Skip to main content
SpringerLink
Log in

Linear Scaling Methodology

  • Chapter
  • First Online:
Quantum-chemical studies on Porphyrins, Fullerenes and Carbon Nanostructures

Part of the book series: Carbon Nanostructures ((CARBON))

  • 887 Accesses

Abstract

In 1991 the elongation method, an efficient method for quantum mechanical calculations of large systems, was originally proposed by Imamura [1] during one of his stays in Heidelberg, Germany. Although in the early 1990s the concept of and need for order-N [O(N)] methods didn’t exist, Prof. Akira Imamura was thinking about “how to avoid direct SCF calculations for large biological systems (biopolymers composed of hundreds if not thousands of residues of amino acids or nucleic acid base pairs in proteins and DNA or RNA) by treating only the local interactions between a few neighbor residues in large systems.” While contemplating how to perform such calculations, he got the idea of “theoretically simulating the synthesis of polymers so as to mimic the chemical reactions which occur in nature during polymerization reactions to form peptides, proteins and polynucleotides.

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

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 84.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 109.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 109.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

References

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

    Google Scholar 

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

    Article  CAS  Google Scholar 

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

    Article  CAS  Google Scholar 

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

    CAS  Google Scholar 

  5. Gu, F.L., Imamura, A., Aoki, Y.: 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, vol. 1, pp. 97–177. Imperial College Press, London (2006)

    Google Scholar 

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

    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.: Linear-scaling implementation of molecular electronic self-consistent field theory. J. Chem. Phys. 126, 114110 (2007)

    Article  Google Scholar 

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

    Article  CAS  Google Scholar 

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

    Article  CAS  Google Scholar 

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

    Article  CAS  Google Scholar 

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

    Article  CAS  Google Scholar 

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

    Article  CAS  Google Scholar 

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

    Article  CAS  Google Scholar 

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

    Article  CAS  Google Scholar 

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

    Article  CAS  Google Scholar 

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

    Article  CAS  Google Scholar 

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

    Article  Google Scholar 

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

    Article  CAS  Google Scholar 

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

    Article  CAS  Google Scholar 

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

    Article  CAS  Google Scholar 

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

    Article  CAS  Google Scholar 

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

    Article  CAS  Google Scholar 

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

    Article  CAS  Google Scholar 

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

    Article  CAS  Google Scholar 

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

    Article  CAS  Google Scholar 

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

    Article  Google Scholar 

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

    Article  Google Scholar 

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

    Article  Google Scholar 

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

    Article  CAS  Google Scholar 

  30. Aoki Y, Gu FL (2009) Generalized elongation method: from one-dimension to three-dimension. In: Champagne, B., Gu, F.L., Luis, J.M., Springborg M (org) International Conference of Computational Methods in Sciences and Engineering 2009 (ICCMSE 2009), pp. 46–49

    Google Scholar 

  31. Makowski, M., Gu, F.L., Aoki, Y.: Elongation-CIS method: describing excited states of large molecular systems in regionally localized molecular orbital basis. J. Comput. Meth. Sci. Eng. 10, 473–481 (2010)

    CAS  Google Scholar 

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

    Article  CAS  Google Scholar 

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

    Article  CAS  Google Scholar 

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

    Article  CAS  Google Scholar 

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

    Article  CAS  Google Scholar 

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

    Article  CAS  Google Scholar 

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

    Article  CAS  Google Scholar 

  38. Gu, F.L., Guillaume, M., Botek, E., Champagne, B., Castet, F., Ducasse, L., Aoki, Y.: 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. Meth. Sci. Eng. 6, 171–188 (2006)

    Google Scholar 

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

    Article  Google Scholar 

  40. Chen, W., Yu, G.-T., Gu, F.L., Aoki, Y.: 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 (2009)

    Article  CAS  Google Scholar 

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

    Article  CAS  Google Scholar 

  42. Schmidt, M.W., Baldridge, K.K., Boatz, J.A., Elbert, S.T., Gordon, M.S.: Jensen, J.H., Koseki, S., Matsunaga, N., Nguyen, K.A., Su, S.J., Windus, T.L., Dupuis, M., Montgomery, J.A.: General atomic and molecular electronic structure system. J. Comput. Chem. 14, 1347–1363 (1993)

    Article  CAS  Google Scholar 

  43. Pomogaeva, A., Gu, F.L., Imamura, A., Aoki, Y.: 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 (2010)

    Article  CAS  Google Scholar 

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

    Article  CAS  Google Scholar 

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

    Article  CAS  Google Scholar 

  46. Gaussian 03, Revision C.02, Frisch M.J., Trucks, G.W., Schlegel, H.B., Scuseria, G.E., Robb, M.A., Cheeseman, J.R., Montgomery, J.A. Jr, Vreven, T., Kudin, K.N., Burant, J.C., Millam, J.M., Iyengar, S.S., Tomasi, J., Barone, V., Mennucci, B., Cossi, M., Scalmani, G., Rega, N., Petersson, G.A., 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, J.E., Hratchian, H.P., Cross, J.B., Adamo, C., Jaramillo, J., Gomperts, R., Stratmann, R.E., Yazyev, O., Austin, A.J., Cammi, R., Pomelli, C., Ochterski, J.W., Ayala, P.Y., Morokuma, K., Voth, G.A., Salvador, P., Dannenberg, J.J., Zakrzewski, V.G., Dapprich, S., Daniels, A.D., Strain, M.C., Farkas, O., Malick, D.K., Rabuck, A.D., Raghavachari, K., Foresman, J.B., Ortiz, J.V., Cui, Q., Baboul, A.G., Clifford, S., Cioslowski, J., Stefanov, B.B., Liu, G., Liashenko, A., Piskorz, P., Komaromi, I., Martin, R.L., Fox, D.J., Keith, T., Al-Laham M.A., Peng, C.Y., Nanayakkara, A., Challacombe, M., Gill, P.M.W., Johnson, B., Chen, W., Wong, M.W., Gonzalez, C., Pople, J.A., Gaussian, Inc., Wallingford, CT (2004)

    Google Scholar 

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

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

    Article  CAS  Google Scholar 

  49. Loboda, O., Zalesny, R., Avramopoulos, A., Papadopoulos, M.G., Artacho, E.: Linear-scaling calculations of linear and nonlinear optical properties of [60]fullerene derivatives. In: Maroulis, G., Simos, T.E. (eds.) Computational Methods in Sciences and Engineering. Advances In Computational Science Book Series: AIP Conference Proceedings, vol 1108, 198–204 (2009)

    Google Scholar 

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

    Article  CAS  Google Scholar 

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

    Article  CAS  Google Scholar 

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

    Article  Google Scholar 

  53. Yeon, K.Y., Jeong, D., Kim, S.K.: 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 (2010)

    Article  CAS  Google Scholar 

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

    Article  CAS  Google Scholar 

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

    Article  Google Scholar 

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

    Article  CAS  Google Scholar 

  57. Meier H, Mühling, B.: 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 (2009)

    Google Scholar 

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

    Article  CAS  Google Scholar 

  59. Tormos, G.V., Nugara, P.N., Lakshmikantham, M.V., Gava, M.P.: Poly(2,5-thienylene ethynylene) and related oligomers. Synth. Met. 53, 271–281 (1993)

    Article  CAS  Google Scholar 

  60. Shirakawa, H., Louis, E.J., MacDiarmid, A.G., Chiang, C.K., Heeger, A.J.: Synthesis of electrically conducting organic polymers: halogen derivatives of polyacetylene, (CH)x. J. Chem. Soc. Chem. Commun. 474, 578–580 (1977)

    Article  Google Scholar 

  61. Chiang, C.K., Druy, M.A., Gau, S.C., Heeger, A.J., Louis, E.J., MacDiarmid, A.G., Park, Y.W., Shirakawa, H.: Synthesis of highly conducting films of derivatives of polyacetylene, (CH)x. J. Am. Chem. Soc. 100, 1013–1015 (1978)

    Article  CAS  Google Scholar 

  62. MacDiarmid, A.G.: Synthetic metals: a novel role for organic polymers. Synth. Metals 125, 11–22 (2001)

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Institute of Colloid and Water Chemistry, Varnadsky blvd 42, Kiev, Ukraine

    Oleksandr Loboda

Authors
  1. Oleksandr Loboda
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Oleksandr Loboda .

Rights and permissions

Reprints and permissions

Copyright information

© 2012 Springer-Verlag Berlin Heidelberg

About this chapter

Cite this chapter

Loboda, O. (2012). Linear Scaling Methodology. In: Quantum-chemical studies on Porphyrins, Fullerenes and Carbon Nanostructures. Carbon Nanostructures. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-31845-0_6

Download citation

Publish with us

Policies and ethics

Search

Navigation