The impact of spurious imaginary phonon modes on thermal properties of Metal-organic Frameworks

Metal-organic Frameworks (MOFs) have emerged as potential candidates for direct air capture (DAC) of green house gases and water. Thermal properties of MOFs, such as their heat capacity, are used to determine the energy penalty associated with the adsorbent retrieval during the Temperature Swing Adsorption process. To aid exploration of the vast experimental design space of MOFs for such applications, computational methods like Density Functional Theory (DFT) or surrogate machine learning models trained on DFT data have been developed for obtaining phonon-derived heat capacities of MOFs. However, the high cost of explicit phonon computation in large and flexible nanoporous MOFs often necessitates the use of small supercells or lower convergence criteria which decrease predictive accuracy. These approximations often result in spurious imaginary phonon modes which are commonly ignored in practice. At present, there is no clear consensus in the literature on what magnitude of negative frequency or what fraction of imaginary modes can be considered acceptable. Here, we systematically demonstrate that spurious imaginary phonon modes can introduce substantial errors in heat capacity estimates, leading to incorrect ranking of MOFs in thermal-property-based screening. We further show that benchmarking machine learning interatomic potentials (MLIPs) against DFT datasets containing spurious imaginary modes can misrepresent models that predict physically meaningful phonon spectra for dynamically stable MOFs. Finally, we introduce a simple, rapid post-processing workflow that can be applied to standard phonon calculations to effectively correct heat capacity estimates and account for spurious imaginary modes in MOFs.

💡 Research Summary

The paper investigates how spurious imaginary phonon modes—numerical artifacts that appear in density‑functional‑theory (DFT) phonon calculations of metal‑organic frameworks (MOFs)—affect the prediction of heat capacity, a key thermal property used to evaluate the energy penalty in temperature‑swing adsorption (TSA) processes for direct air capture and water harvesting. Because MOFs have large unit cells (often > 50 atoms) and many low‑frequency, soft vibrational modes, high‑accuracy phonon calculations are computationally expensive. Researchers therefore frequently resort to small supercells, looser convergence criteria, or broken translational symmetry, which inevitably generate a fraction of imaginary frequencies. The literature reports anywhere from < 2 % to ~6 % imaginary modes, but no consensus exists on an acceptable magnitude or fraction, and practice typically discards all imaginary modes regardless of their absolute values.

Using MOF‑74 (Zn) as a case study, the authors show that even a modest 1.03 % of spurious imaginary modes can cause a >10 % underestimation of the constant‑volume heat capacity (C_v) at low temperatures (below the Debye temperature θ_D ≈ 43 K) and produce errors that are roughly twice the imaginary‑mode fraction at room temperature (≈300 K). The error originates from the fact that the missing modes are primarily the three acoustic branches and the lowest optical branch, which dominate the heat capacity in the low‑frequency regime. Above ≈2 θ_D, optical modes dominate and the relative error diminishes but remains on the order of 2 % because the three acoustic contributions are still omitted.

Extending the analysis to five additional MOFs from the open‑access DFT phonon datasets of Yue et al. and Moosavi et al., the authors demonstrate a systematic relationship: the percentage error in C_v at 300 K grows faster than the percentage of imaginary modes, reaching –12.2 % for a 5.75 % imaginary‑mode fraction. They propose a practical threshold: keeping spurious imaginary modes below 0.5 % limits C_v errors to < 1 %, which is sufficient to discriminate between MOFs that differ by only 1–2 % in heat capacity.

To avoid the prohibitive cost of fully converged DFT calculations, the authors introduce a simple post‑processing correction. The corrected heat capacity is obtained by adding a term proportional to k_B T · (3N · %imag/100), where N is the number of atoms in the primitive cell. This term assumes that each omitted mode would contribute roughly k_B at temperatures above 2 θ_D. Applying the correction to the 1.03 % imaginary‑mode data reduces the C_v error at 300 K from 1.7 % to 0.23 %, achieving accuracy comparable to fully converged DFT results while requiring only seconds of additional computation.

The paper also examines the impact of spurious imaginary modes on benchmarking machine‑learning interatomic potentials (MLIPs). Using the MACE‑MP‑MOF0 MLIP, which reproduces DFT phonon spectra with high fidelity, the authors compare its heat‑capacity predictions against DFT references that contain varying fractions of imaginary modes. When benchmarked against the Moosavi dataset (≈2 % imaginary modes), MACE‑MP‑MOF0 appears to overestimate C_v by ~2.5 %, consistent with the reported mean‑percentage absolute error (MPAE). However, when the reference is the fully converged, 0 % imaginary‑mode DFT data from Wieser et al., or the corrected DFT data, the apparent overestimation drops to < 1 %. This demonstrates that MLIP performance can be mis‑characterized if the reference DFT data retain spurious imaginary modes.

Finally, the authors discuss practical guidelines for large‑scale MOF screening. They recommend either (i) tightening DFT convergence criteria to keep the imaginary‑mode fraction below 0.5 % or (ii) applying the proposed post‑processing correction to existing phonon outputs (e.g., from Phonopy). Both strategies enable rapid, reliable heat‑capacity estimation without the need for expensive, fully converged phonon calculations, thereby facilitating high‑throughput thermodynamic screening, accurate energy‑penalty assessment for TSA processes, and fair benchmarking of MLIPs.

In summary, the study quantifies the detrimental effect of spurious imaginary phonon modes on MOF heat‑capacity predictions, establishes a quantitative error threshold, provides a lightweight correction scheme, and highlights the importance of clean reference data for MLIP validation. These contributions advance the computational toolbox for MOF design and accelerate the deployment of MOFs in energy‑critical applications.