First-Principles Nonequilibrium Green's Function Approach to Ultrafast

Subscriber access provided by GUILFORD COLLEGE

Spectroscopy and Excited States

First-principles nonequilibrium Green's function approach to ultrafast charge migration in glycine Enrico Perfetto, Davide Sangalli, Maurizia Palummo, Andrea Marini, and Gianluca Stefanucci J. Chem. Theory Comput., Just Accepted Manuscript • DOI: 10.1021/acs.jctc.9b00170 • Publication Date (Web): 17 Jul 2019 Downloaded from pubs.acs.org on July 21, 2019

Just Accepted “Just Accepted” manuscripts have been peer-reviewed and accepted for publication. They are posted online prior to technical editing, formatting for publication and author proofing. The American Chemical Society provides “Just Accepted” as a service to the research community to expedite the dissemination of scientific material as soon as possible after acceptance. “Just Accepted” manuscripts appear in full in PDF format accompanied by an HTML abstract. “Just Accepted” manuscripts have been fully peer reviewed, but should not be considered the official version of record. They are citable by the Digital Object Identifier (DOI®). “Just Accepted” is an optional service offered to authors. Therefore, the “Just Accepted” Web site may not include all articles that will be published in the journal. After a manuscript is technically edited and formatted, it will be removed from the “Just Accepted” Web site and published as an ASAP article. Note that technical editing may introduce minor changes to the manuscript text and/or graphics which could affect content, and all legal disclaimers and ethical guidelines that apply to the journal pertain. ACS cannot be held responsible for errors or consequences arising from the use of information contained in these “Just Accepted” manuscripts.

is published by the American Chemical Society. 1155 Sixteenth Street N.W., Washington, DC 20036 Published by American Chemical Society. Copyright © American Chemical Society. However, no copyright claim is made to original U.S. Government works, or works produced by employees of any Commonwealth realm Crown government in the course of their duties.

Page 1 of 41 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Journal of Chemical Theory and Computation

First-principles nonequilibrium Green’s function approach to ultrafast charge migration in glycine E. Perfetto,† D. Sangalli,† M. Palummo,‡ A. Marini,† and G. Stefanucci∗,‡ †CNR-ISM, Division of Ultrafast Processes in Materials (FLASHit), Area della Ricerca di Roma 1, Via Salaria Km 29.3, I-00016 Monterotondo Scalo, Italy ‡Dipartimento di Fisica, Università di Roma Tor Vergata, Via della Ricerca Scientifica 1, 00133 Rome, Italy ΠINFN, Sezione di Roma Tor Vergata, Via della Ricerca Scientifica 1, 00133 Rome, Italy E-mail: [email protected]

Abstract We investigate the photo-induced ultrafast charge migration phenomenon in the glycine molecule using a recently proposed nonequilibrium Green’s functions (NEGF) approach. We first consider the dynamics resulting from the sudden removal of an electron in the valence shells, finding a satisfactory agreement with available data. Then we explicitly simulate the laser-induced photo-ionization process and study the evolution of the system after the pulse. We disentangle polarization and correlation effects in the electron dynamics and assign the main frequencies to specific elements of the reduced one-particle density matrix. We show that electronic correlations renormalize the bare frequencies, redistribute the spectral weights and give rise to new spectral features.

1

ACS Paragon Plus Environment

Journal of Chemical Theory and Computation 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

1

Introduction

The ultrafast charge migration (UCM) process in photo-ionized molecules is attracting increasing interest both theoretically and experimentally. 1–4 Progresses in attosecond laser technology 5 have opened the way to the imaging of the electron dynamics in molecules and molecular complexes, providing a new investigation tool for unraveling the mechanisms at the basis of photo-protection and photo-damage. Naturally, a special attention has been given to biological molecules considering their relevance in genetics, diagnostics, radiotherapy, etc. On the theoretical side the description of UCM is an interdisciplinary challenge at the border between physics and chemistry. Any predictive approach should account for the interaction of the electrons with the external laser pulse (EL), the Coulomb repulsion between electrons (EE), and the electron-nuclei interaction (EN). The EL interaction implies that the continuum (photoelectron) states have to be included in the single-particle basis; this allows for predicting the nonstationary configuration of the parent cationic molecule emerging after the action of the pulse. The EE repulsion requires to develop accurate and feasible approximation schemes to treat nonequilibrium electronic correlations. Equally challenging is dealing with EN interactions, although this is expected to play a role only after a few femtoseconds. 6–8 Notable theoretical and computational advances have been made over the last few years. In Ref. 9,10 the EE interaction has been treated within Time-Dependent Density Functional Theory (TDDFT) at the level of the Adiabatic Local Density Approximation (ALDA) and the EN interaction at the level of the Ehrenfest approximation. The explicit simulation of the ionization process through the inclusion of photoelectron states has been performed in Ref. 11 using TDDFT–ALDA for the EE interaction but keeping the nuclei fixed. The method has been successively improved to account for the EN interaction in the Ehrenfest approximation. 4,6 Discarding EL and EN interactions electronic correlations have been included using the so called Algebraic Diagrammatic Construction (ADC), 12,13 which allows for an approximate propagation of the correlated many-electron wavefunction. 14,15 In the 2


Page 2 of 41

Page 3 of 41 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


context of UCM induced by the ionization of a core electron, the ADC and TDDFT–ALDA methods have been shown to agree to a large extent 16,17 (one of the differences being the steady amount of hole density on the nitroso carbon). Recently, an ADC based approach to simulate both EL and EE interactions has been proposed. 18 The Nonequilibrium Green’s Functions (NEGF) 19 method is an alternative strategy to study the ultrafast electron dynamics of single-photon ionized atoms 20 and molecules. 21 The main advantage of the NEGF approach is the possibility of simulating the photoionization process while including static and dynamical EE correlation effects. In the NEGF language dynamical correlations refer to those interaction effects that cannot be captured by a time-local potential, like the Hartree-Fock potential or the adiabatic exchange-correlation potential in TDDFT, and therefore require a frequency-dependent self-energy. Dynamical correlations are essential to describe, e.g., the retarded EE interaction due to screening, double (or multiple) excitations, 20,22,23 shake-up effects or Auger decays. 24–26 In Ref. 21 we have shown their importance for the description of UCM in the phenylalanine aminoacid. 11 In this work we present an extensive numerical analysis of UCM in the smallest aminoacid, i.e., the glycine molecule. This molecule has been widely studied with different ab-initio methods and it is therefore an ideal test-bed for NEGF. The first study of UCM induced by the sudden ionization from a valence HF orbital has been presented in Ref. 15 using the ADC method. It was found that for outer-valence ionization the time evolution is well described by a simple 1h (one-hole) mixing whereas for inner-valence ionization both 1h and 2h–1p (two-holes and one-particle) mixing come into play. Interestingly, the UCM induced by outer-valence ionization agrees with results from TDDFT-ALDA, 27 pointing to the fact that the inclusion of static correlations is sufficient. On the contrary, the 2h–1p mixing caused by the inner-valence ionization is necessary to describe shake-up processes where an electron in an initially occupied state fills the hole and transfers its energy to another electron which is promoted to an initially unoccupied state. The inclusion of shake-up processes calls for memory-dependent functional in the TDDFT framework 22 or for dynamical correlation

3



self-energies in the NEGF framework. 28 In addition to static and dynamical correlation-induced UCM a laser pulse causes also a polarization induced UCM. This mechanism is absent in the sudden-ionization since the instantaneous removal of an electron from an occupied HF orbital leaves the parent cation unpolarized. However, a physical pulse generates a coherent superposition of cationic states and hence a time-dependent polarization. Clearly, the EL interaction should be explicitly taken into account for the description of this effect. Of course, the polarization-induced UCM is present already at the HF level. We will use the NEGF approach to disentangle its contribution from the UCM induced by correlations. The paper is organized as follows. In Section 2 we characterize the glycine molecule at equilibrium. We compare the HF spectrum generated using a truncated Kohn-Sham (KS) basis with the HF spectrum of an all-electron calculation in GTO-pVTZ basis and of a pseudopotential calculation in plane-waves basis, finding a discrepancy of less than 4%. We also calculate the dipole-dipole response function in the time-dependent HF approximations and show that the results obtained agree with the solution of the (unscreened) Bethe-Salpeter equation. In Section 3 we investigate the glycine molecule out of equilibrium. We first benchmark the NEGF UCM against available results for the sudden ionization. Then, we present a detailed analysis of UCM as generated by an attosecond XUV pulse and highlight the different physical processes involved. Summary and conclusions are drawn in Section 4.

2

Glycine at equilibrium

We examine the most abundant conformer of the aminoacid glycine. Its molecular structure is shown in Fig. 1 and consists of a central CH unit linked to an amino group (NH2), a carboxylic group (COOH) and a single hydrogen atom (the R-group specific to each amino acid). The molecule has a mirror plane perpendicular to the y direction (see Fig. 1) and it is therefore invariant under the two symmetry operations of the Cs point group. In accordance

4


Page 4 of 41

Page 5 of 41 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


z

y

x

Figure 1: Molecular structure of the most abundant conformer of the glycine molecule. Black spheres represent carbon atoms; gray spheres, hydrogen atoms; blue sphere, nitrogen; and red spheres, oxygen. with standard chemistry nomenclature, we label the one-particle orbitals as a0 and a00 where the prime (double-prime) superscript denotes even (odd) parity with respect to the mirror plane. The starting point of the NEGF approach is a suitable one-particle basis. We choose the Kohn-Sham (KS) orbitals of Density Functional Theory (DFT) with PBE exchangecorrelation potential. The DFT calculation has been performed using the Quantum Espresso package 29 with the glycine placed in a cube of side 24 ˚ A. The energy cutoff is set at 50 Ry. Core electrons have been treated with norm-conserving Troullier-Martins pseudopotentials. 30 A total of 18 localized orbitals per spin has been found. The resulting KS spectrum {KS i } is shown in the left panel of Fig. 2. The charge neutral molecule has the lowest fifteen KS-states doubly occupied with the KS-HOMO located 5.9 eV below the onset of the continuum (gray) which we set at zero. The remaining three KS-states above the KS-HOMO have a binding energy lesser than 1 eV. All other KS states (gray) are delocalized and have positive energy.

5



2.1

Page 6 of 41

Hartree-Fock quasi-particle spectrum

In the NEGF theory both static and dynamical correlations are built on top of the HF approximation. Therefore, we preliminary solve the self-consistent HF problem. The equilibrium HF hamiltonian is a functional of the one-particle density matrix ρ and reads

hHF [ρ] = hKS − VHxc + VHF [ρ] ,

(1)

while VHxc is the DFT Hartree-xc potential. The last where, in the KS basis, hKS,ij = δij KS i term in Eq. (1) VHF,ij [ρ] ≡

X mn

(2vimnj − vimjn )ρnm

(2)

is the HF potential written in terms of the Coulomb integrals

vimnj ≡

Z

0 KS 0 KS ϕKS∗ (r)ϕKS∗ i m (r )ϕm (r )ϕj (r) drdr , |r − r0 | 0

(3)

with {ϕKS i } the KS wavefunctions. Both the Coulomb integrals and the matrix elements of VHxc have been extracted using the Yambo code. 31 At self-consistency ρ equals the HF density matrix and satisfies ρ, hHF [ρ] = 0 .

(4)

The HF spectrum {HF i } corresponds to the eigenvalues of hHF [ρ] evaluated at the HF density matrix. In Fig. 3 we report the change of the HF energies of the 18 localized HF states versus the number n of KS states included in the calculation. They have been grouped into valence states (the lowest occupied ones) and resonances (the unoccupied but still localized ones). The HF energies converge already for n = 30 (hence ρ is a 30 × 30 matrix). In the following we refer to the space spanned by these thirty KS states as the active space KS(30). Higher energy states are treated as noninteracting, i.e., we set vimnj = 0 if one or more indices do not belong to KS(30). This prevents the possibility of describing Auger decays, which 6


Page 7 of 41

10

10 continuum resonances

0

HOMO

-10

15

-10

-20

-20

-30

-30

-40

1

HF spectrum (eV)

0 KS spectrum (eV)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


-40

Figure 2: KS spectrum (left) and HF spectrum (right) composed by 15 valence states (colored), 3 empty localized states (magenta), and the continuum delocalized states (gray), represented only up to 10 eV. are relevant recombination channels in a core-level post-ionization dynamics. XUV pulses, however, ionize valence electrons and the filling up of the valence hole (via Auger decay) takes place on time scales much longer than our simulation time. The converged HF spectrum is shown in the right panel of Fig. 2. A comparison with the KS spectrum (left) shows that all occupied HF states shift to lower energies and, in particular, the HF-HOMO increases its binding energy of about 5 eV. On the contrary, the three unoccupied KS states are pushed to higher energy and become resonances (fully embedded in the continuum). In fact, this is a typical trend which can be understood in terms of the different physical meaning of the unoccupied KS and HF orbital energies. In KS theory unoccupied orbital energies are good approximations to neutral excitation energies (measured with respect to a given occupied orbital energy) whereas in HF theory they are good approximations to affinity levels (addition energies) and they are often positive. In Section 3 we show that the unoccupied HF resonances play a crucial role in the UCM through shake-up processes. 7



Page 8 of 41

HF HF Figure 3: Energy variation ∆HF µ = µ (n) − µ (30) of the HF eigenvalues as function of the number n of KS states included in the self-consistent solution of the HF problem.

In Table 1 we report the values of the HF energies in Fig. 2 (fourth column – self-consistent calculation in the active space KS(30)) as well as the HF spectrum of an all-electron calculation in GTO-pVTZ basis 32 (second column) and of a pseudopotential calculation using the plane-wave basis Quantum Espresso package 29 (third column). For all eigenvalues the discrepancy is below 4%.

2.2

Response functions for optical properties

The Yambo code 31 has also been used to solve the Bethe-Salpeter equation (BSE) in the KS(30) basis "

#

Lpq (ω) = lpq (ω) δpn δqm + i

X

mn

Kpq Lrs (ω) , rs

mn

(5)

rs

where the kernel Kpq ≡ vpsqr − 2vpsrq and rs

lpq (ω) = i

fp − fq ω − ˜p + ˜q + iη

8


(6)

Page 9 of 41 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


Table 1: Energies of the HF occupied orbitals (labelled in the first column) as obtained with an all-electron calculation in GTO-pVTZ basis (second column), pseudopotential calculation in plane-wave (pw) basis (third column), and self-consistent solution of Eq. (4) in the active space KS(30) (fourth column). orbital label c1 (1a0 ) c2 (2a0 ) c3 (3a0 ) c4 (4a0 ) c5 (5a0 ) v1 (6a0 ) v2 (7a0 ) v3 (8a0 ) v4 (9a0 ) v5 (10a0 ) v6 (11a0 ) v7 (12a0 ) v8 (1a00 ) v9 (2a00 ) v10 (13a0 ) v11 (14a0 ) v12 (3a00 ) v13 (4a00 ) v14 (15a0 ) v15 (16a0 )

HF eigenvalues (eV) GTO-pVTZ pw basis basis -560.77 – -559.19 – -423.03 – -309.80 – -307.10 – -39.73 -40.38 -37.00 -37.72 -32.30 -32.75 -26.51 -26.64 -22.18 -22.16 -19.15 -19.37 -18.84 -18.87 -18.79 -18.80 -17.03 -16.98 -16.04 -16.38 -15.86 -15.89 -14.47 -14.47 -13.21 -13.23 -12.61 -12.60 -11.08 -11.19

9


KS(30) basis – – – – – -40.29 -37.60 -32.67 -26.51 -22.02 -19.15 -18.74 -18.73 -16.80 -16.09 -15.62 -14.23 -12.93 -12.17 -10.86


1.0 Im χ(ω) (arb. units)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

0.8

Page 10 of 41

BSE-Yambo TDHF-CHEERS@Yambo

0.6 0.4 0.2 0.0

10

15 ω (eV)

20

25

Figure 4: Dipole-dipole response function from the Yambo solution of the BSE and from the CHEERS@Yambo time-dependent HF simulation (full line). is the retarded and bare particle-hole propagator with fp the HF occupations and ˜p = KS p − VHxc,pp + VHF,pp an approximation to the HF single-particle energies. The response function is related to L through

χα (ω) =

X

dα,ij Lij (ω)dα,mn ,

(7)

mn

ijmn

where the quantities dα,ij are the KS matrix elements of the dipole operator along direction α

dα,ij ≡

Z

dr ϕKS∗ (r)(eα · r)ϕKS i j (r),

(8)

with eα the unit vector along α. In Fig. 4 we display the imaginary part of χy (ω) (dashed line). The response function χα (ω) with L the solution of Eq. (5) in which ˜i is replaced by the true HF single-particle energies HF is identical to the Fourier transform of the timei P dependent dipole moment dα (t) = ij dα,ij ρji (t) induced by an infinitesimal electric field E(t) ∝ δ(t) directed along α, provided that ρ(t) is calculated in the time-dependent HF approximation 19 (and all quantities are expressed in the HF basis). In Fig. 4 we show the 10


Page 11 of 41 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


results of a time-dependent HF simulation (in the same subspace as the BSE) performed using the CHEERS@Yambo code, 31,33 see next section for details. The agreement between the two approaches validates the CHEERS@Yambo time-propagation scheme.

3

Charge migration in ionized glycine

In this Section we investigate the UCM initiated by a ionization event. This is done by propagating the one-particle reduced density matrix ρ(t) within the NEGF formalism 19 in the so called Generalized Kadanoff-Baym Ansatz 34 (GKBA). The method has been published elsewhere; 20,21 here we only discuss the physical content of the equations and defer the reader to the Appendix for the mathematical details. The NEGF equation of motion for ρ reads

−i

d ρij (t) + [hHF (t), ρ(t)]ij = i[Icoll (t) + Iion (t)]ij − H.c. , dt

(9)

where the indices i, j run in the active space KS(30). The matrix hHF (t) is the nonequilibrium HF hamiltonian, obtained from Eq. (1) by evaluating VHF at the instantaneous ρ(t) and by including the light-matter interaction:

hHF (t) = hKS − VHxc + VHF [ρ(t)] + E(t) · d ,

(10)

with E(t) the electric field of the ionizing laser pulse and d the vector of dipole matrices with elements defined in Eq. (8). The quantity Icoll in the r.h.s. of Eq. (15) is the collision integral and accounts for correlation and memory effects through a suitable approximation to the correlation self-energy. In this work we have implemented the second Born (2B) approximation, 35 which has been shown to be sufficiently accurate in finite and not too strongly correlated systems. 28,36–44 It is easy to show that the 2B collision integral evaluated at the HF density matrix vanishes. Therefore the HF ground state is stationary in the absence of external fields; we use this state as initial condition to solve Eq. (15).

11



Page 12 of 41

The quantity Iion ∝ E 2 is the ionization integral and accounts for the dipole coupling of the active states with the remaining delocalized ones. This term is responsible for opening the confined glycine molecule to the surrounding empty space, similarly to the embedding integral in quantum transport. 45,46 The NEGF equation of motion has been solved with the CHEERS@Yambo code 33 in the equilibrium HF basis. Hence, we have first rotated the Coulomb integrals, the Hxc potential and the dipole matrices from the original KS basis to the HF basis using the KS unitary transformation Uij = hϕHF i |ϕj i. The UCM has been monitored by extracting the

real-space electron density n(r, t) according to

n(r, t) =

X

HF ϕHF∗ µ (r) ρµν (t) ϕν (r) .

(11)

µν

where the sum runs over the HF states of the active space.

3.1

Sudden selective ionization

To assess merits and limitations of the NEGF approach we benchmark our results against previously published ADC calculations. 15 The ADC method was used to propagate the many-body state of the glycine molecule with an electron less in, say, the µ ¯-th HF orbital (sudden ionization). Therefore, in this calculation the initial state is a Slater determinant of occupied HF wavefunctions with the µ ¯-th orbital missing. In order to simulate a sudden (and selective) ionization using NEGF we set E(t) = Iion = 0 and solve Eq. (15) with initial condition ρµν (0) = fµ δµν (1 − nh δµ¯µ ), where fµ are the ground-state HF occupations and nh is an infinitesimally small hole-density on the µ ¯-th orbital. The hole-density has to be infinitesimally small to ensure that the reduced one-particle density matrix ρ describes a linear superposition of charge-neutral and cationic states. The UCM induced by the ionization from two different HF orbitals are studied, namely 14a0 and 11a0 . These orbitals allow for illustrating two different aspects of the UCM, which

12


Page 13 of 41 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


ADC

2B

HF

0.7 fs

2.1 fs

3.5 fs

Figure 5: Snapshots of the evolution of the natural orbital 14a0 (ADC) and real-space distribution of the molecular charge following the sudden ionization out of orbital 14a0 (2B and HF) at three different times. The ADC calculation from Ref. 15 (left) are compared with the 2B approximation (center) and HF approximation (right). In ADC blue and red colors refer to the sign of the wavefunction whereas in the 2B and HF simulations they refer to hole excess (defect) with respect to the ground-state.

ADC

2B

HF

1.7 fs

5.1 fs

8.4 fs

Figure 6: Same as Fig. 5 but for the sudden creation of a hole in the HF orbital 11a0 .

13



are related to static and dynamical correlations respectively. Sudden ionization out of 14a0 produces only a mixing of 1h (one-hole) configurations, the initial hole being transferred to 13a0 in about 3 fs. In Fig. 5 we compare the time-dependent electron density n(r, t) of the ADC method with the one obtained by solving Eq. (15) in the 2B and HF approximations. The latter performs rather poorly as n(r, t) remains essentially still (no significant charge migration is observed). In fact, the initial state is a quasi-eigenstate, i.e., a state with a very long life-time, of the HF Hamiltonian since the latter changes only little after removing nh electrons. The inclusion of correlation effects through the 2B self-energy improves considerably the agreement with the ADC results in the whole time-window. It is worth observing that the underlying UCM mechanism is exclusively driven by static correlations since the same dynamics has been captured by TDDFT–ALDA simulations. 27 Sudden ionization out of the deeper valence orbital 11a0 provides a more severe benchmark of the NEGF approach. In Fig. 6 we compare snapshots of the electron density from ADC calculations with those obtained using NEGF in the 2B and HF approximation. As expected, see above, the HF density does not change with time. On the contrary, the 2B dynamics correctly captures the ADC sloshing motion along the molecular backbone, although the corresponding frequency is underestimated. An orbital-resolved analysis reveals that the initial hole in 11a0 migrates toward 12a0 (in approximately 10 fs) and to a lesser extent 4a00 , 15a0 and 16a0 . In addition, a sizable amount of charge is promoted to the highest unoccupied state of symmetry a00 , in agreement with Ref. 15 To describe this shake-up process 2h-1p (twoholes and one-particle) configurations have to be included in the ADC description. 15 In the NEGF framework the same process is driven by dynamical correlations and, therefore, is not within reach of the TDDFT-ALDA dynamics. 27 However, in the 2B approximation the energy of the shake-up states is only partially renormalized due to the lack of direct scattering between them: this is the cause of the slower motion along the molecular backbone previously mentioned.

14


Page 14 of 41

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

0

3.2 sha1_base64="qUg8iXI01zUDf6doeaeQGgCgJ1c=">AAAB7XicbVDLSgNBEOyNrxhfUY9eFoPgKeyKoMegF48RzAOSJcxOepMx81hmZoUQ8g9ePCji1f/x5t84SfagiQUNRVU33V1xypmxQfDtFdbWNza3itulnd29/YPy4VHTqExTbFDFlW7HxCBnEhuWWY7tVCMRMcdWPLqd+a0n1IYp+WDHKUaCDCRLGCXWSc2uEjggvXIlqAZz+KskzEkFctR75a9uX9FMoLSUE2M6YZDaaEK0ZZTjtNTNDKaEjsgAO45KItBEk/m1U//MKX0/UdqVtP5c/T0xIcKYsYhdpyB2aJa9mfif18lsch1NmEwzi5IuFiUZ963yZ6/7faaRWj52hFDN3K0+HRJNqHUBlVwI4fLLq6R5UQ2Danh/Wand5HEU4QRO4RxCuIIa3EEdGkDhEZ7hFd485b14797HorXg5TPH8Afe5w+QMY8b

sha1_base64="qUg8iXI01zUDf6doeaeQGgCgJ1c=">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

sha1_base64="Al0DmXOXZP3T18bUEGkWxVi5am4=">AAAB/HicbVDLSgMxFM3UV62v0S7dBItQN2VGBF0WRXBZwT6gM5RMmmlD8xiSjFCG+ituXCji1g9x59+YaWehrQcCh3Pu5Z6cKGFUG8/7dkpr6xubW+Xtys7u3v6Be3jU0TJVmLSxZFL1IqQJo4K0DTWM9BJFEI8Y6UaTm9zvPhKlqRQPZpqQkKORoDHFyFhp4FYDjswYI5bdzuqB5GSEzgZuzWt4c8BV4hekBgq0Bu5XMJQ45UQYzJDWfd9LTJghZShmZFYJUk0ShCdoRPqWCsSJDrN5+Bk8tcoQxlLZJwycq783MsS1nvLITuZR9bKXi/95/dTEV2FGRZIaIvDiUJwyaCTMm4BDqgg2bGoJworarBCPkULY2L4qtgR/+curpHPe8L2Gf39Ra14XdZTBMTgBdeCDS9AEd6AF2gCDKXgGr+DNeXJenHfnYzFacoqdKvgD5/MHX9KUkg==AAAB/HicbVDLSgMxFM3UV62v0S7dBItQN2VGBF0WRXBZwT6gM5RMmmlD8xiSjFCG+ituXCji1g9x59+YaWehrQcCh3Pu5Z6cKGFUG8/7dkpr6xubW+Xtys7u3v6Be3jU0TJVmLSxZFL1IqQJo4K0DTWM9BJFEI8Y6UaTm9zvPhKlqRQPZpqQkKORoDHFyFhp4FYDjswYI5bdzuqB5GSEzgZuzWt4c8BV4hekBgq0Bu5XMJQ45UQYzJDWfd9LTJghZShmZFYJUk0ShCdoRPqWCsSJDrN5+Bk8tcoQxlLZJwycq783MsS1nvLITuZR9bKXi/95/dTEV2FGRZIaIvDiUJwyaCTMm4BDqgg2bGoJworarBCPkULY2L4qtgR/+curpHPe8L2Gf39Ra14XdZTBMTgBdeCDS9AEd6AF2gCDKXgGr+DNeXJenHfnYzFacoqdKvgD5/MHX9KUkg==AAAB/HicbVDLSgMxFM3UV62v0S7dBItQN2VGBF0WRXBZwT6gM5RMmmlD8xiSjFCG+ituXCji1g9x59+YaWehrQcCh3Pu5Z6cKGFUG8/7dkpr6xubW+Xtys7u3v6Be3jU0TJVmLSxZFL1IqQJo4K0DTWM9BJFEI8Y6UaTm9zvPhKlqRQPZpqQkKORoDHFyFhp4FYDjswYI5bdzuqB5GSEzgZuzWt4c8BV4hekBgq0Bu5XMJQ45UQYzJDWfd9LTJghZShmZFYJUk0ShCdoRPqWCsSJDrN5+Bk8tcoQxlLZJwycq783MsS1nvLITuZR9bKXi/95/dTEV2FGRZIaIvDiUJwyaCTMm4BDqgg2bGoJworarBCPkULY2L4qtgR/+curpHPe8L2Gf39Ra14XdZTBMTgBdeCDS9AEd6AF2gCDKXgGr+DNeXJenHfnYzFacoqdKvgD5/MHX9KUkg== (t, t0 )G< (t0 , t) − Σ< (t, t0 )G> (t0 , t)] ,

(13)

where G≶ is the nonequilibrium (lesser/greater) one-particle Green’s function and Σ≶ is the nonequilibrium (lesser/greater) correlation self-energy. In the NEGF language dynamical correlation effects are carried by the time nonlocal contribution of Σ≶ (t, t0 ). In the second Born approximation the self-energy reads

0 Σ≶ ij (t, t ) =

X nm,pq,sr

0 ≶ 0 ≷ 0 virpn (2vmqsj − vmqjs ) G≶ nm (t, t )Gpq (t, t )Gsr (t , t),

(14)

where the sums run over all indices of the KS(30) space. As the system is weakly correlated and the XUV pulse is weak scattering processes with two or more electrons (either before or after the scattering) in initially unoccupied HF states contribute only at higher order (beyond 2B) in the hole dynamics. We have therefore discarded in Eq. (14) all Coulomb integrals with two or more indices in the initially unoccupied sector. An efficient algorithm to evaluate the multiple sums is described in Ref. 35 The lesser/greater Green’s functions are

22


Page 23 of 41 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


evaluated using the Generalized Kadanoff-Baym Ansatz 34 (GKBA)

G< (t, t0 ) = −GR (t, t0 )ρ(t0 ) + ρ(t)GA (t, t0 ),

(15a)

G> (t, t0 ) = GR (t, t0 )¯ ρ(t0 ) − ρ¯(t)GA (t, t0 ),

(15b)

i h Rt where ρ¯ = 1 − ρ and the propagators GR (t, t0 ) = [GA (t0 , t)]† = −iθ(t − t0 ) T e−i t0 dt¯hHF (t¯) are evaluated in the HF approximation. Since hHF is a functional of ρ, see Eq. (10), the collision integral is, through the GKBA, an explicit functional of ρ. The ionization integral in Eq. (15) has the same mathematical structure as the collision integral but the correlation self-energy is replaced by the ionization self-energy. 20,33 The latter has a vanishing lesser component while the greater component reads

0 Σ> ion,ij (t, t ) = −i

X p

KS (t−t0 )

E(t) · dip e−ip

E(t0 ) · dpj ,

(16)

where the sum runs over all continuum KS states not belonging to KS(30). In our simulation we have included the first 2500 states, the highest having an energy of ∼ 45 eV thus covering the entire XUV photon range ∼ (15, 45) eV.

References (1) Krausz, F.; Ivanov, M. Attosecond physics. Rev. Mod. Phys. 2009, 81, 163–234. (2) Kraus, P. M.; Mignolet, B.; Baykusheva, D.; Rupenyan, A.; Horný, L.; Penka, E. F.; Grassi, G.; Tolstikhin, O. I.; Schneider, J.; Jensen, F.; Madsen, L. B.; Bandrauk, A. D.; Remacle, F.; Wörner, H. J. Measurement and laser control of attosecond charge migration in ionized iodoacetylene. Science 2015, 350, 790–795.

23



(3) Rozzi, C. A.; Troiani, F.; Tavernelli, I. Quantum modeling of ultrafast photoinduced charge separation. J. Phys.: Condens. Matter 2018, 30, 013002. (4) Nisoli, M.; Decleva, P.; Calegari, F.; Palacios, A.; Martín, F. Attosecond Electron Dynamics in Molecules. Chemical Reviews 2017, 117, 10760–10825. (5) Hentschel, M.; Kienberger, R.; Spielmann, C.; Reider, G. A.; Milosevic, N.; Brabec, T.; Corkum, P.; Heinzmann, U.; Drescher, M.; Krausz, F. Attosecond metrology. Nature 2001, 414, 509. (6) Lara-Astiaso, M.; Ayuso, D.; Tavernelli, I.; Decleva, P.; Palacios, A.; Martín, F. Decoherence, control and attosecond probing of XUV-induced charge migration in biomolecules. A theoretical outlook. Faraday discussions 2016, 194, 41–59. (7) Lara-Astiaso, M.; Galli, M.; Trabattoni, A.; Palacios, A.; Ayuso, D.; Frassetto, F.; Poletto, L.; De Camillis, S.; Greenwood, J.; Decleva, P.; Tavernelli, I.; Calegari, F.; Nisoli, M.; Martín, F. Attosecond Pump–Probe Spectroscopy of Charge Dynamics in Tryptophan. The Journal of Physical Chemistry Letters 2018, 9, 4570–4577. (8) Despré, V.; Golubev, N. V.; Kuleff, A. I. Charge Migration in Propiolic Acid: A Full Quantum Dynamical Study. Phys. Rev. Lett. 2018, 121, 203002. (9) Rozzi, C. A.; Falke, S. M.; Spallanzani, N.; Rubio, A.; Molinari, E.; Brida, D.; Maiuri, M.; Cerullo, G.; Schramm, H.; Christoffers, J.; Lienau, C. Quantum coherence controls the charge separation in a prototypical artificial light-harvesting system. Nature communications 2013, 4, 1602. (10) Falke, S. M.; Rozzi, C. A.; Brida, D.; Maiuri, M.; Amato, M.; Sommer, E.; De Sio, A.; Rubio, A.; Cerullo, G.; Molinari, E.; Lienau, C. Coherent ultrafast charge transfer in an organic photovoltaic blend. Science 2014, 344, 1001–1005.

24


Page 24 of 41

Page 25 of 41 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


(11) Calegari, F.; Ayuso, D.; Trabattoni, A.; Belshaw, L.; De Camillis, S.; Anumula, S.; Frassetto, F.; Poletto, L.; Palacios, A.; Decleva, P.; Greenwood, J. B.; Martín, F.; Nisoli, M. Ultrafast electron dynamics in phenylalanine initiated by attosecond pulses. Science 2014, 346, 336–339. (12) Schirmer, J. Beyond the random-phase approximation: A new approximation scheme for the polarization propagator. Phys. Rev. A 1982, 26, 2395–2416. (13) Schirmer, J.; Trofimov, A. B. Intermediate state representation approach to physical properties of electronically excited molecules. The Journal of Chemical Physics 2004, 120, 11449–11464. (14) Breidbach, J.; Cederbaum, L. S. Migration of holes: Formalism, mechanisms, and illustrative applications. The Journal of Chemical Physics 2003, 118, 3983–3996. (15) Kuleff, A. I.; Breidbach, J.; Cederbaum, L. S. Multielectron wave-packet propagation: General theory and application. The Journal of Chemical Physics 2005, 123, 044111. (16) Bruner, A.; Hernandez, S.; Mauger, F.; Abanador, P. M.; LaMaster, D. J.; Gaarde, M. B.; Schafer, K. J.; Lopata, K. Attosecond Charge Migration with TDDFT: Accurate Dynamics from a Well-Defined Initial State. The Journal of Physical Chemistry Letters 2017, 8, 3991–3996. (17) Kuleff, A. I.; Kryzhevoi, N. V.; Pernpointner, M.; Cederbaum, L. S. Core Ionization Initiates Subfemtosecond Charge Migration in the Valence Shell of Molecules. Phys. Rev. Lett. 2016, 117, 093002. (18) Ruberti, M.; Decleva, P.; Averbukh, V. Multi-channel dynamics in high harmonic generation of aligned CO2: ab initio analysis with time-dependent B-spline algebraic diagrammatic construction. Phys. Chem. Chem. Phys. 2018, 20, 8311–8325.

25



(19) Stefanucci, G.; van Leeuwen, R. Nonequilibrium Many-Body Theory of Quantum Systems: A Modern Introduction; Cambridge University Press: Cambridge, 2013. (20) Perfetto, E.; Uimonen, A.-M.; van Leeuwen, R.; Stefanucci, G. First-principles nonequilibrium Green’s-function approach to transient photoabsorption: Application to atoms. Phys. Rev. A 2015, 92, 033419. (21) Perfetto, E.; Sangalli, D.; Marini, A.; Stefanucci, G. Ultrafast Charge Migration in XUV Photoexcited Phenylalanine: A First-Principles Study Based on Real-Time Nonequilibrium GreenÕs Functions. The Journal of Physical Chemistry Letters 2018, 9, 1353– 1358. (22) Maitra, N. T.; Zhang, F.; Cave, R. J.; Burke, K. Double excitations within timedependent density functional theory linear response. The Journal of Chemical Physics 2004, 120, 5932–5937. (23) Kümmel, S.; Kronik, L. Orbital-dependent density functionals: Theory and applications. Rev. Mod. Phys. 2008, 80, 3–60. (24) Almbladh, C.-O.; Morales, A. L.; Grossmann, G. Theory of Auger core-valence-valence processes in simple metals. I. Total yields and core-level lifetime widths. Phys. Rev. B 1989, 39, 3489–3502. (25) Almbladh, C.-O.; Morales, A. L. Theory of Auger core-valence-valence processes in simple metals. II. Dynamical and surface effects on Auger line shapes. Phys. Rev. B 1989, 39, 3503–3516. (26) Covito, F.; Perfetto, E.; Rubio, A.; Stefanucci, G. Real-time dynamics of Auger wave packets and decays in ultrafast charge migration processes. Phys. Rev. A 2018, 97, 061401.

26


Page 26 of 41

Page 27 of 41 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


(27) Ayuso, D.; Palacios, A.; Decleva, P.; Martín, F. Ultrafast charge dynamics in glycine induced by attosecond pulses. Physical Chemistry Chemical Physics 2017, 19, 19767– 19776. (28) Säkkinen, N.; Manninen, M.; van Leeuwen, R. The Kadanoff–Baym approach to double excitations in finite systems. New Journal of Physics 2012, 14, 013032. (29) Giannozzi, P. et al. QUANTUM ESPRESSO: a modular and open-source software project for quantum simulations of materials. Journal of Physics: Condensed Matter 2009, 21, 395502. (30) Troullier, N.; Martins, J. L. Efficient pseudopotentials for plane-wave calculations. Phys. Rev. B 1991, 43, 1993–2006. (31) Marini, A.; Hogan, C.; Grüning, M.; Varsano, D. yambo: An ab initio tool for excited state calculations. Computer Physics Communications 2009, 180, 1392 – 1403. (32) Pavlyukh, Y.; Berakdar, J. Electron repulsion integrals for self-energy calculations. Computer Physics Communications 2013, 184, 387 – 395. (33) Perfetto, E.; Stefanucci, G. CHEERS: a tool for correlated hole-electron evolution from real-time simulations. Journal of Physics: Condensed Matter 2018, 30, 465901. (34) Lipavský, P.; Špička, V.; Velický, B. Generalized Kadanoff-Baym ansatz for deriving quantum transport equations. Phys. Rev. B 1986, 34, 6933–6942. (35) Perfetto, E.; Stefanucci, G. The Dissection Algorithm for the Second-Born Self-Energy. physica status solidi (b) 2019, 256, 1800573. (36) Balzer, K.; Bauch, S.; Bonitz, M. Time-dependent second-order Born calculations for model atoms and molecules in strong laser fields. Phys. Rev. A 2010, 82, 033427.

27



(37) Balzer, K.; Hermanns, S.; Bonitz, M. Electronic double excitations in quantum wells: Solving the two-time Kadanoff-Baym equations. EPL (Europhysics Letters) 2012, 98, 67002. (38) Hermanns, S.; Schlünzen, N.; Bonitz, M. Hubbard nanoclusters far from equilibrium. Phys. Rev. B 2014, 90, 125111. (39) Schlünzen, N.; Bonitz, M. Nonequilibrium Green Functions Approach to Strongly Correlated Fermions in Lattice Systems. Contrib. Plasma Phys. 2016, 56, 5–91. (40) Hopjan, M.; Karlsson, D.; Ydman, S.; Verdozzi, C.; Almbladh, C.-O. Merging Features from Green’s Functions and Time Dependent Density Functional Theory: A Route to the Description of Correlated Materials out of Equilibrium? Phys. Rev. Lett. 2016, 116, 236402. (41) Lev, Y. B.; Reichman, D. R. Slow dynamics in a two-dimensional Anderson-Hubbard model. EPL (Europhysics Letters) 2016, 113, 46001. (42) Schlünzen, N.; Joost, J.-P.; Heidrich-Meisner, F.; Bonitz, M. Nonequilibrium dynamics in the one-dimensional Fermi-Hubbard model: Comparison of the nonequilibrium Green-functions approach and the density matrix renormalization group method. Phys. Rev. B 2017, 95, 165139. (43) A.-M. Uimonen,; Khosravi, E.; Stan, A.; Stefanucci, G.; Kurth, S.; R. van Leeuwen,; E.K.U. Gross, Comparative study of many-body perturbation theory and timedependent density functional theory in the out-of-equilibrium Anderson model. Phys. Rev. B 2011, 84, 115103. (44) Boström, E. V.; Mikkelsen, A.; Verdozzi, C.; Perfetto, E.; Stefanucci, G. Charge Separation in Donor–C60 Complexes with Real-Time Green Functions: The Importance of Nonlocal Correlations. Nano Lett. 2018, 18, 785–792.

28


Page 28 of 41

Page 29 of 41 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


(45) Myöhänen, P.; Stan, A.; Stefanucci, G.; van Leeuwen, R. A many-body approach to quantum transport dynamics: Initial correlations and memory effects. EPL (Europhysics Letters) 2008, 84, 67001. (46) Myöhänen, P.; Stan, A.; Stefanucci, G.; van Leeuwen, R. Kadanoff-Baym approach to quantum transport through interacting nanoscale systems: From the transient to the steady-state regime. Phys. Rev. B 2009, 80, 115107. (47) Ruberti, M.; Averbukh, V.; Decleva, P. B-spline algebraic diagrammatic construction: Application to photoionization cross-sections and high-order harmonic generation. The Journal of Chemical Physics 2014, 141, 164126.

29


1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

y

z


x


Page 30 of 41

Page 31 of 41

10

10 continuum resonances

0

HOMO

-10

15

-10

-20

-20

-30

-30

-40

1 ACS Paragon Plus Environment

-40

HF spectrum (eV)

0 KS spectrum (eV)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


Journal of Chemical Theory and Computation 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22


Page 32 of 41

Page 33 of 41

Im χ(ω) (arb. units)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50


1.0 0.8

BSE-Yambo TDHF-CHEERS@Yambo

0.6 0.4 0.2 0.0

10

15 ω (eV)


20

25

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

ADC

2B


0.7 fs

2.1 fs

3.5 fs ACS Paragon Plus Environment

HF

Page 34 of 41

Page 35 of 41 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

ADC

2B


1.7 fs

5.1 fs

8.4 fs ACS Paragon Plus Environment

HF

occupation

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51

0 sha1_base64="qUg8iXI01zUDf6doeaeQGgCgJ1c=">AAAB7XicbVDLSgNBEOyNrxhfUY9eFoPgKeyKoMegF48RzAOSJcxOepMx81hmZoUQ8g9ePCji1f/x5t84SfagiQUNRVU33V1xypmxQfDtFdbWNza3itulnd29/YPy4VHTqExTbFDFlW7HxCBnEhuWWY7tVCMRMcdWPLqd+a0n1IYp+WDHKUaCDCRLGCXWSc2uEjggvXIlqAZz+KskzEkFctR75a9uX9FMoLSUE2M6YZDaaEK0ZZTjtNTNDKaEjsgAO45KItBEk/m1U//MKX0/UdqVtP5c/T0xIcKYsYhdpyB2aJa9mfif18lsch1NmEwzi5IuFiUZ963yZ6/7faaRWj52hFDN3K0+HRJNqHUBlVwI4fLLq6R5UQ2Danh/Wand5HEU4QRO4RxCuIIa3EEdGkDhEZ7hFd485b14797HorXg5TPH8Afe5w+QMY8b

sha1_base64="qUg8iXI01zUDf6doeaeQGgCgJ1c=">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

sha1_base64="Al0DmXOXZP3T18bUEGkWxVi5am4=">AAAB/HicbVDLSgMxFM3UV62v0S7dBItQN2VGBF0WRXBZwT6gM5RMmmlD8xiSjFCG+ituXCji1g9x59+YaWehrQcCh3Pu5Z6cKGFUG8/7dkpr6xubW+Xtys7u3v6Be3jU0TJVmLSxZFL1IqQJo4K0DTWM9BJFEI8Y6UaTm9zvPhKlqRQPZpqQkKORoDHFyFhp4FYDjswYI5bdzuqB5GSEzgZuzWt4c8BV4hekBgq0Bu5XMJQ45UQYzJDWfd9LTJghZShmZFYJUk0ShCdoRPqWCsSJDrN5+Bk8tcoQxlLZJwycq783MsS1nvLITuZR9bKXi/95/dTEV2FGRZIaIvDiUJwyaCTMm4BDqgg2bGoJworarBCPkULY2L4qtgR/+curpHPe8L2Gf39Ra14XdZTBMTgBdeCDS9AEd6AF2gCDKXgGr+DNeXJenHfnYzFacoqdKvgD5/MHX9KUkg==AAAB/HicbVDLSgMxFM3UV62v0S7dBItQN2VGBF0WRXBZwT6gM5RMmmlD8xiSjFCG+ituXCji1g9x59+YaWehrQcCh3Pu5Z6cKGFUG8/7dkpr6xubW+Xtys7u3v6Be3jU0TJVmLSxZFL1IqQJo4K0DTWM9BJFEI8Y6UaTm9zvPhKlqRQPZpqQkKORoDHFyFhp4FYDjswYI5bdzuqB5GSEzgZuzWt4c8BV4hekBgq0Bu5XMJQ45UQYzJDWfd9LTJghZShmZFYJUk0ShCdoRPqWCsSJDrN5+Bk8tcoQxlLZJwycq783MsS1nvLITuZR9bKXi/95/dTEV2FGRZIaIvDiUJwyaCTMm4BDqgg2bGoJworarBCPkULY2L4qtgR/+curpHPe8L2Gf39Ra14XdZTBMTgBdeCDS9AEd6AF2gCDKXgGr+DNeXJenHfnYzFacoqdKvgD5/MHX9KUkg==AAAB/HicbVDLSgMxFM3UV62v0S7dBItQN2VGBF0WRXBZwT6gM5RMmmlD8xiSjFCG+ituXCji1g9x59+YaWehrQcCh3Pu5Z6cKGFUG8/7dkpr6xubW+Xtys7u3v6Be3jU0TJVmLSxZFL1IqQJo4K0DTWM9BJFEI8Y6UaTm9zvPhKlqRQPZpqQkKORoDHFyFhp4FYDjswYI5bdzuqB5GSEzgZuzWt4c8BV4hekBgq0Bu5XMJQ45UQYzJDWfd9LTJghZShmZFYJUk0ShCdoRPqWCsSJDrN5+Bk8tcoQxlLZJwycq783MsS1nvLITuZR9bKXi/95/dTEV2FGRZIaIvDiUJwyaCTMm4BDqgg2bGoJworarBCPkULY2L4qtgR/+curpHPe8L2Gf39Ra14XdZTBMTgBdeCDS9AEd6AF2gCDKXgGr+DNeXJenHfnYzFacoqdKvgD5/MHX9KUkg==

First-Principles Nonequilibrium Green's Function Approach to Ultrafast

Recommend Documents