Block-diagonal Newton-Raphson

Pragmatically, the procedure considers only one atom at a time, computing the 3x3 Hessian matrix associated with that atom and the 3 components of the gradient for that atom and then inverts the 3x3 matrix and obtains new coordinates for the atom according to the Newton-Raphson formula above. It then goes on to the next atom and moves it in the same way, using first and second derivatives for the second atom that include any previous motion of atoms. 

The procedure uses second derivative information and can be quite efficient compared to conjugate gradient methods. However, th e neglect of couplin g in th e Hessian m atrix can lead to situation s where oscillation is possible. Conjugate gradient methods. 

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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. 

HyperChem supplies three types of optimizers or algorithms steepest descent, conjugate gradient (Hetcher-Reeves and Polak-Ribiere), and block diagonal (Newton-Raphson). 

In the block-diagonal Newton-Raphson minimization, the generally small size of the off-diagonal terms is exploited and the matrix describing the curvature is reduced to N 3 3 matrices, i.e., to 9N elements (Fig. 3.8). Due to the approximations... 

Fig. 3.8. Arrangement of the blocks in the block-diagonal Newton-Raphson minimization procedure.

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. 

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