Eigenvalues, eigenvectors and band

It should be mentioned that the matrices A and B are diagonally dominant banded matrices. The solution to the eigenvalue problem for such matrices is numerically stable. Routines from the LAPACK library [37] can be used to obtain the eigenvalues and eigenvectors numerically. 

It is widely accepted that vibrational dynamics of atoms and molecules are reasonably well represented by harmonic force fields. The resolution of the secular equation transforms a set of (say N) coupled oscillators into (N) independent oscillators along orthogonal (normal) coordinates. Eigenvalues of the dynamical matrix are normal frequencies and eigenvectors give atomic displacements for each normal mode [3,4,13]. If band intensities cannot be frilly exploited, as it is normally the case for infrared and Raman spectra, these vectors are unknown and force fields refined with respect to observed frequencies only are largely underdetermined. For complex systems, symmetry consideration or/and isotopic substitutions may remove only partially this under determination. 

Here Bi(k) and Ci(k) are the eigenvalues (bands) and eigenvectors (LCAO expansion coefficients of the Bloch fimction) of the one-electron equation for the infinite periodic system (see Chapter 1)... 

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