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We present a novel approach to efficiently implement thermal expansion in thequasi-harmonic approximation (QHA) for both isotropic and more importantly,anisotropic expansion. In this approach, we rapidly determine a crystal'sequilibrium volume and shape at a given temperature by integrating along thegradient of expansion from zero Kelvin up to the desired temperature. Wecompare our approach to previous isotropic methods that rely on a brute-forcegrid search to determine the free energy minimum, which is infeasible to carryout for anisotropic expansion, as well as quasi-anisotropic approaches thattake into account the contributions to anisotropic expansion from the latticeenergy. We compare these methods for experimentally known polymorphs ofpiracetam and resorcinol and show that both isotropic methods agree to withinerror up to 300 K. Using the Gr\"{u}neisen parameter causes up to 0.04 kcal/moldeviation in the Gibbs free energy, but for polymorph free energy differencesthere is a cancellation in error with all isotropic methods within 0.025kcal/mol at 300 K.  Anisotropic expansion allows the crystals to relax into lattice geometries0.01-0.23 kcal/mol lower in energy at 300 K relative to isotropic expansion.For polymorph free energy differences all QHA methods produced results within0.02 kcal/mol of each other for resorcinol and 0.12 kcal/mol for piracetam, thetwo molecules tested here, demonstrating a cancellation of error for isotropicmethods.  We also find that when expanding in more than a single volume variable, thereis a non-negligible rate of failure of the basic approximations of QHA.Specifically, while expanding into new harmonic modes as the box vectors areincreased, the system often falls into alternate, structurally distinctharmonic modes unrelated by continuous deformation from the original harmonicmode.

A Thermal Gradient Approach for the Quasi-Harmonic Approximation and its Application to Improved Treatment of Anisotropic Expansion