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Scaling Procedures in Vibrational Spectroscopy

  • Olga Bąk
  • Piotr Borowski
Chapter
Part of the Challenges and Advances in Computational Chemistry and Physics book series (COCH, volume 26)

Abstract

This chapter contains a brief review of the up-to-date scaling procedures that are used in the computational vibrational spectroscopy to improve agreement between the calculated harmonic frequencies and the observed fundamentals. Initially, the basics of vibrational spectroscopy are reminded. This includes the concept of potential energy surface, harmonic approximation, and a basic quantum chemistry treatment of the anharmonicity for a diatomic molecule. Brief description of the Wilson–Decius–Cross method for polyatomic molecules is also presented. Then the commonly used scaling procedures are discussed. The distinction between single- and multi-parameter scaling procedures is made. Four scaling procedures are reviewed. First, Pople’s uniform scaling is presented. Second, Yoshida’s wavenumber linear scaling method is discussed. Both methods are simple single-parameter frequency scaling methods. Then basics of two multi-parameter scaling methods, which are much more accurate but less straightforward to use, are given. Thus, Pulay’s scaled quantum mechanical force field method, in which scaling factors are applied directly to the calculated force constants is reviewed. Finally, introduction to quite recently proposed multi-parameter frequency scaling method, called effective scaling frequency factor method, is provided. The relevant sections start with a short description of the theory for a given method. Then a brief literature review on the historical background of methodology development is given.

List of Acronyms

ARPE

Average relative percentage error

ESF

Effective scaling factor

ESFF

Effective scaling frequency factor method

FC

Force constant

FF

Force field

IC

Internal coordinate

LSF

Local scaling (scale) factor

LSMF

Least-squares merit function

NIC

Natural internal coordinate

PEC

Potential energy curve

PES

Potential energy (hyper)surface

PIC

Primitive internal coordinate

QC

Quantum chemistry (methods, calculations…)

RMS

Root-mean-square (deviation)

SF

Scaling factors

SQM(FF)

Scaled quantum mechanical (force field) method; simply SQM

US

Uniform scaling

WDC

Wilson–Decius–Cross method

WLS

Wavenumber linear scaling

X

A general second-row atom

Y

A general third-row atom

ZPVE

Zero-point vibrational energy correction

Symbols Used in Mathematical Formulas

A, B, …

General indices used in summations over atoms

B

Transformation matrix between internal and Cartesian coordinates

\({\mathbf{f}} = \left( {f_{1} ,f_{2} , \ldots ,f_{{N_{{\text{scl}}} }} } \right)\)

A vector with scaling factors

\({\mathbf{f}}^{{\text{eff}}} = \left( {f_{1}^{{\text{eff}}} ,f_{2}^{{\text{eff}}} , \ldots ,f_{K}^{{\text{eff}}} } \right)\)

Effective scaling factors

\({\mathbf{f}}^{{\text{opt}}} = \left( {f_{1}^{{\text{opt}}} ,f_{2}^{{\text{opt}}} , \ldots ,f_{{N_{{\text{scl}}} }}^{{\text{opt}}} } \right)\)

A vector with optimal scaling factors

fx

Cartesian force constant matrix with elements \(f_{ij}^{x}\)

F

Force constant matrix in internal coordinate representation with elements \(F_{\mu \nu }\)

i, j, …

General indices used in summations over Cartesian coordinates

I, J, …

General indices used to indicate types of internal coordinates in multi-parameter scaling procedures

K

Number of vibrational degrees of freedom, K = 3N − 6 (or K = 3N − 5 in the case of linear molecules)

L

Number of redundant primitive internal coordinates (typically L > 3N)

L

Transformation matrix between Cartesian displacements and normal coordinates

N

Number of atoms in a molecule

Nmol

Number of molecules in a training set

Nscl

Number of scaling factors

Ntyp

Number of types (groups) of internal coordinates (=Nscl)

Nvib

Number of vibrational modes used in optimization of scaling factors; additional superscript indicates the molecule the number of modes refers to

p, q, …

General indices used in summations over frequencies of a training set of molecules, basis functions, etc.

s = (s1, s2, …)

Vector with internal coordinates; additional superscripts refer to the type a given coordinate belongs to. Additional subscript “e” denotes the equilibrium values. When used in the potential energy expression the symbols refer to the deviations of coordinates from their equilibrium values

T

Kinetic energy

U

Potential energy

w

Weight used for a given frequency in the scaling factor optimization procedure

α

Transformation matrix between internal coordinates displacements and normal coordinates

\(\theta^{{\text{vib}}}\)

Vibrational temperature

μ, ν, …

General indices used in summations over internal or normal coordinates as well as normal modes of a molecule

\(\nu^{\text{h}}\), \(\nu^{{\text{expt}}}\), \(\nu^{{\text{expt,h}}}\), \(\nu^{{\text{scl}}}\)

Harmonic, experimental, experimental harmonic, and scaled frequencies, respectively (in cm−1)

\(\psi_{p}^{\text{h}}\)

Eigenfunctions of harmonic oscillator

References

  1. 1.
    Pulay P, Fogarasi G, Pang F, Boggs JE (1979) Systematic ab initio gradient calculation of molecular geometries, force constants, and dipole-moment derivatives. J Am Chem Soc 101:2550–2560CrossRefGoogle Scholar
  2. 2.
    Lide DR, Frederikse HPR (eds) (1994) CRC handbook of chemistry and physics. CRC Press, LondonGoogle Scholar
  3. 3.
    Atkins PW (1993) Molecular quantum mechanics. Oxford University Press, New YorkGoogle Scholar
  4. 4.
    Wilson EB Jr, Decius JC, Cross PC (1955) Molecular vibrations. The theory of infrared and Raman vibrational spectra. Dover Publications Inc., New YorkGoogle Scholar
  5. 5.
    Califano S (1976) Vibrational states. Wiley, LondonGoogle Scholar
  6. 6.
    Pulay P (1969) Ab initio calculation of force constants and equilibrium geometries in polyatomic molecules. I Theory Mol Phys 17:197–204Google Scholar
  7. 7.
    Pulay P, Meyer W (1974) Comparison of ab initio force constants of ethane, ethylene and acetylene. Mol Phys 27:473–490CrossRefGoogle Scholar
  8. 8.
    Pople JA, Krishnan R, Schlegel HB, Binkley JS (1979) Derivative studies in Hartree-Fock and Moller-Plesset theories. Int J Quantum Chem 16:225–241CrossRefGoogle Scholar
  9. 9.
    Willetts A, Handy NC, Green WH, Jayatilaka D (1990) Anharmonic corrections to vibrational transition intensities. J Phys Chem 94:5608–5616CrossRefGoogle Scholar
  10. 10.
    Barone V, Bloino J, Guido CA, Lipparini F (2010) A fully automated implementation of VPT2 Infrared intensities. Chem Phys Lett 496:157–161CrossRefGoogle Scholar
  11. 11.
    Pulay P, Fogarasi G, Boggs JE (1981) Force field, dipole moment derivatives, and vibronic constants of benzene from a combination of experimental and ab initio quantum chemical information. J Chem Phys 74:3999–4014CrossRefGoogle Scholar
  12. 12.
    Pulay P, Fogarasi G, Pongor G, Boggs JE, Vargha A (1983) Combination of theoretical ab initio and experimental information to obtain reliable harmonic force-constants—scaled quantum-mechanical (SQM) force-fields for glyoxal, acrolein, butadiene, formaldehyde, and ethylene. J Am Chem Soc 105:7037–7047CrossRefGoogle Scholar
  13. 13.
    Borowski P, Fernández-Gómez M, Fernández-Liencres M-P, Peña Ruiz T, Quesada Rincón M (2009) An effective scaling frequency factor method for scaling of harmonic vibrational frequencies: application to toluene, styrene and its 4-methyl derivative. J Mol Struct 924:493–503CrossRefGoogle Scholar
  14. 14.
    McQuarrie DA (1976) Statistical mechanics. HarperCollinsPublishers, New YorkGoogle Scholar
  15. 15.
    Merrick JP, Moran D, Radom L (2007) An evaluation of harmonic vibrational frequency scale factors. J Phys Chem A 111:11683–11700CrossRefGoogle Scholar
  16. 16.
    Pople JA, Schlegel HB, Krishnan R, Defrees DJ, Binkley JS, Frisch MJ, Whiteside RA, Hout RF, Hehre WJ (1981) Molecular-orbital studies of vibrational frequencies. Int J Quantum Chem 15:269–278Google Scholar
  17. 17.
    Hout RF, Levi BA, Hehre WJ (1982) Effect of electron correlation on theoretical vibrational frequencies. J Comput Chem 3:234–250CrossRefGoogle Scholar
  18. 18.
    Pople JA, Scott AP, Wong MW, Radom L (1993) Scaling factors for obtaining fundamental vibrational frequencies and zero-point energies from HF/6-31 g-asterisk and MP2/6-31 g-asterisk harmonic frequencies. Isr J Chem 33:345–350CrossRefGoogle Scholar
  19. 19.
    Scott AP, Radom L (1996) Harmonic vibrational frequencies: an evaluation of Hartree-Fock, Moller-Plesset, quadratic configuration interaction, density functional theory, and semiempirical scale factors. J Phys Chem 100:16502–16513CrossRefGoogle Scholar
  20. 20.
    Wong MW (1996) Vibrational frequency prediction using density functional theory. Chem Phys Lett 256:391–399CrossRefGoogle Scholar
  21. 21.
    Patton LF, Corchado J, Sanchez ML, Truhlar DG (1999) Optimized parameters for scaling correlation energy. J Phys Chem A 103:3139–3143CrossRefGoogle Scholar
  22. 22.
    Lynch BJ, Truhlar DG (2001) How well can hybrid density functional methods predict transition state geometries and barrier heights? J Phys Chem A 105:2936–2941CrossRefGoogle Scholar
  23. 23.
    Zhao Y, Lynch BJ, Truhlar DG (2004) Doubly hybrid meta DFT: new multi-coefficient correlation and density functional methods for thermochemistry and thermochemical kinetics. J Phys Chem A 108:4786–4791CrossRefGoogle Scholar
  24. 24.
    Zhao Y, Lynch BJ, Truhlar DG (2004) Development and assessment of a new hybrid density functional model for thermochemical kinetics. J Phys Chem A 108:2715–2719CrossRefGoogle Scholar
  25. 25.
    Zhao Y, Truhlar DG (2004) Hybrid meta density functional theory methods for thermochemistry, thermochemical kinetics, and noncovalent interactions: the MPW1B95 and MPWB1K models and comparative assessments for hydrogen bonding and van der Waals interactions. J Phys Chem A 108:6908–6918CrossRefGoogle Scholar
  26. 26.
    Schultz NE, Zhao Y, Truhlar DG (2005) Databases for transition element bonding: metal–metal bond energies and bond lengths and their use to test hybrid, hybrid meta, and meta density functionals and generalized gradient approximations. J Phys Chem A 109:4388–4403CrossRefPubMedCentralGoogle Scholar
  27. 27.
    Curtiss LA, Redfern PC, Raghavachari K, Pople JA (2001) Gaussian-3X (G3X) theory: use of improved geometries, zero-point energies, and Hartree-Fock basis sets. J Chem Phys 114:108–117CrossRefGoogle Scholar
  28. 28.
    Halls M, Velkovski J, Schlegel H (2001) Harmonic frequency scaling factors for Hartree-Fock, S-VWN, B-LYP, B3-LYP, B3-PW91 and MP2 with the Sadlej pVTZ electric property basis set. Theor Chem Acc 105:413–421CrossRefGoogle Scholar
  29. 29.
    Sinha P, Boesch SE, Gu C, Wheeler RA, Wilson AK (2004) Harmonic vibrational frequencies: scaling factors for HF, B3LYP, and MP2 methods in combination with correlation consistent basis sets. J Phys Chem A 108:9213–9217CrossRefGoogle Scholar
  30. 30.
    Andersson MP, Uvdal P (2005) New scale factors for harmonic vibrational frequencies using the B3LYP density functional method with the triple-ζ basis set 6–311 + G(d, p). J Phys Chem A 109:2937–2941CrossRefPubMedCentralGoogle Scholar
  31. 31.
    Csonka GI, Ruzsinszky A, Perdew JP (2005) Estimation, computation, and experimental correction of molecular zero-point vibrational energies. J Phys Chem A 109:6779–6789CrossRefPubMedCentralGoogle Scholar
  32. 32.
    Tantirungrotechai Y, Phanasant K, Roddecha S, Surawatanawong P, Sutthikhum V, Limtrakul J (2006) Scaling factors for vibrational frequencies and zero-point vibrational energies of some recently developed exchange-correlation functionals. J Mol Struc-THEOCHEM 760:189–192CrossRefGoogle Scholar
  33. 33.
    Chan B, Radom L (2016) Frequency scale factors for some double-hybrid density functional theory procedures: accurate thermochemical components for high-level composite protocols. J Chem Theory Comput 12:3774–3780CrossRefPubMedCentralGoogle Scholar
  34. 34.
    Chan B (2017) Use of low-cost quantum chemistry procedures for geometry optimization and vibrational frequency calculations: determination of frequency scale factors and application to reactions of large systems. J Chem Theory Comput 13:6052–6060CrossRefPubMedCentralGoogle Scholar
  35. 35.
    Laury ML, Carlson MJ, Wilson AK (2012) Vibrational frequency scale factors for density functional theory and the polarization consistent basis sets. J Comput Chem 33:2380–2387CrossRefPubMedCentralGoogle Scholar
  36. 36.
    Krasnoshchekov SV, Stepanov NF (2007) Scale factors as effective parameters for correcting nonempirical force fields. Russ J Phys Chem A 81:585–592CrossRefGoogle Scholar
  37. 37.
    Andrade SG, Gonçalves LCS, Jorge FE (2008) Scaling factors for fundamental vibrational frequencies and zero-point energies obtained from HF, MP2, and DFT/DZP and TZP harmonic frequencies. J Mol Struc-THEOCHEM 864:20–25CrossRefGoogle Scholar
  38. 38.
    Sierraalta A, Martorell G, Ehrmann E, Añez R (2008) Improvement of scale factors for harmonic vibrational frequency calculations using new polarization functions. Int J Quantum Chem 108:1036–1043CrossRefGoogle Scholar
  39. 39.
    Alecu IM, Zheng J, Zhao Y, Truhlar DG (2010) Computational thermochemistry: scale factor databases and scale factors for vibrational frequencies obtained from electronic model chemistries. J Chem Theory Comput 6:2872–2887CrossRefGoogle Scholar
  40. 40.
    Basire M, Parneix P, Calvo F (2010) Finite-temperature IR spectroscopy of polyatomic molecules: a theoretical assessment of scaling factors. J Phys Chem A 114:3139–3146CrossRefPubMedCentralGoogle Scholar
  41. 41.
    Friese DH, Törk L, Hättig C (2014) Vibrational frequency scaling factors for correlation consistent basis sets and the methods CC2 and MP2 and their spin-scaled SCS and SOS variants. J Chem Phys 141:194106CrossRefPubMedCentralGoogle Scholar
  42. 42.
    Kesharwani MK, Brauer B, Martin JML (2015) Frequency and zero-point vibrational energy scale factors for double-hybrid density functionals (and other selected methods): can anharmonic force fields be avoided? J Phys Chem A 119:1701–1714CrossRefGoogle Scholar
  43. 43.
    Kashinski DO, Chase GM, Nelson RG, Di Nallo OE, Scales AN, VanderLey DL, Byrd EFC (2017) Harmonic vibrational frequencies: approximate global scaling factors for TPSS, M06, and M11 functional families using several common basis sets. J Phys Chem A 121:2265–2273CrossRefPubMedCentralGoogle Scholar
  44. 44.
    Irikura KK, Johnson RD III, Kacker RN (2005) Uncertainties in scaling factors for ab initio vibrational frequencies. J Phys Chem A 109:8430–8437CrossRefPubMedCentralGoogle Scholar
  45. 45.
    Irikura KK, Johnson RD III, Kacker RN, Kessel R (2009) Uncertainties in scaling factors for ab initio vibrational zero-point energies. J Chem Phys 130:114102CrossRefPubMedCentralGoogle Scholar
  46. 46.
    Irikura KK, Johnson RD III, Kacker RN, Kessel R (2009) Erratum: “Uncertainties in scaling factors for ab initio vibrational zero-point energies” [J. Chem. Phys. 130, 114102 (2009)]. J Chem Phys 131:169902CrossRefGoogle Scholar
  47. 47.
    Teixeira F, Melo A, Cordeiro MNDS (2010) Calibration sets and the accuracy of vibrational scaling factors: a case study with the X3LYP hybrid functional. J Chem Phys 133:114109CrossRefPubMedCentralGoogle Scholar
  48. 48.
    Pernot P, Cailliez F (2011) Comment on “Uncertainties in scaling factors for ab initio vibrational zero-point energies” [J. Chem. Phys. 130, 114102 (2009)] and “Calibration sets and the accuracy of vibrational scaling factors: a case study with the X3LYP hybrid functional” [J. Chem. Phys. 133, 114109 (2010)]. J Chem Phys 134:167101CrossRefPubMedCentralGoogle Scholar
  49. 49.
    Irikura KK, Johnson RD III, Kacker RN, Kessel R (2011) Response to “Comment on ‘Uncertainties in scaling factors for ab initio vibrational zero-point energies’ and ‘Calibration sets and the accuracy of vibrational scaling factors: A case study with the X3LYP hybrid functional’” [J. Chem. Phys. 134, 167101 (2011)]. J Chem Phys 134:167102CrossRefGoogle Scholar
  50. 50.
    Teixeira F, Melo A, Cordeiro MNDS (2011) Response to “Comment on ‘Uncertainties in scaling factors for ab initio vibrational zero-point energies’ and ‘Calibration sets and the accuracy of vibrational scaling factors: A case study with the X3LYP hybrid functional’” [J. Chem. Phys. 134, 167101 (2011)]. J Chem Phys 134:167103CrossRefGoogle Scholar
  51. 51.
    Yoshida H, Ehara A, Matsuura H (2000) Density functional vibrational analysis using wavenumber-linear scale factors. Chem Phys Lett 325:477–483CrossRefGoogle Scholar
  52. 52.
    Yoshida H, Takeda K, Okamura J, Ehara A, Matsuura H (2002) A new approach to vibrational analysis of large molecules by density functional theory: wavenumber-linear scaling method. J Phys Chem A 106:3580–3586CrossRefGoogle Scholar
  53. 53.
    Kudoh S, Takayanagi M, Nakata M (2000) Infrared spectra of Dewar 4-picoline in low-temperature argon matrices and vibrational analysis by DFT calculation. Chem Phys Lett 322:363–370CrossRefGoogle Scholar
  54. 54.
    Baker J, Jarzecki AA, Pulay P (1998) Direct scaling of primitive valence force constants: an alternative approach to scaled quantum mechanical force fields. J Phys Chem A 102:1412–1424CrossRefGoogle Scholar
  55. 55.
    Mills IM (1960) Vibrational perturbation theory. J Mol Spectrosc 5:334–340CrossRefGoogle Scholar
  56. 56.
    Blom CE, Slingerland PJ, Altona C (1976) Application of self-consistent-field ab-initio calculations to organic-molecules. I. Equilibrium structure and force constants of hydrocarbons. Mol Phys 31:1359–1376CrossRefGoogle Scholar
  57. 57.
    Blom CE, Altona C (1976) Application of self-consistent-field ab-initio calculations to organic-molecules. II. Scale factor method for calculation of vibrational frequencies from ab-initio force constants—ethane, propane and cyclopropane. Mol Phys 31:1377–1391CrossRefGoogle Scholar
  58. 58.
    Blom CE, Otto LP, Altona C (1976) Application of self-consistent-field ab-initio calculations to organic-molecules. III. Equilibrium structure of water, methanol and dimethyl ether, general valence force-field of water and methanol scaled on experimental frequencies. Mol Phys 32:1137–1149CrossRefGoogle Scholar
  59. 59.
    Blom CE, Altona C (1977) Application of self-consistent-field ab-initio calculations to organic-molecules. IV. Force constants of propene scaled on experimental frequencies. Mol Phys 33:875–885CrossRefGoogle Scholar
  60. 60.
    Blom CE, Altona C (1977) Application of self-consistent-field ab-initio calculations to organic-molecules. V. Ethene—general valence force-field scaled on harmonic and anharmonic data, infra-red and Raman intensities. Mol Phys 34:177–192CrossRefGoogle Scholar
  61. 61.
    Blom CE, Altona C, Oskam A (1977) Application of self-consistent-field ab-initio calculations to organic-molecules. VI. Dimethylether—general valence force-field scaled on experimental frequencies, infra-red and Raman intensities. Mol Phys 34:557–571CrossRefGoogle Scholar
  62. 62.
    Rauhut G, Pulay P (1995) Transferable scaling factors for density-functional derived vibrational force-fields. J Phys Chem 99:3093–3100CrossRefGoogle Scholar
  63. 63.
    Rauhut G, Pulay P (1995) Transferable scaling factors for density-functional derived vibrational force-fields. J Phys Chem 99:14572CrossRefGoogle Scholar
  64. 64.
    Rauhut G, Pulay P (1995) Identification of isomers from calculated vibrational-spectra—a density-functional study of tetrachlorinated dibenzodioxins. J Am Chem Soc 117:4167–4172CrossRefGoogle Scholar
  65. 65.
    Fogarasi G, Zhou X, Taylor PW, Pulay P (1992) The calculation of ab initio molecular geometries: efficient optimization by natural internal coordinates and empirical correction by offset forces. J Am Chem Soc 114:8191–8201CrossRefGoogle Scholar
  66. 66.
    Borowski P (2012) An evaluation of scaling factors for multiparameter scaling procedures based on DFT force fields. J Phys Chem A 116:3866–3880CrossRefPubMedCentralGoogle Scholar
  67. 67.
    Kalincsák F, Pongor G (2002) Extension of the density functional derived scaled quantum mechanical force field procedure. Spectrochim Acta A 58:999–1011CrossRefGoogle Scholar
  68. 68.
    Borowski P, Drzewiecka A, Fernández-Gómez M, Fernández-Liencres M-P, Peña Ruiz T (2008) An effective scaling frequency factor method for harmonic vibrational frequencies: the factors’ transferability problem. Chem Phys Lett 465:290–294CrossRefGoogle Scholar
  69. 69.
    Borowski P, Drzewiecka A, Fernández-Gómez M, Fernández-Liencres M-P, Peña Ruiz T (2010) A new, reduced set of scaling factors for both SQM and ESFF calculations. Vib Spect 52:16–21CrossRefGoogle Scholar
  70. 70.
    Borowski P, Peña Ruiz T, Barczak M, Pilorz K, Pasieczna-Patkowska S (2012) Application of the multi-parameter SQM harmonic force field, and ESFF harmonic frequencies scaling procedures to the determination of the vibrational spectra of silicon- and sulfur(II)-containing compounds. Spectrochim Acta A 86:571–585CrossRefGoogle Scholar
  71. 71.
    Borowski P, Pilorz K, Pitucha M (2010) An effective scaling frequency factor method for scaling of harmonic vibrational frequencies: application to 1,2,4-triazole derivatives. Spectrochim Acta A 75:1470–1475CrossRefGoogle Scholar
  72. 72.
    Fábri C, Szidarovszky T, Magyarfalvi G, Tarczay G (2011) Gas-phase and Ar-matrix SQM scaling factors for various DFT functionals with basis sets including polarization and diffuse functions. J Phys Chem A 115:4640–4649CrossRefPubMedCentralGoogle Scholar
  73. 73.
    Morino Y, Kuchitsu K (1952) A note on the classification of normal vibrations of molecules. J Chem Phys 20:1809–1810CrossRefGoogle Scholar
  74. 74.
    Borowski P, Fernández-Gómez M, Fernández-Liencres M-P, Peña Ruiz T (2007) An effective scaling frequency factor method for scaling of harmonic vibrational frequencies: theory and preliminary application to toluene. Chem Phys Lett 446:191–198CrossRefGoogle Scholar
  75. 75.
    Borowski P (2010) An effective scaling frequency factor method for scaling of harmonic vibrational frequencies: the use of redundant primitive coordinates. J Mol Spectr 264:66–74CrossRefGoogle Scholar
  76. 76.
    Martı́nez-Torres E (2000) Formulation of the vibrational theory in terms of redundant internal coordinates. J Mol Struct 520:53–61CrossRefGoogle Scholar
  77. 77.
    Borowski P, Pasieczna-Patkowska S, Barczak M, Pilorz K (2012) Theoretical determination of the infrared spectra of amorphous polymers. J Phys Chem A 116:7424–7435CrossRefPubMedCentralGoogle Scholar
  78. 78.
    Reiher M, Neugebauer J (2003) A mode-selective quantum chemical method for tracking molecular vibrations applied to functionalized carbon nanotubes. J Chem Phys 118:1634–1641CrossRefGoogle Scholar
  79. 79.
    Herrmann C, Neugebauer J, Reiher M (2007) Finding a needle in a haystack: direct determination of vibrational signatures in complex systems. New J Chem 31:818–831CrossRefGoogle Scholar
  80. 80.
    Luber S, Neugebauer J, Reiher M (2009) Intensity tracking for theoretical infrared spectroscopy of large molecules. J Chem Phys 130:064105CrossRefPubMedCentralGoogle Scholar
  81. 81.
    Kiewisch K, Luber S, Neugebauer J, Reiher M (2009) Intensity tracking for vibrational spectra of large molecules. Chimia 63:270–274CrossRefGoogle Scholar

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Authors and Affiliations

  1. 1.Faculty of ChemistryMaria Curie-Skłodowska UniversityLublinPoland

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