suhai S, Imamura A (1994) An efficient cluster elongation method in density functional theory and its application to poly-hydrogen-bonding molecules. J Chem Phys 101:10808–10823

    Article  CAS  Google Scholar 

  4. Makowski M, Korchowiec J, Gu FL, Aoki Y (2010) Describing electron correlation effects in the framework of the elongation method—Elongation-MP2: formalism, implementation and efficiency. J Comput Chem 31:1733–1740

    CAS  Google Scholar 

  5. Gu FL, Imamura A, Aoki Y (2006) Elongation method for polymers and its application to nonlinear optics, in atoms, molecules and clusters in electric fields: theoretical approaches to the calculation of electric polarizabilities. In: Maroulis G (ed) Computational numerical and mathematical methods in sciences and engineering, Imperial College Press, London, 1:97–177. http://www.icpress.co.uk/chemistry/p464.html

  6. Goedecker S (1999) Linear scaling electronic structure methods. Rev Mod Phys 71:1085–1123

    Article  CAS  Google Scholar 

  7. Salek P, Hoest S, Thoegersen L, Joergensen P, Manninen P, Olsen J, Jansik B, Reine S, Pawlowski F, Tellgren E, Helgaker T, Coriani S (2007) Linear-scaling implementation of molecular electronic self-consistent field theory. J Chem Phys 126:114110

    Article  Google Scholar 

  8. Goedecker S, Teter M (1995) Tight-binding electronic-structure calculations and tight-binding molecular dynamics with localized orbitals. Phys Rev B 51:9455–9464

    Article  CAS  Google Scholar 

  9. Stephan U, Drabold D (1998) Order-N projection method for first-principles computations of electronic quantities and Wannier functions. Phys Rev B 57:6391–6407

    Article  CAS  Google Scholar 

  10. Li X-P, Nunes W, Vanderbilt D (1993) Density-matrix electronic-structure method with linear system-size scaling. Phys Rev B 47:10891–10894

    Article  CAS  Google Scholar 

  11. Zhao Q, Yang W (1995) Analytical energy gradients and geometry optimization in the divide‐and‐conquer method for large molecules. J Chem Phys 102:9598–9603

    Article  CAS  Google Scholar 

  12. Kim J, Mauri F, Galli G (1995) Total-energy global optimizations using nonorthogonal localized orbitals. Phys Rev B 52:1640–1648

    Article  CAS  Google Scholar 

  13. Hernandez E, Gillan M (1995) Self-consistent first-principles technique with linear scaling. Phys Rev B 51:10157–10160

    Article  CAS  Google Scholar 

  14. Yang W (1991) Direct calculation of electron density in density-functional theory. Phys Rev Lett 66:1438–1441

    Article  CAS  Google Scholar 

  15. Yang W, Lee T-S (1995) A density-matrix divide-and-conquer approach for electronic structure calculations of large molecules. J Chem Phys 103:5674–5678

    Article  CAS  Google Scholar 

  16. Akama T, Kobayashi M, Nakai H (2007) Implementation of divide-and-conquer method including Hartree-Fock exchange interaction. J Comput Chem 28:2003–2012

    Article  CAS  Google Scholar 

  17. Kobayashi M, Imamura Y, Nakai H (2007) Alternative linear-scaling methodology for the second-order Møller-Plesset perturbation calculation based on the divide-and-conquer method. J Chem Phys 127:074103

    Article  Google Scholar 

  18. Kitaura K, Ikeo E, Asada T, Nakano T, Uebayasi M (1999) Fragment molecular orbital method: an approximate computational method for large molecules. Chem Phys Lett 313:701–706

    Article  CAS  Google Scholar 

  19. Fedorov DG, Kitaura K (2004) The importance of three-body terms in the fragment molecular orbital method. J Chem Phys 120:6832–6840

    Article  CAS  Google Scholar 

  20. Fedorov DG, Kitaura K (2004) On the accuracy of the 3-body fragment molecular orbital method (FMO) applied to density functional theory. Chem Phys Lett 389:129–134

    Article  CAS  Google Scholar 

  21. Fedorov DG, Kitaura K (2007) Extending the power of quantum chemistry to large systems with the fragment molecular orbital method. J Phys Chem A 111:6904–6914

    Article  CAS  Google Scholar 

  22. Mochizuki Y, Koikegami S, Nakano T, Amari S, Kitaura K (2004) Large scale MP2 calculations with fragment molecular orbital scheme. Chem Phys Lett 396:473–479

    Article  CAS  Google Scholar 

  23. Mochizuki Y, Fukuzawa K, Kato A, Tanaka S, Kitaura K, Nakano T (2005) A configuration analysis for fragment interaction. Chem Phys Lett 410:247–253

    Article  CAS  Google Scholar 

  24. Mochizuki Y, Ishikawa T, Tanaka K, Tokiwa H, Nakano T, Tanaka S (2006) Dynamic polarizability calculation with fragment molecular orbital scheme. Chem Phys Lett 418:418–422

    Article  CAS  Google Scholar 

  25. Mochizuki Y, Yamashita K, Murase T, Nakano T, Fukuzawa K, Takematsu K, Watanabe H, Tanaka S (2008) Large scale FMO-MP2 calculations on a massively parallel-vector computer. Chem Phys Lett 457:396–403

    Article  CAS  Google Scholar 

  26. White SR (1992) Density matrix formulation for quantum renormalization groups. Phys Rev Lett 69:2863–2866

    Article  Google Scholar 

  27. Kurashige Y, Yanai T (2009) High-performance ab initio density matrix renormalization group method: applicability to large-scale multireference problems for metal compounds. J Chem Phys 130:234114

    Article  Google Scholar 

  28. Mizukami W, Kurashige Y, Yanai T (2010) Communication: novel quantum states of electron spins in polycarbenes from ab initio density matrix renormalization group calculations. J Chem Phys 133:091101

    Article  Google Scholar 

  29. Gu FL, Aoki Y, Korchowiec J, Imamura A, Kirtman B (2004) A new localization scheme for the elongation method. J Chem Phys 121:10385–10391

    Article  CAS  Google Scholar 

  30. Aoki Y, Gu FL (2009) Generalized elongation method: from one-dimension to three-dimension. In Champagne B, Gu FL, Luis JM, Springborg M (org) International conference of computational methods in sciences and engineering 2009 (ICCMSE 2009) 46–49. http://www.uni-saarland.de/fak8/springborg/ICCMSE2009/boax.pdf

  31. Makowski M, Gu FL, Aoki Y (2010) Elongation-CIS method: describing excited states of large molecular systems in regionally localized molecular orbital basis. J Comput Method Sci Eng 10:473–481. doi:10.3233/JCM-2010-0312. http://iospress.metapress.com/content/7t71004j74717106/fulltext.pdf

    Google Scholar 

  32. Korchowiec J, Gu FL, Imamura A, Kirtman B, Aoki Y (2005) Elongation method with cutoff technique for linear SCF scaling. Int J Quantum Chem 102:785–794

    Article  CAS  Google Scholar 

  33. Makowski M, Korchowiec J, Gu FL, Aoki Y (2006) Efficiency and accuracy of the elongation method as applied to the electronic structures of large systems. J Comp Chem 27:1603–1619

    Article  CAS  Google Scholar 

  34. Korchowiec J, Lewandowski J, Makowski M, Gu FL, Aoki Y (2009) Elongation cutoff technique armed with quantum fast multipole method for linear scaling. J Comput Chem 30:2515–2525

    Article  CAS  Google Scholar 

  35. Korchowiec J, Silva P, Makowski M, Gu FL, Aoki Y (2010) Elongation cutoff technique at Kohn–Sham level of theory. Int J Quantum Chem 110:2130–2139

    Article  CAS  Google Scholar 

  36. Gu FL, Aoki Y, Imamura A, Bishop DM, Kirtman B (2003) Application of the elongation method to nonlinear optical properties: finite field approach for calculating static electric (hyper)polarizabilities. Mol Phys 101:1487–1494

    Article  CAS  Google Scholar 

  37. Ohnishi S, Gu FL, Naka K, Imamura A, Kirtman B, Aoki Y (2004) Calculation of static (Hyper) polarizabilities for π-conjugated donor/acceptor molecules and block copolymers by the elongation finite-field method. J Phys Chem A 108:8478–8484

    Article  CAS  Google Scholar 

  38. Gu FL, Guillaume M, Botek E, Champagne B, Castet F, Ducasse L, Aoki Y (2006) Elongation method and supermolecule approach for the calculation of nonlinear susceptibilities. Application to the 3-methyl-4-nitropyridine 1-oxide and 2-methyl-4-nitroaniline crystals. J Comput Method Sci Eng 6:171–188. http://iospress.metapress.com/content/fwabp9hpr2nr7mhh/fulltext.pdf

    Google Scholar 

  39. Ohnishi S, Orimoto Y, Gu FL, Aoki Y (2007) Nonlinear optical properties of polydiacetylene with donor-acceptor substitution block. J Chem Phys 127:084702

    Article  Google Scholar 

  40. Chen W, Yu G-T, Gu FL, Aoki Y (2009) Investigation on the electronic structures and nonlinear optical properties of pristine boron nitride and boron nitride–carbon heterostructured single-wall nanotubes by the elongation method. J Phys Chem C 113:8447–8454

    Article  CAS  Google Scholar 

  41. Yan LK, Pomogaeva A, Gu FL, Aoki Y (2010) Theoretical study on nonlinear optical properties of metalloporphyrin using elongation method. Theor Chem Acc 125:511–520

    Article  CAS  Google Scholar 

  42. Pomogaeva A, Gu FL, Imamura A, Aoki Y (2010) Electronic structures and nonlinear optical properties of supramolecular associations of benzo-2, 1, 3-chalcogendiazoles by the elongation method. Theor Chem Acc 125:453–460

    Article  CAS  Google Scholar 

  43. Schmidt MW, Baldridge KK, Boatz JA, Elbert ST, Gordon MS, Jensen JH, Koseki S, Matsunaga N, Nguyen KA, Su SJ, Windus TL, Dupuis M, Montgomery JA (1993) General atomic and molecular electronic structure system. J Comput Chem 14:1347–1363. http://www.msg.ameslab.gov/gamess/index.html

    Google Scholar 

  44. Blase X, Charlier JC, De Vita A, Car R (1999) Structural and electronic properties of composite BxCyNz nanotubes and heterojunctions. Appl Phys A 68:293–300

    Article  CAS  Google Scholar 

  45. Terrones M, Romo-Herrera JM, Cruz-Silva E, Lopez-Urias F, Munoz-Sandoval E, Velazquez-Salazar JJ, Terrones H, Bando Y, Golberg D (2007) Pure and doped boron nitride nanotubes. Mater Today 10:30–38

    Article  CAS  Google Scholar 

  46. 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 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 (2004) Gaussian 03, Revision C.02. Gaussian, Inc., Wallingford, CT

    Google Scholar 

  47. http://xray.bmc.uu.se/hicup/LYC/index.html#COO

  48. Liao M-S, Watts JD, Huang M-J (2009) Dispersion-corrected DFT calculations on C60-porphyrin complexes. Phys Chem Chem Phys 11:4365–4374

    Article  CAS  Google Scholar 

  49. Loboda O, Zalesny R, Avramopoulos A, Papadopoulos MG, Artacho E (2009) Linear—scaling calculations of linear and nonlinear optical properties of [60]fullerene derivatives. In Maroulis G, Simos TE (ed) Computational methods in sciences and engineering, advances in computational science book series: AIP conference proceedings 1108:198–204. http://link.aip.org/link/?APCPCS/1108/198/1

  50. Loboda O, Zaleśny Avramopoulos A, Luis J-M, Kirtman B, Tagmatarchis N, Reis H, Papadopoulos MG (2009) Linear and nonlinear optical properties of [60]fullerene derivatives. J Phys Chem A 113:1159–1170

    Article  CAS  Google Scholar 

  51. Zaleśny R, Loboda O, Iliopoulos K, Chatzikyriakos G, Couris S, Rotas G, Tagmatarchis N, Avramopoulose A, Papadopoulos MG (2010) Linear and nonlinear optical properties of triphenylamine-functionalized C60: insights from theory and experiment. Phys Chem Chem Phys 12:373–381

    Article  Google Scholar 

  52. Palummo M, Hogan C, Sottile F, Bagalá P, Rubio A (2009) Ab initio electronic and optical spectra of free-base porphyrins: the role of electronic correlation. J Chem Phys 131:084102

    Article  Google Scholar 

  53. Yeon KY, Jeong D, Kim SK (2010) Intrinsic lifetimes of the Soret bands of the free-base tetraphenylporphine (H2TPP) and Cu(II)TPP in the condensed phase. Chem Commun 46:5572–5574

    Article  CAS  Google Scholar 

  54. Li C, Ly J, Lei B, Fan W, Zhang D, Han J, Mayyappan M, Thompson C, Zhou M (2004) Data storage studies on nanowire transistors with self-assembled porphyrin molecules. J Phys Chem B 108:9646–9649

    Article  CAS  Google Scholar 

  55. Kwok KS (2003) Materials for future electronics. Mater Today 6:20–27

    Article  Google Scholar 

  56. Duan X, Huang Y, Lieber CM (2002) Nonvolatile memory and programmable logic from molecule-gated nanowires. Nano Lett 2:487–490

    Article  CAS  Google Scholar 

  57. Meier H, Mühling B (2009) Synthesis and properties of oligo (2,5-thienylene)s. In Waring A (ed) ARKIVOC: Special issue ‘5th Eurasian conference on heterocyclic chemistry’ ix: 57–69. http://www.arkat-usa.org/get-file/25267/

  58. Obara Y, Takimiya K, Aso Y, Otsubo T (2001) Synthesis and photophysical properties of [60]fullerene-oligo(thienylene–ethynylene) dyads. Tetrahedron Lett 42:6877–6881

    Article  CAS  Google Scholar 

  59. Tormos GV, Nugara PN, Lakshmikantham MV, Gava MP (1993) Poly(2,5-thienylene ethynylene) and related oligomers. Synth Met 53:271–281

    Article  CAS  Google Scholar 

  60. Shirakawa H, Louis EJ, MacDiarmid AG, Chiang CK., Heeger AJ (1977) Synthesis of electrically conducting organic polymers: halogen derivatives of polyacetylene, (CH)x. J Chem Soc Chem Commun 1977:578–580

  61. Chiang CK, Druy MA, Gau SC, Heeger AJ, Louis EJ, MacDiarmid AG, Park YW, Shirakawa H (1978) Synthesis of highly conducting films of derivatives of polyacetylene, (CH)x. J Am Chem Soc 100:1013–1015

    Article  CAS  Google Scholar 

  62. MacDiarmid AG (2001) Synthetic metals: a novel role for organic polymers. Synth Met 125:11–22

    Article  Google Scholar 

Download references

Acknowledgments

One of the authors, O. L., acknowledges Japan Society for the Promotion of Science (JSPS) fellowship for financial support during stay at Kyushu University in Japan. This work was partly supported by a grant-in-aid from the Ministry of Education, Culture, Sports, Science and Technology (MEXT) of Japan and by the Asahi Glass Foundation. The calculations were carried out on the Linux cluster system in our laboratory and the high-performance computing system (Hitachi SR16000) of the Research Institute for Information Technology at Kyushu University. The authors are grateful to Prof. K. J. Jalkanen for careful English proofreading and editing of the manuscript.

Author information

Authors and Affiliations

  1. Department of Material Sciences, Faculty of Engineering Sciences, Kyushu University, 6-1 Kasuga-Park, Fukuoka, 816-8580, Japan

    Yuriko Aoki, Oleksandr Loboda & Marcin A. Makowski

  2. Department of Molecular and Material Sciences, Interdisciplinary Graduate School of Engineering Sciences, Kyushu University, 6-1 Kasuga-Park, Fukuoka, 816-8580, Japan

    Kai Liu

  3. Department of Theoretical Chemistry, Jagiellonian University, Ingardena 3, 30-060, Krakow, Poland

    Marcin A. Makowski

  4. Center for Computational Quantum Chemistry, South China Normal University, Guangzhou, 510631, China

    Feng Long Gu

  5. Japan Science and Technology Agency, CREST, 4-1-8 Hon-chou, Kawaguchi, Saitama, 332-0012, Japan

    Yuriko Aoki & Feng Long Gu

Authors
  1. Yuriko Aoki
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Oleksandr Loboda
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Kai Liu
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Marcin A. Makowski
    View author publications

    You can also search for this author in PubMed Google Scholar

  5. Feng Long Gu
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding authors

Correspondence to Yuriko Aoki or Feng Long Gu.

Additional information

Dedicated to Professor Akira Imamura on the occasion of his 77th birthday and published as part of the Imamura Festschrift Issue.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Aoki, Y., Loboda, O., Liu, K. et al. Highly accurate O(N) method for delocalized systems. Theor Chem Acc 130, 595–608 (2011). https://doi.org/10.1007/s00214-011-1011-z

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00214-011-1011-z

Keywords

Search

Navigation