3.320: Lecture 5 (Feb 15 2005) THE MANY-BODY PROBLEMWhen is a particle like a wave ?Time-dependent Schrödinger’s equation(Newton’s 2nd law for quantum objects)Stationary Schrödinger’s Equation (I)Stationary Schrödinger’s Equation (II)Free particle Ψ(x,t)=φ(x)f(t)Interpretation of the Quantum Wavefunction (Copenhagen)A Traveling “Plane” WaveMetal Surfaces (I)Metal Surfaces (II)Infinite Square WellFinite Square WellA Central Potential (e.g. the Nucleus)Solutions in a Coulomb Potential: the Periodic TableOrthogonality, Expectation Values, and Dirac’s <bra|kets>Matrix Formulation (I)Matrix Formulation (II)Variational PrincipleEnergy of an Hydrogen AtomTwo-electron atomEnergy of a collection of atomsElectrons and NucleiComplexity of the many-body ΨMean-field approachHartree EquationsThe self-consistent fieldIterations to self-consistencyDifferential AnalyzerWhat’s missingSpin-StatisticsSlater determinantPauli principleHartree-Fock EquationsShell structure of atomsFaster, or betterRestricted vs. UnrestrictedKoopmans’ TheoremsAtomic Units and Conversion Factors(see Stellar handout)Software3.320: Lecture 5 (Feb 15 2005) THE MANYTHE MANY--BODY PROBLEMBODY PROBLEMFeb 15 2005 3.320 Atomistic Modeling of Materials -- Gerbrand Ceder and Nicola MarzariWhen is a particle like a wave ?Wavelength • momentum = Planck↕λ •p = h( h = 6.6 x 10-34J s )),( trrΨ=ΨFeb 15 2005 3.320 Atomistic Modeling of Materials -- Gerbrand Ceder and Nicola MarzariTime-dependent Schrödinger’s equation(Newton’s 2ndlaw for quantum objects)ttritrtrVtrm∂Ψ∂=Ψ+Ψ∇−),(),(),(),(222rhrrrh1925-onwards: E. Schrödinger (wave equation), W. Heisenberg (matrix formulation), P.A.M. Dirac (relativistic)Feb 15 2005 3.320 Atomistic Modeling of Materials -- Gerbrand Ceder and Nicola MarzariStationary Schrödinger’s Equation (I)ttritrtrVtrm∂Ψ∂=Ψ+Ψ∇−),(),(),(),(222rhrrrh*Feb 15 2005 3.320 Atomistic Modeling of Materials -- Gerbrand Ceder and Nicola MarzariStationary Schrödinger’s Equation (II))()( tfEtfdtdi =h)()()(222rErrVmrrrhϕϕ=⎥⎦⎤⎢⎣⎡+∇−Feb 15 2005 3.320 Atomistic Modeling of Materials -- Gerbrand Ceder and Nicola Marzari)()( tfEtfdtdi =h⎟⎠⎞⎜⎝⎛−= tEitfhexp)(Free particle Ψ(x,t)=φ(x)f(t))()(222xExmϕϕ=∇−h⎟⎟⎠⎞⎜⎜⎝⎛= xmEixh2exp)(ϕFeb 15 2005 3.320 Atomistic Modeling of Materials -- Gerbrand Ceder and Nicola MarzariInterpretation of the Quantum Wavefunction (Copenhagen)2(,)xtΨis the probability of finding an electron in x and t22( ) exp( ) ( )ixEtxϕϕ−=hFeb 15 2005 3.320 Atomistic Modeling of Materials -- Gerbrand Ceder and Nicola MarzariA Traveling “Plane” Wave(,) exp[( )]xt ikx tωΨ∝ −Diagram of plane wave removed for copyright reasons.Feb 15 2005 3.320 Atomistic Modeling of Materials -- Gerbrand Ceder and Nicola MarzariMetal Surfaces (I)Feb 15 2005 3.320 Atomistic Modeling of Materials -- Gerbrand Ceder and Nicola MarzariMetal Surfaces (II)Feb 15 2005 3.320 Atomistic Modeling of Materials -- Gerbrand Ceder and Nicola MarzariInfinite Square WellFeb 15 2005 3.320 Atomistic Modeling of Materials -- Gerbrand Ceder and Nicola Marzari1614121086420n=1n=2n=3n=4-a 0 a xψ1ψ2ψ3ψ4ψ(x)8ma2π2h2EFigure by MIT OCW.Finite Square WellFeb 15 2005 3.320 Atomistic Modeling of Materials -- Gerbrand Ceder and Nicola Marzariaψ2(x)aψ3(x)aψ4(x)aψ1(x)1-110.5-0.5-2 -12 x/a01-110.5-2-12 x/a01-110.5-2-1201-11-2-1 2 x/a x/a0Figure by MIT OCW.A Central Potential (e.g. the Nucleus)222222222ˆ()2HVrmxyz∂∂∂=− ∇ + ∇ = + +∂∂∂h2 2222 22211 1ˆsin ( )2sinsinHrVrmr r r r rϑϑϑ ϑ ϑϕ⎡⎤∂∂ ∂ ∂ ∂⎛⎞ ⎛ ⎞=− + + +⎢⎜⎟ ⎜ ⎟ ⎥∂∂ ∂ ∂ ∂⎝⎠ ⎝ ⎠⎣⎦h),()()(ϕϑψlmElmElmYrRr=r22 2222(1)() () ()22El ElddllVr R r ER rm dr r dr rµ⎡⎤⎛⎞+−+++ =⎢⎥⎜⎟⎝⎠⎣⎦hhFeb 15 2005 3.320 Atomistic Modeling of Materials -- Gerbrand Ceder and Nicola MarzariSolutions in a Coulomb Potential: the Periodic Tablehttp://www.orbitals.com/orb/orbtable.htmCourtesy of David Manthey. Used with permission.Feb 15 2005 3.320 Atomistic Modeling of Materials -- Gerbrand Ceder and Nicola Marzari____________________________________________________________Orthogonality, Expectation Values, and Dirac’s <bra|kets>ψψψ== )(rrijjijirdrrδψψψψ==∫rrr)()(*iiiiiEHrdrrVmr ==⎥⎦⎤⎢⎣⎡+−∫ψψψψˆ)()(2)(2*rrrhrFeb 15 2005 3.320 Atomistic Modeling of Materials -- Gerbrand Ceder and Nicola MarzariMatrix Formulation (I)ψψψψEHrErH =⇔=ˆ)()(ˆrr{}functions orthogonalk ,1nknnncϕϕψ∑==ψϕψϕmmEH =ˆmnknmnEcHc =∑=ϕϕˆ,1Feb 15 2005 3.320 Atomistic Modeling of Materials -- Gerbrand Ceder and Nicola MarzariMatrix Formulation (II)mnmknnEcHc =∑=ϕϕˆ,1mknnmnEccH =∑= ,1⎟⎟⎟⎟⎟⎟⎠⎞⎜⎜⎜⎜⎜⎜⎝⎛=⎟⎟⎟⎟⎟⎟⎠⎞⎜⎜⎜⎜⎜⎜⎝⎛⋅⎟⎟⎟⎟⎟⎟⎠⎞⎜⎜⎜⎜⎜⎜⎝⎛kkkkkkccEccHHHH........................111111Feb 15 2005 3.320 Atomistic Modeling of Materials -- Gerbrand Ceder and Nicola MarzariVariational Principle[]ˆ|||HE<ΦΦ>Φ=<ΦΦ>If , then Φ is the ground state wavefunction, and viceversa…[]0EEΦ≥[]0EEΦ=Feb 15 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=−Feb 15 2005 3.320 Atomistic Modeling of Materials -- Gerbrand Ceder and Nicola MarzariTwo-electron atom),(),(||12121212121212221rrErrrrrZrZelrrrrrrψψ=⎥⎦⎤⎢⎣⎡−+−−∇−∇−Many-electron atom21111( ,..., ) ( ,..., )2||inelniiijiiijZrrErrrrrψψ>⎡⎤−∇− + =⎢⎥−⎢⎥⎣⎦∑∑∑∑rr rrrrFeb 15 2005 3.320 Atomistic Modeling of Materials -- Gerbrand Ceder and Nicola MarzariEnergy of a collection of atomsNeNNeeNeVVVTTH−−−++++=ˆˆˆˆˆˆ•Te: quantum kinetic energy of the electrons•Ve-e: electron-electron interactions•VN-N: electrostatic nucleus-nucleus repulsion•Ve-N: electrostatic electron-nucleus attraction (electrons in the field of all the nuclei)()∑∑∑∑∑>−−−=⎥⎦⎤⎢⎣⎡−=∇−=iijjieeiiIiINeierrVrRVVT||1ˆˆ21ˆ2rrrrFeb 15 2005 3.320 Atomistic Modeling of Materials -- Gerbrand Ceder and Nicola MarzariElectrons and Nuclei),...,,,...,(),...,,,...,(ˆ1111 NntotNnRRrrERRrrHrrrrrrrrψψ=•We