DENSITY-FUNCTIONAL THEORY, AND DENSITY-FUNCTIONAL PRACTICE

Transcription

1 3.320: Lectue 7 Feb DENSITY-FUNCTIONAL THEORY, AND DENSITY-FUNCTIONAL PRACTICE Feb Atomistic Modeling of Mateials -- Geband Cede and Nicola Mazai

2 Hatee-Fock Equations ψ,,..., = 1 2 n 1 n! ϕα 1 ϕβ 1 L ϕν 1 ϕ ϕ L ϕ α 2 β 2 ν 2 M M O M ϕ ϕ L ϕ α n β n ν n 1 2 i + V RI i ϕλ i + 2 I µ µ 1 ϕ ϕ d * µ j µ j j j i ϕλ i * 1 ϕµ j ϕλ j dj ϕµ i = εϕλ i j i Feb Atomistic Modeling of Mateials -- Geband Cede and Nicola Mazai

3 Image emoved fo copyight easons. Sceenshot of online aticle. Nobel Focus: Chemisty by Compute. Physical Review Focus, 21 Octobe Feb Atomistic Modeling of Mateials -- Geband Cede and Nicola Mazai

4 The Thomas-Femi appoach Let s ty to find out an expession fo the enegy as a function of the chage density E=kinetic+extenal+el.-el. Kinetic is the ticky tem: how do we get the cuvatue of a wavefunction fom the chage density? Answe: local density appoximation Feb Atomistic Modeling of Mateials -- Geband Cede and Nicola Mazai

5 Local Density Appoximation We take the kinetic enegy density at evey point to coespond to the kinetic enegy density of the homogenous electon gas 3 5 A T ρ = + + = ] [ d d d v d A E ext Fe Th ρ ρ ρ ρ ρ Feb Atomistic Modeling of Mateials -- Geband Cede and Nicola Mazai

6 It s a poo man Hatee The idea of an enegy functional is not justified It does not include exchange effects - but Diac poposed to add the LDA exchange 4 enegy: C ρ 3 d It scales linealy, and we deal with 1 function of thee coodinates! Feb Atomistic Modeling of Mateials -- Geband Cede and Nicola Mazai

7 The Agon atom Feb Atomistic Modeling of Mateials -- Geband Cede and Nicola Mazai

8 Density-functional theoy The extenal potential V ext and the numbe N of electons completely define the quantum poblem The wavefunctions ae in pinciple! uniquely detemined, via the Schödinge Equation All system popeties follow fom the wavefunctions The enegy and eveything else is thus a functional of V ext and N Feb Atomistic Modeling of Mateials -- Geband Cede and Nicola Mazai

9 The Hohenbeg-Kohn theoems 1965 The density as the basic vaiable: the extenal potential detemines uniquely the chage density, and the chage density detemines uniquely the extenal potential. Feb Atomistic Modeling of Mateials -- Geband Cede and Nicola Mazai

10 1st Theoem: the Density as the Basic Vaiable. Image emoved fo copyight easons. Feb Atomistic Modeling of Mateials -- Geband Cede and Nicola Mazai

11 The univesal functional F[ρ] The gound state density detemines the potential of the Schödinge equation, and thus the wavefunctions F[ ρ ] = Ψ Tˆ + Vˆ e e Ψ It s an emotional moment Feb Atomistic Modeling of Mateials -- Geband Cede and Nicola Mazai

12 E v Second Hohenbeg-Kohn theoem The vaiational pinciple we have a new Schödinge s-like equation, expessed in tems of the chage density only [ ρ ] = F[ ρ ] + v ρ d E ext 0 ρ detemines it s goundstate wavefunction, that can be taken as a tial wavefunction in this extenal potential < Ψ Hˆ Ψ >=< Ψ Tˆ + Vˆ + Ψ >= + e e vext ρ vext F[ ρ ] Feb Atomistic Modeling of Mateials -- Geband Cede and Nicola Mazai

13 The non-inteacting unique mapping The Kohn-Sham system: a efeence system is intoduced the Kohn-Sham electons These electons do not inteact, and live in an extenal potential the Kohn-Sham potential such that thei gound-state chage density is IDENTICAL to the chage density of the inteacting system Feb Atomistic Modeling of Mateials -- Geband Cede and Nicola Mazai

14 Though this be madness, yet thee s method in t Fo a system of non-inteacting electons, the Slate deteminant is the EXACT wavefunction ty it, with 2 obitals The kinetic enegy of the non inteacting system is well defined Feb Atomistic Modeling of Mateials -- Geband Cede and Nicola Mazai

15 Eule-Lagange Equations Feb Atomistic Modeling of Mateials -- Geband Cede and Nicola Mazai

16 The Kohn-Sham equations Feb Atomistic Modeling of Mateials -- Geband Cede and Nicola Mazai

17 Electonic Total Enegy { } [ ] = = d n V n E n E d E ext xc H i i N i i ] [ ] [ * 1 ψ ψ ψ ] [ d d n n n E H = Feb Atomistic Modeling of Mateials -- Geband Cede and Nicola Mazai

18 The Homogeneous Electon Gas Gound State of the Electon Gas by a Stochastic Method Cepeley and Alde Phys. Rev. Lett Feb Atomistic Modeling of Mateials -- Geband Cede and Nicola Mazai

19 The Exchange-coelation Enegy Feb Atomistic Modeling of Mateials -- Geband Cede and Nicola Mazai

20 The Phases of Silicon Feb Atomistic Modeling of Mateials -- Geband Cede and Nicola Mazai

21 GGAs, meta-gga, hybids GGAs: genealized gadient appoximations gadients of the density ae intoduced, peseving analytical scaling featues of the unknown exact xc functional Hybids: a cetain amount of non-local Hatee-Fock exchange is weighed in Feb Atomistic Modeling of Mateials -- Geband Cede and Nicola Mazai

22 Density-functional theoy in pactice: the total-enegy pseudopotential method Remove tightly bound coe electons: the pseudopotential appoach Repesent obitals on a basis plane waves Calculate total enegy fo tial obitals Kinetic, hatee enegy in ecipocal space Exchange-coelation, ext. pot. in eal space FFTs! Sum ove all states: BZ integations Iteate o minimize to self-consistency Feb Atomistic Modeling of Mateials -- Geband Cede and Nicola Mazai

23 Refeences theoy W. Koch, M. C. Holthausen, A Chemist s Guide to Density Functional Theoy R. G. Pa, W. Yang, Density-Functional Theoy of Atoms and Molecules W. Kohn, Nobel lectue F. Jensen, Intoduction to Computational Chemisty J. M. Thijssen, Computational Physics B. H. Bansden and C. J. Joachim, Physics of Atoms and Molecules Feb Atomistic Modeling of Mateials -- Geband Cede and Nicola Mazai

24 Refeences pactice Payne, Tete, Allan, Aias, Joannopoulos, Review of Moden Physics 64, Lectue notes fom L3 Pehlke, L2 Katze, L4 Fuchs Hatee-Fock fo solids, Dovesi et al., Physica Status Solidi b 217, Feb Atomistic Modeling of Mateials -- Geband Cede and Nicola Mazai

25 Softwae Gaussian $$ chemisty, Hatee-Fock, DFT, coelated appoaches Gamess-UK $ chemisty, Hatee-Fock, DFT, coelated appoaches Mateials Studio/Ceius $$ DFT, planewave, ultasoft Cystal $ Hatee-Fock VASP $ DFT, planewave, ultasoft, PAW ESPRESSO fee DFT, planewave, ultasoft, lineaesponse theoy, Ca-Painello ABINIT fee DFT, planewave, linea-esponse theoy, GW CPMD fee DFT, planewave, Ca-Painello, timedependent DFT CASINO fee Quantum Monte Calo Feb Atomistic Modeling of Mateials -- Geband Cede and Nicola Mazai