Orbital forces in the density-functional formalism: Application to the copper dimer

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Abstract
The force field in an atomic cluster, given as the direct gradient of the total energy in the density-functional formalism, is expressed in terms of components corresponding to the solutions of the one-particle Schrödinger equation. The resulting relationship between eigenfunctions and orbital forces provides a useful framework for analysis of the bonding in the system. The expression for the orbital force is given as the sum of a traditional Hellmann-Feynman term and an orbital derivative term which cancels the first-order error due to basis-set incompleteness. The sum of orbital forces gives the total gradient force on the nuclei in the system to essentially the same accuracy as the total-energy surface itself. Results are reported for an all-electron calculation of the orbital forces in the copper dimer. This first local-spin-density calculation of the gradient force for a transition-metal dimer represents a challenging test because of the heavy core. By the introduction of a simple screening force, an orbital cohesive force is defined which provides an interesting and useful framework for quantifying the relative contribution of the molecular orbitals to the chemical bond. The effects of core polarization and valence hybridization and their compensating influence are demonstrated in the results.