从头算分子动力学

18119983PROCRESSINPHYSICSVol.18,No.1March.,1998(310027)CarParrinello,,[1,2],,(observationtime),,(110ps),,,,(epitaxy)(amorphousstructure),,,,,,,,[3,4],[4,5],,[5],,1042106,Lennard2Jones©1995-2005TsinghuaTongfangOpticalDiscCo.,Ltd.Allrightsreserved.,,[6,7],,,,,,(bondingproperties),,,,[8],,1985,CarParrinello,,,(CP)[9],,,,()SCI,:,;CarParrinello;,D.R.Hamann(normconservingpseudopotential)[10,11];CP;,,(Ultrasoftpsuedopotential)[12],CP,CP;CP2.1,Born2Oppenheimer(BO),,,,,,BO,BO[13],,BO[14]BO,v({R1}):v({RI})=50ûHû50(1)8418©1995-2005TsinghuaTongfangOpticalDiscCo.,Ltd.Allrightsreserved.H,50(1),RI,50104,(1),H,,,,,,,,N(O(Nm),mE4),Hartree2Fock,O(N4),,,MD,104,CarParrinello[9],,,,KS2.2,Born2Oppenheimer,,1965,HohenbergKohn[3],,KohnSham[4],,:(1),(He)Q0:70[Q0]ûHeû70[Q0]=FHK[Q0](2)FHK,,(2)Ev(Q),E0=minQEv[Q](3)Ev[Q]=FHK[Q]+drV(r)Q(r),V(r)()Kohn2Sham,,(:Hs=T+V,Es=Ts+drvsQs)Q0=Qs,Q0:Q0(r)=occifiû7i(r)û2(4)7iShrodinger:941:©1995-2005TsinghuaTongfangOpticalDiscCo.,Ltd.Allrightsreserved.HKS7i=-12ý2+Veff7i=Ei7i(5)(c=me=h-=1),,fiVEv:Ev[Q0]=Ts[Q0]+drV(r)Q0(r)+12drdrQ0(r)Q0(r)ûr-rû+Exc[Q0](6)(6),E,2,Veff:Veff=V(r)+drQ(r)ûr-rû+DExc[n]DQ(r)(7)(4),(5),(7),KS,,Exc,,,,,rExc(r),Q(r),:Exc=drQ(r)exc[Q(r)](8)exc,CepeleyAlderMonteCarlo[15],,,(),,,,,,,2.3Kohn-Sham[16],KS:Q0(r),,(7),(5),HdHK,Q0(r),,M3(MKS),KS,7i,[17](steepestdecent):7ai(r,t)=-DED73i(r,t)(9),õt,DED73i=HKS7i,HKS7i(9),,,:0518©1995-2005TsinghuaTongfangOpticalDiscCo.,Ltd.Allrightsreserved.7ai(r,t)=-DED73i(r,t)+j+ij7j(10),((9))7i,7i,+ij,7ai=0,HKS7i=j+ij7j,+ij,KS,,E,,,,,,,[18]HK[19],HdHK7i7i(NM),HKS7iNM2,N2M,,HKS,(-12ý2)(Veff),,7iO(NM)Veff(local),7iFFT,O(NMlnM)M,O(NMlnM),,2.4,{R1}V({R1}),,,MD:(1){R1}KS,(2)Hellman2Feymann[20],(3)MIRbI=-ýRIV,CarParrinello,MDKS,,,,BendtZunger[21],:Lqi7ai=-DED73i+j+ij7j(11)M{IRaI=-9E9RI(12)LqiM{I{7i}{RI}(11)(12){RI},151:©1995-2005TsinghuaTongfangOpticalDiscCo.,Ltd.Allrightsreserved.{7i},E,,BO:V({RI})=min{7i}E[{7i},{RI}],EV{7i}{RI},(11)(12),,(11)(12),[2224],(11)(12),,,V,,CarParrinello[9](CP)1985,CarParrinello[9],,E,V:Lcl=12IMIRa2I-V[{RI}](13):L=occidrLiû7ai(r)û2+12IMIRa2I-E[{RI},{7i}]+i,j+ij(dr73i(r)7j(r)-Dij)(14)L{7i}{RI},t,7iRILi(:2),,Li,,Li7i,LiL,(14),KeKI,E[{RI},{7i}],+ij7i,,(14)Euler:Lq7ai=-DED73i+j+ij7j(15)M{IRbI=-9E9RI(16)RaI7ai,,,,,,,,T=0,[9]2518©1995-2005TsinghuaTongfangOpticalDiscCo.,Ltd.Allrightsreserved.Kirkpatrick[25],MDMonteCarlo,,,,,,,,,,,,,,110Si[26]T=3300K,,T=2000K,,2,Si101Si10(a)T=2000K,(b)T=3300K[26]2Si10[26],,KI,Ke,,(13):M{1Rb1=-9V[{RI}]9RI(17)V[{RI}]=min7iE[{7i},{RI}],(16)(17),,E[{RI},{7i}]7iL{7i}0,{7ai}0,,,E,BO,L{7i}0,{7ai}0,,351:©1995-2005TsinghuaTongfangOpticalDiscCo.,Ltd.Allrightsreserved.,Born2Oppenheimer,,,,(15),2.4MD,,,KS,,CP4.1,Carparrinello[9],(augmentedplanewave)(allelectron)CP[27],CP,(FFT),,,,,,,,,Hellman2Feymann[20],,,,(Ecut),,,,,,,,CP(norm2conserving)[10,11](MDBox),,,,(),,,,,,Bloch,:7ki(r)=eikõrg_cki(g)eigõr(18)g,kBrillouin,cki(g)4518©1995-2005TsinghuaTongfangOpticalDiscCo.,Ltd.Allrightsreserved.Fourier,(18)g,Ecut,g12(k+g)2FEcutk=0,#,,,7i(r),Fourierci(-g)=c3i(g),,4.2,,,,[28],,,1979,HamannSchluterChiang[10],,(all2electron),Hamann(semi2local),,lvl:vps(r)=l=0vl(r)Pdl(19)Pdll,(lE2),vll,,,rc,(rrc),()rrcE[11],(HPu),,Hamannvionl,lvcorel$vionl:vionl=vcore+$vionl(20)vcore=-Zvr{2i=1ccoreierfc[(Acorei)1ö2r]}(21)$vionl=3i=1(Ai+r2Ai+3)e-Air2(22)erfc(x):erfc(x)=1-erf(x)=1-2Px0exp(-z2)dz(21)ccoreiAcorei(22)AiAi+3Ai,,[11],ccoreiAcoreiAi,AiAi+3,(22),3(Au)551:©1995-2005TsinghuaTongfangOpticalDiscCo.,Ltd.Allrightsreserved.3-Zvör,Vdcore(r)[11][11],,,,4.3CP4[29],t=0,E[Q(r)],HFE[Q(r)]RIýRIE,7i,,,,,,,,4(),,FourierHdc(g),:L7bi=-HKS7i(23)(23),,,t=t+$t,,,(15),,,,,BO,4,t=0,7i,E[Q(r)],HFE[Q(r)]RIýRIE,(23),(15),t=t+$t,ýRIE,(16)t=t+$t5E[Q(r)]HF,MDE:E=Ek+Exc+EH+Eps+EM(24)6518©1995-2005TsinghuaTongfangOpticalDiscCo.,Ltd.Allrightsreserved.4[29]EM=12IJZIZJûRI-RJû(25),Ek,Fourier:Ek=12g2c3i(g)ci(g)(26)Exc,(8)exc,(8)EHEpsHartree751:©1995-2005TsinghuaTongfangOpticalDiscCo.,Ltd.Allrightsreserved.5,N3FFTNFFT[29]Eps=drv(r)Qe(r)(27)v,:v(r)=Ivps(r-RI)(28):vps(r)=vloc(r)+lg-1l=0$vl(r)pdl(29)$vl=vl(r)-vlg(r)(r),(local),(nonlocal),,,Eps,(ELps)(ENLps):ELps=drQe(r)Ivloc(ûr-RIû)(30)ENLps=i7iûvNLû7i(31)vNL=Ilg-1l=0$vl(ûr-RIû)Pdl(RI)(32)(32),Pdl(RI)RIl,,EH,EMELps,8518©1995-2005TsinghuaTongfangOpticalDiscCo.,Ltd.Allrightsreserved.,,EM,,(Gauss):QI(r-RI)=-ZI(RCI)3(P)-3ö2exp-ûr-RIû2(RCI)2(33)RCIIEM=12I,JdrdrQI(r-RI)QJ(r-RJ)ûr-rû-Eself-Eovrl(34)(34)I,J,IJ,,(33):Eself=12IdrdrQI(r-RI)QI(r-RI)ûr-rû=12PIZ2I(RCI)(35)Eovrl:Eovrl=12IJZIZJûRI-RJû-drdrQI(r-RI)QJ(r-RJ)ûr-rû(36):Eovrl=ZIZJRIJerfcRIJ(RCI)2+(RCJ)2(37)RIJ=ûRI-RJû,erfc(x):(30),(34),EH,EMELps:EM+EH+ELps=12drdrQ(r)Q(r)ûr-rû+drQe(r)IvIloc(ûr-RIû)-Eself-Eovrl(38)Q=Qe+QI,vIloc(r)=vloc(r)-ZIrerfrRCI(39)-(ZIör)erf(röRCI)drQ(r)öûr-rûerf(x)=2Pxoexp(-z2)dz,,,,(38),Madelung,Hartree,,,g,,Hartree(38),Eself,(35),Eovrl(36),ENLself(29)(31)ENLself:ENLps=iIg,ge-ig.RIc3i(g)vNL(g,g)eig.RIci(g)(40)vNL(g,g)=gûvûg951:©1995-2005TsinghuaTongfangOpticalDiscCo.,Ltd.Allrightsreserved.eigõr=l(2l+l)iljl(gr)Pl(cosH),vNL(g,g)=4Plg-1l=0(2l+1)Pl[cos(Hgg)]drr2jl(gr