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Relaxed ions

A strain moves the atoms that are not fixed by symmetry. To second order in the atomic displacements \(u\) (3N components) and the strain \(\varepsilon\), the energy of a cell of volume \(\Omega\) is

\[ E(u, \varepsilon) = E_0 + \frac{\Omega}{2}\, \varepsilon \cdot C^0 \varepsilon - u \cdot \Lambda \varepsilon + \frac{1}{2}\, u \cdot \Phi u , \]

with the clamped-ion constants \(C^0\), the internal-strain tensor \(\Lambda = \partial F/\partial \varepsilon\) and the \(\Gamma\)-point force constants \(\Phi = -\partial F/\partial u\) (Internal strain). The forces \(F = -\partial E/\partial u = \Lambda\varepsilon - \Phi u\) vanish for

\[ u = \Phi^+ \Lambda\, \varepsilon , \]

where \(\Phi^+\) is the inverse of \(\Phi\) on the complement of the three uniform translations. The stress \(\sigma = \Omega^{-1}\, \partial E/\partial \varepsilon\) at these positions gives the relaxed-ion constants

\[ C = C^0 - \frac{1}{\Omega}\, \Lambda^T \Phi^+ \Lambda . \]

With \(\Lambda\) in eV/Å, \(\Phi\) in eV/Å\(^2\) and \(\Omega\) in Å\(^3\), the second term is in eV/Å\(^3\) (1 eV/Å\(^3\) = 160.2 GPa). minelastic computes \(C\) in one of two ways.

  1. Relaxing the ions (relax_ions=True, --relax-ions). The atoms of every strained cell are relaxed at fixed cell, and the stresses are differentiated as for \(C^0\). The relaxation at finite strain also contains terms beyond linear order; the four-point formula removes them to order \(\delta^4\).
  2. Clamped single points (ionic=True, --ionic), the route of VASP's IBRION = 6. The strained cells give \(C^0\). The symmetry-irreducible atoms, displaced by \(\pm h\) and \(\pm 2h\) (default \(h\) = 0.01 Å), give \(\Phi\) from their forces and \(\Lambda\) from their stresses. \(C\) follows from the formula above, as the exact linear response. If \(\Phi\) has an unstable or soft optical mode, the relaxed-ion response is undefined and minelastic stops with an error.

The second route needs no relaxations. Relaxations are costly, and they must be converged tightly: the forces that a strain of 0.01 induces are only about 0.05 eV/Å.