Exchange energy gradients with respect to atomic positions and cell parameters within the Hartree-Fock Gamma-point approximation

Recently, linear scaling construction of the periodic exact Hartree-Fock exchange matrix within the Gamma-point approximation has been introduced J. Chem. Phys. 122, 124105 (2005)]. In this article, a formalism for evaluation of analytical Hartree-Fock exchange energy gradients with respect to atomic positions and cell parameters at the Gamma-point approximation is presented. While the evaluation of exchange gradients with respect to atomic positions is similar to those in the gas phase limit, the gradients with respect to cell parameters involve the accumulation of atomic gradients multiplied by appropriate factors and a modified electron repulsion integral (ERI). This latter integral arises from use of the minimum image convention in the definition of the Gamma-point Hartree-Fock approximation. We demonstrate how this new ERI can be computed with the help of a modified vertical recurrence relation in the frame of the Obara-Saika and Head-Gordon-Pople algorithm. As an illustration, the analytical gradients have been used in conjunction with the QUICCA algorithm K. Nemeth and M. Challacombe, J. Chem. Phys. 121, 2877 (2004)] to optimize periodic systems at the Hartree-Fock level of theory.