Block-diagonal Newton-Raphson method

Unconstrained optimization methods [W. II. Press, et. ah, Numerical Recipes The An of Scieniific Compulime.. Cambridge University Press, 1 9H6. Chapter 101 can use values of only the objective function, or of first derivatives of the objective function. second derivatives of the objective function, etc. llyperChem uses first derivative information and, in the Block Diagonal Newton-Raphson case, second derivatives for one atom at a time. TlyperChem does not use optimizers that compute the full set of second derivatives (th e Hessian ) because it is im practical to store the Hessian for mac-romoleciiles with thousands of atoms. A future release may make explicit-Hessian meth oils available for smaller molecules but at this release only methods that store the first derivative information, or the second derivatives of a single atom, are used. 

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. 

HyperChem s optimizers (steepest descent, Fletcher-Reeves, and Polak-Ribiere conjugate-gradient methods, and the block diagonal Newton-Raphson) differ in their generality, convergence properties, and computational requirements. They are unconstrained optimization methods however, it is possible to restrain molecular mechanics and quantum mechanics calculations in HyperChem by adding extra restraining forces. 

The Newton-Raphson block diagonal method is a second order optimizer. It calculates both the first and second derivatives of potential energy with respect to Cartesian coordinates. These derivatives provide information about both the slope and curvature of the potential energy surface. Unlike a full Newton-Raph son method, the block diagonal algorithm calculates the second derivative matrix for one atom at a time, avoiding the second derivatives with respect to two atoms. 

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