Orbitals and orbital energies in DFT and TDDFT. E. J. Baerends Vrije Universiteit, Amsterdam

Transcription

1 Orbitals and orbital energies in DFT and TDDFT E. J. Baerends Vrije Universiteit, Amsterdam

2 Common misunderstandings 1) The best orbitals are the HF orbitals or HF is the best one-electron model (because lowest one-det. energy) No: HF orbitals and density are too diffuse! 1a) In HF the one-el. properties are OK (el. density, kinetic energy), the error is in W (el.-el. energy) due to neglect of correlation No: errors in one-el. terms T and V are larger!

3 2) The KS orbitals have no physical meaning, they serve only to build the density. No: the orbitals have a better shape and energy (see 3) than the HF orbitals. They are better suited for qualitative and quantitative MO theory. 3) There is no Koopmans theorem in DFT. The occupied orbital energies (except the first) are meaningless. No: there is a better-than-koopmans relation in DFT between orbital energies and IPs: deviation for valence of ca. 0.1 ev, against HF deviation of ca. 1.1 ev. And theoretically justified!

4 4) the KS band gap (orbital energy gap between HOMO and LUMO in a molecule) is wrong (much too small) No: In molecules the KS gap (HOMO-LUMO orbital energy difference) is much smaller than I A (called the fundamental gap) but it is physically expected to be (and numerically found to be) an excellent approximation for the first excitation energy (optical gap). In solids the fundamental gap (I A) and optical gap (usually close to fundamental gap) are very different from the KS band gap for a reason (not because of the derivative discontinuity).

5 5) Charge-transfer transitions (excitation out of the HOMO of one molecule to the LUMO of another molecule) are not OK in TDDFT because of the derivative discontinuity No: they are more problematic than local excitations because of the physical nature of the KS unoccupied orbitals 6) Computational cost of KS is same as Hartree, much lower than HF. No: higher cost than HF (unless tricks: density fitting to scale Coulomb part down to N 3 scaling)

6 The Hohenberg-Kohn Theorems for non-degenerate ground states HK theorems are a consequence of the variation theorem. Var. theorem: E[Ψ]: E is functional of Ψ: assigns to each Ψ (out of the domain on which E[Ψ] is defined) a real number E. Function f(x) maps real (or complex) variable x on real (or complex) f. Functional F[f] maps function f(x) on real (or complex) F. E[Ψ] = Ψ Ĥ Ψ Ψ Ψ E 0

7 Inequality! Var. Theorem: in particular for a non-degen. ground state Ψ 0 : if Ψ Ψ 0 :, and Ψ is normalized, then Ψ Ĥ Ψ > E 0 Equality only if Ψ = Ψ 0!

8 Formulation of HK theorems-1 HK-1: for non-degenerate ground states Ψ 0 and local external potentials v(r) (usually el.-nucl. pot. v nuc (r)): one-to-one mapping ρ 0 (r) v(r) Ψ 0 Only ground states Ψ 0 and ground state densities ρ 0! if you change v, then Ψ 0 and ρ 0 must change! - two different v s cannot have the same ρ 0 (or Ψ 0 ) - different: v 2 v 1 C ( over a finite domain) map Ψ 0 ρ 0 is invertible: Ψ 0 ρ 0

9 Formulation of HK theorems-2 Conseq.: all gr. state properties are functionals of ρ 0 : A[ρ 0 ] = Ψ 0 [ρ 0 ] Â Ψ 0 [ρ 0 ] e.g. T[ρ 0 ] kinetic energy (not full 1RDM needed) E[ρ 0 ] total energy (not full 2RDM needed) Note: in E[ρ 0 ] the operator Ĥ depends on ρ 0 : ρ 0 determines v, and therefore Ĥ = ˆT + V ˆ + W ˆ ˆT = N (i) V ˆ = v(ri ) W ˆ = i=1 N i=1 i< j 1 r ij

10 Formulation of HK theorems-3 E[ρ 0 ] = Ψ 0 [ρ 0 ] ˆT + ˆ V[ρ 0 ]+ ˆ W Ψ 0 [ρ 0 ] This energy has no obvious lower bound: We can always look for a ρ 0 belonging to a system with a lower energy i.e. with a more attractive v(r). HK-2: a functional with lower bound exists: For a fixed potential v, the functional E v [ρ 0 ] = Ψ 0 [ρ 0 ] ˆT + V ˆ + W ˆ Ψ 0 [ρ 0 ] keep v(r) fixed assumes a minimum for the ρ 0 that corresponds to v(r)

11 Corollary of HK: take v 2 different from v 1 Ψ 2 Ĥ 1 Ψ 2 > Ψ 1 Ĥ 1 Ψ 1 = E 1 T 2 +W 2 + and ρ 2 v 1 dr > T 1 +W 1 + ρ 1 v 1 dr because Ψ 2 Ψ 1 Ψ 1 Ĥ 2 Ψ 1 > Ψ 2 Ĥ 2 Ψ 2 = E 2 T 1 +W 1 + ρ 1 v 2 dr > T 2 +W 2 + ρ 2 v 2 dr sum up ρ 2 v 1 dr + ρ 1 v 2 dr > ρ 1 v 1 dr + ρ 2 v 2 dr or (ρ 2 ρ 1 )(v 2 v 1 )dr < 0, i.e. ρ vdr < 0 (HK: ρ 2 = ρ 1 leads to contrdiction) There is an iterative way to find the unique v(r) belonging to a given ρ 0 : change v(r) locally to increase or decrease ρ.

12 (ρ 2 ρ 1 )(v 2 v 1 )dr < 0, i.e. ρ vdr < 0 If in a small region the poten1al is decreased, v < 0, then ρ must change (cf. HK!), and ρ must be posi1ve over that region, and vice versa. Apply to the KS poten1al v s : by locally adjus1ng v s the density can be made to approach the exact (correlated) density from e.g. CI arbitrarily closely generates the exact KS poten1al v H atom Example: Calculate H atom in Gaussian basis: small devia1ons from exact density. = ρ in % Generate poten1al that produces exactly that Gaussian density: small devia1ons from 1/r, when ρ posi1ve v nega1ve, and vice versa. Schipper, Gritsenko, Baerends, Theor. Chem. Acc. 98 (1997) 16

13 1) What about degenerate ground states? There are many! 2) Only ground states! What about excitations? 3) Can (good approximations to) E v [ρ] or F HK [ρ] ever be found?

14 Degenerate Ground States Suppose set of acceptable potentials V contains potentials v(r) that have a degenerate ground state: HΨ i = Ε 0 Ψ i, i = 1,.. q (nondegenerate case for q =1) v defines a space {Ψ v } of ground state wave functions: $ q q ( & {Ψ v } = Ψ Ψ = c i Ψ i, c i 2 & % =1) '& i=1 i=1 *& Group together ground state densities corresponding to v(r) in set N v : {N v } = { ρ(r) ρ(r) = Ψ(rσ,x 2 x N ) 2 dσdx 2 dx N, Ψ {Ψ v }} Total sets of wavefunctions and of densities by union: {Ψ} = {Ψ v } {N} = {N v } v V v V

15 Mapping of potentials on wavefunctions on densities map C map D v 1 {Ψ v1 } {N v1 } v 2 v 3 {Ψ v2 } {Ψ v3 } {N v2 } {N v3 } V {Ψ} = {Ψ v } v V {N} = {N v } v V

16 properties of maps: C is no longer a proper map: one potential v is associated with more Ψ s. If v 2 v 1 + constant, then each element of {Ψ v1 } differs from each element of {Ψ v2 } (same argument as before). C 1 :{Ψ v1 } v 1 is a proper map (sets {Ψ v1 } and {Ψ v2 } etc. are disjoint) map D: two ground states Ψ 1 and Ψ 2 coming from different potentials v 2 v 1 +constant lead to different densities ρ 2 (r) ρ 1 (r) (same argument as before) sets N v1 and N v2 are disjoint. However, Ψʹ and Ψ coming from the same v may yield the same density: D 1 is not a proper map, Ψ[ρ] is not a unique functional. Example: v(r) = -Z/r, set p states, p + = (1/ 2)(p x +ip y ) etc. p + (r) 2 = p (r) 2 = (1/2)( p x (r) 2 + p y (r) 2 ) p 0 (r) 2 = p z (r) 2

17 Summary for degenerate states: 1) Ground state expectation values of an arbitrary operator A is not a unique functional of ρ, since Ψ A Ψ may differ from Ψ# A Ψ# 2) But energy is still unique functional of ρ, since Ψ and Ψʹ that give the same ρ must belong to same v, characterized by the degenerate ground state energy. Therefore F HK [ρ] may be defined as F HK [ρ] = Ψ i [ρ] T + W Ψ i [ρ] with Ψ i [ρ] {Ψ v }, v = v[ρ] E v [ρ] is now defined for all densities ρ(r) corresponding to a ground state (degenerate or not). It is variational: E v [ρ] = E 0 (the ground state energy of v) if ρ N v E v [ρ] > E 0 if ρ N v

18 Is minimization of E v [ρ] with constraint ρ(r)dr=n viable? (orbital-free DFT) E v [ρ] = Ψ 0 [ρ] ˆT + ˆ V + ˆ W Ψ 0 [ρ] is minimum at ρ 0 of v δ $ δρ(r) E v[ρ] µ ( ρ dr N) ' %& () = 0 δe v δρ(r) µ = 0 δ{t[ρ]+w[ρ]} δρ(r) ρ 0 of v But the HK theorems offer no presciption for obtaining T[ρ] or W[ρ]. Until now no success in finding sufficiently accurate T functional. Therefore: this route has not been successful! = v(r)+ µ

19 The Kohn-Sham molecular orbital model of DFT Kohn-Sham Ansatz: there is an independent particle system of noninteracting electrons, all moving in the same local potential v s (r) such that the density is equal to the exact one (of the interacting electron system in given external potential v ext (r): Kohn-Sham orbitals from: with # 1 & % 2 2 (1)+ v s (r 1 )(ϕ i (r 1 ) = ε i (r 1 ) $ ' ρ s (r) = N i=1 ϕ i (r) 2 = ρ exact (r) The HK theorems also hold for non-interacting electrons (W=0) since proofs do not use form of W. So v s (r) is in one-one correspondence with gr. st. ρ s (r) (and ρ exact (r)). If v s (r) exists (no proof by KS; non-int. v-represent. problem) then it is unique: no other system, with different v and ρ 0, has same v s. It determines KS orbitals and energies: these are system properties.

20 Kohn-Sham model -1 How to obtain approximations to v s (r)? Later. Suppose we have v s (r), obtain the true KS orbitals and ρ s = ρ exact = ρ. Write exact total energy: E = T s + ρv ext dr + (1/2) ρv Coul dr + E xc N i=1 = ϕ i ϕ i everything we don t know if {ϕ i } are known E xc : called exchange-correlation energy of DFT; is defined by above equation: that part of exact E we do not yet know when {ϕ i } and ρ have been obtained. E xc is functional of ρ since other terms are: E xc [ρ] = E[ρ] T s [ρ] ρv ext dr (1/2) ρv Coul dr

21 Kohn-Sham model -2 How does E xc compare to exchange energy of HF, E x HF, and the correlation energy of HF, E c HF? E x HF is exchange energy of det. wavef.: (1/2) ij K ij E c HF = E E HF What is the exchange energy and what is the correlation energy in DFT? Write exact energy in the traditional way: E = T + ρv ext dr + (1/2) ρv Coul dr + W xc exact kinetic energy exchange-correlation part of el.-el. interaction W xc = W (1/2) ρv Coul dr = (1/2) ρ(r 1 ) v hole xc (r1 ) dr Ψ 1 Ψ 0 0 i< j r ij

22 Kohn-Sham model -3 v xc hole (r 1 ) is the potential of the exact hole ρ xc hole (r 1 ; r 2 ) surrounding an electron at position r 1 : if a given electron is at r 1, the probability to find any one of the other electrons at a point r 2 is not the total density ρ(r 2 ) (which gives the Coulomb potential), but is diminished in the neighborhood of r 1 due to the Coulomb repulsion between electrons

23 ρ(r 2 ) ρ hole (r 2 ;r 1 ) ρ cond (r 2 ;r 1 ) = ρ + ρ hole r 1 r 2 An electron traveling through a molecule does not see ρ(r 2 ) with potential v Coul = ρ(r 2 )/r 12 dr 2 but ρ(r 2 ) + ρ hole (r 2 ; r 1 ) W = = 1 2 ˆ W = r 12 Γ(1, 2)d1d2 ρ(1)ρ(2) 1 d1d2 + r 12 2 Γ xc (1, 2) d1d2 r 12 = 1 2 ρ(2) 1 ρ(1) d2d1+ r 12 2 V Coul (1) Γ ρ(1) xc (1, 2) d2 ρ(1)r 12 v hole xc W = W Coul + W xc d1 statistical physics definition

24 Exchange-correlation energy W xc E = Ψ H Ψ = T + % % ρ(r)v(r)dr W XC = 1 2 dr ρ hole 1ρ(r 1 ) XC (r 2 r 1 ) % dr % 2 % r 12 % v hole xc (r1 ) ρ(r 1 )ρ(r 2 ) r 12 dr 1 dr 2 ρ(r 2 ) ρ hole (r 2 ;r 1 ) +W xc ρ cond (r 2 ;r 1 ) Γ(r 1, r 2 ) = ρ(r 1 )ρ(r 2 )+ Γ xc (r 1, r 2 ) ρ cond (r 2 ;r 1 ) = Γ(r 1, r 2 ) ρ(r 1 ) = ρ(r 2 )+ Γ xc (r 1, r 2 ) / ρ(r 1 ) hole (r 2 r 1 ) ρ XC r 1 r 2

25 Definition of E x, E c, E xc E = Ψ Ĥ Ψ = T + ρvdr ρv Coul dr +W xc E in traditional way E = T s + ρvdr ρv Coul dr + E xc E in Kohn-Sham way E xc = T T s T c + W xc E c +W x E c = E xc W x Use KS orbitals to write exchange energy as W x = 1 2 W xc W x W c the correlation part of W xc E c = T c +W c the correlation energy of DFT N K ij ij

26 E c HF = E E HF Compare E c to E c HF = T T HF + ρvdr ρ HF vdr = HF +W Coul W Coul +W xc W x HF ρvdr T HF c V HF c W HF Coul,c W HF c E HF c = T HF c +V HF HF c +W Coul,c +W c HF E c =T c +W c is much simpler!

27 Compare to HF In Hartree-Fock E HF = Ψ HF Ĥ Ψ HF E E HF E c HF And DFT? Put KS orbitals in determinant: E KS = Ψ s Ĥ Ψ s = T s + ρvdr ρv Coul dr +W x Compare to exact E E = T s + ρvdr ρv Coul dr + E xc E E KS = E xc W x E c Correl. energy of DFT is also energy of det. w.r.t. exact energy

28 Definition of correlation energy Ψ s is determinant of KS orbitals Ψ HF is Hartree-Fock determinant E KS = Ψ s ˆ H Ψ s E c DFT E c HF E HF = Ψ HF ˆ H Ψ HF E exact E c DFT > E c HF but by how much?

29 The one-particle model of DFT: Kohn-Sham Minimization of N E = ψ i ψ i + ρv nuc dr i=1 = T s + V + W Coul + E xc ρ(r 1 )ρ(r 2 ) r 12 dr 1 dr 2 + E xc leads to one-el. equations for optimal (KS) orbitals: (1) + v nuc (r 1 ) + v Coul (r 1 ) + v xc (r 1 ) ψ i (x 1 ) = ε i ψ i (x 1 ) ρ(r 2 ) δe dr xc [ρ] 2 r 12 δρ(r 1 ) What about potentials, orbitals and density in the KS model? Common statements: KS orbitals have no physical meaning The only use for the KS orbitals is to build the density We will prove these statements to be totally wrong!

30 Energy density for E xc E xc = ρ(x)ε xc (x)dx Approximations: LDA, GEA, GGA,...: ε xc (x) f(ρ(x), ρʹ (x), ρʹ ʹ (x),...) Exact ε xc (x) from E xc = W xc + T T s = (1/2) ρ(x)v xc hole (x)dx + ρ(x)(vkin (x) v s,kin (x))dx Γ xc γ(1,1ʹ ) γ s (1,1ʹ ) CI CI KS

31 El. correlation is described by Φ(2 N 1) so no wonder that v s can be derived from Φ(2 N 1): hole v s = v ext + v Coul + v xc + (v kin v s,kin ) v cond v c,kin from Φ γ and γ s + vresp from Φ v xc (r) = δe xc [ρ] δρ(r) No information from this expression: Shape? Physics? Even E xc [ρ] is unknown

32 Composition of v xc v xc (r) = δe xc δρ(r) Not very transparent! Therefore use: E xc = T T s +W xc = ρ(r)v c,kin (r)dr ρ(r)v hole xc (r)dr δe xc δρ(r) = 1 2 δρ( r #) δρ(r) δ(r r #) v hole xc ( r #)d r# + δρ( r #) δρ(r) v c,kin( r #)d r# + ρ( r #) δ δρ(r) v xc hole (r) v c,kin (r) $ 1 v resp (r) 2 v xc hole ' ( r #)+ v c,kin ( r #) % & ( ) d r#

33 Holes in H 2 Fermi hole + Coulomb hole = total hole RH-H = 1.4 bohr e e e RH-H = 5.0 bohr e e e e 2 e 1e NB. v xc = v c,kin + v xc hole + v resp

34 In H 2 Fermi hole ρ X hole (r 2 r 1 ) for el. at r 1 is only selfinteraction correction term σ g (r 2 ) 2 So independent of r 1! When R(H-H) is large and r 1 is close to nucleus b, hole is, with σ g (r 2 ) (1/ 2) [1s a (r 2 ) + 1s b (r 2 )], σ g (r 2 ) 2 (1/2) [ 1s a (r 2 ) 2 + 1s b (r 2 ) s a (r 2 ).1s b (r 2 )] 0 Since total ρ is 1s a (r 2 ) 2 + 1s b (r 2 ) 2, the field that the HF electron at r 1 feels is due to ρ(r 2 )+ρ X hole (r 2 r 1 ) = (1/2) [ 1s a (r 2 ) 2 + 1s b (r 2 ) 2 ] Wrong! The other el. should be around nucleus a! The erroneous charge of (1/2) 1s b (r 2 ) 2 that el. around b feels screens nucleus b: the HF orbital becomes too diffuse.

35 Hartree-Fock densities are often poor due to bad HF potential: H 2 corr E total corr T kin corr V el nuc corr W el el H 2 (R=R e ) 1.1 ev H 2 (R=5.0 bohr) H 2 (R=10.0 bohr)

36 Hartree-Fock densities are often poor due to bad potential: He, H 2 O, Ne, N 2 corr E total corr T kin corr V el nuc corr W el el He H 2 O Ne N

37 Hartree-Fock densities are often poor due to bad potential: TM complexes corr E total corr T kin corr V el nuc corr W el el MnO Ni(CO) Cr(CO)

38 Hartree-Fock errors for bond energies (kcal/mol), because of lack of left-right correl. in bond orbitals HF Obs. Error (% of Obs.) N % F % H 2 O % O % H %

39 Holes in H 2 Fermi hole + Coulomb hole = total hole RH-H = 1.4 bohr e e e RH-H = 5.0 bohr e e Localized hole in DFT Slater: uniform depth ρ(r 1 )/2 e e 2 e 1e NB. v xc = v c,kin + v xc hole + v resp

40 E versus R curves (restricted HF/KS) for dissociating H HF=EXX HF Total energy (a.u.) HF HF F 2 case Correl. LDA BP FullCI Error 45 Kcal/mol Correl Bond length (a.u.) 8 10 Grüning, Gritsenko, Baerends, JCP 118 (2003) 7183

41 The anomalous F 2 case: RHF energy above two F atoms! Energy R(F-F) Energy of 2 F atoms

42 The functional Cloud Courtesy of Peter Elliott, Hunter College, New York

43 "exact" KS and HF energies of N 2 D e =0.37 a.u. R (bohr) T s T s T HF = corr T kin (KS) corr T kin (HF) T T s = T T HF = V el-nuc (exact=ks) corr V el nuc (HF) W Coul (exact) corr W Coul (HF) Gritsenko, Schipper, Baerends, J. Chem. Phys. 107 (1997) 5007

44 KS and HF energies of N 2 D e =0.37 a.u. R (bohr) W X (KS orbitals) W X W X HF = W c = W XC W X HF W c (HF) = W XC W X E c E c (HF) E c E c (HF) Gritsenko, Schipper, Baerends, J. Chem. Phys. 107 (1997) 5007

45 Hartree-Fock: good for atoms, not for molecules (bonds) In an electron pair bond: a) HF orbitals will be too diffuse (density too diffuse) kinetic energy too low electron-nuclear energy too high (not negative enough) b) this is worse in case of multiple bonds c) common statement one-particle properties (also the electron density!) are good in the Hartree-Fock model, it is the el.-el. interaction that is wrong, because of lack of electron correlation (electrons do not avoid each other sufficiently, cf. the presence of ionic configurations in the H 2 wavefunction) IS WRONG

46 Conclusion HF versus KS det. - Ψ HF better total energy (marginally) E c E c HF Ψ s better for: V el-nuc no correlation error W Coul T s : (much) smaller correl. error HF "distorts" density (more diffuse) if: gain by lowering T HF is larger (even if barely) than loss by less stable V

47 T s KS V en KS W H KS W fn X KS Energy components for CO at R e =2.132 bohr LDA BLYP EXX HF KS CI Sum CI E fn c ( 0.460) ( 0.454) ( 0.466) E tot Δ CI Baerends, Gritsenko JCP 123 (2005) (T) (W XC )

48 T s KS V en KS W H KS W fn X KS Energy components for CO at R=2.8 bohr LDA BLYP EXX HF KS CI Sum CI (W XC ) E c fn ( 0.518) ( 0.506) ( 0.549) E tot Δ CI Baerends, Gritsenko JCP 123 (2005)

49 Time-dependent DFT Runge-Gross: HK theorem holds for time-dependent case v(r,t) ρ(r,t) Ψ 0 (t) Kohn-Sham (orbital model) in time-dependent case : s + v s (r,t) ψ i (r,t) = i t ψ i s (r,t) N i=1 ρ s (r,t) = ψ i s (r,t) 2 = ρ exact (r,t)

50 LINEAR RESPONSE (1 st order Pert. Theory) δρ(r,ω) = χ(r, & r,ω)δv( & r,ω)dr& χ(r, & r,ω) : first order response function; difficult (requires sum over all excited states) Kohn - Sham :δρ(r,ω) = δρ s (r,ω) = χ s (r,r&,ω)δv s (r&,ω)dr& response function of noninteracting system: simple! δv s (r&,ω) : how related to δv(r&,ω)? (difficult?)

51 Linear response: δv s δv s (r#,ω) = δv(r#,ω) + δv coul (r#,ω) +δv xc (r#,ω) δv(r#,ω) + δv induced (r#,ω) δρ( r #,ω) r# r # d r # δv xc (r#,ω) δρ( r #,ω)d r # δρ( r #,ω) f xc (r#, r #,ω) the xc kernel v xc (r,ω) = δe xc δρ(r,ω) f xc (r,r#,ω) = δe xc δρ(r,ω)δρ(r#,ω) LDA - Xonly : E xc = K ρ 4/3 dr; v xc (r,ω) = 4K 3 ρ1/3 (r,ω); f xc (r,r#,ω) in local, adiabatic appr. (ALDA) = 4K 9 ρ 2 /3 (r)δ(r r#)

52 χ s (r,rʹ ) from first order perturbation theory: Perturbation δv s (r) induces changes in the orbitals: δϕ i (r) = ρ(r) = δρ(r) = = H i=1 p i H i=1 p i H i=1 ϕ i δv s ϕ p ε p ε i n i ϕ i (r)ϕ i (r) ϕ p (r) n ( i ϕ i (r)δϕ i (r)+δϕ i (r)ϕ i (r)) % ϕ i (r) ϕ i δv s ϕ p ϕ p (r) + ϕ i δv s ϕ p ϕ p (r)ϕ ( ' i (r) * ' ε p ε i ε p ε & i * )

53 convention for summation indices: i, j, k, l,... indices for occupied orbitals, N a, b, c, d,...indices for unoccupied orbitals, > N p, q, r, s,... general indices

54 δρ(r) = H i=1 occ.- occ. pairs drop out p i # ϕ i (r) ϕ i δv s ϕ p ϕ p (r) + ϕ i δv s ϕ p ϕ p (r)ϕ & % i (r) ( % ε p ε i ε p ε $ i ( ' if i and p are both occupied orbitals, e.g. k and l, then: for i = k, p = l : ϕ kϕ l V s,lk ε l ε k + V s,klϕ l ϕ k ε l ε k CANCEL! for i = l, p = k : ϕ lϕ k V s,kl ε k ε l + V s,lkϕ k ϕ l ε k ε l Only p-values with p unocc. survive

55 Definition of χ s δρ(r) = = dr* H i=1 i occ a>h # ϕ i (r) ϕ i δv s ϕ a ϕa (r) + ϕ i δv s ϕ a ϕ a (r)ϕ & % i (r) ( % ε a ε i ε a ε $ i ( ' a unocc # ϕ i (r)ϕ i (r*)ϕ a (r*)ϕ a (r) + ϕ i (r*)ϕ a (r*)ϕ a (r)ϕ i (r)& % (δv s (r*) $ ε a ε i ε a ε i ' χ s (r,r*) χ s comes straightforwardly from 1st order pert. theory; only KS orbitals and orbital energies needed

56 Time-dependent case (linear response =1 st order Pert. Th.) δρ(r,t) = i n i ( ψ i (r,t)δψ i *(r,t) +ψ i *(r,t)δψ i (r,t)) Suppose perturbation δv ext (r,t) with single frequency ω: δρ(r,ω) = occ i virt a X ia ω = ψ i (r) δv s (r,ω) ψ a (r) ε i ε a + ω n i ψ i (r)ψ a (r)( ω X ia + ω* Xia ) e iωt + c.c. X ia X ia ω* Yia δv s (r,ω) = δv ext (r,ω) + dr δρ(r,ω) + δv xc [δρ](r,ω) r r v induced

57 δv s (r,ω) = δv ext (r,ω) + dr δρ(r,ω) + δv xc [δρ](r,ω) r r v induced (r,ω) "uncoupled": v ind = 0 "coupled": Coulomb part: δv Coul (r,ω) = dr δρ(r,ω) r r XC part: δv xc (r,ω) = dr δv xc (r,ω) δρ(r,ω) δρ(r,ω) f xc (r,r,ω) usually adiabatic LDA (no ω dependence) for XC kernel f xc (r,rʹ,ω)

58 Shortcut to matrix equations for excitation energies Put in matrix form with basis sets: δρ(r,ω) δp ia (ω) δp(ω) density perturb. occ.unocc supervector Π(ω) ia,jb response + coupling occ.unocc occ.unocc supermatrix δv ia (ω) external perturb. potential occ.unocc supervector δp(ω) = Π(ω)δV(ω) density response due to external perturb. field Π(ω) 1 δp(ω) = δv(ω) Π(ω) 1 often has structure (K-ω1) Then (K-ω1)δP(ω) = δv(ω) is solvable for δv(ω)=0 (no perturb.) when det{k-ω1}=0, or at eigenvalues of K: KδP i =ω i δp i : Excitation energies! A system can have free oscillations δp i ( response ) without perturbation at its eigenfrequencies {ω i }

59 Leads to TDKS (cf. TDHF) equations, dimension n occ n virt n occ n virt : (ε 2 + 2ε 1 Kε ω 2 )ε 1 δv ia ext = ψi (r) v ext (r,ω) ψ a (r) ( ε 2 ) = δ ij δ ab (ε a ε i ) 2 ia, jb ( 2 X + Y) = δvext (ω) Cf. χ 1. δρ = δv F jb K ia, jb X + Y ( ) jb = ψ i (r) v ind ψ a (r) Inhomogeneous equation (δv ext (ω) 0): at each ω: F(ω) δρ(r,ω) polarizability α(ω) etc.

60 Excitation energies: δv ext =0 (ε 2 + 2ε 1 2 Kε 1 2 ω 2 )F = δv ext (ω) = 0 Homogeneous linear equations (δv ext = 0): eigenfrequencies of system (excitation energies) are solutions of ε 2 + 2ε 1 2Kε 1 2 ω 2 F = 0 or ε2 + 2ε 1 2Kε 1 2 F = ω 2 F Excitation energies! Ingredients: orbital energies ε i, ε a orbital shapes ψ i, ψ a xc kernel f xc

61 So we need good orbital shapes and good orbital energies. But: what is the meaning of KS orbital energies? Prevailing view, see e.g. R. G. Parr, W. Yang, DFT of Atoms and molecules, 1989 " one should expect no simple physical meaning for the KS orbital energies. There is none

62 Orbital energies of occupied orbitals: Exact Kohn-Sham: - HOMO orbital energy exactly I 0 (ionization en. to ion ground st.) because of asymptotic density behavior - upper valence orbitals: very close to ionization energies (~ 0.1 ev) - core orbitals: still good, too high lying by ev LDA, GGA: - all orbital energies are shifted up by a molecule-dependent constant of ca. 4 6 ev Gritsenko, Baerends: JCP 116 (2002) 1760 (with Chong); JCP 117 (2003) 9154; JCP 119 (2003) 1937 (with Braïda); JCP 120 (2004) 8364; JCP

63 v xc hole and the interpretation of orbital energies 1º. Long range (asymptotic, r ) behavior = r r = r r r + 1 r 2 sinθ r + 1 r 2 D θ + 1 r 2 D ϕ Consider limit r of KS equation 1 2 ψ i 2 r ψ i r r θ (sinθ θ )+ 1 2 r 2 sin 2 θ ϕ v s (r) ψ i = ε i ψ i + 1 r 2 D θψ i + 1 r 2 D ϕψ i + v s (r)ψ i = ε i ψ i At each point r this must be an identity. At r all terms are negligible compared to: 1 2 ψ i 2 r 2 + v s( )ψ i = ε i ψ i So asymptotic solution is ψ i ~ e 2 ( ε i v s ( )) r All potentials in v s go to 0: ψ i ~ e 2ε i r

64 Long range behavior So asymptotic solution is ψ i ~ e 2 ( ε i v s ( )) r All potentials in v s go to 0: v s ( ) = 0, ψ i ~ e 2ε i r Each orbital has its own exponential decay, so density decays as slowest orbital density decay, i.e. HOMO: ρ(r)~ e 2 2ε H r 2 2I r Katriel-Davidson (1980): density decays like e Conclusion: ε H = I Now take anion: LUMO now occup., slowest decay ε L (M ) = I(M ) = A(M ) I(M ) = E 0 N 1 E 0 N I(M ) = E 0 N E 0 N+1 = A(M ) A(M ) = E 0 N E 0 N+1

65 KS and HF orbital energies and VIPs for H 2 O H 2 O MO HF KS Expt. I k +ε k ε N ε k Average Dev. ε k ε k I k 1b a b Average Dev. 2a a Chong, Gritsenko, Baerends, JCP 116 (2002) 1760

66 CO : KS and HF orbital energies and VIPs CO MO HF KS Expt. I k ( ε k ) ε k ε k I k 5σ Average Dev. Average Dev. 1π σ σ σ σ ε N ε k

67 CO : KS, GGA-BP and HF orbital energies and VIPs CO AAD (val) AAD (inner) MO HF εi GGA-BP εi KS εi Expt. Ii 5σ (4.83) π (16.78) σ (19.10) (0.25) σ (34.29) σ (277.33) σ (518.37) (15.69) 14.39

68 HCl HCl: KS, BP and HF orbital energies and VIPs M O HF ε i GGA-BP ε i KS ε i Expt. I i 2π (4.64) σ (16.53) σ (25.86) AAD(val) (0.04) π (195.62) σ (195.91) σ (255.08)

69 SiO: HF, GGA-BP and KS orbital energies, expt. Ips BP: HOMO 4.02 ev higher than IP; second column: all ε i BB 4.02 SiO MO HF ε i GGA-BP KS Expt. ε i ε i I i 7σ (4.02) π (12.24) σ (14.84) AAD (val) (0.03) σ (27.61) π (99.84) σ (99.63) σ (142.97) σ (514.50) σ ( )

70 N 2 : KS, BP and HF orbital energies and VIPs N2 MO HF KS Expt. I k +ε k ε k ε k I k 3σ g π u σ u Average dev σ g σ u σ g Average dev

71 Virtual orbital energies What are virtuals like in DFT? And in Hartree-Fock? Big difference between HF and KS virtuals: necessary to understand the difference to understand - why TDDFT works so well (in general for molecules); - why there is a problem with charge-transfer transitions - the bandgap problem in solids Difference between KS and HF virtuals are consequence of v xc hole in KS potential, and absence in HF exchange operator v xc hole leads to good shapes and energy of KS virtuals (as it did for KS occupied orbitals) -

72 Meaning of unoccupied orbital energies ε a, ε b,. HF: unocc. orbital represents added electron ε a HF is affinity level; ε a HF ε i HF is NOT excitation energy KS: unocc. orbital represents excited electron ε a KS ε i KS IS good appr. to excitation energy NO KS σ* density H HF σ* density HF virtual orbitals are at (much) higher energy and (way) more diffuse than KS virtual orbitals

73 HF, DFA and exact KS HOMO orbital energies HF LDA BLYP KS = I 0 H H 2 O HF N CO HCN FCN HCl KS HOMO is equal to I 0 ; HF HOMO is appr. equal to I 0 (frozen orbital approx.) LDA, GGA orbital energies are upshifted by ca. 4.5 ev (uniformly: occup. and unoccup valence orbitals)

74 HF, DFA and exact KS LUMO orbital energies HF LDA BLYP KS H H 2 O HF N CO HCN FCN HCl KS LUMO is at negative energy: a bound one-electron state in the KS potential. HF LUMO is most of the time unbound (positive orbital energy) LDA,GGA LUMO: still negative -> therefore bound state

75 KS HOMO-LUMO gaps are excellent approx. to excitation energies HF LDA BLYP KS Expt. excit. energy singlet triplet H H 2 O HF N CO HCN FCN HCl ) The LDA, GGA gaps are similar (slightly smaller) than KS gaps -> the upshift is similar for HOMO and (a bit smaller for) LUMO 2) HF gaps are much larger: they are Koopmans approx. IP EA

76 0.0 A ε L (M ) Orbital energies in exact KS fund = DD L Energy opt ε L (M) ε H (M ) I ε H (M) M M

77 What is the meaning of a (HF) LUMO with positive orbital energy? resonances at specific E

78 What is the meaning of HF LUMO with positive energy? Note: positive one-electron states in a potential (zero at infinity): - there is a continuum of positive states; - most have plane-wave behavior with only a few orthogonality wiggles over the molecular region; - at specific energy (small energy ranges) the one-electron states have large amplitude in the molecular region (small plane-wave like outside) -> scattering resonances with resonance energies corresponding to potential electron capture to form a temporary negative ion, which will decay after some time to molecule + free electron. Since energy at scattering resonance is positive, i.e. higher than free molecule and electron: negative electron affinity! If there are no negative energy unoccupied orbitals (bound states) for the HF operator (frequently!), what is the meaning of the pos. energy orbitals?

79 Orbital energies (ev) of the positive energy HF LUMO of H 2 as function of the basis (STOs) 1σ u 1σ g SZ DZ DZP TZP TZ2P QZ4P ETQZ3P 2D gap

80 Orbital energies (ev) of the positive energy HF LUMO of H 2 as function of the basis (Gaussians) 1σ u 1σ g ccpvdz ccpvqz ccpv5z aug-ccpvdz aug-ccpvtz aug-ccpvqz aug-ccpv5z gap Orbital energies of LUMO are arbitrary; completely determined by the basis set. Go to zero for complete basis. What about shape? Should go to infinitely extended.

81 Shape of the 1σ u LUMO density of H 2 as a function of basis set:

82 Electron affinity (in ev) Positive HF ε LUMO! CCSD(T) negative EAs: basis: G(3df,3dp) HF M06-2X LC-BOP BOP BLYP Lcgau-BOP ε LUMO (ev) Calculated LUMO energies vs EA for 113 molecules. EA from CCSD(T), basis: G(3df,3dp) (Kar, Song, Hirao, JCC 2013) Almost all HF ε LUMO positive!

83 Practical ways to get scattering resonances (negative EAs) with basis set calculations Stabilization method (H. S. Taylor et al.), also called SKT (stabilization Koopmans method): Systematically scan through the spectrum of positive energies by scaling the coefficients of all diffuse basis functions to very low value (diffuse). Then orbital energies go down in energy as function of scaling parameter α. Detect resonance energies by inspecting the orbitals; when getting high amplitude in molecular region, you are at resonance energy. Or by looking at curves of orbital energy as function of α: resonance energies show up as avoided crossing. See K. Jordan et al. (JPC-A 104 (2000) 9605) and Cheng et al. (JPC-A 116 (2012) 12364)

84 Orbital energies and excitation energy calculations (TDDFT) TDDFT: ( ε εk ε )F q = ω 2 q F q ( ε 2 ) = δ ij δ ab (ε a ε i ) 2 ia, jb K is "coupling matrix", see later Suppose i a does not couple to other j b (single pole approximation, SPA), q i a (ε a ε i ) 2 + 2(ε a ε i ) ϕ i (r)ϕ a (r) f xc (r, r )ϕ i (r )ϕ a (r ) drdr F q = ω 2 F q ω = (ε a ε i )+ ϕ i ϕ a f xc ϕ i ϕ a small (ε a ε i ) excitation energy (in molecules!)

85 Acetone: orbital energy differences and excitation energies (ev) Funct. State Weight ε i ε a ε ia ω ω ε ia ω E exp SAOP 1A ( KS) 1B A A B A B B BP86 1A B A A B A B B

86 Orbital energies Rydbergs valence virtuals R V R occupied orbitals V exact KS potential I LDA/GGA potntl: strongly upshifted valence orbs (occup. and virt.); weakly upshifted fewer Rydbergs LDA/GGA small basis LDA/GGA large basis

87 Level diagram of excited states in N 2 bʹ 1Σ u π 1π u 1π 1π g u 1π u 1π g g π u 1π g o 1 u c 1 u cʹ 1Σ u b 1 u σ u 1π g π u 4σ g σ g 2π u σ g 3σ u σ u 1π g 3σ g 2π u 3σ g 3σ u σ u 1π g σ u 1π g 2π u 3σ 1π 4σ u g g ( 2.3) ( 3.3) ( 6.9) Ry val 1π u 4σ g π u 4σ g σ g ( 15.3) 1π u ( 16.5) 2σ ( 18.6) 3σ g 2π u u σ g 3σ u EXP SAOP BP-GRAC LDA BP 3σg 2π u σ g 3σ u 10.35

88 Acetone: orbital energy differences and excitation energies (ev) Funct. State Weight ε i ε a ε ia ω ω ε ia ω E exp HF 1A B A A B A B B M06-2X 1A B A A B A B B

89 Pyrimidine: valence excitations (ev) Funct. State Weight ε i ε a ε ia ω ω ε ia ω E exp SAOP 1B ( KS) 1A B A B A BP86 1B A B A B A

90 Pyrimidine: valence excitations (ev) Funct. State Weight ε i ε a ε ia ω ω ε ia ω E exp TDHF 1B A B A B A M06-2X 1B A B A B A

91 Difference orbital energies in Hartree-Fock and DFT (1) Molecular excitation el. wavef. Why is KS virtual-occup. orbital energy difference a good approx. to excitation energy? The xc hole plays a crucial role here: it is a local hole, exerting a strong attraction at each position r. This attraction mimicks the attraction that in reality would occur by the depletion hole in orbital i where the electron came from. hole wavef. xc hole The virtual orbital in KS theory is a one-electron state for an electron that feels a hole potential, i.e. is like the electron in the electron-hole pair that is created in an excitation.

92 Difference orbital energies in Hartree-Fock and DFT (2) Molecular excitation hole el. wavef. no hole attraction in HF virtual orb. energy wavef. ε a ε i Hartree-Fock det. Φ 0 = ψ 1 ψ 2 ψ N Koopmans' appr. to excitation (frozen orb.): i a Φ i a = ψ 1 ψ 2 ψ i ψ a ψ N E[Φ i a ] E[Φ 0 ] = ε a HF ε i HF J ia electron-hole attraction J ia = i(1)a(2) i(1)a(2) + K ia High HF ε a HF (due to lack in Fock operator for virtual ψ a ) is pulled down by depletion hole (J ia ). ε a KS need not be pulled down, it is already low lying due to pull by xc hole. NB. shape of ψ a HF and ψ a KS is very different! ψ a HF not realistic! small

93 Compare to how TD-DFT works in EXX variant (see Gonze-Scheffler, PRL 1999 EXX: local potential v x (r) appr. to KS pot. with only W KS x as xc functional (also called OPM). v x (r) has X hole, ψ a is pulled down. TD-EXX gives kernel correction to (ε a ε i ) for excitation energy: ω = (ε a ε i )+ ϕ i ϕ a f xc EXX ϕ i ϕ a = (ε a ε i )+ ϕ a ˆK HF v x ϕ a ϕ i ˆK HF v x ϕ i J ia + K ia - 2 d term shifts ε a up from (appr.) KS level ε a KS to (appr.) HF level ε a HF (because ψ a is unoccupied!) - 3 d term has little correction on ε i (ε i KS ε i HF IP i ) anyway - 4 th term ( J ia ) provides electron-hole attraction to correct ε a after upshift from 2d term back to ε a KS. - 5 th term small. TD-EXX benefits from good shape of ψ a KS-EXX compared to ψ a HF!

94 Charge transfer excitations between two remote molecules are much too low in TDDFT. Why? A CT transition should be: I D A A J da ( 1/R) Exact KS: ε d = I D, ε a = A A A TDDFT gives appr. (ε a ε d ) +<ψ a ψ d f xc ψ a ψ d > I D A A A 0 over all space if R large wrong by A! zero contrib. from kernel term

95 Charge transfer excitations between two remote molecules TDDFT gives appr. (ε a ε d ) +<ψ a ψ d f xc ψ a ψ d > I D A A A Now KS orbital energy difference is NOT good! Why? Because xc hole and actual depletion hole are too different. v xc hole : potential due to hole of 1 electron (is +1 charge) around each reference point strongly attractive depletion hole ψ d 2 is far from points r on A where potential is evaluated, so little stabilization should be experienced by electron from far away hole; electron is like an added electron to A. Now HF ε a HF A A (Koopmans) would be better: ε a HF = ψ a v nuc + v Coul + ˆK HF ψ a ˆK HF ( no hole for virtual orb.!)

96 Charge transfer excitations between two remote molecules are much too low in TDDFT. Why? Molecular excitation Charge transfer excitation el. wavef. hole wavef. xc hole hole wavef. Donor transferred el. wavef. Acceptor KS LUMO stabilized by xc hole

97 Comment on bandgap in solids (1) delocalized electron wavef. conduction band of excited electrons exciton weak stabiliz. no stabiliz. KS conduction band, strongly stabilized by local xc hole Fermi level deloc. hole wavef. xc hole valence band Do not expect bottom of conduction band states in DFT to be affinity level

98 Comment on bandgap in solids (2) LDA/GGA bandgap often only 30 50% of the fundamental gap (I A) Surprise? Maybe true KS gap will be close to (I A)? No: Godby, Schlüter, Sham 1986; Grüning, Marini, Rubio (2006): LDA/GGA gap KS gap (as we saw for molecules!) To be expected: Actual hole-electron interaction is different from electron - KS xc hole interaction: - KS xc hole is small (~ atomic size, one unit cell in Si) -> strong pull - fully delocalized electron and hole states: no pull - excitons: Wannier-Mott: still diffuse electron wavefunction, e.g. Si: Bohr radius of exciton ca. 4.3 nm 100 bohr very little electron-hole stabilization [Frenkel excitons in e.g. molecular solid may be more like molecules] NB. explanation for solids very much like for the CT problem

99 References: E. J. Baerends, O. V. Gritsenko, R. Van Meer The Kohn-Sham gap, the fundamental gap and the optical gap: the physical meaning of occupied and virtual Kohn-Sham orbital energies PCCP (Perspective) 15 (2013) R. Van Meer, O. V. Gritsenko, E. J. Baerends Physical meaning of virtual Kohn-Sham orbitals and orbital energies: an ideal basis for the description of molecular excitations J. Chem. Theor. Comp. 10 (2014) 4432

100 Failures of time-dependent DFT a) Wrong Potential Energy Surface (PES) for bonding antibond. excitation b) Failure to treat doubly excited configurations c) Too low charge transfer excitation energy a) and c) are cases where leading term (ε a ε i ) is wrong and usual kernel (ALDA) fails to correct

101 a) Bonding-antibond. excitation problem in TDDFT 1 Σ u + states in H 2 : lowest is (σ g ) 2 (σ g ) 1 (σ u ) Σ u + H 2 cc-pvtz basis excit. energy (a.u.) Σ u Σ u + exact BP86 B3LYP R e R(H-H) bohr Gritsenko, v. Gisb. Görling, Baerends, JCP 113 (2000)

102 N 2 1 Σ u + PECs: exact Gr. St. + TDDFT Σ + u, N 2 energy [a.u.] Exact ground state plus TDDFT excit. energies MRCI excited states and ground state ground state ( 1 Σ + g ) R N-N [a.u.] K. J. H. Giesbertz, E. J. Baerends CPL 461 (2008) 338

103 N 2 1 Σ u + PECs: DFT Gr. St. + TDDFT Σ + u, N 2 energy [a.u.] TDDFT excitation en. added to DFT Gr. St. energy ground state ( 1 Σ + g ) R N-N [a.u.]

104 b)double excitation problem in TDDFT: 1 Σ g + states of H 2 excit. energy (a.u.) R e 4 1 Σ g Σ g Σ + g R(H-H) bohr H 2 cc-pvtz basis exact BP86 B3LYP double excit. character (σ g ) 2 (σ u ) 2

105 Benchmark diabatic potential energy curves from MRCI and fit to expt. (Spelsberg&Meyer, 2001) Ry 1π u 4σ g double excit. 3σ g,1π u (1π g ) 2 valence 2σ u 1π g b c o Ry 3σ g 2π u

106 Benchmark diabatic potential energy curves from MRCI and fit to expt. (Spelsberg&Meyer, 2001) TDDFT (exact KS pot) calculated b 1 Π u : lacks double! valence 2σ u 1π g b c o double excit. 3σ g,1π u (1π g ) 2 TDDFT b 1 Π u (valence 2σ u 1π g ) above the Rydb states because of lack of doubles: always a problem at long distance!

107 c) Charge transfer problem of TDDFT: 1 Σ + excited st. in HeH Σ + HeH + aug-cc-pvqz basis excitation energy (a.u.) Σ Σ + exact BP86 B3LYP bohr

108 TDHF solves CT problem, see Dreuw + Head-Gordon TDHF Cf. hybrid methods TDDFT

109 TDHF does not solve (σ g ) 2 (σ g ) 1 (σ u ) 1 problem 1 Σ u + TDDFT TDHF