Normal Coordinates and Harmonic Frequency Analysis

In mass-weighted coordinates, the hessian matrix becomes the hannonic force constant matrix, from which a normal coordinate analysis may be carried out to yield hannonic frequencies and normal modes, essentially a prediction of the fundamental IR transition 

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A vibrations calculation is the first step of a vibrational analysis. It involves the time consuming step of evaluating the Hessian matrix (the second derivatives of the energy with respect to atomic Cartesian coordinates) and diagonalizing it to determine normal modes and harmonic frequencies. For the SCFmethods the Hessian matrix is evaluated by finite difference of analytic gradients, so the time required quickly grows with system size. 

Harmonic oscillators simplify the analysis. Normal coordinates and frequencies differ in principle from state to state and involve both state rotations and energy renormalization. In practice, unrotated ground-state coordinates are retained even when different frequencies are used in detailed fits [21,91,92]. Displaced harmonic oscillators yield analytical expressions [93,94] for Fp, when ground-state frequencies o>i are kept in excited states. Moreover, since o>i is typically smaller than electronic splittings, some of the sums in Eq. (30) can be evaluated by closure. Closure can readily be checked and holds unless an energy denominator becomes small. Closure leads to Franck-Condon averages [Pg.181]

The MP2/TZDP optimized structures were then used to calculate the stationary state geometry force constants and harmonic vibrational frequencies, also at the MP2 level. These results serve several purposes. Firstly, they test that the calculated geometry is really an energy minimum by showing all real frequencies in the normal coordinate analysis. Secondly, they provide values of the zero-point energy (ZPE) that can be used... 

As the values in Table 18 indicate, the coupling between the benzene ring and the rest of the complex cannot simply be neglected, which was often the practice in the course of the normal-coordinate analysis of this compound, based on observed vibrational frequencies. As mentioned before, the normal-coordinate analysis from the reverse vibrational problem cannot provide all of the significant coupling constants of the harmonic force field this is simply due to the paucity of experimental information. Quantum mechanical calculations are necessary to obtain information about the significant coupling constants. 

Minimized structures gained from MD simulations are also often basis of normal mode analysis (NMA) [41-43]. NMA assumes that all atoms harmonically oscillate around their equilibrium points. The oscillations deflned by frequency and amplitude (normal mode) are extracted and reflect directions of internal protein motions. Given all its normal modes, the entire protein motion can be expressed as a superposition of modes. The modes vith lo vest frequency correspond to rather delocalized motions in proteins in vhich a large number of atoms oscillate in coordinated motion vith considerable amplitude. Modes vith higher frequency represent more localized motions. Linear combinations of the most relevant normal modes can be employed to depict essential protein motions. Stepwise displacement of atoms of the original structure along the modes can be applied to build up an ensemble of relevant protein conformations [44, 45]. 

Vibrational frequencies may be extracted from the PES by performing a normal mode analysis. This analysis of the normal vibrations of the molecular configurations is a difficult topic and can be pursued efficiently only with the aid of group theory and advanced matrix algebra. In essence, the 3 translational, 3 rotational and 3N-6 vibrational modes (2 rotational and 3N-5 vibrational modes for linear molecules) may be determined by a coordinate transformation such that all the vibrations separate and become independent normal modes, each performing oscillatory motion at a well defined vibrational frequency. As a more concrete illustration, assume harmonic vibrations and separable rotations. The PES can thus be approximated by a quadratic form in the coordinates... 

Currently, there are several computational methods for the calculation of the normal modes of a molecule. Harmonic analysis is the direct computation of normal modes by finding the eigenvalues and eigenvectors, which are the solutions to the secular equation of motion involving the second derivatives of the potential function and mass weighted coordinates. It is essential that the calculation be started from a fully energy minimized conformation for the resulting normal mode frequencies to be accurate. 

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