Molecular gradients and hessians implemented in density functional theory

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Abstract
We derive expressions for molecular gradients and hessians for the case when the energy is evaluated using density functional theory. Although derivative expressions have been proposed previously, our derivation is based on the unitary exponential parameterization of the wavefunction, and our expressions are valid for local and non–local potentials. Density functional theory, although similar in implementation to standard SCF theory, differs in that it introduces an exchange–correlation term which is density dependent. The presence of such a quantity introduces additional derivative terms which are not present in standard approaches of electronic structure theory. Expressions are derived for both the exact Coulombic repulsion, as well as the case where the density is expressed as a fitted quantity. Given these choices our final equations offer a computationally tractable expression with particular emphasis on conditions which ensure that the computed quantities are numerically correct. We show that although the use of a fitted density allows significant computational savings in the energy and the first derivatives, it introduces additional computational complexity, beyond that normally encountered in traditional electronic structure methods, once second derivatives are evaluated. The evaluation of second derivatives also introduces derivatives of the exchange–correlation potential which have not been previously considered.The presence of such terms introduces the most serious computational complexity to the evaluation of any second derivative expression based on the density–functional formalism. Our derivation and analysis presents a computationally tractable procedure for the evaluation of all the terms required to compute the first and second derivatives with respect to perturbations such as nuclear coordinates, and external electric fields. Using a general set of response equations for the first order change in the wavefunction, we provide expressions for the evaluation of harmonic frequencies, infrared intensities, and molecular polarizabilities. Our final discussion assesses the computational consequences of using either an exact form for the density, or a fitted form. Although most of our discussion is cast in the form of a closed–shell formalism, extensions to an unrestricted (UHF) formalism are straightforward.

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