Full-matrix second-derivative minimizer

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. 

When the basis size is small enough to store the Hamiltonian matrix in the computer core memory, two things can be said with confidence. First, the method presented in Sec. II based on Eq. (1) and Eqs. (2) and (9) (or better to avoid anomalies, (1) and (21)) are very easy to comprehend and implement. This is especially true when the diagonalization of the full Hamiltonian is the key computational step. Second, there are many other approaches, such as the Kohn variational principle (21), the / -matrix theory (35), and the closely related, log-derivative methods (22, 23), that are easy to implement and anomaly free. The methods which use absorbing potentials clearly have a disadvantage relative to the above methods in the sense that they require larger than minimal basis sets and involve non-Hermitian matrices. 

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