Analytic derivative methods

Pulay P 1987 Analytical derivative methods in quantum chemistry Adv. Chem. Phys. 69 241... 

The various response tensors are identified as terms in these series and are calculated using numerical derivatives of the energy. This method is easily implemented at any level of theory. Analytic derivative methods have been implemented using self-consistent-field (SCF) methods for a, ft and y, using multiconfiguration SCF (MCSCF) methods for ft and using second-order perturbation theory (MP2) for y". The response properties can also be determined in terms of sum-over-states formulation, which is derived from a perturbation theory treatment of the field operator — [iE, which in the static limit is equivalent to the results obtained by SCF finite field or analytic derivative methods. 

Yamaguchi, Y. Osamura, Y. Goddard, J. D. Schaefer H. F. A New Dimension to Quantum Chemistry. Analytic Derivative Methods in Ab Initio Molecular Electronic Structure Theory. Number 29 in International Series of Monographs on Chemistry. Oxford Univ. Press, New York, 1994. 

P. Pulay, Analytical derivative methods in quantum chemistry, in Ab Initio Methods in Quantum Chemistry, (ed. K. P. Lawley), Wiley, New York, 1987. 

Zerner, M.C. (1989). Analytic derivative methods and geometry optimization, in Modern Quantum Chemistry, eds. A. Szabo and N.S. Ostlund (McGraw-Hill, New York), pp. 437 158. 

Pulay, P. Analytical Derivative Methods in Quantum Chemistry, in Advances in Chemical Physics Ab initio Methods in Quantum Chemistry Part II, Lawley, K.P., Ed. Wiley-lnterscience Chichester, 1987. 

Probably even more important to computational quantum chemistry is the development in the analytical evaluation of first, second and higher derivatives of the potential energy with respect to nuclear coordinates . These analytical derivative methods are indispensable to the location and characterization of the stationary points (minima or transition states) on the potential energy surface and have greatly advanced the scope of applicability of ab initio calculations. Ab initio calculations are in a position to predict many new types of the heavier group 15 compounds and provide valuable information for the interpretation of complex experimental data. 

For k = 0, the Fock matrix and its derivatives with respect to the displacements of the nuclei are always block diagonal. Then one can directly apply the analytical derivative methods developed for finite systems to extended systems [69,86,87,88]. But when the displacements break the translational symmetry, the Fock matrix and its derivatives are no longer block diagonal. To solve the CPHF equations, one needs to use the symmetrized (normal mode) coordinates instead of the Cartesian coordinates of the nuclei. Efficient analytical methods have been developed to calculate the energy derivatives for k / 0 with both plane wave [89-90] and general basis functions [85]. The latter can be functions of nuclear coordinates and have linear dependence. These methods reduce the computational cost required to calculate the phonon spectrum with k 7 0 to the same as that needed for the spectrum at k = 0. 

Since their introduction in the late 1960s, gradient methods, or more properly analytical derivative methods, have become one of the most vigorously developing topics on modern quantum chemistry. They have also acquired considerable significance for the solution of practical chemical problems. The first review on this subject was written in 1974-75, although it was published much later (Pulay, 1977) a short chapter, limited to first derivatives,... 

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