第十一讲：现代电子结构计算方法
第十一讲：现代电子结构计算方法
N-Particle System ProblemRememberh222ethe good old days of theH山=E() =E(T)I-electron H-atom??2m4元E0They'reover!Nre2NNNh?"Z,e2h?ZZ12NX2H=N22m2MR-Ri1-1i-l i-i-ir1kinetic energy of ionskinetic energy of electronselectron-ion interactionpotential energy of ionselectron-electroninteractionMulti-Atom-Multi-Electron Schrodinger EquationH(R.,R.ri.r)(R....Rn,r....,)-EY(R....Rnir..,2
N-Particle System Problem 2
近似！：绝热近似Born-OppenheimerApproximation (skinless version)·mass of nuclei exceeds that of theelectrons bya factor of Iooo ormorewe can neglect the kinetic energyof thenucleitreatthejon-ioninteractionclassicallyBornOppenheimersignificantly simplifiestheThis term is just an external potentialV(r)Hamiltonianforthe electrons:N2h?"区1Z22Hr;2m-R-ri-l1i-1
3 近似I：绝热近似
近似Ⅲ：单电子近似InteractingNon-lnteracting体系Hamilton量简化九H=Zh>n2m;riu1h P, =6,单电子薛定方程分子轨道：+.4*非占据轨道O,为单电子波函数or分子轨道(MO)全同粒子本征值8.为对应的分子能级中中中泡利不相容原理占据轨道任何两个粒子不能有完全相同的量子数(n,l,m,s）其中s=±1/2自旋量子数
近似II：单电子近似 体系Hamilton量简化 单电子薛定谔方程 or 任何两个粒子不能有完全相同的量子数 (n, l, m, s) 其中s = ±1/2 自旋量子数
无相互作用多体波函数？Write wavefunction as a simple product of singleHartree积（HP多体波函数）particle states:Y(r1,...,rn) = P1(r1)P2(r2) ... Pn(rn)HardProduct of EasyTotal energyZE=8
无相互作用多体波函数？ Hartree 积 （HP多体波函数） Total energy
近似：平均场近似对HP多体波函数h??Z1e2XV2m;itdr-5jHartree Approximation: the electrons do not interactexplicitly with the others, but each electron interactswith the medium potential given by the other electrons
• 对HP多体波函数 近似III：平均场近似 Hartree Approximation: the electrons do not interact explicitly with the others, but each electron interacts with the medium potential given by the other electrons
变分法NNNZZJ;E=ZH +=2i=1i=1 j=1j#i-单电子积分项ZMW72H; = [Φ;(i)h,Φ;(i)dt; =[Φ,(i)Φ,(i)dt2TiAA=1一库仑双电子积分-代表经典两个电荷密度之间的排斥作用（由Φ；andΦ：描述）J; =JJ(1)二Φ;(2)dt,dt,12
变分法 N i 1 N j i j 1 ij N i 1 E Hi J 21 i i MA 1 iA 2 A i i i i i i i Φ (i)dτ rZ 21 H Φ (i)h Φ (i)dτ Φ (i) - 单电子积分项 1 2 2j 12 2 ij i Φ (2)dτ dτ r1 J Φ (1) - 库仑双电子积分 - 代表经典两个电荷密度之间的排斥作用（由 Φi and Φj 描述）
Hartree方程MNZAN>Z[Φ, ()]dt:Φ.;(i)=8,Φ, (i)X2TiAi=1 j=1A=1j#iN-分子轨道能量8; =H, +ZJji=1j+iNNN-1总能量E=&-ZJi=1i=1 j=i+1
Hartree 方程 dτ Φ (i) ε Φ (i) r 1 Φ (j) r Z 2 1 i i i N i 1 N j i j 1 j ij 2 j M A 1 iA 2 A i N j i j 1 i Hi Jij ε - 分子轨道能量 N-1 i 1 N j 1 ij N i 1 E i J i 总能量
(0In order to find , we needΦ, = SCF procedure-中(ildtJ,Jodtdt,Ci次送代电子密度分布(i(i)L江P;(r)=Φ;(r)1NNN总电子密度B=ZEEJE-二NNPtot(r)= p;(r)=Φ;(r)i=1i=1No(Converged?Solution: Self-ConsistentYesField(SCF)Stop
Solution: Self-Consistent Field （SCF ） In order to find Φi we need Φi SCF procedure 2 i i ρ (r) Φ (r) i 次迭代 电子密度分布 N i 1 2 i N i 1 tot i ρ (r) ρ (r) Φ (r) 总电子密度
N个电子体系Slater波函数HP波函数不满足电子（费米子）全同性要求的波函数反对成性如：双粒子体系PROBLEM=P。(1)（2）是两电子体系本征方程的解也是本征解交换算符P12P[(1)(2)]=(2)(1)-(1)(2)wa则满足交换反对称条件的波函数：Slater行列式形式（SD）(b) (1)(1)Forfermions the negativesignmustbeused,sothat方[9. (1)%(2)-9.(2)g (1)] —ysDthewavefunction goesto.(2) (2)identicallyzero ifthestatesaand bareidentical
N个电子体系Slater波函数 HP 波函数