Excitation energies for molecules by Time-Dependent. based on Effective Exact Exchange Kohn-Sham potential

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1 Excitation enegies fo molecules by Time-Dependent Density-Functional Theoy based on Effective Exact Exchange Kohn-Sham potential Fabio Della Sala National Nanotechnology Laboatoies Lecce Italy A. Göling Technische Univesität München Gaching Gemany F. Della Sala Excitation Enegies with Effective-Exact Exchange Pag.

2 Density-Functional-Theoy: Kohn-Sham(KS Scheme Schödinge Equation: KS obitals KS eigenvalues 2 h 2m 2 + v ext ( + v coul Classical Electostatic Potential ( + v xc ( ϕ ( = ε i i ϕi ( Exchange-Coelation(XC Potential Fom the solution of a single-paticle equation the exact gound-state density fo N inteacting electons is obtained N ρ( = ϕ ( i= i 2 F. Della Sala Excitation Enegies with Effective-Exact Exchange Pag. 2

3 Exchange-Coelation Potential v ( = v ( ;[ ρ] = xc xc δexc[ ρ] δρ( =? The XC-enegy functional is still unknown! DFT focuses only on the gound-state density: thus KS eigenvalues ae only auxiliay quantities and only occupied obitals ae equied. Cuent appoximations (LDA GGA ae made in ode to obtain accuate gound-state density and total-enegy. KS specta and vitual obitals ae not consideed. Howeve KS eigenvalues and unoccupied obitals ae of fundamental impotance fo Time-Dependent Density-Functional Theoy (TD-DFT. F. Della Sala Excitation Enegies with Effective-Exact Exchange Pag. 3

4 Time-Dependent Density-Functional Theoy(TD-DFT Excitation enegies ω ae obtained fom the eigenvalue equation: ( 2 2 δ stδ ab( εs εa + 2 εs εa K sa tb εt εb Tsa tb = ω Tsa tb (Fo singlet-excitation K XC-kenel = * 2 * 3 φ s ( φ a ( + f xc[ ρ ]( ω φt ( φb ( d d sa tb f xc [ ρ]( 0 vxc ( = ρ( ab: occupied st: vitual 3 2 Exc[ ρ] = ρ( ρ( Small off-diagonal tems: ω ε ε + s a K sa sa Fo TD-DFT accuate KS specta ae equied! F. Della Sala Excitation Enegies with Effective-Exact Exchange Pag. 4

5 Asymptotical-Coected XC-potentials Cuent appoximations (LDA GGA have wong KS specta due to a wong asymptotic behavio. The exact XC satisfied : v xc( PRA (984 PRB (985 (fo atoms Asymptotical-Coected (AC XC-potentials with an / asymptotic behavio have been intoduced (van Leeuwen-Baends Handy-Toze Casida Gitsenko... and successfully applied to molecules. Howeve they contains seveal empiical paametes and ae not self-inteaction fee. E xc Self-inteaction fee F. Della Sala Excitation Enegies with Effective-Exact Exchange Pag. 5 X v xc(

6 EXX equation: Exact-Exchange(EXX Teating the KS exchange exactly will solve the self-inteaction poblem: exchange and coulomb inteactions ae of the same ode (Göling-Levy KS Petubation-Theoy PRA 50 ( Coelation is of highe ode. occ. n φ i ( φ j ( φ i ( φ j ( s 3 δe [ ρ ] E x = d v ( x x = 2 i j δρ ( occ. unocc. X s ( v x ( d = 2n S φ a ( φ s ( a s X s :KS density-density esponse The EXX equation can be solved fo atoms o peiodic systems. Fo molecules stong numeical poblems ae pesent. Othe appoach ae needed! F. Della Sala Excitation Enegies with Effective-Exact Exchange Pag. 6 φ a ε v a HF x ε φ s s HF v x : Hatee-Fock Exchange

7 Localized Hatee-Fock We assume that exists a local potential that geneate a KS Slate deteminant Φ identical to Hatee-Fock one. Della Sala Göling JCP (200 occ. ψ = i ( ψ i ( HF v x ( i Localization v LHF ( x An obital-dependent potential is obtained: SCF equied! occ ( ( ( ( occ. ( ( LHF 3 v ( = ψ i ψ j ψ i ψ j i j LHF HF x d + i vx vx j ij ( ψ ψ ρ ij ( N N ρ ( Slate Potential Coection Tem Without off-diagonal tems equivalent to the KLI potential (Kiege Li and Iafate PRA 450 (992 F. Della Sala Excitation Enegies with Effective-Exact Exchange Pag. 7

8 Results Enegy (ev System HF LHF-HF EXX-HF Ne LiH NH H CO CH Ethylene Benzene Napthalene Li The LHF and HF deteminant deviates in enegy less than 0.02% close to numeical accuacy meaning that the hypothesis Φ KS =Φ HF is well justified Enegy LHF EXX HF The LHF can be called effective EXX F. Della Sala Excitation Enegies with Effective-Exact Exchange Pag. 8

9 Example of of LHF Potential: Benzene Shell stuctues Diffeent asymptotic behavio No stuctue aound hydogens LDA v x 3 ( = ρ ( π 3 F. Della Sala Excitation Enegies with Effective-Exact Exchange Pag. 9

10 Compaison of of one-electon eigenvalues In HF no vitual bound obitals (VBOs ae obtained. LDA(GGA gives incoect(shifted absolute eigenvalues; few VBOs ae pesent. Ionization Pot. In LHF the HOMO is close to the HF HOMO. Rydbeg seies of VBOs ae obtained. Vitual unbound (ε>0 obitals ae difficult to epesent in any LCAO basis-set (plane-wave equied F. Della Sala Excitation Enegies with Effective-Exact Exchange Pag. 0

11 Non-Locality Benzene LHF is a functional of obitals and thus is a stong non-local functional of the density The non-locality incease in the asymptotic egion due to the coection tem v slate x ( F. Della Sala Excitation Enegies with Effective-Exact Exchange Pag.

12 Anisotopy LHF coection tem HOMO H C C H The coection tem appoach a constant along HOMO Nodal Sufaces (H.N.S.! v co. x ( 0 Const on H.N.S. Diffeent behavio than in spheically symmetic systems(i.e.atoms F. Della Sala Excitation Enegies with Effective-Exact Exchange Pag. 2

13 v x ϕ M ( Asymptotic Popeties fo Molecues Poof fo exact-exchange ϕ M vˆ x vˆ HF x ϕm on H.N.S. Highest MO which doesn t vanish on the H.N.S. Poof fo exchange-coelation v ( xc ϕ ˆ Σˆ M vxc ( ω M ϕm on H.N.S. LHFKLI exhibit the same asymptotic behavio! Della Sala Göling PRL (2002 in pess JCP 6 ( Σˆ ( ω Self-Enegy F. Della Sala Excitation Enegies with Effective-Exact Exchange Pag. 3

14 Asymptotic Baie-Wells The HOMO nodal suface(s ae of zeo measue. Howeve due to the continuity of exchange potential asymptotic baie-wells ae pesent. Della Sala Göling PRL (2002 in pess JCP 6 ( AC z= z=0 F. Della Sala Excitation Enegies with Effective-Exact Exchange Pag. 4

15 Influence of of exchange potential on on KS eigenvalues LDA/GGA wong specta LB94 is an AC method LHF-AC:= LHF with the asymptotic behavio eplaced by / Stong impotance of asymptotic baie-wells on vitual obitals Della Sala Göling PRL (2002 in pess THEORICAL FAILURE OF THE AC METHODS! F. Della Sala Excitation Enegies with Effective-Exact Exchange Pag. 5

16 F. Della Sala Excitation Enegies with Effective-Exact Exchange Pag. 6 Excitation enegies using LHF Excitation enegies using LHF In TD-DFT the XC-kenel has only a small contibution(0%-20% compaed to the KS eigenvalue diffeences. Fo molecules and excitation enegies without chage-tansfe the kenel is equied only in a egion whee occupied obital ae located i.e. only nea the molecule: hee it is appoximatively local and simila to the LDA one. ( ( ( ]( [ ω ω ω ω ρ = s X s EXX X H X f We use LHF KS eigenvalue and obitals as input fo the TD-DFT. The adiabatic exchange-only LDA is used fo the kenel. The EXX kenel is fequency dependent and stongly non-local Fo molecules the EXX kenel is still unde study: the computational cost will be vey high and numeical eos have to be solved. + = 3 3 * * ( ( ]( [ 2 ( ( d d f K b t xc a s tb sa φ φ ω ρ φ φ

17 Excitation enegies F. Della Sala Excitation Enegies with Effective-Exact Exchange Pag. 7

18 Excitation Enegy: Summay Mean Absolute Values fo singlet/tiplet excitation fo N 2 ethylene benzene. Method BP HCTH-AC LHFX MAE LDA/GGA fails fo Rydbeg excitation enegies: it can be used only fo lowlying excitation enegy. AC method ae bette but use seveal empiical paametes and ae no moe theoetically justified. LHFX is quite good despite the coelation is completely neglected. Pue exchange-only calculation without any empiical paamete. It can teat both valence and Rydbeg excitations. Della Sala Göling IJQC (2002 in pess F. Della Sala Excitation Enegies with Effective-Exact Exchange Pag. 8

19 Conclusions The use of an (effective exact-exchange potential is of fundamental impotance to descibe KS obitals and eigenvalues. With LHF potential the same total enegy of the HF method can be obtained: howeve the eigenvalue specta is completely diffeent. The discovey of asymptotic baie-wells invalidate the use of the AC method fo molecules: a tue self-inteaction fee potential is equied. LHF obitals and eigenvalues lead to impooved excitation enegy. TD-DFT F. Della Sala Excitation Enegies with Effective-Exact Exchange Pag. 9

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