Quadratic function geometry

Fig. 2.17 The potential energy of a diatomic molecule near the equilibrium geometry is approximately a quadratic function of the bond length. Given an input structure (i.e. given the bond length qi), a simple algorithm would enable the bond length of the optimized structure to be found in one step, if the function were strictly quadratic...

The principle of equating a second derivative with a stretching or bending force constant is not exactly correct. A second derivative d2E/ dq1 would be strictly equal to a force constant only if the energy were a quadratic function of the geometry, i.e. if a graph of E versus q were a parabola. However vibrational curves are not exactly parabolas (Fig. 5.32). For a parabolic Elq relationship, and considering a diatomic molecule for simplicity, we would have ... 

In 1970, Bender and Schaefer " reported afc initio computations of triplet methylene. Employing the CISD/DZ method, they computed the energy of triplet methylene at 48 different geometries, varying the C-H distance and H-C-H angle. Fitting this surface to a quadratic function, they predicted that the H-C-H angle is 135.1°, and emphatically concluded that the molecule is not linear. 

The Newton-Raphson approach is another minimization method.f It is assumed that the energy surface near the minimum can be described by a quadratic function. In the Newton-Raphson procedure the second derivative or F matrix needs to be inverted and is then usedto determine the new atomic coordinates. F matrix inversion makes the Newton-Raphson method computationally demanding. Simplifying approximations for the F matrix inversion have been helpful. In the MM2 program, a modified block diagonal Newton-Raphson procedure is incorporated, whereas a full Newton-Raphson method is available in MM3 and MM4. The use of the full Newton-Raphson method is necessary for the calculation of vibrational spectra. Many commercially available packages offer a variety of methods for geometry optimization. 

If U were accurately a quadratic function of the coordinates in the region near (A l, y,), then the second partial derivatives (the elements of the Hessian matrix) would be constants in this region, and the subscript 1 on the second partials would be unnecessary. Accurate ab initio SCF calculation of the second derivatives is very time-consuming, so one usually uses a quasi-Newton method, meaning that one starts with an approximation for the Hessian and improves this approximation as the geometry optimization proceeds. We therefore write... 

Determination of the paiameters entering the model Hamiltonian for handling the R-T effect (quadratic force constant for the mean potential and the Renner paiameters) was carried out by fitting special forms of the functions [Eqs. (75) and (77)], as described above, and using not more than 10 electronic energies for each of the X H component states, computed at cis- and toans-planai geometries. This procedure led to the above mentioned six parameters... 

Abstract. A smooth empirical potential is constructed for use in off-lattice protein folding studies. Our potential is a function of the amino acid labels and of the distances between the Ca atoms of a protein. The potential is a sum of smooth surface potential terms that model solvent interactions and of pair potentials that are functions of a distance, with a smooth cutoff at 12 Angstrom. Techniques include the use of a fully automatic and reliable estimator for smooth densities, of cluster analysis to group together amino acid pairs with similar distance distributions, and of quadratic progrmnming to find appropriate weights with which the various terms enter the total potential. For nine small test proteins, the new potential has local minima within 1.3-4.7A of the PDB geometry, with one exception that has an error of S.SA. 

Geometry of a quadratic objective function of two independent variables—elliptical contours. If the eigenvalues are equal, then the contours are circles. 

Geometry of a quadratic objective function of two independent variables—saddle point. 

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