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We make use of the existing formalism for calculating atomic forces within the local-density approximation (LDA) to determine forces in an all-electron, local-orbital framework. The forces are calculated as the proper total-energy derivatives, including the necessary basis-set corrections. In our computational technique the LDA potential is evaluated exactly on a variationally determined integration mesh which allows all integrals relevant to the electronic-structure problem to be computed to any desired accuracy. We apply our technique to the molybdenum dimer, the ethylene molecule, and the NaCl monomer to demonstrate the high accuracy of the calculated forces. In addition, we investigate directly the effects of integration and self-consistency errors on the accuracy of the forces, and use the results of these tests to suggest strategies for the efficient use of force information in structural studies.