Potential energy functions Hessian

While Eq. (9.49) has a well-defined potential energy function, it is quite difficult to solve in the indicated coordinates. However, by a clever transfonnation into a unique set of mass-dependent spatial coordinates q, it is possible to separate the 3 Ai-dirncnsional Eq. (9.49) into 3N one-dimensional Schrodinger equations. These equations are identical to Eq. (9.46) in form, but have force constants and reduced masses that are defined by the action of the transformation process on the original coordinates. Each component of q corresponding to a molecular vibration is referred to as a normal mode for the system, and with each component there is an associated set of harmonic oscillator wave functions and eigenvalues that can be written entirely in terms of square roots of the force constants found in the Hessian matrix and the atomic masses. 

The Hessians of potential energy functions, for example, separate naturally into local terms (among atom pairs involved in bonds, bond angles, and dihedral angles) and nonlocal terms (among nonbonded atom pairs). The number of local terms increases linearly with n, whereas the nonlocal terms increase as n2. Thus, a preconditioner from the local terms is a good choice that has performed well in practice.23-82... 

Figure 3 Sparse matrix structures resulting from the Hessian of the potential energy function of the protein BPTI (a) when 8 A cutoffs are used for the nonbonded terms, 13% nonzeros and (b) when only bond length, bond angle, and dihedral angle terms are included, with insets showing enlarged submatrices...

The strategies for saddle point optimizations are different for electronic wave functions and for potential energy surfaces. First, in electronic structure calculations we are interested in saddle points of any order (although the first-order saddle points are the most important) whereas in surface studies we are interested in first-order saddle points only since these represent transition states. Second, the number of variables in electronic structure calculations is usually very large so that it is impossible to diagonalize the Hessian explicitly. In contrast, in surface studies the number of variables is usually quite small and we may easily trans-... 

Density functional theory (DFT) calculations have been carried out to elucidate the structure and energetics of the various isomeric (Si-N) rings as well as the corresponding anions and dianions. The local minima on the potential energy surfaces were verified by computation of the eigenvalues of the respective Hessian matrices. From these, harmonic vibrational frequencies and the zero-point vibration corrected energetics were calculated. The calculations were carried out for isolated molecules in the gas phase. The theoretical results are expected to be reliable for molecules in non-polar or weakly aprotic polar solvents. 

This transition-state-like point is called a bond critical point. All points at which the first derivatives are zero (caveat above) are critical points, so the nuclei are also critical points. Analogously to the energy/geometry Hessian of a potential energy surface, an electron density function critical point (a relative maximum or minimum or saddle point) can be characterized in terms of its second derivatives by diagonalizing the p/q Hessian([Pg.356]

In the present work, we report DFT calculations of hydrocarbons adsorption on Au2o cluster [4]. All calculations were carried out with the nonempirical local PBE (Perdew—Burke—Ernzerhof) functional, which we have used earlier in the study of gold complexes [5]. Calculations were performed with a PRIRODA software [6]. The basis set with the SBK pseudopotential was used [7]. In this pseudopotential, the outer electronic shells are described by the following basis sets H [311/1], C [311/311/11] and Au [51111/51111/5111]. The types of stationary points on potential energy surfaces were determined from analysis of Hessians. The second derivatives were calculated analytically. 

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