Full-matrix second-derivative method

Although vibrational frequencies have been calculated and compared with experimental data since the early days of molecular mechanics refinements using full-matrix second-derivative procedures196,971, there are few reports on the application of this method in the field of transition metal chemistry. One of the few examples is a recent study on linear metallocenes [981. Here, the molecular mechanics force constants were obtained by adjusting initially assumed values by fitting the calculated vibrations to thoroughly analyzed experimental spectra. The average difference (rms) between experimental and calculated vibrations were of the order of ca. 30 cm-1.2 Table... 

Th c Newton-Raph son block dingotial method is a second order optim izer. It calculates both the first and second derivatives of potential energy with respect to Cartesian coordinates. I hese derivatives provide information ahont both the slope and curvature of lh e poten tial en ergy surface, Un like a full Newton -Raph son method, the block diagonal algorilh m calculates the second derivative matrix for one atom at a lime, avoiding the second derivatives with respect to two atoms. 

Besides the above-mentioned problems with step control, there are also other computational aspects which tend to make the straightforward NR problematic for many problem types. The true NR method requires calculation of the full second derivative matrix, which must be stored and inverted (diagonalized). For some function types a calculation of the Hessian is computationally demanding. For other cases, the number of variables is so large that manipulating a matrix the size of the number of variables squared is impossible. Let us address some solutions to these problems. 

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. 

Usually, p is chosen to be a number between 4 and 10. In this way the system moves in the best direction in a restricted subspace. For this subspace the second-derivative matrix is constructed by finite differences from the stored displacement and first-derivative vectors and the new positions are determined as in the Newton-Raphson method. This method is quite efficient in terms of the required computer time, and the matrix inversion is a very small fraction of the entire calculation. The adopted basis Newton-Raphson method is a combination of the best aspects of the first derivative methods, in terms of speed and storage requirements, and the more costly full Newton-Raphson technique, in terms of introducing the most important second-de-... 

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. 

There are a number of variations on the Newton-Raphson method, many of which aim to eliminate the need to calculate the full matrix of second derivatives. In addition, a family of methods called the quasi-Newton methods require only first derivatives and gradually construct the inverse Hessian matrix as the calculation proceeds. One simple way in which it may be possible to speed up the Newton-Raphson method is to use the same Hessian matrix for several successive steps of the Newton-Raphson algorithm with only the gradients being recalculated at each iteration. 

The full second derivative matrix calculated analytically or numerically. This is the most costly and most accurate option, but may be the method of choice for difficult cases. 

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