Stress Concepts and Applications

Xerox Palo Alto Research Center, 3333 Coyote Hill Road Palo Alto, CA 94304 

The stress theorem determines the stress from the electronic ground state of any quantum system with arbitrary strains and atomic displacements. We derive this theorem in reciprocal space, within the local-density-functional approximation. The evaluation of stress, force and total energy permits, among other things, the determination of complete stress-strain relations including all microscopic internal strains. We describe results of ab-initio calculations for Si, Ge, and GaAs, giving the equilibrium lattice constant, all linear elastic constants Cy and the internal strain parameter t,. 

Total energy calculations of the quantum mechanical ground state have advanced significantly in recent years, and have been applied to an ever-increasing number of different systems and physical properties. The core of most of this work is the density-functional theory of Hohenberg, Kohn and Sham, and in particular the local-density approximation (see, e.g., Lundqvist and March, 1983). This theory is based on the variational principle and applies to the ground-state of a quantum system. It is, however, not restricted to the state of globally lowest 

This general result is well known as the Hellmann-Feynman theorem when X represents the position x of a nucleus. The force F that the system exerts on the nucleus is the expectation value of minus the gradient of V(x), where V is the potential that acts on the nucleus. This theorem was originally derived by Ehrenfest (1927), and was used in Hellmann s (1937) treatise to establish the forces in a molecule. Feynman (1939) independently derived the result for molecules. We will refer to the result simply as the force theorem . 

A different type of structural parameter was considered recently by the present authors, namely X representing a homogeneous macroscopic strain defined as the linear scaling of all particle positions as x- 1+e)x. The e is a constant 3x3 strain tensor, and e=0 corresponds to some reference configuration. The conjugate force is in this case defined as the macroscopic stress a, and an explicit general expression denoted the stress theorem is derived by Nielsen and Martin (1983). The result is a generalization of the quantum virial theorem (Born et al., 1926), 

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