Pulay

Pulay P 1995 Analytioal derivative teohniques and the oaloulation of vibrational speotra Modem Electronic Structure Theory yo 2, ed D R Yarkony (Singapore World Soientifio) pp 1191-240 Muoh of the early work is desoribed in ... 

Pulay P 1977 Direot use of the gradient for investigating moleoular energy surfaoes Modern Theoretical Chemistry yo ... 

Pulay P 1987 Analytioal derivative methods in quantum ohemistry Advances in Chemical Physics vol LXIX, ed K P Lawley (New York Wiley-Intersoienoe) pp 241-86... 

Saebo S and Pulay P 1993 Loeal treatment of eleetron eorrelation Ann. Rev. Phys. Chem. 44 213-36... 

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

Pulay P 1969 Ab initio calculation of force constants and equilibrium geometries in polyatomic molecules. I. Theory/Wo/. Phys. 17 197... 

Pulay P 1977 Direct use of the gradients for investigating molecular energy surfaces Appiications of Eiectronic Structure Theory ed H F Schaefer III (New York Plenum) p 153... 

Pulay P 1980 Convergence acceleration in iterative sequences the case of SCF iteration Chem. Phys. Lett. 73 393... 

Pulay P 1982 Improved SCF convergence acceleration J. Comput. Chem. 3 556... 

Csaszar P and Pulay P 1984 Geometry optimization by direct inversion in the iterative subspace J. Moi. Struct. (Theochem) 114 31... 

Hamilton T P and Pulay P 1986 Direct Inversion In the Iterative subspace (DNS) optimization of open-shell, excited-state and small multiconfiguratlonal SCF wavefunctlons J. Chem. Phys. 84 5728... 

Palzs B, FogarasI G and Pulay P 1998 An efficient direct method for geometry optimization of large molecules J. Chem. Phys. 109 6571... 

Baker J, KInghorn D and Pulay P 1999 Geometry optimization In delocalized Internal coordinates An efficient quadratically scaling algorithm for large molecules J. Chem. Phys. 110 4986... 

Pulay P and Meyer W 1971 Ab initio calculation of the force field of ethylene J. Moi. Spectrosc. 40 59... 

Pulay P, FogarasI G, Pang F and Boggs J E 1979 Systematic ab initio gradient calculation of molecular geometries, force constants and dipole moment derivatives J. Am. Chem. Soc. 101 2550... 

FogarasI G, Zhou X, Taylor P W and Pulay P 1992 The calculation of ab initio molecular geometries efficient optimization by natural Internal coordinates and empirical correction by offset forces J. Am. 

Eckert F, Pulay P and Werner H-J 1997 Ab initio geometry optimization for large molecules J. Comput. Chem. 18 1473... 

Pulay P and FogarasI G 1992 Geometry optimization In redundant Internal coordinates J. Chem. Phys. 96 2856... 

Pulay P 1995 Analytical derivative teclmiques and the calculation of vibrational spectra Modern Electronic Structure Theory ed D Yarkony (Singapore World Scientific) pp 1191-240... 

This, the well-known Hellmann-Feynman theorem [128,129], can then be used for the calculation of the first derivatives. In nonnal situations, however, the use of an incomplete atom-centered (e.g., atomic orbital) basis set means that further terms, known as Pulay forces, must also be considered [130]. 

P. Pulay, in Modem Electronic Structure Theory, D. Yarkony, ed.. World Scientific, Singapore, 1995, pp. 1191-1240. 

Unfortunately, this only holds for the exact wavefunction and certain other types ol leavefuiiction (such as at the Hartree-Fock limit). Moreover, even though the Hellmarm-Feynman forces are much easier to calculate they are very unreliable, even for accurate wavefunctions, giving rise to spurious forces (often referred to as Pulay forces [Pulay l )S7]). 

When three-point interpolation fails to yield a convergent calculation, you can request a second accelerator for any SCFcalculation via the Semi-empirical Options dialog box and the Ab Initio Options dialog box. This alternative method. Direct Inversion in the Iterative Subspace (DIIS), was developed by Peter Pulay [P. Pulay, Chem. Phys. Lett., 73, 393 (1980) J. Comp. Chem., 3, 556(1982)]. DIIS relies on the fact that the eigenvectors of the density and Fock matrices are identical at self-consistency. At SCF convergence, the following condition exists... 

A pioneering paper in gradient evaluation is that of Pulay. 

Ab Initio Calculation of Force Constants and Equilibrium Geometries in Polyatomic Molecules. I. Theory P. Pulay... 

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. 

The only remaining question is the nature of the error function. Pulay suggested the difference FDS — SDF (S is the overlap matrix), which is related to the gradient of the SCF energy with respect to the MO coefficients. This has been found to work well in practice. 

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