A problem with the stress theorem commonly used in DFT codes

The change in energy when an affine transformation (strain) is applied to a lattice of ions can be obtained by evaluating the algebraic derivative of the DFT energy (in practice a local or other approximation) of an electron density that has been similarly strained [1]. Because the DFT energy is stationary in the density, it is only required that the strained density reduces to the exact density at zero strain; it does. The algebraic derivatives of the Hartree and exchange energies are straightforward. The derivative with respect to strain of the non-interacting kinetic energy depends on two assumptions: 1) the modulus squared of the strained orbitals equals the strained electron density, and 2) the strained orbitals minimize the non-interacting kinetic energy. The first assumption is correct. I find that the second assumption applies only in special cases. The limitations and possible modifications of the stress theorem are discussed.\\[0pt] [1] Nielsen, O. H. \& Martin, R. M. 1983 First-Principles Calculation of Stress. Physical Review Letters 50, 697.\\[0pt] [2] D. M. Nicholson, Madhusudan Ojha, and T. Egami, Journal of Physics Condensed Matter 10/2013; 25(43):435505.