Table 2 Summary of computational components and available approaches for DFT modeling of HEMs

Exchange-Correlation Functional

Local density approximation (LDA), Generalized gradient approximation (GGA-PBE, GGA-PW91), meta-GGA (SCAN, r²SCAN), Hybrid functionals (HSE06, B3LYP)

GGA and GGA + U are widely used for magnetic high-entropy materials^{72,212,213,214}, but LDA struggles with strong electronic correlations in d and f orbital systems, leading to incorrect magnetic moments and crystal structures²¹⁵. GGA improves accuracy over LDA but struggles to fully capture strong electronic correlations. The accuracy of GGA + U depends heavily on careful tuning of the U parameter²¹⁶. Hybrid functionals (e.g., HSE06) improve accuracy through Hartree-Fock exchange but are computationally intensive for large systems²¹⁷. Despite DFT convergence challenges, r2SCAN meta-GGA excels for structural, mechanical, and thermophysical properties in high-throughput alloy studies²¹⁶.

Hubbard U Correction

Standard DFT without +U, DFT + U (Dudarev, Liechtenstein), Self-consistent U calculations

The Hubbard U correction is applied in HEM modeling to address strong electronic correlations in d and f orbitals, which standard GGA fails to capture accurately, leading to errors in magnetic and electronic properties⁴⁷. Dudarev’s approach²¹⁸, add an on-site Coulomb term, improving predictions but requiring careful tuning of the U parameter. Liechtenstein’s method separately treats Coulomb and exchange interactions for enhanced accuracy²¹⁹. Self-consistent U calculations dynamically determine U, reducing empirical tuning.

Pseudopotential

Norm-conserving pseudopotentials, Ultrasoft pseudopotentials (USPP), Projector augmented wave (PAW)

Norm-conserving pseudopotentials ensure accurate wavefunctions but are computationally demanding. USPPs reduce computational cost by softening the potential, though they may compromise some accuracy. PAW combines high accuracy with efficiency, reconstructing all-electron wavefunctions near nuclei, making it ideal for complex HEMs with d and f orbitals²²⁰.

Dispersion Correction

No dispersion, DFT-D2, DFT-D3, Tkatchenko-Scheffler (TS)

No dispersion correction may lead to inaccurate structural and energetic predictions. DFT-D2²²¹ provides a simple pairwise correction but can overestimate dispersion in some systems. DFT-D3²²² improves accuracy with environment-dependent terms, better handling complex HEMs. The Tkatchenko-Scheffler (TS) method offers further refinement by incorporating self-consistent polarizability, enhancing precision for diverse compositions²²³.