Hessian matrix frequencies

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. 

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

An additional advantage of second-derivative methods is that frequencies of infrared vibrations can be calculated from the final Hessian matrix. This is only likely to be of relevance to small-molecule systems where good-quality spectra can be obtained. However, in such cases there is the potential to predict spectra and so characterize an unknown compound (see Chapter 9, Section 9.1). The ability to reproduce infrared frequencies should also provide a good test of the force field parameters, but little use has been made so far of this approach [43 5]. 

TABLE I. The two highest frequencies of the short bridge carbonate adsorbed on Pt4 and Pt18 surface cluster models (a) frequencies obtained from explicit diagonalization of the full hessian matrix (b) frequencies obtained using the normal coordinate approach... 

The computation of molecular vibrations is possible with all methods for structure refinement which compute the Hessian matrix (for MM this is the case for optimizers based on second derivatives such as the Newton-Raphson method18). The computed frequencies may then be used for comparison with experimental data90. Recent developments in this area are novel QM-based approaches for the efficient computation of specific vibrational frequencies in large molecules.177... 

The leading quantum correction to the static JT energetics is given by the zero-point energy gain due to the softening of the vibrational frequency at the JT-distorted minima [5]. To obtain this information, by finite differences we compute the Hessian matrix of the second-order derivatives of the lowest adiabatic potential sheet, at one of the static JT minima Q,mn... 

To extract Hessian (matrix of second derivatives of total energy on atoms coordinates) from the results of calculations performed in frames of PC Gamess version of PM3-method one can compute the frequencies of all self vibration modes as good as intensities of infrared (IR) spectra of T-junctions and compare this spectra with computed IR spectra of the ideal nanotubes. 

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