Pulay correction

The first term on the right hand side is what we had before. The second term is the new ingredient, and is the Pulay correction to the force. It clearly vanishes with basis functions independent of atomic position. 

The first term is the HF expression and the other terms are the Pulay corrections. The second and third terms describe the dependence of 4>t on... 

This approximation often gives good results. Pulay corrections for the forces with pure distributions are particularly difficult to calculate. The analytic expression for the derivatives of i o are unknown and hence approximations are usually introduced, similar to the one mentioned for the mixed estimator. Badinski et al. were able to find an expression for these terms on basis of an integral over the nodal surface. The transformation back to a volume integral introduced an error of order ( r- which they judged to be much smaller than the error in the direct approximation of... 

All calculations have been performed taking fixed positions for the atoms in the structures, so that no relaxations are considered. The Siesta method, however, allows relaxations and molecular dynamics by calculating the forces on the atoms and the stress tensor from the Hellman-Feynman theorem with Pulay corrections The computational cost would be much higher than in the frozen geometry calculations, but the calculations for complex magnetic systems are cheaper than with other ab initio methods (KKR, FLAPW, LMTO). Apart from the use of pseudopotentials and a minimal basis set, the supercell construction does not require the inclusion of empty spheres as it is the case for the LMTO-ASA method. Besides, going from 2D to 3D supported clusters of similar size does not increase the computational cost like in the KKR-GF method where Green-Functions have to be computed. 

However, this relation holds only if the variationally exact solution of the KS equation (eq. 2.5.) has been achieved. Otherwise, the Hellmann-Feynman term for each orbital has to be completed by the following contribution [54] - Pulay-correction ... 

The most common form of AIMD simulation employs DFT (see section First Principles Electronic Structure Methods ) to calculate atomic forces, in conjimction with periodic boundary conditions and a plane wave basis set. Using a plane wave basis has two major advantages over atom-centered basis functions (1) there is no basis set superposition error (Boys and Bernardi 1970 Marx and Hutter 2000) and (2) the Pulay correction (Pulay 1969,1987) to the HeUmann-Feynman force, due to basis set incompleteness, vanishes (Marx and Hutter 2000, 2009). 

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. 

The first choice seems to be more natural since, H() being invariant, the partitioning scheme remains untouched of Moeller-Plesset type. The price to be paid for this principal simplicity, however, is high in calculational details, as the well-developed, systematic many-body graphical algorithms are not applicable if the unperturbed eigenfunctions bear a complicated structure. In a series of papers [44-48], Pulay and Ssebo developed formulas for the second- and third -and fourth-order perturbative corrections with localized orbitals using a CEPA-... 

The arguments of W are explicitly shown here to stress that this equation is valid only for the correct variational solution C(R). Eq. (5) shows the important fact that the first-order changes in the variational parameters are not needed for the evaluation of the gradients of a variational energy expression. This was already stressed by an early publication of Pulay (1969). The importance of this lies in the fact that the solution of the first-order response equations... 

A. Finite-basis sets The molecular wave functions are represented as a finite sum of atom-centred four-component basis functions, which causes a spurious force often called orbital basis correction (OBC) (also known as Pulay force (Pulay 1983)). For an atom centred at Ra it reads... 

Sellers, H., Pulay, P. The adiabatic correction to molecular potential surfaces in the SCF approximation, Chem. Phys. Lett. 1984,103,463-5. 

Estimators of operators other than energy suffer from substantial fluctuations and the zero-variance principle does not hold for them. Variance can be reduced by construction of new estimators that decrease fluctuations without biasing the resulting estimate [99, 100]. Calculations of forces are particularly sensitive to the magnitude of the variance because the estimators are related to derivatives of the total energy. Various aspects of force calculations have been investigated, including sensitivity to electron-nucleus cusp quality [101], effect of Pulay s correction on calculations based on the Hellmann-Feynman theorem [102], and applicability of... 

From the computational experiments of Kristydn and Pulay and P6rez-Jorda and Becke on the potential energy curves of rare gas dimers (Hea, Nea...), it has been shown that local and gradient-corrected functionals fail to describe the dispersion interaction properly. Local functionals severely overestimate the dissociation, while gradient-corrected functionals either display very weak binding, or yield a purely repulsive potential. Much effort must be invested in the future in the study of functionals that can describe dispersion forces properly. 

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