Capturing thermal effects beyond the zero-temperature approximation using the uniform electron gas

Abstract: Density functional theory at finite temperatures often relies on the zero-temperature approximation, which uses a ground-state exchange-correlation functional with thermalized densities. This approach, however, neglects the explicit temperature dependence of the exchange-correlation free energy -- a key factor in regimes such as warm dense matter, where both electronic and thermal effects are significant. In this work, we introduce the entropy-corrected zero-temperature approach, in which the exchange-correlation entropy is extracted using the generalized thermal adiabatic connection formula to construct a thermal correction to the standard zero-temperature approximation. Using a uniform electron gas parametrization, we compare this approach to the finite-temperature adiabatic connection and demonstrate that it performs best at lower densities. This provides a useful complement to zero-temperature density functional approximations, which generally perform better at moderate-to-large densities. We further identify a density-dependent intersection between the adiabatic connection curves, revealing a dependence on the ground state correlation energy and correlation potential. Additionally, extension of the entropy corrected approach applied as a local density approximation--like temperature correction to the zero temperature approximation is discussed.

• The paper develops an entropy-corrected zero-temperature approach that uses the generalized thermal adiabatic connection and UEG exchange-correlation entropy to restore explicit temperature dependence missing from standard ZTA calculations.
• The method agrees closely with the GDSM benchmark across 0.1 ≤ rₛ ≤ 20 and 0.5 ≤ Θ ≤ 8, achieving a 0.0027 Ha mean absolute error and roughly 1.4% worst-case relative error.
• The eZT and FTAC integrands intersect at a temperature-independent but density-dependent coupling strength, while the proposed correction remains limited by its UEG foundation and untested performance for inhomogeneous systems and experimental observables.

Motivation and context

Thermal density functional theory (DFT) provides the formal framework for modeling warm dense matter (WDM), where electron degeneracy $\Theta = k_BT/E_F$ and Coulomb coupling $\Gamma$ are both near unity. In practice, most finite-temperature DFT calculations employ the zero-temperature approximation (ZTA): a ground-state exchange-correlation (XC) functional evaluated on thermally weighted densities. The ZTA captures only the implicit temperature dependence carried by the density, discarding the explicit temperature dependence of the XC free energy $A_\mathrm{xc}^\tau$. This omission is known to produce significant errors at temperatures above roughly 10,000 K (2603.24544), a regime directly relevant to planetary interiors and inertial confinement fusion diagnostics.

The paper by Aguilar-Solis, Harding, and Pribram-Jones addresses this deficiency within the adiabatic connection formalism. Building on the generalized thermal adiabatic connection (GTAC), which permits simultaneous variation of interaction strength $\lambda$ and a fictitious temperature parameter $\tau'$, the authors construct an entropy-corrected zero-temperature (eZT) approach: the XC entropy is extracted via a Maxwell-style relation and used to build an explicit thermal correction to the ZTA, formulated for the uniform electron gas (UEG) using the Groth et al. (GDSM) parametrization of $a_\mathrm{xc}^\tau$.

Derivation of the eZT integrand

The GTAC expresses the finite-temperature XC free energy as

$$A_\mathrm{xc}^\tau[n] = E_\mathrm{xc}[n] + \int_0^1 d\lambda \int_0^\tau d\tau' \frac{\partial}{\partial \tau'}\frac{U_\mathrm{xc}^{\tau',\lambda}[n]}{\lambda},$$

where $E_\mathrm{xc}$ is evaluated on the Fermi-weighted density. Rather than working with the potential XC directly, the authors exploit the Maxwell relation $$(\partial U_\mathrm{xc}^{\tau,\lambda}/\partial \tau)_\lambda = -\lambda(\partial S_\mathrm{xc}^{\tau,\lambda}/\partial \lambda)_\tau$$ to route the correction through the XC entropy. Applying simulated scaling, $$a_\mathrm{xc}^{\tau,\lambda}(n)=\lambda^2a_\mathrm{xc}^{\tau/\lambda^2}(\lambda r_\mathrm{s})$$, to the GDSM parametrization yields the entropy per particle, from which the eZT adiabatic connection integrand follows:

$\Gamma$0

Two deliberate choices define the scheme: exact UEG exchange, and the Perdew–Wang (PW) ground-state correlation parametrization in place of the GDSM correlation. The result is therefore a purpose-built correction to the ZTA rather than a fully independent free-energy functional. Integration over $\Gamma$1 recovers $\Gamma$2; the derivation confirms analytically that this reproduces $\Gamma$3, i.e., the FTAC result, with the geometric interpretation that the area above the eZT curve corresponds to $\Gamma$4 times the correlation entropy.

The density-dependent intersection point

A central analytical result concerns the intersection of the eZT and FTAC adiabatic connection curves. Numerically, the curves cross at an interaction strength $\Gamma$5 that is temperature-independent but varies with the Wigner-Seitz radius $\Gamma$6, decreasing monotonically with increasing $\Gamma$7 (e.g., $\Gamma$8 at $\Gamma$9 Bohr across all degeneracies considered). Separating exchange and correlation via coordinate scaling and the high-density limit shows that the exchange free energies cancel identically at the crossing, leaving

$A_\mathrm{xc}^\tau$0

The intersection condition thus depends only on the ground-state correlation energy and correlation potential: at $A_\mathrm{xc}^\tau$1, the zero-temperature limit of the $A_\mathrm{xc}^\tau$2-scaled correlation-only FTAC integrand equals its average value, which coincides with the ground-state correlation energy. This is a clean structural statement about the relationship between the two adiabatic connections, and it holds independent of temperature — a non-obvious property given the explicit entropic content of the eZT integrand.

Performance against the GDSM benchmark

The eZT free energies were computed over $A_\mathrm{xc}^\tau$3 and $A_\mathrm{xc}^\tau$4 Bohr, matching the GDSM fitting domain. Representative values illustrate the agreement:

$A_\mathrm{xc}^\tau$5 (Bohr)	GDSM $A_\mathrm{xc}^\tau$6 ($A_\mathrm{xc}^\tau$7)	eZT $A_\mathrm{xc}^\tau$8 ($A_\mathrm{xc}^\tau$9)	Relative error
0.1	-2.58651	-2.59600	-0.0037
1	-0.39506	-0.39591	-0.0021
4	-0.13063	-0.13039	-0.0018
20	-0.03355	-0.03346	-0.0028

Decomposition of the error shows that all deviation resides in the correlation component; the eZT exchange free energy matches GDSM exactly across all densities, as required by construction. The correlation error tracks the difference between PW and GDSM ground-state correlations: the largest discrepancy, 0.01 Ha, occurs at $\lambda$0 Bohr, outside the PW fit range of $\lambda$1. Across the full sampled domain the mean absolute error in $\lambda$2 is 0.0027 Ha (~1.69 kcal/mol), with worst-case relative errors of about 1.4% confined to the edges of the parameter space ($\lambda$3, $\lambda$4).

The accuracy trend is complementary to standard practice: eZT performs best at lower densities (large $\lambda$5), whereas beyond-LDA zero-temperature functionals are typically more reliable at moderate-to-large densities. This complementarity is the practical motivation for deploying eZT as a post-hoc, LDA-like temperature correction on top of existing ZTA calculations, preserving the accuracy of higher rungs of Jacob's ladder for the implicit temperature dependence while supplying the missing explicit entropy term.

Limitations and open questions

Several caveats bound the results. First, the entire analysis is restricted to the UEG; extension to inhomogeneous systems requires the approximate eZT-LDA functional sketched in the conclusion,

$\lambda$6

with $\lambda$7, which the authors note will be "imperfect" when transferred from the model system. Second, the observed low-density errors are inherited entirely from the PW ground-state correlation parametrization; improved parametrizations could reduce them, but this remains untested here. Third, no comparison against established thermal functionals such as corrKSDT or against experimental observables such as x-ray Thomson scattering spectra has yet been performed, so the relative merit of the eZT correction in realistic WDM simulations is undetermined. Finally, the choice among zero-temperature DFAs to pair with the eZT-LDA correction is left as an open empirical question.

Conclusion

This work derives and validates an entropy-based thermal correction to the zero-temperature approximation for the UEG, extracting the XC entropy through the generalized thermal adiabatic connection and demonstrating sub-percent mean agreement with quantum Monte Carlo-based parametrizations across the warm dense regime. The identification of a temperature-independent, density-dependent intersection between the eZT and FTAC integrands, governed solely by the ground-state correlation energy and potential, provides new structural insight into the relationship between thermal and ground-state adiabatic connections. The framework's principal value lies in its compatibility with existing zero-temperature functionals, positioning it as a modular route to restoring explicit temperature dependence in WDM-scale simulations pending validation on inhomogeneous systems.