Restricted or Unrestricted HFRestricted HFrestricted wave-functioni(x)=P(r)a(w)重2(×)=Φi(r)β(w)Restricted wave-function for Li atomSS[YRHF)=[15P1,P2spf1sBut: K1s(a)2s(α )0 and K1s(β)2s(a)=01s(α) and 1s(p) electrons will experience differentpotentials so that it will be more convenient to describethe two kind of electrons by different wave-functions
Restricted or Unrestricted HF Φ 1(x)= φ 1(r) α ( ω) Φ 2(x)= φ 1(r) β ( ω ) restricted wave-function Restricted wave-function for Li atom RHF 1 s 1 s 2 s But: K1s( )2s( ) ≠0 and K1s( )2s( )=0 1s( ) and 1s( ) electrons will experience different potentials so that it will be more convenient to describe the two kind of electrons by different wave-functions Restricted HF
Restricted or Unrestricted HFUnrestricted HFNorestriction on spatial wavefunctionfor spinorbit;(×x)=pi(r)a(w)Φ2(×)=pβ(r)p(w)Unrestricted wave-function for Li atomP2[4uHF) =0% % 02spppa15f1s(o% oP)=Sgpap)=8)=8j(0
Restricted or Unrestricted HF No restriction on spatial wavefunction for spin orbit Unrestricted HF Unrestricted wave-function for Li atom UHF 1s 1s 2 s Φ 1(x)= φ α 1(r) α ( ω) Φ 2(x)= φ β 1(r) β ( ω ) δij α j α i ij β j β i δ αβij β j α i S
UHF Ssolution: possible spin containmentClosedShellSystemRHF is goodRHFandUHFpresentsameresultsOpenShellSystemUHF优点：Highefficient withtwo setsof spatialfunction缺点：单行列式波函数不是<S2>的本征波函数高自旋态的混合>自旋污染
UHF solution: possible spin containment Closed Shell System Open Shell System RHF is good RHF and UHF present same results UHF 优点：High efficient with two sets of spatial function 缺点：单行列式波函数不是<S 2 >的本征波函数 高自旋态的混合 自旋污染
For an UHF wave-function,the expectation value of S2is:+NP-2sP(00%)=8(010)=815(00100)=SgYwhere:22SUHFexactHow to check?Gaussiansoftware，useiop(5/14)=2输出<S2<S2)Numberofunpairedelectronss00010.50.75如果误差<10%ok212.031.53.75426.052.58.75
How to check? 如果误差<10% ok Gaussian software, use iop(5/14)=2 输出 <S2>
RecommendedSolutionRestrictedOpen-Shell HF(ROHF)波函数由多个slater行列式线性组合1. Good for energy and wavefunction2. Bad for spin-dependent properties&24)=c|2)+c|4)+c|6)+..[2>-exactdoubletstate|4>-exactquartetstate-approximatelyadoublet[6>-exactsextetstate对比Unrestricted HF (UHF)波函数由单个slater行列式线性组合1. Energy: EuHF ≤ERHF Or EROHF2. Good for spin-dependent propertiesM.W.Wong,L.Radom,J.Phys.Chem.99,8582(1995).J.Baker, A. Scheiner, J.Andzelm, Chem. Phys. Lett. 216, 380 (1993)
Recommended Solution Restricted Open-Shell HF (ROHF) 波函数由多个slater行列式线性组合 1. Good for energy and wavefunction 2. Bad for spin-dependent properties 对比 Unrestricted HF (UHF) 波函数由单个slater行列式线性组合 1. Energy: EUHF ≤ ERHF or EROHF 2. Good for spin-dependent properties