Mass-weighted Hessian matrix

These transformation eoeffieients Crj,k can be used to earry out a unitary transformation of the 9x9 mass-weighted Hessian matrix. In so doing, we need only form bloeks... 

The eigenvalues (coa of the mass weighted Hessian matrix (see below) are used to compute, for each of the 3N-7 vibrations with real and positive cOa values, a vibrational partition function that is combined to produce a transition-state vibrational partition function ... 

In general, the mass-weighted Hessian matrix A has 3N eigenvectors,... 

No first derivative terms appear here because the transition state is a critical point on the energy surface at the transition state all first derivatives are zero. This harmonic approximation to the energy surface can be analyzed as we did in Chapter 5 in terms of normal modes. This involves calculating the mass-weighted Hessian matrix defined by the second derivatives and finding the N eigenvalues of this matrix. 

Now we set up what s called the mass-weighted Hessian matrix, a square, symmetric matrix whose elements are the second derivatives U/ dZjdZj) divided by Vm ... 

Diagonalize the projected, mass-weighted Hessian matrix. 

The standard analytic procedure involves calculating the orthogonal transformation matrix T that diagonalizes the mass weighted Hessian approximation H = M 2HM 2, namely... 

As a result, the 9x9 mass-weighted Hessian eigenvalue problem ean be sub divided into two 3x3 matrix problems ( of ai and b2 symmetry), one 2x2 matrix of bi symmetry... 

A different approach comes from the idea, first suggested by Helgaker et al. [77], of approximating the PES at each point by a harmonic model. Integration within an area where this model is appropriate, termed the trust radius, is then trivial. Normal coordinates, Q, are defined by diagonalization of the mass-weighted Hessian (second-derivative) matrix, so if... 

The Hessian matrix is useful in others ways, too. The square root of the mass-weighted Hessian eigenvalue is proportional to the vibrational frequency < ,. 

INMs are obtained by diagonalizing the Hessian matrix generated by expanding the system potential energy to quadratic order in mass-weighted coordinates. 

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... 

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