Sandia National Laboratories is a multimission laboratory managed and operated by National Technology and Engineering Solutions of Sandia LLC, a wholly owned subsidiary of Honeywell International Inc. for the U.S. Department of Energy’s National Nuclear Security Administration under contract DE-NA0003525. SAND2017-6201 C

TheoryappliedtoWarmDenseMatterLukeShulenburger

WhatisWarmDenseMatter?§ WarmDenseMatterisgenerallyassociatedwithstronglycoupledions

(Gii >1)andmoderatelydegenerateelectrons(q ~1)

§ Itistypicallyfoundatthejunctionofsolid,liquid,gas,andplasma.ThecomplicatedinterplayofthephysicalprocessesthatWDMshareswithitsneighborscreatesconsiderabledifficultiesfortheory.

§ q >>1isequivalenttowhereL isthethermaldeBrogliewavelength§ Fermi-Diracstatisticsforelectronicdegreesoffreedomstarttakingon

moreofaMaxwell-Boltzmanncharacter

WhereisWarmDenseMatterfound?

is closely connected with (P > 1 Mbar)

Somehistoryofthefield§ Conferencesdevotedtonon-idealplasmasandtheirstudy

havebeenaroundforalongtime§ ThePhysicsofNon-IdealPlasmasMeetingseriesstartedin1980informer

EastGermany§ TheStronglyCoupledCoulombSystemsmeetingshavealsohadalong-

standingcontingentinterestedinwhatisnowcalledWarmDenseMatter

§ Thefirst“WarmDenseMatter”meetingwasorganizedbyAndrewNg(whocoinedthename)andheldinVancouver,BCin2000.

4

Whatmakesitdifficult?§ Thereisnosmallparameter

§ Plasmaexpansionsing,theplasmaparameter,failforWDM

§ Manydifferentaspectsofthephysicscontributeatacomparablelevelandmustbeincluded§ strongcorrelations§ ionization§ bondformationandbreaking§ complexpressureandtemperaturedependentchemistry

§ Computations/simulationsareoftenquitedemanding§ Massivelyparallelcomputationsarethenorm§ Shortcutsareverytempting

§ Experimentalconditionsareshortlivedandhardtodiagnose§ Whichexperimentalresultsshouldyoubelieve?

5

6

Better EOSPhase diagrams

Time/Path dependence Approach

to phase equilibrium

Transport and mechanical properties

Mixtures

Material structure, Electronic, Ionic

Non-LTE

Electronic Excitation

Heating the Material

Ionic ExcitationCompressing the

Material

ScientificneedsinWarmDenseMatterResearch

Experimentalapproaches

Considersomethingas“simple”asthedeuteriumHugoniot

7Pierre Henri Hugoniot

One of these thingsis not like the others

Theprogresscontinuestothisday

§ Experimentaltechniquescontinuetoimproveinprecisionandaccuracy§ Understandingofmaterial

propertiesbootstrapsitself

§ Startingtodiscriminatethelimitationsofcurrenttheoreticalapproximations

8

Knudson and Desjarlais, PRL 118, 035501 (2017)

Pres

sure

• Equations governing the properties of a material under any conditions are known

• Just need to solve the 3N dimensional partial differential equations• Approximations are necessary for real materials

QuantumCalculationsofferanappealingroute

9

HΨ(r1…rN ) = EΨ(r1…rN )

ååå-

+-

+Ñ

-=¹ Ii iI

I

ji jii

i

rReZ

rre

mH

,

222

21

2ˆ !!!!

• Equations governing the properties of a material under any conditions are known

• Just need to solve the 3N dimensional partial differential equations• Approximations are necessary for real materials

• If we could do this accurately and efficiently, we could calculate any physical property

QuantumCalculationsofferanappealingroute

10

HΨ(r1…rN ) = EΨ(r1…rN )

ååå-

+-

+Ñ

-=¹ Ii iI

I

ji jii

i

rReZ

rre

mH

,

222

21

2ˆ !!!!

Quantumcalculationsarenottrivial• RecastSchrodingerequationasanintegralproblemin3N

dimensions

● Massiveparallelismavailable,eachpointcanbecalculatedindependently

• Poorscaling

11

òò

YY

YY>=<

RRR

RRRR

d

dHH

)()(

)()(ˆ)(ˆ*

*

Quantumcalculationsarenottrivial• RecastSchrodingerequationasanintegralproblemin3N

dimensions

● Massiveparallelismavailable,eachpointcanbecalculatedindependently

• Poorscaling� 3dimensionsperelectron� 20pointsineachdirection� 209 ≈512billionpointsfor3electrons

� 3.8TBjusttostore!

12

òò

YY

YY>=<

RRR

RRRR

d

dHH

)()(

)()(ˆ)(ˆ*

*

DFT– TheMostCommonApproximation§ Threeinsightsunderpinthedevelopmentofthemost

commonlyusedtheory§ PhysicalInsight

§ Wavefunctionisnotanobservablebutthedensityis§ Replacethe3Ndimensionalwavefunctionwiththe3dimensionaldensity

§ Canapproximatekineticenergyanddevelopasensibledensitybysolvingfornoninteracting electronsinaneffectivepotential

§ Areasonableapproximationistomaketheeffectivepotentialasimplefunctionofthedensity

13

N(r) = ∫Ψ(r,r2..rN)dr2���drN

V(r) ∝ 1/r

Ion

DFT– TheMostCommonApproximation§ Threeinsightsunderpinthedevelopmentofthemost

commonlyusedtheory§ PhysicalInsight

§ Wavefunctionisnotanobservablebutthedensityis§ Replacethe3Ndimensionalwavefunctionwiththe3dimensionaldensity

§ Canapproximatekineticenergyanddevelopasensibledensitybysolvingfornoninteracting electronsinaneffectivepotential

§ Areasonableapproximationistomaketheeffectivepotentialasimplefunctionofthedensity§ Thisisthedensityfunctional

14

N(r) = ∫Ψ(r,r2..rN)dr2���drN

V(r) ∝ 1/r

Ion

(Kohn-Sham)DFTinpractice§ Startbychoosingapproximation(functional)§ MakeBorn-Oppenheimerapproximation

§ Chooseachunkofmaterialtostudy(generallyperiodicboundaryconditions)

§ Solveforelectronsinpresenceofstaticions§ Note:Mermin approximationmeansthermalgroundstate

§ Hamiltonianisnowset.Solveforsingleparticlesolutionsinsomebasis(physiciststendtopreferplanewaves,chemistsgaussians)§ Note:allofthesesolutionsneedtobeorthogonalforFermionslike

electrons,soneedO(N3)diagonalizationofamatrix

§ Combinesingleparticlewavefunctionstogetdensityandthusinteractionpotential§ Usethistogetnewapproximationofpotentialanditeratetoself-

consistency 15

WhatdoesDFTallowustocalculate?

§ Formally,wegetE(R)andrelatedquantitieslikeF(R)§ Ifweareintegratingovertheelectrons,thisisgenerally

enough§ Structure,Diffusion…

§ Forelectronicproperties,thingsaremorecomplicated§ Formallyspeaking,needanewtypeoffunctional§ Practically,weoftenuse(abuse)thesingleparticlewavefunctions

§ Conductivity,Opacity…

16

DFTisaverysuccessfultechniqueforstudyingWDM

§ CarefulDFT/QMDcalculationscancomplementexperimentbyprovidingadditionalinformation

§ Thisisespeciallypowerfulwhenexperimentscanvalidateapproximations

17

Root, LNS, Lemke, Dolan, Mattsson and Desjarlais, PRL 115, 198501 (2015)

Shock melting of diamond Phase diagram of MgO

Knudson, Desjarlais and Dolan, Science 322, 1823 (2008)

DFTisnotperfect

§ Scalingwithtemperature§ Memory~T3

§ CPUtime~T4.5

§ Approximationsarenotoriouslydifficulttoimprove

§ Theseshortcomingsarenotjustacademic!

18

Knudson, Desjarlais, Becker, Lemke, Cochrane, Savage, Bliss, Mattsson and

Redmer, Science 348, 1455 (2015)

D2 liquid-liquid phase transition

FRONTIERSINCALCULATINGPROPERTIESOFWDM:EXTENDINGRANGEOFVALIDITY

19

AverageAtomApproximation

§ Athightemperatures,thepresenceofnearbyatomsislessrelevant§ Solveforanisolated“averageatom”in

abackgroundofelectrons§ Selfconsistentlychangeionizationstate

withtemperature§ Fewerelectronson”atom”moreinbackground

§ InadditiontoEOS,ageneralizationtohandlevariousionicstatesinaplasmacanbeusedtocalculatetransportproperties

20

Bailey et al. Nature 517, 56 (2015)

Fe opacity at stellar conditions

AverageAtomApproximation

§ Athightemperatures,thepresenceofnearbyatomsislessrelevant§ Solveforanisolated“averageatom”in

abackgroundofelectrons§ Selfconsistentlychangeionizationstate

withtemperature§ Fewerelectronson”atom”moreinbackground

§ InadditiontoEOS,ageneralizationtohandlevariousionicstatesinaplasmacanbeusedtocalculatetransportproperties

21

Bailey et al. Nature 517, 56 (2015)

Fe opacity at stellar conditions

What about higher densities / lower temperatures?See for example talk by C. Starrett

OrbitalFreeDFT§ Kohn-Shamapproximationisnotonly

approachtoDFT§ Cansolveeverythingwithoutcalculating

singleparticlestates§ Approximationbecomesmuchmore

difficult§ Needtogetkineticenergyandelectronic

entropyfromdensityalone!§ Notoriouslydifficulttogetmoleculesto

bind§ However,thisisanoldtechnique

§ Thomas-Fermiapproximationisperhapsearliestexample

§ Typicallyworkswellathightemperatures

22

SiO2 from OFDFT and KSDFT

Sjostrom and Crockett, PRB 92, 115104 (2015)

OrbitalFreeDFT§ Kohn-Shamapproximationisnotonly

approachtoDFT§ Cansolveeverythingwithoutcalculating

singleparticlestates§ Approximationbecomesmuchmore

difficult§ Needtogetkineticenergyandelectronic

entropyfromdensityalone!§ Notoriouslydifficulttogetmoleculesto

bind§ However,thisisanoldtechnique

§ Thomas-Fermiapproximationisperhapsearliestexample

§ Typicallyworkswellathightemperatures

23

SiO2 from OFDFT and KSDFT

Sjostrom and Crockett, PRB 92, 115104 (2015)

Work continues to improve functionals and implementationSee for example work of E. Carter, T. Sjostrom, S. Trickey and K. Burke

PathIntegralMonteCarlo

§ ReturntodirectsolutionofmanybodySchrodingerequation

§ UseFeynmanformulationtocastasseriesofpathintegrals

§ SampleoverpathintegralsusingMonteCarlotechniques

§ Veryexpensive,potentiallyveryaccurate§ Difficultiesduetosymmetryof

particlesandergodicity(pathsgettangledatlowtemperatureorhighdensity)

24

Hugoniot of Deuterium

Militzer and Ceperley, PRL, 85, 1890 (2000)

PathIntegralMonteCarlo

§ ReturntodirectsolutionofmanybodySchrodingerequation

§ UseFeynmanformulationtocastasseriesofpathintegrals

§ SampleoverpathintegralsusingMonteCarlotechniques

§ Veryexpensive,potentiallyveryaccurate§ Difficultiesduetosymmetryof

particlesandergodicity(pathsgettangledatlowtemperatureorhighdensity)

25

Hugoniot of Deuterium

Militzer and Ceperley, PRL, 85, 1890 (2000)

Work continues to improve approximations and expand validity for WDMSee for example work of Bonitz, Ceperely and talk by B. Militzer

FRONTIERSINCALCULATINGPROPERTIESOFWDM:IMPROVINGAPPROXIMATIONS

26

GroundstateMonteCarloApproaches§ Improvetreatmentof

electronicinteractionbystochasticsamplingofmanybodySchrodingerequation

§ Canbehighlyaccurate,butveryexpensive

§ Inaccuraciesareoftenduetoapproximationsnecessarytoimprovecomputationalcost§ Methodisverywellsuited

tosupercomputersthough,sothisshouldimprove

27

Mazzola and Sorella, PRL 114, 105701 (2015)

BetterDFTfunctionals

§ DFTgets10’softhousandsofcitationsyearly§ Lotsofworkonimproving

functionalsforgroundstate/ambientproperties

§ Functionalsforfinitetemperature§ OlderworkbyDharma-

Wardana andPerrot(1984)§ RecentPIMCcalculationsfor

referencesystemhavespurredwork

§ Trickey groupandBurkegroupbothhavenewapproximations

28

Browne et al. PRL 110, 146405 (2013)

FRONTIERSINCALCULATINGPROPERTIESOFWDM:BEYONDEQUATIONOFSTATE

29

TransportProperties:Kubo-Greenwood

§ CancalculateelectronicandthermalconductivityusingDFT

§ CalculateimaginarypartofdielectricfunctionusingtheKubo-Greenwoodrelation§ Linearresponse§ Matrixelementsofsingle

particlewavefunctions

§ WorkswellinWDMregime§ Issuesifbandgapisnot

closed(Helium?)

30

Desjarlais, Kress, and Collins, Phys. Rev. E 66, 025401 (2002)

Aluminum electrical conductivity

Transportproperties:Energytransfer§ Stoppingafastmovingion§ Understandinghowionsareslowedisessentialtounderstanding

theenergybalanceininertialconfinementfusion§ Asatestproblem,wedragahydrogenionthroughaluminumat

constantvelocityandmeasuretheforceontheion§ Generationofplasmons necessarytocapturetheproperbehavior

31

Born-Oppenheimer TDDFT

Transportproperties:Energytransfer§ Stoppingafastmovingion§ Understandinghowionsareslowedisessentialtounderstanding

theenergybalanceininertialconfinementfusion§ Asatestproblem,wedragahydrogenionthroughaluminumat

constantvelocityandmeasuretheforceontheion§ Generationofplasmons necessarytocapturetheproperbehavior

32

Born-Oppenheimer TDDFT

Transportproperties:Energytransfer§ Stoppingafastmovingion§ Understandinghowionsareslowedisessentialtounderstanding

theenergybalanceininertialconfinementfusion§ Asatestproblem,wedragahydrogenionthroughaluminumat

constantvelocityandmeasuretheforceontheion§ Generationofplasmons necessarytocapturetheproperbehavior

33

Born-Oppenheimer TDDFT

FRONTIERSINCALCULATINGPROPERTIESOFWDM:WHERETOFROMHERE

34

Whatproblemsseemripe?

§ WhiteDwarfs§ Cosmochronology – ageofwhitedwarfsthroughcoolingrate

§ EquationsofstateforC,O,andHeatextremeconditions§ Thermalconductivities,viscosities,diffusioncoefficients

§ Asteroseismology – modelpulsatingwhitedwarf,toinfer§ Totalmassandmasscomposition§ Interiorrotationprofile§ Surfacetemperature§ Structuraldetails

35

SummaryComments

§ WarmDenseMatterresearchisgrowingrapidlyandaddressingimportantproblemsininertialfusion,planetaryscience,andmanyotherfields

§ WarmDenseMatterhasbeenveryslowtoyieldtopurelytheoreticaldescription

§ Advancesinelectronicstructureandmoleculardynamicsmethodshaveplayedanenormousroleinadvancingourunderstandingofwarmdensematter

§ Thisisgoingtobeaveryinterestingweek!36