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Electronic Structure Calculations and Density Functional Theory

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Electronic Structure Calculations and Density Functional Theory Electronic Structure Calculations and Density Functional Theory Rodolphe Vuilleumier P^olede chimie th´eorique D´epartement de chimie de l'ENS CNRS { Ecole normale sup´erieure{ UPMC Formation ModPhyChem { Lyon, 09/10/2017 Rodolphe Vuilleumier (ENS { UPMC) Electronic Structure and DFT ModPhyChem 1 / 56 Many applications of electronic structure calculations Geometries and energies, equilibrium constants Reaction mechanisms, reaction rates... Vibrational spectroscopies Properties, NMR spectra,... Excited states Force fields From material science to biology through astrochemistry, organic chemistry... ... Rodolphe Vuilleumier (ENS { UPMC) Electronic Structure and DFT ModPhyChem 2 / 56 Length/,("-%0,11,("2,"3,4)(",3"2,"1'56+,+&" and time scales 0,1 nm 10 nm 1µm 1ps 1ns 1µs 1s Variables noyaux et atomes et molécules macroscopiques Modèles Gros-Grains électrons (champs) ! #$%&'(%')$*+," #-('(%')$*+," #.%&'(%')$*+," !" Rodolphe Vuilleumier (ENS { UPMC) Electronic Structure and DFT ModPhyChem 3 / 56 Potential energy surface for the nuclei c The LibreTexts libraries Rodolphe Vuilleumier (ENS { UPMC) Electronic Structure and DFT ModPhyChem 4 / 56 Born-Oppenheimer approximation Total Hamiltonian of the system atoms+electrons: H^T = T^N + V^NN + H^ MI me approximate system wavefunction: ! ~ ~ ~ ΨS (RI ; ~r1;:::; ~rN ) = Ψ(RI )Ψ0(~r1;:::; ~rN ; RI ) ~ Ψ0(~r1;:::; ~rN ; RI ): the ground state electronic wavefunction of the ~ electronic Hamiltonian H^ at fixed ionic configuration RI n o Energy of the electrons at this ionic configuration: ~ E0(RI ) = Ψ0 H^ Ψ0 h j j i ionic Hamiltonian for the evolution of the nuclei wavefunction: ~^ H^N = T^N + V^NN + E0(RI ) Rodolphe Vuilleumier (ENS { UPMC) Electronic Structure and DFT ModPhyChem 5 / 56 Table of contents 1 Interacting electrons 2 The Hartree-Fock method 3 Post Hartree-Fock methods 4 Density Functional Theory Rodolphe Vuilleumier (ENS { UPMC) Electronic Structure and DFT ModPhyChem 6 / 56 Interacting electrons Table of contents 1 Interacting electrons 2 The Hartree-Fock method 3 Post Hartree-Fock methods 4 Density Functional Theory Rodolphe Vuilleumier (ENS { UPMC) Electronic Structure and DFT ModPhyChem 7 / 56 Interacting electrons Interacting electrons Hamiltonian (in atomic units) for an N electrons system H^ = T^ + V^ee + V^ext kinetic energy operator 1 N T^ = ^ 2 −2 ri i=1 X electron-electron repulsion 1 1 V^ = ee 2 ^ ^ i6=j ~ri ~rj X j − j ^ where ~ri is the position operator for electron i Rodolphe Vuilleumier (ENS { UPMC) Electronic Structure and DFT ModPhyChem 8 / 56 Interacting electrons Ground state wavefunction Ψ0 (~r1;:::; ~rN ) function of 3N variables satisfies (fermions) Ψ0 (~r1;:::; ~ri ;:::; ~rj ;:::; ~rN ) = Ψ0 (~r1;:::; ~rj ;:::; ~ri ;:::; ~rN ) − We have H^Ψ0 (~r1;:::; ~rN ) = E0Ψ0 (~r1;:::; ~rN ) or, alternatively, Ψ0 can be obtained from a variational principle: E0 = min Ψ H^ Ψ : Ψ(~r1;:::;~rN )h j j i The minimization of Ψ H^ Ψ is realized for Ψ = Ψ0 (non-degenerate groundh j state).j i Rodolphe Vuilleumier (ENS { UPMC) Electronic Structure and DFT ModPhyChem 9 / 56 Interacting electrons Early calculations Linear Combination of Atomic Orbitals (LCAO) for one electron atomic orbitals to construct many-electron wavefunctions Two fundamental papers Boys, S. F., G. B. Cook, C. M. Reeves, et I. Shavitt. 1956. "Automatic Fundamental Calculations of Molecular Structure". Nature 178 (4544): 1207-9. doi:10.1038/1781207a0. Boys, S. F., et G. B. Cook. 1960. "Mathematical Problems in the Complete Quantum Predictions of Chemical Phenomena". Reviews of Modern Physics 32 (2): 285-95. doi:10.1103/RevModPhys.32.285. Rodolphe Vuilleumier (ENS { UPMC) Electronic Structure and DFT ModPhyChem 10 / 56 Interacting electrons Gaussian basis set for one electron orbitals (Pople) Primitive cartesian gaussian functions: 2 lx ly lz −η(~r−A~) g(~r; η;~l; A~) = Nc (x Ax ) (y Ay ) (z Az ) e (1) − − − ~r: Electron coordinate A~: Atomic coordinate ~l: Angular momentum Contracted cartesian gaussian functions: 'µ(~r) = dµi gµi (~r) (2) i X Rodolphe Vuilleumier (ENS { UPMC) Electronic Structure and DFT ModPhyChem 11 / 56 The Hartree-Fock method Table of contents 1 Interacting electrons 2 The Hartree-Fock method 3 Post Hartree-Fock methods 4 Density Functional Theory Rodolphe Vuilleumier (ENS { UPMC) Electronic Structure and DFT ModPhyChem 12 / 56 The Hartree-Fock method Hartree-Fock approximation Wavefunction ansatz φ1(~r1) φ2(~r1) : : : φN (~r1) 1 HF . Ψ (~r1;:::; ~rN ) = . pN! φ1(~rN ) :::::::::: φN (~rN ) single Slater determinant (φi single electron orbitals) ΨHF obtained by minimization of ΨHF H^ ΨHF 0 h j j i However Ψ0 is not in general a single determinant except for non-interacting particles (correlation effects) Rodolphe Vuilleumier (ENS { UPMC) Electronic Structure and DFT ModPhyChem 13 / 56 The Hartree-Fock method Solution of the Hartree-Fock method (restricted case) For an even number of electrons being paired in each orbital, the orbitals φi (~r) are solutions of N=2 1 2 ^ + V^eN + 2J^j K^j φi (~r) = F^φi (~r) = i φi (~r) 2−2r − 3 j=1 X 4 5 with 1 J^φ (~r) = φ∗(~r 0) φ (~r 0)φ (~r) d3~r 0 (Coulomb operator) j i j ~r 0 ~r j i Z j − j and 1 K^ φ (~r) = φ∗(~r 0) φ (~r 0)φ (~r) d3~r 0 (Exchange operator) j i j ~r 0 ~r i j Z j − j Self-Consistent Field method Rodolphe Vuilleumier (ENS { UPMC) Electronic Structure and DFT ModPhyChem 14 / 56 Post Hartree-Fock methods Table of contents 1 Interacting electrons 2 The Hartree-Fock method 3 Post Hartree-Fock methods 4 Density Functional Theory Rodolphe Vuilleumier (ENS { UPMC) Electronic Structure and DFT ModPhyChem 15 / 56 Post Hartree-Fock methods Full Configuration Interaction (FCI) Construct all possible Slater determinants for a given basis set, DI Construct the electronic Hamiltonian HIJ on this basis set of wavefunctions and diagonilize Computer cost grows very quickly with system size and basis-set size Rodolphe Vuilleumier (ENS { UPMC) Electronic Structure and DFT ModPhyChem 16 / 56 Post Hartree-Fock methods Perturbation theory { Møller-Plesset method The Hartree-Fock Slater determinant DHF is an eigenvector of the Fock operator F^ We are looking for the eigenvector of H^ Rewrite H^ as H^ = F^ + H^ F^ − and treat H^ F^ as a perturbation − Rodolphe Vuilleumier (ENS { UPMC) Electronic Structure and DFT ModPhyChem 17 / 56 Post Hartree-Fock methods Other methods { recipes to define a basis of Slater determinant Active space methods: CASSCF, CASPT2 Specified by m active electrons in n active orbitals Coupled-cluster methods: CCSD, CCSD(T) T^ Ψ = e DHF j i j i with T^ = 1 + T^1 + T^2 + ::: Multi-reference conference interaction (MRCI), Valence Bond... ... Rodolphe Vuilleumier (ENS { UPMC) Electronic Structure and DFT ModPhyChem 18 / 56 Post Hartree-Fock methods Stochastic methods: Quantum Monte-Carlo Here, one flavor: Variational Monte-Carlo Rewritting of the ground-state energy: 3 ∗ Πi d ~ri Ψ ( ~r )H^ Ψ0( ~r ) E = 0 f g f g 0 Π d3~r Ψ ( ~r ) 2 R i i 0 j f g^ j 3 2 H Ψ0(f~rg) ΠiRd ~ri Ψ0( ~r ) j f g j Ψ0(f~rg) = 3 2 Πi d ~ri Ψ0( ~r ) R j f g j ^ R Sample the local energy H Ψ0(f~rg) with the probability density Ψ0(f~rg) 2 Ψ0( ~r ) with a Monte-Carlo method j f g j Rodolphe Vuilleumier (ENS { UPMC) Electronic Structure and DFT ModPhyChem 19 / 56 Post Hartree-Fock methods Stochastic methods: Quantum Monte-Carlo Here, one flavor: Variational Monte-Carlo Allows complicated Ansatz for the wavefunction Ψ0 : j i J^ Ψ = e cI DI j i j i I X J^= J( RK ri ; ri rj ): Jastrow factor f − g f − g VMC: optimize variationally all parameters of the Ansatz Note that for the exact ground state the local energy is constant: H^ Ψ0(f~rg) = E0 Ψ0(f~rg) Other falvors: diffusive Monte-Carlo... Rodolphe Vuilleumier (ENS { UPMC) Electronic Structure and DFT ModPhyChem 20 / 56 Post Hartree-Fock methods Stochastic methods: Full-CI Quantum Monte-Carlo Diagonalization of a large matrix HIJ in the basis of all possible Slater determinants DI : j i Ψ0 = CI DI j i j i I X HIJ CJ = CI Stockastic diagonalisation using a set of walkers such that the average population nI for determinant DI is the eingvector coefficient: j i nI = CI h i Evolution equation: ∆nI (β + τ) = τ HIJ nJ (β) τ(HII S)nI (β) − 2 3 − − J6=I X Rodolphe Vuilleumier (ENS { UPMC) Electronic4 Structure and DFT5 ModPhyChem 21 / 56 Post Hartree-Fock methods Outline of Density Functional Theory Instead of using the full electronic wavefunction Ψ0 (~r1;:::; ~rN ), the electronic ground state of a system can be entirely described by its electron density n(~r ) (Hohenberg-Kohn theorem) or, alternatively, by N one-electron orbitals φ1(~r ); : : : ; φN (~r ) giving rise to the same density n(~r ): N 3 3 2 2 N d ~r2 ::: d ~rN Ψ(~r; :::; ~rN ) = n(~r ) = φi (~r ) ··· j j j j i=1 Z Z X (Kohn-Sham theorem) n(~r ) or φ1(~r ); : : : ; φN (~r ) are obtained by minimization of a density functional Rodolphe Vuilleumier (ENS { UPMC) Electronic Structure and DFT ModPhyChem 22 / 56 Density Functional Theory Table of contents 1 Interacting electrons 2 The Hartree-Fock method 3 Post Hartree-Fock methods 4 Density Functional Theory Rodolphe Vuilleumier (ENS { UPMC) Electronic Structure and DFT ModPhyChem 23 / 56 Density Functional Theory The Hohenberg-Kohn theorem The Hohenberg-Kohn theorem Existence of a map n(~r ) v(~r ) ! v(~r ) Ψ0 n(~r ) ! ! −−−−−−−−−−−−−−−−−−−−−−HK theorem Knowledge of n(~r ) gives knowledge of v(~r ) and of Ψ0, thus all properties of the electronic system and in particular any expectation value of an observable O^ = Ψ O^ Ψ , are functionals of n(~r ) (O[n] for the observable O^). h j j i Rodolphe Vuilleumier (ENS { UPMC) Electronic Structure and DFT ModPhyChem 24 / 56 Density Functional Theory The Hohenberg-Kohn theorem Energy functional The total energy of the system is now a functional E[n] of the density.
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