Thermal density functional theory is commonly used in simulations of warm dense matter, a highly energetic phase characterized by substantial thermal effects and by correlated electrons demanding quantum mechanical treatment. Methods that account for temperature dependence, such as Mermin–Kohn–Sham finite-temperature density functional theory and free energy density functional theory, are now employed with more regularity and available in many standard code packages. However, approximations from zero-temperature density functional theory are still often used in temperature-dependent simulations using thermally weighted electronic densities as an input to exchange–correlation functional approximations, a practice known to miss temperature-dependent effects in the exchange–correlation free energy of these systems. In this work, the temperature-dependent adiabatic connection is demonstrated and analyzed using a well-known parameterization of the uniform electron gas free energy. Useful tools based on this formalism for analyzing and constraining approximations of the exchange–correlation at zero temperature are leveraged for the finite-temperature case. Inspired by the Lieb–Oxford inequality, which provides a lower bound for the ground-state exchange–correlation energy, bounds for the exchange–correlation at finite temperatures are approximated for various degrees of electronic correlation.

Recent decades have seen a rapid growth of interest in the study of matter under conditions of extreme excitation and/or compression. This interest includes growing focus on a warm dense matter (WDM), a highly energetic phase characterized by the simultaneous existence of strongly correlated electrons, thermal effects, and quantum effects of electrons.1–5 Astrophysical examples of WDM include white dwarf atmospheres and planetary cores,6,7 while more down-to-earth examples include laser-excited solids and inertial confinement fusion capsules.8–10 There have been corresponding advances in the use of density functional theory11,12 (DFT) calculations of WDM. Aside from describing planetary interiors,13–15 these calculations are useful for predicting material properties,4,5,16 developing experimental standards,3,17 and supplementing experiments that push the boundaries of accessible conditions.18–21 

In many thermal DFT calculations, a small-but-crucial free energy component, called the exchange–correlation (XC), is approximated by a ground-state approximation. However, in principle, the XC free energy depends explicitly on temperature.22–25 If the exact temperature-dependent XC free energy were known, the free energy and density for a given distribution of the nuclei and any properties that could be extracted from these, such as Hugoniot shock curves and equations of state of materials, could be predicted exactly.26 Although using a ground-state approximation to the XC free energy has not been shown to be a fatal flaw, its shortcomings continue to be investigated via comparison with existing temperature-dependent XC approximations and through analysis of model systems.4,5,21,27–37

Approximations to the XC functional can be informed by the adiabatic connection formalism (ACF), which is an important analytical and interpretive tool in DFT.38–40 The ACF expresses EXC[n] exactly as an integral over the coupling constant, λ, which smoothly connects the fictitious non-interacting Kohn–Sham reference system (λ = 0) with the real physical interacting system (λ = 1) while holding the density fixed. A similar theoretical construction with an explicit temperature dependence for an ensemble at finite temperature exists,25 but it has only been numerically demonstrated as it has for zero-temperature systems using model systems, such as the Hubbard dimer.30,41

A useful tool in the quest for better functional approximations is the list of known exact properties and constraints on the formally exact XC functional.42,43 The Lieb–Oxford (LO) bound44–51 is an exact inequality that provides a strict bound on the XC energy, and it has been used to constrain some of the most successful XC approximations. Perdew derived looser lower bounds on the XC and exchange energies using the LO lower bound on the indirect Coulomb energy,52 which was subsequently used in the development of the Perdew–Burke–Ernzerhof approximation.53 While GGAs must violate this constraint for realistic energetic predictions,54 a meta-GGA that satisfies important constraints, including the LO bound, was constructed by Sun and collaborators.55 The few temperature-dependent XC approximations developed in the literature56 are constructed to satisfy the LO bound, but the effect of temperature on this formal relationship has not been explored in a detailed manner.

In this work, finite-temperature adiabatic connection (FTAC) curves are generated using a parameterization57 of the XC free energy of the uniform electron gas (UEG) for various temperatures and densities in the WDM regime. In order to validate the finite-temperature simulated scaling approach used in this work, calculated XC free energy values are compared to XC free energy values generated from the Fortran 90 program provided by Groth and collaborators,57 upon which the simulated scaling results also rely. In addition, the behavior of simulated curves is evaluated in light of known mathematical conditions on the exchange free energy. The calculated adiabatic connection curves are used to explore how the balance between exchange and correlation changes in different regimes. Subsequently, adherence to the LO bound is analyzed, and an approach to estimating temperature-dependence is evaluated. Approximate bounds are generated numerically using these results and compared to those known at zero-temperature (ZT).

Hohenberg and Kohn have shown11 that there exists a functional of the density such that

Edrv(r)n(r)+F[n(r)]
(1)

is equal to the ground-state energy when n(r) is the ground-state density.11 DFT relies on the fact that the ground-state energy is expressed as a functional of the ground-state density. The core idea of Kohn–Sham DFT,12 one of the most successful approaches to the many-body problem, is to approach the interacting problem of interest by mapping it to a more tractable non-interacting problem. This mapping would be exact if the correct exchange–correlation (XC) functional for arbitrary physical systems was available. However, this is not the case in the vast majority of systems, and so one must use approximations for practical calculations.

The ZT adiabatic connection gives an exact expression for the exchange–correlation functional,

EXC[n]=01dλWXCλ[n],
(2)

expressed as an integral over a coupling constant λ, which, while holding the density n(r) fixed, continuously connects the KS reference system (λ = 0) with the real physical interacting system (λ = 1).38–40 The integrand, WXCλ[n], is generated by introducing λ into the universal functional, Fλ[n]=minΨnΨ|T̂+λV̂ee|Ψ, where Ψ is the minimizing wave function for a given λ. For all values of λ, the density remains that of the real physical system (the superscript λ is often omitted when λ = 1). An illustration of the adiabatic connection, which provides a geometrical representation of the components of EXC, is shown in Fig. 1.

FIG. 1.

A diagram of the adiabatic connection at zero temperature. The area in red, EX, represents the exchange energy component of EXC, while the gray area, EC, depicts the correlation energy component. The blue shaded region is related to the kinetic correlation component of EXC.

FIG. 1.

A diagram of the adiabatic connection at zero temperature. The area in red, EX, represents the exchange energy component of EXC, while the gray area, EC, depicts the correlation energy component. The blue shaded region is related to the kinetic correlation component of EXC.

Close modal

Moving to non-zero temperatures requires shifting our focus from ground-state energy functionals to free energy functionals. Mermin22 generalized the Hohenberg–Kohn (HK) theorems to equilibrium systems at finite temperatures. At non-zero temperatures, many-electron systems can be described by the grand-canonical ensemble. The grand potential25 can be written as

Ω̂=ĤτŜμN̂,
(3)

where Ĥ represents the electronic Hamiltonian operator, τ is the temperature, Ŝ is the entropy operator, μ is the chemical potential, and N̂ is the particle-number operator. The entropy operator is defined via

Ŝ=kBlnΓ̂,
(4)

where Γ̂ is the statistical operator,

Γ̂=N,iwN,i|ΨN,iΨN,i|.
(5)

Orthonormal N-particle states are denoted by |ΨN,i⟩, and the normalized statistical weights, wN,i, satisfy N,iwN,i = 1. One obtains a set of thermal KS equations through a mapping similar to the ZT one, but this time, keeping both the equilibrium density and temperature fixed. In the ZT case, we focused on the ground-state density, but here, the fundamental object, generated by the equilibrium statistical operator, is the thermally weighted, equilibrium density. This mapping and energy decomposition defines the XC free energy, which now includes an entropic term. The Mermin–Kohn–Sham (MKS) equations12,22 are similar to the ground-state KS equations but are complicated by temperature-dependent eigenvalues, eigenstates, and effective potential. This yields the MKS density, which is equal to the exact equilibrium density by definition,

nτ(r)=ifiτ|ϕiτ(r)|2,
(6)

where ϕiτ(r) is the ith eigenstate and fiτ is the state’s corresponding Fermi occupation. The free energy of the physical system is written in the usual way,26 

A=TτS+Vee+Vext,
(7)

where T is the kinetic energy, S is the entropy, Vee is the electron–electron repulsion, and Vext is the external nuclear potential.58 To emphasize the tied coordinate-temperature scaling relationships in thermal DFT, we can write the kentropy25 as K = TτS. The same equation for the free energy of the systems can be written in terms of the corresponding KS quantities,

A=TsτSs+U+AXCτ+Vext,
(8)

where the subscript s denotes the KS quantities, U is the Hartree energy, and Axc is the XC free energy. Mirroring the interacting case, the non-interacting kentropy is written as Ks = TsτSs.

While the operators of quantum mechanics scale in a simple way, the conversion to density functionals invites some complexity. One commonly exploited relationship in modern DFT is that between coordinate scaling and the coupling constant.59 When the length scale of our system is changed by a factor γ, we can maintain the normalization of the system via the following definition of the scaled ZT density,

nγ(r)=γ3n(γr).
(9)

Using this scaled density as the input, straightforward scaling of energy quantities results, unless the quantities include correlation.

In the adiabatic connection, a coupling constant, λ, is introduced into the universal functional,42,59

Fλ[n]=minΨnΨ|T̂+λV̂ee|Ψ,
(10)

where, again, λ = 0 yields the fictitious KS system and λ = 1 gives the real interacting system of interest. For all values of λ, the density is that of the physical system. The universal functional can be combined with coordinate and density scaling relationships, yielding the relation

Fλ[n]=λ2F[n1/λ].
(11)

Here, the fully interacting functional evaluated on a scaled density is multiplied by the square of the inverse of the density scaling factor, giving access to the universal functional at a scaled interaction strength. Explicitly known conditions on other energy components can be written under concurrent coordinate and interaction strength scaling, as we will see later.39,42

The same concepts applied in ZT simulated interaction strength scaling can be extended to the finite-temperature adiabatic connection (FTAC),25,60

AXCτ[n]=01dλWXCτ,λ[n],
(12)

where, here, the adiabatic connection integrand is both temperature- and interaction strength-dependent as follows: WXCτ,λ[n]=UXCτ,λ[n]/λ. The extension of Eq. (11) to finite temperature is written as Fτ,λ[n]=λ2Fτ/λ2[n1/λ]. Throughout this work, we follow the standard practice: the temperature at which a functional is evaluated appears as a superscript, subscripts refer to coordinate scaling, and superscripts indicate interaction strength scaling. Lack of a λ superscript implies that λ = 1. An illustration of the adiabatic connection generalized to finite temperatures is shown in Fig. 2. The red arrow indicates the monotonic behavior of the exchange free energy component, AX[n], which increases with an increase in the temperature. On the other hand, the correlation free energy component, AC, shows a more complex behavior and varies non-monotonically as the temperature increases.

FIG. 2.

A diagram of the finite-temperature adiabatic connection. The area in red depicts the exchange free energy component of AXC, the gray area represents the correlation free energy, and KC is the kentropic correlation [the finite-temperature (FT) analog of the kinetic correlation]. While the behavior of AX for a fixed rs with an increase in the temperature is predictable, the temperature-dependent behavior of AC is more complex.

FIG. 2.

A diagram of the finite-temperature adiabatic connection. The area in red depicts the exchange free energy component of AXC, the gray area represents the correlation free energy, and KC is the kentropic correlation [the finite-temperature (FT) analog of the kinetic correlation]. While the behavior of AX for a fixed rs with an increase in the temperature is predictable, the temperature-dependent behavior of AC is more complex.

Close modal

When discussing the uniform gas, constraints on the density functions describing the system are similar to those on density functionals42 and can be used to extract useful information about properties such as limiting behavior. For instance, coordinate scaling can be expressed in terms of interaction strength scaling in the expression for the XC free energy per particle,

aXCτ,λ(n)=λ2aXCτ/λ2n1/λ.
(13)

The exchange free energy per particle, now expressed in terms of the Wigner–Seitz radius, rs, can be extracted by scaling to the high-density limit of the FT uniform gas,53,61,62

aXτ(rs)=limγaXCγ2τrsγγ.
(14)

The expression for the correlation component of the free energy per particle is then obtained using the simple relationship between exchange and correlation,

aCτ(rs)=aXCτ(rs)aXτ(rs).
(15)

These results of tied coordinate–temperature–interaction strength scaling can be combined to extract any component of the correlation free energy from any other piece.60 We extract the potential contribution from the full correlation free energy per particle via

uCτrsγ=γdaCγ2τrsγdγ+2aCγ2τrsγ
(16)

and combine it with the definition of the FT adiabatic connection integrand for the uniform gas,

WCτ,λ(rs)=uCτ,λ(1/rs)λ=λ2uCτ/λ2(λrs)λ=λuCτ/λ2(λrs).
(17)

This yields the finite-temperature adiabatic connection integrand in terms of aCτ(rs) evaluated at carefully scaled densities and temperatures. This process is usually used for the generation of ACF curves at zero-temperature and is called simulated scaling.63,64 The final expression for WXCτ,λ is the sum of WCτ,λ and the λ-independent, FT exchange component from Eq. (14),

WXCτ,λ(rs)=λuCτ/λ2(λrs)+limγaxcγ2τ(rs/γ)γ.
(18)

This ACF allows us to calculate the temperature-dependent correlation free energy per particle at any coupling constant strength and temperature. To do so, we use the parameterization in Ref. 57 for the FTUEG and construct FTAC curves under varying density and temperature conditions using Mathematica version 11.2.65Table I compares values of the XC free energy per particle obtained from the Fortran 90 code provided by Groth and collaborators to values obtained using the FTAC formalism. The agreement between the values obtained from the formalism in this work and the Fortran code, and its controllable accuracy via increased interaction strength resolution, verifies the validity of the implemented simulated scaling technique.

TABLE I.

Comparison of the XC free energy per particle values generated using the Fortran 90 program57 provided by Groth and collaborators and the values obtained via integration of simulated scaling curves (FTAC) at three different temperatures and values of the Wigner–Seitz radius. Improved agreement with the Fortran values (reported in Hartree) can be obtained by increasing the resolution in interaction strength, λ, in the numerical implementation of simulated scaling.

rs = 1rs = 2rs = 4
Groth et al.FTACGroth et al.FTACGroth et al.FTAC
τ = 0.1 −0.516 68 −0.516 67 −0.271 94 −0.271 92 −0.133 75 −0.133 74 
τ = 0.5 −0.500 31 −0.500 27 −0.223 35 −0.223 35 −0.087 79 −0.087 79 
τ = 1.0 −0.459 19 −0.459 22 −0.181 67 −0.181 67 −0.067 31 −0.067 31 
rs = 1rs = 2rs = 4
Groth et al.FTACGroth et al.FTACGroth et al.FTAC
τ = 0.1 −0.516 68 −0.516 67 −0.271 94 −0.271 92 −0.133 75 −0.133 74 
τ = 0.5 −0.500 31 −0.500 27 −0.223 35 −0.223 35 −0.087 79 −0.087 79 
τ = 1.0 −0.459 19 −0.459 22 −0.181 67 −0.181 67 −0.067 31 −0.067 31 

Another verification that our application of simulated scaling is sound relies on a known behavior of the exchange free energy for the uniform gas. The XC free energy in this work is expressed as a function of the electron degeneracy parameter or reduced temperature,

θ=2τ3π2n2/3=2τrs249π2/3.
(19)

The ground-state exchange can be related to aXτ via a thermal reduction factor, RX, where RX can only depend on the temperature and density through θ, the electron degeneracy parameter,24,26

aXτ(rs)=εXunif(rs)RX(θ).
(20)

The thermal reduction factor decreases with an increase in θ, meaning that as the electron degeneracy increases, the exchange free energy per particle decreases in magnitude.

To further validate the use of simulated scaling at finite temperatures, FTAC curves were generated with the same θ value. These should have the same ratio WXCτ,λ/εXunif at λ = 0. Since θ is a function of τ and rs, individual characteristics can be scaled while the electron degeneracy remains constant. Dividing WXC by the exact ground-state exchange demonstrates this equivalency. If the electron degeneracy is constant for a set of curves, then the ratio aXτ/εXunif should also be equivalent for all three curves. As the exchange is constant over the interaction strength, plotting the FTAC in this manner exposes the correlation component of the XC free energy for pairs of rs and τ that yield the same electron degeneracy. Figure 3 shows sets of curves for three different electron degeneracy values, both at low interaction strengths and up to λ = 1, the physical interaction strength. These are plotted against a logarithmic scale in λ to demonstrate the expected behavior at λ = 0. The complicated dependence of correlation on the electron degeneracy arises at relatively low interaction strength, with the set of curves having three times the reference degeneracy crossing those of the 2θ0 set almost immediately, before even a tenth of the physical interaction strength is reached.

FIG. 3.

WXCτ,λ(rs)/εXunif(rs) is plotted against a logarithmic scale in λ for an arbitrary reference electron degeneracy, θ0, using combinations of rs and τ to produce three curves with equivalent electron degeneracies [Eq. (18)]. At low interaction strength, λ, all three curves in a set with the same electron degeneracy meet at the y axis. Curves plotted for two additional values of θ, which are simply scaled values of θ0, demonstrate the same behavior and further validate the use of simulated scaling at FT. Plotting curves with equivalent electron degeneracies isolates the mixed temperature-density effect on the correlation free energy and demonstrate how the simple grouping of equivalent electron degeneracy curves is disrupted by correlation effects as the interaction approaches realistic strength.

FIG. 3.

WXCτ,λ(rs)/εXunif(rs) is plotted against a logarithmic scale in λ for an arbitrary reference electron degeneracy, θ0, using combinations of rs and τ to produce three curves with equivalent electron degeneracies [Eq. (18)]. At low interaction strength, λ, all three curves in a set with the same electron degeneracy meet at the y axis. Curves plotted for two additional values of θ, which are simply scaled values of θ0, demonstrate the same behavior and further validate the use of simulated scaling at FT. Plotting curves with equivalent electron degeneracies isolates the mixed temperature-density effect on the correlation free energy and demonstrate how the simple grouping of equivalent electron degeneracy curves is disrupted by correlation effects as the interaction approaches realistic strength.

Close modal

The FTAC curves are observed at low, intermediate, and high temperatures and densities. In dimensionless units, typical parameters in the WDM regime are the previously introduced Wigner–Seitz radius, rs = r/aB, and the reduced temperature, θ = kBτ/EF, both being of order one for WDM (typically in the range of 0.1–10). Here, kB is the Boltzmann constant, aB is the Bohr radius, and EF is the system’s Fermi energy, defined as the zero-temperature limit of the chemical potential. By holding the density fixed as the KS system is smoothly connected to the physical system, the FTAC formalism demonstrates how the proportions of the exchange and correlation free energy components vary with the temperature, density, and interaction strength. In addition, these curves invite investigation of the LO bound and its relationship to the exchange free energy.

As an electronic system’s temperature rises at a fixed density, it may be expected that the importance of the electron–electron interaction will decrease. However, this oversimplified picture hides the influence of both the temperature and density on the components of exchange–correlation. Figure 4 displays the adiabatic connection curves generated for a uniform gas with moderate density at temperatures varying up to the level of the Fermi energy. The solid curves of the top figure demonstrate the correlation energy, while the dashed line corresponds to the exchange component of the highest-temperature curve. For clarity, the exchange component is only explicitly shown for the high-temperature curve, although similar exchange curves could be drawn, starting at λ = 0 for the exchange energy per particle for τ = 0.1 and τ = 0.5.

FIG. 4.

FTAC curves for the UEG at a range of temperatures (top). The exchange (dashed line) is shown for the high-temperature system only. Purely correlation contributions are shown to exhibit opposite ordering with an increase in the temperature (bottom).

FIG. 4.

FTAC curves for the UEG at a range of temperatures (top). The exchange (dashed line) is shown for the high-temperature system only. Purely correlation contributions are shown to exhibit opposite ordering with an increase in the temperature (bottom).

Close modal

The ordering of these curves with rising temperature demonstrates that under these conditions, the XC free energy does decrease. The bottom plot of Fig. 4, however, isolates the correlation component for all three temperatures. Under these conditions, the relative contribution of the correlation to aXCτ varies greatly with the temperature. This is most easily observed in the reversal of the exchange and XC curves’ ordering with the temperature, but it can also be seen clearly in the flattening of the curves as we move to colder temperatures.

The curves in Fig. 5 demonstrate the adiabatic connection for a uniform gas with a relatively large Wigner–Seitz radius at low, intermediate, and high temperatures. Again, the exchange free energy component is only explicitly shown for the high-temperature curve. Since the Wigner–Seitz radius is inversely proportional to the density, a large Wigner–Seitz radius corresponds to a small density. As expected, the low-temperature curve, which was nearly linear in Fig. 4, now exhibits a fairly dramatic curvature at low λ values under these conditions. It, and all other calculated curves, obeys the exact constraint of a negative first derivative.42,66

FIG. 5.

FTAC curves for a low-density UEG is demonstrated for a range of temperatures. The exchange is shown only for the highest temperature. At this high rs value, the correlation clearly dominates the XC.

FIG. 5.

FTAC curves for a low-density UEG is demonstrated for a range of temperatures. The exchange is shown only for the highest temperature. At this high rs value, the correlation clearly dominates the XC.

Close modal

In contrast to the previous figures, Fig. 6 shows FTAC curves at a fixed temperature while the density is varied. Here, the XC curves show similar curvature at all densities when at the Fermi temperature, and the magnitude of the XC free energy per particle increases with an increase in the density. This behavior is expected since the magnitude of the exchange energy is inversely related to the Wigner–Seitz radius. The bottom plot of Fig. 6 shows the correlation contribution of each curve and demonstrates that the behavior of aXCτ is dominated by the exchange. Without the exchange component, the ordering of the curves is disrupted at this temperature, and we do not observe the same trend. However, at a high enough temperature, the magnitude of the correlation free energy consistently decreases with an increase in rs, or a decrease in the density. The differences in behavior of the correlation free energy at different temperatures demonstrate that the correlation contribution to the XC free energy fluctuates non-monotonically, as predicted via the asymmetric Hubbard dimer.41 

FIG. 6.

FTAC curves for three different densities at a fixed temperature of τ = 1 (top). Purely correlation contributions demonstrate how the ordering of the curves becomes non-monotonic when the density is varied at a fixed temperature (bottom).

FIG. 6.

FTAC curves for three different densities at a fixed temperature of τ = 1 (top). Purely correlation contributions demonstrate how the ordering of the curves becomes non-monotonic when the density is varied at a fixed temperature (bottom).

Close modal

Electron degeneracy, as defined in Eq. (19), can be related to the Lieb–Oxford bound,44,45 commonly expressed as an exact inequality for the potential energy contribution to the XC functional.59 In other words, for wavefunctions that are ground states of an N-electron Hamiltonian, the adiabatic connection integrand, WXCλ, is limited by the LO bound, where CLO is the lowest possible number that makes this inequality true for all wavefunctions.50 The LO inequality provides a strict bound on how negative the XC energy can become

WXCλCLOd3rn4/3(r),
(21)

where CLO is a constant. This is often rewritten in terms of the local density approximation (LDA) to the exchange energy,

WXCλCLOCXLDAEXLDA,
(22)

where CXLDA is the usual constant for the LDA exchange expression. Lieb and Oxford originally established that CLO ≤ 1.68, but these bounds have since been tightened by others46,51 to 1.444 ≤ CLO ≤ 1.636.

Reference 45 states that the original LO proof holds not only for pure states but also for density matrices. Some preliminary evidence, based on definitions of the Hartree-exchange energy and low-density limits of AXC, indicates that although the ZT LO bound should still hold for the FTAC, bounds similar to the LO bound may be temperature-dependent in thermal DFT. In Fig. 7, FTAC curves for a fixed density are shown while the temperature is varied along with the LO bound and a naive approach to include what might be called the implicit temperature dependence. This “quick and dirty” second approach is built on the zero-temperature expression for the LO bound for the uniform gas,

WXCλ(rs)CLOCXLDAεX(rs),
(23)

but uses the exchange–correlation free energy per particle of our system, aXτ(rs), in place of the zero-temperature ɛX(rs),

WXCτ,λ(rs)CLOCXLDAaXτ(rs).
(24)

This approach would reduce the magnitude of the integral of Eq. (21) as temperatures increase for non-uniform systems. At higher densities for the uniform gas, the FTAC curves appear to satisfy this inequality at lower temperatures. However, at a low enough density, WXCτ,λ is not bound by this attempt to temperature-adjust the LO inequality, even at low temperatures. As shown in Fig. 7, only the true, zero-temperature LO bound is satisfied, whether using the 1999 value for the CLO constant or the more recently tightened 2019 value. This suggests that although LO-type bounds may exhibit some temperature dependence, it is not appropriately captured by the implicit temperature dependence of the thermally weighted density combined with the electron-degeneracy dependence of the thermal reduction factor.24,26 In addition, the adherence of the FTAC curves to the tighter LO bound for these high-rs (i.e., low-density) conditions indicates the density-dominated behavior of the low-temperature, low-density limit. Observing this predominately at low densities is related to the well-known dominance of correlation over exchange that holds in finite-temperature systems as well.25 Since the exchange comprises a much smaller fraction of the total XC free energy, the correlation is large enough in relative magnitude that it dips well below the exchange-based dotted curves. As temperatures increase and the curves approach the λ axis, it seems likely that inclusion of the temperature dependence in the bound could provide a tighter bound than the ZT value and for the full range of rs and θ.

FIG. 7.

ZT LO bound and temperature-adjusted quantities for a low density UEG at three different temperatures. Even at low temperatures, the XC free energy is not bounded by the LO-like line calculated from the FT exchange. The shaded region indicates the range of possible values of CLO in the Lieb–Oxford inequality.

FIG. 7.

ZT LO bound and temperature-adjusted quantities for a low density UEG at three different temperatures. Even at low temperatures, the XC free energy is not bounded by the LO-like line calculated from the FT exchange. The shaded region indicates the range of possible values of CLO in the Lieb–Oxford inequality.

Close modal

One approach to investigate the temperature-dependence of the LO bound is to look at situations where the LO bound is most likely to hold; for example, when |aXτ||εXunif|. The exchange free energy is most likely to be smaller than the ground state exchange at large values of θ. Since θτrs2, a large value of theta can be achieved with a high value of τ and low value of rs, in which the exchange component of the XC free energy dominates. A large θ may also be achieved with a low value of τ and a high value of rs in which the correlation component dominates.

The parameterization provided by Groth and collaborators is valid within 0 ≤ θ ≤ 8 and 0.1 ≤ rs ≤ 20. Within this range of θ values provided by Groth and co-workers,57WXCτ,λ=0 can be as small in magnitude as −0.0016 when rs = 20 and τ ≈ 0.04, and Wτ,λ=1 can be as low as − 4.778 0 when rs = 0.1 and τ = 0. Writing the inequality in terms of the uniform gas exchange per particle,50 

WXCτ,λ(rs)Crs,λCXLDAεXunif(rs),
(25)

we define two approximate bounding parameters for our finite-temperature curves, Ctop=Crs,λ=0 and Cbottom=Crs,λ=1. The rs dependence of these two parameters is rooted in the exchange free energy’s dependence on the electron degeneracy, θ, through the reduction factor. This means that maximal and minimal values are constant with respect to θ, but the value shifts depending on how one constructs this degeneracy for a given selection of rs and its corresponding τ at that specific value of θ.

Figure 8 shows approximated upper and lower bounds for the FTAC for a system of rs = 10 at three different temperatures. For context, the Lieb–Oxford bound for the ground-state XC energy is also shown using both the 1999 and 2019 values for CLO. As expected, the ZT bounds are more negative than either of the FT approximated constraints.

FIG. 8.

Bounds are shown for FTAC curves at rs = 10 and three different temperatures. The Lieb–Oxford bound, which constrains the exchange–correlation energy at ZT, is also shown for the rs = 10 case.

FIG. 8.

Bounds are shown for FTAC curves at rs = 10 and three different temperatures. The Lieb–Oxford bound, which constrains the exchange–correlation energy at ZT, is also shown for the rs = 10 case.

Close modal

Figure 9 shows approximated upper and lower bounds for the FTAC formalism within the allowable range of conditions for the parameterized curves. The curves correspond, from the top to the bottom, to θ = 8, three intermediate electron degeneracies, and θ = 0. The curve reaching the highest magnitude corresponds to θ = 0, which agrees with the relationship between the exchange free energy and the thermal reduction factor, RX(θ). This relationship, along with the dependence of the thermal reduction factor on θ, predicts that the exchange free energy decreases with an increase in θ. Thus, when the electron degeneracy is low, the exchange free energy is expected to have a large magnitude relative to the exchange free energy at a high electron degeneracy.

FIG. 9.

Approximate upper and lower bounds generated for the FTAC formalism within the constraints of the parameterization provided by Groth and collaborators are shown for a low electron degeneracy (bottom curve), three intermediate electron degeneracy values, and a large electron degeneracy (top curve).

FIG. 9.

Approximate upper and lower bounds generated for the FTAC formalism within the constraints of the parameterization provided by Groth and collaborators are shown for a low electron degeneracy (bottom curve), three intermediate electron degeneracy values, and a large electron degeneracy (top curve).

Close modal

We have presented finite-temperature adiabatic connection curves based on an accurate parameterization of the exchange–correlation free energy per particle for the uniform electron gas under warm dense matter conditions.57 Free energies obtained from the simulated scaling process60,63,64 reproduce the directly calculated results57 with controllable accuracy. This demonstrates the usefulness of simulated scaling for a wide range of temperatures and densities and also the reliability of the parameterization in applications beyond those for which it was originally designed.

The temperature- and density-dependent AC curves presented in this work behave as expected, in line with the ZT curve for the uniform gas in many ways. However, the relationship between the density, interaction strength, and temperature is complex, and combinations of conditions exist that can disrupt the naive ordering of curves based on the temperature alone. As is well understood in the community, this is due to the dependence of the adiabatic connection integrand on the electron correlation. This is similar to previous calculations using the asymmetric Hubbard dimer30,41 and supports careful use of this model system for DFT analysis. In general, the complicated behavior of the correlation free energy with the temperature can be hidden by the exchange free energy due to its strong, monotonic dependence on the electron degeneracy parameter. For careful examination of the correlation, we have demonstrated that scaling to a correlation-dominated limit can expose these details.

All curves obey the ZT Lieb–Oxford bound, as expected. However, the FT exchange free energy for the UEG does not satisfy the same relationship to this bound that the ZT exchange does, particularly in regimes of a large electron degeneracy. Even at low temperatures, in low-density regimes, scaling the FT exchange free energy using the ZT Lieb–Oxford constant fails to contain the FTAC curves. In other regimes, where the magnitude of correlation is small, the exchange’s dependence on the temperature is strong enough to disrupt this ZT relationship. Some approximate bounds on the FTAC formalism have been generated, demonstrating the tied density–temperature effects on exchange–correlation and showing a large discrepancy between the adiabatic connection curves’ most negative values and the ZT LO bound, in line with ZT analysis of similar issues. It has been shown50 that uniform densities are not the most challenging for the LO bound at ZT, so using thermal densities near or approaching the strong-interaction limit would be a reasonable approach for better estimating any temperature dependence to be found for the LO bound.

The investigations within this work have all been undertaken within the allowed range of applicability given by Groth and co-workers, save for the initial extraction of the exchange free energy by taking a high-density (i.e., low-rs) limit. Although this represents a deviation from the recommended usage of the parameterization, our tests using a known behavior of the exchange for the UEG with the electron degeneracy demonstrate the robustness to extension beyond the low-rs regime into the “very low” regime. Our group continues work to extend our studies of the interplay between the temperature, density, and interaction strength via new approaches approximating these curves and investigating the connections between zero-temperature and finite-temperature exchange–correlation approximations. Although these methods do not rely on constraints such as the Lieb–Oxford bound, work examining the effect of temperature on bounding relationships is ongoing in hopes that connections between the two areas of study will provide insight into both the entropic contributions to the exchange–correlation free energy and the exchange–correlation as a whole.

We thank Dr. Juri Grossi, Dr. Sara Giarrusso, and Dr. Arnold Kim for useful discussions. This work was supported by the U.S. Department of Energy, National Nuclear Security Administration, Minority Serving Institution Partnership Program, under Award No. DE-NA0003866. We thank all indigenous peoples local to the site of University of California, Merced, including the Yokuts and Miwuk. We embrace their continued connection to this region and thank them for allowing us to live, work, learn, and collaborate on their traditional homeland.

The authors have no conflicts to disclose.

The data that support the findings of this study are available from the corresponding author upon reasonable request.

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