Pressure Correction in Density Functional Theory Calculations

First-principles calculations based on density functional theory have been widely used in studies of the structural, thermoelastic, rheological, and electronic properties of earth-forming materials. The exchange-correlation term, however, is implemented based on various approximations, and this is believed to be the main reason for discrepancies between experiments and theoretical predictions. In this work, by using periclase MgO as a prototype system we examine the discrepancies in pressure and Kohn-Sham energy that are due to the choice of the exchange-correlation functional. For instance, we choose local density approximation and generalized gradient approximation. We perform extensive first-principles calculations at various temperatures and volumes and find that the exchange-correlation-based discrepancies in Kohn-Sham energy and pressure should be independent of temperature. This implies that the physical quantities, such as the equation of states, heat capacity, and the Gr"{u}neisen parameter, estimated by a particular choice of exchange-correlation functional can easily be transformed into those estimated by another exchange-correlation functional. Our findings may be helpful in providing useful constraints on mineral properties %at thermodynamic conditions compatible to deep Earth. at deep Earth thermodynamic conditions.

💡 Research Summary

This paper investigates how the choice of exchange‑correlation (XC) functional in density‑functional theory (DFT) calculations influences the predicted pressure and Kohn‑Sham (KS) energy, using periclase MgO as a prototypical Earth‑forming mineral. The authors begin by separating the total internal energy of a crystal into three contributions: the classical kinetic energy of the ions, the ion‑ion Coulomb interaction, and the KS electronic energy, which depends on the XC functional. By assuming harmonic ionic vibrations and classical treatment of ionic motion, they show analytically that the difference in internal energy between two XC functionals (e.g., LDA and GGA) reduces to the difference in the zero‑temperature KS energies, ΔE_KS, which is a function of volume only and does not contain any temperature‑dependent term. Consequently, the pressure difference ΔP = P_GGA – P_LDA is also independent of temperature and depends solely on volume (Eq. 8).

To test these theoretical predictions, the authors performed extensive Car‑Parrinello molecular dynamics (CPMD) simulations on a 64‑atom MgO supercell (32 MgO units) over a wide range of temperatures (300 K to 4000 K) and volumes that correspond to pressures up to ~170 GPa, i.e., conditions relevant to the deep Earth. For each temperature‑volume point they carried out two separate simulations, one with LDA‑generated pseudopotentials and one with GGA‑generated pseudopotentials, keeping all other computational parameters identical (plane‑wave cutoff 30 Ry, charge‑density cutoff 240 Ry, fictitious electron mass µ_e = 400 m_e, time step ≈0.3 fs, simulation length >4 ps). The pressure data were fitted to a third‑order Birch‑Murnaghan equation of state (EOS) to obtain LDA and GGA EOS curves.

The results confirm the analytical expectations. The energy difference ΔE(V) = E_GGA – E_LDA is essentially flat with respect to temperature, showing only a systematic dependence on volume. The pressure difference ΔP(V) = P_GGA – P_LDA is likewise temperature‑independent; curves for all temperatures collapse onto a single line with a maximum deviation of about 1 GPa, which is smaller than typical statistical uncertainties in DFT‑based pressure estimates. ΔP(V) can be accurately represented by a series in 1/V (Eq. 9), indicating that the discrepancy grows as the volume shrinks but vanishes both at infinite volume (isolated atoms) and at vanishing volume (free‑electron‑gas limit).

Because ΔE is temperature‑independent, the authors argue that any thermodynamic quantity derived from the internal energy—such as the constant‑volume heat capacity C_V—will be identical for LDA and GGA (ΔC_V = 0). They extend this reasoning to the Grüneisen parameter γ = α K_T T V / C_V, showing that the difference Δγ also vanishes because the pressure derivative with respect to temperature, ∂ΔP/∂T, is zero. This conclusion is consistent with previous quasiharmonic approximation (QHA) studies, which have shown that high‑temperature thermodynamic properties of MgO and related silicates are insensitive to the XC functional as long as the QHA remains valid.

The paper distinguishes its “pressure correction” from the more common experimental‑based EOS adjustment (e.g., Wu et al.), which shifts an entire EOS to match measured data. Instead, the present correction quantifies the systematic bias introduced solely by the XC functional, allowing one to transform results obtained with one functional into those that would be obtained with another without re‑running expensive simulations. Combining both corrections could make first‑principles predictions from different codes or functional choices mutually comparable, a valuable asset for high‑pressure mineral physics.

Finally, the authors acknowledge limitations: the derivation assumes classical ionic motion and small vibrational amplitudes, so it may break down at very low temperatures where quantum zero‑point motion dominates, or for highly anharmonic phases. Nevertheless, their extensive MgO simulations, together with supporting calculations on MgSiO₃ perovskite and post‑perovskite, provide strong evidence that XC‑induced pressure and energy discrepancies are volume‑dependent but temperature‑independent, offering a practical pathway to reconcile disparate DFT results in Earth‑science applications.