castep最全的资料 clark3

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1、Exchange-CorrelationFunctionals;Successes,failuresandprospectsStewartClarkUniversityofDurhamOutline•Introduction•Exchangeandcorrelationinteractions•WhatareGGA’s•Implementation•Someexamples•Whenshouldtheybeused•Futuredevelopments•ClosingstatementsXCFunct

2、ionalsStewartClark-Universityof2DurhamDFTandexchange-correlation•Basicallycomesfromourattempttomap1N-bodyQMproblemontoN1-bodyQMproblems•Attempttoextractsingle-electronpropertiesfrominteractingN-electronsystem•Thesearequasi-particles“DFTcannotdo…”:Thisst

3、atementisdangerousandusuallyendsincorrectly(inmanypublications!)Shouldread:“DFTusingthe???XC-functionalcanbeusedtocalculate???,butthatparticularfunctionalintroducesanderrorof???becauseof???XCFunctionalsStewartClark-Universityof3DurhamIntroductionExactXC

4、interactionisunknownWithinDFTwecanwritetheexactXCinteractionas1n(r,r')xcE[n]=n(r)drdr'xc!!2

5、r"r'

6、Thiswouldbeexcellentifonlyweknewwhatnwas!xcThisrelationdefinestheXCenergy.ItissimplytheCoulombinteractionbetweenanelectronanrandthevalueofitsXCholen(r,r’)atr

7、’.xcXCFunctionalsStewartClark-Universityof4DurhamAlternativeXCdefinition•Exchangeenergyisdefinedas1EX="$$#i(r1)#j(r2)#i(r2)#j(r1)dr1dr2r12•Hartree-FockenergyisE=KE+EE+EN+EHFHFHFHFX!CorrelationenergyisdefinedasthedifferencebetweentheexactenergyandtheHartr

8、ee-Fockenergy!XCFunctionalsStewartClark-Universityof5DurhamTheXChole•FromthePauliexclusionprinciplewehaveforagivenrn(r,r')dr'="1!xcWhichisknownasthesumrule.XCFunctionalsStewartClark-Universityof6DurhamNon-localNatureofXCHoleXCFunctionalsStewartClark-Uni

9、versityof7DurhamXCInteractionatSurfaces•Asanelectronapproachesasurface(distancexaway)then:•Coulombinteractionappearstohave1/xsingularity.•FixedbypropertiesofXChole.XCFunctionalsStewartClark-Universityof8DurhamApproximationstoExcAswehavealreadyseenasimpl

10、e,buteffectiveapproximationtotheexchange-correlationinteractionisLDAhomE[n(r)]=n(r)"[n(r)]drxc!xcThegeneralisedgradientapproximationcontainsthenextterminaderivativeexpansionofthechargedensity:GGAGGAE[n(r)]=n(r)#[n(r),"n(r)]drxc!x