3.320: Lecture 7 (Feb 24 2005) DENSITY-FUNCTIONAL THEORY,AND DENSITY-FUNCTIONAL PRACTICEHartree-Fock EquationsThe Thomas-Fermi approachLocal Density ApproximationIt’s a poor man Hartree…The Argon atomDensity-functional theoryThe Hohenberg-Kohn theorems (1965)The universal functional F[ρ]Second Hohenberg-Kohn theoremThe non-interacting unique mappingThough this be madness, yet there’s method in’tEuler-Lagrange EquationsThe Kohn-Sham equationsElectronic Total EnergyThe Homogeneous Electron GasThe Exchange-correlation EnergyThe Phases of SiliconGGAs, meta-GGA, hybridsDensity-functional theory in practice: the total-energy pseudopotential methodReferences (theory)References (practice)Software3.320: Lecture 7 (Feb 24 2005) DENSITYDENSITY--FUNCTIONAL THEORY,FUNCTIONAL THEORY,AND DENSITYAND DENSITY--FUNCTIONAL PRACTICEFUNCTIONAL PRACTICEFeb 24 2005 3.320 Atomistic Modeling of Materials -- Gerbrand Ceder and Nicola MarzariFeb 24 2005 3.320 Atomistic Modeling of Materials -- Gerbrand Ceder and Nicola MarzariHartree-Fock Equations2**1()21() ()||1() ()()()() ()||iIiIjjjjijiiiijj ijVR rrrdrrrrdrrrrrrrrµµµµµλµλλλϕϕϕϕϕϕεϕϕ⎡⎤−∇+ − +⎢⎥⎣⎦⎡⎤−⎢⎥−⎢⎥⎣⎦⎡⎤=⎢⎥−⎢⎥⎣⎦∑∑∫∑∫rrrrrrrrrrrrrrrr11 122 212() () ()() () ()1(, ,..., )!() () ()nnn nrr rrr rrr rnrr rαβ ναβ ναβ νϕϕ ϕϕϕ ϕψϕϕ ϕ=rrrLrrrLrr rMMOMrrrLImage removed for copyright reasons.Screenshot of online article.“Nobel Focus: Chemistry by Computer.” Physical Review Focus, 21 October 1998.http://focus.aps.org/story/v2/st19Feb 24 2005 3.320 Atomistic Modeling of Materials -- Gerbrand Ceder and Nicola MarzariFeb 24 2005 3.320 Atomistic Modeling of Materials -- Gerbrand Ceder and Nicola MarzariThe Thomas-Fermi approach• Let’s try to find out an expression for the energy as a function of the charge density• E=kinetic+external+el.-el.• Kinetic is the tricky term: how do we get the curvature of a wavefunction from the charge density ?• Answer: local density approximationFeb 24 2005 3.320 Atomistic Modeling of Materials -- Gerbrand Ceder and Nicola MarzariLocal Density Approximation• We take the kinetic energy density at every point to correspond to the kinetic energy density of the homogenous electron gas)()(35rArTrrρ=∫∫∫∫−++=− 21212135||)()(21)()()(][ rdrdrrrrrdrvrrdrAEextFeThrrrrrrrrrrrρρρρρIt’s a poor man Hartree…• The idea of an energy functional is not justified• It does not include exchange effects - but Dirac proposed to add the LDA exchange energy:• It scales linearly, and we deal with 1 function of three coordinates !∫− rdrCrr34)(ρFeb 24 2005 3.320 Atomistic Modeling of Materials -- Gerbrand Ceder and Nicola MarzariFeb 24 2005 3.320 Atomistic Modeling of Materials -- Gerbrand Ceder and Nicola MarzariThe Argon atomDensity-functional theory• The external potential Vextand the number N of electrons completely define the quantum problem• The wavefunctions are – in principle ! – uniquely determined, via the Schrödinger Equation• All system properties follow from the wavefunctions• The energy (and everything else) is thus a functional of Vextand NFeb 24 2005 3.320 Atomistic Modeling of Materials -- Gerbrand Ceder and Nicola MarzariThe Hohenberg-Kohn theorems (1965)• The density as the basic variable: the external potential determines uniquely the charge density, and the charge density determines uniquely the external potential.Feb 24 2005 3.320 Atomistic Modeling of Materials -- Gerbrand Ceder and Nicola MarzariFeb 24 2005 3.320 Atomistic Modeling of Materials -- Gerbrand Ceder and Nicola Marzari1st Theorem: the Density as the Basic Variable.Image removed for copyright reasons.The universal functional F[ρ]• The ground state density determines the potential of the Schrödinger equation, and thus the wavefunctions• It’s an emotional moment…Ψ+Ψ=−eeVTrFˆˆ)]([rρFeb 24 2005 3.320 Atomistic Modeling of Materials -- Gerbrand Ceder and Nicola MarzariSecond Hohenberg-Kohn theoremThe variational principle – we have a new Schrödinger’s-like equation, expressed in terms of the charge density only 0)()()]([)]([ ErdrrvrFrEextv≥′+′=′∫rrrrrρρρ(ρ’ determines it’s groundstate wavefunction, that can be taken asa trial wavefunction in this external potential)∫′+′>=Ψ′++Ψ′>=<Ψ′Ψ′<−][|ˆˆ||ˆ|ρρFvvVTHextexteeFeb 24 2005 3.320 Atomistic Modeling of Materials -- Gerbrand Ceder and Nicola MarzariThe non-interacting unique mapping• The Kohn-Sham system: a reference system is introduced (the Kohn-Sham electrons)• These electrons do not interact, and live in an external potential (the Kohn-Sham potential) such that their ground-state charge density is IDENTICAL to the charge density of the interacting systemFeb 24 2005 3.320 Atomistic Modeling of Materials -- Gerbrand Ceder and Nicola MarzariFeb 24 2005 3.320 Atomistic Modeling of Materials -- Gerbrand Ceder and Nicola MarzariThough this be madness, yet there’s method in’t• For a system of non-interacting electrons, the Slater determinant is the EXACT wavefunction (try it, with 2 orbitals)• The kinetic energy of the non interacting system is well definedFeb 24 2005 3.320 Atomistic Modeling of Materials -- Gerbrand Ceder and Nicola MarzariEuler-Lagrange EquationsFeb 24 2005 3.320 Atomistic Modeling of Materials -- Gerbrand Ceder and Nicola MarzariThe Kohn-Sham equationsFeb 24 2005 3.320 Atomistic Modeling of Materials -- Gerbrand Ceder and Nicola MarzariElectronic Total Energy212121||)()(21)]([ rdrdrrrnrnrnEHrrrrrrr∫∫−={}[]∫∫∑++++∇−==rdrnrVrnErnErdrrEextxcHiiNiirrrrrrrr)()()]([)]([)()(212*1ψψψThe Homogeneous Electron GasGround State of the Electron Gas by a Stochastic MethodCeperley and Alder Phys. Rev. Lett. (1980)Feb 24 2005 3.320 Atomistic Modeling of Materials -- Gerbrand Ceder and Nicola MarzariThe Exchange-correlation EnergyFeb 24 2005 3.320 Atomistic Modeling of Materials -- Gerbrand Ceder and Nicola MarzariFeb 24 2005 3.320 Atomistic Modeling of Materials -- Gerbrand Ceder and Nicola MarzariThe Phases of SiliconFeb 24 2005 3.320 Atomistic Modeling of Materials -- Gerbrand Ceder and Nicola MarzariGGAs, meta-GGA, hybrids• GGAs: generalized gradient approximations (gradients of the density are introduced, preserving analytical – scaling – features of the