Concept harmonic force constants

The connecting link between ab initio calculations and vibrational spectra is the concept of the energy surface. In harmonic approximation, usually adopted for large systems, the second derivatives of the energy with respect to the nuclear positions at the equilibrium geometry give the harmonic force constants. For many QM methods such as Hartree-Fock theory (HF), density functional methods (DFT) or second-order Moller-Plesset pertiubation theory (MP2), analytical formulas for the computation of the second derivatives are available. However, a common practice is to compute the force constants numerically as finite differences of the analytically obtained gradients for small atomic displacements. Due to recent advances in both software and computer hardware, the theoretical determination of force field parameters by ab initio methods has become one of the most common and successful applications of quantum chemistry. Nowadays, analysis of vibrational spectra of wide classes of molecules by means of ab initio methods is a routine method [85]. 

From a harmonic o.scillator. the frequencies may be calculated according to Equation 28-42. where k is the suetching constant and m is the mass. Extending the concept from a single mass held to a surface by a spring to N particles requires an extension of the Taylor series expansion (Eq. 28-13) to a matrix formulation of partial second derivatives. Each mode has associated its own force constant, frequency, and 3N relative displacements. The normal modes are assigned to the expcrimenlal IR or Raman spectrum. 

The following first section of this appendix describes quantities that are measured when registering spectra obtained using various experimental set-ups and their relations with molecular quantities. These relations form the basis of the interpretations of molecular spectra. The second section describes some general properties of a distribution that are used in various chapters of this book when this distribution is the band of a spectram. The third section deals with such concepts as normal modes in the harmonic approximation, while the fourth section deals with force constants, reduced masses, etc., and offers comparisons of these various quantities. The last section provides a more specific calculation of the first and second moment of a band such as which corresponds to a normal mode characterized by a strong anharmonic coupling with a much slower mode. 

The concept of the vibrational force field was initially taken from vibrational spectroscopy where the potential energy of a molecule, upon deformation, could be described by a Taylor series with respect to the potential energy at the equilibrium structure. In this case it turns out that the second derivative gives the force constant, and if the harmonic approximation is assumed, then all higher terms may be neglected. 

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