Quadratic Expansions Normal Equations

A truncated Taylor expansion of the objective function S(6) around the starting point 6 of iteration k gives the approximating function 

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The form of the symmetric matrix of coefficients in Eq. 3-20 for the normal equations of the quadratic is very regular, suggesting a simple expansion to higher-degree equations. The coefficient matrix for a cubic fitting equation is a 4 x 4... 

NORDIO - According to the Rotational Isomeric State approximation, the geometry dependent interactions are simply expressed by additional terms in the total energy. On the other hand, the hydrodynamic interactions give rise to configuration dependent friction coefficients. Under these assumptions, the stationary distribution function is uniquely defined, but the construction of the site functions requires identification of a reactive path. This is done by quadratic expansion of the multivariate diffusion equation about the saddle point connecting two stable conformers, followed by a normal mode analysis. 

The quadratic force constants in the normal co-ordinates are, of course, diagonal, i.e. H = eor<5J<5 .) The expansion (49) is then obtained by substituting (42) and (44) into (48). In fact it is possible to write closed formulae for the force constants in terms of the force constants /, as is done in equations (11) of ref. 12. It is an important property of this transformation that a purely quadratic force field in the curvilinear co-ordinates 8 gives rise to quadratic,... 

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