Internal strain and force constants¶
The internal-strain tensor is the force on each atom per unit strain at fixed fractional coordinates,
for atom \(n\), Cartesian direction \(\alpha\) and Voigt strain component \(J\) (engineering shear), in eV/Å per unit strain. The net force vanishes, so \(\sum_n \Lambda_n = 0\). minelastic obtains it in two ways.
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From the strained cells, without further calculations. The force responses of each pattern, \(f_k = \delta^{-1} \sum_m g_m F(m\delta u_k)\), use the same finite-difference formula as the stresses. If a space-group operation \(\{R|t\}\) maps atom \(n\) onto atom \(g(n)\), the forces obey
\[ F_{g(n)}(R \varepsilon R^T) = R\, F_n(\varepsilon) . \]The computed patterns therefore give the response of every atom to the whole orbit of strains \(\{R^T \varepsilon_k R\}\). This orbit spans all six strain components whenever the patterns determine \(C\), so the minimal set gives the complete tensor; \(\Lambda_n\) follows by least squares over the orbit.
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From displaced atoms (
ionic=True). The mixed second derivative of the energy taken in the other order gives\[ \Lambda_{n\beta, J} = -\Omega\, \frac{\partial \sigma_J}{\partial u_{n\beta}} , \]from the stresses of the cells in which atom \(n\) is displaced along \(\beta\) (the route of VASP's
IBRION = 6).
The displaced atoms also give the \(\Gamma\)-point force constants
in eV/Å\(^2\). Only the symmetry-irreducible displacements are computed: for the first atom of each orbit, the Cartesian directions x, y, z are kept as long as they add a new direction to the images under its site group, as VASP chooses them. The other atoms and directions follow from \(F_{g(m)} = R F_m\), and \(\Phi\) is symmetrized as a \(3N \times 3N\) matrix. The relaxation of the atoms per unit strain is
in Å, with \(\Phi^+\) the inverse of \(\Phi\) on the complement of the uniform translations. It enters the relaxed-ion constants (Relaxed ions) and the piezoelectric tensor (Born charges and piezoelectricity).