Journal of Molecular Modeling

, Volume 19, Issue 7, pp 2865–2877 | Cite as

Relating normal vibrational modes to local vibrational modes: benzene and naphthalene

  • Wenli Zou
  • Robert Kalescky
  • Elfi Kraka
  • Dieter Cremer
Original Paper


Local vibrational modes can be directly derived from normal vibrational modes using the method of Konkoli and Cremer (Int J Quant Chem 67:29, 1998). This implies the calculation of the harmonic force constant matrix F q (expressed in internal coordinates q) from the corresponding Cartesian force constant matrix f x with the help of the transformation matrix U = WB (BWB )−1 (B: Wilson’s B-matrix). It is proven that the local vibrational modes are independent of the choice of the matrix W. However, the choice W = M 1 (M: mass matrix) has numerical advantages with regard to the choice W = I (I: identity matrix), where the latter is frequently used in spectroscopy. The local vibrational modes can be related to the normal vibrational modes in the form of an adiabatic connection scheme (ACS) after rewriting the Wilson equation with the help of the compliance matrix. The ACSs of benzene and naphthalene based on experimental vibrational frequencies are discussed as nontrivial examples. It is demonstrated that the local-mode stretching force constants provide a quantitative measure for the C–H and C–C bond strength.


Normal vibrational modes Local vibrational modes Adiabatic connection scheme Local mode analysis Benzene Naphthalene 



This work was financially supported by the National Science Foundation, grant CHE 1152357. We thank Southern Methodist University for providing computational resources.


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Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  • Wenli Zou
    • 1
  • Robert Kalescky
    • 1
  • Elfi Kraka
    • 1
  • Dieter Cremer
    • 1
  1. 1.Department of ChemistrySouthern Methodist UniversityDallasUSA

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