Benchmarking Nonequilibrium Green's Functions against Configuration Interaction for time-dependent Auger decay processes
BBenchmarking Nonequilibrium Green’s Functions against Configuration Interactionfor time-dependent Auger decay processes
F. Covito, E. Perfetto,
2, 3
A. Rubio,
1, 4, 5 and G. Stefanucci
3, 6 Max Planck Institute for the Structure and Dynamics of Matter and Center for Free-Electron Laser Science,Luruper Chaussee 149, 22761 Hamburg, Germany CNR-ISM, Division of Ultrafast Processes in Materials (FLASHit),Area della ricerca di Roma 1, Monterotondo Scalo, Italy Dipartimento di Fisica, Universit`a di Roma Tor Vergata,Via della Ricerca Scientifica, 00133 Rome, Italy Center for Computational Quantum Physics (CCQ),The Flatiron Institute, 162 Fifth avenue, New York NY 10010 Nano-Bio Spectroscopy Group, Universidad del Pa´ıs Vasco, 20018 San Sebastin, Spain INFN, Sezione di Roma Tor Vergata, Via della Ricerca Scientifica 1, 00133 Roma, Italy (Dated: October 4, 2018)We have recently proposed a Nonequilibrium Green’s Function (NEGF) approach to includeAuger decay processes in the ultrafast charge dynamics of photoionized molecules. Within the socalled Generalized Kadanoff-Baym Ansatz the fundamental unknowns of the NEGF equations arethe reduced one-particle density matrix of bound electrons and the occupations of the continuumstates. Both unknowns are one-time functions like the density in Time-Dependent Functional Theory(TDDFT). In this work we assess the accuracy of the approach against Configuration Interaction(CI) calculations in one-dimensional model systems. Our results show that NEGF correctly capturesqualitative and quantitative features of the relaxation dynamics provided that the energy of theAuger electron is much larger than the Coulomb repulsion between two holes in the valence shells.For the accuracy of the results dynamical electron-electron correlations or, equivalently, memoryeffects play a pivotal role. The combination of our NEGF approach with the Sham-Schl¨uter equationmay provide useful insights for the development of TDDFT exchange-correlation potentials with ahistory dependence.
I. INTRODUCTION
Photo-ionized many-body systems relax to lower en-ergy states through nuclear rearrangement and chargeredistribution. Nuclear dynamics does typically play arole on longer time scales, although there are situationswhere electron-nuclear and electron-electron interactionscompete on the same timescale, e.g., in the vicinity of aconical intersection. At the (sub)femtosecond timescale,however, the most relevant relaxation channel of core-ionized molecules is the Auger decay which is exclusivelydriven by the Coulomb interaction [1].Recent advances in pump-probe experiments made itpossible to follow the attosecond dynamics of atoms afterthe sudden expulsion of a core electron [2–6]. Theoreticalframeworks describing the Auger decay have been pro-posed, the more accurate being the ones based on many-body wavefunctions, see also Ref. [7]. Although thesemethods are in principle applicable to atoms as well asmolecules, they quickly become prohibitive for systemswith more than a few active electrons. For instance,Auger decays in ionized small molecules or molecules ofbiological interest are extremely difficult to cope withwavefunction approaches due to the large number ofstates involved in the process. Still, Auger decays con-tribute to the relaxation dynamics of these more com-plex systems, which are currently attracting an increas-ing interest and attention [8–12]. It is therefore crucial todevelop first-principles approaches capable of capturing the (sub)femtosecond relaxation mechanisms induced byelectronic correlations and applicable to atoms as well asmolecules.The most widely used method for large scale real-timesimulations is Time-Dependent Density Functional The-ory [13–15] (TDDFT), which gives an adequate and com-putationally affordable tool for the description of systemsconsisting of up to thousands of atoms. The most efficientand extensively used functionals for TDDFT calculationsare the space-time local exchange-correlation (xc) func-tionals. It has been shown numerically in Ref. [16] thatthese approximate functionals fail in capturing Auger de-cays, the fundamental reason being that they lack mem-ory effects – the xc potential depends on the instanta-neous density only.We have recently proposed a first-principles NonEqui-librium Green’s Function (NEGF) approach [17] whichovercomes the limitation of adiabatic functionals andthat may inspire new ideas for the inclusion of memoryeffects in the TDDFT functionals. The method is ap-plicable to molecules with up to tens of atoms and at itscore there is an equation to simulate the electron dynam-ics in the parent cation without dealing explicitly withthe Auger electrons. The idea is similar in spirit to theembedding scheme in time-dependent quantum transportwhere the electron dynamics in the molecular junction issimulated without dealing explicitly with the electrons inthe leads [18–22]. However, whereas in quantum trans-port the integration out of electrons in the leads gives a r X i v : . [ c ond - m a t . o t h e r] O c t an embedding self-energy which is independent of thedensity in the junction, the integration out of the Augerelectrons gives an Auger self-energy which is a functionalof the density in the molecule.In order to assess the quality of the NEGF approachin this work we use the time-dependent charge distri-bution of the bound electrons to reconstruct the Augerwavepacket in free space, and then benchmark the resultsagainst exact configuration interaction (CI) calculations.We perform NEGF and CI simulations in a model one-dimensional (1D) system and study the real space-timeshape of the Auger wavepacket as well as the Auger spec-trum. The main outcome of this investigation is that theresults of the NEGF approach are in excellent agreementwith those from CI provided that the repulsion betweenthe valence holes is much smaller than the energy of theAuger electron. II. DESCRIPTION OF THE SYSTEM ANDTHEORY
Let us consider a 1D finite system described by the one-particle Hartree-Fock (HF) basis { ϕ i , ϕ µ } , where romanindices run over bound states and greek indices run overcontinuum states. The equilibrium Hamiltonian can beconveniently written as the sum of three termsˆ H eq = ˆ H bound + ˆ H Auger + ˆ H cont , (1)where ˆ H bound is the bound electrons Hamiltonian, ˆ H Auger is the Auger interaction and ˆ H cont is the free-continuumpart. In our basis, these are written asˆ H bound = (cid:88) ij h ij ˆ c † i ˆ c j + 12 (cid:88) ijmn v ijmn ˆ c † i ˆ c † j ˆ c m ˆ c n , (2a)ˆ H Auger = (cid:88) ijm (cid:88) µ (cid:16) v Aijmµ ˆ c † i ˆ c † j ˆ c m ˆ c µ + H . c . (cid:17) , (2b)ˆ H cont = (cid:88) µ (cid:15) µ ˆ c † µ ˆ c µ , (2c)where c † i ( c i ) is the creation (annihilation) operator forthe state ϕ i (the same convention applies to the con-tinuum index µ ), h ij are the one-electron integrals, (cid:15) µ are the continuum single-particle HF energies and v ijmn ( v Aijmµ ) are the two-electron Coulomb integrals respon-sible for intra-molecular (Auger) scatterings. The one-and two-electron integrals are defined as h ij ≡ (cid:90) dxϕ (cid:63)i ( x )[ − ∇ x + V n ( x )] ϕ j ( x ) , (3a) v ijmn ≡ (cid:90) dxdx (cid:48) ϕ (cid:63)i ( x ) ϕ (cid:63)j ( x (cid:48) ) V e ( x, x (cid:48) ) ϕ m ( x (cid:48) ) ϕ n ( x ) , (3b)with V n ( x ) and V e ( x, x (cid:48) ) the nuclear and electron-electron potential. Note that the Auger Coulomb inte-grals v Aijmµ are defined according to Eq. (3b) with n = µ . In Eq. (1) we discard all the off-diagonal contribution h iµ , h µµ (cid:48) as well as all Coulomb integrals with more than oneindex in the continuum. This approximation does not af-fect the physical description of the dynamics as demon-stated by comparisons against full grid calculations inRef. [17]. In fact, in the HF basis both h iµ and h µµ (cid:48) aremuch smaller than h ij and (cid:15) µ whereas Coulomb integralswith two or more indices in the continuum are respon-sible for scattering process that are highly suppressedby phase-space arguments if the photoelectron energy ismuch larger than the kinetic energy of the Auger electron.Henceforth, this condition is assumed to be fulfilled.The explicit simulation of the ionization process witha laser field does not represent a complication for theNEGF method. In fact, the general framework pre-sented in Ref. [17] accounts for the coupling of exter-nal fields with the bound-bound and bound-continuumdipole matrix elements. Instead, the framework dis-cards the coupling of external fields with the continuum-continuum dipole matrix elements and, therefore, light-field streaking experiments relevant to, e.g., attosecondmetrology [23], or multuphoton ionization processes areleft out.In this work we focus on the dynamics induced by thesudden removal of a core electron, thus the ionizationprocess is not simulated. An additional simplificationused for the simulations below (which is however not es-sential for the approach) consists in keeping only inte-grals of the form v Acµv v , where c labels the state of thesuddenly created core hole, v and v label two valencestates and µ an arbitrary continuum state. We also ob-serve that the HF wavefunctions are real since the Hamil-tonian is invariant under time-reversal. This implies thatthe Coulomb integrals have the following symmetries v ijmn = v jinm = v imjn = v njmi (4)and the like with n → µ . A. NEGF equations
The derivation of the NEGF equations within the socalled Generalized Kadanoff-Baym Ansatz [24] (GKBA)has been presented elsewhere [17]; here we only describethe structure of these equations without entering into thecomplex mathematical and numerical details.Let ρ be the one-particle reduced density matrix in thebound sector and f µ be the occupations of the continuumstates. Then the NEGF equations read ˙ ρ = − i [ h HF [ ρ ] , ρ ] − I [ ρ, f ] − I † [ ρ, f ]˙ f µ = −J µ [ ρ, f ] − J ∗ µ [ ρ, f ] , (5)where the single-particle HF Hamiltonian is defined ac-cording to h HF ,ij = h ij + (cid:88) mn ( v imnj − v imjn ) ρ nm . (6)The matrix I and the scalar J µ at time t are explicitfunctionals of ρ and f at all previous times. They areevaluated using the so-called second-Born (2B) approxi-mation which has been shown to contain the fundamentalscattering of the Auger process [25, 26]. The dependenceon ρ and f occurs through the lesser and greater GKBAGreen’s functions [24] G ≶ ( t, ¯ t ) = ∓ (cid:104) G R ( t, t (cid:48) ) ρ ≶ ( t (cid:48) ) − ρ ≶ ( t ) G A ( t, t (cid:48) ) (cid:105) , (7)and the like for G ≶ with indices in the continuum. Here,the retarded ( G R ) and advanced ( G A ) Green’s functionsare evaluated in the HF approximation (and hence theyare functionals of ρ and f too). The functional I ( J µ )is linear in G ≶ with indices in the continuum and quar-tic (cubic) in G ≶ with indices in the bound sector. Theircalculation requires to perform an integral from some ini-tial time, say t = 0, up to time t . The implementation ofEqs. (5) does therefore scale quadratically with the num-ber of time steps. Notice that by setting I = J µ = 0is equivalent to perform time-dependent HF simulations.Like the adiabatic approximations in TDDFT, HF is lo-cal in time and therefore it is unable to describe Augerdecays.The scaling of the calculation of I and J µ with the number of basis functions ismax[( N bound ) p , ( N bound ) q N cont ], where N bound isthe number of bound states, N cont the number of con-tinuum states and the exponents 3 ≤ p ≤
5, 2 ≤ q ≤ I and J µ are implementedin the CHEERS code [27] which, for J µ = 0, has beenrecently used to study the charge transfer dynamicsin a donor-C model dyad [28] and the ultrafastcharge migration in the phenylalanine aminoacid up to40 fs [29]. Since the calculation of J µ is not heavierthan the calculation of I , the NEGF approach can beused to study time-dependent Auger processes drivenby XUV or X-ray pulses in molecules with up to tens ofatoms. B. CI calculation
Let us consider the simplest possible case of a systemwith one occupied core state, one occupied valence stateand a continuum of empty states. We are interested indescribing the evolution of the system starting from theinitial state | φ x (cid:105) = c † c ↑ c † v ↓ c † v ↑ | (cid:105) , (8)representing a core-hole of down spin. The evolution op-erator defined by the Hamiltonian in Eq. (1) mixes | φ x (cid:105) with (we recall that only Coulomb integrals of the form v cµvv and the like related by symmetries are nonvanish- ing, see Section II) | φ g (cid:105) = c † c ↑ c † c ↓ c † v ↑ | (cid:105) , (9a) | φ µ (cid:105) = c † c ↑ c † c ↓ c † µ ↑ | (cid:105) , (9b)where | φ g (cid:105) is the “intermediate” state with the filled core,i.e., the ground state of the parent cation, and | φ µ (cid:105) is thestate describing the dication with an Auger electron inthe continuum state µ . Carrying out the calculationsit is easy to show that these states are coupled by theHamitonian as followsˆ H eq | φ x (cid:105) = E x | φ x (cid:105) + T | φ g (cid:105) + (cid:88) µ V µ | φ µ (cid:105) , (10a)ˆ H eq | φ g (cid:105) = E g | φ g (cid:105) + T | φ x (cid:105) , (10b)ˆ H eq | φ µ (cid:105) = E µ | φ x (cid:105) + V µ | φ x (cid:105) , (10c)where the energies E x , E g , E µ , T and V µ are given by E x = h cc + 2 h vv + 2 v cvvc + v vvvv − v cvcv , (11a) E g = 2 h cc + h vv + 2 v cvvc + v cccc − v cvcv , (11b) E µ = 2 h cc + (cid:15) µ + v cccc , (11c) T = h cv + v ccvc + v cvvv , (11d) V µ = v vvcµ . (11e)The simplification brought about by the HF basis isnow evident. The HF Hamiltonian h HF ,ij = h ij + (cid:80) occ k (2 v ikkj − v ikjk ) is diagonal in the HF basis, therefore0 = h HF ,cv = h cv + (2 v cccv − v ccvc ) + (2 v cvvv − v cvvv )= h cv + v cccv + v cvvv ≡ T. (12)Thus the “intermediate” state | φ g (cid:105) decouples from thedynamics.We write the three-body wave function at time t as | ψ ( t ) (cid:105) = a x ( t ) | φ x (cid:105) + (cid:88) µ a µ ( t ) | φ µ (cid:105) , (13)with initial condition | ψ (0) (cid:105) = | φ x (cid:105) . Taking into ac-count Eqs. (10), the time-dependent Schr¨odinger equa-tion yields a set of coupled equations for the coefficientsof the expansion (cid:40) i ˙ a x ( t ) = E x a x ( t ) + (cid:80) µ V µ a µ ( t ) i ˙ a µ ( t ) = V µ a x ( t ) + E µ a µ ( t ) (14)to be solved with boundary conditions a x (0) = 1 and a k (0) = 0.From the definitions in Eqs. (11) it follows that for thecontinuum three-body state to have the same energy ofthe initial state, i.e., E µ = E x , the energy (cid:15) µ of the Augerelectron has to be (cid:15) µ ≡ (cid:15) CIAuger = 2 (cid:15) HF v − (cid:15) HF c − v vvvv , (15)where (cid:15) HF c = h cc + v cccc + 2 v cvvc − v cvcv , (16a) (cid:15) HF v = h vv + v vvvv + 2 v vccv + v vcvc , (16b)are the core and valence HF energies, respectively. It istherefore reasonable to expect a peak in the continuumoccupations f µ for the µ corresponding to an energy closeto the value in Eq. (15).In the next Section we solve numerically Eqs. (14).However, in order to get some physical insight into the so-lution we here make a “wide-band-limit approximation”(WBLA) and carry on the analytic treatment a bit fur-ther. Integrating the second equation (14) we have a µ ( t ) = − i (cid:90) t dt (cid:48) e − iE µ ( t − t (cid:48) ) V µ a x ( t (cid:48) ) , (17)which correctly satisfies the boundary conditions a µ (0) =0. Substituing this result into the first equation (14) weget i ˙ a x ( t ) = E x a x ( t ) + (cid:90) ∞ dt (cid:48) K ( t − t (cid:48) ) a x ( t (cid:48) ) , (18)where K ( t − t (cid:48) ) = − iθ ( t − t (cid:48) ) (cid:88) µ V µ e − iE µ ( t − t (cid:48) ) ≡ (cid:90) dω π e − iω ( t − t (cid:48) ) (cid:20) Λ( ω ) − i ω ) (cid:21) , (19)and Λ( ω ) − i ω ) = (cid:88) µ V µ ω − E µ + i + . (20)The real function Λ is connected to Γ through a Hilberttransform, i.e., Λ( ω ) = (cid:90) dω (cid:48) π Γ( ω (cid:48) ) ω − ω (cid:48) , (21)and from Eq. (20) it is easy to show thatΓ( ω ) = 2 π (cid:88) µ V µ δ ( ω − E µ ) . (22)For systems in a box of lenght L the continuum wave-functions are proportional to 1 / √ L and hence V µ scaleslike 1 /L , see definition in Eq. (3b). In the limit L → ∞ the discrete sum in Eq. (22) becomes an integral andΓ( ω ) becomes a smooth function of ω . Assuming that E x is a few times larger than Γ( E x ) and that Γ( ω ) is aslowly varying function for ω (cid:39) E x , we can then neglectthe frequency dependence in Γ:Γ( ω ) (cid:39) Γ( E x ) ≡ γ, (23)which implies, see Eq. (21), that we can approximate Λ (cid:39)
0, see Eq. (21). This is the so called WBLA, according to which the kernel K in Eq. (19) can be approximatedas K ( t − t (cid:48) ) = − i γ δ ( t − t (cid:48) ) . (24)Substituing this result into Eq. (18) and then usingEq. (17) it is straighforward to find the following ana-lytic solution a x ( t ) = e − iE x t − γ t , (25a) a µ ( t ) = − V µ e − i ( E x − i γ ) t − e − iE µ t E µ − E x + i γ . (25b)From Eqs. (25) we infer that the occupation of the con-tinuum states is peaked at E µ = E x or, equivalently,at (cid:15) µ = (cid:15) CIAuger , in agreement with the discussion aboveEq. (15). We emphasize that this conclusion is basedon the WBLA. The exact solution contains a small cor-rection which is proportional to the Hilbert transform ofΓ( ω ) at frequency ω (cid:39) E x . C. Comparing NEGF with CI
In the NEGF approach at the 2B level of approxima-tion two holes, in addition to feel an average (HF) po-tential generated by all other electrons, scatter directlyonce. However, for a strong enough repulsion v vvvv itis necessary to include multiple valence-valence scatter-ings to predict the correct energy of the Auger electron.In fact, the red shift v vvvv in Eq. (15) can be capturedonly by summing multiple scatterings to infinite order(T-matrix approximation) [30, 31]. Since the 2B approx-imation includes just a single scattering, the predictedAuger energy is (cid:15) = 2 (cid:15) HF v − (cid:15) HF c . (26)In 3D molecules the neglect of v vvvv has only a minorimpact on the internal (bound-electrons) dynamics since v vvvv is typically less than 1 eV and Γ( ω ) varies ratherslowly on this energy scales. In this work, however,we are also interested in the description of the Augerwavepacket. Taking into account that the repulsion v vvvv in 1D systems is larger than in 3D ones, a sizable differ-ence between the CI and 2B results has to be expected.To demonstrate that such a difference does not affectthe overall physical picture nor the details of the Augerwavepacket but only the speed at which the Auger elec-tron is expelled, we isolate the effects of multiple valence-valence scatterings from the CI formulation. Let us ex-press the energy E x defined in Eq. (16a) in terms of HFenergies E x = 2 (cid:15) HF v + (cid:15) HF c − v cccc − v vccv + 2 v vcvc − v vvvv . (27)The HF energy (cid:15) HF v is blue shifted by v vvvv , see Eq. (16),an effect captured by the 2B approximation. The effectof multiple scatterings manifests in the red shift givenby the last term of Eq. (27). In the next Section weshow that solving Eqs. (14) using for E x the value inEq. (27) with v vvvv = 0 one recovers the NEGF results(notice that this is not equivalent to set v vvvv = 0 in theHamiltonian since this Coulomb integral renormalizes theHF energy (cid:15) HF v ). We will refer to this CI approximationas CI2B. III. RESULTS
We consider a one-dimensional (1D) atom with softCoulomb interactions. This particular example is a se-vere test for the NEGF method since the continuum spec-trum has a strong frequency dependence and the valence-valence repulsion energy is of the same order of magni-tude of the Auger energy.The 1D atom is defined on the points x n = na of a1D grid, with | n | ≤ N grid /
2. In our model the Coulombinteraction is different from zero only in a box of radius R centered around the nucleus. The one-body Hamiltonianon the grid reads h ( x n , x m ) = δ n,m [2 κ + V n ( x n )] − δ | n − m | , κ (28)with V n ( x ) = U en / √ x + a the nuclear potential and κ the hopping integral between neighbouring points. Elec-trons interact through v ( x, x (cid:48) ) = ZU ee / (cid:112) ( x − x (cid:48) ) + a .We analyze the system using N grid = 1601 grid-pointsand choose the parameters according to (atomic unitsare used throughout): a = 0 . κ = 2, Z = 4, U en = 2, U ee = U en / R = 10 a . With four electrons the HFspectrum has five bound states (per spin), the lowesttwo of which are occupied. The energies of the occu-pied levels are (cid:15) HF c = − .
33 and (cid:15) HF v = − .
65 for thecore and valence respectively, yielding a 2B Auger en-ergy (cid:15) = 1 .
02. We work in the sudden creation ap-proximation, according to which the system is perturbedby suddenly removing a core electron. In the NEGF ap-proach this is simulated by subtracting to the equilibriumdensity matrix ρ eq ij an infinitesimal amount of charge fromthe core, hence ρ ij (0) = ρ eq ij − δ ic δ jc n h . In the results be-low the hole density n h = 0 . n c ( t ) is pre-dicted in both CI and 2B calculations to have the follow-ing behavior n c ( t ) = 1 − n h e − Γ t , where n h is the corehole created and Γ is the inverse lifetime of the Augerdecay. Due to the neglect of multiple scatterings, theAuger decay is faster in 2B and the corresponding Γ isoverestimated by a factor 1.5. As already pointed out,this discrepancy is expected to be much smaller in 3Dmolecules since the valence-valence repulsion is not aslarge. . . . . . t = 50 a.u.t = 100 a.u.t = 150 a.u.t = 200 a.u.t = 250 a.u. . . . . n A u g e r x . . . . FIG. 1: Snapshots of the density of the Auger wavepacketleaving the atom (nucleus is situated in x = 0) calculatedusing CI (top), NEGF approach (middle) and CI2B (bottom).The vertical axes have been rescaled by a factor 10 for allcurves. In Fig. 1 we display snapshots at different times of thereal-space density of the Auger wavepacket as obtainedby performing CI (top), NEGF (middle) and CI2B cal-culations (bottom). The results in the NEGF approachclosely resemble the ones in the CI2B treatment, in agree-ment with the discussion in Section II C. The CI calcula-tion, as expected, shows a slower wavepacket. However,the overall shape, i.e., asymmetric packet with superim-posed accumulating ripples on the tail, is common to allmethods. We mention that the amplitude of the ripplesas well as the wavefront of the Auger wavepacket changeif, instead of the sudden creation of a core-hole, we wouldhave simulated the ionization process using an externallaser pulse. In fact, these features are not universal anddepend on the intensity and duration of the perturbingfield [17]. On the other hand, the time T r elapsing be-tween two consecutive maxima at any fixed position isan intrinsic feature of the Auger decay, following the law T r = 2 π(cid:15) Auger . (29)In the top panel of Fig. 2 we show the time-dependentdensity n Auger ( x , t ) of the Auger wavepacket at a certaindistance x from the nucleus. The densities exhibit rip-ples of different frequency since the energy of the Augerelectron is different in CI, NEGF and CI2B. The smalldiscrepancy between NEGF and CI2B is due to the factthat the solution in Eqs. (25) is valid only in the WBLA.Taking into account the frequency dependence of Γ onewould find a small correction to E µ − E x proportional tothe Hilbert transform of Γ. From the top panel of Fig. 2 t . . . . . n x A u g e r NEGFCICI2B0 5 10 15 20 25
Period Number T r × (cid:15) A u g e r (cid:15) Auger ’ .
02 a.u. (cid:15)
Auger ’ .
76 a.u. (cid:15)
Auger ’ .
66 a.u. (cid:15)
CIAuger ’ .
51 a.u. (cid:15)
CI2BAuger ’ .
02 a.u.
FIG. 2: The top panel shows the time-dependent density ofthe Auger wavepacket at a fixed distance x = 30 from thenucleus for NEGF, CI and CI2B. The bottom panel displaysthe period of the ripples at x versus the number of elapsingperiods for the three calculations of the top panel and for twomore NEGF calculations, see main text. we see that this correction is rather small and thereforethe WBLA is an excellent approximation in this case.In the bottom panel of Fig. 2 we show the valueof the time T r elapsing between two consecutive max-ima of the wavepacket versus the number of maxima(counted starting from the left most maximum in thetop panel). In the figure T r is rescaled by the Augerenergy. In all cases, after a short transient phase, T r attains the value 2 π . In addition to the values of T r corresponding to the three curves of the top panel, inthe bottom panel we also report the trend of T r cal-culated in Ref. [17] for two more NEGF simulations.More specifically, we considered two different combina-tions of range and strengths of the Coulomb interactions( R, U en , U ee ) = (100 a, . , . , (10 a, . , . (cid:15) = 1 . , .
66 re-spectively. As we can see, the quantity T r × (cid:15) Auger re-mains independent of the system.Finally, in Fig. 3 we display the snapshots of the time-dependent occupations f µ ( t ) of the continuum states ϕ µ .After the creation of the core-hole, occurring at t = 0,the continuum states start to get populated and, as timepasses, gradually get peaked around the Auger energy (cid:15) CIAuger (cid:39) .
51 for the CI calculation and (cid:15) (cid:39) . . . . . . . . . (cid:15) µ . . . . . f µ
50 a.u.150 a.u.250 a.u.
FIG. 3: Snapshots of the occupations f µ of the continuumstates versus their energy (cid:15) µ for CI (blue), CI2B (green) andNEGF (orange). The times of the snapshots (from light todark) are given by the color bars. IV. CONCLUSIONS
To summarize, we have benchmarked a recently pro-posed NEGF approach [17] against configuration inter-action calculations in a simple 1D model atom. With theexception of the quantitative discrepancies due to the ne-glect of multiple valence-valence scatterings, good agree-ment is found for the qualitative features of the Augerprocess. In fact, NEGF correctly predicts an exponentiallaw for the core-hole refilling and an asymmetric shape ofthe Auger wavepacket characterized by a long tail withsuperimposed ripples of period T r = 2 π/(cid:15) Auger . Thequantitative difference is only related to the red shift ofthe energy of the Auger electron, as demonstrated by theagreement between NEGF and CI2B results. We pointout that for the systems that we are interested to studyin the future, i.e., organic molecules and molecules of bio-logical interest, the valence-valence repulsion is less than1 eV; therefore the neglect of multiple scatterings for thedescription of the internal dynamics is expected to be lessrelevant.The NEGF equations (5) are equations of motion forthe one-particle density matrix in the bound sector andfor the occupations of the continuum states, not for theGreen’s function. Both quantities are one-time functionslike the charge density of TDDFT n ( r , t ). In particular,in a real space basis ρ ( r , r , t ) = n ( r , t ). Given the tightrelation between ρ and n it would be interesting to usethe explicit form of the functionals I [ ρ, f ] and J µ [ ρ, f ] asa guide to generate approximate xc TDDFT potentialswith memory. One possibility would be to combine thelinearized Sham-Schl¨uter equation [32, 33] with NEGFusing the Generalized Kadanoff-Baym Ansatz [24]. Akcknowledgements
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