密度泛函理论新进展及应用

密度泛函理论新进展及应用杨金龙中国科学技术大学ComputationExperimentTheoryScienceResearch计算机模拟已经与理论与实验并列，成为三种基本的科学研究手段之一Time•空间尺度：电子机构•时间尺度：动力学电子结构计算：预言材料性质、验证理论猜想、理解实验观测现象。动力学模拟：预言反应过程、验证理论猜想、理解实验观测现象。MaterialsPropertiesfromFirst-principles“Supercomputer”Giganticcomputerprograms01020304050600246810...)34()3;42()14()31()12(][||||16)()()(]21[,22222dGWifvirrVVViffiiiixcextHTop500SupercomputersintheworldA“small”PCclustertodayFourordersofmagnitudein15years计算量随体系大小急剧增长MaterialPropertiesfromFirst-PrinciplesFromfirstprinciples!Predictnewbehaviors/propertiesofexistingmaterialsDesignmaterialswithdesiredpropertiesUnderstandandexplainmaterialsproperties√√Becomingreality内容•密度泛函理论新进展•石墨烯条带体系的第一性原理计算研究密度泛函理论新进展•理论体系交换相关泛函、含时密度泛函、动力学平均场、密度泛函微扰理论•数值方法基组、格点、线性标度•应用物理、化学、生物、材料、纳米科学、光谱学PartI:理论体系PerdewPRL2003+Localdensity+Densitygradient+Inexplicitoccupiedorbitalinformation+Explicitoccupiedorbitalinformation+Unoccupiedorbitalinformation交换相关泛函jacob'sladder•LDAunderestimatesEcbutoverestimatesEx,resultinginunexpectedlygoodvaluesofExc.•TheLDAhasbeenappliedin,calculationsofbandstructuresandtotalenergiesinsolid-statephysics.•Inquantumchemistry,itismuchlesspopular,becauseitfailstoprovideresultsthatareaccurateenoughtopermitaquantitativediscussionofthechemicalbondinmolecules.局域密度近似(LDA)•Anyrealsystemisspatiallyinhomogeneous,ithasaspatiallyvaryingdensityn(r),itwouldclearlybeusefultoalsoincludeinformationontherateofthisvariationinthefunctional.•Inthisapproximation,onetriestosystematicallycalculategradient-correctionsofgeneralfunctionsofn(r)and∇n(r)•DifferentGGAsdifferinthechoiceofthefunctionf(n,∇n).rnrnrfdnEGGAxc,3广义梯度近似(GGA)AlexD.Becke“一切都是合法的”剑宗JohnP.Perdew一定的物理规律（如标度关系和渐进行为）为基础，PBE气宗•GGAsusedinquantumchemistrytypicallyproceedbyfittingparameterstotestsetsofselectedmolecules.•NowadaysthemostpopularGGAsarePBEinphysics,andBLYPinchemistry.•CurrentGGAsseemtogivereliableresultsforallmaintypesofchemicalbonds(covalent,ionic,metallicandhydrogenbridge).•Inadditiontothedensityanditsderivatives,Meta-GGAsdependalsoontheKohn-Shamkinetic-energydensity:•SothatExccanbewrittenasExc[n(r),∇n(r),τ(r)].TheadditionaldegreeoffreedomprovidedbyτisusedtosatisfyadditionalconstraintsonExc.•Meta-GGAshavegivenfavorableresults,evenwhencomparedtothebestGGAs.•Thefullpotentialofthistypeofapproximationisonlybeginningtobeexploredsystematically.22()|()|2iirrmMeta-GGA•CommonhybridfunctionalmixafractionofHartree-FockexchangeintotheDFTexchangefunctional.HybridFunctionalsB3LDAexactLDAGGAGGAEEa(EE)bcxcxcxxxcEEa0.20,b0.72,c0.81)Ea(EEEDFTxexactxDFTxc0xc25.0a(Becke,1993)(Perdew,1998)B3PW91,B3LYPPBE0B3LYPisthemainworking-horseincomputationalchemistryLDA:SlaterexchangeVosko-Wilk-Nusaircorrelation,etcGGA:Exchange:B88,PW91,PBE,OPTX,HCTH,etcCorrelations:LYP,P86,PW91,PBE,HCTH,etcHybridGGA:B3LYP,B3PW91,B3P86,PBE0,B97-1,B97-2,B98,O3LYP,etcMeta-GGA:VSXC,PKZB,TPSS,etcHybridmeta-GGA:HCTHh,TPSSh,BMK,etcL(S)DA+U•Mott绝缘体，Hubbard模型•Anisimovetal.：StonerI--HubbardU•轨道序：•Dudarevetal.：惩罚泛函PartII:数值方法数值离散方法•基组展开–LCAO基组（Gaussian基组、数值基组）•实空间网格平面波基组：从OPW到PP•平面波展开•正交化平面波（OPW）•赝势（PP）方法–经验赝势–模守恒赝势–超软赝势Muffin-tin势场与分波方法•Muffin-tin势场近似–缀加平面波（APW）–格林函数方法（KKR）•线性化方法–LAPW–LMTO•分波方法的发展–FP-LAPW–third-generationMTO,NMTO,EMTO平面波基组：从USPP到PAW•投影缀加波（PAW）方法•赝波函数空间•USPPorPAW?(VASP,ABINIT,...)实空间网格•简单直观•允许通过增加网格密度系统地控制计算收敛精度•线性标度•可以方便的通过实空间域分解实现并行计算•处理某些特殊体系（带电体系、隧穿结。。。）有限差分•从微分到差分•提高FD方法的计算效率–对网格进行优化，如曲线网格（适应网格）和局部网格优化（复合网格）–结合赝势方法–多尺度（multiscale）或预处理（preconditioning）有限元•变分方法•处理复杂的边界条件•矩阵稀疏程度及带状结构往往不如有限差分好•广义的本征值问题多分辨网格上的小波基组•多分辨分析•半取样（semicardinal）基组PartIII:应用物理学：强相关体系•模型哈密顿量•LDA++•电子结构：CrO2•点阵动力学:钚化学：弱作用体系•松散堆积的软物质、惰性气体、生物分子和聚合物，物理吸附、Cl+HD反应•用传统的密度泛函理论处理弱作用体系•一个既能产生vdW相互作用系数又能产生总关联能的非局域泛函：无缝的（seamless）方法•GW近似•密度泛函加衰减色散（DFdD）生命科学：生物体系•困难（尺寸问题、时间尺度）•QM/MM方法（饱和原子法、冻结轨道法）•简单势能面方法–线性同步过渡（LST）–二次同步过渡（QST）•完全的分子动力学–并行复制动力学（parallelreplicadynamics）–超动力学（hyperdynamics,metadynamics）–温度加速的动力学（temperatureaccelerateddynamics）–快速蒙特卡罗（on-the-flykinericMonteCarlo）方法纳米和材料科学：输运性质及其他•输运：非平衡态第一性原理模拟•材料力学：运动学MonteCarlo（KMC）--点阵气体和元胞自动机--连续方程的有限差分有限元求解光谱学：激发态和外场•系综密度泛函理论•考虑系统对称性，用求和方法计算多重态激发能•多体微扰理论，GW近似Bethe-Salpeter方程•TDDFT，线性响应石墨烯体系的第一性原理研究Graphene•Introductiontographeneandgraphenenanoribbon(GNR)•GNRbasedspintronics•Nearlyfreeelectron(NFE)statesingatedGNRsuperlattice•Cuttingmechanismingrapheneoxide(GO)Graphene:amonolayeroftwo-dimensionalcarbonatoms198519912004CrystalstructureofgrapheneEnergybandsKorK’Siliconout,Graphenein?RVanNoorden,Nature442,228(2006)WhatareGraphenenanoribbons(GNRs)?UnlimitedLimitedZigzagGNRsUnlimitedLimitedArmchairGNRsArmchairGNRsZigzagGNRs•ArmchairGNRsarePM.•ZigzagGNRsfavorAFM.BandGapsinGNRsY.-W.Sonetal.,Phys.Rev.Lett.2006,97,216803Half-metallicity(HM)•100%spinpolarization•Applications:–Spininjection–Spintransport•SomeHMmaterials:–CrO2,NiMnSb,Fe3O4•TransitionMetalEncapsula