Quantified-accuracy extension of the average-atom two-component-plasma (AA+TCP) model of warm dense matter to strong ionic coupling

The AA+TCP model of Saumon, Starrett, Anta, Daughton and Chabrier (2013/2014) computes the electronic and ionic structure of warm dense matter by coupling a finite-temperature density-functional average atom to Ornstein-Zernike integral equations for the surrounding two-component ion-electron plasma; it has no adjustable parameters and reproduces ab initio pair distribution functions remarkably well, but its authors explicitly flagged its unresolved accuracy issues: the hypernetted-chain closure (zero ion-ion bridge function) that degrades at strong ionic coupling, the unquantified arbitrariness of the non-unique 'green-box' modeling choices (the ideal-electron chemical potential criterion and the partition of pseudo-atom electrons into ion versus screening populations), missing core-overlap, relativistic and gradient-level exchange-correlation refinements, and undelineated applicability boundaries in $(Z,T,\rho)$. Later work bypassed several of these items along other routes -- pseudo-atom molecular dynamics replaces the integral equations by exact classical molecular dynamics, and relativistic or first-principles-derived average atoms exist in other formulations -- but no published work extends the integral-equation model itself with a bridge function, quantifies its intrinsic choice sensitivity, or maps its validity domain quantitatively. The problem of making the parameter-free integral-equation average-atom model quantitatively accurate at strong ionic coupling, with known error bars and a mapped domain of validity, remains open.

Background

Warm dense matter (WDM) designates materials at near-solid densities and temperatures from about 1 eV to several keV, where electrons are partially degenerate and partially ionized; it occurs in inertial-confinement fusion experiments, giant planets, and white-dwarf envelopes. A central microscopic observable is the ion-ion pair distribution function $g_{II}(r)$, which describes the probability of finding two ions a distance $r$ apart and underlies equations of state, transport coefficients, and the elastic feature of X-ray Thomson scattering (XRTS) diagnostics. Ab initio quantum molecular dynamics (QMD, i.e., finite-temperature density-functional theory for the electrons plus classical dynamics for the ions) provides benchmark $g_{II}(r)$ but becomes prohibitively expensive for metals above roughly 10-20 eV and at large system sizes.

A much cheaper alternative is the average-atom (AA) approach: the plasma is replaced by one representative nucleus embedded in the average plasma environment, and the electrons are solved for self-consistently in finite-temperature (Mermin-Kohn-Sham) density functional theory. 'One-ion' or atom-in-a-cell AA models of this type (going back to Thomas-Fermi and Liberman-type models) ignore the correlations of the plasma surrounding the central ion. The model at issue here, introduced by Saumon, Starrett, Anta, Daughton and Chabrier (arXiv:1311.3986, published as a 2014 Springer chapter, with the journal-length companion Starrett and Saumon, Phys. Rev. E 87, 013104 (2013) and precursor Starrett and Saumon, Phys. Rev. E 85, 026403 (2012)), couples such an AA model to the Ornstein-Zernike (OZ) integral-equation theory of fluids. The OZ equations relate the total correlation functions $h_{ab}(r)=g_{ab}(r)-1$ of a classical fluid mixture to its direct correlation functions $c_{ab}(r)$; here the fluid is a two-component plasma (TCP) of classical ions and a classical representation of the screening electrons, closed at the pair level. The central ion's electronic structure determines the effective ion-ion interaction and the electron screening clouds, while the TCP solution determines the ionic structure $g_{II}(r)$ that feeds back into the AA calculation; the inputs are only the nuclear charge $Z$, the temperature $T$, and the ion density $n_I$, with no adjustable parameters. A key original feature is the recognition that the ion charge $Z^{*}$ computed in the AA cell differs from the TCP charge $\bar{Z}$ that enters observables.

Closing the OZ equations requires a bridge function $B_{II}(r)$, a formally exact but generally unknown contribution that summarizes many-body direct correlations. The AA+TCP model sets $B_{II}(r)=0$, the hypernetted-chain (HNC) approximation. This is known to degrade as the ionic coupling parameter $\Gamma=(\bar{Z}e)^{2}/(a\,k_{B}T)$ grows (here $a$ is the ion-sphere radius, the radius of a sphere containing on average one ion): Wünsch, Vorberger and Gericke benchmarked HNC ion structure against first-principles simulations and quantified this strong-coupling error in related systems. The source chapter itself reports visible HNC-related deviations for aluminum at 1 eV, solid density, and names bridge functions from hard-sphere, one-component-plasma (OCP), Yukawa, or variationally optimized reference systems as the remedy needed to reach liquid-metal and very strongly coupled regimes.

The model also contains elements the authors call 'green-box' choices (so labeled in the paper's flow chart): places where several different closure rules are admissible, in particular the criterion that fixes the ideal-electron chemical potential $\mu_e^{\mathrm{id}}$ entering the AA calculation, and the rule that partitions the pseudo-atom electron density into an ion part $n_e^{\mathrm{ion}}$ (bound electrons attributed to the ion) and a screening part $n_e^{\mathrm{scr}}$ (valence-electron contribution to the TCP). These choices are not unique -- earlier ion-correlation average-atom formulations, such as Chihara's quantum hypernetted chain (QHNC, an average-atom integral-equation formulation in which the bound-electron structure is treated quantum-mechanically within the hypernetted-chain framework) and the variational formulation of Starrett and Saumon, resolve them differently -- and their effect on observables was acknowledged by the authors to be non-negligible but was never quantified. Further refinements left unimplemented in the published model are core-overlap corrections to the direct ion-ion potential, a relativistic (Dirac) treatment of the electrons for high-$Z$ or high-energy conditions, and exchange-correlation functionals beyond the local density approximation (LDA), such as generalized gradient approximations (GGA).

The model spawned a successful research program. The same pseudo-atom construction was coupled to classical molecular dynamics instead of integral equations, giving pseudo-atom molecular dynamics (PAMD; Starrett, Daligault and Saumon, Phys. Rev. E 91, 013104 (2015)), which the authors describe as a solution to the problem of consistently including ionic structure in the average-atom methodology; because molecular dynamics is exact for a given pair potential, PAMD avoids the bridge-function closure question entirely, at the cost of finite simulation cells and statistical noise. The integral-equation model itself was extended to mixtures (Starrett, Saumon, Daligault and Hamel, Phys. Rev. E 90, 033110 (2014)) and remains in active use for opacities and electron-ion coupling under the name 'the Starrett and Saumon average-atom model with ion correlations' (Ovechkin, Loboda and Falkov; Ovechkin, Smirnov, Loboda and Falkov). Independent average-atom-plus-HNC formulations also continue to appear (e.g., Hou, Bredow, Yuan and Redmer), sharing the same closure limitation. Relativistic Dirac average atoms exist in the Tartarus line (Gill and Starrett; Starrett, Gill, Sjostrom and Greeff), but as in-cell Green's-function codes not coupled to ion correlations through the TCP integral equations. A recent systematic status summary is Section 7 ('Average Atom and Chemical Models') of the multi-author 'Roadmap for warm dense matter physics' (Vorberger et al.).

Problem Statement

Extend the parameter-free average-atom-plus-two-component-plasma (AA+TCP) model of Saumon, Starrett, Anta, Daughton and Chabrier so that its accuracy for the ionic and electronic structure of warm dense matter is quantitatively characterized and controlled over the strong-coupling regime in which the published model is known to degrade. A complete answer delivers a model that simultaneously satisfies the following jointly supporting requirements:

1. (Closure at strong coupling) Replace the hypernetted-chain closure $B_{II}(r)=0$ (which neglects the ion-ion bridge function, the formally exact but generally unknown many-body contribution to the closure of the Ornstein-Zernike integral equations) with a physically appropriate ion-ion bridge function -- for example drawn from hard-sphere, one-component-plasma, Yukawa, or variationally optimized reference systems, or constructed anew -- such that the ion-ion pair distribution function $g_{II}(r)$ and derived structure observables remain accurate at strong ionic coupling (lower temperatures and/or higher densities, including liquid-metal-like conditions), where the published model shows visible degradation.

2. (Quantified 'green-box' choices) Quantify the sensitivity of the model's observables -- $g_{II}(r)$, the average ion charge, equation-of-state quantities, and quantities entering opacity and transport applications -- to the non-unique modeling choices, namely the criterion fixing the ideal electron chemical potential $\mu_e^{\mathrm{id}}$ and the rule partitioning pseudo-atom electrons between the ion population $n_e^{\mathrm{ion}}$ and the screening population $n_e^{\mathrm{scr}}$, so that the model's intrinsic arbitrariness becomes a known error bar rather than an unquantified caveat.

3. (Physical refinements where needed) Incorporate each of the refinements the authors list as missing -- core-overlap effects in the direct ion-ion potential, a relativistic (Dirac) treatment of the electrons for high-$Z$ or high-energy regimes, and exchange-correlation treatment beyond the local density approximation (e.g., generalized gradient approximations) -- in those $(Z,T,\rho)$ regions where the error metrics reported under Requirement 4 show that omitting that refinement makes a dominant contribution to the model's total error, and report, for each refinement adopted, the change it produces in those error metrics.

4. (Validity domain) Delineate, with quantified error metrics against independent ab initio simulations (e.g., quantum molecular dynamics) and, where available, experimental structure data such as X-ray Thomson scattering measurements, the boundaries in $(Z,T,\rho)$ space (nuclear charge $Z$, temperature $T$, density $\rho$) within which the model's assumptions (spherical symmetry, pairwise correlations, classical ions) remain accurate.

The single scientific objective uniting these deliverables is to convert the AA+TCP model from a parameter-free model with acknowledged but unquantified approximations into one with demonstrated, benchmarked accuracy across the warm-dense-matter regime including strong ionic coupling. A claim of accuracy counts as demonstrated only through reported error metrics: for Requirements 1 and 4 the deviation of $g_{II}(r)$ and of the derived structure observables from independent quantum molecular dynamics (QMD) benchmarks (and, where available, experimental structure data) must be reported over an explicit set of $(Z,T,\rho)$ state points spanning from the weakly coupled regime into the strongly coupled regime in which the published model degrades, and Requirement 1 additionally requires that the reported deviation at the strongly coupled state points be smaller than that of the published $B_{II}(r)=0$ result; a demonstration confined to an isolated state point or to the weak-coupling regime alone does not complete Requirements 1 or 4. Throughout, the model must remain free of parameters fitted to the benchmark data against which it is validated.

References

1. D. Saumon, C. E. Starrett, J. A. Anta, W. Daughton, G. Chabrier, 'The Structure of Warm Dense Matter Modeled with an Average Atom Model with Ion-Ion Correlations', in Frontiers and Challenges in Warm Dense Matter, Lecture Notes in Computational Science and Engineering (Springer, 2014), pp. 151-176, DOI: 10.1007/978-3-319-04912-0_6; preprint arXiv:1311.3986, https://arxiv.org/abs/1311.3986
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