Introduction to density-functional theory. Emmanuel Fromager

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1 Institut de Chimie, Strasbourg, France Page 1 Emmanuel Fromager Institut de Chimie de Strasbourg - Laboratoire de Chimie Quantique - Université de Strasbourg /CNRS M2 lecture, Strasbourg, France.

2 Institut de Chimie, Strasbourg, France Page 2 Electronic Hamiltonian N-electron Hamiltonian (in atomic units) within the Born-Oppenheimer approximation: Ĥ = ˆT + ˆV + Ŵee ˆT = ˆV = N ˆt(i) where ˆt(i) r i kinetic energy N ˆv(i) where ˆv(i) v(r i ) nuclei A Z A r i R A electron-nuclei attraction Ŵ ee = 1 2 N ŵ ee (i, j) where ŵ ee (i, j) i j 1 electron-electron repulsion r i r j

3 Walter Kohn, John Pople Share this: S ( S Introduction US&s=google_plusone_share&url=http%3A%2F to density-functional theory %2Fwww.nobelprize.org%2Fnobel_prizes%2Fchemistry%2Flaureates%2F1998%2Fpople-facts.html& title=john%20pople%20-%20facts&ate=at-nobelprize/-/-/ /2&frommenu=1& uid= bfb5966&ct=1&pre=http%3a%2f %2Fwww.nobelprize.org%2Fnobel_prizes%2Fchemistry%2Flaureates%2F1998%2Fkohn-facts.html& tt=0&captcha_provider=nucaptcha) TheS MNobel Prize in Chemistry S John Pople - Facts John A. Pople Born: 31 October 1925, Burnham-on-Sea, United Kingdom Died: 15 March 2004, Chicago, IL, USA Affiliation at the time of the award: Northwestern University, Evanston, IL, USA Prize motivation: "for his development of computational methods in quantum chemistry" Field: theoretical chemistry Prize share: 1/2 Institut de Chimie, Strasbourg, France Page 3 1 of 2 16/05/14 17:48 Share this:

4 Institut de Chimie, Strasbourg, France Page 4 Density-functional theory (DFT) The exact ground-state energy E 0 can be obtained by calculating the associated ground-state wavefunction Ψ 0 (X 1, X 2,..., X N ) explicitly. For that purpose, one should solve the Schrödinger equation or apply the variational principle. FCI is applicable only to very (very) small molecular systems since it is computationally expensive. Note that the nuclear-electron attraction contribution to E 0 can be expressed in terms of a much simpler quantity n 0 than the wavefunction, which is called the ground-state electron density: E 0 = Ψ 0 ˆT + Ŵee Ψ 0 + dr v(r)n 0 (r) R } 3 {{} (v n 0 ) where n 0 (r) = N σ 1 dx 2 dx 3... Ψ0 dx N (rσ 1, X 2, X 3,..., X N ) 2 Note that both kinetic and two-electron repulsion contributions to the energy cannot be expressed explicitly in terms of n 0.

5 Institut de Chimie, Strasbourg, France Page 5 Expectation value of any one-electron potential interaction energy The following expression for the expectation value of the one-electron potential energy in terms of the electron density will be used intensively in the rest of this lecture: N Ψ v(r i ) Ψ = dr v(r)n Ψ (r) = (v n Ψ ) R 3 where n Ψ (r) = N σ 1 dx 2 dx 3... Ψ(rσ1 dx N, X 2, X 3,..., X N ) 2. In the following we will write the electronic Hamiltonian as Ĥ ˆT + Ŵee + N v(r i ) ˆT and Ŵee are universal contributions (i.e. they do not depend on the nuclei) while the local potential energy v(r) is molecule-dependent. where Note that if Ψ is an N-electron eigenfunction of Ĥ with energy E, it remains eigenfunction when the local potential is shifted by a constant i.e. v(r) v(r) µ: ( N ( ˆT + Ŵee + v(ri ) µ ) ) ) ( ) Ψ = (ĤΨ Nµ Ψ = E Nµ Ψ.

6 Institut de Chimie, Strasbourg, France Page 6 First Hohenberg Kohn theorem Note that v Ψ 0 E 0 n 0 HK1: Hohenberg and Kohn have shown that, in fact, the ground-state electron density fully determines (up to a constant) the local potential v. Therefore n 0 v Ψ 0 E 0 In other words, the ground-state energy is a functional of the ground-state density: E 0 = E[n 0 ]. Proof (part 1): Let us consider two potentials v and v that differ by more than a constant, which means that v(r) v (r) varies with r. In the following, we denote Ψ 0 and Ψ 0 the associated ground-state wavefunctions with energies E 0 and E 0, respectively. P. Hohenberg and W. Kohn, Phys. Rev. 136, B864 (1964).

7 Institut de Chimie, Strasbourg, France Page 7 First Hohenberg Kohn theorem If Ψ 0 = Ψ 0 then N ( ) v(r i ) v (r i ) Ψ 0 = = N v(r i ) Ψ 0 v (r i ) Ψ 0 ( ˆT + Ŵee + ) ( N v(r i ) Ψ 0 ˆT + Ŵee + ) N v (r i ) Ψ 0 = E 0 Ψ 0 E 0 Ψ 0 = ( E 0 E 0) Ψ0 so that, in the particular case r 1 = r 2 =... = r N = r, we obtain v(r) v (r) = ( E 0 E 0) /N constant (absurd!) Therefore Ψ 0 and Ψ 0 cannot be equal.

8 Institut de Chimie, Strasbourg, France Page 8 First Hohenberg Kohn theorem Proof (part 2): Let us now assume that Ψ 0 and Ψ 0 have the same electron density n 0. According to the variational principle E 0 < Ψ 0 ˆT N N + Ŵee + v(r i ) Ψ 0 and E 0 Ψ < 0 ˆT + Ŵ ee + v (r i ) Ψ 0 }{{}}{{} E 0 + (v v n 0 ) E 0 (v v n 0 ) thus leading to 0 < E 0 E 0 (v v n 0 ) < 0 absurd! P. Hohenberg and W. Kohn, Phys. Rev. 136, B864 (1964).

9 Institut de Chimie, Strasbourg, France Page 9 Second Hohenberg Kohn theorem HK2: The exact ground-state density n 0 (r) of the electronic Hamiltonian Ĥ[v ne ] ˆT N + Ŵee + v ne (r i ) minimizes the energy density functional E[n] = F [n] + dr v ne (r)n(r), R 3 where the Hohenberg Kohn universal functional F [n] is defined as F [n] = Ψ[n] ˆT + Ŵee Ψ[n], and the minimum equals the exact ground-state energy E 0 : min n E[n] = E[n 0 ] = E 0 Comment: we know from HK1 that n(r) v[n](r) Ψ[v[n]] = Ψ[n] }{{} ground-state wavefunction with density n.

10 Institut de Chimie, Strasbourg, France Page 10 Second Hohenberg and Kohn theorem Proof: for any density n(r), Ψ[n] is well defined according to HK1 and thus leading to E[n] E 0 Ψ[n] Ĥ[v ne] Ψ[n] E 0 Ψ[n] ˆT + }{{ Ŵee Ψ[n] } + dr v ne (r) n Ψ[n] (r) R 3 }{{} E 0 F [n] n(r) When n(r) equals the exact ground-state density n 0 (r): n 0 (r) v ne (r) Ψ[n 0 ] = Ψ[v ne ] = Ψ 0 E[n 0 ] = Ψ 0 ˆT + Ŵee Ψ 0 + dr v ne (r)n 0 (r) = Ψ 0 ˆT + Ŵee + ˆV ne Ψ 0 = E 0 R 3

11 Institut de Chimie, Strasbourg, France Page 11 Kohn Sham DFT The Hohenberg Kohn theorem is also valid for "non-interacting" electrons (Ŵee = 0) : let us consider the particular case of two non-interacting electrons with Hamiltonian ˆT + ˆV. The associated ground-state wavefunction can be written exactly as a Hartree product ( Φ(r 1, r 2 ) = ϕ(r 1 )ϕ(r 2 ) where 1 ) 2 2 r + v(r) ϕ(r) = εϕ(r). For two electrons, the non-interacting ground-state density equals n(r) = 2 dr 2 Φ 2 (r, r 2 ) = 2ϕ 2 (r). R 3 Idea of Kohn and Sham (KS) : consider the local potential v KS (r) such that the non-interacting electrons have exactly the same density n 0 as the physical (interacting) electrons... W. Kohn and L. J. Sham, Phys. Rev. A 140, 1133 (1965).

12 Institut de Chimie, Strasbourg, France Page 12 Kohn Sham DFT... which can be written as ( 1 ) 2 2 r + v KS(r) ϕ KS (r) = ε KS ϕ KS (r) where n 0 (r) = 2 ϕ 2 KS (r). Since the exact energy E 0 of the physical system is a functional of n 0, it can be written as follows, in analogy with the Hartree Fock energy expression, E 0 = 2 ( 1 2 ) dr ϕ KS (r) 2 rϕ KS (r) + dr v(r)ϕ 2 R 3 R 3 KS (r) + +E c [n 0 ] density-functional correlation energy R 3 dr dr ϕ2 KS (r)ϕ2 KS (r ) R 3 r r W. Kohn and L. J. Sham, Phys. Rev. A 140, 1133 (1965).

13 accordance with our Cookie Policy. Institut de Chimie, Strasbourg, France Page 13 Work The Nobel Prize in Chemistry 1998 Walter Kohn, John Pople Share this: 6 The Nobel Prize in Chemistry 1998 Walter Kohn - Facts Walter Kohn Born: 9 March 1923, Vienna, Austria Died: 19 April 2016, Santa Barbara, CA, USA Affiliation at the time of the award: University of California, Santa Barbara, CA, USA Prize motivation: "for his development of the density-functional theory" Field: theoretical chemistry Prize share: 1/2

14 Non-interacting electrons For an even number N of electrons the KS determinant with density n can be formally written as where the KS orbitals ϕ i fulfill Φ KS [n] = ϕ 2 1 ϕ ϕ2 N r ϕ i(r) + v KS [n](r)ϕ i (r) = ε i ϕ i (r) i = 1,..., N/2. Note that the local potential v KS [n](r) is an implicit functional of n ensuring that N/2 n(r) = n Φ KS [n] (r) = 2 ϕ 2 i (r). Once this potential is obtained, the "non-interacting" kinetic energy simply equals T s [n] = Φ KS [n] ˆT Φ KS [n] = 2 N R 3 dr ϕ i (r) 2 rϕ i (r) Institut de Chimie, Strasbourg, France Page 14

15 Institut de Chimie, Strasbourg, France Page 15 Kohn Sham DFT (KS-DFT) The non-interacting kinetic energy density functional T s [n] being unknown but somehow simpler, Kohn and Sham proposed the following decomposition: F [n] = T s [n] + What are the physical effects contained in F [n] T s [n]? ( F [n] T s [n] }{{}? ) Let us have a look at the MP2 energy expression for analysis purposes... E 0 = Φ HF ˆT + ˆV ne Φ HF + N/2 i,j=1 2 ij ij }{{} N/2 i,j=1 }{{} ij ji + ( ) ab ij 2 ab ij ab ji +... ε a,b,i,j i + ε j ε a ε b }{{} E H E x E c

16 Institut de Chimie, Strasbourg, France Page 16 Kohn Sham DFT (KS-DFT) Hartree exchange correlation energy functional: E Hxc [n] = F [n] T s [n] Hartree (or Coulomb) energy in Hartree Fock (HF): E H = 1 2 R 3 drdr nφhf(r)n ΦHF(r ) R 3 r r universal Hartree functional E H [n] = 1 2 Exchange energy in HF: R 3 drdr n(r)n(r ) R 3 r r E x = R 3 R 3 drdr N/2 ϕ i(r)ϕ i (r ) N/2 j=1 ϕ j(r)ϕ j (r ) r r = Φ HF Ŵee Φ HF E H universal (implicit) exchange functional E x [n] = Φ KS [n] Ŵee Φ KS [n] E H [n]

17 Institut de Chimie, Strasbourg, France Page 17 Kohn Sham DFT (KS-DFT) Universal correlation functional: E c [n] = E Hxc [n] E H [n] E x [n] According to HK2 the exact ground-state energy equals E 0 = F [n 0 ] + dr v ne (r)n 0 (r) R 3 = T s [n 0 ] + E Hxc [n 0 ] + dr v ne (r)n 0 (r) R 3 = Φ KS ˆT Φ KS + E Hxc [n 0 ] + dr v ne (r)n 0 (r) R 3 where Φ KS = Φ KS [n 0 ] is the KS determinant with the exact physical (fully-interacting) density n 0. Important conclusion: the exact energy is obtained with one single determinant in KS-DFT! How do we find Φ KS? All we need to know is actually the Hxc functional E Hxc [n] W. Kohn and L. J. Sham, Phys. Rev. A 140, 1133 (1965)

18 Institut de Chimie, Strasbourg, France Page 18 Levy Lieb constrained search formalism For a given density n there is a unique potential v KS [n](r), if it exists..., such that Φ KS [n] is the ground state of ˆT N + v KS [n](r i ) with density n. For all normalized wavefunctions Ψ with density n the following inequality is fulfilled: ( ( Φ KS N N [n] ˆT + v KS [n](r i ) ) ΦKS [n] Ψ ˆT + v KS [n](r i ) ) Ψ T s [n] Ψ ˆT Ψ T s [n] = min Ψ n Ψ ˆT Ψ Therefore Φ KS ˆT Φ KS = T s [n 0 ] = min Ψ ˆT Ψ but we do not know n 0... Ψ n 0 Note that, as a consequence of the previous equality, T s [n 0 ] Ψ 0 ˆT Ψ 0!

19 Institut de Chimie, Strasbourg, France Page 19 For any normalized wavefunction Ψ, Kohn Sham DFT (KS-DFT) Ψ ˆT Ψ T s [n Ψ ] Ψ ˆT + ˆV ne Ψ T s [n Ψ ] + dr v ne (r)n Ψ (r) R 3 Ψ ˆT + ˆV ne Ψ + E Hxc [n Ψ ] T s [n Ψ ] + E Hxc [n Ψ ] + dr v ne (r)n Ψ (r) R }{{ 3 } E[n Ψ ] E 0 The exact ground-state energy E 0 is recovered when Ψ = Φ KS thus leading to E 0 = min Ψ { Ψ ˆT + ˆV } ne Ψ + E Hxc [n Ψ ] Note that the minimization can be restricted to single determinantal wavefunctions Φ

20 Institut de Chimie, Strasbourg, France Page 20 E 0 = min Ψ Comparing wavefunction theory with KS-DFT { Ψ ˆT + ˆV } ne + Ŵee Ψ = min Φ { Φ ˆT + ˆV } ne Φ + E Hxc [n Φ ] Ψ = Φ HF + C k det k k }{{} multideterminantal wave function Φ = ϕ 2 1 ϕ ϕ2 N 2 }{{} single determinant Standard approximations to E xc [n]: E LDA xc [n] = E GGA xc [n] = E meta GGA xc [n] = dr f ( n(r) ) local density approximation (LDA) R 3 ( ) dr f n(r), n(r) generalized gradient approximation R 3 R 3 dr f where f denotes a function. ( N/2 n(r), n(r), 2 n(r), ϕ i (r) 2) meta-gga

21 Institut de Chimie, Strasbourg, France Page 21 Basic idea: The uniform electron gas as a model system Let us first consider a single free particle in a box with volume L L L and L +. The Schrödinger equation is rϕ k (r) = ε k ϕ k (r) and the solutions are ϕ k (r) = 1 L 3 eik.r and ε k = k 2 /2 with k (k x, k y, k z ). Note that the wavefunction is complex In this case the electron density equals n(r) = ϕ k (r) 2 = 1 L 3 constant! A large number of electrons N is then introduced into the box. The electron density n(r) = N/L 3 is held constant as N + and L + The KS system is simply obtained when neglecting the electron-electron repulsions ( ) Analytical expression for the exchange energy: Ex LDA [n] = 3 1/3 3 dr n 4/3 (r) 4 π R 3 Numerical calculation of the correlation energy (Coupled Cluster, Quantum Monte Carlo)

22 Institut de Chimie, Strasbourg, France Page H 2 [1 1 Σ g +, aug cc pvqz] 0.9 Energy [E h ] Interatomic distance [units of a 0 ] FCI HF LDA PBE B3LYP CAM B3LYP

23 He 2 [aug cc pvtz] Interaction energy [µe h ] LDA MP2 CCSD(T) Expt Interatomic distance [units of a 0 ] Institut de Chimie, Strasbourg, France Page 23

24 He 2 [aug cc pvtz] Interaction energy [µe h ] PBExPBEc PBExPW91c BLYP BP86 BPW91 MP2 CCSD(T) Expt Interatomic distance [units of a 0 ] Institut de Chimie, Strasbourg, France Page 24

25 He 2 [aug cc pvtz] Interaction energy [µe h ] PBE0 BHandHLDA B3LYP CAM B3LYP BHandHLYP MP2 CCSD(T) Expt Interatomic distance [units of a 0 ] Institut de Chimie, Strasbourg, France Page 25