第一性原理整理总结 英文版适合组会.ppt

1 First principlescalculationsXueliSun 2 Contents Whatarefirstprinciplescalculations Howdowedofirstprinciplescalculations WhatisDFT Hohenberg KohntheoremKohn ShamequationLocal DensityApproximation LDA Pseudopotentials 3 Whatarefirstprinciplescalculations Amaterialissimplyacollectionofatomsthatareboundbychemicalreactions Chemicalreactions inturn aresimplyinteractionsbetweenelectrons 4 Thismeansthatallmaterialproperties chemical mechanical electrical optical can inprinciple bepredictedfromnothingmorethantheatomicnumberandmassoftheatomicspeciesinvolved withtheaidofquantumphysics Thisispreciselywhatfirstprinciplescalculationsattempttodo 5 Howdowedofirstprinciplescalculations SolvingtheSchr dingerequationbybrutemathematicalforceisextremelydemandingcomputationallyandnotpracticalforallbuttinysystems 6 Instead weuseacombinationoftwophysicalapproximations 1 Weusedensityfunctionaltheory DFT 2 Weusefirstprinciplespseudopotentialtheory 7 WhatisDFT DFTmapstheoriginalmany electronproblemintoanequivalentsingle electronproblem Itdoessobylumpingallthemany bodyquantumphenomena suchasPauli sexclusionprincipleandelectroncorrelation intoasingleadditive exchange correlation potential whichisafunctionalofthechargedensityalone 8 Inpractice theexactfunctionalisunknownandpeopleuseapproximateformsforthefunctional usually butnotalways derivedfrompropertiesofauniformelectrongas 9 1 Hohenberg Kohntheorem ThefirstH Ktheoremdemonstratesthatthegroundstatepropertiesofamany electronsystemareuniquelydeterminedbyanelectrondensitythatdependsononly3spatialcoordinates 10 ThesecondH Ktheoremdefinesanenergyfunctionalforthesystemandprovesthatthecorrectgroundstateelectrondensityminimizesthisenergyfunctional 11 TheoriginalH Ktheoremsheldonlyfornon degenerategroundstatesintheabsenceofamagneticfield althoughtheyhavesincebeengeneralizedtoencompassthese 12 2 Kohn Shamequation WithintheframeworkofKohn ShamDFT KSDFT theintractablemany bodyproblemofinteractingelectronsinastaticexternalpotentialisreducedtoatractableproblemofnon interactingelectronsmovinginaneffectivepotential 13 TheeffectivepotentialincludestheexternalpotentialandtheeffectsoftheCoulombinteractionsbetweentheelectrons e g theexchangeandcorrelationinteractions 14 15 3 Local DensityApproximation LDA Theexchange correlationpartofthetotal energyfunctionalremainsunknownandmustbeapproximated ModelingthelattertwointeractionsbecomesthedifficultywithinKSDFT Thesimplestapproximationisthelocal densityapproximation LDA whichisbaseduponexactexchangeenergyforauniformelectron 16 where istheelectricdensityand xcistheexchange correlationenergydensity 17 Pseudopotentials Theperiodictabletellsusthatchemicalreactivityisgovernedbyvalenceelectrons withcoreelectronsbeingchemicallyinert Pseudopotentialsmakeuseofthisbasicfactbyreplacingtheinertcoreelectronswithaneffectivepotential Thisreduces sometimesdrastically thenumberofelectronsweneedtosolvefor 18 19 Evenmoreimportantly thisresultsinmuchsmootherwavefunctionsfortheremainingvalenceelectrons makingtheproblemmucheasiertosolvenumerically 1 Norm conservingPseudopotentials2 UltrasolfPseudopotentials 20 Conclusion 1 Ourapproximationsaresystematic 2 All hid