Density Functional Theory: from theory to Applications

Transcription

1 Density Functional Theory: from theory to Applications Uni Mainz November 29, 2010

2 The self interaction error and its correction Perdew-Zunger SIC Average-density approximation Weighted density approximation All electrons vs pseudopotentials Classes of Basis-set Condensed phase: Bloch s th and PBC

3 The self interaction error The self interaction error and its correction Perdew-Zunger SIC In Thomas Fermi theory the electron electron interaction is: V TF ee [ρ] = J [ρ] = 1 2 ρ(r1 )ρ(r 2 ) r 12 dr 1 dr 2 (1) If we consider the one electron system described by φ(r), we get J [ φ(r) 2] 0 (2) while the exact potential energy functional must give 0 for one electron system: V ee [ φ(r) 2 ] = 0 (3) In 1934 Fermi and Amaldi proposed the simple self-interaction corrected formula: Vee FA = N 1 ρ(r1 )ρ(r 2 ) dr 1 dr 2 (4) 2N r 12

4 The self interaction error and its correction Perdew-Zunger SIC No SIE in the Hartree Fock Approximation! E HF = Ψ HF Ĥ Ψ HF = N H i i=1 N (J ij K ij ) (5) i,j=1 where H i = J ij = K ij = this explain the double sum in (5). ψ i (x)[ v(x)]ψ i (x)dx (6) ψ i (x 1 )ψ i (x 1 ) 1 r 12 ψ j (x 2 )ψ j (x 2 )dx 1 dx 2 (7) ψ i (x 1 )ψ j (x 1 ) 1 r 12 ψ i (x 2 )ψ j (x 2 )dx 1 dx 2 (8) J ii = K ii (9)

5 The self interaction error and its correction Perdew-Zunger SIC In DFT the direct (self-interacting) Coulomb energy is large, e.g. for Hydrogen is about 8.5 ev, but about 93% is cancelled by LSD exchange-correlation energy. In LSD a spurious self interaction remains as the price to be paid for a simple, local one-electron potential. The SIE is more severe for localized systems, while it vanishes for orbitals delocalized over extended systems.

6 The self interaction error and its correction Perdew-Zunger SIC Perdew and Zunger 1 Self Interaction Corrections (SIC). In approximate DFT (including LDA, the exact functional V ee [ρ α, ρ β] = J [ρ α, ρ β] [ + E xc ρ α, ρ β] (10) is approximated by the functional: Ṽ ee [ρ α, ρ β] = J [ ρ α, ρ β] + Ẽxc [ ρ α, ρ β] (11) The requirement to exclude the self interaction can be written as Or in a more detailed form: V ee [ρ α i, 0] = J [ρ α i ] + E xc [ρ α i, 0] = 0 (12) J [ρ α i ] + E x [ρ α i, 0] = 0 (13) 1 Perdew and Zunger (1981) Phys Rev B 23, E c [ρ α i, 0] = 0 (14)

7 The self interaction error and its correction Perdew-Zunger SIC Perdew-Zunger self interaction corrected (SIC) version of a given approximate exchange and correlation functional is [ Exc SIC = Ẽ xc ρ α, ρ β] ( ) J [ρ α i ] + Ẽ x [ρ α i, 0] (15) iσ The SIC one-electron equation become [ 1 ρ(r ) v(r) + β e b(r) + r r + v xc iα,sic [ 1 ρ(r ) v(r) β e b(r) + r r + v xc jβ,sic (r) ] (r) ] φ SIC iα (r) = ɛ SIC iα φ SIC iα (r) φ SIC jβ (r) = ɛsic NOTE: the one electron equation for SIC have different potentials for different orbitals, which causes the orbital to be non-orthogonal. jβ φsic jβ (r)

8 The self interaction error and its correction Perdew-Zunger SIC SIC improves the LSD approximation considerably. For the exchange energies we have:

9 The exchange-energy functional [ E x ρ α, ρ β] = 1 1 [ ρ αα 1 (r 1, r 2 ) 2 + ρ ββ 1 2 r (r 1, r 2 ) 2] dr 1 dr 2 (18) 12 can be explicitly written in terms of the orbitals for the system of non-interacting electrons with density ρ α and ρ β [ [ T s ρ α, ρ β] = Min n iσ drφ iσ(r) ( 12 ) ] 2 φ iσ (r) iσ (19) E [ ρ α, ρ β] = n iσ drφ iσ(r)( )φ iσ (r) + J [ ρ α + ρ β] iσ 1 1 [ ρ αα 1 (r 1, r 2 ) 2 + ρ ββ 1 2 r (r 1, r 2 ) 2] dr 1 dr dr [ (v(r) + β e b(r))ρ α (r) + (v(r) β e b(r))ρ β (r)(20) ]

10 Minimization through the orbitals imposing the orthonormalization gives the equations: φ iσ (r) + veff σ (r, r )φ iσ (r )dr = ɛ iσ φ iσ (r) (21) where ɛ iσ are the Lagrange multiplier for the constraint φiσ (r)φ jσ (r) = δ ij. The spin-dependent effective (non local) potentials are: [ veff α ρ(r ) = v(r) + β e b(r) + r r dr + δe [ c ρ α, ρ β] ] δρ α δ(r r ) ραα 1 (r, r ) (r) r r (22 v β eff [v(r) = β e b(r) + ρ(r ) r r dr + δe [ c ρ α, ρ β] ] δρ β (r) δ(r r ) ρββ 1 (r, r ) r r (23 The spin density are: ρ σ (r) = i φ iσ(r) 2 and N σ = drρ σ (r)dr and N = N α + N β.

11 The HFKS equations differs from the KS equations in having a non-local exchange potential; the exchange potential is exact and explicit. The correlation potential is included, if we knew the exact form of E c [ ρ α, ρ β] we would have the exact solution.

12 [ What to use for E c ρ α, ρ β]? One can take the local spin density: [ ρ α, ρ β] = E LSD c ρɛ c (ρ, ζ)dr (24) but it overestimate the the true value by a great deal. Stoll, Pavlidou, and Preuss 2 suggested to use LSD only for electron with different spin. E SPP c [ ρ α, ρ β] [ = Ec LSD ρ α, ρ β] [ ] Ec LSD [ρ α, 0] Ec LSD ρ β, 0 (25) = ρɛ c (ρ, ζ)dr ρ α ɛ c (ρ α, 1)dr ρ β ɛ c (ρ β, 1)dr (26) NOTE: E SPP c [ φi (r) 2, 0 ] = 0. This approximation excludes the SI 2 Theor. Chem. Acta (1978).

13 Vosko and Wilk 3 proposed a another improvement over LSD. [ Ec VW ρ α, ρ β] [ = Ec LSD ρ α, ρ β] (27) ( [ ]) N α Ec LSD [ρ α /N α, 0] + N β Ec LSD ρ β /N β, 0 (28) = ρɛ c (ρ, ζ)dr (29) ρ α ɛ c (ρ α /N α, 1)dr ρ β ɛ c (ρ β /N β, 1)dr (30) For N = 1, E VW c = 0 For N and slowly-varying densities, E VW c = E LSD c. 3 J. Phys. B 16, 3687 (1983)

14 Average-density approximation Weighted density approximation Let s recall the expression for the E xc : E xc [ρ] = T [ρ] T s [ρ] + V ee [ρ] J [ρ] E xc is the sum of two unrelated contributions: T [ρ] T s [ρ] is associated to first order reduced density matrix V ee [ρ] J [ρ] is associated with second order reduced density matrix

15 Average-density approximation Weighted density approximation The diabatic connection was introduced by Langreth and Perdew 4 switch on the electron-electron interaction throght the parameter λ, so that the density ρ is unchanged along the path. F λ [ρ(r)] = Min Ψ ρ Ψ ˆT + λ ˆV ee Ψ (31) = Ψ λ ρ ˆT + λ ˆV ee Ψ λ ρ (32) where Ψ λ ρ is the N-electron wave function that minimize ˆT + λ ˆV ee and yields the density ρ. 4 Phys. Rev. B (1977).

16 Average-density approximation Weighted density approximation The parameter λ characterize the strength of the electron-electron interaction. F 1 [ρ] = F [ρ] = T [ρ] + V ee [ρ] (33) F 0 [ρ] = T s [ρ] (34) E xc [ρ] = T [ρ] T s [ρ] + V ee [ρ] J [ρ] = F 1 [ρ] F 0 [ρ] J [ρ] (35) = 1 0 dλ F λ [ρ] λ We want to evaluate E xc from (36). The condition that ψ ρ(r) can be expressed as: J [ρ] (36) ρ(r) = Ψ ˆρ(r) Ψ = Ψ n δ(r r i ) Ψ (37) i

17 Average-density approximation Weighted density approximation It is necessary that Ψ λ ρ make stationary the functional: ( Ψ ˆT + λ ˆV ee Ψ + v λ (r) E ) λ Ψ ˆρ(r) Ψ dr N (38) N = Ψ ˆT + λ ˆV ee + v λ (r i ) E λ Ψ (39) where v λ (r) E λ N is the Lagrangian multiplier for the constraint (37). Ψ λ ρ has to be an eigenstate of an Hamiltonian Ĥλ Ĥ λ Ψ λ ρ = ( with eigenvalue E λ. ˆT + λ ˆV ee + i ) N v λ (r i ) Ψ λ ρ = E λ Ψ λ ρ (40) i

18 Average-density approximation Weighted density approximation E λ λ Considering that: E λ = Ψ λ ρ ˆT + λ ˆV ee + We obtain: = Ψ λ ρ Ĥ λ λ Ψλ ρ (41) = Ψ λ ρ ˆV ee Ψ λ ρ + Ψ λ ρ N v λ (r i ) Ψ λ λ ρ (42) i = Ψ λ ρ ˆV ee Ψ λ ρ + ρ(r) λ v λ(r)dr. (43) N v λ (r i ) Ψ λ ρ = F λ [ρ] + i F λ [ρ] λ N v λ (r i ) (44) = Ψ λ ρ ˆV ee Ψ λ ρ (45) i

19 Finally we can now use F λ [ρ] λ Average-density approximation Weighted density approximation = Ψ λ ρ ˆV ee Ψ λ ρ into the exchange and correlation functional: E xc [ρ] = dλ F λ [ρ] λ J [ρ] (46) = dλ Ψ λ ρ ˆV ee Ψ λ ρ J [ρ] (47) 0 1 = ρ 2 (r 1, r 2 )dr 1 dr 2 J [ρ] (48) r 12 = 1 1 ρ(r 1 )ρ(r 2 ) h(r 1, r 2 )dr 1 dr 2 (49) 2 r 12 = 1 1 ρ(r 1 ) ρ xc (r 1, r 2 )dr 1 dr 2 (50) 2 r 12 where λ 0 dλρλ 2 (r 1, r 2 = ρ 2 (r 1, r 2 ) = 1 2 ρ(r 1)ρ(r 2 )[1 + h(r 1, r 2 )].

20 Average-density approximation Weighted density approximation where the average exchange and correlation hole ρ xc (r 1, r 2 ) = ρ(r 2 ) h(r 1, r 2 ) (51) and the average pair correlation functional is given by: 1 0 dλρ λ 2(r 1, r 2 ) = ρ 2 (r 1, r 2 ) = 1 2 ρ(r 1)ρ(r 2 ) [ 1 + h(r 1, r 2 ) ] (52) The exchange and correlation energy can be viewed as the classical Coulomb interaction between the electron density ρ(r) and a charge ρ xc, the exchange and correlation hole, averaged over λ

21 Average-density approximation Weighted density approximation The sum rule for the exchange and correlation hole is: ρ xc (r 1, r 2 )dr 2 = ρ(r 2 ) h(r 1, r 2 )dr 2 = 1 (53) This can be obtained from ρ(r 1 ) = 2 N 1 N 1 ρ(r 1 ) = 2 ρ 2 (r 1, r 2 )dr 2 (54) ρ 2 (r 1, r 2 )dr 2 (55) Inserting ρ 2 (r 1, r 2 ) = 1 2 ρ(r 1)ρ(r 2 ) [1 + h(r 1, r 2 )] (56) into eq.55 we obtain: N 1 1 ρ(r 1 ) = 2 2 ρ(r 1)ρ(r 2 ) [1 + h(r 1, r 2 )] dr 2 (57) N 1 ρ(r 1 ) = 1 [ ] 2 2 ρ(r 1) N + ρ(r 2 )h(r 1, r 2 )dr 2 (58) 1 = ρ(r 2 )h(r 1, r 2 )dr 2 (59)

22 Average-density approximation Weighted density approximation ρ x c(r 1, r 2 )dr 2 = ρ(r 2 ) h(r 1, r 2 )dr 2 = 1 From here we see that ρ x c(r 1, r 2 ) represent a hole around r 1 with unit positive charge. Such condition can be a test for DFT where an approximation of ρ xc (r 1, r 2 ) is given in term of electron density.

23 Average-density approximation Weighted density approximation The second consequence of 1 E xc [ρ] = ρ(r 1 ) ρ xc (r 1, r 2 )dr 1 dr 2 r 12 is that E xc only depends on certain spherically averaged behavior of ρ xc, namely E xc [ρ] = 1 drρ(r) 4πsdsρ SA xc (r, s) (60) 2 where ρ SA xc (r, s) = 1 4π 0 ρ xc (r, r )dr (61) The sum rule can be written as: 4π s 2 dsρ SA xc (r, s) = 1 (62)

24 Average-density approximation Weighted density approximation We can decompose ρ xc into exchange and correlation contribution. We can define the exchange hole (for spin-compensated) as: ρ x (r 1, r 2 ) = 1 ρ 1 (r 1, r 2 ) 2. (63) 2 ρ(r 1 ) thus the exchange energy is given by E x [ρ] = 1 1 ρ(r 1 )ρ x (r 1, r 2 )dr 1 dr 2 (64) 2 r 12 The exchange hole satisfies We define the correlation hole: ρ x (r 1, r 2 )dr 2 = 1; (65) ρ xc = ρ x (r 1, r 2 ) + ρ c (r 1, r 2 ). (66) As consequence of the sum rule and of eq.65 we have that ρ c (r 1, r 2 )dr 2 = 0. (67)

25 Average-density approximation Weighted density approximation The exchange energy equals the Coulomb interaction energy of the electrons with a charge distribution contains one unit charge The correlation energy comes from the interaction of the electrons with a neutral charge distribution For the local density approximation Exc LDA [ρ] = ρ(r)ɛ xc (ρ(r))dr (68) Recalling: 1 E xc = ρ(r 1 )ρ(r 2 ) h(r 1, r 2 )dr 1 dr 2 (69) r 12 The LDA formula corresponds to: ρ LDA xc = ρ(r 1 ) h 0 ( r 1 r 2 ; ρ(r 1 )) (70) Note the difference with the exact formula: ρ xc (r 1, r 2 ) = ρ(r 2 ) h(r 1, r 2 )

26 Average-density approximation Weighted density approximation The ρ LDA xc obeys the sum rule: ρ LDA xc (r 1, r 2 )dr 2 = 1 (71) Indeed for every r 1, ρ LDA xc is the exact exchange-correlation hole of a homogeneous electron gas with density ρ(r 1 ).

27 Average-density approximation Weighted density approximation

28 Average-density approximation Weighted density approximation

29 Average-density approximation Average-density (AD) approximation 5 Average-density approximation Weighted density approximation ρ AD xc = ρ(r 1 ) h 0 ( r 1 r 2 ; ρ(r 1 )) (72) where the average density is given by ρ(r) = w(r r ; ρ(r))ρ(r )dr (73) The corresponding exchange and correlation energy is equal to Exc AD [ρ] = ρ(r)ɛ xc ( ρ(r))dr (74) LDA correspond to a weighting factor w(r r ; ρ) that is a Dirac delta function. 5 Gunnarsson, Jonson, and Lundqvist, Phys Lett 59 A177 (1976)

30 Weighted density approximation Average-density approximation Weighted density approximation Other approximations include: weighted density (WD) ρ WD xc (r 1, r 2 ) = ρ(r 2 ) h 0 ( r 1 r 2 ; ρ(r 1 )) (75) where ρ(r 1 ) is determined by the sum rule: ρ WD xc (r 1, r 2 )dr 2 = ρ(r 2 ) h 0 ( r 1 r 2 ; ρ(r 1 ))dr 2 = 1 (76)

31 All electrons vs pseudopotentials Classes of Basis-set Condensed phase: Bloch s th and PBC

32 All electrons vs pseudopotentials Classes of Basis-set Condensed phase: Bloch s th and PBC All electrons vs pseudopotentials There are two classes of electrons: valence electrons (participate to chemical bonds) and core electrons (tightly bound to the nuclei). Eventually semi-core electrons (close in energy to valence states to feel the presence of the environment) All-electron methods fixed orbital basis set: core electron minimal number of basis function to reproduce atomic features, valence and semi-core more complete basis set to describe the chemical bond. augmented basis set. Divide the space into spherical regions around the atoms and interstitial regions and requesting that the basis functions are continuous and differentiable across the boundaries.

33 All electrons vs pseudopotentials Classes of Basis-set Condensed phase: Bloch s th and PBC Pseudopotential methods Core electrons are eliminated. Nuclei effective charge Z V = Z Z core. Number of electron treated explicitly is reduced The bare Coulomb potential is replaced by a screened Coulomb potential Inner solution, inside the core radius, is replaced with a smooth, node-less pseudo-wave function Pseudopotentials are usually chosen to be dependent on the angular momentum. E.g. for Pt 6p orbitals are quite external and peaked at around 3.9Å, the 6s peak at around 2.4 Åand the main peak of 5d is located ate 1.3 Å.

34 All electrons vs pseudopotentials Classes of Basis-set Condensed phase: Bloch s th and PBC

35 All electrons vs pseudopotentials Classes of Basis-set Condensed phase: Bloch s th and PBC Classes of Basis-set Extended basis set: delocalized, such as plane waves, useful for condensed phase systems. Tends to be inefficient for molecular systems. Localized basis set: mainly centered at the atomic positions (but also at position of ghost atoms). Mainly used for molecular systems Mixed basis set: designed to take best of the two worlds (delocalized + localized). There can be some technical issues (over-completeness). Augmented basis set: where an extended or atom centered basis set is augmented with atomic like wf in spherical regions around the nuclei.

36 Condensed phase: Bloch s th and PBC All electrons vs pseudopotentials Classes of Basis-set Condensed phase: Bloch s th and PBC Condensed phase: problem of choosing the size of the simulation cell. For periodic system: unit of Wigner-Seitz cell, the minimal choice that contains the whole symmetry of the system. Sometimes it is convenient to choose a larger cell to simplify description of symmetry properties. In an external periodic potential v(r) = v(r + a i ) the wf can be written as: ψ k (r) = e ik r u k (r) (77) with u k (r) = u k (r + a i ). ψ k (r + a i ) = e ik a i ψ k (r) (78) So that the probability density is ψ k (r) 2 is exactly the same.

37 All electrons vs pseudopotentials Classes of Basis-set Condensed phase: Bloch s th and PBC Looking at ψ k (r + a i ) = e ik a i ψ k (r) we notice that there is a a class of vectors k such that e ik a i = 1 (79) The reciprocal lattice vectors are defined by a i b j = 2πδ ij (80) and b 1 = 2π a 2 a 3 Ω ; b 2 = 2π a 3 a 1 Ω ; b 3 = 2π a 1 a 2 Ω (81) The reciprocal lattice vectors define the first Brillouin Zone (BZ).

38 All electrons vs pseudopotentials Classes of Basis-set Condensed phase: Bloch s th and PBC

39 All electrons vs pseudopotentials Classes of Basis-set Condensed phase: Bloch s th and PBC Bloch s theorem indicates that it is not necessary to determine the electronic wavefunction everywhere in space. It is sufficient to know the solution in the unit cell. Using the fact that a periodic function can be represented by a Fourier series: ψ k (r) = e ik r C k+g e ig r (82) G where the sum is over G = n 1 b 1 + n 2 b 2 + n 3 b 3, the reciprocal lattice vectors. k is restricted to all the vectors in the first Brillouin zone. In practice calculations are done only for a finite number of k. The number of k points depends on the systems we want to study.

40 All electrons vs pseudopotentials Classes of Basis-set Condensed phase: Bloch s th and PBC Aperiodic systems: molecules, surfaces and defects supercell approach with PBC, making sure that required physical and chemical properties are converged with respect to the size of the supercell. For surfaces and molecules, e.g., introduce a a vacuum region large enough that there so interaction between images. For charged systems difficulties due to the electrostatic interactions (long range). A uniform neutralizing background is introduced.