3.320: Lecture 6 (Feb 17 2005) HARTREE-FOCK AND DFTMatrix Formulation and Variational PrincipleEnergy of an Hydrogen AtomTwo-electron atomAtomic Units and Conversion Factors(see Stellar handout)Molecules and Solids:Electrons and NucleiEnergy of a collection of atomsComplexity of the many-body ΨMean-field approachHartree EquationsThe self-consistent fieldIterations to self-consistencyDifferential AnalyzerWhat’s missingSpin-StatisticsSlater determinantPauli principleHartree-Fock EquationsShell structure of atomsRestricted vs. UnrestrictedThe Dissociation of H2Koopmans’ TheoremsWhat is missingFaster, or betterConfiguration InteractionDensity-functional theoryThe Thomas-Fermi approachLocal Density ApproximationIt’s a poor man Hartree…The Argon atomThe Hohenberg-Kohn theorems (1965)The universal functional F[ρ]Second Hohenberg-Kohn theoremEuler-Lagrange equationsReferencesSoftware3.320: Lecture 6 (Feb 17 2005) HARTREEHARTREE--FOCK AND DFTFOCK AND DFTPhotos of Hartree, Fock, Hohenberg, Kohn, and Sham removed for copyright reasons.Feb 17 2005 3.320 Atomistic Modeling of Materials -- Gerbrand Ceder and Nicola MarzariMatrix Formulation and Variational PrinciplemnmknnEcHc =∑=ϕϕˆ,1[]0ˆ|||HEE<Φ Φ>Φ= ≥<Φ Φ>Feb 17 2005 3.320 Atomistic Modeling of Materials -- Gerbrand Ceder and Nicola MarzariFeb 17 2005 3.320 Atomistic Modeling of Materials -- Gerbrand Ceder and Nicola MarzariEnergy of an Hydrogen AtomˆHEαααααΨΨ=ΨΨ()expCrααΨ= −22223 211,22CCCrαα α α α απππαααΨΨ= Ψ−∇Ψ= Ψ−Ψ=−Two-electron atom),(),(||12121212121212221rrErrrrrZrZelrrrrrrψψ=⎥⎦⎤⎢⎣⎡−+−−∇−∇−Feb 17 2005 3.320 Atomistic Modeling of Materials -- Gerbrand Ceder and Nicola MarzariMany-electron atom21111( ,..., ) ( ,..., )2||inelniiijiiijZrrErrrrrψψ>⎡⎤−∇− + =⎢⎥−⎢⎥⎣⎦∑∑∑∑rr rrrrFeb 17 2005 3.320 Atomistic Modeling of Materials -- Gerbrand Ceder and Nicola MarzariFeb 17 2005 3.320 Atomistic Modeling of Materials -- Gerbrand Ceder and Nicola MarzariAtomic Units and Conversion Factors(see handout)1 a.u. = 2 Ry = 1 Ha1 Ry = 13.6057 eV1 eV = 23.05 kcal/molFeb 17 2005 3.320 Atomistic Modeling of Materials -- Gerbrand Ceder and Nicola MarzariMolecules and Solids:Electrons and Nuclei),...,,,...,(),...,,,...,(ˆ1111 NntotNnRRrrERRrrHrrrrrrrrψψ=• We treat only the electrons as quantum particles, in the field of the fixed (or slowly varying) nuclei• This is generically called the adiabatic or Born-Oppenheimer approximation• “Adiabatic” means that there is no coupling between different electronic surfaces; “B-O” implies there is no influence of the ionic motion on one electronic surfaceFeb 17 2005 3.320 Atomistic Modeling of Materials -- Gerbrand Ceder and Nicola MarzariEnergy of a collection of atoms•Te: quantum kinetic energy of the electrons•Ve-e: electron-electron interactions•Ve-N: electrostatic electron-nucleus attraction (electrons in the field of all the nuclei)•VN-N: electrostatic nucleus-nucleus repulsion()∑∑∑∑∑>−−−=⎥⎦⎤⎢⎣⎡−=∇−=iijjieeiiIiINeierrVrRVVT||1ˆˆ21ˆ2rrrrˆˆˆ ˆeeeeN NNHTV V V−−−=+ + +Feb 17 2005 3.320 Atomistic Modeling of Materials -- Gerbrand Ceder and Nicola MarzariComplexity of the many-body Ψ“…Some form of approximation is essential, and this would mean the construction of tables. The tabulation function of one variable requires a page, of two variables a volume and of threevariables a library; but the full specification of a single wavefunction of neutral iron is a function of 78 variables. It would be rather crude to restrict to 10 the number of values of each variable at which to tabulate this function, but even so, full tabulation would require 1078entries.”Feb 17 2005 3.320 Atomistic Modeling of Materials -- Gerbrand Ceder and Nicola MarzariMean-field approach• Independent particle model (Hartree): each electron moves in an effective potential, representing the attraction of the nuclei and the average effect of the repulsive interactions of the other electrons• This average repulsion is the electrostatic repulsion of the average charge density of all other electronsFeb 17 2005 3.320 Atomistic Modeling of Materials -- Gerbrand Ceder and Nicola MarzariHartree EquationsThe Hartree equations can be obtained directly from the variationalprinciple, once the search is restricted to the many-body wavefunctions that are written – as above – as the product of single orbitals (i.e. we are working with independent electrons))()(||1|)(|)(2122iiiijIijijjjiIirrrdrrrrRVrrrrrrrrϕεϕϕ=⎥⎥⎦⎤⎢⎢⎣⎡−+−+∇−∑∑∫≠)()()(),...,(22111 nnnrrrrrrLrrrrϕϕϕψ=Feb 17 2005 3.320 Atomistic Modeling of Materials -- Gerbrand Ceder and Nicola MarzariThe self-consistent field• The single-particle Hartree operator is self-consistent ! It depends on the orbitals that are the solution of all other Hartree equations• We have n simultaneous integro-differential equations for the n orbitals• Solution is achieved iteratively2211( ) | ( )| () ()2||i I i j j j ii iiIjijiVR r r dr r rrrϕϕεϕ≠⎡⎤−∇+ − + =⎢⎥−⎢⎥⎣⎦∑∑∫rrrrrrrrIterations to self-consistency• Initial guess at the orbitals• Construction of all the operators• Solution of the single-particle pseudo-Schrodinger equations• With this new set of orbitals, construct the Hartree operators again• Iterate the procedure until it (hopefully) convergesFeb 17 2005 3.320 Atomistic Modeling of Materials -- Gerbrand Ceder and Nicola MarzariDifferential AnalyzerVannevar Bush and the Differential Analyzer.Courtesy of the MIT Museum. Used with permission.Feb 17 2005 3.320 Atomistic Modeling of Materials -- Gerbrand Ceder and Nicola MarzariFeb 17 2005 3.320 Atomistic Modeling of Materials -- Gerbrand Ceder and Nicola MarzariWhat’s missing• It does not include correlation• The wavefunction is not antisymmetric• It does remove nl accidental degeneracy of the hydrogenoid atomsFeb 17 2005 3.320 Atomistic Modeling of Materials -- Gerbrand Ceder and Nicola MarzariSpin-Statistics• All elementary particles are either fermions (half-integer spins) or bosons (integer)• A set of identical (indistinguishable) fermions has a wavefunction that is antisymmetric by exchange• For bosons it is symmetric12 12( , ,..., ,..., ,..., ) ( , ,..., ,..., ,..., )kkj njnrr