Lecure 9: Advanced DFT conceps: The Exchange-correlaion funcional and ime-dependen DFT Marie Curie Tuorial Series: Modeling Biomolecules December 6-11, 2004 Mark Tuckerman Dep. of Chemisry and Couran Insiue of Mahemaical Science 100 Washingon Square Eas New York Universiy, New York, NY 10003
Ψ 0λ is he ground sae wavefuncion of a Hamilonian H=T+λW+V. Noe ha when λ=0. V=V KS and when λ=1, V=V ex.
λ λ λ λ
n( r1 ) [ ] ( ) 1 g ( ; ( )) 0 r r n r Exc n = dr n r d 2 r r r Where f xc can be obained by inegraing over r wih a Change of variables s=r-r.
SIC E [ n] = E [ n] ( J[ n ] + E [ n ]) i xc xc i xc i i n () r = ψ () r i 2
Gradien Correcions Exend he LDA form o include densiy gradiens: E n d r n r f n r n r n r 2 xc[ ] = ( ) xc ( ( ), ( ), ( )) Example: Becke, Phys. Rev. A (1988) χ () r 2 4/3 4/3 x[] = x r () r β r () r 1 6 sinh 1 + β χ ( r ) E n C d n d n χ() r = n() r n 4/3 () r Funcional form chosen o have he correc asympoic behavior: 1 1 Ex = d n( ) ex( ) lim ex( ) n( ) e 2 r r r r r r r αr
Moivaion for TDDFT Phooexciaion processes Aomic and nuclear scaering Dynamical response of inhomogeneous meallic sysems.
The ime-dependen Hamilonian Consider an elecronic sysem wih a Hamilonian of he form: H () = T+ V + V () Where V() is a ime-dependen one-body operaor. e ee Our ineres is in he soluion of he ime-dependen Schrödinger equaion: H () Ψ () = i Ψ() Ψ ( ) =Ψ 0 0 Le V be he se of ime-dependen poenials associaed and le N be he se of densiies associaed wih ime-dependen soluions of he Schrödinger equaion. There exiss a map G such ha G : V N
The Hohenberg-Kohn Theorem Since V() is a one-body operaor: Ψ() V() Ψ () = d r n( r,) V ( r,) Assume he poenial can be expanded in a Taylor series: 1 Vex (,) r = vk ()( r 0) k! k = 0 k ex Suppose here are wo poenials such ha V (,) r V (,) r c() ex Then, here exiss some minimum value of k such ha ex k v () () [ (,) (,)] k r v k r = Vex r V ex r cons k = 0
The Hohenberg-Kohn Theorem For ime-dependen sysems, we need o show ha boh he densiy n(r,) and he curren densiy j(r,) are differen for he wo differen poenials, where he coninuiy equaion is saisfied: {} s {} s n( r, ) = dr dr Ψ( r, s,..., x, ) 2 N 1 e n(,) r + ij(,) r = 0 N e 2 * * jr (, ) = dr2 dr N (, 1,..., x, ) (, 1,..., x, ) (, 1,..., x, ) (, 1,..., x, ) e Ψ r s N Ψ rs e N Ψ r s e N Ψ rs e N e For any operaor O(), we can show ha: d i Ψ() O() Ψ () = Ψ () i O() + [ O(), H()] Ψ() d
The Hohenberg-Kohn Theorem From equaion of moion, we can show ha i in V V [ (,) (,)] = (, ) [ (, ) (, )] ex ex 0 jr j r r r r = 0 0 0 And, in general, for he minimal value of k alluded o above: k+ 1 i [ (,) (,) ] = in(, 0) i [ Vex (,) V ex (,) ] 0 = 0 = 0 jr j r r r r k Hence, even if j and j are differen iniially, hey will differ for imes jus laer han 0.
For he densiy, since The Hohenberg-Kohn Theorem [ n r n r ] i[ j r j r ] (,) (,) + (,) (,) = 0 I follows ha: k+ 2 [ n(,) n (,) ] = n(, 0) [ Vex (,) V ex (,) ] 0 = 0 = 0 r r r i r r k Therefore, even if n and n are iniially he same, hey will differ for imes jus laer han 0. Hence, any observable can be wrien as a funcional of n and a funcion of. Ψ[ n]( ) O( ) Ψ [ n]( ) = O[ n]( )
Acions in quanum mechanics and DFT Consider he acion inegral: A = d ' Ψ( ') i H ( ') Ψ( ') 0 ' Schrödinger equaion resuls requiring ha he acion be saionary according o: δ A δ Ψ() = 0 Hence, if we view A as a funcional of he densiy, A[ n] = d' Ψ[ n]( ') i H( ') Ψ[ n]( ') 0 ' An [ ] = Bn [ ] d' dr n( r, V ) ( r, ) 0 B[ n] = d' Ψ[ n]( ') i T Vee Ψ[ n]( ') 0 ' ex
Hohenberg-Kohn: Hohenberg-Kohn and KS schemes δ A δ B = Vex (,) r = 0 δn(,) r δn(,) r Kohn-Sham formulaion: Inroduce a non-ineracing sysem wih effecive poenial V KS (r,) ha gives he same ime-dependen densiy as he ineracing sysem. For a non-ineracing sysem, inroduce single-paricle orbials ψ i (r,) such ha he densiy is given by N e n(,) r = ψ i (,) r i= 1 2 KS acion: 1 n( r, ') n( r', ') AKS[ n] = d' Ψ( ) i Ts Ψ( ') d n(, ') Vex (, ') d d ' Axc[ n] 0 ' r r r 2 r r r r'
Time-dependen Kohn-Sham equaions δ A / δ n( r, ) = 0 From : KS 1 2 i ψi(,) r = VKS(,) i(,) + r 2 ψ r n( r ', ) δ Axc VKS (,) r = Vex (,) r + dr' + r r' δ n( r, ) Adiabaic LDA/GGA: A [] n = d ' d n (,) f ((,), n n (,)) xc r r r r xc 0
Linear response soluion for he densiy Sraegy: Solve he Liouville equaion for he densiy marix o linear order. H () = H + V() 0 Ψ() V() Ψ () = d r n(,) r V (,) r ex Quanum Liouville equaion for he densiy operaor ρ(): Time-dependen densiy: i () [ H(), ()] ρ = ρ n(,) r = Ψ() ρ() Ψ()
Linear response soluion for he densiy Wrie he densiy operaor as: To linear order, we have Soluion: ρ() = ρ + δρ() 0 δρ() = i[ H0, δρ()] iv [ (), ρ0] δρ() = i d' e [ V('), ρ ] e 0 ih0( ') ih0( ') 0
To linear order: Linear response soluion for he densiy = Ψ 0 Ψ = 0 δn(,) r δρ() dr d χ(,, r r, ) V ( r,') 0 ex Where he Fourier ransform of he response kernel is: Ψ0 ρ0() r Ψm Ψm ρ0( r ) Ψ0 Ψ0 ρ0( r ) Ψm Ψm ρ0() r Ψ0 χ(, rr, ω) = m ω ( Em E0) + iε ω+ ( Em E0) + iε Hence, poles of he response kernel are he elecronic exciaion energies.
from Appel, Gross and Burke, PRL 93, 043005 (2003).
Lecure Summary Adiabaic connecion formula provides a rigorous heory of he exchange-correlaion funcional and is he saring poin of many approximaions. Generalizaion of densiy funcional heory o ime-dependen sysems is possible hrough generalizaion of he Hohenberg-Kohn heorem. In linear response heory, he response kernel (or is poles) is he objec of ineres as i yields he exciaion energies.