Density Functional Theory (DFT) in a nutshell.
Élise Dumont
ENS Lyon
Élise Dumont Density Functional Theory (DFT) in a nutshell.
Lecture M2 (RFCT, 1st week and ATOSIM) 25th Oct. 2010
Outline
Why resorting to DFT ? From a 3N -D to a 3D view : RDM theory... Fundations : Hohenberg-Kohn and Kohn-Sham theorems Climbing Jacob's Ladder : LD(S)A, GGA, meta-GGA, hybrid and AC Pitfalls of DFT DFT shortcomings Most recent developments... Further reading
Élise Dumont Density Functional Theory (DFT) in a nutshell.
Lecture M2 (RFCT, 1st week and ATOSIM) 25th Oct. 2010
Why resorting to DFT ? 0. Key advantages of DFT over post HF-methods both treating (more or less accurately) the dynamical electron correlation : Cost O(N3 ) also the need for large basis sets is reduced... Conceptutal DFT (density easier to visualize and manipulate... ) Ability to treat excited states (DNA photostability) AIMD (because, after all, molecules do vibrate) DFT is rstly justi ed by a computational bottleneck. → "Get a correlated calculation at the price of a simple HF calculation". For a (rather) recent review, for non-specialists : R. A. Friesner, "Ab initio quantum chemistry: Methodology and applications", PNAS, 2005, 102, 19, 66486653.
Élise Dumont Density Functional Theory (DFT) in a nutshell.
Lecture M2 (RFCT, 1st week and ATOSIM) 25th Oct. 2010
Why resorting to DFT ? An alternative
→ RI : a neat & universal mathematical trick!
Key idea: approximate calculations of many (ia|jb)... P 1 Use a projector I = K |K i hK | P K 2 Auxiliary basis expansion (ABE): |jb i = K Cjb |K i K coe cients... (density tting DF) 3 Fit of Cjb X (ia|jb ) = (ia|L)(L|K )−1 (K |jb ) K ,L Main advantages: Error ≈ 60 µH/atom, mostly from cores Small prefactor (4) → speed-up by 10 or 100 ! Still O(N5 ) Resolution of Coulomb operator... Fairly universal: RI-HF, DFT, MP2, CC ...
Élise Dumont Density Functional Theory (DFT) in a nutshell.
Lecture M2 (RFCT, 1st week and ATOSIM) 25th Oct. 2010
From a 3N-D to a 3D view : RDM theory... 1. Reduced density matrices (RDM) theory Key idea : Ψ(~ri ) fully describes a N particles system (SE), but is a many-complicated object... (1 ≤ i ≤ N, a priori 4N ) Let's write down equations for the ground-state total energy, as functional of the total electronic density ρ(~r ) or more generally a reduced density matrix.
E [Ψ] → E [ρ]
(1)
The RDM theory allows us to condensate Ψ(~ri ) into a more handy quantity, e.g. a molecular descriptor. The rst one we shall de ne is an overall density matrix (DM) :
γN (~ r1 , r~2 , . . . , r~N , r~10 , r~20 , . . . , r~N0 ) = Ψ(~ri ) ∗ Ψ? (r~j0 )
Élise Dumont Density Functional Theory (DFT) in a nutshell.
(2)
Lecture M2 (RFCT, 1st week and ATOSIM) 25th Oct. 2010
From a 3N-D to a 3D view : RDM theory... By integrating over a subset of ~ri (p +1,. . .,N) , one de nes a reduced DM, in a p -particles con guration space :
γp (~ r1 , r~2 , . . . , r~p , r~10 , r~20 , . . . , r~p0 ) = CpN R
R . . . γN (~ r1 , r~2 , . . . , r~N , r~10 , r~20 , . . . , r~N0 )d rp~+1 . . . d r~N (3)
Assuming the electronic dance (condensed into a single number, i.e. the dynamic correlation energy) can be expressed when considering a one- or two-electron informations, one will mostly refer to: 1
the one-particle DM Z Z 0 ~ γ1 (~ r1 , r1 ) = N . . . γN (~ r1 , r~2 , . . . , r~N , r~10 , r~20 , . . . , r~N0 )d r~2 . . . d r~N
2
the two-particles DM
Élise Dumont Density Functional Theory (DFT) in a nutshell.
γ2 (~ r1 , r~2 , r~10 , r~20 )
Lecture M2 (RFCT, 1st week and ATOSIM) 25th Oct. 2010
From a 3N-D to a 3D view : RDM theory... From 1-RDM to density... By restricting the one-particle RDM to its diagonal terms, one gets the probability to nd a particule (an electron here) in an elementary volume d r~1 centered on r~1 . Z Z γ1 (~ r1 , r~1 ) = N . . . |Ψ(~ r1 , r~2 , . . . , r~p )|2 d r~2 . . . d r~N = ρ1 (~ r1 ) (4) At the end of this "distillation" one has expressed a well-celebrated one-electron quantity . . . the electron density in r~1 , which veri es R ρ1 (~ r1 )d r~1 = N . One can already intuite that it will be hard to infer a correct correlation functional... (How to describe a two-electron dance starting with a one-electron averaged information... It is possible: we don't know how.) see slide xx.
Élise Dumont Density Functional Theory (DFT) in a nutshell.
Lecture M2 (RFCT, 1st week and ATOSIM) 25th Oct. 2010
From a 3N-D to a 3D view : RDM theory... From 2-RDM to electron pair probability Similarly, one may privilege the use of a two-particule DM. It is associated to a diagonal part : Z Z γ2 (~ r1 , r~2 , r~1 , r~2 ) = ρ2 (~ r1 , r~2 ) = C2N . . . |Ψ(~ r1 , r~2 , . . . , r~p )|2 d r~3 . . . d r~N
r1 , r~2 ) modulo a division by 2. which gives the probability P2 (~ This is the starting point quantity used by Baerends and coworkers to derive a DMDF theory. Gill and coworkers chose a Wigner intracule, and relative position ~u and momenta ~v . We will point out later the intrinsic advantages of such a choice... and why the two of them abandon DFT !
Élise Dumont Density Functional Theory (DFT) in a nutshell.
Lecture M2 (RFCT, 1st week and ATOSIM) 25th Oct. 2010
From a 3N-D to a 3D view : RDM theory... 2. Expression of the energy implying ρ2 For a given system, E can be expressed as a function of r~1 , r~10 and r~2 : # N " X X ZA X 1 1 2 ˆ = H − ∇i − + (5) 2 i A riA i 2
(14)
The later experiences an external potential Vs (~r ). KS orbitals φi must mimimize the energy of this ctitious system, respecting the orthonormality. They are solution of the following Euler-Lagrange system (one introduces an operator FKS ) because of the constraint R ρ(~r )d~r = N . 1 2 − ∇ + Vs φi = εi φi 2
FKS .C = S .C .E
Élise Dumont Density Functional Theory (DFT) in a nutshell.
Lecture M2 (RFCT, 1st week and ATOSIM) 25th Oct. 2010
Fundations : Hohenberg-Kohn and Kohn-Sham theorems These are known as Kohn-Sham equations. " # Z M 1 2 X ZA ρ(r~0 ) ~0 ∂ EXC (ρ(~r ) + dr + φi (~r ) = i φi (~r ) − ∇ − 2 |r − r 0 | ∂ρ(~r ) A |r − rA | (15) In perfect line with the Hartree-Fock equations : M X X ZA − 1 ∇2 − + (Jij − Kij ) φi (~r ) = i φi (~r ) 2 |r − rA | A j
(16)
where J is a local operator, unlike K . A key dierence : VXC includes exchange and correlation components... at the price of a "simple" HF calculation :) One stands with the equality : Vs = V +
Élise Dumont Density Functional Theory (DFT) in a nutshell.
R
2 r12 d r~2 + VXC
ρ(~ r)
Lecture M2 (RFCT, 1st week and ATOSIM) 25th Oct. 2010
Fundations : Hohenberg-Kohn and Kohn-Sham theorems A short reminder in terms of E... The energy associated to the ctitious system is expressed as independent particles : Z E [ρ] = Ts [ρ] + Vs (~r )ρ(~r )d~r
(17)
... to be compared to the real system energy where ee stands for the inter-electronic repulsion. Z E [ρ] = T [ρ] + Eee [ρ] + V (~r )ρ(~r )d~r (18) The key idea is to build up the ctitious system such that both the density ρ(~r ) and the energy E [ρ] of the pseudo-particles system are identical to density and energy of the real system. Z Z Ts [ρ] + Vs (~r )ρ(~r )d~r = T [ρ] + Eee [ρ] + V (~r )ρ(~r )d~r (19)
Élise Dumont Density Functional Theory (DFT) in a nutshell.
Lecture M2 (RFCT, 1st week and ATOSIM) 25th Oct. 2010
Fundations : Hohenberg-Kohn and Kohn-Sham theorems What's beyound EXC ?
VXC is the 'bin' of DFT... In the exchangecorrelation energy, one gathers : 1
Fermi correlation for electrons of same spin
2
Coulomb between electrons of opposite spin
3
self-interaction correction
4
dierence of kinetic energy between virtual and real systems
E [ρ] =
Z
V (~r )ρ(~r ) + Ts [ρ] + J [ρ] + T [ρ] − Ts [ρ] + Eee [ρ] − J [ρ] (20) EXC [ρ] = T [ρ] − Ts [ρ] − Vee [ρ] − J [ρ]
(21)
Nota It is a complete non-sense to consider separately X and C energies ! One really relies on an inner cancellation of errors. Second order for T ...
Élise Dumont Density Functional Theory (DFT) in a nutshell.
Lecture M2 (RFCT, 1st week and ATOSIM) 25th Oct. 2010
Fundations : Hohenberg-Kohn and Kohn-Sham theorems Three eects of exchangecorrelation Fermi hole : the non-independance of motion arising from the Pauli exclusion particle (particles of same spin). One has exactly : Z Z ρ2 (~ r1 , r~2 ) Eee = d r~1 d r~2 (22)
r12
It is important to note that this implies ρ2 and we will have to approximate it as ρ2 (~ r1 , r~2 ) 1 ρ1 (~ r1 )ρ1 (~ r2 ) ≈ r12 2 r12
(23)
which is correct at the limit r → ∞ but is not when r12 → 0. Integrating ρ2 over the entire space leads to N(N-1) against N2 for the rho1 product, hence a correction factor for self-interaction.
Élise Dumont Density Functional Theory (DFT) in a nutshell.
Lecture M2 (RFCT, 1st week and ATOSIM) 25th Oct. 2010
Fundations : Hohenberg-Kohn and Kohn-Sham theorems Coulomb hole : purely electrostatic (opposite spin) A cusp for ρ2 (this time σ1 6= σ2 that is also not reproduced by the oversimpli ed ρ1 product this leads us to de ne hXC as an exchangecorrelation hole To enforce a correct description of the two holes, one has to enforce a purely-ρ1 description and to modify ρ1 (~ r2 ). 2ρ2 (~ r1 , r~2 ) = ρ1 (~ r1 )[ρ1 (~ r2 ) + hXC (~ r1 , r~2 )]
(24)
hXC is a negative quantity, the so-called exchangecorrelation hole. It
accounts for three eects (Fermi, Coulomb and the self-interaction error) : physically speaking, it corresponds to the density deformation induced in r2 by an electron placed in r1 .
One splits hXC into two components depending on the spin, one for the exchange (X) and one for the correlation (C).
Élise Dumont Density Functional Theory (DFT) in a nutshell.
hXC = hX + hC
(25)
Lecture M2 (RFCT, 1st week and ATOSIM) 25th Oct. 2010
Fundations : Hohenberg-Kohn and Kohn-Sham theorems
Interlude : physical interpretation of the εi Dierent than within the HF framework
→ "Janak theorem" instead of Koopman's one. ∂E = εi ∂ ni
(26)
Electroa nity : -AE ≈ εLUMO and Ionization potential -IP = ≈ εHOMO Anyway, too qualitative and barely used.
Élise Dumont Density Functional Theory (DFT) in a nutshell.
Lecture M2 (RFCT, 1st week and ATOSIM) 25th Oct. 2010
Fundations : Hohenberg-Kohn and Kohn-Sham theorems A fourfold taxonomy Reminder: one likes to partition the total energy as :
E = ET + EV + EJ + EX + EC
(27)
whose respective magnitudes are very dierent. Depending on whether ET / EXC are expressed as a function of Ψ or ρ... 1 Hartree-Fock based theory, where both come from Ψi . 3
Adiabatic connection (AC) theories : ET ← ρ and EXC ← Ψi ,ρ Kohn-Sham theories : ET ← Ψi and EXC ← ρ
4
Pure density functional theories : both from ρ
2
Élise Dumont Density Functional Theory (DFT) in a nutshell.
Lecture M2 (RFCT, 1st week and ATOSIM) 25th Oct. 2010
Fundations : Hohenberg-Kohn and Kohn-Sham theorems Adiabatic connexion (AC) 1976,1977 The remaining dierence to be taken into account is the dierence of kinetic energy T − TS ... A formal way to build EXC is to consider a continuum of ctitious systems. described by : λ H λ = T + Vext + λVee
Vee =
X
i
