Real-Space Green's functions for Warm Dense Matter
M. Laraia, C. Hanson, N. R. Shaffer, D. Saumon, D. P. Kilcrease, C. E. Starrett
RReal-Space Green’s functions for Warm Dense Matter
M. Laraia,
1, 2
C. Hanson, N. R. Shaffer, D. Saumon, D. P. Kilcrease, and C. E. Starrett ∗ Los Alamos National Laboratory, P.O. Box 1663, Los Alamos, NM 87545, U.S.A. School of Physics and Astronomy, University of Minnnesota, Minneapolis, MN 55455, U.S.A. (Dated: January 7, 2021)Accurate modeling of the electronic structure of warm dense matter is a challenging problemwhose solution would allow a better understanding of material properties like equation of state,opacity, and conductivity, with resulting applications from astrophysics to fusion energy research.Here we explore the real-space Green’s function method as a technique for solving the Kohn-Shamdensity functional theory equations under warm dense matter conditions. We find the method tobe tractable and accurate throughout the density and temperature range of interest, in contrast toother approaches. Good agreement on equation of state is found when comparing to other methods,where they are thought to be accurate.
Keywords: warm dense matter, real-space Green’s function, equation of state
I. INTRODUCTION
The term “warm dense matter” refers to states of mat-ter with densities from a fraction of solid to several timessolid density, and temperatures of a fraction of an eV tohundreds of eV. The electronic structure of warm densematter is strongly density- and temperature-dependent,seen in effects like pressure ionization as well as therelocalization of electrons with increasing temperature(caused by reduced screening of the nucleus by hot elec-trons). This state of matter appears in inertial fusionexperiments [1–3], basic science experiments [4], and inastrophysical objects like white dwarf stars [5, 6], giantplanets [7, 8], and the interior of the sun [9, 10].The electronic structure of warm dense matter is rou-tinely calculated using Kohn-Sham density functionaltheory (DFT)[11, 12], most often using a plane-wave ex-pansion of the electron orbitals. However, this methodhas certain drawbacks that limit its applicability acrossthe warm dense matter regime. The first drawback isa limitation to relatively low temperatures due to pro-hibitive scaling with temperature when the degeneracyparameter Θ = k B T /E F exceeds unity, where T is thetemperature and E F is the Fermi energy [13]. A seconddrawback is the use of pseudo-potentials to represent thescreening of the nucleus by core electrons. These mustbe carefully created and results checked for sensitivitywith respect to them [14, 15], since even core electronsare affected by the plasma environment in warm densematter.In this work, we consider an alternative approach tothe electronic structure of warm dense matter based ona real-space Green’s Functions (RSGF) solution of theDFT equations. The RSGF method sidesteps some ofthe problems encountered with plane-wave DFT whenapplied to warm dense matter. Notably, it is based onthe one-body Green’s function rather than one-body or-bitals, which allows more favorable scaling at elevated ∗ [email protected] temperature. It also uses a multi-center expansion of theGreen’s function with spherical harmonics, which circum-vents the need for pseudo-potentials and allows genuineall-electron calculations.RSGF is a method with a long history, and in particu-lar has been very useful for the optical properties of ma-terials at standard temperature and pressure, includingsolids and molecular systems[16–22]. The x-ray spectraof warm dense matter using RSGF and related methodshave also been explored [23–25].Two of us (CES and NRS) recently explored the useof the closely related Korringa-Kohn-Rostoker Green’sFunction (KKR-GF) [26–28] method for the electronicstructure of warm dense matter [29]. Both methods,KKR-GF and RSGF, fall under the broader theoreticalconcept known as multiple scattering theory [28]. Thedifference between KKR-GF and RSGF comes down towhether one approximates the infinite plasma using aperiodically repeating system (KKR-GF) or by a finitecluster (RSGF). While RSGF is appealing for its sim-plicity and avoidance of artificially imposed periodicity,it is not obvious a priori how large the cluster must beto attain converged electronic structure. Further, if con-verged calculations are possible for warm dense matter,it is not obvious if the required cluster sizes are so largeto make the method impractical.In this contribution we demonstrate that RSGF is infact a viable and accurate method for predicting theequation of state (EOS) and electronic structure of warmdense matter. We also discuss the advantages of RSGFover KKR-GF. In short, these are simplicity in imple-mentation and avoidance of k -space integrations, whichmakes the method better suited for calculations of spec-tral quantities like densities of state (DOS) and opticalconstants. Convergence of the EOS with respect to clus-ter size is demonstrated, and shown to be more computa-tionally efficient than equivalent KKR-GF calculations.Comparison is then made with an average atom modelon EOS and DOS. It is found that while RSGF and theaverage atom models agree on EOS at low densities andhigh temperatures, the DOS remains qualitatively dif- a r X i v : . [ phy s i c s . p l a s m - ph ] J a n ferent due to broadening of states by the plasma envi-ronment treated in RSGF but neglected in the averageatom. We develop a perturbative model of this effect tobroaden the average atom DOS. This improves the quali-tative agreement between the models, mainly by destroy-ing the average atom’s artificially sharp valence statesand continuum resonances. II. REAL-SPACE GREEN’S FUNCTIONS
In multiple scattering theory, space is divided intocells, each containing a special point called its center.Each of these centers is taken to be the origin of itscell, and the multi-center electronic structure problemis recast as many single-center ones. Done na¨ıvely, theboundary conditions on each cell would be very com-plicated due to both the cell geometry and the needto match wavefunctions in neighboring cells. By repre-senting the electronic structure in terms of the complex-energy one-body Green’s function, G ( x , x (cid:48) , z ), one caninstead solve each cell with simple boundary conditions,(e.g., free-particle). By also calculating the scattering t -matrix for each cell, these single-center solutions can becorrected to obtain the full multi-center Green’s functionby solving a Dyson equation.In practice, the method uses a double spherical har-monic expansion of the Green’s function, one for eachposition argument. If x and x (cid:48) lie respectively in cells n and n (cid:48) with centers R n and R n (cid:48) , then one writes x = r + R n and x (cid:48) = r (cid:48) + R n (cid:48) and expands the r and r (cid:48) dependence in spherical harmonics about their respec-tive centers. The full Green’s function decomposes intosingle-site and multi-site terms G ( r + R n , r (cid:48) + R n (cid:48) , z ) = G ss ( r + R n , r (cid:48) + R n (cid:48) , z )+ G ms ( r + R n , r (cid:48) + R n (cid:48) , z ) (1)The single site Green’s function is given by G ss ( r + R n , r (cid:48) + R n (cid:48) , z ) =2 m e δ nn (cid:48) ∞ (cid:88) L =0 H n, × L ( r > , z ) R nL ( r < , z ) (2)which represents the electronic structure of the systemas if each cell were considered separately and then super-posed. The multi-site Green’s function G ms ( r + R n , r (cid:48) + R n (cid:48) , z ) =2 m e ∞ (cid:88) LL (cid:48) R nL ( r , z ) G nn (cid:48) LL (cid:48) ( z ) R n (cid:48) × L (cid:48) ( r (cid:48) , z ) (3)then corrects the electronic structure to account for in-terference from other sites. Here, R nL ( r , z ) and H nL ( r , z )are the complex-energy solutions of the Kohn-ShamSchr¨odinger or Dirac equations in cell n , separated in spherical coordinates about R n . Non-relativistically,these are Pauli spinors with angular momentum num-bers L = ( l, m ); relativistically, these are Dirac spinorswith angular momentum numbers L = ( κ, m ). Thesuperscript × means to take the complex conjugate ofthe angular part of the non-relativistic solution or totake the Hermitian conjugate of the spin-angular partof the relativistic solution. In the following, we onlyshow results for non-relativistic calculations. The no-tation r > ( r < ) means to take r or r (cid:48) according to whichone is greater (lesser) in magnitude. Throughout, we useHartree atomic units with ¯ h = 4 π(cid:15) = e = 1, leaving m e symbolic for easy conversion to Rydberg units. For othernormalization and sign conventions, see Reference [29].The key quantities of multiple scattering theory arethe G nn (cid:48) LL (cid:48) in Eq. (3), called the structural Green’s functionmatrix elements. These make up the structural Green’sfunction matrix G , which is calculated from the struc-ture constants matrix G and the t -matrix using Dyson’sequation G ( z ) = G ( z ) [ I − t ( z ) G ( z )] − (4)This expression is known as the fundamental equation ofmultiple scattering theory and is solved by matrix inver-sion. The size of the matrix is N ( l max + 1) , where N is the number of cells, and l max is the maximum valueof the orbital angular momentum quantum number thatoccurs in Eq. (3). While in principle l max goes to infinity,in practice it is accurate to use small values like 2 or 3[29].The structure constant matrix elements G nn (cid:48) ,LL (cid:48) dependon the positions of the expansion centers { R n } and theenergy z . For periodic systems, one uses the Ewald tech-nique and a Fourier space integration and the result isthe so-called Korringa-Kohn-Rostoker Green’s function(KKR-GF) method [26, 27]. For a finite cluster of cen-ters one can directly evaluate the structure constants inreal space [30] G nn (cid:48) ,LL (cid:48) ( z ) = − πip (cid:88) L (cid:48)(cid:48) i l − l (cid:48) − l (cid:48)(cid:48) h + l (cid:48)(cid:48) ( pR nn (cid:48) ) × C L (cid:48) LL (cid:48)(cid:48) Y L (cid:48)(cid:48) ( ˆ R nn (cid:48) ) (5)which is the Real-Space Green’s function (RSGF)method. Here, Y L is a spherical harmonic including theCondon-Shortley phase, R nn (cid:48) = R n − R n (cid:48) is the vectorconnecting centers n and n (cid:48) , p = √ mz is the complexmomentum lying in the upper-right quadrant of the z plane, h + l is the spherical Hankel function, and C L (cid:48) LL ” ≡ (cid:90) d ˆ r Y L (ˆ r ) Y ∗ L (cid:48) (ˆ r ) Y L (cid:48)(cid:48) (ˆ r ) (6)are Gaunt coefficients, defined here to be real-valued. Asthere is no Ewald sum or Fourier space integration, theRSGF expression, Eq. (5), is much simpler to implementthan the corresponding KKR expression (c.f. the Ap-pendix of Ref. [29]). FIG. 1. (Color online) Illustration of centers included in thereal-space Green’s function cluster approximation. The com-putational cube is shown as the shaded box, centers are shownas blue (dark) circles, and those inside the box are surroundedby orange (light) shaded spheres.
The RSGF method is strictly valid only for finite clus-ters of centers, whereas the KKR-GF method is validfor any periodic system. For the electronic structure ofbulk disordered systems like plasmas, either method isapproximate and introduces finite size effects dependingrespectively on the number of particles in the cluster or ineach periodic image. Intuitively, it would seem the finite-size effects of the KKR-GF method might be less severein general. However, if one takes the cluster to be suf-ficiently large and focuses on the electronic structure ofcells near the center of the cluster, then the RSGF G canbe considered to be an approximation to that of KKR-GF. The essence of the approximation is that long-rangestructure far from the center of the cluster does not sig-nificantly influence the electronic structure there. Thiscluster approximation should be especially well-suited todisordered plasmas, as opposed to ordered crystals, be-cause coherent Bloch states that are possible in crystalsare destroyed by the ionic disorder. III. RESULTSA. Physical Model, Convergence, and Timing
We adopt the same physical model as presented in ref-erence [29]. We consider a periodic computational cubewith N nuclei of charge Z and N Z electrons. The posi-tions of the nuclei are provided by the pseudoatom molec-ular dynamics (MD) model [31]. In the results shownhere, we use non-relativistic DFT in the temperature-dependent local density approximation (LDA) [32]. Thetessellation of space is carried out using the power tes-sellation technique [33], and centers not corresponding to
T [eV] ρ [g/cm ] RSGF r c No MS10 2.7 15.7 1.3 2.710 0.027 11.4 0.8 2.9100 2.7 14.4 1.0 3.3100 0.027 9.5 0.8 2.6TABLE I. Wall time in minutes to convergence for one MDframe, run on an 18-core CPU. Times shown for RSGF solu-tion with l max = 2 and with no multiple scattering solution(No MS). The size of the correlation sphere radius ( r c ) is alsogiven, in units of the ion-sphere radius. nuclear positions are added to minimize the volume notin the muffin-tin spheres, as in [29].The difference to the model presented in reference [29]is that the structural Green’s function matrix is calcu-lated using the RSGF cluster approximation. To definethe cluster, we include at a minimum all centers in thecomputational cube. We further include any centers out-side that cube but within a fixed distance from any cen-ter in the computational cube, illustrated in Figure 1.This fixed distance we call the correlation radius . Thelarger the correlation radius, the more centers enter intothe calculation of the structural Green’s function matrix,Eq. (4), and the better converged the multi-site contri-bution to the electronic structure. The correlation radiusis one of the main parameters to check convergence within RSGF, taking the place of k -point sampling density inKKR-GF.In Figure 2, we show numerical convergence of the EOSwith respect to the correlation radius for a range of alu-minum plasmas. It is seen that the size of the correlationradius necessary for convergence of the pressure dependson the density and temperature. At higher temperatures,the multiple scattering correction is less important, aspointed out in Ref. [29]. This is because the electrons ofhigh-temperature plasmas have higher kinetic energies onaverage and therefore it is reasonably accurate to assumefree-electron boundary conditions on each site, equivalentto ignoring G ms in Eq. (1). At low densities we also findthe multi-center Green’s function to be less importantbecause free-electron boundary conditions become exactin the limit of isolated atoms.Table I shows timing results for the same conditionsshown in Figure 2. For each density, the correlationsphere radius was taken to be the smallest convergedvalue. In all cases, the time to obtain an SCF solu-tion is modest. Comparing the solid density timings tothe KKR-GF timings reported in Ref. [29], we see ap-proximately a fourfold speed up in the RSGF methoddue its more economical calculation of the structure con-stants. This is helped by the fact the correlation radiusin RSGF is a fine-grained parameter to determine con-vergence, compared to refining the k -space integration
10 eV2.7 g/cm
100 eV 0.027 g/cm
10 eV0.027 g/cm
100 eV0.5 0.8 1.0 1.20.5 0.8 1.0 1.20.5 0.8 1.0 1.3 1.51.41.31.21.00.80.6 t o t a l p r e ss u r e [ M b a r] number of centers in cluster FIG. 2. (Color online) Convergence of the total pressure of aluminum plasmas with respect to the number of particles in thecorrelation sphere. For each temperature and density condition, four distinct lines are shown corresponding to four MD frames.On one line the size of the correlation radii in units of the ion-sphere radius are given. The computational cube contains 8nuclei and 35 extra centers, therefore the minimum number of centers in this cluster approximation is 43. mesh in KKR-GF. That is, one can gradually increase r c in RSGF in order to obtain converged results with-out wasted effort, whereas going from a 2 -point meshto a 3 -point one in KKR-GF represents a big jump inexpense. B. Equation of State
Table II presents total pressures from the RSGFmethod for aluminum plasmas. The pressure is evalu-ated as a time average over MD configurations with 8 nu-clei and 35 extra centers in the computational volume.First, we compare to the KKR-GF results of Ref. [29] and See Ref. [29] for a convergence study on the number of extracenters. find good agreement, validating that the RSGF finite-cluster approximation is a good one. Note that we haveused correlation sphere radii sufficiently large that largervalues do not change the results significantly (1.3 at 2.7g/cm and 0.8 at 0.027 g/cm , each units of the ion-sphere radius for that density). The column labeled “NoMS” in the table refers to a calculation in which we haveset the multiple scattering Green’s function to be zero.Comparing this calculation to the RSGF results revealsexcellent agreement at 100 eV, for all densities, affirmingthat multiple scattering effects are small at high temper-ature. At 10 eV, differences increase with density as ex-pected due to the increasing overlap of valence electronsof nearby atoms.In the table we also compare to the T artarus averageatom model. The link between the multiple scatteringmodel and the average atom was explored in Ref. [35].In short, the average atom model is obtained from RSGFby ignoring the multi-site Green’s function, setting the T [eV] ρ [g/cm ] RSGF KKR-GF No MS difference T artarus difference10 2.7 3.35 (0.20) 3.36 (0.18) 3.64 (0.25) 8.7% 2.93 -13%10 0.27 0.291 (2.3 × − ) 0.300 (2.3 × − ) 3.1% 0.271 -6.8%10 0.027 3.12 × − (5.7 × − ) 3.17 × − (3.3 × − ) 1.6% 3.11 × − -0.3%100 2.7 76.7 (0.51) 76.6 (0.35) 77.0 (0.53) 0.3% 75.4 -1.7%100 0.27 9.11 (2.2 × − ) 9.12 (2.3 × − ) 0.1% 9.03 -0.9%100 0.027 1.05 (7.5 × − ) 1.05 (3.4 × − ) 0% 1.05 0%TABLE II. Total pressures in Mbar for aluminum plasmas from the RSGF method, compared to KKR-GF results of reference[29], and to the T artarus average atom model [34], as well as to a calculation with the multiple scattering Green’s function setto zero (No MS). Also shown, beside the No MS and T artarus columns are the percentage differences of that model’s pressurecompared to the RSGF result. The numbers in brackets are one standard deviation. -4 -2 0 2 4010203040-1 -0.5 0 0.5 1050010001500-4 -2 0 2 4010203040 TartarusperturbationtheoryRSGFfree electron-2 0 20100020003000 energy [E H ] d e n s it y o f s t a t e s p e r a t o m [ E H - ]
10 eV2.7 g/cm
100 eV0.027 g/cm
10 eV100 eV
FIG. 3. (Color online) Density of states for aluminum plasmas. Shown are the results from the RSGF method, where the“noise” is due to averaging over MD frames. Also shown is the DOS from the T artarus average atom model, and the DOSresulting from using a microfield and perturbation theory to broaden the T artarus result. For the T artarus result, the verticallines for negative energy values correspond to the discrete core states. cells to be identical spheres with the ion-sphere radius,and solving the Poisson equation inside each sphere inde-pendently. The average atom approximation is more se-vere than just neglecting multiple scattering. Indeed, theagreement of total pressure between T artarus and RSGFis generally worse than the agreement between RSGF and“No MS”, but it is still quite reasonable at higher tem-peratures or lower densities. The fair agreement at highdensity and temperature might be surprising, since theMD simulation allows nuclei to get closer together thanthe radius of the ion-sphere. On the one hand, the av-erage atom approximation ought to break down due itsneglect of electron overlap between atoms. On the otherhand, the EOS at high temperature is controlled mainlyby high-energy, nearly free electrons, whose contributionto the EOS is essentially correct even in the average atommodel. The same cannot be said for electrons in the lowerenergy valence states, which are poorly treated by theaverage atom model. Quantities that interrogate the va-lence, such as the density of states or absorption spectra,will display strong disagreement between average atomand RSGF calculations, even at conditions where the twoapproaches agree in the EOS. C. Density of States
In addition to the simplicity and rapid convergenceof RSGF, another advantage it holds over the KKR-GFmethod comes in computing spectral quantities like thedensity of states (DOS). One is generally interested inthe DOS at real energies, for which one needs to evaluatethe Green’s function on or very near the real energy axis,where it is a rapidly varying function of energy. In theKKR-GF method, this requires many k -space integrationpoints to accurately calculate the structural Green’s func-tion (more than needed for EOS), which is very costly.In contrast, we find the RSGF cluster sizes needed forconverged DOS calculations to be no greater than whatis needed for EOS in practice. This makes RSGF well-suited to the calculation not only of DOS, but for opticalconstants as well, which are intrinsically more expensiveto evaluate [17].In Figure 3, the density of states in the valence regionis shown for aluminum plasmas. Looking first at theRSGF results at 10 eV and solid density, we see two dis-tinct bands at negative energy, corresponding to the 2 s and 2 p bound states, as well as a free-electron-like contin-uum for positive energies. For lower densities or highertemperatures, the valence states broaden considerably tothe point of merging with the continuum.It is useful to compare the DOS with that from the T artarus average atom model. In average atom mod-els, the bound orbitals are at specific energies and are δ -distributed in the DOS, no matter how weakly bound.The eigenvalues of the average atom bound spectrum dotend to correspond to bands in the RSGF calculation,but because of the lack of an explicit plasma environ- ment, there is no broadening of these states. The caseof 0.027 g/cm and 100 eV highlights this defect of theaverage atom model, showing series of atomic-like boundstates and multiple continuum resonances, all of whichare completely washed out in the RSGF calculation.The average atom model is missing physics that wouldlead to the broadening of these bound state features seenin RSGF. The missing effect is the electric potential fluc-tuations from atom to atom as well as in time due to theplasma environment. If one knew the statistical distri-bution of electric potential felt by a bound electron, onecould use perturbation theory to approximate the distri-bution of eigenvalue shifts due to the fluctuating poten-tial, which would appear as a broadening of the boundstate.The distribution of electric potentials can be approx-imated using the same pseudoatom MD model used togenerate the RSGF configurations [31]. In that model,the total potential is given by a superposition of spheri-cally symmetric pseudoatom potentials V P A ( r ), V ( x ) = (cid:88) j V P A ( | x − R j | ) (7)As a model for the plasma’s perturbing effect on the aver-age atom model, we consider the deviation of the poten-tial near atom i from the average atom model potential V AA ( r )∆ V i ( r ) = V ( r + R i ) − V AA ( r ) (8)= V P A ( r ) − V AA ( r ) + (cid:88) j (cid:54) = i V P A ( | r + R i − R j | )(9)where r is the position vector measured from the atom’slocation. For r < | R i − R j | , the first term in a multipoleexpansion of this potential near the atom is a constant∆ V i ( r ) ≈ ∆ V i (0)= V P A (0) − V AA (0) + (cid:88) j (cid:54) = i V P A ( | R i − R j | )(10)We take this constant as an approximation to the plasmapotential experienced by the electrons within atom i inthe configuration { R j } . A statistical distribution ofplasma potentials is accumulated over each atom andtime step of an MD simulation. We further assume thatthe potential distribution is centered about the averageatom potential. Treating ∆ V as a random perturbationto the average atom Hamiltonian leads, at first order, toa shift of the average atom eigenvalues by the constant∆ V . The distribution of these shifts leads to a broaden-ing of the average atom DOS, calculated by convolvingthe normalized ∆ V distribution with the average atomDOS.In Fig. 3, the red dashed lines show the result of ap-plying the monopole broadening perturbation theory tothe average atom DOS. The qualitative agreement withRSGF is much improved, with δ -distributed bound statesand sharp continuum resonances broadened into bands ofstates. The model is not quantitative, but still gives rea-sonably realistic valence structure in spite of the crudeapproximations involved.The monopole approximation could be improved ei-ther by going to higher orders in the expansion of ∆ V ( r )or by expanding about a point other than the origin,for instance, each orbital’s mean radius. Either ap-proach would predict more broadening, and this broad-ening would be orbital-dependent rather than constant.For relatively deep valence states, like the 2 s and 2 p fea-tures at 2.7 g/cm and 10 eV, this would likely improveagreement with RSGF. However, for the other conditionsshown in Fig. 3, where the valence structure is closer tothe threshold, the accuracy of the broadening model isprobably limited by perturbation theory rather than thedetails of the plasma potential approximation. This isbecause perturbation theory is likely not convergent fornear-threshold valence states, where the calculated bandsare wider than the spacing between eigenstates. A di-rect re-diagonalization of the average atom Hamiltonianis possible in principle, but to do self-consistently is notvery practical, especially if considering higher multipoleeffects that break the spherical symmetry of the averageatom. IV. CONCLUSIONS AND DISCUSSION
In this work, the real-space Green’s function (RSGF)method has been demonstrated to be an effective tech- nique for solving the Kohn-Sham DFT equations forwarm dense matter. The method is tractable for plasmasat any temperature, since it avoids the calculation of ex-plicit eigenstates that makes orbital-based DFT methodsexpensive at high temperature. RSGF relies on a clusterapproximation, where it is assumed that the positionsof the nuclei far from the center of the cluster do not af-fect the electronic structure near the center. A numericalstudy of the convergence of the pressure with respect tothe cluster size affirms the accuracy of this approxima-tion, even with modest cluster sizes. Comparisons of theRSGF model’s predicted equation of state showed excel-lent agreement with KKR-GF results at high densities,while at low densities we find agreement with the aver-age atom model T artarus. An examination of the densityof states and comparison with the T artarus model high-lighted important broadening physics lacking from theaverage atom model but accounted for in RSGF, even atconditions where the equation of state is in good agree-ment. This led us to propose a simple model for thebroadening of average atom bound states and resonancesbased on the qualitative effect of plasma disorder lack-ing in average atom models. We conclude that RSGF isan accurate and practical method for solving the Kohn-Sham DFT equations in warm dense matter conditionsand is also useful for highlighting shortcomings and pos-sible refinements to the average atom model. ACKNOWLEDGMENTS
This work was performed under the auspices of theUnited States Department of Energy under contract DE-AC52-06NA25396. [1] S. W. Haan, J. D. Lindl, D. A. Callahan, D. S. Clark,J. D. Salmonson, B. A. Hammel, L. J. Atherton, R. C.Cook, M. J. Edwards, S. Glenzer, A. V. Hamza, S. P.Hatchett, M. C. Herrmann, D. E. Hinkel, D. D. Ho,H. Huang, O. S. Jones, J. Kline, G. Kyrala, O. L. Lan-den, B. J. MacGowan, M. M. Marinak, D. D. Meyerhofer,J. L. Milovich, K. A. Moreno, E. I. Moses, D. H. Munro,A. Nikroo, R. E. Olson, K. Peterson, S. M. Pollaine, J. E.Ralph, H. F. Robey, B. K. Spears, P. T. Springer, L. J.Suter, C. A. Thomas, R. P. Town, R. Vesey, S. V. We-ber, H. L. Wilkens, and D. C Wilson. Point design tar-gets, specifications, and requirements for the 2010 igni-tion campaign on the national ignition facility.
Physicsof Plasmas , 18(5):051001, 2011.[2] S. Le Pape, L. F. Berzak Hopkins, L. Divol, A. Pak, E. L.Dewald, S. Bhandarkar, L. R. Bennedetti, T. Bunn, J. Bi-ener, J. Crippen, D. Casey, D. Edgell, D. N. Fittinghoff,M. Gatu-Johnson, C. Goyon, S. Haan, R. Hatarik,M. Havre, D. D-M. Ho, N. Izumi, J. Jaquez, S. F. Khan,G. A. Kyrala, T. Ma, A. J. Mackinnon, A. G. MacPhee,B. J. MacGowan, N. B. Meezan, J. Milovich, M. Millot,P. Michel, S. R. Nagel, A. Nikroo, P. Patel, J. Ralph, J. S. Ross, N. G. Rice, D. Strozzi, M. Stadermann, P. Volegov,C. Yeamans, C. Weber, C. Wild, D. Callahan, and O. A.Hurricane. Fusion energy output greater than the kineticenergy of an imploding shell at the national ignition fa-cility.
Phys. Rev. Lett. , 120:245003, Jun 2018.[3] M. R. Gomez, S. A. Slutz, A. B. Sefkow, D. B. Sinars,K. D. Hahn, S. B. Hansen, E. C. Harding, P. F. Knapp,P. F. Schmit, C. A. Jennings, T. J. Awe, M. Geissel, D. C.Rovang, G. A. Chandler, G. W. Cooper, M. E. Cuneo,A. J. Harvey-Thompson, M. C. Herrmann, M. H. Hess,O. Johns, D. C. Lamppa, M. R. Martin, R. D. McBride,K. J. Peterson, J. L. Porter, G. K. Robertson, G. A.Rochau, C. L. Ruiz, M. E. Savage, I. C. Smith, W. A.Stygar, and R. A. Vesey. Experimental demonstrationof fusion-relevant conditions in magnetized liner inertialfusion.
Phys. Rev. Lett. , 113:155003, Oct 2014.[4] S. H. Glenzer, L. B. Fletcher, E. Galtier, B. Nagler,R. Alonso-Mori, B. Barbrel, S. B. Brown, D. A. Chap-man, Z. Chen, C. B. Curry, F. Fiuza, E. Gamboa,M. Gauthier, D. O. Gericke, A. Gleason, S. Goede,E. Granados, P. Heimann, J. Kim, D. Kraus, M. J.MacDonald, A. J. Mackinnon, R. Mishra, A. Ravasio,
C. Roedel, P. Sperling, W. Schumaker, Y. Y. Tsui, J. Vor-berger, U. Zastrau, A. Fry, W. E. White, J. B. Hasting,and H. J. Lee. Matter under extreme conditions exper-iments at the Linac Coherent Light Source.
Journal ofPhysics B Atomic Molecular Physics , 49(9):092001, May2016.[5] P. M. Kowalski, S. Mazevet, D. Saumon, and M. Challa-combe. Equation of state and optical properties of warmdense helium.
Phys. Rev. B , 76(7):075112, August 2007.[6] Simon Blouin, Nathaniel R. Shaffer, Didier Saumon, andCharles E. Starrett. New Conductive Opacities for WhiteDwarf Envelopes.
Astrophys. J. , 899(1):46, August 2020.[7] Ravit Helled, Guglielmo Mazzola, and Ronald Redmer.Understanding dense hydrogen at planetary conditions.
Nature Reviews Physics , 2(10):562–574, September 2020.[8] Mandy Bethkenhagen, Bastian B. L. Witte, MaximilianSch¨orner, Gerd R¨opke, Tilo D¨oppner, Dominik Kraus,Siegfried H. Glenzer, Philip A. Sterne, and Ronald Red-mer. Carbon ionization at gigabar pressures: An ab initioperspective on astrophysical high-density plasmas.
Phys-ical Review Research , 2(2):023260, June 2020.[9] James E. Bailey, Taisuke Nagayama, Guillaume PascalLoisel, Gregory Alan Rochau, C. Blancard, J. Colgan,Ph. Cosse, G. Faussurier, C. J. Fontes, F. Gilleron, et al.A higher-than-predicted measurement of iron opacity atsolar interior temperatures.
Nature , 517(7532):56–59,2015.[10] T. Nagayama, J. E. Bailey, G. P. Loisel, G. S. Dun-ham, G. A. Rochau, C. Blancard, J. Colgan, Ph. Coss´e,G. Faussurier, C. J. Fontes, F. Gilleron, S. B. Hansen,C. A. Iglesias, I. E. Golovkin, D. P. Kilcrease, J. J. Mac-Farlane, R. C. Mancini, R. M. More, C. Orban, J. C.Pain, M. E. Sherrill, and B. G. Wilson. SystematicStudy of L -Shell Opacity at Stellar Interior Tempera-tures.
Phys. Rev. Lett. , 122(23):235001, June 2019.[11] W. Kohn and L. J. Sham. Self-consistent equations in-cluding exchange and correlation effects.
Phys. Rev. ,140:A1133–A1138, Nov 1965.[12] N. David Mermin. Thermal properties of the inhomoge-neous electron gas.
Phys. Rev. , 137:A1441–A1443, Mar1965.[13] Travis Sjostrom and J´erˆome Daligault. Fast and accuratequantum molecular dynamics of dense plasmas acrosstemperature regimes.
Phys. Rev. Lett. , 113:155006, Oct2014.[14] B. B. L. Witte, L. B. Fletcher, E. Galtier, E. Gam-boa, H. J. Lee, U. Zastrau, R. Redmer, S. H. Glenzer,and P. Sperling. Warm Dense Matter DemonstratingNon-Drude Conductivity from Observations of Nonlin-ear Plasmon Damping.
Phys. Rev. Lett. , 118(22):225001,June 2017.[15] B. B. L. Witte, P. Sperling, M. French, V. Recoules,S. H. Glenzer, and R. Redmer. Observations of non-linearplasmon damping in dense plasmas.
Physics of Plasmas ,25(5):056901, May 2018.[16] A. L. Ankudinov, B. Ravel, J. J. Rehr, and S. D. Conrad-son. Real-space multiple-scattering calculation and inter-pretation of x-ray-absorption near-edge structure.
Phys.Rev. B , 58(12):7565–7576, September 1998.[17] J. J. Rehr and R. C. Albers. Theoretical approachesto x-ray absorption fine structure.
Reviews of ModernPhysics , 72(3):621–654, July 2000.[18] S. I. Zabinsky, J. J. Rehr, A. Ankudinov, R. C. Albers,and M. J. Eller. Multiple-scattering calculations of x-ray- absorption spectra.
Phys. Rev. B , 52(4):2995–3009, July1995.[19] J. J. Rehr, R. C. Albers, and S. I. Zabinsky. High-ordermultiple-scattering calculations of x-ray-absorption finestructure.
Phys. Rev. Lett. , 69(23):3397–3400, December1992.[20] I. A. Abrikosov, S. I. Simak, B. Johansson, A. V. Ruban,and H. L. Skriver. Locally self-consistent green’s functionapproach to the electronic structure problem.
Phys. Rev.B , 56:9319–9334, Oct 1997.[21] Yang Wang, G. M. Stocks, W. A. Shelton, D. M. C.Nicholson, Z. Szotek, and W. M. Temmerman. Order-nmultiple scattering approach to electronic structure cal-culations.
Phys. Rev. Lett. , 75:2867–2870, Oct 1995.[22] H. Ebert, V. Popescu, and D. Ahlers. Fully relativistictheory for magnetic exafs: Formalism and applications.
Phys. Rev. B , 60:7156–7165, Sep 1999.[23] Brian A. Mattern, Gerald T. Seidler, Joshua J. Kas,Joseph I. Pacold, and John J. Rehr. Real-space green’sfunction calculations of compton profiles.
Physical Re-view B , 85(11):115135, 2012.[24] Brian A. Mattern and Gerald T. Seidler. Theoreticaltreatments of the bound-free contribution and experi-mental best practice in X-ray Thomson scattering fromwarm dense matter.
Physics of Plasmas , 20(2):022706,February 2013.[25] O. Peyrusse. Theoretical calculations of K-edge absorp-tion spectra in warm dense Al.
Journal of Physics Con-densed Matter , 20(19):195211, May 2008.[26] J. Korringa. On the calculation of the energy of a blochwave in a metal.
Physica , 13(6-7):392–400, 1947.[27] W. Kohn and N. Rostoker. Solution of the schr¨odingerequation in periodic lattices with an application to metal-lic lithium.
Physical Review , 94(5):1111, 1954.[28] Hubert Ebert, Diemo Koedderitzsch, and Jan Minar.Calculating condensed matter properties using the KKR-green’s function method-recent developments and appli-cations.
Reports on Progress in Physics , 74(9):096501,2011.[29] C. E. Starrett and N. Shaffer. Multiple scattering theoryfor dense plasmas.
Phys. Rev. E , 102:043211, Oct 2020.[30] Jan Zabloudil, Robert Hammerling, Laszlo Szunyogh,and Peter Weinberger.
Electron Scattering in Solid Mat-ter: a theoretical and computational treatise , volume 147.Springer Science & Business Media, 2006.[31] C. E. Starrett, J. Daligault, and D. Saumon. Pseudoatommolecular dynamics.
Phys. Rev. E , 91:013104, Jan 2015.[32] Valentin V. Karasiev, Travis Sjostrom, James Dufty,and S. B. Trickey. Accurate homogeneous electron gasexchange-correlation free energy for local spin-densitycalculations.
Phys. Rev. Lett. , 112:076403, Feb 2014.[33] Aftab Alam and D. D. Johnson. Optimal site-centeredelectronic structure basis set from a displaced-center ex-pansion: Improved results via a priori estimates of saddlepoints in the density.
Phys. Rev. B , 80:125123, Sep 2009.[34] C.E. Starrett, N.M. Gill, T. Sjostrom, and C.W. Greeff.Wide ranging equation of state with Tartarus: A hybridGreen’s function/orbital based average atom code.
Com-puter Physics Communications , 235:50 – 62, 2019.[35] C. E. Starrett. High-temperature electronic struc-ture with the Korringa-Kohn-Rostoker green’s functionmethod.