First-principles thermodynamics of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msub> <mml:mi>Al</mml:mi> <mml:mn>10</mml:mn> </mml:msub> <mml:mi mathvariant="normal">V</mml:mi> </mml:mrow> </mml:math> : An analytical tre

Many complex intermetallic structures possess cagelike environments that can host additional guest atoms. In ${\mathrm{Al}}_{10}\mathrm{V}$, these atoms give rise to low-frequency, localized vibrations (Einstein modes) that dominate the thermodynamic response at low temperature. They become imaginary under volume expansion as temperature rises, invalidating the harmonic approximation. We develop a framework to incorporate these strongly anharmonic vibrational modes into first-principles thermodynamic calculations. By explicitly modeling the cage potential and solving the associated Schr"odinger equation numerically, we compute the full anharmonic free energy contribution and demonstrate its impact on the thermodynamic behavior of ${\mathrm{Al}}_{10}\mathrm{V}$. This allows us to examine structures with different cage fillings and construct the Al-V phase diagram in the relevant composition range. Our results reproduce key experimental signatures, including the anomalous rise in the thermal expansion coefficient and heat capacity at low temperatures, and reveal that the presence and the extent of cage filling by guest atoms is essential to stabilizing the ${\mathrm{Al}}_{10}\mathrm{V}$ phase at elevated temperatures.