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 − 𝑲𝒋 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 interorbital); 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
zH,molecule112Molecules1Ra- e.g. H2Y(r,r,R.,R,)-energyEandwavefunctionincludingallfourparticlesMolecular Schrodinger equation-22++2RV2RaβM2m=EY(r,..,In,R,..,Rn)
H2 molecule • Molecules – e.g. H2 – energy E and wavefunction including all four particles • Molecular Schrödinger equation 7
Hamiltonianforamoleculee?Z-h2-h2electronsnucleielectronsnucleielectronsnuclei77H=BZ2ZZN72me2marisYABrjAii>jA>Bkinetic energy of the electronskinetic energy of the nucleielectrostatic interaction between the electrons andthe nucleielectrostatic interaction between the electronselectrostatic interaction between the nuclei8
Hamiltonian for a molecule • kinetic energy of the electrons • kinetic energy of the nuclei • electrostatic interaction between the electrons and the nuclei • electrostatic interaction between the electrons • electrostatic interaction between the nuclei nuclei A B AB A B electrons i j ij nuclei A iA A electrons i A nuclei A A i electrons i e r e Z Z r e r e Z m m 2 2 2 2 2 2 2 2 2 ˆ H 8
Born-OppenheimerapproximationM>> melectronnuclei The nuclei move more slowly than the electrons;.Thenucleiinstantaneouslywillappearimmobile-Theelectronandnuclearmotionscanthusbeapproximately decoupled;》separation of variablesY(ri. ,,R,..R) Ye(ri. ,,R..R.)u(R..,R.)O
Born-Oppenheimer approximation • Mnuclei >> melectron – The nuclei move more slowly than the electrons; • The nuclei instantaneously will appear immobile; – The electron and nuclear motions can thus be approximately decoupled; »separation of variables 9
TheelectronicSchrodingerequationDepends only parametrically on the locations of thenuclei[2+2岁]Yele(r.. ,; R,., R) - Eee Ylee (ri., ,; R,..Rn)Z.e>27Orbital approximation, again!Y(.r)中(r),(r)Applying variational principle,again![i, + Ccouomb[p] + Uexchane[ + Uonehtio [ ] ; fiw; = );Ucoml[pl = JPCdr, p(r)=Z;(r)1210
The electronic Schrödinger equation • Depends only parametrically on the locations of the nuclei 10 • Orbital approximation, again! • Applying variational principle, again!
Potentialenergysurface(PEs). Defines a potential energy surface (adiabatic surface)+2Z.ZnEpes(R,..,Rv) Ele + Eachpoint onthePES isasolutiontoelectronicSchrodingerequationNucleican bethought to“"travel"onAthis PES(R)·3N degree offreedomWaystodescribenucleimotion:1.quantum mechanically, e.g.to capture"tunneling." Expensive andspecialized.2. classical particles rolling along the PEs. Essence of classical ab initiomoleculardynamics.3.focus on locating“critical points" along PES, like stable minima("molecules")and saddle points ("transition states"). Least expensive and11most common
Potential energy surface (PES) • Defines a potential energy surface (adiabatic surface) 11 Each point on the PES is a solution to electronic Schrodinger equation Ways to describe nuclei motion: 1. quantum mechanically, e.g. to capture “tunneling.” Expensive and specialized. 2. classical particles rolling along the PES. Essence of classical ab initio molecular dynamics. 3. focus on locating “critical points” along PES, like stable minima (“molecules”) and saddle points (“transition states”). Least expensive and most common. • Nuclei can be thought to “travel” on this PES • 3N degree of freedom