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$160,(DFT).(LDA)Kohn-Sham(KS),DFT.DFT,,,,.,1987Hartree-Fock(HF).,DFT.HF.W.KohnDFT1998.DFT..,.,(observationtime).,(1-10ps).,,.DFT(MD).,,,,,,,,,,.,..DFT,(¯rst-principal)..DFT,(LDA),,DFT.LDA.DFT.(GGA),n(~r),LDA.,(CI),DFT+MD.,CI.,DFT+MD.DFT,.2,Kohn-ShamHamiltonian,DFTLDA12DFTLDALDALDA+ULDA++ON$23BOBorn{Oppenheimer).^H=^T+^V+^Vext(2:1)$2.1Hohenberg-Kohn1964HKHK1)Vext½(~r),^O½(~r).ªj^Ojª=O[½](2:2)2)^O^HH[½]=EVext[½]EVext[½]=ªj^T+^Vjª+ªj^Vextjª(2:3)FHK[½]FHK[½]=ªj^T+^VjªEVext[½]=FHK[½]+R½(~r)Vext(~r)d~r(2:4)31½onetoonecorrespondence)2)FHK[½]3)HK$2.2Kohn-ShamKS1965,KSDFTEe[½](Hartree-FockEHF[½]Ee=T+V(2:5)EHF=T0+(VH+VX|{z}V)(2:6)T,VT0VHVXVC.VC=Ee¡EHF=T-T0(2:7)HKFHKFHK=T+V+T0¡T0=T0+V+(T¡T0|{z}VC)=T0+V+VC+VH¡VH=T0+VH+VC+(V¡VH|{z}VX)4=T0+VH+(VX+VC|{z}VXC)(2:8)VXC{EVext[½]=T0[½0]+(1=2)Zd~rd~r0½o(~r)½o(~r0)j~r¡~r0j+ZVext[½]d~r+EXC[½o](2:9)^VXC=±EXC[½]±½KS^HksÁi=[¡12r2i+^Veff]Ái=EiÁi(2:10)½(~r)=NXi=1Á¤i(~r)Ái(~r)(2:11)^Veff=Vext(~r)+Zd~r½(~r0)j~r¡~r0j+±EXC[½]±½(~r)(2:12)(2.9){(2.12)KSEXC.LDA~r²xc(~r)½(~r),EXC=Zd~r½(~r)²xc[½(~r)](2:13)EXCCepeleyAlderMonteCarlo(Phys.Rev.lett(566-568)Vol.45.No.7)LDA1996Perdew,Burke,ErnzerhofGGAGeneralizedGradientApproximation.Phys.Rev.lettVol.77.No.8(3895-3897)),(LSD)2.13)ELSDXC[n;n#]=Zd3rn²unifxc[n;n#](2:14)GGAELSDXC[n;n#]=Zd3rf(n;n#;rn;rn#)(2:15)5GGA60$2.3Kohn-ShamKS½o(~r),(2.12)Veff,(2.10)^HKSÁi(~r)½o(~r)M3(MKS)KSÁi_Ái(~r;t)=¡±E±Á¤i(~r;t)(2:16)¢±E±Á¤i=HKSÁi,Ái_Ái(~r;t)=¡±E±Á¤i(~r;t)+Xj^ijÁj(2:17)(2.16)ÁiÁi^ij_Ái=0,HKSÁi=Pj^ijÁj,^ijKSE^HKSÁiÁiNMHKSÁiNM2N2M,HKS6ÁiO(NM)VeffÁi(FFT),O(NMlnM)O(NMlnM),,.$2.4,fRIg,V[fRIg],,,MD:(1)fRIgKS.(2)Hellman{Feymam.(3)MIÄRI=¡rRIV(Bendt,Zunger).,.¹i_Ái=¡±E±Á¤i+Xj^ijÁj(2:18a)MI_RI=¡@E@RI(2:18b)¹iMifÁigfRIg.(2.18).fRIg,fÁig,E.BO;VfRIg=min|{z}[Ái]E[fÁig,fRIg],EV.fÁig,fRIg,(2.18).,.,(2.18)..,V,,1985,Car,Parrinello.$3(CP)$3.1CPCar,Parrinello,,E.V.{.7:LCL=12PIMI_R2I¡V[fRIg]()(3.1)L=T-V,:L=occXiZd~r¹ij_Áij2+12XIMI_RI2¡E[fRIg;fÁig]+Xi;j^ijZd~r(Á¤iÁj¡±ij)(3:2),LfÁig,fRIg,t,Ái,RI.¹i.(:££),.¹i=¹.E[fÁig,fRIg],,(Hartree),..^ijÁi-@L@qi¡ddt[@L@_qi]=0i=1:2:3¢¢¢n:(3.2)¹_Ái=¡±E±Á¤i+Xj^ijÁj(3:3a)MIÄRI=¡@E@RI(3:3b)_RI_ÁiT!0K.,,,.,,.,..,.,.,,,.,.,,8.:MIÄRI=¡@V[fRIg]@RI(3:4)V[fRIg]=min|{z}[Ái]E[fÁig,fRIg],(3.3b)(3.4),,E[fRIg,fÁig]Ái.¹fÁig,f_Áig,(BO),,E,BO.¹fÁig,f_Áig,,,.,KS,..,Cluster,,.$3.2CP.((augmentedplanewave).WIN2K.,FFT,,,.,..,,,,Hellman-Feymann,..,,(ECUT)..,.,,,,,..(MDBox),,,,.(),,,,MakovPayne:,,9L¡5,L.,,,.,Bloch,Ái(~k)=exp(i~k~r)X~GC(~k;~G)exp(i~G~r)(3:4)~G,~k,C(~k;~G).(3.4)~G,ECUT,~G(1/2)(~k+~G)2·ECUT.¡~k=0,,Ái(~r),C(~k;~G)=C¤(¡~k;~G),.$3.2(Norm-ConservingPseudopotential,NCPP)1979,Hamann,Schlecter,Chiang(Norm-ConservingPseudopotential,NCPP).HSCNCPP..,.HSCNCPP(Semi-Local),,jlmVe,Vps(~r)=Xl=0Vl(~r)jlmlmj(3:5),(l¸2),l¸2,Vll,.NCPP:1).2)rrc(),fÁn=Án().,r=rc.3)rrc,,.fÁnjfÁnrc=ÁnjÁnrc4)¡(1=2)[(rfÁn)2d2dEdrlnfÁn]r=rc=ÁnjÁnr=rc1.2,3.4NCPP.10HSCVionl,lVcorel4VioneVionl=Vcore+4Vione(3:6)Vcore=¡Zvr[2Xi=1Ccoreierfc[[®corei]12r](3:7)erfc(x),erf(x),erfc(x)=1¡erf(x)=1¡2p¼Zx0e¡z2dz(3:8)4Vione=3Xi=1(Ai+r2Ai+3)e¡®ir2(3:9)Ccorei,®corei,Ai,Ai+3,®i,NCPP.NCPP,.2.,,FFT.NCPP,.Vanderbilt.$3.4(UltrasoftPseudopotentialUSPP)HSC,.().O2P,3,,,(),.90,D.Vanderbilt,.,.,,.$3.4.1USPPD.Vanderbilt(KB),.,Schrodinger[^T+VAE(~r)]'i(~r)=²i'i(~r)(3:10)11i,i=f²i;l;mg,,,'i(~r)²i,VAE(~r).jÁi,rcl,Ái(~r)'i(~r)r=rcl,.ÁijÁiR='ij'iR.R.jÂi=(²i¡^T¡VLOC)jÁi(3.11)12(r¸R)jÂi´0,(r¸rC),VLOC=VAE,r=rcl,VAE.Phys,Rev.Vol.41.No.11(1892),,VNLVNL=jÂiÂijÂijÁi(3:12)Bij=ÁijÂj,j¯i=Xj(B¡1)jijÂj(3:13)VNLj¯iVNL=Xi;jBi;jj¯i¯jj(3:14)(3.13)(3.14)(3.12)VNLjÁi=Xi;jBi;jj¯i¯jjÁi=Xi;jBi;jXm(B¡1)mijÂmXn(B¡1)+jnÂnjÁi13=XijXm(B¡1B)mjjÂmXn(B¡1)+jnB+ni=Xij±mjjÂm±ji=jÂi(^T+VLOC+VNL)jÁi=²ijÁiD.Vanderbilt(scatteringproperties)²iÁiQij´0Qij='ij'jR¡ÁijÁjR(3.15)Qij=0BijVNL²i,Ái(r)Ui(r)rjÂj=(²j¡T¡VLOC)jÁijÂj=[²j+12rd2dr2(r¡l(l+1)2r2¡VLOC]Uj(r)rBij=ÁijÂj=ZR0r2drU¤i(r)r[²j+12rd2dr2r¡l(l+1)2r2¡VLOC]Uj(r)r=ZR0drU¤i(r)[²j+12d2dr2¡l(l+1)2r2¡VLOC(r)]Uj(r)Bij¡B¤ji1,2,3,41ZR0u¤i(r)uj(r)(²j¡²i)dr=ZR0u¤i(r)uj(r)r2(²j¡²i)r2dr=ÁijÁjR(²j¡²i)340212ZR0dru¤i(r)d2dr2uj(r)¡12ZR0druj(r)d2dr2u¤i(r)=12[u¤i(R)u0j(R)¡u0i(R)uj(R)]14Bij¡B¤ji=(²j¡²i)ÁijÁjR+12[u¤i(R)u0j(R)¡u¤i(R)uj(R)](3:16)AE'i(3.16)r=RBij¡B¤ji=(²i¡²j)Qij;Qij='ij'jR¡ÁijÁjR(3:17)Qij=0,Bij(3.12)(3.14)VNLD.VanderbiltQij=0S=1+Xi;jQi;jj¯i¯jj(3:18)VNL=Xi;jDijj¯i¯jj(3:19)Dij=Bij+²jQij(3.20)'ij'jR=ÁijSjÁjR(3:21)ÁijSjÁjR=ÁijÁjR+Xm;nÁijQm;nj¯m¯njÁj=ÁijÁjR+Xm;nÁijQm;nXl(B¡1)lmjÂlXR(B¡1)+knÂkjÁjR=ÁijÁjR+Xm;nQm;nXlBil(B¡1)lmXk(B¡1)+knB+jk=ÁijÁjR+Xm;nQm;n±im±jn=ÁijÁjR+(Qij)R='ij'jRDijDij¡D¤ji=Bij+²jQij¡(Bji+²iQji)¤=Bij¡B¤ji+(²j¡²i)Qij=0D.(^H¡²iS)jÁi=015[^T+VLOC+Xm;n(Bmn+²nQm;n)j¯m¯nj]jÁi¡²i(1+Xm;nQm;nj¯m¯nj)=[^T+VLOC+Xm;n(Bmnj¯m¯nj¡²i]jÁi+Xm;n(²n¡²i)Xm;nQm;nj¯m¯njjÁi0,ÁijXm;nQmn²n±mi±mj¡²iXm;nQm;n±mi±mj=00=[dd²Á²j^T+VLOC+VNL¡²SjÁ²R]²=²i(3:22)(T+VLOC+VNL)jÁ²=²SjÁ²(T+VLOC+VNL)jÁ²+4²=(²+4²)SjÁ²+4²Á²+4²jÁ²jÁ²jSjÁ²+4²=Á²+4²jSjÁ²Á²j¡12r2+VLOC+VNLjÁ²+4²¡Á²+4²j¡12r2+VLOC+VNLjÁ²=4²Á²jSjÁ²+4²¡12Zd~s[Á¤²@Á²+4²@r¡Á²+4²@Á¤²