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The self-consisted harmonic approximation (SCHA) allows the computation offree energy of anharmonic crystals considering both quantum and thermalfluctuations. Recently, a stochastic implementation of the SCHA has beendeveloped, tailored for applications that use total energy and forces computedfrom first principles. In this work, we extend the applicability of thestochastic SCHA to complex crystals with many degrees of freedom, with theoptimisation of both the lattice vectors and the atomic positions. To thisgoal, we provide an expression for the evaluation of the pressure and stresstensor within the stochastic SCHA formalism. Moreover, we develop a more robustfree energy minimisation algorithm, which allows us to perform the SCHAvariational minimisation very efficiently in systems having a broad spectrum ofphonon frequencies and many degrees of freedom. We test and illustrate the newapproach with an application to the phase XI of water ice usingdensity-functional theory. We find that the SCHA reproduces extremely well theexperimental thermal expansion of ice in the whole temperature range between 0K and 270 K, in contrast with the results obtained within the quasi-harmonicapproximation, that underestimates the effect by about 25 %.

Pressure and stress tensor of complex anharmonic crystals within the stochastic self-consistent harmonic approximation