《分子电子结构》研究生课程教学资源（Electronic Structure of Molecules）5-BOA-LCAO-Semi-ROHF

LasttimeHartree-Fock method1.Theproblemofmanyelectrons2.Hartree atom3.Self-consistent field approach4.Pauli principle5.Slater determinants6.Coulomb integral7.Exchange integral8.Hartree-Fock equation9.Local densityapproximation10.Correlation

Last time 2 Hartree-Fock method 1. The problem of many electrons 2. Hartree atom 3. Self-consistent field approach 4. Pauli principle 5. Slater determinants 6. Coulomb integral 7. Exchange integral 8. Hartree-Fock equation 9. Local density approximation 10. Correlation

TheHartree-FockmethodSlater determinants (the Pauli principle + the one-electron Hartreeorbitals)areapproximationforthe“"real"wavefunctionY(.,,)(),(r)Restricted Hartree-Fock (RHF)equation(h + 2x-(2); -R,)(r1) = fib;(r1) = eib;(r1)j,=Jv,(r)=dr,K(r)=[ v(r,);(z)二dr, [;(r)ri212Local density approximation (LDA) (1951)33p(r)(r)= fHFS (r)=8W (r)2元12Theexchange potential depends onlyon the valueof the density at a point;3

The Hartree-Fock method • Slater determinants (the Pauli principle + the one-electron Hartree orbitals) are approximation for the “real” wavefunction 3 • Restricted Hartree-Fock (RHF) equation 𝒉෡ 𝒊 + σ𝒋=𝟏 𝑵 𝟐 𝟐𝑱෠ 𝒋 − 𝑲෡𝒋 𝝍𝒊 𝒓𝟏 = 𝒇෡ 𝒊𝝍𝒊 𝒓𝟏 = 𝜺𝒊𝝍𝒊(𝒓𝟏) • Local density approximation (LDA) (1951) • The exchange potential depends only on the value of the density at a point;

CorrelationCorrelation: Ecorr = Etrue -EHFInstantaneous,dynamical Coulomb interaction between allelectrons;Prevents electrons of opposite spin from being in the sameplace at the same time (Coulomb hole: intra-orbital and inter-orbital);H山 = EY-Y+Etaya+Etabyab+Etabeyabe+..Wavefuntion-basedmethodsiajabjkabeConfiguration Interaction (Cl)Coupled Cluster (CC)Many-body perturbation theory (MPn)Density-basedmethodsDensity Functional Theory (DFT)-LDA, GGA, MetaGGA[h; + Ucoulomb[p] + Vexchange[p] + Ucorrelation[p])b;(r1) = fib;(r1) = &;;(r1)4Jensen,Chp4

Correlation 4 • Correlation: Ecorr = Etrue – EHF • Instantaneous, dynamical Coulomb interaction between all electrons; • Prevents electrons of opposite spin from being in the same place at the same time (Coulomb hole: intra-orbital and inter￾orbital); 𝐻𝜓 = 𝐸𝜓 • Wavefuntion-based methods • Configuration Interaction (CI) • Coupled Cluster (CC) • Many-body perturbation theory (MPn) • Density-based methods • Density Functional Theory (DFT) – LDA, GGA, MetaGGA Jensen, Chp 4 E d d t t t t abc ijk ijkabc abc ijk ab ij ijab ab ij a i ia a i / with respect to minimize ˆ * * 0                       H  𝒉෡ 𝒊 + 𝝊𝐂𝐨𝐮𝐥𝐨𝐦𝐛 𝝆 + 𝝊𝐞𝐱𝐜𝐡𝐚𝐧𝐠𝐞 𝝆 + 𝝊𝐜𝐨𝐫𝐫𝐞𝐥𝐚𝐭𝐢𝐨𝐧[𝝆] 𝝍𝒊 𝒓𝟏 = 𝒇෡ 𝒊𝝍𝒊 𝒓𝟏 = 𝜺𝒊𝝍𝒊(𝒓𝟏)

Phenomenal successofhybridmethodsHybridDFTmethodsincludesomeflavorofHartree-Fockexchange (exact exchange)e.g.B3LYP, PBEO, MN15, APFD, TPSSh, HSE,DM21Workhorse of computational chemistry/physics/materialsB3LYPEB3LYP = (1 - a)ELSDA + aEHF + bAEB8 + (1 - c)ELSDA + cELYPXCBecke,A.D.J.Chem.Phys.98,5648(1993)Lee.C., Yang,W.&Parr, R.G.Phys.Rev.B37,785 (1988)Citations of both papers > 100,000 timesTop 10 most cited papersObituary: Density Functional Theory (1927-1993), Gill, P.M.WAust. J. Chem., 54, 661 (2002),5

Phenomenal success of hybrid methods 5 • Hybrid DFT methods include some flavor of Hartree-Fock exchange (exact exchange) • e.g. B3LYP, PBE0, MN15, APFD, TPSSh, HSE, DM21 • Workhorse of computational chemistry/physics/materials Becke, A. D. J. Chem. Phys. 98, 5648 (1993). Lee. C., Yang, W. & Parr, R. G. Phys. Rev. B 37, 785 (1988). • Citations of both papers > 100,000 times • Top 10 most cited papers • Obituary: Density Functional Theory (1927-1993), Gill, P.M.W. Aust. J. Chem., 54, 661 (2002). • B3LYP

Contents1.Born-Oppenheimerapproximation;2. Linear Combination Atomic Orbitals;3. Semi-empirical methods;4.Open-shell systems;5. SCF detailsJensen, chp36

Contents 6 Jensen, chp 3 1. Born-Oppenheimer approximation; 2. Linear Combination Atomic Orbitals; 3. Semi-empirical methods; 4. Open-shell systems; 5. SCF details