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Classical Representation of Quantum Systems at Equilibrium

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Title:
Classical Representation of Quantum Systems at Equilibrium
Creator:
Dutta, Sandipan
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[Gainesville, Fla.]
Florida
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University of Florida
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english
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1 online resource (111 p.)

Thesis/Dissertation Information

Degree:
Doctorate ( Ph.D.)
Degree Grantor:
University of Florida
Degree Disciplines:
Physics
Committee Chair:
Dufty, James W
Committee Co-Chair:
Cheng, Hai Ping
Committee Members:
Muttalib, Khandker A
Reitze, David H
Ladd, Anthony J
Graduation Date:
8/10/2013

Subjects

Subjects / Keywords:
Approximation ( jstor )
Chemicals ( jstor )
Correlations ( jstor )
Effective temperature ( jstor )
Eggshells ( jstor )
Low temperature ( jstor )
Particle density ( jstor )
Pseudopotentials ( jstor )
Quantum field theory ( jstor )
Thermodynamics ( jstor )
Physics -- Dissertations, Academic -- UF
dft -- fluids -- hnc -- quantum -- rpa
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bibliography ( marcgt )
theses ( marcgt )
government publication (state, provincial, terriorial, dependent) ( marcgt )
born-digital ( sobekcm )
Electronic Thesis or Dissertation
Physics thesis, Ph.D.

Notes

Abstract:
A quantum system at equilibrium is represented by an effective classical system, chosen to reproduce thermodynamic and structural properties. The motivation is to allow application of classical strong coupling theories and classical simulations like molecular dynamics and Monte Carlo to quantum systems at strong coupling. The correspondence is made at the level of the grand canonical ensembles for the two systems. The effective classical system is defined in terms of an effective temperature, local chemical potential, and pair potential. These are determined formally by requiring the equivalence of the grand potentials and their functional derivatives of the quantum and representative classical systems. The mapping is inverted using the classical density functional theory to solve for these three parameters. Practical forms of these formal solutions are obtained using the classical liquid state theories like hypernetted chain approximation (HNC). The mapping is applied to the ideal Fermi gas is demonstrated and the details of the thermodynamics of the effective system is derived explicitly. As the next application we consider the uniform electron gas and an explicit form for the effective interaction potential is obtained in the weak coupling limit. The pair correlation functions are calculated using the HNC equations and compared with path integral Monte Carlo data and other theoretical models like Perrot Dharma-wardana. Excellent agreement is obtained over a wide range of temperatures and densities. The last application is to the shell structure of harmonically bound charges. We show that in the mean field limit, the quantum effects of degeneracy and diffraction produce shells at very low temperatures. ( en )
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In the series University of Florida Digital Collections.
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Includes vita.
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Includes bibliographical references.
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Description based on online resource; title from PDF title page.
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This bibliographic record is available under the Creative Commons CC0 public domain dedication. The University of Florida Libraries, as creator of this bibliographic record, has waived all rights to it worldwide under copyright law, including all related and neighboring rights, to the extent allowed by law.
Thesis:
Thesis (Ph.D.)--University of Florida, 2013.
Local:
Adviser: Dufty, James W.
Local:
Co-adviser: Cheng, Hai Ping.
Statement of Responsibility:
by Sandipan Dutta.

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Copyright Dutta, Sandipan. Permission granted to the University of Florida to digitize, archive and distribute this item for non-profit research and educational purposes. Any reuse of this item in excess of fair use or other copyright exemptions requires permission of the copyright holder.
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CLASSICALREPRESENTATIONOFQUANTUMSYSTEMSATEQUILIBRIUMBySANDIPANDUTTAADISSERTATIONPRESENTEDTOTHEGRADUATESCHOOLOFTHEUNIVERSITYOFFLORIDAINPARTIALFULFILLMENTOFTHEREQUIREMENTSFORTHEDEGREEOFDOCTOROFPHILOSOPHYUNIVERSITYOFFLORIDA2013

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c2013SandipanDutta 2

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Tomyparents 3

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ACKNOWLEDGMENTS Firstandforemost,IwouldliketothankmyadvisorandmentorProfessorJimDuftywithoutwhoseguidanceandencouragementthisworkwouldnothavebeenpossible.IwouldalsoliketoacknowledgeJefferyWrightonforhelpingmewiththecomputations.Further,IwouldliketothankmycommitteemembersProfessorCheng,ProfessorLadd,ProfessorMuttalibandProfessorReitzefortheirsupport.ThisworkwasmadepossiblebythegrantsfromDOE. 4

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TABLEOFCONTENTS page ACKNOWLEDGMENTS .................................. 4 LISTOFFIGURES ..................................... 7 ABSTRACT ......................................... 9 CHAPTER 1INTRODUCTION ................................... 11 2CORRELATIONSINQUANTUMSYSTEMS .................... 18 2.1QuantumTheoriesForElectronSystems .................. 18 2.1.1FermiLiquidTheory .......................... 18 2.1.2RandomPhaseApproximationWithLocalFieldCorrections .... 19 2.1.3DynamicalMeanFieldTheory ..................... 21 2.1.4HohenbergKohnDensityFunctionalTheory ............. 21 2.2ClassicalMethods:QuantumPotentials ................... 22 2.2.1ClassicalDFT .............................. 25 2.2.2PerrotDharma-wardanaMethod ................... 26 2.3WhyWeChoseClassicalApproachInsteadOfQuantumtheories(OurMotivation) ................................... 27 3THEEFFECTIVECLASSICALSYSTEM ..................... 29 3.1DenitionOfTheRepresentativeClassicalSystem ............. 29 3.1.1ThermodynamicsFromStatisticalMechanics ............ 29 3.1.2ConstructionOfTheEffectiveClassicalSystem:Classical-Quan-tumCorrespondenceConditions ................... 33 3.1.3InversionOfTheCorrespondenceConditions ............ 35 3.1.4HypernettedChainApproximation .................. 38 3.2PeculiarityOfTheThermodynamicsOfTheEffectiveSystem ....... 40 3.3Example-IdealFermiGas .......................... 41 3.4Summary .................................... 49 4UNIFORMELECTRONGAS ............................ 50 4.1ThermodynamicsOfTheEffectiveClassicalSystem ............ 50 4.1.1ClassicalPotentialcc(r) ....................... 52 4.1.2ClassicalEffectiveTemperatureAndChemicalPotential ...... 56 4.2RadialDistributionFunctionAndThermodynamics ............. 57 4.2.1Thermodynamics ............................ 61 4.3KelbgFittingForTheEffectivePotentialcc ................ 62 4.4Summary .................................... 65 5

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5SHELLFORMATIONINNONUNIFORMSYSTEMS ............... 66 5.1ZeroTemperatureClassicalDensity ..................... 66 5.2ShellModels .................................. 67 5.3FiniteTemperatureFormalism ......................... 68 5.3.1ShellFormationInAPolynomialPotential .............. 70 5.4ShellFormationInQuantumSystems .................... 74 5.4.1EffectiveLocalChemicalPotentialForIdealFermiGas ....... 74 5.4.2TheQuantumEffectsOnTheShellFormationInTheMeanFieldTheory .................................. 75 5.4.3NDependenceOfRadiusOfTheTrap ................ 77 5.5Summary .................................... 78 6CONCLUSION .................................... 80 7OTHERAPPLICATIONSOFTHEFORMALISM ................. 83 7.1DensityProleOfCoulombSystemsInAHarmonicTrap ......... 83 7.2SpinPolarizedUniformElectronGas ..................... 83 7.3MagneticSusceptibilityAtFiniteTemperaturesOfTheIdealFermiGas 84 7.4CrystalLatticeSystems ............................ 85 7.52DElectronGas ................................ 86 APPENDIX AEXACTCOUPLEDEQUATIONSFORnc(r)ANDgc(r,r0) ............ 87 BINHOMOGENEOUSIDEALFERMIGAS ..................... 90 CQUANTUMPRESSURE ............................... 94 DSTATICSTRUCTUREFACTORINRPA ...................... 96 EPROPERTIESOFEFFECTIVEINTERACTIONPOTENTIALFORRPASYS-TEMS ......................................... 101 FIDEALFERMIGASINAHARMONICTRAP ................... 104 REFERENCES ....................................... 107 BIOGRAPHICALSKETCH ................................ 111 6

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LISTOFFIGURES Figure page 1-1Thetemperaturedensityplotshowingtheregimeofthewarmdensematter[ 12 ]. .......................................... 12 1-2ThecomparisonbetweenthedensityproleobtainedbyWrightonetal.andthatoftheclassicalMCisshown.ThebridgetermsneedtobeaddedtotheHNCequationstogetgoodagreement[ 46 ]. .................... 16 3-1Thecomparisonoftheexactdensity(asinEquation( 3 ))fortheidealFermigasfor100particlesinaharmonictrapwiththatobtainedusingLDA(Equa-tion( 3 )). ...................................... 44 3-2IdealgasPaulipairpotentialasafunctionofr=r=r0fort=0,0.1,1,10[ 35 ]. 46 3-3Idealgasreducedclassicaltemperaturetc=Tc=TFasafunctionoft=T=TF.AlsoshownistheresultofPDW[ 35 ]. ................... 47 3-4Idealgasdimensionlesschemicalpotentialc=Fasafunctionoft.Alsoshownisthecorrespondingquantumchemicalpotential=F+tln2[ 35 ]. ....... 47 4-1Demonstrationofcrossoverforr4(t,rs,r)toCoulombwitheffectivecou-plingconstant)]TJ /F4 7.97 Tf 6.78 -1.8 Td[(e(t,rs)givenbyEquation( 4 ),forrs=5andt=0.5,1and10[ 48 ].Alsoshownarethecorrespondingresultsforr4PDW(t,rs,r). .... 55 4-2QuantumRPApressurepRPAatt=0asafunctionofrs[ 48 ]. ......... 57 4-3ClassicalreducedtemperatureTc=TFasafunctionoftforrs=0,1,3and4[ 48 ]. .......................................... 58 4-4Dimensionlessclassicalchemicalpotentialc=EFasafunctionoftforrs=1,3,5[ 48 ]. ...................................... 58 4-5Radialdistributionfunctiong(r)forrs=6att=0.5,1,4,8.AlsoshownaretheresultsofPIMC[ 48 ]. .............................. 59 4-6Radialdistributionfunctiong(r)forrs=6att=0.5,1,4,8.AlsoshownaretheresultsofPDW[ 48 ]. ............................... 60 4-7Radialdistributionfunctiong(r)forrs=5att=0.5,1and10.AlsoshownaretheresultsofTanaka-Ichimaru[ 48 ]. ...................... 60 4-8Radialdistributionfunctiong(r)fort=0atrs=1,5,10.AlsoshownareresultsfromPIMCanddiffusionMonteCarlo.ThePIMCanddiffusionMonteCarloplotsareindistinguishable[ 11 48 ]. ..................... 61 4-9Dimensionlessclassicalpressurepc=(nF)asafunctionoftforrs=1,3,5.AlsoshownarethecorrespondingmodiedRPAresults[ 48 ]. .......... 62 7

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4-10ThepaircorrelationfunctioniscalculatedusingthemodiedKelbgpotentialatt=8forrs=1,6,10and40.ComparisonwiththePIMCisalsoshown[ 59 ]. .......................................... 62 4-11ThepaircorrelationfunctioniscalculatedusingthemodiedKelbgpotentialatt=0.5forrs=1,6,10and40.ComparisonwiththePIMCisalsoshown[ 59 ]. 63 4-12ThepaircorrelationfunctioniscalculatedusingthemodiedKelbgpotentialatt=1forrs=1,6,10and40.ComparisonwiththePIMCisalsoshown[ 59 ]. 63 4-13ComparisonofthepaircorrelationfunctionsforthemodiedKelbgpotentialatt=0forrs=1,6,10and40withPIMCatt=0.0625anddiffusionMonteCarloatt=0[ 59 ]. .................................. 64 5-1MeaneldclassicaldensityproleforN=100particlesforvariousvaluesof)]TJ /F1 11.955 Tf 6.77 0 Td[(.ThemeaneldCoulomblimit)]TJ /F2 11.955 Tf 10.1 0 Td[(!1isastepfunction[ 46 ]. ......... 69 5-2ThedensityproleinthemeaneldlimitofCoulombsystemsindifferentpoly-nomialtrappotentialsfor)-277(=40 .......................... 72 5-3Thevariationofthedensityprolewiththecouplingconstant)]TJ /F1 11.955 Tf 10.1 0 Td[(for100parti-clesinthetrap[ 46 ]. ................................. 72 5-4Thevariationofthedensityprolewiththenumberofparticlesinthetrapfor)-277(=100[ 46 ]. ..................................... 73 5-5Thellingoftheshellsandthedependenceofthepopulationineachshellonthetotalnumberofparticlesinthetrapisshown.Alsoshownarethecorre-spondingMCandMDdata[ 46 ]. .......................... 73 5-6Effectiveclassicaltrappotential4cc(r)=)-404(=(cc(r))]TJ /F3 11.955 Tf 12.56 0 Td[(cc(0))=)]TJ /F1 11.955 Tf 10.09 0 Td[(asafunctionofrfort=0.5,10[ 48 ].Alsoshownistheharmonicpotential. .... 75 5-7Diffractionmeaneldapproximatedensityprolefor)-328(=3andt=0.1,0.5,1[ 48 ]. .......................................... 76 5-8Comparisonof)]TJ /F6 11.955 Tf 9.3 0 Td[(c(r)andVK(r)att=0.1,0.27bothcorrespondingto)-351(=3[ 48 ].AlsoshownistheCoulomblimitq2=r. .................. 76 5-9Exchangemeaneldapproximatedensityproleforrs=5andt=0.5,1,2,5,10[ 48 ]. .......................................... 77 8

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AbstractofDissertationPresentedtotheGraduateSchooloftheUniversityofFloridainPartialFulllmentoftheRequirementsfortheDegreeofDoctorofPhilosophyCLASSICALREPRESENTATIONOFQUANTUMSYSTEMSATEQUILIBRIUMBySandipanDuttaAugust2013Chair:JamesWDuftyCochair:Hai-PingChengMajor:PhysicsAquantumsystematequilibriumisrepresentedbyaneffectiveclassicalsystem,chosentoreproducethermodynamicandstructuralproperties.ThemotivationistoallowapplicationofclassicalstrongcouplingtheoriesandclassicalsimulationslikemoleculardynamicsandMonteCarlotoquantumsystemsatstrongcoupling.Thecor-respondenceismadeatthelevelofthegrandcanonicalensemblesforthetwosystems.Theeffectiveclassicalsystemisdenedintermsofaneffectivetemperature,localchemicalpotential,andpairpotential.Thesearedeterminedformallybyrequiringtheequivalenceofthegrandpotentialsandtheirfunctionalderivativesofthequantumandrepresentativeclassicalsystems.Themappingisinvertedusingtheclassicaldensityfunctionaltheorytosolveforthesethreeparameters.Practicalformsoftheseformalsolutionsareobtainedusingtheclassicalliquidstatetheorieslikehypernettedchainapproximation(HNC).ThemappingisappliedtotheidealFermigasisdemonstratedandthedetailsofthethermodynamicsoftheeffectivesystemisderivedexplicitly.Asthenextapplicationweconsidertheuniformelectrongasandanexplicitformfortheeffectiveinteractionpotentialisobtainedintheweakcouplinglimit.ThepaircorrelationfunctionsarecalculatedusingtheHNCequationsandcomparedwithpathintegralMonteCarlodataandothertheoreticalmodelslikePerrotDharma-wardana.Excellentagreementisobtainedoverawiderangeoftemperaturesanddensities.Thelastappli-cationistotheshellstructureofharmonicallyboundcharges.Weshowthatinthemean 9

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eldlimit,thequantumeffectsofdegeneracyanddiffractionproduceshellsatverylowtemperatures. 10

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CHAPTER1INTRODUCTIONTheliteratureonthethermodynamicsofquantumsystemsatzeroandveryhightemperaturesisvastandwelldeveloped.Manyexperimentsandsimulationshavebeenperformedaroundroomtemperatureandveryhightemperatures.SinceroomtemperatureisusuallysmallcomparedtotheFermienergyofmostsystems,whichisintheorderoftenthousandkelvin,thesesystemscanbesuccessfullydescribedbyzerotemperaturetheoriesliketheFermiliquidtheory[ 1 3 ]orthezero-temperaturedensityfunctionaltheory(DFT)[ 1 4 6 ].Ontheotherextreme,atveryhightemperatures,weakcouplingtheorieslikeDebye-Huckelarequitesuccessful.Veryrecentlyanewclassofquantumsystemscalledwarmdensematter(complexion-electronsystems)[ 7 10 ],hasemergedforwhichthetraditionalzeroandhightemperaturetheoriesfail.Althoughtheionsaretypicallysemi-classical,theelectronscanhavemoderatetostrongcouplinganddegeneracy.ThisisroughlytheregimeatwhichthetemperaturesarecomparabletotheFermitemperatureandthedensitiesareclosetosolid.ThetypicaltemperaturesanddensitiesforwarmdensematterareshownbelowinFigure 1-1 .AsofnowwehaveahostofexperimentaldataandsimulationmethodslikethepathintegralMonteCarlo(PIMC)[ 11 ],butwelackaneffectivetheory.Thebasicproblemliesintheconditionsforwhichboththestrongcouplingandthestrongquantumeffectsoccurtogetheratnitetemperatures.Newtheoreticalideasandmethodsareneedednowtoexplainthephysicsofsomeoftheinterestingphenomenaassociatedwithsuchsystems.Thegeneralobjectivehereistoaddressthisproblemwithanewmethodbasedonexploitingstrongcouplingclassicalmany-bodymethodsforapplicationtothesequantumsystems.Thezerotemperaturetheoriesaredifculttoextendtonitetemperatures.TheFermiliquidtheoryisbasedonthepicturethattheenergylevelsoftheinteractingsystemcanbeadiabaticallymodiedintotheweaklyinteractingsystem[ 1 3 ],whichisusuallytrueonlyatverylowtemperatures.Thequasi-particles,onwhichtheFermiliquid 11

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Figure1-1. Thetemperaturedensityplotshowingtheregimeofthewarmdensematter[ 12 ]. theoryisbased,haveshortlifetimesatnitetemperaturesandthisiswherethetheorybreaksdown.AnothersuccessfulzerotemperaturetheoryistheKohnShamformulationofdensityfunctionaltheory[ 1 4 6 ],whichisbasedonthellingupofthelowestenergyKohn-Shamorbitalsatzerotemperature.Atnitetemperaturesmoreandmoreoftheseorbitalsarelledandthecalculationsbecomeincreasinglydifcult.Atmoderatelyhightemperaturesthepracticalcalculationsdonotconverge.Furthermore,theformoftheexchange-correlationpartofthefreeenergyisnotknownatnitetemperatures,hencetheorbitalsdonotcapturethephysicsaccurately.Othermethodslikethelocaleldcorrectiontheories[ 1 13 ],whichaddscorrelationstoweaklycoupledtheoriesliketherandomphaseapproximation(RPA),failatthemetallicdensitiesandlowtemperatureswheretheygivenegativepaircorrelationsatshortdistances.Dynamicalmeaneldtheoryassumesrstthattheproblemofinterestcanberepresented(approximately)byalatticemodel[ 14 18 ].Thislatticesystemisthenmappedexactlyontoanimpurityproblemthatcanbesolvedapproximately.Itisnotclearhowthisapproachcanbeappliedtothedisordered,uid-likestatesofWDM. 12

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Acompletelydifferentapproachistheattempttoapplyeffectivestrongcouplingmethodsofclassicalstatisticalmechanicstoquantumsystemsusingeffectivequan-tumpotentials[ 19 33 ].Thequantumpotentialsaredenedtoincorporateselectedquantumeffectsintotheclassicalsystems.MolecularDynamics(MD)isverysuccessfulindescribingthedynamicsandthecorrelationsforclassicalsystems[ 32 ].MDisalsobeusedwiththequantumpotentialstostudystrongcorrelationsinquantumsystems.MolecularDynamicsandclassicalMonteCarlomethodsarefarlesscomputationallyintensivethanquantumsimulationslikepathintegralMonteCarlo(PIMC).Inaddition,thereexistsalargeliteratureinliquidstatetheorylikehypernettedchainapproximation(HNC)andPercusYevick(PY)whichsuccessfullydescribethestrongcorrelationsinclassicalsystems[ 32 ].Theseequationsareeasytosolvenumerically,whichledustoadoptthemodiedclassicalmethodsinsteadofthetraditionalquantumtheories.RecentlyPerrotandDharma-wardana(PDW)introducedaneffectivetemperatureaswellasaquantumpotentialtoprovideamodelthatcandescribetheequilibriumpropertiesofhydrogenandtheuniformelectrongasforawiderangeoftemperaturesanddensities[ 29 33 34 ].Theysplittheireffectivepotentialfortheuniformelectrongasintoanoninteractingterm,alsocalledthePaulipotential,andaregularizedCoulombterm,forwhichtheyusetheDeustchpotential.Writingthepotentialinthiswayimpliesthattheexactidealgasexchangeeffectsarerecoveredinthelimittheinteractionsarezero.TheyintroducedtheconceptoftheeffectivetemperatureTc=p T2+T20whichremainsniteatT=0.ThequantityT0isdeterminedbyequatingtheexchange-correlationenergyofthequantumsystematT=0obtainedfromsimulationdatatotheexcessenergyoftheclassicalsystems.ThiseffectivetemperatureTccapturesadditionalquantummany-bodyeffectsbeyondthoseoftheeffectivepairpotential.Theirotherideawastousetheireffectivepotentialintheclassicalliquidstatetheorylikethehypernettedchainapproximation(HNC)topredictthepaircorrelationfunctionsforthe 13

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uniformelectrongasatequilibrium.Theirphenomenologicalworkprovidedanadditionalmotivationformoresystematicwork.Thegeneralobjectiveaddressedinthisworkistodevelopasystematictheorytoexplainthethermodynamicsofthestronglycorrelatedquantumplasmasthatwillworkforawiderangeoftemperaturesanddensities.Ourapproachistoconstructaneffectiveclassicalsystemthatwillrepresentthethermodynamicsofthequantumsystemwewanttodescribe[ 35 ].Theparticlesinthequantumsysteminteractthroughapotential(r,r0)inthepresenceofanexternalpotentialext(r).InthegrandensembletherelevantthermodynamicvariablesarethetemperatureTandchemicalpotential.Theeffectiveclassicalsystemhasthecorrespondingeffectivepotentialsc(r,r0)andc,ext(r)attheeffectivethermodynamicvariablesTcandcrespectively.Theunknowneffectivepotentialsandthermodynamicvariablesfortheeffectivesystemaredeterminedbymappingthethermodynamicsoftheclassicalsystemtothatofthequantumsystem.Sincewehavetodeterminethreeunknownparametersweneedthreeequivalentconditions.Wesetthepressure,densityandpaircorrelationfunctionsforthequantumandtheclassicalsystemequaltoeachother.Tosolvefortheeffectiveparameters,weinvertthemappingdenedabove.Theinversionisdonewithintheclassicaldensityfunctionaltheorywhichrelatestheexternalpotentialandtheinteractionpotentialtothedensitiesandthepaircorrelationfunctionsrespectively[ 44 45 ].Exactinversionisverydifcultsincethatwouldinvolvethesolutionoftheclassicalmany-bodyproblem.HoweverwithinapproximationsliketheintegralequationsinliquidstatetheorylikeHNCandPercus-Yevickthiscanbedonetoagoodapproximation[ 32 ].Theeffectivetemperatureisdeterminedfromthevirialequationforpressureandtheequivalenceconditionsofthepressures.Inthiswaywehavetheeffectivepotentialsandtheclassicalthermodynamicalvariablesintermsofthequantumquantities:pressure,densityandpaircorrelationfunction.Itseemslikecalculatingthepressure,densityandpaircorrelationsthemselves 14

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wouldinvolvesolvingthefullquantummanybodyproblemandhencelittleprogresshasbeenmade.Howeverthereexistsomeknownlimitsfromwhichtheexactexpressionsforthesequantitiescanbeevaluatedasapproximations.Theeffectiveparameters,calculatedwiththeseapproximatequantumquantities,arethenputintosomestrongcouplingclassicaltheoriestoobtainthedesiredthermodynamicalquantities.Thusthecorrelationsintheresultantquantitiesareincorporatedintwosteps:thequantumcorrelationsareincorporatedintheeffectiveparametersinsomesimplequantumapproximation,suchasweakcouplingRPA,andthenstrongercorrelationsareobtainedthroughtheevaluationofpropertiesbyachosenstrongcouplingclassicalmethodlikeHNCequationsorMDsimulations.AsarstapplicationweconsidertheidealFermigas.AlthoughtheidealFermigasisasimplenon-interactingsystem,thecorrespondingeffectiveclassicalsystemiscomplicatedbecauseoftheinteractionscomingfromthePauliexclusionstatistics.TheeffectivetemperaturestaysnitebecauseoftheniteenergiesoftheFermionsatzerotemperatures.SomeresultsfortheaverageinternalenergyandtheentropywhicharetruefortheidealFermisystemsisrecoveredforitsclassicalcounterpart.Thenextapplicationistotheuniformelectrongasorjellium.TheeffectivepotentialisconstructedtoincludetheexchangeeffectsthroughthePaulipotentialandtheCoulombeffectsthroughtheweakcouplingRPAlimit.ThepaircorrelationfunctionsarethencalculatedwiththiseffectivepotentialusingtheHNCequations.TheagreementbetweentheseresultsandthequantumsimulationdatafromPIMCandPDWmodelisquitegoodoverawiderangeoftemperaturesanddensities.Theseresultsarenextappliedtothechargesconnedinaharmonictrap.Suchsystemsareimportantformodelinglasercooledionsinaharmonictrapandelectronsinquantumdots.TheHNCequationsweresuccessfullyusedbyWrightonetal.[ 46 ]topredicttheshellformationinclassicalCoulombsystemsinaharmonictrap.TheirresultsagreedquitewellwiththeclassicalMCsimulationsasshownbelowinFigure 1-2 andprovidedmotivationto 15

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extendourworktoconnedsystems.ForclassicalCoulombsystemsinaharmonictraponlythestrongcorrelationsproduceshells.Consequentlywedonotgetanyshellstructureinthemeaneldlimitforanyvalueofthecouplingconstant[ 46 ].Hereweexplorethepossibleoriginsofshellformationduetothequantumeffectsofdiffractionanddegeneracy.WeusetheeffectivepotentialstoincludeappropriatequantumeffectsandsolveforthedensityusingtheHNCequations.Evenatthemeaneldlevelwegetshellsatverylowtemperaturesanddensities(largecouplingconstant). Figure1-2. ThecomparisonbetweenthedensityproleobtainedbyWrightonetal.andthatoftheclassicalMCisshown.ThebridgetermsneedtobeaddedtotheHNCequationstogetgoodagreement[ 46 ]. TherestoftheChaptersarearrangedinthefollowingway: 1. InChapter 2 weexploreinfurtherdetailsomeoftheavailabletheoriesthattreatstrongcorrelationsinboththeclassicalandquantumsystems.FirstwelookattheFermiliquidtheory,itscentralconcepts,formalismandlimitations.TheRPAwiththelocaleldcorrectionsisbrieydiscussedinparticulartheearliestone,theSTLSscheme.Webrieylookintothedensityfunctionaltheory(DFT)forbothclassicalandquantumasourmodelisbasedontheseconcepts.Lastlywelookintothehistoryanddevelopmentofquantumpotentialsandthemostsuccessfulapplicationsofthisapproach.WediscussindetailtherecentclassicaltheorybyPerrotandDharma-wardana. 2. Chapter 3 setsupthetheoreticalframeworkonwhichtherestofthisworkisbuilt.Weconstructaneffectiveclassicalsystemthatdescribesthethermodynamicsofagivenquantumsystem.Wedenetheeffectivepotentialsandeffectivethermodynamicvariablesfortheeffectiveclassicalsystembyequatingthethreethermodynamicquantitiesoftheclassicalandthequantumsystem.Themapis 16

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invertedusingtheclassicaldensityfunctionaltheoryandthevirialequation.Thepeculiarityofthethermodynamicsoftheeffectivesystemisdiscussed.FinallytheclassicalmappingisappliedtotheidealFermigas. 3. InChapter 4 weexploretheeffectivesystemfortheuniformelectrongasorjellium.FortheuniformelectrongasweseparatetheinteractionpotentialintoanidealgasandaCoulombterm,whichisdeterminedusingweakcouplingRPA.WeusetheHNCequationstocalculatethepaircorrelationfunctions.WecomparethesepaircorrelationswiththoseofthePerrot-DharmawardanamodelandthesimulationdatafromPIMC.Theregimeofthevalidityofthismodelisbrieydiscussed.Otherthermodynamicquantitieslikethepressureandchemicalpotentialarealsopredictedusingthismodelandcomparedwithotherdata. 4. InChapter 5 ,rsttheexistingclassicaltheoreticalmodelsforshellformationarebrieyreviewed.Theclassicalmapisthengeneralizedtotheconnedquantumsystems.Thedensityprolefortheeffectivesystemiscalculatedandthequantumeffectsofdiffractionanddegeneracyontheshellformationisdiscussedinthemeaneldlimit.Inthemeaneldlimit,theclassicalCoulombsystemsdonotproduceshellsatanyvalueofthecouplingconstantinaharmonictrap.However,weshowthatquantumdiffractionandexchangeeffectsleadtoshellformationinthislimit. 5. InChapter 7 ,webrieymentionsomepossibleapplicationsofthismaptothecorrelatedquantumsystemslikethespinpolarizedelectrongasandthelatticesystems.Thecompleteresultsforthesesystemshavenotbeenfullyworkedoutwithourmapyet. 17

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CHAPTER2CORRELATIONSINQUANTUMSYSTEMSThisChapterisdividedintotwosections.Intherstsection 2.1 webrieyreviewthequantumtheoriesliketheFermiliquidtheory,theRPAwiththelocaleldcorrections,thedensityfunctionaltheory(DFT)andthedynamicalmeaneldtheory(DMFT).Wediscusshowthecorrelationsandmany-bodyeffectsarebuiltintothesetheories.Althoughthesetheoriesareverysuccessfulattheirrespectivedomains,inthewarmdensematterconditionswhereweareinterestedintheyfail.Inthesecondsection 2.2 welookattheclassicalmethodtodescribethequantumcorrelationsusingthequantumpotentials.ThequantumpotentialsareusedinclassicalstrongcouplingtheorieslikeclassicalDFT,liquidstatetheories,MolecularDynamicsandclassicalMonteCarlosimulationstodescribecorrelationsinquantumsystems.Therecentsuccessofthesemi-classicalformalismbyPerrotandDharma-wardanaforuniformquantumsystemsandtheclassicalstronglycoupledliquidstateformalismbyWrightonetal.providedmotivationtolooktoclassicalmethodsforourwork.WediscussbrieythePerrotDharma-wardanamodel. 2.1QuantumTheoriesForElectronSystemsInthissectionwebrieylookintosomeofthequantumtheoriesthatarecommonlyusedtostudyelectronsystems. 2.1.1FermiLiquidTheoryFermiLiquidtheoryiswidelyusedtostudycorrelationsinFermisystemsatverylowtemperatures[ 1 3 ].Westartwithanon-interactingsystemandthenturnontheinteractionsadiabatically.Incaseofnoenergylevelcrossing,thisinteractingsystemwillhaveasetofeigenvalueswhichisinone-to-onecorrespondencewiththoseofthenon-interactingsystem.Thusthetheorymapsstronglycoupledsystemstoweaklycoupledsystems,withthestrongcorrelationsabsorbedintotheeffectivemassesandotherparametersofthequasi-particles.Thesharplydenedsingleparticleenergy 18

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statesgetbroadenedbyswitchingontheinteractions,thewidthofwhicharegivenbytheinverselifetimeofthequasi-particles.TheFermiliquidtheorysuccessfullydescribesstronglycorrelatedquantumsystemsatverylowtemperatures.Howeveritbreaksdownatnitetemperaturesduetotworeasons:i)thehigherenergystatesgetlledandtheFermisurfaceisnotwelldenedii)thequasi-particlesbecomeveryunstable. 2.1.2RandomPhaseApproximationWithLocalFieldCorrectionsTheresponsefunctionsarewidelyusedtostudythecollectiveandmany-particleexcitationsandtheirassociatedequilibriumstructurefactors,inandaroundtheequilib-rium[ 1 ].Theuctuationdissipationtheorem[ 37 38 ]relatestheresponsefunctionstothestructurefactor,fromwhichtheotherthermodynamicquantitiesarecalculated.Thesimplestmethodfortheresponsefunctioncalculationistherandomphaseapproxima-tion(RPA)whichholdsforweaklycoupledsystems[ 51 ].ToincludestrongcorrelationsthelocaleldcorrectionsareaddedtotheRPA.Whenatestchargemovesthroughanelectrongas,itisnotonlyinuencedbytheexternalpotentialextbutalsobythedensityuctuationsduetothemotionofthetestcharge.Hencetheparticleseesaneffectivepotential:Rdr0ext(r0,!)=(r,r0,!),i.e.theexternalpotentialmodiedbythemedium.TheFouriertransformofthedielectricfunction,intheeffectivepotentialforanuniformelectrongasisgivenby:(k,!)=1)]TJ /F3 11.955 Tf 11.96 0 Td[((k)nn(q,!) (2)wherennistheproperdensity-densityresponsefunctionandtheinteractionpotential.Undertherandomphaseapproximation[ 1 ],wekeeponlythenon-interactingpartoftheproperresponsefunctionalsocalledtheLindhardfunction.FurtherdetailsaboutthestructurefactorandotherthermodynamicquantitiesintheRPAapproximationaregivenintheAppendix D .AsaresultRPAworksforweaklyinteractingsystems.ThesituationisremediedbyintroducingthelocaleldcorrelationsG(q,!)toaddthemany-bodyeffectsmissinginRPAduetoneglectingthenon-idealgasterms.Thisisaccomplished 19

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byreplacingtheinteractionpotential(q)ofthesystemby(q)(1)]TJ /F6 11.955 Tf 12.36 0 Td[(G(q,!))inalltheRPAcomputations.ThisnotonlypreservesthesimpleformoftheRPAbutatthesametimeaddstheexchangeandcorrelationeffects.Thelocaleldcorrectionsarerelatedtotheresponsefunctionsbytheidentity:(q)G(q,!)=1 (q,!))]TJ /F5 11.955 Tf 31 8.09 Td[(1 0(q,!) (2)wherethe0isthenon-interactingresponsefunction.TherstattempttoestimatetheformoftheeldcorrectionswasbyHubbard[ 40 ].HeassumedafrequencyindependentformforGforthesamespins:G(q)=q2 q2+k2F (2)andG=0fordifferentspins.Theeldcorrectionshavesomeexactanalyticallimits,suchasq!0,G(q,!)!0andq!1,G(q,!)!1.AmongtheearliestmethodstocalculatetheeldtechniquesistheSTLSscheme[ 13 ].InSTLStheeldcorrectionsarecalculatedselfconsistentlyfromthestaticstructurefactorandtheuctuationdissipationtheorematzerotemperature.S(q)=)]TJ /F12 11.955 Tf 14.04 8.09 Td[(~ nZd!=nn(q,!)nn(q,!)=0(q,!) 1)]TJ /F6 11.955 Tf 11.95 0 Td[(vq[1)]TJ /F6 11.955 Tf 11.95 0 Td[(G(q,!)]0(q,!)G(q)=)]TJ /F5 11.955 Tf 10.59 8.09 Td[(1 nZdq0 (2)3q.q0 q2vq0 vq[S(q0)]TJ /F7 11.955 Tf 11.96 0 Td[(q))]TJ /F5 11.955 Tf 11.96 0 Td[(1] (2)WhiletheSTLSimprovesthepaircorrelationfunctionsfromtheRPA,theystillgonegativeclosetotheoriginatmetallicdensities.ThecorrelationenergyisingoodagreementwithMonteCarlosimulationsandexhibitthecorrectr)]TJ /F9 7.97 Tf 6.58 0 Td[(1sbehavioratlargevaluesofrs.Theplasmondispersionrelationwhichisobtainedfromthepolesofthedensity-densityresponsefunctiongoesnegativeatmetallicdensities.Alsothecompressibilitysumruleisviolatedatthoseregimes.ThezerotemperatureEquations 20

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( 2 )havebeenextendedtonitetemperaturesbyTanakaandIchimaru[ 39 41 ].TheonlydifferenceatnitetemperaturesisthestaticstructurefactorequationgetsmodiedbyaBosedistributionterm:S(q)=)]TJ /F12 11.955 Tf 14.04 8.08 Td[(~ nZd!coth(~!)=nn(q,!) (2)ThenitetemperatureSTLSagainbreaksdownatmetallicdensitiesandsuffersthesameproblemastheSTLS[ 39 ]. 2.1.3DynamicalMeanFieldTheoryDynamicalmeaneldtheory(DMFT)isusedtostudystronglycorrelatedlatticesystems[ 14 17 ].TheDMFTassociatesasingle-siteeffectivedynamicsthroughalocalGreen'sfunctionG0(i!n)whichdoesnotdependonthemomentum.G0isrelatedtotheon-siteGreen'sfunctionG(i!n),whichiscalculatedlater,throughG0(i!n)=i!n++G(1!n))]TJ /F9 7.97 Tf 6.59 0 Td[(1)]TJ /F6 11.955 Tf 11.95 0 Td[(R[G(i!n)]] (2)AHubbardHamiltonian[ 42 ]andanimaginary-timeactionfunctionalSedenedforthelatticesystem.ThisformalismissolvedbymappingittoanAndersonimpuritymodel(AM)[ 18 ].DMFTcanbeusedtodeterminethestabilityoflongrangedsystems,theunderstandingtheHubbardmodelandtheMottinsulatortransitionandhightemperaturesuperconductors.Howeveritisnotclearhowthistheorycanbeappliedtothewarmdensemattersystems. 2.1.4HohenbergKohnDensityFunctionalTheoryThethermodynamicsofamany-particlequantumsystemisdeterminedfromthegrandpotentialthrough()=)]TJ /F8 7.97 Tf 15.86 14.95 Td[(1XN=0zNDNje)]TJ /F14 7.97 Tf 6.59 0 Td[((cKN+bV+Rdrbn(r)ext(r)jNE (2)whereN,KN,VandextaretheN-particlemany-bodywavefunction,kinetic,inter-actionandtheexternalpotentialrespectively.TheHohenberg-Kohn-Mermintheorem 21

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providesanalternatedescriptionofthethermodynamicsintermsoftheequilibriumdensityofthesystem[ 4 6 ].Thetheoremclaimsthatthegroundstatedensityuniquelydeterminestheexternalpotentialthatgivesrisetoit[ 43 ].Thetheoremhasbeenex-tendedtonitetemperaturesbyMermin[ 6 ].Theproofofthetheoremisbasedonminimizingthegrandpotentialwithrespecttothedensitymatrix.Thegrandpotentialfunctionalisgivenby:[]=)]TJ /F6 11.955 Tf 9.3 0 Td[(Tr^^H)]TJ /F3 11.955 Tf 11.96 0 Td[(^N+ln^ (2)Thegrandpotentialisgivenby:=)]TJ /F9 7.97 Tf 10.86 4.71 Td[(1 lnTrexp()]TJ /F3 11.955 Tf 9.29 0 Td[((^H)]TJ /F3 11.955 Tf 11.95 0 Td[(^N))andtheequilibriumdensitymatrixby:^0=exp()]TJ /F3 11.955 Tf 9.3 0 Td[((^H)]TJ /F3 11.955 Tf 12 0 Td[(^N))=Trexp()]TJ /F3 11.955 Tf 9.3 0 Td[((^H)]TJ /F3 11.955 Tf 11.96 0 Td[(^N)).TheMermintheoremstatesthat[^]>[^0]foralldensitymatrices^.ThepracticalimplementationofMermin'stheoremhoweverisverycomplicatedasitinvolvesthesolutionoftheoperatorequations.KohnandShamdevelopedanelegantrepresentationoftheDFTwhichgivestheexactresultatthelevelofanon-interactingsystem[ 5 ].TheirapproachinvolvessolvingsingleparticleorbitalequationswithKohn-ShampotentialsandprovidesapracticalimplementationoftheDFTtheoryatzerotemperature.KohnandShamwrotetheirpotentialasaHartreepartandtherestcalledtheexchange-correlationpart.Howevertheexchange-correlationpotentialisunknown.TheKohn-Shamapproachisnotpracticalatnitetemperaturessincethecomputationsdonotconverge. 2.2ClassicalMethods:QuantumPotentialsInthissectionwelookintovarioustechniquestoincorporatequantumeffects,inparticularquantumexchangeanddiffraction,intoclassicalsystems.Thisisdoneusingthequantumpotentials.Thequantumpotentialsarethepotentialsusedinclassicalsystemsthatincorporatequantumeffectsbyappropriateparameterizationorformsforthesepotentials.ForexampleaquantumpotentialreplacingtheclassicalCoulombpotentialisaDeutschpotential[ 21 ]thatregularizestheCoulombpotentialattheorigin, 22

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removingtheclassicalsingularityduetothediffractioneffects.Theparametersinthequantumpotentialsaredeterminedusingsimulationdataorexactknownlimitsforthequantumsystems.Intherestofthesectionwelookintovariouswaystobuildthequantumpotentials.ThesepotentialsarethenputintosomeclassicalstrongcouplingtheoriesliketheclassicalDFT,classicalliquidstatetheoriesandtheclassicalsimulationstostrongcorrelations.TheeffectivepotentialeisacombinationofNparticlepotentialseN,whicharedenedintermsoftheNparticleSlatersumsWNforthequantumsystemsby:1 N!NZdr(N)WN(r(N))=Trhexp()]TJ /F3 11.955 Tf 9.3 0 Td[(^H)ieN(r(N))=)]TJ /F6 11.955 Tf 9.3 0 Td[(ln(WN(r(N)))e=NXi
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whereijisthetwoparticledensitymatrixincoordinaterepresentation.Intheweakcouplinglimitthediagonalelementofthedensitymatrixcanbecalcu-latedexactly:4K(xij)=qiqj ijxij1)]TJ /F6 11.955 Tf 11.95 0 Td[(exp()]TJ /F6 11.955 Tf 9.3 0 Td[(x2ij)+p xij(1)]TJ /F6 11.955 Tf 11.95 0 Td[(erf(xij)) (2)whichiscalledtheKelbgpotential,wherexij=rij=ij.TheKelbgformcanbegeneral-izedbyaddinganextraparameter,ijtogivesomeeffectsoftheboundstates.AddingthisextraparameterpreservestherstderivativeoftheKelbgpotentialbutatthesametimecontrolstheheightofthepotentialattheorigin.Todeterminetheparameterij,theeffectivepotentialisdenedas:exp()]TJ /F3 11.955 Tf 9.3 0 Td[(ij)Sij,whereSijisthebinarySlatersumofparticlesiandj.Fromthisweobtain:ij=)]TJ 10.49 16.7 Td[(p ijqiqj ln[Sij(rij=0,)] (2)Inparticulartheelectron-electronSlatersumattheoriginisgivenby:See(ree=0,)=2p eeZ10exp()]TJ /F6 11.955 Tf 9.3 0 Td[(x2)xdx 1)]TJ /F6 11.955 Tf 11.96 0 Td[(exp()]TJ /F3 11.955 Tf 9.3 0 Td[(ee=x) (2)whereee=qiqj=ij.WiththisnewparameterijthemodiedKelbglookslike:4K(xij)=qiqj ijxij1)]TJ /F6 11.955 Tf 11.95 0 Td[(exp()]TJ /F6 11.955 Tf 9.3 0 Td[(x2ij)+p xij ij(1)]TJ /F6 11.955 Tf 11.96 0 Td[(erf(xijij)) (2)UsingthispotentialFilinov[ 27 28 ]couldexplaintheformationofhydrogenatomsusingclassicalstatisticalmechanics.AsdiscussedabovethequantumpotentialsusedinMDorclassicalMCsimulationscanbeusedtodescribestrongcorrelationsforquantumsystems.Theclassicaldensityfunctionaltheoryisanothertheoreticaltooltodothesame.SinceourformalismisbasedonclassicalDFTwewilldiscussthistheoryindetailinthesubsection 2.2.1 24

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2.2.1ClassicalDFTJustlikethequantumDFT,theclassicalDFTprovidesaformalismtoincludestrongcorrelationsinclassicalsystems[ 44 45 ].ThereasonforthediscussionoftheclassicalDFTinthisChapteristobroadenthescopeoftheclassicalmethodsofthequantumpotentialsandthePDWformalism.ThissectionprovidesaquickoverviewoftheclassicalDFTthatwillbediscussedindetailinChapter 3 .ThequantumeffectswillbeincorporatedthroughtheappropriatethermodynamicquantitiesintheclassicalDFTequations.InclassicalDFTthegrandpotentialfunctionaliswrittenas:[n;ext]=F[n]+Zdrn(r)(ext(r))]TJ /F3 11.955 Tf 11.96 0 Td[() (2)whereF[n]=P1N=0R^KN+^VN+kbTln(N!hDN)+kBTlnf[ext[n]]dpNdqNinDdimensions.TheEuler-Lagrangeequationis:F[n] n(r)=)]TJ /F3 11.955 Tf 11.95 0 Td[(ext(r) (2)NexttheF[n]iswrittenastheidealgaspartFidandtheexcesspartFex[ 45 ].ItcanbeshownthatFid=n(r)=ln(n(r)D).TheNthordercorrelationfunctionisdenedas:cN(r1,r2,...,rN)=)]TJ /F3 11.955 Tf 45.41 8.09 Td[(NFex[n] n(rN)n(rN)]TJ /F9 7.97 Tf 6.58 0 Td[(1)...n(r1) (2)ThesecondordercorrelationfunctioniscalledthedirectcorrelationfunctionandisrelatedtothepaircorrelationfunctiongthroughtheOrnsteinZernikeequation:g(r1,r2))]TJ /F5 11.955 Tf 11.96 0 Td[(1=c2(r1,r2)+Zdr3(g(r1,r3))]TJ /F5 11.955 Tf 11.95 0 Td[(1)n(r3)c2(r3,r2) (2) 25

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Usingtheaboveexpressionstheexcessfreeenergyatdensityn(r)canbewrittenintermsofthatatdensityn0(r)as:Fex[n]=Fex[n0]+Zdr1[)]TJ /F6 11.955 Tf 11.95 0 Td[(ln(n0(r1)))]TJ /F3 11.955 Tf 11.95 0 Td[(ext(r1;n0)(n(r1)]TJ /F6 11.955 Tf 11.96 0 Td[(n0(r1))))]TJ /F5 11.955 Tf 10.49 8.09 Td[(1 2Z10dZ0d0Zdr1dr2c2(r1,r2;n0)@n0(r1) @0@n0(r2) @0 (2)wheren(r)=(1)]TJ /F3 11.955 Tf 12.58 0 Td[()n0(r)+n(r).Theaboveexpressionforexcessfreeenergyisstilldifculttocalculatebecauseofthenon-localdependenceondensity.Inmostcasesthereferencedensityischosentobeuniformdensityn0(r)=n0andalsothenonlocaldensityisreplacedbythelocaldensity.Onesuchtheories,Ramakrishnan-Yussouff,canpredictthefreezingofliquids.Thereplacementofthenon-localdensitydependencebythecorrespondinglocaloneisalsocalledhypernettedchainapproximation.FurtherdetailsisgiveninAppendix A .TheclassicalDFTprovidesausefulformalismtoincludemany-bodyeffectsatnitetemperature.ThecalculationsareeasieraswedonothavetosolvetheeigenvalueequationasKohn-ShamDFT. 2.2.2PerrotDharma-wardanaMethodPerrotandDharma-wardanaprovidedacomputationallysimpleandnovelmethodforcalculatingthepaircorrelationfunctionsandotherthermodynamicquantitiesfortheuniformelectrongas(UEG)usingonlytheinformationaboutthecorrelationenergyEcobtainedfromPIMCatzerotemperature[ 29 33 34 ].Theyusedclassicalliquidstatetheory,theHNCapproximationtoobtainthepaircorrelationfunctiong(r)fortheUEGbyputtinginquantummany-bodyeffectsthroughaquantumpotentialc(r)andaneffectivetemperatureTc.TheeffectivepotentialiswrittenasacombinationofPaulipotential(0)andtheDeustchpotentialcc(r)=(cc)(0)(r)+)]TJ /F4 7.97 Tf 6.78 -1.8 Td[(c r(1)]TJ /F6 11.955 Tf 11.96 0 Td[(exp()]TJ /F6 11.955 Tf 9.3 0 Td[(r=c)) (2)TheDeustchpotentialisasimplerttingfunctionthantheKelbgpotentialinEquation( 2 ).ThePaulipotential(0)containstheexchangeeffectsandtheDeustchpotential 26

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containstheCoulombeffectsalongwithdiffraction.ThePDWpotentialisparameterizedbytheeffectivetemperatureTcthroughtheeffectivecouplingconstant)]TJ /F4 7.97 Tf 6.77 -1.8 Td[(candthethermalwavelengthc.Sincetheleadingenergydependenceontemperatureisquadratic,theeffectivetemperatureisassumedoftheformTc=p T2+T20.TisthetemperatureoftheactualquantumsystemandT0isthetemperatureoftheclassicaluidwhenthetemperatureofthequantumsystemiszero.Atagivendensityn,T0isdeterminedbysettingthecorrelationenergyEc(n)ofUEGequaltotheexcessenergyoftheclassicalsystematthetemperatureT0.T0isttedtoaform1=(a+bp rs+crs),wherers=(3=(4n))1=3.Thecoefcientsa,bandcareobtainedfromvariationalMonteCarloanddiffusionMonteCarlottingdataforEcforrsrangingfrom1and10.Usingthiseffectivepotentialc(r)andtemperatureTcintheHNCequationsg(r)=exp()]TJ /F3 11.955 Tf 9.3 0 Td[(cc(r)+g(r))]TJ /F5 11.955 Tf 11.95 0 Td[(1)]TJ /F6 11.955 Tf 11.95 0 Td[(c(r)) (2)theywereabletosuccessfullyreproducethepaircorrelationfunctionsforUEGintwoandthreedimensionsatzerotemperature.Thefunctionc(r)inEquation( 2 )isobtainedusingtheOrnstein-Zernikeequationg(r))]TJ /F5 11.955 Tf 11.96 0 Td[(1=c(r)+nZdr0(g(jr)]TJ /F7 11.955 Tf 11.95 0 Td[(r0j))]TJ /F5 11.955 Tf 11.96 0 Td[(1)c(r0) (2)ThedetailsabouttheHNCequationsarediscussedfurtherinChapter 3 .Thepaircorrelationfunctions,thespecicheatandotherthermodynamicquantitiescalculatedusingthismodelagreequitewellwiththedatafromthequantumsimulations. 2.3WhyWeChoseClassicalApproachInsteadOfQuantumtheories(OurMotivation)Allthequantumtheoriesdiscussedinthesection 2.1 failintheregimeofwarmdensematter.EithertheconceptstheyarebasedoncannotbeeasilyextendedtonitetemperaturesliketheFermiliquidtheoryandDFTortheirpredictionofsomeofthefundamentalthermodynamicquantitiesiswrongliketheSTLStheory.Quantum 27

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simulationslikePIMCiscomputationallymoreintensivethantheclassicalMolecularDynamicsandMonteCarlo.Intheclassicaltheorieswedonothavetosolvetheeigenvaluesastheoperatorformalismofthequantumtheories.TheformalismbyPerrotandDharma-wardanashowthatusingappropriatequantumpotentialsandtheconceptofeffectivetemperature,theclassicaltheoriescansuccessfullypredicttheuniformelectrongasoverawiderangeoftemperatureanddensities.Wrightonetal.usedtheHNCequationstostudytheshellformationinstronglycorrelatedclassicalsystems.ThedensityprolepredictedbythemisinexcellentagreementwiththeclassicalMCdata.Thesetwoworksprovideduscondencethattheclassicalmethodsaregoingtoworkforthewarmdensematterconditionsforjelliumandcanbefurtherextendedtoconnedquantumsystems.ThePerrowDharma-wardanamodelishoweverphenomenologicalanditisnotclearwhatkindofphysicsgivessuchgoodagreement.InthenextChapterwebuildasystematictheorybasedontheclassicalapproachincorporatingsomeexactlimitsofthequantumsystems.Asisseenlaterourclassicalformalismsuccessfullypredictstheessentialthermodynamicsoftheuniformelectrongasandsomeinterestingphysicsfortheconnedsystems. 28

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CHAPTER3THEEFFECTIVECLASSICALSYSTEMInthisChapterweconstructaformalismtodescribequantumcorrelations,developpracticalwaysofimplementingitandthenapplyittothecaseoftheidealFermigas.Weexplicitlybuildaneffectiveclassicalsystemhavingthethermodynamicsandtheequilib-riumstructureofthequantumsystemwewanttodescribe[ 35 ].Thethermodynamicsoftheeffectivesystemdependsonthreeparameters:effectiveinteractionpotentialc,effectivelocalchemicalpotentialcandeffectivetemperatureTc.Theseparametersaredeterminedbymappingselectedquantumpropertiesontotheclassicalsystem.Theexplicitcalculationoftheseparametersisonlyformalbecauseitrequiresasolutiontotheclassicalmanybodyproblem.ForpracticalapplicationsweintroduceapproximationsliketheweakcouplingclassicalapproximationandtherandomphaseapproximationforthequantumsystemsasillustratedinChapters 4 .InthisChapterweapplythisformalismtotheidealFermigasandinterprettheresults,asarstillustration. 3.1DenitionOfTheRepresentativeClassicalSystem 3.1.1ThermodynamicsFromStatisticalMechanicsConsideraquantumsystemofNparticlesinsideavolumeVinanexternalpoten-tialext.Theparticlesareinteractingthoughapair-potential.WechosetheGrandcanonicaldescriptionforthesystem,hencethethermodynamicsofthesystemneedstwoparameters,theinversetemperatureandthechemicalpotential.TheHamilto-nianofthesystemdoesnotdependontheinternaldegreesoffreedomlikespin.Itisoftheform:H=K++ext (3)whereKdenotesthetotalkineticenergy,K=NXi=1p2i 2m (3) 29

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isthetotalinteractionpotentialenergyamongparticles,extisthetotalexternalpotentialactingonthesystem.=1 2NXij(qi,qj),ext=NXi=1ext(qi) (3)Thesecanbewrittenintheequivalentforms:^ext=NXi=1ext(^qi)=Zdrext(r)^n(r)^=NXi6=j(^qi,^qj)=Zdrdr0(r,r)^g(r,r0) (3) (3)where^n(r)=PNi=1(^qi)]TJ /F7 11.955 Tf 11.14 0 Td[(r)and^g(r,r0)=^n(r)^n(r0))]TJ /F5 11.955 Tf 11.25 0 Td[(^n(r)(r)]TJ /F7 11.955 Tf 11.15 0 Td[(r0).IntheGrandensembletheHamiltonianoccursincombinationwiththechemicalpotential^H)]TJ /F3 11.955 Tf 11.96 0 Td[(N=^K+^+^ext (3)=^K+Zdrdr0(r,r)^g(r,r0))]TJ /F10 11.955 Tf 11.95 16.27 Td[(Zdr()]TJ /F3 11.955 Tf 11.95 0 Td[(ext(r))^n(r) (3)Thequantityintheaboveequation)]TJ /F3 11.955 Tf 12.22 0 Td[(ext(r)iscalledthelocalchemicalpotentialandisdenotedbythesymbol(r).Allthermodynamicquantitiescanbederivedfromthegrandpotential[ 47 ].Itdependsonthreethermodynamicparameters:theinversetemperature,thelocalchemicalpotential(r),andthevolumeV,andisgivenby(j,)=)]TJ /F3 11.955 Tf 9.3 0 Td[()]TJ /F9 7.97 Tf 6.59 0 Td[(1lnXNTrNe)]TJ /F14 7.97 Tf 6.58 0 Td[((^H)]TJ /F4 7.97 Tf 6.58 0 Td[(N) (3)ThesymbolTrNdenotesatraceovertheNparticleantisymmetricwave-functionsforFermionsorsymmetricwave-functionsforBosons.FromtheEquations( 3 )and( 3 )weseethatisafunctionoftheinversetemperatureandfunctionalsoftheinteractionpotentialandlocalchemicalpotential.Fromnowonweusethe 30

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(parametersjfunctionals)notationtomakethedistinctionbetweentheparameterandfunctionaldependence.ThedependenceonthevolumeVisleftimplicit.Inthegrandensemblethepressurep(j,)isrelatedtothegrandpotentialby:p(j,)V=)]TJ /F5 11.955 Tf 9.29 0 Td[((j,) (3)TherstorderderivativesofthegrandpotentialgivetheinternalenergyE(j,)andaveragenumberdensityn(r;j,)andaveragepaircorrelationfunctiong(r,r0;j,)E(j,)=@(j,) @j(r) (3)n(r;j,)V=)]TJ /F3 11.955 Tf 10.49 8.09 Td[((j,) (r)j (3)(j,) (r,r0)j,n(r;j,)n(r0;j,)g(r,r0;j,)=hbn(r)bn(r0);j,i)]TJ /F6 11.955 Tf 19.26 0 Td[(n(r;j,)(r)]TJ /F7 11.955 Tf 11.95 0 Td[(r0) (3)ascanbeseenfromEquation( 3 ).Higherorderderivativesprovidetheuctuations(susceptibilities)andstructurefunctions.Inparticular,thesecondfunctionalderivativewithrespectto(r)isrelatedtotheresponsefunction(r,r0;j)1 2(j,) (r)(r0)j=)]TJ /F3 11.955 Tf 9.3 0 Td[((r,r0;j,)=)]TJ /F5 11.955 Tf 10.99 8.08 Td[(1 Z0d0De0Hbn(r)e)]TJ /F14 7.97 Tf 6.59 0 Td[(0Hbn(r0);j,E (3)wherebn(r)=bn(r))]TJ /F6 11.955 Tf 12.69 0 Td[(n(r),andhX;j,idenotesanequilibriumgrandcanonicalaverageofthequantityX.NextweconsideraclassicalsystemwithNparticlesinteractingviaapairpotentialcandactedonbyanexternalpotentialext,candconnedtothesamevolumeV.Thesystemhasachemicalpotentialcandinversetemperaturec.TheHamiltonianHc 31

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hasthesameformasinEquation( 3 )exceptthatthepotentialenergyfunctionsinEquation( 3 )aredifferent,anddenotedbyc=1 2NXi6=jc(qi,qj),c,ext=NXi=1c,ext(qi) (3)Thelocalchemicalpotentialisc(r)c(r)c)]TJ /F3 11.955 Tf 11.95 0 Td[(c,ext(r) (3)Theclassicalgrandpotentialisdenedintermsofthesequantitiesbyc(cjc,c)=)]TJ /F5 11.955 Tf 11.3 0 Td[(lnXN1 3NcN!Zdq1..dqNe)]TJ /F14 7.97 Tf 6.59 0 Td[(c(c)]TJ /F15 7.97 Tf 6.59 6.42 Td[(Rdrc(r)bn(r)) (3)Here,c=)]TJ /F5 11.955 Tf 5.48 -9.69 Td[(2c~2=m1=2isthethermaldeBrogliewavelengthassociatedwiththetemperaturec.TheintegrationforthepartitionfunctionistakenovertheNparticlecongurationspace.ThethermodynamicalquantitiesfortheclassicalsystemaredeterminedinthesamewayasinEquations( 3 )-( 3 )forthequantumsystem.pc(cjc,c)V=)]TJ /F5 11.955 Tf 9.3 0 Td[(c(cjc,c) (3)Ec(cjc,c)=@cc(cjc,c) @cjcc,c (3)nc(r;cjc,c)=)]TJ /F3 11.955 Tf 10.49 8.09 Td[((cjc,c) c(r)jc,c (3)1 c2(cjc,c) c(r)c(r0)jc,c)]TJ /F3 11.955 Tf 21.92 0 Td[(c(r,r0;cjc,c)=)-166(hbn(r)bn(r0);cjc,cic (3) 32

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(cjc,c) c(r,r0)jc,cnc(r;cjc,c)nc(r0;cjc,c)gc(r,r0;cjc,c)=hbn(r)bn(r0);cjc,cic)]TJ /F6 11.955 Tf 11.95 0 Td[(nc(r;cjc,c)(r)]TJ /F7 11.955 Tf 11.96 0 Td[(r0) (3)ThedifferencefromtheEquations( 3 )-( 3 ),istheensembleaveragingforaclassicalsystemisdonebyintegratingoverthephasespace.TheEquations( 3 )-( 3 )haveanadditionalconstraintthatthederivativesaretakenatconstantpairpotentialc.Thisisnecessarybecausewewillseelaterthattheinteractionpotentialoftheclassicalsystemcdependsonthetemperaturecandc.SeeSection 3.2 belowforfurtherelaboration. 3.1.2ConstructionOfTheEffectiveClassicalSystem:Classical-QuantumCorrespondenceConditionsOurgoalinthissectionistoconstructaneffectiveclassicalsystemthatwouldhavesomeselectedthermodynamicpropertiesasthequantumsystemwewanttodescribe.Thethermodynamicsofthiseffectivesystemisunknownatthisstage,becausethethreeingredientsofthegrandpotential:theeffectiveinversetemperature,c,thelocalchemicalpotentialc(r),andtheinteractionpotentialamongtheparticlesc(r,r0)arestillunknown.Determiningthesequantitiesinvolveexpressingthemasfunctionsorfunctionalsofthecorrespondingquantumquantities:,(r),and(r,r0).Sincewehavethreeunknownquantitiesfortheeffectivesystem,weneedthreeequivalentconditionsrelatingtheeffectiveclassicalandquantumsystem.Thisisaccomplishedbyrequiringthenumericalequivalenceoftwoindependentthermodynamicpropertiesandonestructuralpropertyfortheclassicalandquantumsystems.Onechoiceistheequalityofthegrandpotentials,sincethegrandpotentialscompletelydeterminethethermodynamicsofanysystem.Thetwoadditionalchoicesareequivalenceofitsrstfunctionalderivativeswithrespecttothelocalchemicalpotentialandtheinteraction 33

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potential.c(cjc,c)(j,) (3)c(cjc,c) c(r)jc,c(j,) (r)j (3)c(cjc,c) c(r,r0)jc,c(j,) (r,r0)j, (3)Theequivalenceconditionscanbeexpressedintermsofphysicalquantities.Inthegrandcanonicalensemble,thegrandpotentialisproportionaltothepressure,anditsrstderivativeswithrespecttothepairandexternalpotentialsarethedensityandthepaircorrelationfunctionsrespectively,asseenfromtheEquations( 3 )and( 3 ).SotheEquations( 3 )and( 3 )canbereinterpretedas:pc(cjc,c)p(j,) (3)nc(r;cjc,c)n(r;j,) (3)gc(r,r0;cjc,c)g(r,r0;j,) (3)Theequivalenceofthepaircorrelationfunctionsabovefollowsintwosteps:i)fromtheEquation( 3 ) c(r,r0)=n(r)n(r0)g(r,r0)andii)theequivalenceofthedensities.TheEquation( 3 )canalsobeinterpretedintermsoftheequivalenceofdensityuctuationshbn(r)bn(r0);cjc,cichbn(r)bn(r0);j,i (3)Insummary,theclassical-quantumcorrespondenceconditionsaretheequivalenceofthepressures,densities,andpaircorrelationfunctions.Therearesomealternatescenariosforthechoiceoftheequivalenceconditions.AnalternativechoicetoEquation( 3 )wouldbetoequatethesecondfunctionalderivativesofthegrandpotentialwithrespecttothelocalchemicalpotential,whichfrom 34

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Equations( 3 )-( 3 )implies:c(r,r0;cjc,c)(r,r0;j,).However,werunintoproblemsbecausetheclassicalresponsefunctionhasasingularcontributionproportionalto(r)]TJ /F7 11.955 Tf 11.96 0 Td[(r0)thatisnotpresentinthequantumresponsefunction,whichmakesthemapcomplicated.Usingtheequivalenceoftheclassicalandquantumformsforthepaircorrelationfunctionsdonothavethisproblem.InthiswaythethreeEquations( 3 )determine,formally,theclassicalparametersc,c,andcasfunctionsof,andfunctionalsof(r),and(r,r0)c=c(j,),c=c(r;j,),c=c(r,r0;j,) (3)Sofarwehavedenedaneffectiveclassicalsystemandprovidedequivalentconditionstocalculatethedetailsofitsthermodynamics.ItisnecessarytoinverttheseequivalenceconditionstodeterminethethreeparametersasisdoneinSection 3.1.3 3.1.3InversionOfTheCorrespondenceConditionsTheclassicaldensityfunctionaltheoryprovidesawaytoinvertthetwoequivalentconditionsinEquations( 3 )fordensityandpaircorrelationfunctions,tosolvefortheeffectivepotentialsinEquation( 3 ).Thisformalismrelatestheinteractionpotentialandtheexternalpotentialtothepaircorrelationfunctionanddensity,respectively.SeeAppendix A forfurtherdetails.lngc(r,r0)=)]TJ /F3 11.955 Tf 9.3 0 Td[(cc(r,r0)+Z10dZdr00c(2)c(r,r00jnc+nc(gc)]TJ /F5 11.955 Tf 11.96 0 Td[(1))nc(r00)(gc(r00,r0))]TJ /F5 11.955 Tf 11.95 0 Td[(1), (3)ln)]TJ /F6 11.955 Tf 5.48 -9.69 Td[(nc(r)3c=cc(r)+Z10dZdr00c(2)c(r,r00jnc)nc(r00) (3)wheretheconstantcinEquation( 3 )isthethermaldeBrogliewavelengthevaluatedattheclassicaltemperaturecdenedbelow,c=)]TJ /F5 11.955 Tf 5.48 -9.68 Td[(2~2c=m1=2.ThefunctionccinEquations( 3 )and( 3 )isthedirectcorrelationfunctionanditisdeterminedfrom 35

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thepaircorrelationfunctiongcviatheOrnstein-Zernikeequation:(gc(r,r0))]TJ /F5 11.955 Tf 11.96 0 Td[(1)=cc(r,r0jnc)+Zdr00cc(r,r00jnc)nc(r00)(gc(r00,r0))]TJ /F5 11.955 Tf 11.95 0 Td[(1), (3)Thevirialequationfortheclassicalpressure(obtainedbydifferentiatingc(cjc,c)withrespecttothevolume)iscpc(cjc,c)=1 VZdrnc(r)1+1 3rrcc(r))]TJ /F5 11.955 Tf 13.15 8.09 Td[(1 6Zdr0nc(r0)gc(r,r0)r0r0cc(r,r0) (3)ItprovidesthethirdinversionofthepressureequivalenceinEquation( 3 )tosolvefortheeffectivetemperaturec.SeeAppendix A forfurtherelaboration.InSection 3.1.4 wegetexplicitformsfortheeffectivepotentialsandtheeffectivetemperatureintermsofthequantumquantities.WerewritetheEquations( 3 )and( 3 )andthenusetheequivalenceconditionsinEquations( 3 )toreplacealltheclassicalpaircorrelationsgcanddensitiesncbythecorrespondingquantumquantitiesgandn:cc(r)=3 2lnc +ln)]TJ /F6 11.955 Tf 5.48 -9.69 Td[(n(r)3)]TJ /F10 11.955 Tf 11.95 16.27 Td[(Z10dZdr00c(2)(r,r00jn)n(r00) (3)cc(r,r0)=)]TJ /F5 11.955 Tf 11.29 0 Td[(lng(r,r0)+Z10dZdr00c(2)(r,r00jn+n(g)]TJ /F5 11.955 Tf 11.95 0 Td[(1))n(r00)(g(r00,r0))]TJ /F5 11.955 Tf 11.96 0 Td[(1), (3)c(2)(r,r0jn)=(g(r,r0))]TJ /F5 11.955 Tf 11.95 0 Td[(1))]TJ /F10 11.955 Tf 11.95 16.27 Td[(Zdr00c(2)(r,r00jn)n(r00)(g(r00,r0))]TJ /F5 11.955 Tf 11.95 0 Td[(1) (3)OnthelefthandsideoftheEquations( 3 )and( 3 )wehavetheeffectivelocalchemicalpotentialandinteractionpotentialfortheclassicalsystemsthatwewantedtocalculate.Therighthandsidesoftheseequationsdependonlyonthequantumquantities,nandg.Thec(2)isdenedbytheOrnstein-Zernickeequation,Equation 36

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( 3 ),intermsofthequantumpaircorrelationfunctiongbytheequivalenceconditioninEquation( 3 ).Finally,anequationforc=isobtainedbyusingtheequivalenceoftheclassicalandquantumpressurespc=p,Equation( 3 ),towritec =cpc p (3)orwithEquation( 3 )c =1 pVZdrn(r)1+1 3rrcc(r))]TJ /F5 11.955 Tf 13.15 8.09 Td[(1 6Zdr0n(r0)g(r,r0)r0r0cc(jr)]TJ /F7 11.955 Tf 11.96 0 Td[(r0j) (3)Equations( 3 ),( 3 ),( 3 ),and( 3 )thendeterminetheclassicalpa-rametersasc=,cc(r),andcc(r,r0)fromthepropertiesofthegivenquantumsystemc=c(jn,g),c(r)=c(r,jn,g),c(r,r0)=c(r,r0,jn,g) (3)whichisequivalenttoEquation( 3 )(correspondingtoachangeofvariablesfrom,ton,gforthequantumsystem).Althoughtheseequationslooksimple,thecalculationofcc,ccandcisdifcultbecausewestillneedtosolvetheclassicalmany-bodyproblemhiddeninsidethefunctionalc(2)(r,r00j).Anexactcalculationwouldinvolvesolvingthehigherordercorrelationfunctionsandthereisnosimplewaytodeterminethesefunctions.Hencewewillhavetointroducesomeapproximationsforpracticalapplications.Tillnowtheformalismisexact.InSection 3.1.4 approximationsareintroducedtosolvetheclassicalmany-bodyproblem.Thesearecalledthehypernettedchainapproximation(HNC)whichformsaclosedsetofequationsreplacingEquations( 3 )and( 3 ).Thegeneralcoursewewillfollowis:i)Firstsimplifythemany-bodytermsinthecorrelationfunctionsc(2)(r,r00j)usingtheHNCequations.ii)useapproximationstocalculatethequantuminputsinthemap:pressure,densityandpaircorrelations. 37

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Thetypeofapproximationwillbespecictothequantumsystemwewanttodescribe.ThenputtheseintotheEquations( 3 ),( 3 )and( 3 )togetc,candc.iii)Usetheseeffectiveparameterstocalculatepropertiesofinterestusingvarioustoolsinclassicalstatisticalmechanics(liquidstatetheory,densityfunctionaltheory,moleculardynamics,classicalMonteCarlo). 3.1.4HypernettedChainApproximationHypernettedchainapproximatinisaclosurerelationthatrelatesthepaircorrelationfunctionanddensitytotheinteractionpotentialandtheexternalpotentialthroughtheuseoftheOrnstein-Zernikeequation,Equation( 3 ).Thisinvolvesreplacingthenon-localdensitydependenceofthecorrelationfunctionalsinEquations( 3 )and( 3 )bythelocaldensitydependence:c(2)c(r,r00jnc)!c(2)c(r,r00jnc),c(2)c(r,r00jnc+nc(gc)]TJ /F5 11.955 Tf 11.96 0 Td[(1))!c(2)c(r,r00jnc) (3)ThenEquations( 3 )and( 3 )becomeln)]TJ /F6 11.955 Tf 5.48 -9.69 Td[(nc(r)3c=cc(r)+Zdr0c(2)c(r,r0jnc)nc(r00). (3)lngc(r,r0jnc)=)]TJ /F3 11.955 Tf 9.3 0 Td[(cc(r,r0)+Zdr00c(2)c(r,r00jnc)nc(r00)(gc(r00,r0jnc))]TJ /F5 11.955 Tf 11.95 0 Td[(1) (3)TogetherwiththeOrnstein-Zernickeequation,Equation( 3 ),theyformaclosedsetofequationsthatdeterminethedensityandpaircorrelationfunctionforthegivenpotentials.ThisistheHNCofliquidstatetheory,generalizedtospatiallyinhomogeneoussystems[ 54 ].ItisknowntogiveverygoodresultsforuniformCoulombsystems[ 32 ]andforinhomogenous,connedCoulombsystemsevenatstrongcouplingconditions[ 46 49 ]. 38

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TheeffectivepotentialsinEquations( 3 )and( 3 )havethefollowingformwithinHNCapproximation:cc(r)=ln)]TJ /F6 11.955 Tf 5.48 -9.69 Td[(n(r)3c)]TJ /F10 11.955 Tf 11.96 16.28 Td[(Zdr00c(2)(r,r00jn)n(r00) (3)cc(r,r0)=)]TJ /F5 11.955 Tf 11.29 0 Td[(ln(1+h(r,r0jn))+h(r,r0jn))]TJ /F6 11.955 Tf 11.96 0 Td[(c(r,r0jn) (3)andtheOrnstein-Zernickeequation,Equation( 3 a),isunchangedc(2)(r,r0jn)=h(r,r0jn))]TJ /F10 11.955 Tf 11.95 16.27 Td[(Zdr00c(2)(r,r00jn)n(r00)h(r00,r0jn) (3)Theholefunction,h(r,r0jn)hasbeenintroducedfornotationalsimplicityh(r,r0jn)=g(r,r0jn))]TJ /F5 11.955 Tf 11.96 0 Td[(1 (3)SimilarapproximationshavetobemadeinEquation( 3 )forc=,toprovidepracticalformstodeterminec=forgivenappropriatequantuminput.Foruniformsystems,thesecondtermofEquation( 3 )issimplyrelatedtothestaticstructurefactorS(k)Zdr00c(2)(r,r00jn)n(r00)!nZdr00c(2)(r)]TJ /F7 11.955 Tf 11.95 0 Td[(r00,n)=1)]TJ /F5 11.955 Tf 33.47 8.08 Td[(1 S(k=0) (3)FortheuniformsystemslikejelliumandtheidealFermiuidsatzerotemperature,S(k)vanishesask!0andthiscontributiontoccdiverges.Thereforeforsuchsystems,insteadofEquation( 3 )analternativeformderivedusingacouplingconstantintegrationandtheHNCapproximation[ 45 ]isused,cc=ln)]TJ /F6 11.955 Tf 5.48 -9.68 Td[(nc3c)]TJ /F6 11.955 Tf 11.96 0 Td[(nZdrc(r,n)+cc(r))]TJ /F5 11.955 Tf 13.15 8.08 Td[(1 2h(r,n)(h(r,n))]TJ /F6 11.955 Tf 11.96 0 Td[(c(r,n)) (3) 39

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3.2PeculiarityOfTheThermodynamicsOfTheEffectiveSystemThedenitionoftheequivalentclassicalsystemassuresthatthepressureanddensityinthegrandensemblegivethecorrectquantumresults(inprinciple),e.g.pc(cjc,c)V=)]TJ /F5 11.955 Tf 9.3 0 Td[(c(cjc,c)=p(j,)V=)]TJ /F5 11.955 Tf 9.3 0 Td[((j,) (3)Howeverpropercareshouldbetakenwiththefunctionalderivativesfortheeffectiveclassicalsystem.Theeffectiveparametersc,candcarecoupledandarenotindependentvariablesasthoseofthequantumsystem.AsaresultwhiletakingthederivativesinEquations( 3 )-( 3 )wehavetoholdthepotentialcconstant.Sinceallthreequantitiesarecoupled,asmallvariationinonewouldchangetheothers.Therefore,thevariationofthepressureleadsto(inthefollowingthevolumeVisalwaysheldconstant)(pcV)=@pcV @cjc,c+Zdrdr0pcV c(r,r0)jc,c@c(r,r0) @cjcc+ZdrpcV c(r)jc,c+Zdr0dr00pcV c(r0,r00)jc,cc(r0,r00) c(r)jcc(r)eScdTc+Zdrenc(r)c(r) (3)ThesecondequalitydenestheclassicalthermodynamicentropyandthermodynamicdensityintermsofthegrandpotentialTceSc=)]TJ /F3 11.955 Tf 9.3 0 Td[(c@pcV @cjc,c+Zdrdr0pcV c(r,r0)jc,c@c(r,r0) @cjc=c@c @cjc (3)enc(r)=pcV c(r)jc,c+Zdr0dr00pcV c(r0,r00)jc,cc(r0,r00) c(r)jc=)]TJ /F3 11.955 Tf 16.68 8.09 Td[(c c(r)jc (3) 40

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Similarly,theclassicalinternalenergyisdenedbyeEcTceSc+Zdrenc(r)c(r))]TJ /F6 11.955 Tf 11.95 0 Td[(pcV=@c @cjc)]TJ /F10 11.955 Tf 11.29 16.27 Td[(Zdrcc(r)c cc(r)jc (3)=@cc @cjcc (3)NotethatderivativesinthelastequalitiesofEquations( 3 )-( 3 )donothavetherestrictionofconstantc.Thesesamerelationshipsholdforthequantumpropertiesaswell,sincetheyarethegeneraldenitionsofthermodynamicsforthechosenvariables,and.Inthequantumcaseisindependentofthethermodynamicvariablesandhenceisconstantinthevariations.Thisleadstotheequivalentexpressionsintermsofequilibriumaveragesn(r)=)]TJ /F3 11.955 Tf 16.68 8.09 Td[( (r)j=hbn(r)i,E=@ @j=DbHE (3)However,intheclassicalcasetheaboveleadstoenc(r)=hbn(r)ic)]TJ /F10 11.955 Tf 11.96 16.28 Td[(Zdr0dr00c c(r0,r00)jc,cc(r0,r00) c(r)jc (3)eE=hHic+Zdr0dr00cc c(r0,r00)jccc(r0,r00) cc(r)jc (3)Forexample,thethermodynamicdensity,enc(r),differsfromtheaveragedensityofthetextabove,nc(r),becausethelatterisdenedasaderivativeofthegrandpotentialatconstantc. 3.3Example-IdealFermiGasAsarstapplicationoftheabovemap,weconsiderthesimplestquantumsystem,theidealFermigas.FornowweconsideraninhomogeneousunpolarizedidealFermi 41

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systemandcalculateallthequantuminputsthatgointotheeffectivequantities.H)]TJ /F3 11.955 Tf 11.96 0 Td[(N!Zdr^p2 2m)]TJ /F3 11.955 Tf 11.96 0 Td[((r)bn(r) (3)Sinceforanon-interactingsystemthehamiltonianisthesumofsingleparticleHamil-tonians,allthethermodynamicquantitiesofthesystemcanbewrittenintermsofthecorrespondingthermodynamicquantitiesofasingleparticlesystem[ 47 ].p(j)Vp(j,=0)V=)]TJ /F9 7.97 Tf 6.59 0 Td[(1(2s+1)Zdrhrjln1+e)]TJ /F14 7.97 Tf 6.59 0 Td[(bp2 2m)]TJ /F14 7.97 Tf 6.58 0 Td[((br)jri (3)n(r,j)n(r,j,=0)=(2s+1)hrjexpbp2 2m)]TJ /F3 11.955 Tf 11.96 0 Td[((br)+1)]TJ /F9 7.97 Tf 6.58 0 Td[(1jri (3)g(r,r0;j)g(r,r0;j,=0) (3)=1)]TJ /F5 11.955 Tf 26.5 8.09 Td[(1 2s+1n(r,r0)n(r0,r) n(r,r)n(r0,r0),n(r,r0)=hrjexpbp2 2m)]TJ /F3 11.955 Tf 11.96 0 Td[((br)+1)]TJ /F9 7.97 Tf 6.58 0 Td[(1jr0i (3)wherehrjXjr0idenotesamatrixelementincoordinaterepresentation,andsisthespin.Tofurthersimplifytheoperatorequations,Equations( 3 )-( 3 ),wehavetointroduceacompletesetofwave-functions nandsolvetheenergyeigenvaluesnforthesingleparticlehamiltonian(bp2=2m))]TJ /F3 11.955 Tf 11.95 0 Td[((^q).p(j)V=)]TJ /F9 7.97 Tf 6.58 0 Td[(1(2s+1)1Xn=0ln(1+exp()]TJ /F3 11.955 Tf 9.29 0 Td[((n)]TJ /F3 11.955 Tf 11.95 0 Td[())) (3)n(r,j)=(2s+1)(exp((n)]TJ /F3 11.955 Tf 11.95 0 Td[())+1))]TJ /F9 7.97 Tf 6.58 0 Td[(1j n(r)j2 (3)g(r,r0;j)=1)]TJ /F5 11.955 Tf 26.5 8.09 Td[(1 2s+1n(r,r0)n(r0,r) n(r,r)n(r0,r0),n(r,r0)=1Xn=0(exp((n)]TJ /F3 11.955 Tf 11.96 0 Td[())+1))]TJ /F9 7.97 Tf 6.59 0 Td[(1 ?n(r) n(r0), (3) 42

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Forslowlyvaryingexternalpotentialsapracticalapproximationisobtainedbyreplacingtheoperator(^q)byitseigenvalue(r),alsoknownasalocaldensityapproximation,whichisequivalenttoreplacingthenon-localdensitydependencebythelocaldensityasseenbelow.Thisapproximationusuallyworkswellforsystemswhennodiscreteboundstatesareformed.Theexpectationvaluesabovethencanbecalculatedassimpleintegralsp(j)!1 VZdr(2s+1)1 h3Zdpp2 2me(p2 2m)]TJ /F14 7.97 Tf 6.59 0 Td[((r))+1)]TJ /F9 7.97 Tf 6.59 0 Td[(1 (3)n(r,j)!(2s+1)h)]TJ /F9 7.97 Tf 6.59 0 Td[(3Zdpe(p2 2m)]TJ /F14 7.97 Tf 6.59 0 Td[((r))+1)]TJ /F9 7.97 Tf 6.59 0 Td[(1 (3)n(r,r0)!1 h3Zdpei ~p(r)]TJ /F9 7.97 Tf 6.59 0 Td[(r0)e(p2 2m)]TJ /F14 7.97 Tf 6.59 0 Td[((R))+1)]TJ /F9 7.97 Tf 6.59 0 Td[(1,R=r+r0 2 (3)TheresultsinEquation( 3 )arethefamiliarnitetemperatureThomas-Fermiap-proximationsforthethermodynamics,whileEquation( 3 )isitsextensiontostructure[ 55 ].Theexpressionsforn(r,j)andn(r,r0)arenolongerfunctionalsof(r),butratherlocalfunctionsof(r)and(R),respectively.Practicaltsforn(r,j)andthisinversionaregiveninAppendix B ,alongwithsimplicationofn(r,r0).ThecomparisonofthedensityobtainedusingtheLDAwiththeexactdensitycalculationinEquation( 3 )isshowninFigure( 3-1 )atrs=10for100particles.Forsmallrsandlargenum-berofparticlesinthetrap,LDAagreeswellwiththeexactdensitycalculation.Atverylowtemperaturesandlargevaluesofrs,LDAdensitiesdifferfromtheactualdensitiesclosetotheorigin.ThedetailsofthewavefunctionsusedtogeneratetheplotisgiveninAppendix F .Eventhoughthequantumsystemisnon-interacting,thecorrespondingclassicalsystemisnon-trivialbecausec(qi,qj)6=0.Thisisbecausetheclassicalpairinterac-tionsarerequiredtoreproducetheeffectsofthequantumstatistics.Wearestillrequired 43

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Figure3-1. Thecomparisonoftheexactdensity(asinEquation( 3 ))fortheidealFermigasfor100particlesinaharmonictrapwiththatobtainedusingLDA(Equation( 3 )). tosolvethecomplicatedclassicalmanybodyproblem.ThisisaddressedwithintheHNCapproximationdescribedabovebyEquations( 3 )and( 3 )togetherwiththeOrnstein-Zernickeequation,Equations( 3 )and( 3 )forc=.Theeffectivepotentialandtheeffectivetemperaturecanbeexplicitlycalculatednumerically,howevercandcbothdependontheexternalpotentialthroughthelocalchemicalpotentialc(r).ToillustratetheabovediscussioninmoredetailwerestrictourselvestoauniformunpolarizedidealFermigas((r)=).FortheuniformgastheEquations( 3 )-( 3 )fortheeffectiveparameterssimplifytoc =n p1)]TJ /F5 11.955 Tf 13.15 8.09 Td[(1 6nZdrg(r)rrcc(r) (3)cc=ln)]TJ /F6 11.955 Tf 5.48 -9.68 Td[(nc3c)]TJ /F6 11.955 Tf 11.95 0 Td[(nZdrc(r)+cc(r))]TJ /F5 11.955 Tf 13.15 8.09 Td[(1 2h(r)(h(r))]TJ /F6 11.955 Tf 11.95 0 Td[(c(r)) (3)cc(r)=)]TJ /F5 11.955 Tf 11.3 0 Td[(ln(1+h(r))+h(r))]TJ /F6 11.955 Tf 11.95 0 Td[(c(r) (3)c(r)=h(r))]TJ /F6 11.955 Tf 11.95 0 Td[(nZdr0c(jr)]TJ /F7 11.955 Tf 11.96 0 Td[(r0j)h(r0) (3) 44

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Thesuperscript(2)onc(2)(r)andthedependenceofc(r),h(r)onthermodynamicvariableshasbeensuppressedforsimplicity.Theselasttwoequationscanbesolvedforcc(r)usingh(r)fromEquations( 3 )and( 3 )intheuniformlimit.Withthatresult,c=canbecalculatedfromEquation( 3 ),andthenccdeterminedfromEquation( 3 ).ThedimensionlesspotentialccwillbereferredtoasthePaulipotential.Fromnowonweuseonlydimensionlessparameters.Thedimensionlessdistanceisgivenbyr=r=r0,wherer0isthemeandistancebetweenparticlesrelatedtotheuniformdensityby4r30=3=1=n.TheplotsofthePaulipotentialatdifferenttemperaturesareshowninFigure 3-2 asafunctionofthedimensionlesscoordinater.ThethermodynamicsofaquantumsystemisparameterizedbythetemperatureTandthedensityn.Inourdiscussionsweusedimensionlesstemperaturetanddimensionlessaverageinter-particledistancersinstead.Theyaredenedbyrs=r0=aBwhereaBistheBohrradius,andt=F=forthetemperaturerelativetotheFermitemperature()]TJ /F9 7.97 Tf 6.59 0 Td[(1F=F=~2)]TJ /F5 11.955 Tf 5.48 -9.68 Td[(32n2=3=2m).Forexample,intheseunitsn3=8=)]TJ /F5 11.955 Tf 5.48 -9.68 Td[(31=2t3=2andtheclassicallimitoccursfort>>1wherethedistancebetweenparticlesislargecomparedtothethermaldeBrogliewavelengthandthedegeneracyeffectsarelow.Thethermodynamicsoftheidealgaspropertiesisindependentofrsandcanbeobtainedasthers!0limitofanyquantumsystem.Figure 3-2 showsthePaulipotentialfort=0,10)]TJ /F9 7.97 Tf 6.59 0 Td[(1,1,and10.ThePaulipotentialisrepulsivebecauseofthePauliexclusionprinciple,niteatr=0forspin-lesssystems,andmonotonicallydecreasing.Thebehaviorisexponentialathightemperatures,butanr)]TJ /F9 7.97 Tf 6.58 0 Td[(2algebraictaildevelopsforlowtemperaturest,arisingfromthedirectcorrelationfunctionc(2)(r)inEquation( 3 ).Thegrandpotentialdoesnotexistforsuchapotentialwhichdecaysslowerthanr)]TJ /F9 7.97 Tf 6.58 0 Td[(3inthreedimension[ 47 ]anditwouldappearthattheequivalentclassicalsystemproposedherefailsevenforthissimplestcaseofanidealFermigas.However,weadoptthesametechniquethatis 45

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Figure3-2. IdealgasPaulipairpotentialasafunctionofr=r=r0fort=0,0.1,1,10[ 35 ]. usedtocurethelongrangedproblemfortheCoulombsystemsbyaddingauniformneutralizingbackground(onecomponentplasma).Thesameprocedurecanbeusedhere,i.e.,aclassicalsystemisconsideredwhereauniformcompensatingbackgroundtermhasbeenaddedtotheHamiltonian.ChangingtheHamiltonianimpliesreplacinggc(r,r0)bygc(r,r0))]TJ /F5 11.955 Tf 11.93 0 Td[(1inthepressureequationEquation( 3 ),sothatEquation( 3 )becomesc =n p1)]TJ /F5 11.955 Tf 13.15 8.09 Td[(1 6nZdrh(r)rrcc(r) (3)Figure 3-3 showstheclassicaltemperaturerelativetotheFermitemperature,F=ctc,asafunctionoftobtainedfromEquation( 3 ).ItisseenthattheclassicaltemperatureTcremainsniteatT=0,andcrossesovertoTc=Tathightemperatures.ThenitenessofTcwhenT=0resultsbecauseaquantumsystemhasaniteaveragekineticenergyatT=0.ThePDWmodelpostulatesthephenomenologicalformTc=)]TJ /F6 11.955 Tf 5.48 -9.68 Td[(T2+T201=2.ThemodeloriginallyusestheaverageenergyperparticleatT=0toevaluateT0=2TF=5.TheresultfromEquation( 3 )isquitecloseTc(t=0)0.43TF.Tocomparethedependenceatnitet,thePDW 46

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Figure3-3. Idealgasreducedclassicaltemperaturetc=Tc=TFasafunctionoft=T=TF.AlsoshownistheresultofPDW[ 35 ]. formisalsoshowninFigure 3-3 withT0=Tc(t=0).ItisseenthattheresultsarequitesimilaralthoughthePDWformhasasomewhatfastercrossovertotheclassicallimit. Figure3-4. Idealgasdimensionlesschemicalpotentialc=Fasafunctionoft.Alsoshownisthecorrespondingquantumchemicalpotential=F+tln2[ 35 ]. Sincethereisnoexternalpotential,thelocalchemicalpotentialisaconstantc.Figure 3-4 showssimilarresultsforc=FasafunctionoftobtainedfromEquation( 3 ).Alsoshownistheresultforthequantum=F.Bothformsdependonlyont(independentofrs).Athightemperaturesthechemicalpotentialoftherepresentativesystemgoesovertoln(n3)asinEquation( 3 )whilethequantumchemicalpotential 47

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goesovertoln(n3=2).Theydifferbyaconstantamountofln(2)athightempera-turesduetotheinternalspindegreesoffreedominthequantumcalculation.ThisisaccommodatedinthecomparisonshowninFigure 3-4 .NowthatwehavedeterminedtheeffectivepotentialandtheeffectivetemperaturefortheuniformidealFermigas,itwouldbeusefultolookattheaverageinternalenergyofthesystem.FortheidealFermigas,theaverageenergyisrelatedtothepressureasE=3 2pV.First,notethatallthedimensionlessthermodynamicquantitiesforboththeidealFermigasanditseffectiveclassicalsystemdependononlyonethermodynamicparameterz=e.Inparticularcpc3c=G(z) (3)ThespecicformforG(z)isnotrequiredforthepresentdiscussion.Thecorrespondingquantumresultisp3=f3=2(z),wheref3=2(z)istheFermiintegralofAppendix B ,( B ).AlthoughG(z)andf3=2(z)arequitedifferent,thefactthattheybothdependonlyonzimpliesthattherelationshipamongdifferentthermodynamicpropertiesisthesameinbothclassicalandquantumcases.Forexample,theenergyperparticleisdeterminedfromthepressureviathethermodynamicdenitionEquation( 3 )withEquation( 3 )eEc=@cpcV @cjc,cc=@G(z)V=3c @cjc,cc=)]TJ /F6 11.955 Tf 9.3 0 Td[(VG(z)@)]TJ /F9 7.97 Tf 6.59 0 Td[(3c @cjzc=2 3pcV (3)Thecontributionfrom@G(z)=@cjzc,Vvanishessincez=z(zc).Thisfollowsondimensionalgroundssincezisdimensionless,andthereisnoadditionalenergyscaletomakecdimensionless.Hencetheclassicalcalculationgivestheknownexactquantumresult.Thisnon-trivialresultforaclassicalinteractingsystemisastrongconrmationoftheeffectiveclassicalmapdenedhere.AsnotedinSection 3.2 ,itcanbeveriedthattheclassicalaverageoftheclassicalHamiltoniandoesnotgivethecorrectrelationship 48

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tothepressure.Insteaditisnecessarytodenetheinternalenergythermodynamically,asisdoneinEquation( 3 ).InasimilarwayitisveriedthattherelationshipoftheclassicalentropytothepressureanddensityisthesameasinthequantumcaseTcSc=5 2pcV)]TJ /F3 11.955 Tf 11.95 0 Td[(c Nc (3)ThisisaformoftheSackur-Tetrodeequationvalidforthequantumcase.Itisem-phasizedthatthesesimpleidealquantumgasresultsarebeingretainedforthemorecomplexeffectiveclassicalsystemwithpairinteractionsviathePaulipotential. 3.4Summary 1. Wedevelopedaformalismtodescribethequantumcorrelationsusingclassicalmethods. 2. Wedevelopaformalismthatconstructsaneffectiveclassicalsystembymappingitsthermodynamicstothatofthequantumsystemwewanttodescribe.Thethermodynamicsoftheeffectivesystemisdescribedcompletelybytheeffectiveinteractionpotential,theeffectiveexternalpotentialandtheeffectivetemperature. 3. Sincetherearethreeunknowns,thetwoeffectivepotentialsandtheeffectivetemperature,weneedthreeequivalentconditionstosolvethem.Wedothatbyequatingthepressure,densityandthepaircorrelationsfortheeffectiveclassicalandthequantumsystem. 4. ThemapisinvertedusingtheclassicaldensityfunctionaltheoryandfurthersimpliedusingtheHNCapproximation. 5. ThemapisappliedtotheidealFermigas.Theeffectivequantitiesforitseffectiveclassicalsystemhavebeencalculatedandplotted.TheresultE=3=2pV,whichistrueforthequantumsystem,isrecoveredfortheeffectivesystemalso. 49

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CHAPTER4UNIFORMELECTRONGASTheformalismdevelopedinChapter 3 isnowappliedtotheuniformelectrongasorjellium[ 48 ].Thecalculationofthethreeparametersoftheeffectiveclassicalsystemforjelliumneedsthreequantuminputs.However,gettingtheexactexpressionsforthequantuminputsforjellium(pressuresandpaircorrelationfunctions)isconsiderablymoredifcultthanfortheidealFermigas.HenceinthisChapterthesequantitiesarecalculatedwithintherandomphaseapproximation.Moreprecisely,wedenetheeffectiveparameterssuchthattheypreservetheidealgasandtheexactweakcouplinglimits.Oncewedeterminethethreeparameters,theyareusedwiththeclassicalstrongcouplingHNCequationstoobtainthepaircorrelationfunctionsandotherthermodynamicfunctions.ThepaircorrelationfunctionsarecomparedwithothertheorieslikePDWandSTLSandsimulationmethodslikePIMC.TheagreementwiththesimulationdataandPDWmodelisquitegood.Otherthermodynamicfunctionslikethepressureandthechemicalpotentialarealsocalculated. 4.1ThermodynamicsOfTheEffectiveClassicalSystemTheinteractingelectrongasorjelliumisasystemofelectronsinteractingviatheCoulombinteractionsembeddedinanuniformneutralizingbackground.Theneutralizingbackgroundisrequiredforthethermodynamicstabilityofasysteminteractingwithlongrangedforces[ 47 ].Theuniformelectrongasisthestartingmodelforthestudyofelectroninteractionsinthemetals,lattices,andplasmas[ 52 ].Thecorrespondingclassicalsystemiscalledtheonecomponentplasma.InthisChapterwewilllookintoboththethermodynamicsandstructureofjellium.Weconstructtheparametersoftheeffectiveclassicalsystem,c=1=kBTc,c(r),andc(r,r0),whichareexactintheweakcouplinglimit.TheeffectiveinteractionpotentialisthenusedinHNCequationstocalculatethepaircorrelationfunctiontofurtherincludeclassicalstrongcouplingeffects. 50

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Onceweobtainthepaircorrelationfunctions,thepressureandotherquantitiescanbederivedusingtheclassicalstatisticalmechanics.TosummarizetheresultsofChapter 3 ,theclassicalparametersforanuniformquantumsystemaredenedasfollowing.FirstthepairpotentialisobtainedfromtheinversionoftheHNCequationcc(r)=)]TJ /F5 11.955 Tf 11.29 0 Td[(ln(1+h(r))+h(r))]TJ /F6 11.955 Tf 11.95 0 Td[(c(r) (4)wherethedirectcorrelationfunctioncintheequationaboveisdeterminedfromtheOrnstein-Zernikeequation[ 32 ]:c(r)=h(r))]TJ /F6 11.955 Tf 11.95 0 Td[(nZdr0c(jr)]TJ /F7 11.955 Tf 11.96 0 Td[(r0j)h(r0) (4)Theclassicaltemperatureisobtainedfromthecorrespondenceconditionofequalpressures,Equation( 3 ),andtheclassicalvirialequationc =cpc p=n p1)]TJ /F6 11.955 Tf 13.15 8.09 Td[(n 6Zdrh(r)rrcc(r) (4)whereweusedtheeffectivepotentialderivedinEquation( 4 ).Thereplacementofthepaircorrelationfunctiong(r)bytheholefunctionh(r)=g(r))]TJ /F5 11.955 Tf 12.14 0 Td[(1occursbecauseoftheuniformneutralizingbackground[ 32 ].Theclassicalactivityccisgivenby[ 45 ]cc=ln)]TJ /F6 11.955 Tf 5.48 -9.68 Td[(n3c)]TJ /F6 11.955 Tf 11.95 0 Td[(nZdrc(r)+cc(r))]TJ /F5 11.955 Tf 13.15 8.09 Td[(1 2h(r)(h(r))]TJ /F6 11.955 Tf 11.96 0 Td[(c(r)) (4)wherec=(2c~2=m)1=2.Thefunctionsgareallthequantumpaircorrelationfunctions.TheclassicalpaircorrelationfunctionshavebeenreplacedbythequantumpaircorrelationsbyusingtheequivalenceconditionsinEquations( 3 ).Inthefollowingsectionsweconsiderauniformelectrongaswithdimensionlesstemperaturetandaverageinterparticledimensionlessdistancers.Theobjectivehereistondawaytoputappropriateapproximationsintothesystemforpracticaluse. 51

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4.1.1ClassicalPotentialcc(r)ThedominantexchangeeffectshavealreadybeenincludedinthequantitiescalculatedinChapter 3 fortheidealFermigas.Thereforeitisconvenienttowrite[ 35 ]cc(r)intheformcc(r)=(cc(r))(0)+(r) (4)wherethersttermisthePaulipotentialand(r)istheCoulombinteractionterm.ThePaulipotentialcapturestheidealgasexchangeeffectsandtheCoulombtermcontainsthemodicationsoftheCoulombinteractionbyboththeexchangeandthediffractioneffects.Writingtheeffectivepotentialinthiswayimplieswerecovertheidealgasresultsinthelimitofzerointeractions.Intheclassicallimit,thePaulipotentialvanishesandtheCoulombpartbecomestheCoulombpotential.Intheweakcouplinglimit,thedirectcorrelationfunctionbecomesproportionaltothepotentialcc(r)!)]TJ /F6 11.955 Tf 24.58 0 Td[(c(r),(cc(r))(0)!)]TJ /F6 11.955 Tf 24.58 0 Td[(c(0)(r) (4)ThisgivesanexplicitformfortheCoulombterm(r)=)]TJ /F10 11.955 Tf 11.29 9.68 Td[()]TJ /F6 11.955 Tf 5.48 -9.68 Td[(c(r))]TJ /F6 11.955 Tf 11.95 0 Td[(c(0)(r).Thusweproposeanapproximationfortheeffectivepotential:cc(r)!(cc(r))(0))]TJ /F10 11.955 Tf 11.95 9.68 Td[()]TJ /F6 11.955 Tf 5.48 -9.68 Td[(c(r))]TJ /F6 11.955 Tf 11.96 0 Td[(c(0)(r)(w) (4)where)]TJ /F6 11.955 Tf 5.48 -9.69 Td[(c(r))]TJ /F6 11.955 Tf 11.96 0 Td[(c(0)(r)(w)denotesaweakcouplingcalculationofthedirectcorrelationfunctionsfromtheOrnstein-Zernickeequation,Equation( 4 ).Thisapproximationclearlyincorporatestheidealgasandweakcouplinglimits.FortheclassicalOCP(Coulombpotential)thisyieldstheDebye-Huckelapproximationtog(r).Inthequan-tumcase,theweakcouplinglimitgRPA(r)canbecalculatedexplicitlyusingtherandomphaseapproximation[ 1 ].TodeterminecweusegRPAintheOrnstein-Zernikeequation, 52

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Equation( 4 ).c(r)(w)=1 nZdk (2)3e)]TJ /F4 7.97 Tf 6.58 0 Td[(ikrSRPA(k))]TJ /F5 11.955 Tf 11.96 0 Td[(1 SRPA(k) (4)whereSRPA(k)istheRPAstaticstructurefactorSRPA(k)=1+nZdreikrhRPA(r) (4)FinallythemodiedCoulombpotential(r)inEquation( 4 )becomes(r)!1 nZdk (2)3e)]TJ /F4 7.97 Tf 6.59 0 Td[(ikr1 SRPA(k))]TJ /F5 11.955 Tf 28.27 8.09 Td[(1 S(0)(k) (4)ThedenitionofSRPA(k)intermsoftheLindhardfunctionandtheRPAdielectricfunctionisgiveninAppendix D .Theproofofsomeofthelimitsof(r)=(t,rs,r)isgiveninAppendix D .TheRPAstructurefactorbehavesasSRPA(k)/k2becauseoftheperfectscreeningsumrule.Aconsequenceofthisisthatforlarger=r=r0,(r)hasaCoulombtailasshownintheAppendix D .limr(t,rs,r)!)]TJ /F4 7.97 Tf 6.78 -1.79 Td[(e(t,rs)r)]TJ /F9 7.97 Tf 10.82 0 Td[(1 (4)where)]TJ /F4 7.97 Tf 6.77 -1.79 Td[(e(t,rs)isaneffectiveCoulombcouplingconstant)]TJ /F4 7.97 Tf 6.77 -1.79 Td[(e(t,rs)=2 ~!pcoth(~!p=2),)]TJ /F2 11.955 Tf 36.75 0 Td[(q2 r0 (4)Here!p=p 4nq2=mistheplasmafrequency.Thedimensionlessparameteris~!p=(4=3))]TJ /F5 11.955 Tf 5.48 -9.68 Td[(2p 3=21=3p rs=tsoforxedrsthehighandlowtemperaturelimitsare)]TJ /F4 7.97 Tf 6.78 -1.8 Td[(e!8><>:,~!p<<1)]TJ /F9 7.97 Tf 6.68 -4.97 Td[(4 3rs1=2,~!p>>1 (4) 53

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ThecoefcientoftheCoulombtailfollowsfromthefactthattheRPAincorporatestheexactperfectscreeningsumrule[ 1 50 51 ].Itisillustratedforr(t,rs,r)inFigure 4-1 atrs=5forseveralvaluesoft.AlsoshowninthisgurearetheresultsfromthePDWclassicalpotentialwhichwebrieydiscussedinSection 2.2.2 (cc(r))PDW=(cc(r))(0)+PDW(r) (4)ThePaulipotential(cc(r))(0)asinisthesameasinEquation( 4 )butitsCoulombpartPDW(r)isgivenbytheDeutschpotential[ 21 ]PDW(r)=PDWq2 r1)]TJ /F6 11.955 Tf 11.96 0 Td[(e)]TJ /F4 7.97 Tf 6.59 0 Td[(r=PDW,PDW=PDW~2 m1=2 (4)wherePDW=1=kBTPDWandTPDW)]TJ /F6 11.955 Tf 5.48 -9.69 Td[(T2+T201=2.ThusthePDWpotentialdependsononlyoneunspeciedparameterT0.SincethePaulipotentialisindependentofrs,thedensitydependenceinthepotentialoccursthroughT0.TheT0=T0(rs)isobtainedbyrequiringthattheclassicalexcessenergyatT0matchesthequantumexchange-correlationenergyobtainedfromquantumsimulationatT=0.T0isttedtoaform[ 33 34 ]T0'TF a+bp rs+crs (4)witha=1.594,b=)]TJ /F5 11.955 Tf 9.3 0 Td[(0.3160,andc=0.0240whendiffusionMonteCarlodataisused.ForvariationalMonteCarlodata,thettingparametersarea=1.3251,b=)]TJ /F5 11.955 Tf 9.3 0 Td[(0.1779andc=0.0.ThePDWpotentialisquitesimilartotheapproximateformofourpotentialinEquation( 4 )atrs=5.Greaterdiscrepanciesoccurforbothlargerandsmallerrsexceptathighertemperatures.Furthercommentsonthiscomparisonaregivenbelow. 54

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Figure4-1. Demonstrationofcrossoverforr4(t,rs,r)toCoulombwitheffectivecouplingconstant)]TJ /F4 7.97 Tf 6.78 -1.8 Td[(e(t,rs)givenbyEquation( 4 ),forrs=5andt=0.5,1and10[ 48 ].Alsoshownarethecorrespondingresultsforr4PDW(t,rs,r). Closetotheoriginr<<1,(t,rs,r)approachesanitevaluesotheCoulombpotentialisregularized.(t,rs,0)=1 nZdk (2)31 SRPA(k))]TJ /F5 11.955 Tf 28.27 8.08 Td[(1 S(0)(k) (4)TheintegralconvergesbecausetheRPAstaticstructurefactorsforlargekapproach1ask)]TJ /F9 7.97 Tf 6.58 0 Td[(4duetoquantumeffects(cuspcondition[ 57 ]).Finally,anotherlimitobtainedintheAppendix D isthatforlargersandlarget(lowdensity,hightemperature)inwhichcasetheKelbgpotential[ 20 ]isrecoveredlimt,rs>>1(t,rs,r)!K(t,r)=)]TJ ET q .478 w 180.86 -461.63 m 191.1 -461.63 l S Q BT /F6 11.955 Tf 180.86 -472.82 Td[(r)]TJ /F5 11.955 Tf 5.48 -9.68 Td[(1)]TJ /F5 11.955 Tf 11.96 0 Td[(exp()]TJ /F6 11.955 Tf 9.29 0 Td[(r2=a2)+p r=a(1)]TJ /F5 11.955 Tf 11.96 0 Td[(erf(r=a)) (4)witha==(p 2r0).TheKelbgpotentialisobtainedastheweakcouplinglimitoftheeffectivepotentialdenedusingthetwoparticledensitymatrix.ThedetailsaregiveninChapter 2 .Thislimitisapproachedtowithin10percentatt=10and1rs10.NOTE:i)TherearetwowaystocalculatetheRPAstaticstructurefactorSRPA(k).Onewayistousethedenitionofthestaticstructurefactorasafrequencyintegralofthedynamicstructurefactor.Atsmallmomentumkvalues,theintegrand(thedynamicstructurefactor)startsdevelopingsharppeaks.Thepeaksoccurbecauseoftheplasma 55

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oscillationswherethedielectricfunction(thedenominatoroftheintegrand)becomesverysmall.Thenumericalestimationoftheintegrationisverydifcultatsmallmomen-tum.Toovercomethisproblemweusedthefrequencysummationrepresentationofthestaticstructuregivenin[ 39 ].TheexplicitformoftheintegralandthefrequencysumsaregivenintheAppendix D .ii)Whenexpressedintermsofthedensity,thedegeneracyparameterzthatoccursintheFermifunctioninthestructurefactorcalculationdependsonthechemicalpotentialoftheidealFermigas,notthatforjellium.Thischoiceofzpreservestheexactscreeningsumrule. 4.1.2ClassicalEffectiveTemperatureAndChemicalPotentialTheapproximatetemperatureandchemicalpotentialequationsareobtainedfollowingthesamelogicofsplittingthethermodynamicquantitiesintoanidealgastermandaninteractionterm.Theinteractiontermisdeterminedfromtheweakcouplingformsofthesequantities.c=(0)c+)]TJ /F3 11.955 Tf 5.48 -9.69 Td[(RPAc)]TJ /F3 11.955 Tf 11.96 0 Td[(RPA,(0)c (4)cc=(cc)(0)+(cc)RPA)]TJ /F5 11.955 Tf 11.95 -.17 Td[((cc)RPA,(0) (4)where(0)cand(cc)(0)denotetheidealgasresultsof[ 35 ],andfromEquations( 4 )and( 4 )RPAc=nh1)]TJ /F4 7.97 Tf 13.15 4.71 Td[(n 6RdrhRPA(r)rr(cc(r))RPAi pRPA (4)(cc)RPA=3 2lnRPAc +ln)]TJ /F6 11.955 Tf 5.48 -9.68 Td[(n3RPA+1 2nZdrhRPA(r)hRPA(r)+(cc(r))RPA (4)TheRPAresultsforpRPAand)]TJ /F6 11.955 Tf 5.48 -9.69 Td[(n3RPAarecomputedfromthePadetsofreference[ 52 ]. 56

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Figure4-2. QuantumRPApressurepRPAatt=0asafunctionofrs[ 48 ]. Thepressureforjelliumbecomesnegativeatlargersandsmallt[ 52 53 ].Forrealsystems,thepressureispositiveasfollowsfromtheconvexityofthefreeenergyasafunctionofthevolume.Thisconvexitydoesnotholdforjellium[ 45 47 ].Thepressureoftheeffectiveclassicalsystematthecorrespondingtemperatureanddensitiesisstillpositive.FromtheEquation( 4 )wegetunphysicalresultslikenegativetemperatures.SincepRPAintheEquation( 4 )isamonotonicfunctionoft,forsometemperaturet=t0,pRPA=0.Thissituationoccursforrs4.Thereforeforjellium,thisisoneoftheequivalenceconditionsthatshouldbereplacedbyadifferentcondition(e.g.,equivalenceofinternalenergies).Instead,theanalysishereisrestrictedtot>t0(rs)toassurepositivepressure.Figure 4-3 showstc=Tc=TFasafunctionoftcalculatedfromEquation( 4 )forrs=0,1,3,4,and5.Figure 4-4 showsthecorrespondingresultsforc=FcalculatedfromEquation( 4 ). 4.2RadialDistributionFunctionAndThermodynamicsWecanusetheapproximateeffectivepotentialgivenbytheEquations( 4 )and( 4 )inMDorusetheEquations( 4 )and( 4 )tocalculatethepaircorrelationfunctionsg(r).Theresultingfunctionghastwokindsofcorrelationsbuiltintoit:i)theweakcouplingcorrelationsinRPA(ringdiagrams)throughSRPAandii)classicalstrongcouplingeffects(e.g.,moleculardynamicssimulation,HNCequations).Herewewill 57

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Figure4-3. ClassicalreducedtemperatureTc=TFasafunctionoftforrs=0,1,3and4[ 48 ]. Figure4-4. Dimensionlessclassicalchemicalpotentialc=EFasafunctionoftforrs=1,3,5[ 48 ]. solveforgusingtheHNCapproximationEquations( 4 )and( 4 )lng(r)=)]TJ /F3 11.955 Tf 9.3 0 Td[(cc(r)+h(r))]TJ /F6 11.955 Tf 11.96 0 Td[(c(r),c(r)=h(r))]TJ /F6 11.955 Tf 11.95 0 Td[(nZdr0c(jr)]TJ /F7 11.955 Tf 11.95 0 Td[(r0j)h(r0) (4)TheeffectivepotentialintheformalisminChapter 3 isdenedusingtheHNCequationswhichtakesthequantumpaircorrelationfunctionasinput.ButtheEquation( 4 )producesthepaircorrelationfunctionastheoutput.Althoughitlookslikeacircularlogic,thefactisweconstructedccusingonlytheweakcouplingformoftheHNCequations.Thequantuminputforccandg(r)iscalculatedusingtheRPAtheory, 58

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hencethenalpaircorrelationfromtheEquation( 4 )containstheeffectsfromtheweakquantumcorrelationsandthestrongclassicalcorrelations.TheformoftheHNCequationsguaranteesg(r)alwaysstayspositivejustasnotedin[ 33 ].IncontrastthepaircorrelationfunctionbecomesnegativeatshortdistancesforlargevaluesofrsandlowtemperaturesintypicalcalculationsbasedonstaticlocaleldcorrectionstoRPA.Thecomputationofg(r)fromEquation( 4 )hasbeendoneusingNg'smethod[ 58 ]bysplittingtheinteractionpotentialintoalongrangedandashortrangedpartfollowingthatreference.TheEquations( 4 )and( 4 )donotusethepressureortemperaturecalculations.Hencetheydonothavetherestrictiontopositivepressuresandtheassociatedrestrictiononrs.TheresultsareshowninFigure 4-5 forthecaseofrs=6att=0.5,1,4and8.AlsoshownaretheresultsfromtherecentrestrictedPIMC[ 11 ].Theagreementisquitegood.Figure 4-6 showsthesameconditionsasFigure 4-5 forcomparisonwiththeresultsofPDW.Theagreementisremarkableeventhoughtheyhaveverydifferentorigins.Thisagreementbetweenthepredictionshere,PIMC,andPDWextendstootherstateconditionsaswell,exceptforsmalltandverylargers. Figure4-5. Radialdistributionfunctiong(r)forrs=6att=0.5,1,4,8.AlsoshownaretheresultsofPIMC[ 48 ]. Othertheoreticalmodelsforg(r)arebasedonmodifyingthedielectricformalismoftheRPAbyincludinglocaleldcorrections.TheSTLSmodel[ 13 ]isoneofthe 59

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Figure4-6. Radialdistributionfunctiong(r)forrs=6att=0.5,1,4,8.AlsoshownaretheresultsofPDW[ 48 ]. earliestself-consistentatT=0,latergeneralizedtonitetemperaturebyTanakaandIchimaru(TI)[ 39 ].WebrieylookedatboththeRPAandSTLSschemeinSection 2.1.2 .AtmetallicdensitiesandlowtemperaturesTIfoundthatthepaircorrelationfunctionsbecomenegativeasshowninFigure 4-7 .TheRPAresultsaresignicantlymorenegativeinthisrange.Figure 4-7 showsthecomparisonofthepaircorrelationfunctionfromourmodelwithTIatrs=5att=0.5,1and10.BothRPAanditsimprovedTIoverestimatethesizeoftheelectroncorrelationhole[ 1 ]atlargervaluesforrs. Figure4-7. Radialdistributionfunctiong(r)forrs=5att=0.5,1and10.AlsoshownaretheresultsofTanaka-Ichimaru[ 48 ]. 60

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Figure4-8. Radialdistributionfunctiong(r)fort=0atrs=1,5,10.AlsoshownareresultsfromPIMCanddiffusionMonteCarlo.ThePIMCanddiffusionMonteCarloplotsareindistinguishable[ 11 48 ]. ThePDWg(r)isingoodagreementwithdiffusionMCdataatt=0[ 33 ]forrs=1,5,10.ThePDWmodeldependsononeparameterT0whichcontainsallthequantummany-bodyeffects.Itisdeterminedbyttingtheclassicalexcessenergytothet=0exchange/correlationenergyfromMCdata.Itisimpressivethatthisprovidesgoodresultsforg(r)acrossarangeofbothrsandt.Figure 4-8 showsacomparisonoftheresultsofthepresentanalysiswiththesameT=0diffusionMCdata[ 56 ],andalsotherecentPIMCforT=0.065atrs=1,10.ThegoodagreementisquitesurprisingsincethereisnoMCparametrizationinthepresentanalysisandallquantuminputisviatheRPAandidealgasexchange.However,itisrecalledthattheRPApreservestheexactquantummechanicsoftheperfectscreeningsumrulethatgovernsthecrossovertotheexactlargerCoulomblimit.ThisisdiscussedfurtherinSection 4.1.1 4.2.1ThermodynamicsThepredictedpressure,pcinatomicunits,fortheeffectiveclassicalsystemisobtainedfromcpc nc=1)]TJ /F5 11.955 Tf 13.15 8.09 Td[(1 6nZdrh(r)rrcc(r) (4) 61

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usingtheeffectivetemperature,Equation( 4 ),andthefullpaircorrelationfunctionobtainedfromtheEquation( 4 ).Figure 4-9 showsthisasafunctionoftforrs=1,3and5.AlsoshownarethecorrespondingresultsformodiedRPA(usingthetsfromreference[ 53 ]). Figure4-9. Dimensionlessclassicalpressurepc=(nF)asafunctionoftforrs=1,3,5.AlsoshownarethecorrespondingmodiedRPAresults[ 48 ]. Figure4-10. ThepaircorrelationfunctioniscalculatedusingthemodiedKelbgpotentialatt=8forrs=1,6,10and40.ComparisonwiththePIMCisalsoshown[ 59 ]. 4.3KelbgFittingForTheEffectivePotentialccThecalculationoftheeffectivepotentialcintheEquation( 4 )iscomplicatedasitinvolvescomputingtheRPAstructureandthenthedirectcorrelationfunction. 62

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Figure4-11. ThepaircorrelationfunctioniscalculatedusingthemodiedKelbgpotentialatt=0.5forrs=1,6,10and40.ComparisonwiththePIMCisalsoshown[ 59 ]. Figure4-12. ThepaircorrelationfunctioniscalculatedusingthemodiedKelbgpotentialatt=1forrs=1,6,10and40.ComparisonwiththePIMCisalsoshown[ 59 ]. Forpracticalapplicationitwouldbemoreconvenienttoobtainattingfunctionforthepotential.chasanexactKelbgformathightemperaturesasprovedinAppendix D .TheeffectivepotentialisrequiredtohaveaCoulombtailwhichhasthecouplingcoefcientdeterminedbytheexactscreeningsumrule.TheeffectivecouplingconstantisgivenbytheformintheEquation( 4 ).First,theweakcouplingKelbgformEquation 63

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Figure4-13. ComparisonofthepaircorrelationfunctionsforthemodiedKelbgpotentialatt=0forrs=1,6,10and40withPIMCatt=0.0625anddiffusionMonteCarloatt=0[ 59 ]. ( 4 )isimprovedbyincludinganotherparameter[ 27 ]4K(r)=)]TJ ET q .478 w 146.05 -294.28 m 156.29 -294.28 l S Q BT /F6 11.955 Tf 146.05 -305.47 Td[(r1)]TJ /F6 11.955 Tf 11.96 0 Td[(exp()]TJ /F5 11.955 Tf 9.3 0 Td[((ar)2)+p (ar) erfc(ar) (4)where=)]TJ /F9 7.97 Tf 11.93 5.48 Td[(()]TJ /F4 7.97 Tf 4.82 0 Td[(rs)1=2 ln(S()]TJ /F4 7.97 Tf 8.27 0 Td[(rs))anda=(rs=\1=2.ThefunctionSisthetwoelectronSlatersumattheorigin.S(x)=)]TJ /F5 11.955 Tf 9.3 0 Td[((4x)1=2Z10dye)]TJ /F4 7.97 Tf 6.59 0 Td[(y2y 1)]TJ /F6 11.955 Tf 11.95 0 Td[(ep x=y (4)ThenewparameteraffectstheheightofthemodiedpotentialinEquation( 4 )attheoriginwhileitkeepstherstderivativeofthepotentialthesameastheoriginalKelbgpotential,Equation( 4 ).Second,inorderforthiseffectivepotentialtoagreewiththecorrectlargedistancebehavioroftheoriginaleffectivepotential,Equation( 4 ),theCoulombcoefcient)]TJ /F1 11.955 Tf -436.52 -23.91 Td[(inEquation( 4 )shouldbereplacedbytheeffectivecouplingconstant)]TJ /F4 7.97 Tf 6.78 -1.79 Td[(e,Equation( 4 ).Weproposetheformoftheeffectivepotential[ 59 ]cc(r)=(cc)(0)(r)+4K(r) (4) 64

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ThiseffectivepotentialcanbeusedintheHNCequations,Equations( 4 ),toobtainthepaircorrelationfunctionsg(r)atvarioustemperaturesanddensities.ThecomparisonofthepaircorrelationfunctionswiththePIMCdataisshownbelow.Theagreementisgoodatrelativelyhightemperaturet=8for1rs40asshowninFigure 4-10 .Figures 4-11 and 4-12 showthecomparisonatt=0.5and1respectively.Againtheagreementisgoodexceptatverylowdensitiesrs=40.Itestimatesthelocationofthepeakofthepaircorrelationfunctiongcorrectlybutunderestimatesitsheight.AlsoshowninFigure 4-13 isthecomparisonofgwiththatofPIMCatt=0.0625anddiffusionMonteCarloatzerotemperature.Fromtheguresitisseenthatghasaweakdependenceonthetemperatureforrs>1andt0.5. 4.4Summary 1. Weconstructedtheeffectiveclassicalsystemforjellium. 2. Theeffectiveinteractionpotentialciswrittenassumoftheexchangeterm(Paulipotential)andCoulombinteractionterm.TheexplicitformoftheCoulombtermisobtainedusingweakcouplingclassicaltheoryandquantumrandomphaseapproximation.TheresultisaregularizedCoulombpotentialatsmallr,andaCoulombtailwhosecoefcientisdeterminedbytheexactscreeningsumrule. 3. ThepaircorrelationswerecalculatedusingtheHNCequationsandtheeffectivepotentialandcomparedwiththePDWtheoryandPIMC. 4. Theeffectivetemperatureandtheeffectivechemicalpotentialarealsocalculatedwithintheweakcouplingapproximations.ThenthefullpressureandchemicalpotentialwerecalculatedfromtheHNCequations. 65

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CHAPTER5SHELLFORMATIONINNONUNIFORMSYSTEMSThisChaptergivesabriefoverviewoftheclassicaltheoriesoftheshellformationduetoclassicalcorrelations.Thisisthenusedasthebasisforstudyingquantumeffectsontheshellformationviatheeffectiveclassicalsystem.ZerotemperatureclassicalDFTisusedtoobtainthegroundstatedensitybyminimizingtheenergyfunctional[ 60 ].ThecontributionsofthecorrelationsareincludedusingLDAandtsforthecorrelationenergiesforuniformsystem[ 61 ].Theshellmodelsphenomenologicallyconstructtheenergyfunctionalthatareparameterizedbythenumberofshells,theirpositionandthenumberofchargesineachshellandthenminimizedittosolvefortheseparameters.Wrightonetal.[ 46 49 ]successfullyusedclassicalDFTandHNCtoincludethecorrelationsatnitetemperatures.Theshellformationduetostrongquantumcorrelationsinultra-coldatomsinatrapandquantumdotsisstudiedonlyatzerotemperature.Sincetherearenonitetemperaturetheoriestoexplaintheshellsinthesesystems,wewillextendourformalisminChapter 3 totheconnedquantumsystems.Webrieylookintothemechanismoftheshellformation:quantumeffectsandquantumcorrelations.WeprimarilyusethemodelofWrightonetal.andtheclassicalmapofChapter 3 and 4 tostudyquantumeffectsintheshellformationinaharmonictrap.TheeffectivepotentialsandcorrelationfunctionsofthepreviousChaptersareusedintheHNCequationstocalculatethedensityproleoftheeffectivesystem.Thedensityofclassicalsystemsdependsontwoparameters,thenumberofparticlesinthetrapandthestrengthofthepotential(couplingconstant).Thisdependencewillbediscussedfortheclassicalsystemsfollowingthediscussioninreference[ 46 49 ]andwillbeextendedtothequantumsystemsinthemeaneldtheory. 5.1ZeroTemperatureClassicalDensityJustasinthezerotemperaturequantumDFT,theaverageenergyofaclassicalsystemisminimizedattheequilibriumdensityproleinclassicalDFT[ 60 ].Toobtain 66

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thedensityprolewewritetheaverageenergyasafunctionalofthedensity.LetusconsiderasystemofNparticlesinanexternalpotentialext(r)andinteractingvia(r,r0).TheHamiltonianisgivenby:^H=NXi=1^p2i 2m+Zdrext^n(r)+Zdrdr0(r,r0)^g(r,r0) (5)Theaverageenergyisgivenby:E=h^Hi=3 2kBT+NZdrextn(r)+N(N)]TJ /F5 11.955 Tf 11.96 0 Td[(1) 2Zdrdr0(r,r0)g(r,r0)n(r)n(r0) (5)whereh^n(r)i=n(r)istheaveragedensityandh^g(r,r0)i=g(r,r0)n(r)n(r0)istheaveragepaircorrelationfunction.Sincethepaircorrelationfunctionsaredifculttocalculate,Henningetal.[ 61 ]includedcorrelationsthroughalocaldensityapproximation.Thisisdoneinthefollowingsteps:Startwithenergyexpressionfortheuniformsystem:Rdrdr0(r,r0)g(r,r0)n(r)n(r0)!n2Rdrdr0(r)]TJ /F7 11.955 Tf 12.6 0 Td[(r0)(h(r)]TJ /F7 11.955 Tf 11.96 0 Td[(r0)+1)=Vn2(k)+Ecorr.ThecorrelationpartofenergyEcorrisgivenforYukawapotentialbyanapproximateformbyTotsujietal[ 62 ]:Ecorr=)]TJ /F5 11.955 Tf 9.3 0 Td[(1.44q2n4=3exp()]TJ /F5 11.955 Tf 9.3 0 Td[(0.375n)]TJ /F9 7.97 Tf 6.59 0 Td[(1=3+0.000074(n)]TJ /F9 7.97 Tf 6.59 0 Td[(1=3)4).UsingLDAtheuniformdensitynisreplacedbynon-uniformdensityn(r).Theenergyisnowminimizedtosolveforthegroundstatedensity.EvenafterincludingthecorrelationsthroughLDA,itfailstoproduceanyshellstructure.Howeveritagreesquitewellwiththeaveragedshelldensitiesfromthesimulations. 5.2ShellModelsShellmodelsattempttoexplainthelocationandnumberoftheshellsandthenumberofparticlesinashellbyminimizingtheenergyfunction[ 63 65 ].ItassumesadensityrepresentedbyasetofnestedshellsnSM(r)=LX=1N(jrj)]TJ /F6 11.955 Tf 17.93 0 Td[(R) 4R2 (5) 67

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whereLrepresentsthetotalnumberofthinshells,locatedataradiusRandwithnumberofparticlesNinthethshell.TheenergyisdenedasafunctionofthisshellmodeldensitynSM.ForYukawasystems(q2 rexp()]TJ /F3 11.955 Tf 9.3 0 Td[(r)),theaveragetotalenergyisthesumofthreeterms[ 64 65 ]ESM=Eext+Eintra+Einter (5)Eext=LXnu=1Next(R) (5)Eintra=LXnu=1Nq2eR Rsinh(R) RN 2 (5)Einter=LXnu=1Nq2eR RLX
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bytheuniformcorrelationsofOCPsystemscOCP(r)]TJ /F7 11.955 Tf 10.32 0 Td[(r0).TheresultsherearespecializedtotheirworkontheCoulombsystemsinharmonictraps.Thustheinteractionpotentialis(r)=)]TJ ET q .478 w 54 -56.78 m 62.17 -56.78 l S Q BT /F4 7.97 Tf 54 -63.89 Td[(randthelocalchemicalpotential)]TJ /F9 7.97 Tf 13.7 4.71 Td[()]TJ ET q .478 w 268 -56.78 m 272.82 -56.78 l S Q BT /F9 7.97 Tf 268.19 -63.89 Td[(2(r)2,r=r=r0,)-391(=q2=r0.TheuniformcorrelationsforsuchsystemareobtainedfromtheHNCequationsforthepaircorrelationfunctionsfortheonecomponentplasmaandtheOrnstein-Zernikeequation,Equation( 4 ).Atthemeaneldlevel,whichmeansreplacingc(r)!)]TJ /F5 11.955 Tf 26.27 0 Td[()]TJ /F3 11.955 Tf 6.77 0 Td[(=r,theydidnotgetanyshellsforanyvalueofthecouplingconstant)]TJ /F1 11.955 Tf 6.78 0 Td[(,asshowninFigure 5-1 .HoweverincludingthecorrelationsintheHNCequationsproducedshellsforstrongcoupling)]TJ /F3 11.955 Tf 10.1 0 Td[(>>1asshowninFigure 5-3 Figure5-1. MeaneldclassicaldensityproleforN=100particlesforvariousvaluesof)]TJ /F1 11.955 Tf 6.78 0 Td[(.ThemeaneldCoulomblimit)]TJ /F2 11.955 Tf 10.1 0 Td[(!1isastepfunction[ 46 ]. Wegeneralizetheirresultstothequantumsystems.Wewillusethesameequation,Equation( 5 ),butthequantuminformationisincorporatedviathetwoinputs,theeffectivelocalchemicalpotentialcandthedirectcorrelationscforjelliumcalculatedinChapter 4 .IntherestofthisChapter,weaddressthequestionwhetherthequantumeffectscanproduceshellstructureinthemeaneldlimit.Theeffectivelocalchemicalpotentialcisnolongerharmonic,butismodiedbyquantumeffectsasisshownbelow.Henceaninterestingpreliminaryexplorationwouldbetogeneratethedensityproleforsomegeneralexternalpolynomialpotentialsatmeaneldandseethe 69

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effectsoftheirshapeonthepossibilityofshellformation.Theeffectivelocalchemicalpotentialcorrespondingtotheclassicalharmonictrapchangesitsshapeatdifferenttemperaturesanddensities.Knowingtheeffectsoftheshapeoftheexternalpotentialontheshellformationwouldhelpusstudytheinuenceofthequantumeffectsatdifferenttemperatureanddensityrange. 5.3.1ShellFormationInAPolynomialPotentialConsideraclassicalsystemofNparticlesinteractingviaapotentialconnedbyatrappotentialext.ThetotalpotentialenergyoftheHamiltonianisgivenbytheEquation( 5 ).TheradiusofthetrapR,whichistheaveragepositionoftheoutermostparticle,isdenedtobethepositionwheretheforcesduetotheinteractionpotentialofotherparticlesontheoutermostparticleisbalancedbytheforceduetothetrappotentialrext(r).FortheparticularcaseofCoulombinteractions(r)=q2=randaclassofexternalpotentialslabeledbyaparameter,ext(r,),theforcebalanceconditiongives0ext(R,)=(N)]TJ /F5 11.955 Tf 11.96 0 Td[(1)q2 R2, (5)Usingthefollowingidentities:4nr30=3=14nR3=3=N)R3=r30N, (5)andthelastequationfromEquation( 5 ),wegetr20N2=30ext(r0N1=3,)=(N)]TJ /F5 11.955 Tf 11.96 0 Td[(1)q2 (5)Thedimensionlesspotentialenergyforthesystemis:(ri)=)]TJ /F10 11.955 Tf 29.59 20.45 Td[( N1=3 r0Xiext(rir0,) 0ext(r0N1=3,)+1 2Xi6=j1 jri)]TJ /F7 11.955 Tf 11.95 0 Td[(rjj!, (5) 70

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where)-281(=q2=r0.Inparticularforapolynomialexternalpotentialoftheformext(r,)=rn,thetotalpotentialbecomes:=)]TJ /F10 11.955 Tf 21.38 20.44 Td[( N(2)]TJ /F4 7.97 Tf 6.59 0 Td[(n)=3 nXirin+1 2Xi6=j1 jri)]TJ /F7 11.955 Tf 11.96 0 Td[(rjj!, (5)Inparticularforaharmonictrapwehavethetotalpotential:=)]TJ /F10 11.955 Tf 21.39 20.45 Td[( 1 2Xiri2+1 2Xi6=j1 jri)]TJ /F7 11.955 Tf 11.95 0 Td[(rjj! (5) (5)WecannowcalculatethedensityprolefromtheEquation( 5 ).Formostexper-imentsandsimulationsitismoreconvenienttousetheaveragenumberofparticlesinthetrapNinsteadofthechemicalpotential.TheconversionbetweenthetwoisdoneusingtheequationRdrn(r,)=N.Theresultis:n(r)=Nexp()]TJ /F5 11.955 Tf 9.29 0 Td[()]TJ /F6 11.955 Tf 6.78 0 Td[(U(r)) Rdr0exp()]TJ /F5 11.955 Tf 9.3 0 Td[()]TJ /F6 11.955 Tf 6.77 0 Td[(U(r0)))]TJ /F6 11.955 Tf 6.78 0 Td[(U(r)=ext(r)+NRdr0exp()]TJ /F5 11.955 Tf 9.3 0 Td[()]TJ /F6 11.955 Tf 6.77 0 Td[(U(r0))cOCP(jr)]TJ /F7 11.955 Tf 11.95 0 Td[(r0j) Rdr0exp()]TJ /F5 11.955 Tf 9.3 0 Td[()]TJ /F6 11.955 Tf 6.77 0 Td[(U(r0)) (5)where)-318(=q2=r0.Alltheaboveequationscanbewrittenintermsofthedimensionlessdistancex=r=r0.WecancalculatethedensityproleforvariouspolynomialpotentialsasinEquation( 5 ):ext(x)=)]TJ /F4 7.97 Tf 34.86 4.7 Td[(N(2)]TJ /F16 5.978 Tf 5.76 0 Td[(n)=3 nxninthemeaneldlimiti.e.replacingcOCP(r)]TJ /F7 11.955 Tf 11.95 0 Td[(r0)by)]TJ /F3 11.955 Tf 9.29 0 Td[((r)]TJ /F7 11.955 Tf 11.96 0 Td[(r0),oftheEquation( 5 ).TheresultsareshowninFigure 5-2 .TheplotsshowthatinthemeaneldlimitwedonotgetanyshellforthelinearandharmonictrapsasfoundbyWrightonetal.[ 46 ].Howeverhigherordertrappolynomialsproduceshellsi.e.whentheeffectivelocalchemicalpotentialissteeperthantheharmonicpotentialwegetshells.Fortherestofthesection,wefocusononlytheharmonictrappotentials.Theaboveresultscanbegeneralizedbeyondthemeaneldbyincludingthecorrelations 71

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Figure5-2. ThedensityproleinthemeaneldlimitofCoulombsystemsindifferentpolynomialtrappotentialsfor)-277(=40 fromtheuniformOCP,cOCP.Thedensityproledependsontwoparameters)]TJ /F1 11.955 Tf 10.1 0 Td[(andN,soitisinterestingtoseehowthedensitychangesastheseparametersarechanged.ForxednumberofparticlesinthetrapN,thesharpnessoftheshellsincreasesasweincrease)]TJ /F1 11.955 Tf 10.1 0 Td[(butthenumberandthepositionofshellsremainsunchanged[ 46 ]asshowninFigure 5-3 Figure5-3. Thevariationofthedensityprolewiththecouplingconstant)]TJ /F1 11.955 Tf 10.09 0 Td[(for100particlesinthetrap[ 46 ]. Intheothercaseataconstant)]TJ /F1 11.955 Tf 6.77 0 Td[(,thenumberofshellsincreasesaswechangingNasisshowninFigure 5-4 72

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Figure5-4. Thevariationofthedensityprolewiththenumberofparticlesinthetrapfor)-278(=100[ 46 ]. Figure 5-5 showshowtheshellsarelledandwhennewshellsareformed.ThenumberofparticlesineachshellvariesasN2=3.Fromtheseresultsweconcludethat Figure5-5. Thellingoftheshellsandthedependenceofthepopulationineachshellonthetotalnumberofparticlesinthetrapisshown.AlsoshownarethecorrespondingMCandMDdata[ 46 ]. onlythestrongcorrelations)]TJ /F3 11.955 Tf 12.61 0 Td[(>>1produceshellstructure.ThesingularityoftheCoulombpotentialattheorigindestroystheshellsforclassicalparticlesinthemeaneldlimit.Theshellsinaharmonictrapareequi-spacedandthepositionoftheshellsvariesasN2=3.TheHNCequationsprovideagooddescriptionofthestrongcorrelationsintheclassicalsystemsasshowninFigure 1-2 andFigure 5-5 73

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5.4ShellFormationInQuantumSystemsForquantumsystemswestillcanusetheclassicalequationstoobtainthedensityprole[ 46 ].Usingtheeffectivelocalchemicalpotentialc(r)fromChapter 3 andtheuniformcorrelationscforjelliumcalculatedinChapter 4 wecanincorporateappropriatequantumeffects.ln)]TJ /F6 11.955 Tf 5.48 -9.68 Td[(n(r)3c=cc(r)+Zdr0c(jr)]TJ /F7 11.955 Tf 11.96 0 Td[(r0j)n(r0) (5)AnequivalentrepresentationisintermsoftheaveragenumberofparticlesinthetrapNsimilartothediscussioninEquation( 5 ).Thedensityisthengivenby:n(r)=Nexp()]TJ /F3 11.955 Tf 9.3 0 Td[(U(r)) Rdr0exp()]TJ /F3 11.955 Tf 9.3 0 Td[(U(r0))U(r)=cc(r))]TJ /F10 11.955 Tf 11.96 16.27 Td[(Zdr0c(jr)]TJ /F7 11.955 Tf 11.96 0 Td[(r0j)n(r0) (5)Withoutthediffractionandexchangeeffects,thelocalchemicalpotentialinEquation( 5 )isharmonic.Asarstestimateofquantumeffectsontheeffectivelocalchemicalpotential,weneglecttheCoulombinteractionsamongtheparticlesandconsidertheeffectivelocalchemicalpotentialwithonlytheexchangeeffects. 5.4.1EffectiveLocalChemicalPotentialForIdealFermiGasThelocalchemicalpotentialfortheidealFermigasisgivenby:(cc)(0)(r)=ln)]TJ /F6 11.955 Tf 5.47 -9.69 Td[(n(0)(r)((0)c)3)]TJ /F10 11.955 Tf 11.95 16.27 Td[(Zdr0c(0)(jr)]TJ /F7 11.955 Tf 11.96 0 Td[(r0j)n(0)(r0) (5)wheren(0)andc(0)arethedensityanddirectcorrelationfunctionsoftheidealFermigascalculatedinChapter 3 .Thedensityn(0)canbecalculatedintwoways:bysummingoveralltheeigenfunctionsoftheharmonicHamiltonian,orbyalocaldensityapproxima-tion.SomettingfunctionsforthelatterisgiveninAppendix B .ThedetailsofthedensityexpressionsareprovidedinChapter 3 .ThedensityprolecalculatedintwodifferentwaysdonotdiffermuchasdiscussedinSection 3.3 74

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Thelocalchemicalpotential()(0)iscalculatedusingEquation( 5 )withthedensitiescalculatedusingLDAEquations( 3 3 3 ).Theplotsof()(0)forrsandtemperaturest=0.5and10forN=100particlesareshowninFigure 5-6 : Figure5-6. Effectiveclassicaltrappotential4cc(r)=)-277(=(cc(r))]TJ /F3 11.955 Tf 11.96 0 Td[(cc(0))=)]TJ /F1 11.955 Tf 10.1 0 Td[(asafunctionofrfort=0.5,10[ 48 ].Alsoshownistheharmonicpotential. ItcanbeseenfromFigure 5-6 that()(0)differsfromtheharmonicpotentialatintermediatedistancesatverylowtemperatures.Atlargedistancesithasaharmonicasymptote. 5.4.2TheQuantumEffectsOnTheShellFormationInTheMeanFieldTheoryForclassicalCoulombsystemsinaharmonictrapwedonotgetshellsinthemeaneldforanyvalueofthecouplingconstantasdiscussedinSection 5.3 .Inthissectionwestudythequantumeffectsontheshellformationinthemeaneldlimit.Firstweconsiderthediffractioneffect.DiffractioneffectsareputintoclassicalsystemsbyusingaregularizedCoulombpotentialfortheinteractionpotential.InourcasewechoosetheKelbgpotentialfortheinteractionpotentialandtheharmonicpotentialfortheexternalpotential.Thusoursystemhasonlydiffractioneffects.InthemeaneldlimitwegetshellsatverylowtemperaturesasshowninFigure 5-7 for)-363(=3att=0.1,0.5and1.TheshellformationisbecauseofthefactthattheKelbgpotentialisregularizedatthe 75

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Figure5-7. Diffractionmeaneldapproximatedensityprolefor)-277(=3andt=0.1,0.5,1[ 48 ]. originandatverylowtemperaturesitcanmimicthestrongcorrelationsoftheclassicalOCPasshowninFigure 5-8 Figure5-8. Comparisonof)]TJ /F6 11.955 Tf 9.3 0 Td[(c(r)andVK(r)att=0.1,0.27bothcorrespondingto)-278(=3[ 48 ].AlsoshownistheCoulomblimitq2=r. Tostudythedegeneracyeffects,weneglectthediffractioneffectsbyusingtheCoulombpotentialastheinteractionpotential,butkeeptheexchangeeffectsintheexternalpotentiali.e.use()(0).Againwegetshellsatlowtemperaturesforrs=5att=0.5,1,2,5and10for100particlesasshowninFigure 5-9 .Tounderstandthemechanismoftheshellformationfordegeneracyeffects,wenotethatinthecaseofthe 76

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polynomialpotentials,rn,wegetshellsforn>2asshowninFigure 5-2 .Thusasthesteepnessofthepotentialsincreasewegetshells.Atlowtemperaturesanddensitiestheclassicalharmonicpotentialismodiedbythestrongdiffractioneffects,makingitsteeperthantheharmonicpotential.Alsothecouplingconstantislargeatverylowtemperatureswhichisanotherfactorfortheshellformation.Thusthequantumeffectsofdiffractionanddegeneracyproduceshellsatverylowtemperatureseveninthemeaneldlimit. Figure5-9. Exchangemeaneldapproximatedensityproleforrs=5andt=0.5,1,2,5,10[ 48 ]. 5.4.3NDependenceOfRadiusOfTheTrapInthissectionwestudyhowtheradiusofthetrapvariesaswechangetheaveragenumberofparticlesinthetrap.Inparticularweareinterestedinhowthequantumsystemswoulddifferfromtheclassicalsystemsinthewaytheshellsarelled.Theforcebalanceconditioncanbeusedtocalculatethepositionoftheoutermostshell.ThedependenceoftheradiusofthetrapRonthenumberofparticlesNrelatestothepositionoftheothershellsalso.ForclassicalCoulombsystemsinaharmonictraptheradiusofthetrap(positionoftheoutermostparticles)variesasN1=3.ThesamemethodcanbeappliedtotheeffectivepotentialtopredicttheRvsNbehaviorforthequantumsystemsinharmonictraps.Thesepredictionscanbecomparedwiththeexperimental 77

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datafortheultra-coldionsandthequantumdotsinharmonictrapstocheckthevalidityofourformalism.TokeepthingssimpleweconsidertheKelbginteractionpotentialK(r)=)]TJ ET q .478 w 50.46 -56.78 m 55.28 -56.78 l S Q BT /F4 7.97 Tf 50.95 -63.89 Td[(r1)]TJ /F6 11.955 Tf 11.96 0 Td[(exp()]TJ /F6 11.955 Tf 9.3 0 Td[(r2=2K)+p r Kerfc(r=K)fortheeffectiveinteractionpotentialandtheharmonicpotentialastheexternalpotential.ThiswouldgiveustheforcebalanceconditionfromtheEquation( 5 )forthediffractioneffectsonly:0ext(r)=N)]TJ /F5 11.955 Tf 11.95 0 Td[(1 NZdr0n(r0)r(K(jr)]TJ /F7 11.955 Tf 11.96 0 Td[(r0j))=0 (5)Atlowtemperatures!1,andasaresultK(r)!1 2.Atthelimitofhightemper-atures!0weget0K(r)!)]TJ /F5 11.955 Tf 25.73 0 Td[(1+1 22Kr2.UsingtheseresultsinEquation( 5 )weget!0,m!2R=(N)]TJ /F5 11.955 Tf 11.95 0 Td[(1)q2 R2 (5)!1,m!2R=Nq2 2K (5)Fromtheaboveequations,weseethatinthequantumlimit,theradiusofthetrapRdependsonthetotalnumberofparticlesinthetrapNasRN,whileintheclassicallimitthedependenceisRN1=3.ThusthetrappotentialisstrongcomparedtotheCoulombrepulsionathightemperatureswhileatlowtemperaturestheparticlesspreadoutwardduetothestrongerCoulombrepulsion. 5.5Summary 1. Welookedatclassicaltheoriesthatstudystrongcorrelationsinshellformation:zerotemperatureclassicalDFT,shellmodelandnitetemperatureformalismbyWrightonetal. 2. TheexplanationbehindshellformationintheclassicalCoulombsystemsisdiscussed.Thedependenceofthenumberofparticlesintheshellandthepositionoftheshellsonthetotalnumberofparticlesinthetrapandthecouplingconstanthasalsobeendiscussed. 3. Weextendedthemaptoconnedquantumsystemsusingtheeffectivepotentialsandthecorrelationsofjelliumfromthepreviouschapters. 78

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4. WestudiedthemodicationsduetothequantumexchangeeffectstotheharmonicpotentialincaseoftheidealFermigas. 5. Welookedintothepossiblemechanismforshellformationinthequantumsys-tems.Inparticular,toconsiderthequantumeffectsofdiffractionweconsiderKelbgpotentialinaharmonictrap.SimilarlyfordegeneracyeffectsweconsidertheCoulombinteractionsamongtheparticlesandanexternalpotentialmodiedfromtheharmonicformbythediffractioneffects.Inbothcasesweobtainshellsatthelowtemperaturesanddensitieseveninthemeaneldlimit.Weexplainedthepossiblemechanismbehindtheshellformationduetothequantumeffects. 79

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CHAPTER6CONCLUSIONThereexistanumberofquantummany-bodytheoriestostudythecorrelatedquantumsystemsatzerotemperature.Eithertheirextensiontobothnitetemperaturesandstrongcouplingarenotstraightforwardortheyarecomputationallyintensive.PerrotandDharma-wardanacouldsuccessfullypredictsomeofthethermodynamicsoftheuniformelectrongasbyusingasimpleclassicalmapping.Howevertheirmodelisphenomenologicalandthefunctionalformsoftheireffectiveparameterswereguessedfromthephysicsofthesystems.Ourobjectivehasbeentoprovideasystematicandgeneralformalismthatwouldincorporatesomeoftheirideasbutwouldbeapplicabletoalargeclassofsystems.Inthisthesiswehaveconstructedanoveltechniquetodescribethequantumcorrelationsusingtheclassicalmany-bodytheory,inparticularusingtheclassicalDFT.Theprocedurehasbeendevisedtoworkforanyquantumsystemswithanygeneralinteractionandexternalpotentialoverawiderangeoftemperaturesanddensities.Wehaverstconstructedaclassicalrepresentativesystemthatdescribesthethermodynamicsofagivenquantumsystem.Itsthermodynamicsisdescribedbythreeparameters:aneffectiveinteractionpotential,localchemicalpotentialandtemperature.Theseparametersaredeterminedbyamapfromthethermodynamicsoftheclassicalrepresentativesystemtothatofthequantumsystem.TheyaredeterminedbyinvertingthemapusingclassicalDFTandthevirialequations.Thedependenceoftheclassicalparametersoneachothergiverisetosomepeculiarthermodynamicsfortheeffectiveclassicalsystem.TheidealFermigasandweakcouplinglimitsoftheuniformelectrongashavebeenworkedoutexplicitly.Theidealgasresultsarerecoveredbytakingthers!0limitoftheinteractingsystem.SincebothourandPDWmodelhavethesameidealgaslimits,theagreementbetweenthetwoisgoodatsmallrsvalues.Howeverthedensity 80

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dependenceof)]TJ /F4 7.97 Tf 6.77 -1.79 Td[(eofthePDWmodeleffectivepotentialatT=0is)]TJ /F2 11.955 Tf 10.1 0 Td[(/rs(a+bp rs+crs)andforourmodel)]TJ /F4 7.97 Tf 6.77 -1.8 Td[(e/r1=2sdependence.Hencetheresultsfromthetwomodelsdifferatlargedensitiesandlowtemperatures.TheeffectiveinteractionpotentialforourmodelisrequiredtohaveaCoulombtailfromtheperfectscreeningsumrule.TheKelbgformofthepotentialisrecoveredathightemperaturesandlowdensities.ThepaircorrelationfunctionshavebeencalculatedusingtheHNCequationsandcomparedwiththePIMCandPDWdata.Theagreementisgoodinthedensityrangers=1and10foralltemperatures.Thusthismodelwouldbeabletodescribewellallmetallicsystemsandwarmdensemattersystems.Manyofthequantumtheoriesfailtogivepositivepaircorrelationfunctionsattheserangeofdensities.Itappearsthatbyimposingthesumruleconstraintontheeffectivepotentialwecancapturetheessentialmanybodyeffectsofthejellium.Thepaircorrelationfunctions,calculatedusingtheHNCequationsforbothourmodelandPDWtheory,havegoodagreementeventhoughtheeffectivepotentialsaredifferent.Thismaybebecausethenon-linearequationsofHNCmayhavemultiplesolutionsforthesamepaircorrelationfunction.Thepressureofjelliumbecomesnegativeatmetallicdensitiesandlowtemperatures,theequivalenceofthequantumandclassicalpressuresbreaksdownatthatregime.Thiswouldimplyanegativeeffectivetemperatureandwemightneedtoreneitusingsomeotherequivalencecondition.Anotherapplicationofourmodelhasbeentotheconnedchargesinaharmonictrap.Wrightonetal.successfullypredictedtheshellformationforstrongcoupling.TheirdensityprolecalculatedusingtheHNCequationisingoodagreementwiththeclassicalMonteCarlodata.WehaveappliedtheeffectivepotentialsfromourapproachintheHNCequationstogeneratethedensityproleforthequantumsystems.Thisprocedurehasbeenusedtostudythequantumeffectsontheshellformation.Inthemeaneldlimitthereisnoshellformationintheclassicalsystem,butthereisforthequantumsystem.Welookedintothediffractionanddegeneracyeffectsseparately. 81

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Forthediffraction,weconsideredtheKelbgpotentialinaharmonictrap.Asecondmechanismforshellformationisfoundfordegeneracyeffectswithaneffectiveexternalpotential,whichistheharmonicpotentialmodiedbythequantumeffectsofdegeneracyandCoulombinteractions.Atlowtemperaturesanddensitiesbotheffectsproduceshellseveninthemeaneldlimit.TheradiusofthetrapfortheclassicalquantumsystemhasN1=3dependence,forNparticlesinthetrap.Forthequantumsystem,theradiusofthetrapatlowtemperatureshasNdependencewhileathightemperatureswerecovertheclassicalresult.Thisformalismcanbeimmediatelyappliedtospinpolarizedsystems,magneticsystemsandlatticesystemsasisshowninthenextchapter. 82

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CHAPTER7OTHERAPPLICATIONSOFTHEFORMALISMInthisChapterweconsidersomeadditionalapplicationsoftheformalismforfuturestudy. 7.1DensityProleOfCoulombSystemsInAHarmonicTrapInChapter 5 weseethatinthemeaneldlimitthequantumeffectsproduceshellstructurewhichismissingtheclassicalsystems.Thisdoesnotimplythatwewillalwayshavemoreshellsinthequantumsystemthanaclassicalsystemwiththesamepotential.InfactwefoundthatthePaulipotentialbroadenstheshellsandthustheexchangeeffectsmaydestroytheshellformation.InChapter 5 wehavetreatedthetwoquantumeffects,degeneracyanddiffractionseparately.Ifweputthemalltogetherwendthatthenalshellstructurehasastronginuencefromthediffractionthanthedegeneracythanthediffractioneffect.Ifweincreasethenumberofparticlesintheshell,inthemeaneldlimit,wedonotnecessarilygetmoreshells. 7.2SpinPolarizedUniformElectronGasWhenthereisamagneticeld,theHamiltonianacquiresaspindependentterm.Thespininclassicalsystemsistreatedasdifferentspeciesofparticles.Thuswehavedifferentinteractionpotentialsamongtheparticlesdependingonthespeciestheybelongto.Thusfor"and#spinsinthequantumsystemwehavefourcomponentsforallthethermodynamicalquantities"","#,#"and##.TheOrnstein-Zernikeequationsplitsintocomponentstoohij(r)=cij(r)+XknkZdr0hik(jr)]TJ /F7 11.955 Tf 11.96 0 Td[(r0j)ckj(r0) (7)wherenkisthedensityofthespinspeciesk.Theeffectiveinteractionpotentialsimilarlyhasfourcomponents[ 33 ]:(cc)ij(r)=(cc)(0)ij(r)+4ij(r) (7) 83

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whereij="","#,#",##.Forsamespinspeciesi=j,(cc)(0)ij(r)issingularattheoriginbecauseofthesingularityofthesamespinPaulipotential,otherwisetheyarenite.AsaresultwhenusedintheHNCequations,thepaircorrelationfunctioniszeroattheoriginasexpectedforaquantumsystemduetothePauliexclusionprinciple.The4ijistheaboveequationiscalculatedusingtheRPAjustlikethediscussioninChapter 4 .PDWassumedaspinindependentformfor4ij.FromtheseresultsthemagneticsusceptibilitycanbecalculatedasshowninSection 7.3 fortheidealFermigas. 7.3MagneticSusceptibilityAtFiniteTemperaturesOfTheIdealFermiGasLetthedensitiesofthetwospincomponents,1and2withspinsSi=S(S=1 2),beni=n 2(1), (7)wherenistheaveragedensityofthesystem.TheHNCequationforthedensitybecomesln(ni3)=)]TJ /F3 11.955 Tf 11.96 0 Td[(BSi+XjZdr0njcji(r0) (7)=)]TJ /F6 11.955 Tf 11.96 0 Td[(BSi+Xjnjcji(k=0) (7)hij(r)=cij(r)+XknkZdr0cik(r)]TJ /F7 11.955 Tf 11.96 0 Td[(r0)hkj(r0) (7) (7) 84

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h12(r)=h21(r)=0h11(r)=h22(r))c12(r)=c21(r)=0cii(r)=hii(r) 1+nihii(r), (7)Subtractionofdensityofonecomponentfromanother:lnn2 n1=B+1 1+n1h11(k=0))]TJ /F5 11.955 Tf 53.22 8.09 Td[(1 1+n2h11(k=0) (7)Themagnetizationisgivenby:M=S(n1)]TJ /F6 11.955 Tf 11.96 0 Td[(n2)=1 2n, (7)HencetheEquation( 7 )alongwiththeEquation( 7 )giveln(1)]TJ /F5 11.955 Tf 11.95 0 Td[(2M=n 1+2M=n)=B+1 1+n1h11(k=0))]TJ /F5 11.955 Tf 53.21 8.08 Td[(1 1+n2h11(k=0)=B)]TJ /F6 11.955 Tf 82.71 8.09 Td[(nh11(k=0) (1+1 2nh11(k=0))2)]TJ /F5 11.955 Tf 11.95 0 Td[((1 2nh11(k=0))2=B)]TJ /F5 11.955 Tf 76.16 8.08 Td[(2Mh11(k=0) (1+1 2nh11(k=0))2)]TJ /F5 11.955 Tf 11.95 0 Td[((Mh11(k=0))2, (7)whereh11(k=0)=)]TJ /F9 7.97 Tf 10.49 4.71 Td[(1 2Rdkn2(k) n2fromthediscussioninChapter 3 .Thisdiscussioncanbeeasilyextendedtointeractingsystems,wherewecanusedthespinpolarizedpaircorrelationfunctiondiscussedinSection 7.2 7.4CrystalLatticeSystemsThetheoriesthatexistforquantumsystemsinaperiodicpotential,ext(r+R)=ext(r)withlatticevectorsR,usuallyuseindependentelectronmodel.Theinteractions 85

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amongtheelectronsareelectronsareneglected.ThesolutionofthesingleparticleSchroedingerequationinperiodicpotentialisaBlockwavevector n,kwiththeproperty: n,k(r)=exp(ik.r)un,k(r)withun,k(r)=un,k(r+R)andkthecrystalmomentum.Theenergyeigenvaluesaregivenby:n(k)=n(k+K)whereKisthereciprocalvector.Thetightbindingmodelassumesthatlatticeismadeofweaklyoverlappingatomicsystems.Hencethesingleparticlewave-functionsinthelatticecanbewrittenasasumofsingleparticleatomicorbitals.Theeffectiveclassicalsystemstillhasthreeparametersweneedtodetermine.ConsideraquantumCoulombsysteminaperiodicpotentialext.Weassumethelatticesymmetryispreservedinthemap.Alsoforsimplicityweconsidertheeffectivepairpotentialtodependontherelativecoordinates.Hencecc(r)]TJ /F7 11.955 Tf 10.4 0 Td[(r0)=PR(cc)UEG(r)]TJ /F7 11.955 Tf 10.4 0 Td[(r0)where(cc)UEGistheeffectiveinteractionpotentialdeterminedinChapter 4 usingRPAtheory.TheHNCequationforthisnon-uniformsystemishighlycomplicated.Howeverevenifwereplacethenon-uniformdensitybyanuniformaveragedensityofthesystem,thenthepaircorrelationsdoesnothavetheperiodicityofthelattice:g(r)=XRg0(jr)]TJ /F7 11.955 Tf 11.96 0 Td[(Rj) (7)whereg0aresomeperiodicfunctionwiththeperiodicityofthelattice.ThisisbecausetheHNCequationisanon-linearequationinc.Thesameconclusioncanbedrawnfromthedensity. 7.52DElectronGasFormaterialswhereelectronsareconnedtotwodimensionslikegraphene,theanalysisof2Dsystemsbecomesimportant.Importantapplicationsaretheshellformationinthequantumdots.For2DsystemstheFouriertransformoftheCoulombpotentialisdifferentthanthe3Dsystems.Thesumrulesarealsodifferentandtheformoftheeffectivepotentialarecompletelydifferentthanthe3Dcase. 86

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APPENDIXAEXACTCOUPLEDEQUATIONSFORNC(R)ANDGC)]TJ /F7 11.955 Tf 5.48 -9.68 Td[(R,R0Thisappendixoutlinestheoriginoftheexactequations,Equations( 3 )-( 3 ),fortheclassicaldensityandpaircorrelationfunctionfromtheclassicaldensityfunctionaltheory(DFT).WegofromlocalchemicalpotentialrepresentationsofthermodynamicsthroughthegrandpotentialtothedensityrepresentationsthoughfreeenergybytheuseoftheLegenderetransform.Fc(cjnc)=c(cjc)+Zdrc(r)nc(r) (A)wherethefreeenergyisafunctionaloftheclassicaldensityandthegrandpotentialofc(r).ThedensityandlocalchemicalpotentialareconjugatetooneanotherFc nc(r)=c(r). (A)ThefreeenergyisnowdividedintoitsidealclassicalgascontributionF(0)c=)]TJ /F3 11.955 Tf 9.3 0 Td[()]TJ /F9 7.97 Tf 6.58 0 Td[(1cRdr[1)]TJ /F5 11.955 Tf 11.29 0 Td[(ln)]TJ /F6 11.955 Tf 5.48 -9.69 Td[(n(r)3n(r),wherec=)]TJ /F5 11.955 Tf 5.48 -9.69 Td[(2~2c=m1=2,andtheremainderFc,extheinteractionpart,sothattheEquation( A )becomesanequationforthedensitycc(r)=ln)]TJ /F6 11.955 Tf 5.48 -9.69 Td[(nc(r;c)3c)]TJ /F6 11.955 Tf 11.95 0 Td[(c(1)c(r;cjnc) (A)Herec(1)c(rjnc)istherstofafamilyoffunctions(directcorrelationfunctions)denedbyderivativesofthefreeenergyc(m)c(r1,..,r1;cjnc))]TJ /F3 11.955 Tf 21.92 0 Td[(mFc nc(r1)..nc(rm) (A)Equation( A )relatesthedensitync(r)toagivenexternalpotential(recallc(r)=c)]TJ /F3 11.955 Tf 12.63 0 Td[(c,ext,(r)).Nowaninterestingcaseiswhenoneoftheparticlesinthesystemistreatedasanimpurityatsomepointr0.Thentheinteractionpotentialc(r)becomestheexternalpotentialfortheotherparticles.Thusthetotalexternalpotentialisc,ext+candthedensityoftheparticlesinthenewexternalpotentialisrelatedtothepaircorrelations 87

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anddensityoftheoriginalsystembync(r,r0)=nc(r)gc(r,r0). (A)TheequationcorrespondingtoEquation( A )forthisnewexternalpotentialisln)]TJ /F6 11.955 Tf 5.48 -9.68 Td[(nc(r)gc(r,r0)3c=cc(r))]TJ /F3 11.955 Tf 11.96 0 Td[(cc(r,r0)+c(1)c(r;cjncgc). (A)Finally,subtractingEquation( A )fromEquation( A )givesthedesiredequationforgc(r,r0)ln)]TJ /F6 11.955 Tf 5.48 -9.69 Td[(gc(r,r0)3c=)]TJ /F3 11.955 Tf 9.3 0 Td[(cc(r,r0)+c(1)c(r;cjncgc))]TJ /F6 11.955 Tf 11.96 0 Td[(c(1)c(r;cjnc). (A)Thenotationusedimpliesthatthefunctionalc(1)c(r;cj)inbothEquations( A )and( A )arethesame.Thisfollowsfromdensityfunctionaltheorywhereitisdemonstratedthatthefreeenergyisauniversalfunctionalofthedensity,thesameforallexternalpotentials.Equations( 3 )and( 3 )nowfollowdirectlyfromEquations( A )and( A )andtheidentityc(1)c(r;cjX)=c(1)c(r;cjY)+Z10d@c(1)c(r;cjX+(1)]TJ /F3 11.955 Tf 11.96 0 Td[()Y)=c(1)c(r;cjY)+Z10dZdr0c(1)c(r;cjX+(1)]TJ /F3 11.955 Tf 11.95 0 Td[()Y) nc(r0)(X(r0))]TJ /F6 11.955 Tf 11.95 0 Td[(Y(r0))=c(1)c(r;cjY)+Z10dZdr0c(2)c(r,r0;cjX+(1)]TJ /F3 11.955 Tf 11.95 0 Td[()Y)(X(r0))]TJ /F6 11.955 Tf 11.96 0 Td[(Y(r0)) (A)withappropriatechoicesforXandY.TheOrnstein-Zernickeequation,Equation( 3 ),isanidentityobtainedasfollows.Thesecondfunctionalderivativeofthegrandpotentialisrelatedtothepaircorrelationfunctionby 88

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2()]TJ /F3 11.955 Tf 9.29 0 Td[(cc) c(r)c(r0)=nc(r0) c(r)=cnc(r0)[(r)]TJ /F7 11.955 Tf 11.95 0 Td[(r0)+nc(r)(gc(r,r0))]TJ /F5 11.955 Tf 11.96 0 Td[(1)], (A)Similarly,thesecondderivativeofthefreeenergyis2Fc nc(r0)nc(r)=c(r) nc(r0)=)]TJ /F9 7.97 Tf 6.59 0 Td[(1cnc(r))]TJ /F9 7.97 Tf 6.59 0 Td[(1(r)]TJ /F7 11.955 Tf 11.96 0 Td[(r0))]TJ /F6 11.955 Tf 11.96 0 Td[(nc(r)c(2)c(r,r0jn). (A)ThenthechainruleZdr00nc(r) c(r00)c(r00) nc(r0)=(r)]TJ /F7 11.955 Tf 11.95 0 Td[(r0) (A)canbewrittenZdr00[(r)]TJ /F7 11.955 Tf 11.96 0 Td[(r00)+nc(r)(gc(r,r00))]TJ /F5 11.955 Tf 11.96 0 Td[(1)](r00)]TJ /F7 11.955 Tf 9.3 0 Td[(r0))]TJ /F6 11.955 Tf 11.96 0 Td[(nc(r00)c(2)c(r00,r0jn)=(r)]TJ /F7 11.955 Tf 11.96 0 Td[(r0). (A)ThisgivestheOrnstein-Zernickeequation,Equation( 3 ). 89

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APPENDIXBINHOMOGENEOUSIDEALFERMIGASThethermodynamicandstructuralpropertiesofaninhomogeneousidealFermigasarestraightforwardtocalculateinarepresentationthatdiagonalizestheeffectivesingleparticleHamiltonianbp2 2m)]TJ /F3 11.955 Tf 11.96 0 Td[((br) k(r)=k k(r) (B)whereklabelsthecorrespondingquantumnumbers.ForFermionswithspins,thequantumnumbersarelabeledby=(s,k).TheHamiltonianinsecondquantizedformisthensimplyH=Xkaya, (B)whereay,aarethecreationandannihilationoperatorsforoccupationofthestatesf kg.Thenthepressureisfounddirectlyfromevaluationofthegrandpotentialcp(j)V=Xln)]TJ /F5 11.955 Tf 5.48 -9.69 Td[(1+e)]TJ /F14 7.97 Tf 6.59 0 Td[(k=Tr(1)ln1+e)]TJ /F14 7.97 Tf 6.59 0 Td[(bp2 2m)]TJ /F14 7.97 Tf 6.59 0 Td[((br)=(2s+1))]TJ /F9 7.97 Tf 6.59 0 Td[(1Zdrhrjln1+e)]TJ /F14 7.97 Tf 6.59 0 Td[(bp2 2m)]TJ /F14 7.97 Tf 6.58 0 Td[((br)jri (B)whereacoordinaterepresentationhasbeenusedinthelastexpression.Thelocaldensityandpaircorrelationfunctionareobtainedfromtheoneatwoparticledensitymatricies.Inthediagonalrepresentationtheseare(1)(1;2)=ay1a2=ay1a11,2 (B)(2)(1,2;3,4)=ay1ay2a4a3=(1,32,4)]TJ /F3 11.955 Tf 11.95 0 Td[(1,42,3)ay1a1ay2a2 (B)wherethemeanoccupationnumberisaya0=,0)]TJ /F6 11.955 Tf 5.48 -9.69 Td[(ek+1)]TJ /F9 7.97 Tf 6.58 0 Td[(1. (B) 90

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Thecoordinaterepresentationsare(1)(r,1;r0,2)=Xk1,k2 1(r) 2(r0)ay1a2=1,2hrjlnebp2 2m)]TJ /F14 7.97 Tf 6.59 0 Td[((br)+1)]TJ /F9 7.97 Tf 6.59 0 Td[(1jr0i1,2n(r,r0) (B)(2)(r1,1,r2,2;r01,3,r02,4)=Xk1..k6 1(r1) 2(r2) 3(r01) 4(r02)ay1ay2a4a3=1,32,4Xk1 1(r1) 1(r01)ay1a1Xk2 2(r2) 2(r02)ay2a2)]TJ /F3 11.955 Tf 11.96 0 Td[(1,42,3Xk1 1(r1) 1(r02)ay1a1Xk2 2(r2) 2(r01)ay2a2 (B)Thediagonalelementsare(1)(r,1;r,1)=n(r,r) (B)(2)(r1,1,r2,2;r1,1,r2,2)=(1)(r1,1;r1,1)(1)(r2,2;r2,2))]TJ /F3 11.955 Tf 11.96 0 Td[(1,2(1)(r1,1;r2,1)(1)(r2,2;r1,2) (B)Finally,thedensityandpaircorrelationfunctionareidentiedfromthesummationoverspinstatesn(r)=X1(1)(r,1;r,1)=(2s+1)n(r,r) (B)n(r1)n(r2)g(r1,r2)=X1,2(2)(r1,1,r2,2;r1,1,r2,2)=n(r1)n(r2))]TJ /F5 11.955 Tf 11.96 -.16 Td[((2s+1)n(r1,r2)n(r2,r1) (B) 91

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ThisgivestheresultsEquations( 3 )and( 3 ).Thelocaldensityandpaircorrelationfunctionaredeterminedfromthefunctionn(r,r0)obtainedfromthesingleparticledensitymatrixfromtheEquation( B ),n(r,r0)=hrjebp2 2m)]TJ /F14 7.97 Tf 6.59 0 Td[((br)+1)]TJ /F9 7.97 Tf 6.59 0 Td[(1jr0i. (B)Inthelocaldensityapproximationofthetext,(br)!(R),whereR=(r+r0)=2,thisbecomesEquation( 3 )n(r,r0)=1 h3Zdpei ~p(r)]TJ /F9 7.97 Tf 6.59 0 Td[(r0)e(p2 2m)]TJ /F14 7.97 Tf 6.59 0 Td[((R))+1)]TJ /F9 7.97 Tf 6.58 0 Td[(1. (B)Furthersimplicationispossibletogetn(r,r0)3=2 jr)]TJ /F7 11.955 Tf 11.96 0 Td[(r0jZ10dxxz)]TJ /F9 7.97 Tf 6.59 0 Td[(1(R)ex2+1)]TJ /F9 7.97 Tf 6.58 0 Td[(1sin)]TJ /F5 11.955 Tf 5.48 -9.68 Td[(2p xjr)]TJ /F7 11.955 Tf 11.96 0 Td[(r0j=, (B)withz(R)=e(R) (B)Accordinglythedensityandpressuresimplifyton(r)3=(2s+1)n(r,r)3=(2s+1)f3=2(z(r)), (B)p3=1 VZdr(2s+1)f5=2(z(r)), (B)withthedenitionsf3=2(z)=4 p Z10dxx2z)]TJ /F9 7.97 Tf 6.59 0 Td[(1ex2+1)]TJ /F9 7.97 Tf 6.58 0 Td[(1,f5=2(z)=8 3p Z10dxx4z)]TJ /F9 7.97 Tf 6.58 0 Td[(1ex2+1)]TJ /F9 7.97 Tf 6.59 0 Td[(1. (B) 92

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Attingfunctionforthedensityintermsofthelocalchemicalpotentialisgivenbyf3=2(e)=e)]TJ /F14 7.97 Tf 6.59 0 Td[(+1=2())]TJ /F9 7.97 Tf 6.59 0 Td[(11=2()=3r 2h(+2.13)+)]TJ /F2 11.955 Tf 5.48 -9.68 Td[(j)]TJ /F5 11.955 Tf 11.96 0 Td[(2.13j2.4+9.65=12i)]TJ /F9 7.97 Tf 6.59 0 Td[(3=2 (B) 93

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APPENDIXCQUANTUMPRESSUREHerewederivetheexpressionforpressureforaninhomogeneousFermigas.WeconsideraquantumsysteminavolumeVwiththeinteractionpotential(r)andexternalpotentialext.pV=lnTr(e)]TJ /F14 7.97 Tf 6.58 0 Td[(bH) (C)wherebH=bK+b+bextP=1 @VlnTr(e)]TJ /F14 7.97 Tf 6.58 0 Td[(cHV)=D@VcHVE (C)Tre)]TJ /F14 7.97 Tf 6.59 0 Td[(bH=ZidqiDq1,q2..,qNje)]TJ /F14 7.97 Tf 6.59 0 Td[(bHbSjq1..qNE (C)wherewehaveintroducedcompletesetofketsqisuchthatjqij=1.Hencethewavefunctionsareindependentofthevolumeofthesystem.Wecandenethehamiltonianintermsofdimensionlessmomentumandpositionoperatorsbp=V1=3bpandbq=bq=V1=3:bH=NXibpi2 2m+1 2Zdr1dr2(r1,r2)bg(r1,r2))]TJ /F10 11.955 Tf 11.96 16.27 Td[(Zdr(r)b(r)=1 V2=3NXibpi2 2m+1 2Zdq1dq2(q1V1=3,q2V1=3)bg(q1,q2))]TJ /F10 11.955 Tf 11.96 16.28 Td[(Zdq(qV1=3)b(q) (C) 94

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whereri=V1=3qi.e)]TJ /F14 7.97 Tf 6.59 0 Td[(bHV+V=e)]TJ /F14 7.97 Tf 6.59 0 Td[(cHV)]TJ /F5 11.955 Tf 11.95 0 Td[(VZ0d0e)]TJ /F9 7.97 Tf 6.58 0 Td[(()]TJ /F14 7.97 Tf 6.59 0 Td[(0)cHV@VcHVe)]TJ /F14 7.97 Tf 6.59 0 Td[(0cHV+O(2V) (C)Usingthiswecanprove:@VTre)]TJ /F14 7.97 Tf 6.58 0 Td[(cHV=D@VcHVE (C)@VcHV=)]TJ /F5 11.955 Tf 15.49 8.09 Td[(2 3VNXibpi2 2m+1 3VZdr1dr2r1@r1(r1,r2)bg(r1,r2))]TJ /F5 11.955 Tf 18.16 8.09 Td[(1 3VZdrr@r(r)b(r), (C)p=)]TJ /F5 11.955 Tf 10.5 8.09 Td[(2n 3*bpi2 2m++1 3VZdr1dr2r1@r1(r1,r2)n(r1)n(r2)g(r1,r2))]TJ /F5 11.955 Tf 18.15 8.09 Td[(1 3VZdrr@r(r)n(r), (C) 95

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APPENDIXDSTATICSTRUCTUREFACTORINRPATherearetwoequivalentrepresentationsoftheRPAstaticstructurefactorasgiveninreferences[ 1 ]:SRPA(k)=)]TJ /F12 11.955 Tf 10.51 8.09 Td[(~ nZd! 1 1)]TJ /F6 11.955 Tf 11.95 0 Td[(e)]TJ /F14 7.97 Tf 6.58 0 Td[(~!I(!,k) (D)where(!,k)=1 V(k)(1)]TJ /F4 7.97 Tf 6.59 0 Td[(V(k)(0)(!,k))andV(k)=4e2 k2.The(0)(!,k)isthenon-interactingresponse:(0)(!,k)=lim!0(2s+1) 4p 3Z1dxln(1+ze)]TJ /F4 7.97 Tf 6.59 0 Td[(x2)1 )]TJ /F6 11.955 Tf 11.95 0 Td[(x)]TJ /F3 11.955 Tf 11.96 0 Td[(+i)]TJ /F5 11.955 Tf 46.93 8.09 Td[(1 )]TJ /F6 11.955 Tf 11.96 0 Td[(x++i (D)where=q 4p and=~! 4.AnotherrepresentationoftheRPAstaticstructurefactorfollowsfromconvertingthefrequencyintegralintheaboveequationtoasumofdiscretefrequencies[ 39 ].SRPA(x=k=kF)=3 2l=1Xl=(x,l) 1+(2)]TJ /F3 11.955 Tf 38.82 0 Td[(=x2)(x,l) (D)where(x,l)=1 2xZ10dyy exp[(y2=))]TJ /F3 11.955 Tf 11.95 0 Td[(]+1ln(2l)2+(x2+2xy)2 (2l)2+(x2)]TJ /F5 11.955 Tf 11.95 0 Td[(2xy)2 (D)Theconstantisdeterminedfromthenormalizationconditions.I1=2()=2 3)]TJ /F9 7.97 Tf 6.59 0 Td[(3=2withI()=R10dtt exp(t)]TJ /F14 7.97 Tf 6.59 0 Td[()+1.ToprovethattheRPAstructurefactorsatisestheexactscreeningsumrule,welookatSRPA(k)forsmallvaluesofk.TheintegrandinEquation( D )forsmallkvaluesbecomessharplypeakedaroundtwospecicfrequenciesp[ 1 ].Wewilltrytondoutthefunctionalformoftheintegrandaroundthosefrequenciesanddothefrequencyintegraltoprovethesumrule.Forsmallk,theintegrandissharplypeakedbecausethe 96

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denominatorofinEquation( D ),(,)=(1)]TJ /F6 11.955 Tf 12.12 0 Td[(V(k)(0)(!,k))becomesverysmall.Thesefrequenciesareactuallythezerosoftherealpartof(,p)=0.Thereasonisthatsincetheimaginarypartofisafastdecayingfunction,iftherealpartiszerothedenominatorbecomessmall.<(,p)=0)1)]TJ /F3 11.955 Tf 13.15 8.09 Td[(V(k)(2s+1) 3<(0)(k,p)=0=(,p)= (D)Forsmallkvaluessomesimplicationscanbemadetothefunctionalformof(0)lim!0(0)(,))]TJ /F5 11.955 Tf 31.63 8.09 Td[(1 2p Zdxln(1+ze)]TJ /F4 7.97 Tf 6.59 0 Td[(x2)1 ()]TJ /F6 11.955 Tf 11.96 0 Td[(x)2+i<(0)(,))]TJ /F5 11.955 Tf 28.49 8.09 Td[(1 p PZdxln(1+ze)]TJ /F4 7.97 Tf 6.58 0 Td[(x2)1 ()]TJ /F6 11.955 Tf 11.96 0 Td[(x)2=(0)(,)p n (D) (D)Wecanexpand(,)aboutp(,)(,p)+@ @j=p()]TJ /F3 11.955 Tf 11.96 0 Td[(p)+O(()]TJ /F3 11.955 Tf 11.96 0 Td[(p)2)=i)]TJ /F3 11.955 Tf 13.15 8.09 Td[(V(k)(2s+1) 3@ @<(0)(k,p)()]TJ /F3 11.955 Tf 11.95 0 Td[(p)@ @<(0)(k,p)2 p PZdxln(1+ze)]TJ /F4 7.97 Tf 6.59 0 Td[(x2) (p)]TJ /F6 11.955 Tf 11.96 0 Td[(x)32 pp PZdxln(1+ze)]TJ /F4 7.97 Tf 6.58 0 Td[(x2) (p)]TJ /F6 11.955 Tf 11.95 0 Td[(x)2+2 pp PZdxxln(1+ze)]TJ /F4 7.97 Tf 6.58 0 Td[(x2) (p)]TJ /F6 11.955 Tf 11.95 0 Td[(x)3=2<(,p) p+O(1 4p)V(k)(2s+1) 3@ @<(0)(k,p)=@ @j=p2 p(,)2 p(i+()]TJ /F3 11.955 Tf 11.96 0 Td[(p)) (D) (D) 97

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Forsmallkvaluesthemaincontributiontotheintegrandinequation( D )comesfromaroundthepeaksaroundpSRPA(k)=)]TJ /F5 11.955 Tf 24.19 8.09 Td[(4 nV(k)Z1d 1 1)]TJ /F6 11.955 Tf 11.96 0 Td[(e)]TJ /F9 7.97 Tf 6.59 0 Td[(4=(1 (,))=)]TJ /F5 11.955 Tf 24.19 8.08 Td[(4 nV(k)Zp+p)]TJ /F14 7.97 Tf 6.58 0 Td[(d 1 1)]TJ /F6 11.955 Tf 11.96 0 Td[(e)]TJ /F9 7.97 Tf 6.59 0 Td[(4=(1 (,)))]TJ /F5 11.955 Tf 24.19 8.09 Td[(4 nV(k)Zp)]TJ /F14 7.97 Tf 6.58 0 Td[()]TJ /F14 7.97 Tf 6.58 0 Td[(p+d 1 1)]TJ /F6 11.955 Tf 11.96 0 Td[(e)]TJ /F9 7.97 Tf 6.59 0 Td[(4=(1 (,)) (D)Letslookattheintegralaroundthepeakaroundp.S(k)=)]TJ /F5 11.955 Tf 24.19 8.08 Td[(4 nV(k)Zp+p)]TJ /F14 7.97 Tf 6.59 0 Td[(d 1 1)]TJ /F6 11.955 Tf 11.95 0 Td[(e)]TJ /F9 7.97 Tf 6.59 0 Td[(4=1 (,)=2 nV(k)Zp+p)]TJ /F14 7.97 Tf 6.59 0 Td[(d 1 1)]TJ /F6 11.955 Tf 11.96 0 Td[(e)]TJ /F9 7.97 Tf 6.59 0 Td[(4p()]TJ /F3 11.955 Tf 11.96 0 Td[(p)=2 nV(k)p 1)]TJ /F6 11.955 Tf 11.96 0 Td[(e)]TJ /F9 7.97 Tf 6.59 0 Td[(4p (D) (D)Similarlywegetcontributionfromanotherpeakat)]TJ /F3 11.955 Tf 9.3 0 Td[(pbyreplacingp!)]TJ /F3 11.955 Tf 26.94 0 Td[(pandaddingthetwoweget:S(k).around.peaks=2 nV(k)p 1)]TJ /F6 11.955 Tf 11.95 0 Td[(e)]TJ /F9 7.97 Tf 6.58 0 Td[(4p)]TJ /F5 11.955 Tf 30.22 8.09 Td[(2 nV(k)p 1)]TJ /F6 11.955 Tf 11.95 0 Td[(e4p=1 nV(k)2pcoth(2p) (D)Theexactscreeningsumrule:nZdrdr0S(rjr0) jrj=~p 2coth(~p 2)V(k)nS(k)=2pcoth(2p) (D)wherethedenitionofdimensionlessfrequency:p=~!p=(4)hasbeenusedabove.Toproveathightemperaturet>>1thestructurefactorgoestotheDebyelimit,wedene0= 2p tandthentakelimit0!0for(0).Wechangethemomentumscaling 98

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from=k 4p tok=krsandredenethedimensionlessfrequencyas=~! 4kin(0)(0)(,)=)]TJ /F5 11.955 Tf 22.38 8.09 Td[(1 4p Zdxln(1+ze)]TJ /F4 7.97 Tf 6.59 0 Td[(x2)1 )]TJ /F5 11.955 Tf 11.96 0 Td[((x)]TJ /F3 11.955 Tf 11.95 0 Td[()+i)]TJ /F5 11.955 Tf 51.82 8.09 Td[(1 )]TJ /F5 11.955 Tf 11.95 0 Td[((x+)+i(0)(k0,0=0)=)]TJ /F5 11.955 Tf 31.01 8.09 Td[(1 4p k0Zdxln(1+ze)]TJ /F4 7.97 Tf 6.59 0 Td[(x2)1 =0)]TJ /F5 11.955 Tf 11.95 0 Td[((x)]TJ /F6 11.955 Tf 11.95 0 Td[(k0)+i)]TJ /F5 11.955 Tf 69.56 8.08 Td[(1 =0)]TJ /F5 11.955 Tf 11.96 0 Td[((x+k0)+i (D)Thesusceptibilitycanbeexpressedintermsofthedimensionlesssusceptibilityasshownbelow(0)(k,!)=(2s+1) 3(0)(k0,=0) (D)Inthehightemperaturelimitz<<1,hencethedimensionlesssusceptibilityinEquation( D )canbewrittenaslim0!0(0)(k0,=0)=)]TJ /F5 11.955 Tf 19.01 8.09 Td[(1 2p Zdxze)]TJ /F4 7.97 Tf 6.58 0 Td[(x21 (=0)]TJ /F6 11.955 Tf 11.96 0 Td[(x)2+i=z p Zdxxe)]TJ /F4 7.97 Tf 6.59 0 Td[(x21 =0)]TJ /F6 11.955 Tf 11.95 0 Td[(x+i=z p PZdxxe)]TJ /F4 7.97 Tf 6.59 0 Td[(x21 =0)]TJ /F6 11.955 Tf 11.96 0 Td[(x)]TJ /F6 11.955 Tf 11.95 0 Td[(izp =0e)]TJ /F9 7.97 Tf 6.58 0 Td[((=0)2=)]TJ /F6 11.955 Tf 9.29 0 Td[(zG(=0))]TJ /F6 11.955 Tf 11.95 0 Td[(izp =0e)]TJ /F9 7.97 Tf 6.58 0 Td[((=0)2 (D)TheRPAstructurefactor( D )inthislimitbecomesSRPA(k)=)]TJ /F5 11.955 Tf 10.49 8.08 Td[(4k3 3)]TJ /F10 11.955 Tf 11.32 24.47 Td[(Zd1 1)]TJ /F6 11.955 Tf 11.95 0 Td[(e)]TJ /F9 7.97 Tf 6.59 0 Td[(4k=(k0,=0)=)]TJ /F5 11.955 Tf 10.49 8.09 Td[(4k30 3)]TJ /F10 11.955 Tf 17.51 24.47 Td[(Zd1 1)]TJ /F6 11.955 Tf 11.95 0 Td[(e)]TJ /F9 7.97 Tf 6.58 0 Td[(4k0=(k0,))]TJ /F6 11.955 Tf 24.89 8.08 Td[(k2 3)]TJ /F10 11.955 Tf 9.96 24.48 Td[(Zd1 =(k0,)=)]TJ /F6 11.955 Tf 12.27 8.09 Td[(k2 3)]TJ /F10 11.955 Tf 9.96 24.48 Td[(Zd1 = 1 1+3)]TJ ET q .478 w 256.88 -546.71 m 269.85 -546.71 l S Q BT /F4 7.97 Tf 256.88 -554.43 Td[(k2G()+i3)]TJ ET q .478 w 316.83 -546.71 m 329.8 -546.71 l S Q BT /F4 7.97 Tf 316.83 -554.43 Td[(k2p e)]TJ /F14 7.97 Tf 6.58 0 Td[(2!=)]TJ /F6 11.955 Tf 12.27 8.09 Td[(k2 3)]TJ /F2 11.955 Tf 7.97 8.2 Td[(< 1 1+3)]TJ ET q .478 w 218.15 -586.57 m 231.12 -586.57 l S Q BT /F4 7.97 Tf 218.15 -594.28 Td[(k2G(0))]TJ /F5 11.955 Tf 11.95 0 Td[(1!=k2 k2+3)]TJ ET BT /F1 11.955 Tf 431.47 -618.02 Td[((D) 99

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wheretheKramer-Kronigequations[ 51 ],zn3 2s+1andG(0)=1havebeenusedinthelastequation.HereweshowthatatlargektheRPAstructurefactorbehaveslikek)]TJ /F9 7.97 Tf 6.59 0 Td[(4[ 57 ].AtthislimitV(k)issmall,andwecanexpandinpowersofV(k):SRPA(k)=)]TJ /F12 11.955 Tf 10.5 8.09 Td[(~ nZ1d! (1)]TJ /F6 11.955 Tf 11.95 0 Td[(e)]TJ /F14 7.97 Tf 6.58 0 Td[(~!)V(k)Im1 1)]TJ /F6 11.955 Tf 11.95 0 Td[(V(k)(2s+1)(0)(k,!)=)]TJ /F12 11.955 Tf 10.5 8.08 Td[(~ nZ1d! (1)]TJ /F6 11.955 Tf 11.95 0 Td[(e)]TJ /F14 7.97 Tf 6.58 0 Td[(~!)Im(2s+1)(0)(k,!)+V(k)(2s+1)2(0)2(k,!)+..1)]TJ /F12 11.955 Tf 13.16 8.09 Td[(~ nV(k)(2s+1)2Z1d! (1)]TJ /F6 11.955 Tf 11.95 0 Td[(e)]TJ /F14 7.97 Tf 6.59 0 Td[(~!)=((0)(k,!))21)]TJ /F12 11.955 Tf 13.16 8.09 Td[(~ nV(k)(2s+1)24 ~Z1d (1)]TJ /F6 11.955 Tf 11.96 0 Td[(e)]TJ /F9 7.97 Tf 6.59 0 Td[(4)=((0)(,))21)]TJ /F5 11.955 Tf 13.25 8.08 Td[(1 nV(k)(2s+1)21 Z1d =((0)(,))21)]TJ /F5 11.955 Tf 13.25 8.09 Td[(1 nV(k)(2s+1)21 <((0)(,0))2=1)]TJ /F5 11.955 Tf 13.25 8.08 Td[(1 nV(k)(2s+1)21 (<(0)(,0))2Re(,0,z)=)]TJ /F5 11.955 Tf 22.38 8.09 Td[(1 2p g()=)]TJ /F5 11.955 Tf 22.38 8.09 Td[(1 2p P(Z1dxln(1+ze)]TJ /F4 7.97 Tf 6.59 0 Td[(x2) )]TJ /F6 11.955 Tf 11.96 0 Td[(x=1 2p P(Z1dxln(1+ze)]TJ /F4 7.97 Tf 6.59 0 Td[(x2) x)]TJ /F5 11.955 Tf 21.67 8.09 Td[(1 2p P(Z1dxln(1+ze)]TJ /F4 7.97 Tf 6.59 0 Td[(x2) )]TJ /F6 11.955 Tf 11.96 0 Td[(xf3=2(z) 4SRPA()1)]TJ /F5 11.955 Tf 13.15 10.21 Td[(\(2s+1)2f23=2(z) 43n31 4 (D)AgaintheKramers-Kronigrelationshasbeenused. 100

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APPENDIXEPROPERTIESOFEFFECTIVEINTERACTIONPOTENTIALFORRPASYSTEMSTheintegralrepresentationoftheRPAstructurefactorwillbeusedinthissectiontoprovethatatverylowdensitiesandhightemperatures,inEquation( 4 )wouldlooklikeaKelbgpotential.Inthislimitthefugacity,z,isverysmall,z<<1.Thuswecanexpandtheresponsefunction,Equation( D ),inpowersofz.SRPA(k)=)]TJ /F12 11.955 Tf 10.5 8.09 Td[(~ nZ1d! (1)]TJ /F6 11.955 Tf 11.95 0 Td[(e)]TJ /F14 7.97 Tf 6.58 0 Td[(~!)V(k)Im1 1)]TJ /F6 11.955 Tf 11.96 0 Td[(V(k)(2s+1)(0)(k,!)=)]TJ /F12 11.955 Tf 10.5 8.09 Td[(~ nZ1d! (1)]TJ /F6 11.955 Tf 11.95 0 Td[(e)]TJ /F14 7.97 Tf 6.58 0 Td[(~!)Im(2s+1)(0)(k,!)+V(k)(2s+1)2(0)2(k,!)+..=S(0)(k))]TJ /F12 11.955 Tf 13.16 8.08 Td[(~ nV(k)(2s+1)2Z1d! (1)]TJ /F6 11.955 Tf 11.96 0 Td[(e)]TJ /F14 7.97 Tf 6.59 0 Td[(~!)Im((0)(k,!))2=S(0)(k))]TJ /F5 11.955 Tf 13.15 8.09 Td[(2~ nV(k)(2s+1)2Z1d! (1)]TJ /F6 11.955 Tf 11.95 0 Td[(e)]TJ /F14 7.97 Tf 6.58 0 Td[(~!)Im(0)(k,!)Re(0)(k,!)=S(0)(k)+V(k)f(k) (E)wheref(k)=)]TJ /F9 7.97 Tf 10.98 4.7 Td[(2~ n(2s+1)2R1d! (1)]TJ /F4 7.97 Tf 6.59 0 Td[(e)]TJ /F20 5.978 Tf 5.76 0 Td[(~!)Im(0)(k,!)Re(0)(k,!).Theexpressionforfcanbesimplied:f()=)]TJ /F5 11.955 Tf 11 8.08 Td[(2~ n(2s+1)24 ~2 6Z1d (1)]TJ /F6 11.955 Tf 11.96 0 Td[(e)]TJ /F9 7.97 Tf 6.59 0 Td[(4)Im(0)(,)Re(0)(,)=2~ n(2s+1)24 ~2 6Z1d (1)]TJ /F6 11.955 Tf 11.95 0 Td[(e)]TJ /F9 7.97 Tf 6.58 0 Td[(4)1 4p z(e)]TJ /F9 7.97 Tf 6.59 0 Td[((+)2)]TJ /F6 11.955 Tf 11.96 0 Td[(e)]TJ /F9 7.97 Tf 6.59 0 Td[(()]TJ /F14 7.97 Tf 6.59 0 Td[()2) (E)p 4z(g(+))]TJ /F6 11.955 Tf 11.96 0 Td[(g()]TJ /F3 11.955 Tf 11.95 0 Td[())=n 2(2s+1)2Z1d e)]TJ /F9 7.97 Tf 6.59 0 Td[(()]TJ /F14 7.97 Tf 6.59 0 Td[()2(g(+))]TJ /F6 11.955 Tf 11.96 0 Td[(g()]TJ /F3 11.955 Tf 11.95 0 Td[()) (E) 101

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e()=SRPA() SRPA())]TJ /F5 11.955 Tf 11.96 0 Td[(1)]TJ /F6 11.955 Tf 23.6 8.08 Td[(S(0)() S(0)())]TJ /F5 11.955 Tf 11.96 0 Td[(1=S(0)()+V()f())]TJ /F5 11.955 Tf 11.95 0 Td[(1 S(0)()+V()f())]TJ /F6 11.955 Tf 23.59 8.09 Td[(S(0)() S(0)())]TJ /F5 11.955 Tf 11.95 0 Td[(1V()f() S(0)()=V()f() (S(0))2() (E)sinceS(0)()=1+n3h()1+O(z).Hence:e()=V()f() (E)f()Z1de)]TJ /F14 7.97 Tf 6.59 0 Td[(2(g(+2))]TJ /F6 11.955 Tf 11.96 0 Td[(g())=Z1d e)]TJ /F14 7.97 Tf 6.59 0 Td[(2ReZ1e)]TJ /F4 7.97 Tf 6.59 0 Td[(x21 +2)]TJ /F6 11.955 Tf 11.95 0 Td[(x+i)]TJ /F5 11.955 Tf 36.26 8.09 Td[(1 )]TJ /F6 11.955 Tf 11.96 0 Td[(x+i=Z1de)]TJ /F14 7.97 Tf 6.59 0 Td[(2ReZ1e)]TJ /F9 7.97 Tf 6.59 0 Td[((x+)21 2)]TJ /F6 11.955 Tf 11.96 0 Td[(x+i)]TJ /F5 11.955 Tf 36.13 8.08 Td[(1 0)]TJ /F6 11.955 Tf 11.96 0 Td[(x+i=Z1dxe)]TJ /F4 7.97 Tf 6.59 0 Td[(x2=2ReZ1de)]TJ /F9 7.97 Tf 6.58 0 Td[((x=2+)21 2)]TJ /F6 11.955 Tf 11.95 0 Td[(x+i)]TJ /F5 11.955 Tf 36.14 8.09 Td[(1 0)]TJ /F6 11.955 Tf 11.95 0 Td[(x+i=Z1dxe)]TJ /F4 7.97 Tf 6.59 0 Td[(x2=2p p 2Re1 2)]TJ /F6 11.955 Tf 11.96 0 Td[(x+i)]TJ /F5 11.955 Tf 36.13 8.09 Td[(1 0)]TJ /F6 11.955 Tf 11.96 0 Td[(x+i=r 2g(p 2))]TJ /F6 11.955 Tf 11.96 0 Td[(g(0)=r 2g(p 2) (E)HencewehavefromEquation( E ):e()=V()f()g(p 2) 3 (E)whichcanbeshowntobetheFouriertransformoftheKelbgpotential4K(x)=1 x(1)]TJ /F6 11.955 Tf 12.66 0 Td[(exp()]TJ /F6 11.955 Tf 9.3 0 Td[(x2=2K)+p x=K(1)]TJ /F6 11.955 Tf 12.66 0 Td[(erf(x=K))).Hencewehaveprovedthatathightemperaturesandlowdensities,behaveslikeaKelbgpotential. 102

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Inthissection,wewillshowthattheeffectiveinteractionpotentialhasaCoulombtailandhowitfollowsfromtheperfectscreeningsumrule.Wearegoingtouseanimportantfact:thelargedistancebehaviorofafunctionisgovernedbythesmallmomentumbehaviorofitsFouriertransform.FromEquation( 4 ):](k)=1 n1 SRPA(k))]TJ /F5 11.955 Tf 28.28 8.09 Td[(1 S(0)(k) (E)InthesmallklimitS(k)=~k2 2m!pcoth(~!p=2),where!pistheplasmafrequency.ButS(0)(k)goestoanitenumber.Thus1 SRPA(k)>>1 S(0)(k)forsmallkvalues.Hence:e(k)2m!p n~k2coth(~!p=2) (E)Writinge(k)=4r0)]TJ /F16 5.978 Tf 4.82 -1.4 Td[(e k2,wegetaswastobeprovedinEquation( 4 ):)]TJ /F4 7.97 Tf 6.78 -1.79 Td[(e=2 ~!pcoth(~!=2))]TJ -117.47 -32.96 Td[()-277(=q2 r0 (E) 103

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APPENDIXFIDEALFERMIGASINAHARMONICTRAPHerewederivetheLDAformEquation( 3 )ofthedensityprolefortheidealFermigasinaharmonictrapatzerotemperature.n(0)(x)r30=1 (2)3Zdk1 exp(q2a0(k)2 2r20+1 2q2 r0x2)=z+1=1 (2)3Zdk(k)2 2r2s+1 2rsx2)]TJ /F3 11.955 Tf 11.96 0 Td[(F=1 22Zdkk2(k)2 2r2s+1 2rsx2)]TJ /F3 11.955 Tf 11.95 0 Td[(F=p 2r3s 32(F)]TJ /F6 11.955 Tf 14.85 8.08 Td[(x2 2rs)3=4 (F) (F)TheparameterFisobtainedbyintegratingthedensityinEquation( F )togetthetotalnumberofparticlesinthetrap.f3=2()=4 3p 3=2n(0)(r)=2)]TJ /F9 7.97 Tf 6.58 0 Td[(3f3=2(exp((0)(r)))8 3p ((0)(r))3 3Zn(0)(r)dr=N)N=16 3p 2(m! ~)3ZF0(F=(m!2))]TJ /F6 11.955 Tf 11.96 0 Td[(r2)3=2r2dr)F=~!3N 1=3 (F) 104

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Theexactequationfordensityforanon-interactingFermisystemisgivenintermsofthesingleparticlewavefunctionsbyn(r)=hXi(qi)]TJ /F7 11.955 Tf 11.96 0 Td[(r)i=X,0h0j(bq)]TJ /F6 11.955 Tf 11.95 0 Td[(r)jiha+0ai=X,0Zdr0h0jr0i(r0)]TJ /F7 11.955 Tf 11.96 0 Td[(r)hr0jiha+ai,0=Xj (r)j2n (F) (F)where isthesingleparticlenon-interactingwavefunctionandnistheFermidistribu-tionwithenergy.AtT=0,thedensityequationbecomes:n(r)=Xklmr 23 2k+2l+3k!lr2le)]TJ /F9 7.97 Tf 6.58 0 Td[(2r2 (2k+2l+1)!!(L(l+1=2)k(2r2))2Y2lm(,), (F)where=m!2 2~.Usingtheforcebalanceconditionwegetanequality2r20=p rs.Henceintermsofdimensionlessdistancex=r=r0,thedensityequationbecomesn(r)=Xklm,k0l0m0s r3=2s s 2k+l+2k!rl=2sx2le)]TJ 6.59 5.41 Td[(p rsx2 (2k+2l+1)!!s 2k0+l0+2k0!rl0=2sx2l0e)]TJ 6.58 5.41 Td[(p rsx2 (2k0+2l0+1)!!L(l+1=2)k(p rsx2)L(l0+1=2)k0(p rsx2)Ylm(,)Yl0m0(,) (F)Iftheangularsymmetryisnotbroken,theangularintegralscanbeperformed:n(r)=Xklms r3=2s 2k+l+2k!rl=2sx2le)]TJ 6.59 5.42 Td[(p rsx2 (2k+2l+1)!!(L(l+1=2)k(p rsx2))2=Xkls r3=2s 2k+l+2k!rl=2sx2le)]TJ 6.59 5.41 Td[(p rsx2 (2k+2l+1)!!(2l+1)(L(l+1=2)k(p rsx2))2=Xnls r3=2s 2n=2+l=2+2(n=2)]TJ /F5 11.955 Tf 11.95 0 Td[(1=2)!rl=2sx2le)]TJ 6.58 5.42 Td[(p rsx2 (n+l+1)!!(2l+1)(L(l+1=2)n=2)]TJ /F9 7.97 Tf 6.59 0 Td[(1=2(p rsx2))2 (F) 105

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Intheaboveequationwereplacequantumnumberkbythequantumnumbernbytherelationship2k=n)]TJ /F6 11.955 Tf 12.81 0 Td[(l.Togureoutwhatvaluesof(n,l)tosumover,weconsiderthecaseofN=100particlesandtrytogureouthowmanyofthese(n,l)arelledup.DuetospindegeneracyweneedtoconsiderN=2=50particlesllingupthestates.Theconstraintsonlandmare:)]TJ /F6 11.955 Tf 9.3 0 Td[(mlmandl=0,2,..,n)]TJ /F5 11.955 Tf 12.63 0 Td[(2,nfornevenandl=1,3,..,n)]TJ /F5 11.955 Tf 12.73 0 Td[(2,n.Thedegeneracycalculationgives:n=0,1,..4arecompletelylledandtherestofthe15electronsllupthefollowingquantumnumbers(k=(n)]TJ /F6 11.955 Tf 11.95 0 Td[(l)=2,l)!(1,3),(2,1)inthen=5level.TheFermienergyinthiscaseistheenergyofthehighestelectron:F=~!(n+3=2)=~!(5+3=2)=~!6.5.ThisishigherthanpredictedbytheLDAinEquation( F ).Thenumberofparticlestheshellwithquantumnumberncanholdis(n+1)(n+2) 2.Henceifshellstilln0arecompletelylled,thenthetotalnumberofparticlesinthesystemisPn0n=0(n+1)(n+2) 2=1 6(n0+1)(n0+2)(n0+3).IfwehaveNparticlesinthesystem,thenforlargenumberofparticles:n0(6N)1=3.HencetheFermienergyforlargenon-interactingsystemisF~!(6N)1=3.LDAinEquation( F ):F=(3N )1=3~!underestimatestheFermienergy. 106

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BIOGRAPHICALSKETCH SandipanDuttawasborninIndia.HehadhisnalexamatUniversityofFloridaonApril15,2013. 111