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

Technical Summary: The Impact of Spurious Imaginary Phonon Modes on Thermal Properties of Metal-organic Frameworks

Problem Statement
Metal-organic frameworks (MOFs) are promising candidates for direct air capture (DAC) and gas separation, where thermal properties like heat capacity ($C_p$) are critical for determining the energy penalty of adsorbent regeneration via Temperature Swing Adsorption (TSA). Accurate computational prediction of these properties relies on phonon spectra derived from Density Functional Theory (DFT) or machine learning interatomic potentials (MLIPs). However, MOFs possess large unit cells with hundreds of atoms and flexible linkers, making high-fidelity phonon calculations computationally prohibitive. To manage costs, researchers often employ smaller supercells or looser convergence criteria, which frequently introduce spurious imaginary phonon modes (negative frequencies) that do not represent true dynamical instabilities but rather numerical artifacts.

Currently, there is no consensus on the acceptable magnitude or fraction of these imaginary modes. Standard practice involves simply omitting all modes with imaginary frequencies from thermal property calculations. The authors argue that this approach lacks systematic assessment and may introduce substantial, unquantified errors in heat capacity estimates, potentially leading to the mis-ranking of MOF candidates and the mis-evaluation of MLIP performance.

Methodology
The study systematically investigates the impact of spurious imaginary modes on heat capacity ($C_v$) using a combination of DFT data and MLIPs:

1. Case Study (MOF-74): The authors analyzed MOF-74 (Zn-based) using DFT data containing 1.03% spurious imaginary modes (primarily acoustic and low-frequency optical branches) compared to a reference dataset with 0% imaginary modes obtained via tight convergence and large supercells.
2. Error Quantification: They calculated the percentage deviation in $C_v$ between the "imaginary-mode" data and the "0% imaginary" baseline across temperatures from 0 to 1000 K.
3. MLIP Benchmarking: The study utilized the MACE-MP-MOF0 MLIP, which was trained to produce physically meaningful phonon spectra (0% imaginary modes) even with standard settings. This model served as a high-fidelity surrogate for the 0% imaginary DFT baseline to test a broader set of five diverse MOFs from the QMOF database.
4. Correction Workflow: A simple post-processing correction was proposed. Based on the observation that low-frequency modes (which are often the ones becoming spurious imaginary) reach their full classical contribution ($k_B$ per mode) well below 300 K, the authors added a correction term to the standard $C_v$ calculation:
   $$C_v(T)_{corrected} = C_v(T)_{imaginary} + k_B T \left( \frac{3N \times \% \text{ imaginary modes}}{100} \right)$$
   This assumes a full $k_B$ contribution from the missing modes at temperatures where $T \gg \theta_D$ (Debye temperature).
5. Comparative Analysis: The corrected DFT data was compared against MLIP predictions (MACE-MP-MOF0 and a model by Yue et al.) to demonstrate how uncorrected imaginary modes distort benchmarking results.

Key Results

• Magnitude of Error: Neglecting spurious imaginary modes leads to a systematic underestimation of heat capacity. For MOF-74, a mere 1.03% fraction of imaginary modes resulted in a $C_v$ underestimation of >10% at low temperatures, saturating at approximately 2× the percentage of imaginary modes (e.g., ~2% error for 1% modes) at 300 K and above.
• Temperature Dependence: The error is most pronounced at low temperatures (0–50 K) where acoustic modes dominate. However, even at 300 K (relevant for TSA), the error remains significant and non-negligible compared to the experimental variability of MOF heat capacities (typically 1–3%).
• Impact on Screening: The study demonstrates that errors from spurious modes can match or exceed the intrinsic differences in $C_v$ between distinct MOFs. This risks mis-ranking materials based on convergence criteria rather than intrinsic properties.
• Benchmarking Distortion: When benchmarking MLIPs against DFT data containing spurious imaginary modes, models that correctly predict stable spectra (0% imaginary) appear to overestimate heat capacity relative to the "flawed" DFT reference. For example, MACE-MP-MOF0 showed a 2.53% apparent overestimation against uncorrected DFT data, which dropped to 0.77% when compared against corrected or 0% imaginary DFT data.
• Effectiveness of Correction: The proposed post-processing correction reduced the error in $C_v$ by an order of magnitude across all tested MOFs, bringing results into near-quantitative agreement with high-fidelity 0% imaginary baselines.

Significance and Claims
The paper claims to provide a systematic demonstration that spurious imaginary phonon modes are not merely negligible artifacts but introduce substantial, temperature-dependent errors in thermal property screening.

• Correction Utility: The authors introduce a rapid, low-cost post-processing workflow that can be applied to standard phonon outputs (e.g., from Phonopy) to salvage heat capacity estimates from existing, underconverged datasets where full recomputation is prohibitive.
• Benchmarking Integrity: The work highlights that benchmarking MLIPs against DFT datasets containing spurious imaginary modes can artificially penalize models that predict physically meaningful spectra. The authors argue that future benchmarking must explicitly verify the physical validity of phonon spectra, not just property-level agreement.
• Limitations: The authors modestly note that their correction is valid only in the classical regime ($T \gg \theta_D$) and does not address other spectral inaccuracies (e.g., missing density of states in real frequency regions) or genuine dynamical instabilities (true phase transitions). They emphasize that this correction is specific to heat capacity and should not be applied to other properties like vibrational entropy or elastic constants, which remain sensitive to the full quality of the phonon spectrum.
• Future Path: The paper posits that the development of MLIPs capable of producing fully converged, physically meaningful phonon spectra is the most promising path forward for high-throughput MOF screening, potentially eliminating the need for such corrections in the future.