content_image/340410.jpg content_image/340412.jpg content_image/340414.jpg content_image/340416.jpg content_image/340418.jpg
form: 

<img src='content_image/340410.jpg'>

where z i = √ m$_{i}$ ( r i − r 0 ) are the mass-scaled position coordinates of a particle i . The first and second derivatives of U ( r ) with respect to the vector z are the force and the Hessian matrix, denoted by F and H respectively. The eigenvalues of the Hessian H are ( { ω 2 i } , i = 1 , 3 N ) representing the squares of normal mode frequencies, and W ( r ) are the corresponding eigenvectors. In a stable solid, r 0 can be conveniently taken as the global minimum of the potential energy surface U ( R ), which implies that F = 0 and H has only positive eigenvalues corresponding to oscillatory modes. The INM approach for liquids interprets r as the configuration at time t relative to the configuration r 0 at time t$_{0}$ . Since typical configurations, r 0 are extremely unlikely to be local minima, therefore F ̸ = 0 and H will have negative eigenvalues. The negative eigenvalue modes are those which sample negative curvature regions of the PES, including barrier crossing modes. The ensemble- averaged INM spectrum, 〈 f ( ω ) 〉 , is defined as 

<img src='content_image/340412.jpg'>

Quantities that are convenient for characterizing the instantaneous normal mode spectrum are: (i) the fraction of imaginary frequencies, namely 

<img src='content_image/340414.jpg'>

where the subscript in means that the integral is performed only in the imaginary branch; (ii) the fraction of real frequencies, that is 

<img src='content_image/340416.jpg'>

where the subscript r indicates that the integral is performed only in the real branch and (ii) the mean square or Einstein frequency, ω$_{E}$ , given by 

<img src='content_image/340418.jpg'>

where the last equality comes from using Eq. (7) and 〈 T r H 〉 is the ensemble-averaged value of the trace of the Hessian. 

