Pulay’s method

Pulay s method is very powerful for slowly convergent cases, while at the same time it does not normally slow down the rate of convergence in well-behaved systems. 

Note Pulay s mention of the Hellman-Feynman theorem. We have moved on since 1969, especially in the development and application of analytical methods for evaluating the gradients. 

Transition states for rate-limiting elimination of nitrogen on unimolecular thermal decomposition of methyl and ethyl azide have been defined by application of Pulay s SQMFF method.68... 

The general approach used in LSA methods to introduce electron correlation is based on Meyer s [22] self-consistent electron pair (SCEP) theory, later extended by Ahlrichs [23] and Dykstra et al. [24-27]. Additional important improvements in SCEP theory were made by Saebo and Pulay [28-36] leading ultimately to their very successful local correlation treatment. We have adopted many of the specific features devised by Saebo and Pulay. Indeed, at first glance, the similarity between their treatment and ours might be more evident than the differences. However, the LSA deals particularly with the situation where a calculation of the entire system is impractical at any level. Thus, in contrast with Saebo and Pulay s procedure, our method applies even for the HF model. Moreover, because we start with isolated fragments the LSA method can be utilized to treat fully delocalized problems like chemisorption on a metal surface which are not amenable to a local correlation treatment. 

A more sophisticated method that is often very successful is Pulay s direct inversion of the iterative subspace (DIIS) [Pulay 1980]. Here, the energy is assumed to vary as a quadratic function of the basis set coefficients. In DUS the coefficients for the next iteration are calculated from their values in the previous steps. In essence, one is predicting where the minimum in the energy will lie from a knowledge of the points that have been visited and by assuming that the energy surface adopts a parabolic shape. 

Pulay s paper is an early landmark in the explosive growth in computational chemistry that we have seen in the past quarter century. The method for calculating first derivatives as outlined in the article forms the basis for the subsequent development of first, second and higher energy derivatives for many different theoretical methods (for reviews, see Refs. [1-5]). The advances brought about by energy derivative methods have enabled theoretical calculations to become practical and efficient... 

Pulay demonstrated that analytic first derivatives with respect to geometric parameters can be calculated easily and efficiently for HF energies. Derivatives of correlated methods followed a number of years after SCF derivatives [4, 5]. Extensions of the SCF derivatives to density functional theory methods were straightforward. In the three decades since Pulay s article, hundreds of papers on energy derivatives have been published, and all can trace their roots back to his paper. Energy derivatives have become so useful for calculating molecular structures and properties that, almost universally, first derivatives are formulated and coded soon after a new theoretical method is developed for the energy. 

Pulay s paper opened the way for analytic second and higher derivatives of the SCF energy. Earlier papers had suggested that this might be prohibitively expensive [7], but the development of an efficient method to solve the couple perturbed HF (CPHF) equations, made the calculation of SCF second derivatives practical [8]. As a consequence, vibrational force constants and frequencies could be calculated routinely and efficiently. Third and fourth geometric derivatives of the SCF energy followed after a few years [9-12]. The solution of the CPHF equations (in their full or reduced Z-vector form [13]) also made post-SCF first derivatives practical and cost-effective. 

A method to speed up MP2 calculations on large molecules is the local MP2 (LMP2) method of Saebp and Pulay [S. Saebd and P. Pulay, Annu. Rev. Phys. Chem., 44, 213 (1993)]. Here, instead of using canonical SCE MOs in the Hartree-Fock reference determinant d>o. one transforms to localized MOs (Section 15.8). Also, instead of using the virtual orbitals found in the SCF calculation as the orbitals a and b in (16.13) to which electrons are excited, one uses atomic orbitals that are orthogonal to the localized occupied MOs. Also, in (16.13), one includes only unoccupied orbitals a and b that are in the neighborhood of the localized MOs i and j. 

Saebe, S., Tong, W. and Pulay, P. Efficient elimination of basis set superposition errors by the local correlation method Accurate ab initio studies of the water dimer, J. Chem. Phys., 98, 2170-2175. 

S. S b0 and P. Pulay,/. Chem. Phys., 86, 914 (1987). Fourth-Order Moller-Plesset Perturbation Theory in the Local Correlation Treatment. I. Method. 

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