Excitation energies for molecules by TimeDependent. based on Effective Exact Exchange KohnSham potential


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1 Excitation enegies fo molecules by TimeDependent DensityFunctional Theoy based on Effective Exact Exchange KohnSham potential Fabio Della Sala National Nanotechnology Laboatoies Lecce Italy A. Göling Technische Univesität München Gaching Gemany F. Della Sala Excitation Enegies with EffectiveExact Exchange Pag.
2 DensityFunctionalTheoy: KohnSham(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 ( ExchangeCoelation(XC Potential Fom the solution of a singlepaticle equation the exact goundstate density fo N inteacting electons is obtained N ρ( = ϕ ( i= i 2 F. Della Sala Excitation Enegies with EffectiveExact Exchange Pag. 2
3 ExchangeCoelation Potential v ( = v ( ;[ ρ] = xc xc δexc[ ρ] δρ( =? The XCenegy functional is still unknown! DFT focuses only on the goundstate 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 goundstate density and totalenegy. KS specta and vitual obitals ae not consideed. Howeve KS eigenvalues and unoccupied obitals ae of fundamental impotance fo TimeDependent DensityFunctional Theoy (TDDFT. F. Della Sala Excitation Enegies with EffectiveExact Exchange Pag. 3
4 TimeDependent DensityFunctional Theoy(TDDFT 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 singletexcitation K XCkenel = * 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 offdiagonal tems: ω ε ε + s a K sa sa Fo TDDFT accuate KS specta ae equied! F. Della Sala Excitation Enegies with EffectiveExact Exchange Pag. 4
5 AsymptoticalCoected XCpotentials 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 AsymptoticalCoected (AC XCpotentials with an / asymptotic behavio have been intoduced (van LeeuwenBaends HandyToze Casida Gitsenko... and successfully applied to molecules. Howeve they contains seveal empiical paametes and ae not selfinteaction fee. E xc Selfinteaction fee F. Della Sala Excitation Enegies with EffectiveExact Exchange Pag. 5 X v xc(
6 EXX equation: ExactExchange(EXX Teating the KS exchange exactly will solve the selfinteaction poblem: exchange and coulomb inteactions ae of the same ode (GölingLevy KS PetubationTheoy 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 densitydensity 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 EffectiveExact Exchange Pag. 6 φ a ε v a HF x ε φ s s HF v x : HateeFock Exchange
7 Localized HateeFock We assume that exists a local potential that geneate a KS Slate deteminant Φ identical to HateeFock one. Della Sala Göling JCP (200 occ. ψ = i ( ψ i ( HF v x ( i Localization v LHF ( x An obitaldependent 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 offdiagonal tems equivalent to the KLI potential (Kiege Li and Iafate PRA 450 (992 F. Della Sala Excitation Enegies with EffectiveExact Exchange Pag. 7
8 Results Enegy (ev System HF LHFHF EXXHF 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 EffectiveExact 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 EffectiveExact Exchange Pag. 9
10 Compaison of of oneelecton 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 basisset (planewave equied F. Della Sala Excitation Enegies with EffectiveExact Exchange Pag. 0
11 NonLocality Benzene LHF is a functional of obitals and thus is a stong nonlocal functional of the density The nonlocality incease in the asymptotic egion due to the coection tem v slate x ( F. Della Sala Excitation Enegies with EffectiveExact 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 EffectiveExact Exchange Pag. 2
13 v x ϕ M ( Asymptotic Popeties fo Molecues Poof fo exactexchange 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 exchangecoelation 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 SelfEnegy F. Della Sala Excitation Enegies with EffectiveExact Exchange Pag. 3
14 Asymptotic BaieWells The HOMO nodal suface(s ae of zeo measue. Howeve due to the continuity of exchange potential asymptotic baiewells ae pesent. Della Sala Göling PRL (2002 in pess JCP 6 ( AC z= z=0 F. Della Sala Excitation Enegies with EffectiveExact Exchange Pag. 4
15 Influence of of exchange potential on on KS eigenvalues LDA/GGA wong specta LB94 is an AC method LHFAC:= LHF with the asymptotic behavio eplaced by / Stong impotance of asymptotic baiewells on vitual obitals Della Sala Göling PRL (2002 in pess THEORICAL FAILURE OF THE AC METHODS! F. Della Sala Excitation Enegies with EffectiveExact Exchange Pag. 5
16 F. Della Sala Excitation Enegies with EffectiveExact Exchange Pag. 6 Excitation enegies using LHF Excitation enegies using LHF In TDDFT the XCkenel has only a small contibution(0%20% compaed to the KS eigenvalue diffeences. Fo molecules and excitation enegies without chagetansfe 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 TDDFT. The adiabatic exchangeonly LDA is used fo the kenel. The EXX kenel is fequency dependent and stongly nonlocal 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 EffectiveExact Exchange Pag. 7
18 Excitation Enegy: Summay Mean Absolute Values fo singlet/tiplet excitation fo N 2 ethylene benzene. Method BP HCTHAC 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 exchangeonly 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 EffectiveExact Exchange Pag. 8
19 Conclusions The use of an (effective exactexchange 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 baiewells invalidate the use of the AC method fo molecules: a tue selfinteaction fee potential is equied. LHF obitals and eigenvalues lead to impooved excitation enegy. TDDFT F. Della Sala Excitation Enegies with EffectiveExact Exchange Pag. 9
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