【文章內(nèi)容簡(jiǎn)介】 iscussion of the chemical bond in molecules.局域密度近似 (LDA)? Any real system is spatially inhomogeneous, it has a spatially varying density n(r), it would clearly be useful to also include information on the rate of this variation in the functional.? In this approximation ,one tries to systematically calculate gradientcorrections of general functions of n(r) and ? n(r)? Different GGAs differ in the choice of the function f(n,? n).廣義梯度近似 (GGA)Alex D. Becke “一切都是合法的 ” 劍宗John P. Perdew一定的物理規(guī)律（如標(biāo)度關(guān)系和漸進(jìn)行為）為基礎(chǔ)， PBE 氣宗? GGAs used in quantum chemistry typically proceed by fitting parameters to test sets of selected molecules.? Nowadays the most popular GGAs are PBE in physics, and BLYP in chemistry.? Current GGAs seem to give reliable results for all main types of chemical bonds (covalent, ionic, metallic and hydrogen bridge).? In addition to the density and its derivatives, MetaGGAs depend also on the KohnSham kiicenergy density:? So that Exc can be written as Exc [n(r),? n(r), τ (r)]. The additional degree of freedom provided by τ is used to satisfy additional constraints on Exc.? MetaGGAs have given favorable results, even when pared to the best GGAs.? The full potential of this type of approximation is only beginning to be explored systematically.MetaGGA? Common hybrid functional mix a fraction of HartreeFock exchange into the DFT exchange functional.Hybrid Functionals(Becke, 1993)(Perdew,1998)B3PW91, B3LYPPBE0B3LYP is the main workinghorse in putational chemistryLDA: Slater exchange VoskoWilkNusair correlation, etcGGA: Exchange: B88, PW91, PBE, OPTX, HCTH, etc Correlations: LYP, P86, PW91, PBE, HCTH, etcHybrid GGA: B3LYP, B3PW91, B3P86, PBE0, B971, B972, B98, O3LYP, etcMetaGGA: VSXC, PKZB, TPSS, etcHybrid metaGGA: tHCTHh, TPSSh, BMK, etc? Describing the behavior of many electrons interacting via coulombs’law: ? It will vanish for one electron system because of the selfinteraction in it.? So we have: Fermi and Amaldi 1934(the first version of SIC)自相互作用修正 (SIC)? Now it bees:? This correction can be applied on top of any approximate density functional, and ensures that the resulting corrected functional satisfies: for a oneelectron system.? Since orbital functionals depend on the density only implicitly, we can not directly calculate the functional derivative via orbitals .? In the case of kiicenergy functional, we use the KohnSham scheme to minimize E[n].? In the case of orbital expressions for Exc, the corresponding indirect scheme is known as the optimized effective potential.Optimized Effective Potential (OEP)? The minimization of the orbital functional with respect to the density is achieved by repeated application of the chain rule for functional derivatives:? Further evaluation of Eq. above gives rise to an integral equation that determines the belonging to the chosen orbital functional ? KLI(Krieger, Li and Iafrate) approximation to solve the full OEP integral equation. The application of the OEP methodology to the Fock term is as known as the EXX (extraexchange method).GW近似? 以自能代替密度泛函局域近似中的交換關(guān)聯(lián)能? 固體能隙問(wèn)題? 準(zhǔn)粒子方程? 零級(jí)近似， plasmonpole模型，自洽含時(shí)密度泛函? RungeGross定理? 作用量泛函? TDKS方程TDDFT中的線性響應(yīng)? 外場(chǎng)微擾? 一階密度響應(yīng)? KS響應(yīng)函數(shù)TDDFT中的線性響應(yīng)? 線性響應(yīng)方程? 交換相關(guān)核，絕熱局域密度近似L(S)DA+U? Mott絕緣體， Hubbard模型? Anisimov et al.： Stoner I Hubbard U? 軌道序：? Dudarev et al.：懲罰泛函動(dòng)力學(xué)平均場(chǎng)理論? 量子多體問(wèn)題局域動(dòng)力學(xué)（把點(diǎn)陣模型映射到自洽的量子雜質(zhì)模型）? 凍結(jié)空間漲落，考慮局域量子漲落? Hubbard模型哈密頓量? 單格點(diǎn)動(dòng)力學(xué)? 自洽方程? Anderson雜質(zhì)模型? DFTDMFT流密度泛函理論? 處理任意強(qiáng)度磁場(chǎng)下相互作用電子體系（1987）? 一套規(guī)范不變且滿(mǎn)足連續(xù)性方程的自洽方程組? 交換相關(guān)能量不僅依賴(lài)于電荷密度還依賴(lài)于順磁流密度? 原子分子對(duì)磁場(chǎng)的響應(yīng)，自發(fā)磁化，磁場(chǎng)中的二維量子點(diǎn)，造新的交換相關(guān)近似相對(duì)論性密度泛函理論? 量子電動(dòng)力學(xué)的單粒子方程： Dirac方程? DiracCoulomb（ DC）哈密頓量? DiracCoulombBreit（ DCB）哈密頓量相對(duì)論性密度泛函理論? 相對(duì)論情形的 HK定理，四分量 DiracKohnSham（ DKS）方程，數(shù)值旋量基組，縮并 Gaussian型旋量基組? 兩分量準(zhǔn)相對(duì)論方法–BreitPauli近似–ZORA近似? 有效核勢(shì)（ ECP）方法密度泛函微擾理論? 晶格振動(dòng)理論? 線性響應(yīng) Hessian矩陣， 2n+1定理? 凍聲方法，分子動(dòng)力學(xué)譜分析方法幾何 Berry位相? 電介質(zhì)極化，介電常數(shù)? 偶極矩 宏觀極化；流 極化變化? 電荷密度（波函數(shù)的模）；流（波函數(shù)的位相）? 零電場(chǎng)情況下，任意兩個(gè)晶體態(tài)之間的極化變化對(duì)應(yīng)著 一個(gè)幾何量子位相? 晶格振動(dòng)、鐵電、壓電效應(yīng)、自發(fā)極化、靜態(tài)介電張量、電子介電常數(shù)。? 不如傳統(tǒng)的微擾理論方法普適，但實(shí)現(xiàn)簡(jiǎn)單、計(jì)算量小Part II:數(shù)值方法數(shù)值離散方法? 基組展開(kāi)– LCAO基組（ Gaussian基組、數(shù)值基組）? 實(shí)空間網(wǎng)格平面波基組：從 OPW到 PP? 平面波展開(kāi)? 正交化平面波（ OPW）? 贗勢(shì)（ PP）方法– 經(jīng)驗(yàn)贗勢(shì) – 模守恒贗勢(shì) – 超軟贗勢(shì)Muffintin勢(shì)場(chǎng)與分波方法? Muffintin勢(shì)場(chǎng)近似– 綴加平面波（ APW） – 格林函數(shù)方法（ KKR）? 線性化方法– LAPW – LMTO? 分波方法的發(fā)展– FPLAPW – thirdgeneration MTO, NMTO, EMTO平面波基組：從 USPP到 PAW? 投影綴加波（ PAW）方法 ? 贗波函數(shù)空間? USPP or PAW? (VASP, ABINIT, ...)實(shí)空間網(wǎng)格? 簡(jiǎn)單直觀 ? 允許通過(guò)增加網(wǎng)格密度系統(tǒng)地控制計(jì)算收斂精度? 線性標(biāo)度 ? 可以方便的通過(guò)實(shí)空間域分解實(shí)現(xiàn)并行計(jì)算? 處理某些特殊體系（帶電體系、隧穿結(jié)。）有限差分? 從微分到差分? 提高 FD方法的計(jì)算效率– 對(duì)網(wǎng)格進(jìn)行優(yōu)化，如曲線網(wǎng)格（適應(yīng)網(wǎng)格）和局部網(wǎng)格優(yōu)化（復(fù)合網(wǎng)格）– 結(jié)合贗勢(shì)方法 – 多尺度（ multiscale）或預(yù)處理（ preconditioning）有限元? 變分方法 ? 處理復(fù)雜的邊界條件? 矩陣稀疏程度及帶狀結(jié)構(gòu)往往不如有限差分好 ? 廣義的本征值問(wèn)題多分辨網(wǎng)格上的小波基組? 多分辨分析? 半取樣（ semicardinal）基組線性標(biāo)度與量子力學(xué)中的局域性? “近視原理 ” ? 局域化的 Wannier函數(shù)或密度矩陣– 絕緣體：指數(shù)衰減，能隙越大衰減越快– 金屬：零溫下按冪率衰減，在有限溫度下可出現(xiàn)指數(shù)衰減? 局域區(qū)域 ? 線性標(biāo)度系數(shù)， crossover線性標(biāo)度算法? 分治方法 ? 費(fèi)米算符展開(kāi)和費(fèi)米算符投影方法? 直接最小化方法– 密度矩陣最小化– 軌道最小化– 優(yōu)基組密度矩陣最小化線性標(biāo)度算法? 基于格林函數(shù)的遞歸方法 ? 脫離軌道的（ orbitalfree,OF）算法 ? 對(duì)角化以外的線性標(biāo)度– 構(gòu)造有效哈密頓量的算法 – 幾何優(yōu)化與分子動(dòng)力學(xué)– TDDFTPart III:應(yīng)用物理學(xué)：強(qiáng)相關(guān)體系? 模型哈密頓量 ? LDA++ ? 電子結(jié)構(gòu)： CrO2? 點(diǎn)陣動(dòng)力學(xué) : 钚化學(xué)：弱作用體系? 松散堆積的軟物質(zhì)、惰性氣體、生物分子和聚合物，物理吸附、 Cl+HD反應(yīng)? 用傳統(tǒng)的密度泛函理論處理弱作用體系? 一個(gè)既能產(chǎn)生 vdW相互作用系數(shù)又能產(chǎn)生總關(guān)聯(lián)能的非局域泛函：無(wú)縫的（seamless）方法? GW近似 ? 密度泛函加衰減色散（ DFdD）生命科學(xué)：生物體系? 困難（尺寸問(wèn)題、時(shí)間尺度） ? QM/MM方法（飽和原子法、凍結(jié)軌道法） ? 簡(jiǎn)單勢(shì)能面方法– 線性同步過(guò)渡（ LST ） – 二次同步過(guò)渡（ QST ）? 完全的分子動(dòng)力學(xué)– 并行復(fù)制動(dòng)力學(xué)（ parallel replica dynamics） – 超動(dòng)力學(xué)（ hyperdynamics, metadynamics）– 溫度加速的動(dòng)力學(xué)（ temperature accelerated dynamics ）– 快速蒙特卡羅（ onthefly kineric Monte Carlo）方法納米和材料科學(xué)：輸運(yùn)性質(zhì)及其他? 輸運(yùn)：非平衡態(tài)第一