Hessian matrix normal mode analysis

In view of the Hessian character (10.20) of the thermodynamic metric matrix M(c+2), the eigenvalue problem for M(c+2) [(10.23)] can be usefully analogized with normal-mode analysis of molecular vibrations [E. B. Wilson, Jr, J. C. Decius, and P. C. Cross. Molecular Vibrations (McGraw-Hill, New York, 1955)]. The latter theory starts from a similar Hessian-type matrix, based on second derivatives of the mechanical potential energy Vpot (cf. Sidebar 2.8) rather than the thermodynamic internal energy U. 

The most common method for determining vibrational frequencies is the normal mode analysis, based on the harmonic force constant matrix of energy second derivatives (Hessians). Of course, vibrations are not truly harmonic, and the anharmonicity generally increases as the frequency of the vibration (steepness of the potential) decreases. That is, the more anharmonic a motion is, the less applicable is the traditional approach to... 

Chemical reactions are usually pictured as changes in a molecule s structure, as defined by the positions of its atomic nuclei. In this electron-following picture [1, 2], the rearrangement of the atomic nuclei is the essential feature of a chemical process, and the most favorable types of rearrangement can be revealed by normal mode analysis of the Hessian or the compliance matrix [3-5],... 

All the quantities contributing to the partition function in Eqs [39-41] can be computed once the intermolecular potentials are known. The calculation proceeds as follows. Several proton-disordered hydrate structures with zero net dipole are chosen. The minimum energy of each gas hydrate structure is computed from a geometry optimization, usually using the steepest descent method. The vibrational entropies are from normal mode analysis of lattice dynamics results. The host and guest contributions are determined from the Hessian matrix V (second derivative of the intermolecular potential with respect to the displacements). The Hessian matrix can be divided into four sub-matrices, V, ... 

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... 

On the other hand, one can calculate the approximate vibrational spectra via the instantaneous normal mode (INM) analysis [32-35]. The l-th INMs are defined by diagonalizing an instantaneous Hessian ( INM-Hessian ) matrix at the Z-th solvent conformation = Rm in the sequence of conformations. The corresponding l-th eigenvalues (Z = 1, , 3N) are obtained by... 

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