A Rigorous and Direct Route to Gibbs Free Energies of Solids

By: Karel L. K. De Witte, Tom Braeckevelt, Massimo Bocus, Sander Vandenhaute, and Veronique Van Speybroeck

What is the issue?

Accurate predictions of Gibbs free energies are central to understanding the stability of crystals under realistic temperature and pressure conditions. Thermodynamic integration (TI) is the gold-standard computational approach for this task in solid-state systems. However, conventional TI frameworks rely on a harmonic reference in the NVT ensemble. Hereafter, three corrections are required to recover the true Gibbs free energy, including an approximate NVT-to-NPT conversion. Critically, this step neglects full cell flexibility, which can become important for materials with complex structural fluctuations, potentially limiting accuracy.

"By moving entirely to the NPT ensemble, we eliminate approximation steps and provide a more rigorous route to predicting the Gibbs free energy of solids." 

What solution did we come up with?

We developed a new, fully rigorous thermodynamic integration scheme that operates entirely in the NPT ensemble. The key innovation is a novel reference system that explicitly incorporates full cell fluctuations, eliminating the need for the approximate NVT-to-NPT correction. This reduces the conventional three-step workflow to a simpler and theoretically consistent two-step process.

Figure 1: Left: Schemes for computing Gibbs free energies via thermodynamic integration (TI). Purple arrows indicate the conventional scheme, in which the NVT harmonic reference is first corrected towards the true potential energy surface (PES). A second approximate correction is performed from isochoric to isobaric conditions and, finally, a third from low to high temperature. Green arrows indicate the new scheme, where the reference is a newly derived NPT harmonic PES. Subsequent corrections  account for anharmonicity and temperature effects, all performed under constant-pressure conditions. Right: Illustration of case study materials. The first case study covers three ice phases (Ice XI shown). The second case study comprises the black and yellow phase of CsPbI3 (black phase shown).

We validated the method using two complementary case studies:

• Ice polymorphs (simple cell-shape behavior): the strong dipole moment and bend geometry of water molecules cause them to organize into complex hydrogen-bonded networks. As a result, ice exhibits over 20 different crystalline phases.

• CsPbI₃ (complex cell-shape behavior, due to the black phase exhibiting 6 different minima): a metal halide perovskite (MHP) that is investigated for its use in solar cells, as its black phase exhibits favorable optoelectronic properties. Unfortunately, this phase is only stable at high temperature and quickly converts to the inactive yellow phase at ambient conditions. The development of physicochemical techniques to stabilize the black phase at low temperatures is a very active field of research.

Main findings

• For systems with simple structural behavior (e.g., ice), the new method reproduces conventional TI results with excellent agreement.

• For materials with complex cell dynamics (e.g., CsPbI₃), the new approach provides more accurate Gibbs free energy differences than the conventional one.

• The new workflow is more transparent and reproducible, and has equal computational cost (compared to the conventional one).

"Our approach matches conventional methods for simple materials while delivering higher accuracy for systems with complex structural behavior." 

Future potential

Our novel thermodynamic integration approach shows great potential for designing (nanoporous) materials for timely applications. In pharmaceuticals, many drugs exist in multiple crystalline forms (polymorphs), and the stability ranking of these forms can subtle to unravel. Our approach can reliably predict which polymorph is thermodynamically favored under specific storage or physiological conditions. In materials chemistry, our method is particularly well suited to studying the flexible behavior of metal–organic frameworks (MOFs). These porous materials exhibit functions that depend critically on their ability to morph, or “breathe,” between two states in response to external conditions. Because our NPT-based approach captures large, anisotropic structural fluctuations, it provides a direct way to compute free energy differences between these flexible states - something conventional methods struggle to do rigorously. This could accelerate the design of MOFs for applications like gas storage and separation, where pore-opening transitions are central to performance.

Key take-away

By moving entirely to the NPT ensemble, our novel Thermodynamic Integration framework establishes a rigorous, direct, and more robust path to Gibbs free energy predictions of solids under realistic thermodynamic conditions.

Role of VSC and HPC

"More than 10,000 GPU hours on VSC infrastructure made it possible to develop and validate this rigorous free energy framework."

Molecular dynamics simulations form the backbone of this work, requiring hundreds of thousands of time steps across multiple thermodynamic conditions and material systems. Even with the use of machine-learned interatomic potentials to accelerate simulations, the total computational demand exceeded 10,000 GPU hours. Access to the Vlaams Supercomputer Centrum (VSC) infrastructure was therefore essential, enabling large-scale simulations that made this rigorous free energy framework practically feasible.

Original Publication

Read the original research article, "A Novel NPT Thermodynamic Integration Scheme to Derive Rigorous Gibbs Free Energies for Crystalline Solids," published in ACS Publications, here.

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