The zero static internal stress approximation in lattice dynamics, and the calculation of isotope effects on molar volumes

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Abstract
For crystals in which relative positions of atoms within the unit cell are not wholly determined by symmetry, a complete application of the quasiharmonic approximation requires the minimization of the free energy with respect to both external (ηλ) and internal (εj) strains. The zero static internal stress approximation first minimizes the static lattice energy with respect to the εj for each state of external strain; the total free energy is then minimized only with respect to the ηλ. We show that although this gives an incorrect internal strain, to the first order it gives the correct external strain at each temperature; in principle, errors are thus of the same order as those due to the use of the quasiharmonic approximation. In particular, recent calculations by Lacks [D. J. Lacks, J. Chem. Phys. 103, 5085 (1995)] of the effect of deuteration on the molar volume of polyethylene are shown by the present analysis to include indirectly the effect of vibrational stretching of the C–H(D) bonds, and their reasonable agreement with room temperature measurements may after all not be fortuitous.