Ab initio calculation of nonlinear optical properties for chiral carbon nanotubes. Second harmonic generation and dc-Pockels effect

  • Michel Rérat
  • Panaghiotis Karamanis
  • Bartolomeo Civalleri
  • Lorenzo Maschio
  • Valentina Lacivita
  • Bernard Kirtman
Regular Article
Part of the following topical collections:
  1. In Memoriam of Claudio Zicovich


Although the static first hyperpolarizability \(\beta\) of graphene and chiral carbon nanotubes (CNTs) vanishes by symmetry, that is no longer true for frequency-dependent fields in which case there is a unique nonzero component. We evaluate that component for the dc-Pockels effect and second harmonic generation of several CNTs by means of the coupled perturbed Kohn–Sham method with the CAM-B3LYP functional as implemented in the CRYSTAL code. Both electronic and vibrational contributions are considered. Of particular interest is the frequency dependence in the region below the first resonance. Our results for the role of band gap, type of chirality, CNT radius and effect of orbital relaxation are analyzed using the conventional sum-over-states formulation in conjunction with the calculated band structure and density of states.


Chiral carbon nanotubes dc-Pockels effect Second harmonic generation Coupled perturbed Kohn–Sham method CRYSTAL code 



This work would not have been possible without the invaluable contribution given by our colleague and friend Claudio M. Zicovich-Wilson to the exploitation of symmetry in periodic systems, starting from his two seminal papers on symmetry-adapted crystalline orbitals [47, 48] to the use of symmetry for large crystalline solids, nanotubes, helices and fullerenes. Yet, his work was of great relevance to dramatically reduce the computational cost in the CRYSTAL code (i.e., one and two-electron integrals, SCF process, and many other properties).

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  1. 1.
    Zheng X, Feng M, Zhan HJ (2014) Enhanced nonlinear optical properties of nonzero-bandgap graphene materials in glass matrices. Mater Chem C 2:4121–4125CrossRefGoogle Scholar
  2. 2.
    Wang J, Hernandez Y, Lotya M, Coleman JN, Blau WJ (2009) Broadband nonlinear optical response of graphene dispersions. Adv Mater 21:2430–2435CrossRefGoogle Scholar
  3. 3.
    Wu R, Zhang Y, Yan S, Bian F, Wang W, Bai X, Lu X, Zhao J, Wang E (2011) Purely coherent nonlinear optical response in solution dispersions of graphene sheets. Nano Lett 11:5159CrossRefGoogle Scholar
  4. 4.
    Karamanis P, Otero N, Pouchan C (2014) Unleashing the quadratic nonlinear optical responses of graphene by confining white-graphene (h-BN) sections in its framework. J Am Chem Soc 136(20):7464–7473CrossRefGoogle Scholar
  5. 5.
    Avouris P (2010) Graphene: electronic and photonic properties and devices. Nano Lett 10:4285–4294CrossRefGoogle Scholar
  6. 6.
    Gu T, Petrone N, McMillan JF, van der Zande A, Yu M, Lo GQ, Kwong DL, Hone J, Wong CW (2012) Regenerative oscillation and four-wave mixing in graphene optoelectronics. Nat Photon 6:554–559CrossRefGoogle Scholar
  7. 7.
    Konishi A, Hirao Y, Matsumoto K, Kurata H, Kishi R, Shigeta Y, Nakano M, Togunaga K, Kamada K, Kubo TJ (2013) Synthesis and characterization of quarteranthene: elucidating the characteristics of the edge state of graphene nanoribbons at the molecular level. J Am Chem Soc 135:1430CrossRefGoogle Scholar
  8. 8.
    Bonaccorso F, Sun Z, Hasan T, Ferrari AC (2010) Graphene photonics and optoelectronics. Nat Photon 4:611CrossRefGoogle Scholar
  9. 9.
    Novoselov KS, Geim AK (2007) The rise of graphene. Nat Mater 6:183CrossRefGoogle Scholar
  10. 10.
    Boyd RW (1992) Nonlinear Optics, Academic Press, San DiegoGoogle Scholar
  11. 11.
    Zyss J (1994) Molecular nonlinear optics: materials physics and devices. Academic Press, San DiegoGoogle Scholar
  12. 12.
    Saleh BEA (1991) Fundamental of photonics. Wiley, New YorkCrossRefGoogle Scholar
  13. 13.
    Kanis DR, Ratner MA, Marks TJ (1994) Design and construction of molecular assemblies with large second-order optical nonlinearities. Quantum chemical aspects. J Chem Rev 94:195CrossRefGoogle Scholar
  14. 14.
    Otero N, Pouchan C, Karamanis P (2017) Quadratic nonlinear optical (NLO) properties of borazino (B3N3)-doped nanographenes. J Mater Chem C 5(32):8273–8287CrossRefGoogle Scholar
  15. 15.
    Guo GY, Chu KC, Wang DS, Duan CG (2004) Linear and nonlinear optical properties of carbon nanotubes from first-principles calculations. Phys Rev B 69:205416CrossRefGoogle Scholar
  16. 16.
    Lacivita V, Rérat M, Orlando R, Dovesi R, D’Arco P (2016) Longitudinal and transverse hyperpolarizabilities of carbon nanotubes: a computational investigation through the coupled-perturbed Hartree–Fock/Kohn–Sham scheme. Theor Chem Acc 135:81CrossRefGoogle Scholar
  17. 17.
    Dovesi R, Erba A, Orlando, Zicovich-Wilson CM, Civalleri B, Maschio L, Rérat M, Casassa S, Baima J, Salustro S, Kirtman B (2017a) Quantum-mechanical condensed matter simulations with crystal. Submitted to Wiley Interdisciplinary Reviews Google Scholar
  18. 18.
    Dovesi R, Orlando R, Erba A, Zicovich-Wilson CM, Civalleri B, Casassa S, Maschio L, Ferrabone M, De La Pierre M, D’Arco P, Noël Y, Causà M, Rérat M, and Kirtman B (2014) Int J Quant Chem, 114: 1287. URL
  19. 19.
    Becke AD (1993) Density-functional thermochemistry. III. The role of exact exchange. J Chem Phys 98:5648. CrossRefGoogle Scholar
  20. 20.
    Lee C, Yang W, Parr RG (1988) Development of the Colle-Salvetti correlation-energy formula into a functional of the electron density. Phys Rev B 37:785. CrossRefGoogle Scholar
  21. 21.
    Stephens PJ, Devlin FJ, Chabalowski CF, Frisch MJ (1994) Ab initio calculation of vibrational absorption and circular dichroism spectra using density functional force fields. J Phys Chem 98(45):11623. CrossRefGoogle Scholar
  22. 22.
    Iikura H, Tsuneda T, Yanai T, Hirao K (2001) A long-range correction scheme for generalized-gradient-approximation exchange functionals. J Chem Phys 115:3540CrossRefGoogle Scholar
  23. 23.
    Yanai T, Tew DP, Handy NC (2004) A new hybrid exchange-correlation functional using the coulomb-attenuating method (CAM-B3LYP). Chem Phys Lett 393:51–57CrossRefGoogle Scholar
  24. 24.
    Dovesi R, Saunders VR, Roetti C, Orlando R, Zicovich-Wilson CM, Pascale F, Civalleri B, Doll K, Harrison NM, Bush IJ, D’Arco P, Llunell M, Causà M, Noël Y, Maschio L, Erba A, Rérat M, Casassa S (2017b) CRYSTAL17 User’s Manual, University of Torino. URL
  25. 25.
    Schlegel BH (1984) Estimating the hessian for gradient-type geometry optimizations. Theor Chim Acta 66:333–340. CrossRefGoogle Scholar
  26. 26.
    Wittbrodt JM, Schlegel HB (1997) Estimating stretching force constants for geometry optimization. J Mol Struct THEOCHEM 398–399:55–61. CrossRefGoogle Scholar
  27. 27.
    Broyden CG (1970a) The convergence of a class of double-rank minimization algorithms 1. General considerations. IMA J Appl Math 6:76. CrossRefGoogle Scholar
  28. 28.
    Broyden CG (1970b) The convergence of a class of double-rank minimization algorithms: 2. The new algorithm. IMA J Appl Math 6:222. CrossRefGoogle Scholar
  29. 29.
    Goldfarb D (1970) A family of variable-metric methods derived by variational means. Math Comput 24:23–26. CrossRefGoogle Scholar
  30. 30.
    Shanno DF (1970) A family of variable-metric methods derived by variational means. Math Comput 24:647–656. CrossRefGoogle Scholar
  31. 31.
    Rérat M, Maschio L, Kirtman B, Civalleri B, Dovesi R (2016) Computation of second harmonic generation for crystalline urea and KDP. An ab initio approach through the coupled perturbed hartree fock/kohn sham scheme. J Chem Theor Comput 12:116CrossRefGoogle Scholar
  32. 32.
    Maschio L, Rérat M, Kirtman B, Dovesi R (2015) Calculation of the dynamic first hyperpolarizability \(\beta (-~\omega _\sigma; \omega _1,\omega _2)\) of periodic systems. implementation in the crystal code. J Chem Phys 143:244102CrossRefGoogle Scholar
  33. 33.
    Canepa P, Hanson RM, Ugliengo P, Alfredsson M (2011) J Appl Cryst 44:225CrossRefGoogle Scholar
  34. 34.
    Beata G, Perego G, and Civalleri B (2017) in preparation. URL
  35. 35.
    Movlarooy T, Kompany A, Hosseini SM, Shahtahmasebi N (2010) Optical absorption and electron energy loss spectra of single-walled carbon nanotubes. Comput Mater Sci 12:450CrossRefGoogle Scholar
  36. 36.
    Ferrari AM, Orlando R, Rérat M (2015) Ab initio calculation of the ultraviolet visible (uv-vis) absorption spectrum, electron-loss function, and reflectivity of solids. J Chem Theor Comput 11:3245CrossRefGoogle Scholar
  37. 37.
    Demichelis R, Noël Y, D’Arco P, Rérat M, Zicovich-Wilson CM, Dovesi R (2011) Properties of carbon nanotubes: an ab initio study using large gaussian basis sets and various dft functionals. J Phys Chem C 115:8876CrossRefGoogle Scholar
  38. 38.
    Kirtman B, Lacivita V, Dovesi R, Reis H (2011) Electric field polarization in conventional density functional theory: from quasilinear to two-dimensional and three-dimensional extended systems. J Chem Phys 135:154101CrossRefGoogle Scholar
  39. 39.
    Tsuneda T, Hirao K (2014) Long-range correction for density functional theory. WIREs Comput Mol Sci 4:375–390CrossRefGoogle Scholar
  40. 40.
    Bulik IW, Zalesny R, Bartkowiak W, Luis JM, Kirtman B, Scuseria GE, Avramopoulos A, Reis H, Papadopoulos MG (2013) Performance of density functional theory in computing nonresonant vibrational (hyper)polarizabilities. J Comput Chem 34:1775CrossRefGoogle Scholar
  41. 41.
    David Bishop M, De Kee DW (1996) The frequency dependence of nonlinear optical processes. J Chem Phys 104(24):9876–9887CrossRefGoogle Scholar
  42. 42.
    David Bishop M, De Kee DW (1996) The frequency dependence of hyperpolarizabilities for noncentrosymmetric molecules. J Chem Phys 105(18):8247–8249CrossRefGoogle Scholar
  43. 43.
    Bishop DM, Kirtman B (1991) A perturbation method for calculating vibrational dynamic dipole polarizabilities and hyperpolarizabilities. J Chem Phys 95:2646CrossRefGoogle Scholar
  44. 44.
    Kirtman B, Luis JM (2011) On the contribution of mixed terms in response function treatment of vibrational nonlinear optical properties. Int J Quantum Chem 111:839CrossRefGoogle Scholar
  45. 45.
    Maschio L, Kirtman B, Orlando R, Rérat M (2012) Ab Initio analytical infrared intensities for periodic systems through a coupled perturbed Hartree–Fock/Kohn–Sham method. J Chem Phys 137:204113CrossRefGoogle Scholar
  46. 46.
    Maschio L, Kirtman B, Rérat M, Orlando R, Dovesi R (2013) Ab initio analytical Raman intensities for periodic systems through a coupled perturbed Hartree–Fock/Kohn–Sham method in an atomic orbital basis. I. Theory. J Chem Phys 139:164101CrossRefGoogle Scholar
  47. 47.
    Zicovich-Wilson CM, Dovesi R (1998a) On the use of symmetry adapted crystalline orbitals in SCF–LCAO periodic calculations. I. The construction of the symmetrized orbitals. Int J Quantum Chem 67:299–310CrossRefGoogle Scholar
  48. 48.
    Zicovich-Wilson CM, Dovesi R (1998b) On the use of symmetry adapted crystalline orbitals in SCF–LCAO periodic calculations. II. Implementation of the self-consistent-field scheme and examples. Int J Quantum Chem 67:311–320CrossRefGoogle Scholar
  49. 49.
    Reich S, Thomsen C, Maultzsch J (2004) Carbon nanotubes. Basic concepts and physical properties. Wiley, WeinheimGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.IPREMCNRS / Université de Pau et des Pays de l’Adour, UMR5254PauFrance
  2. 2.Dipartimento di Chimica and NIS (Nanostructured Interfaces and Surfaces) CentreUniversità di TorinoTorinoItaly
  3. 3.Materials Sciences DivisionLawrence Berkeley National LaboratoryBerkeleyUSA
  4. 4.Department of Chemistry and BiochemistryUniversity of CaliforniaSanta BarbaraUSA

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