form: U ( r ) = U ( r 0 ) − F • z + 1 2 r T • H • z (6) 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 f ( ω ) = 〈 1 3 N 3 N ∑ i =1 δ ( ω − ω$_{i}$ ) 〉 . (7) Quantities that are convenient for characterizing the instantaneous normal mode spectrum are: (i) the fraction of imaginary frequencies  namely F$_{im}$ = ∫ im f ( ω ) dω (8) where the subscript in means that the integral is performed only in the imaginary branch; (ii) the fraction of real frequencies  that is F$_{r}$ = ∫ r f ( ω ) dω (9) 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 ω 2 E = ∫ ω $^{2}$f ( ω ) dω = 〈 T r H 〉 m (3 N − 3) (10) where the last equality comes from using Eq. (7) and 〈 T r H 〉 is the ensemble-averaged value of the trace of the Hessian. 7