Harmonic approximation Hessian

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. 

The Mode-Tracking idea originated from the fact that the standard quantum chemical calculation of vibrational spectra within the harmonic approximation requires the calculation of the complete Hessian matrix [92]. The Hessian is the matrix of all second derivatives of the electronic energy E i with respect to the nuclear coordinates R. Its calculation gets more computer time demanding the larger the molecule is. However, many if not most vibrations of a supramolecular assembly are of little importance for the function and chemical behavior of this assembly. 

Finally, for this section we note that the valence interactions in Eq. [1] are either linear with respect to the force constants or can be made linear. For example, the harmonic approximation for the bond stretch, 0.5 (b - boV, is linear with respect to the force constant If a Morse function is chosen, then it is possible to linearize it by a Taylor expansion, etc. Even the dependence on the reference value bg can be transformed such that the force field has a linear term k, b - bo), where bo is predetermined and fixed, and is the parameter to be determined. The dependence of the energy function on the latter is linear. [After ko has been determined the bilinear form in b - bo) can be rearranged such that bo is modified and the term linear in b - bo) disappears.] Consequently, the fit of the force constants to the ab initio data can be transformed into a linear least-squares problem with respect to these parameters, and such a problem can be solved with one matrix inversion. This is to be distinguished from parameter optimizations with respect to experimental data such as frequencies that are, of course, complicated functions of the whole set of force constants and the molecular geometry. The linearity of the least-squares problem with respect to the ab initio data is a reflection of the point discussed in the previous section, which noted that the ab initio data are related to the functional form of empirical force fields more directly than the experimental data. A related advantage in this respect is that, when fitting the ab initio Hessian matrix and determining in this way the molecular normal modes and frequencies, one does not compare anharmonic and harmonic frequencies, as is usually done with respect to experimental results. 

In practice, of course, the surface is only quadratic to a first approximation and so a number of steps will be required, at each of which the Hessian matrix must be calculated and inverted. The Hessian matrix of second derivatives must be positive definite in a Newton-Raphson minimisation. A positive definite matrix is one for which all the eigenvalues are positive. When the Hessian matrix is not positive definite then the Newton-Raphson method moves to points (e.g. saddle points) where the energy increases. In addition, far from a mimmum the harmonic approximation is not appropriate and the minimisation can become unstable. One solution to this problem is to use a more robust method to get near to the minimum (i.e. where the Hessian is positive definite) before applying the Newton-Raphson method. 

A common method for computing frequencies is to calculate the second derivative matrix of the potential energy. Once a minimum energy configuration is obtained, the harmonic approximation (i.e., F = -k(r - req) and dV= Fdr, so V= V2 k(r - reqf where F is the force, k is the force constant, and req is an equilibrium position) may be employed to calculate the vibrational frequencies. The elements of the vibrational matrix, V, are related to the potential energy second derivative matrix, the Hessian, by... 

The calculation of the vibrational spectrum from an (AI)MD trajectory involves Fourier-transforming the time-dependent velocity autocorrelation function [60] an alternative approach involves calculating the phonon frequencies by diagonalizing the Hessian matrix of a model obtained by structural optimization of the classical MD structure [53]. The AIMD-VACF approach naturally include finite-temperature anharmonic effects missing in the Hessian-harmonic approximation, but it does not produce accurate IR intensities (for which an autocorrelation function based on the exact dipole moments would be needed [61-63]). Despite these issues, it turns out that, in the case of 45S5 Bioglass , the two methods give similar frequencies of the individual modes [53]. 

As mentioned in the introduction, most of the QM calculations of vibrational frequencies are performed within the double-harmonic approximation, that is, the truncation of the expansion of the potential energy as a function of the nuclear coordinates to the quadratic term (mechanical harmonic approximation) and the consideration of the hnear term only in the expansion of the dipole moment as a function of the nuclear coordinates (electric harmonic approximation). In such a framework, the QM calculation of vibrational frequencies can be reduced to the evaluation of the components of the Hessian matrix followed by diagonalization of the corresponding mass-weighted matrix [1]. Let us start by writing out the energy of a... 

Using the normal modes that diagonalize the mass weighted Hessian both the expressions q5[q - q 0[q ) and 0[q -q ) appearing in (32) can be easily calculated analytically as canonical averages, yielding the TST-rate constant in the harmonic approximation. 

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... 

Vibrational frequencies, used to predict IR and Raman spectra, are computed from the Hessian matrix, assuming a harmonic oscillator approximation. Errors in the... 

So far, we have only separated out the HX vibrational motion. Generally, such a solution of the stationary Schiodinger equation is not computationally feasible for clusters with more atoms. Therefore, other approximations have to be employed. At the same time, all phenoniena important for the cluster structure have to be properly included. We have performed an adiabatic separation of the HX libra-tional motion from the motion of the heavy particles, i.e., from the cage modes. Moreover, the cage modes have been calculated within the harmonic approach, i.e., by a diagonalization of the Hessian matrix. Formally, the wavefunction is expressed as... 

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