Ultrafast charge dynamics in glycine induced by attosecond pulses

David Ayuso a, Alicia Palacios a, Piero Decleva b and Fernando Martín *acd
aDepartamento de Química, Módulo 13, Universidad Autónoma de Madrid, 28049 Madrid, Spain
bDipartimento di Scienze Chimiche e Farmaceutiche, Universita di Trieste, and CNR-Istituto Officina dei Materiali, I-34127 Trieste, Italy
cInstituto Madrileño de Estudios Avanzados en Nanociencia (IMDEA-Nanociencia), Cantoblanco, 28049 Madrid, Spain
dCondensed Matter Physics Center (IFIMAC), Universidad Autónoma de Madrid, 28049 Madrid, Spain. E-mail: fernando.martin@uam.es

Received 23rd March 2017 , Accepted 1st June 2017

First published on 2nd June 2017


The combination of attosecond pump–probe techniques with mass spectrometry methods has recently led to the first experimental demonstration of ultrafast charge dynamics in a biomolecule, the amino acid phenylalanine [Calegari et al., Science, 2014, 346, 336]. Using an extension of the static-exchange density functional theory (DFT) method, the observed dynamics was explained as resulting from the coherent superposition of ionic states produced by the broadband attosecond pulse. Here, we have used the static-exchange DFT method to investigate charge migration induced by attosecond pulses in the glycine molecule. We show that the observed dynamics follows patterns similar to those previously found in phenylalanine, namely that charge fluctuations occur all over the molecule and that they can be explained in terms of a few typical frequencies of the system. We have checked the validity of our approach by explicitly comparing with the photoelectron spectra obtained in synchrotron radiation experiments and with the charge dynamics that follows the removal of an electron from a given molecular orbital, for which fully correlated ab initio results are available in the literature. From this comparison, we conclude that our method provides an accurate description of both the coherent superposition of cationic states generated by the attosecond pulse and its subsequent time evolution. Hence, we expect that the static-exchange DFT method should perform equally well for other medium-size and large molecules, for which the use of fully correlated ab initio methods is not possible.


1 Introduction

The study of ultrafast charge migration occurring in biomolecules after ionization has become a topic of increasing interest in the last few years.1,2 In contrast with conventional electron or hole transfer phenomena, which are due to nuclear rearrangements and therefore observed at larger time scales (from femto- to picoseconds),3–5 the so-called charge migration phenomena are associated with purely electron dynamics and take place in the sub-femtosecond time scale.1,2,6 While the former allows one to eventually control chemical reactions by driving nuclear motion, the latter pursues to manipulate them by acting at the core of the process, the electronic wave packet. The experimental observation and subsequent possibility of manipulation of ultrafast charge migration thus require the availability of well-characterized attosecond pulses. Isolated attosecond pulses were first produced in 2001,7,8 opening the door to this and many other processes in which the electron motion is the driving force.9,10 Interest in charge migration phenomena was raised by the experimental findings of Weinkauf and coworkers in the nineties. They showed that charge-directed reactivity could be induced in peptide chains by means of UV irradiation.11,12 The molecular target was ionized, localizing the charge at a given site of the molecular target, and bond-breaking was observed at the remote site of the chain after charge transfer. Later experiments performed on smaller biological systems led to the conclusion that charge transfer should occur in a few tens of femtoseconds.13,14 In 2014, the use of stable isolated attosecond pulses combined with mass spectrometry methods provided the first experimental evidence of charge dynamics in the amino acid phenylalanine with attosecond time resolution.15

Along with these experimental efforts, Cedeberbaum and coworkers published in 1999 a theoretical study of ultrafast electron dynamics in an organic molecule, difluoropropadienone,6 introducing for the first time the term ultrafast charge migration. A charge redistribution was predicted to occur in the sub-femtosecond time scale after the sudden ionization of the molecule. This result was explained as a consequence of electron correlation, which was explicitly included by using ab initio quantum chemistry methods. The observation of sub-femtosecond dynamics was further confirmed in an extension of this work to a number of organic molecules, ranging from small amino acids such as glycine16,17 to larger systems containing aromatic rings.18 In these and all subsequent theoretical works reported until 2014, the ionization step responsible for the creation of the charge was ignored and the initial charge distribution was assumed to be that resulting from the removal of an electron from a specific Hartree–Fock (HF) orbital.

However, to interpret the charge dynamics induced by an attosecond pulse, one has to go a step farther, since such dynamics is the result of the coherent superposition of cationic states induced by such a broadband pulse. Thus, any realistic description of charge dynamics in this context requires the evaluation of the ionization amplitudes that define this superposition. For large molecules, this is only possible by using extensions of density functional theory (DFT) that incorporate a description of the electronic continuum. These methods have been proven to provide a reasonable description of ultrafast electronic processes in amino acids15,19,20 and tetrapeptides21 by using generalized-gradient approximations to describe the exchange–correlation terms. In particular, the static-exchange DFT method22,23 has been successfully used to interpret attosecond pump–probe experiments performed in phenylalanine.15 The results of these calculations were crucial to conclude that the temporal evolution of the measured fragmentation yields reflect the dynamics of the electronic wave packet created by the attosecond pulse. This has led to an intense debate in the literature,2,20,24 since, in the context of pump–probe schemes, the comparison between theory and experiment is rather indirect, due to the neglect of both the probe pulse and the nuclear motion in the theoretical modelling. Some of these aspects have already been addressed in previous work.20 However, it still remains to be proved that the accuracy of the static-exchange DFT approach is good enough to provide a realistic description of the coherent superposition of cationic states and its subsequent time evolution.

In this work, we have checked the validity of the static-exchange DFT approach to describe ultrafast charge dynamics in the glycine molecule by explicitly comparing with (i) the available experimental photoelectron spectrum25 and (ii) the evolution of the charge density resulting from fully correlated ab initio calculations in which ionization is assumed to be sudden and the ejected electron is selectively removed from a well-defined molecular orbital.16,17 From the good agreement with these earlier results, we conclude that, in the spectral range of the UV pulse used in ref. 15, our calculated ionization amplitudes are sufficiently accurate to describe the coherent superposition of states generated by the attosecond pulse and the subsequent evolution of the corresponding electronic wave packet. We show that although the charge density resulting from the interaction with the attosecond pulse is significantly delocalized, charge fluctuations associated with a few typical frequencies are clearly visible, in agreement with the behavior previously observed in phenylalanine. A similar response is observed for different molecular conformations and orientations with respect to the light polarization direction.

2 Theoretical approach

The description of the electronic wave packet created by an attosecond pulse first requires the evaluation of the ionization amplitudes of the energetically accessible electronic continua. A few theoretical methods have been proposed to evaluate ionization amplitudes in large molecules.15,19,26–28 Here, we will use the static-exchange DFT method,22,23,29–31 which has been previously used to describe ionization and charge migration in the amino acid phenylalanine.15,19,32 The method makes use of the Kohn–Sham DFT formalism to describe bound states and of the Galerkin approach to evaluate photoelectron wave functions in the field of the corresponding Kohn–Sham density. Over the years, this methodology has provided accurate photoionization cross sections for small molecules (N2, CO, CH4, etc.)33–37 as well as for medium and large size systems (fullerenes).23,38–42 The method resides in two approximations. Firstly, it is a single-active electron approximation, i.e. shake-up processes are neglected, which is a reasonable approximation taking into account the minor contribution of excitation–ionization processes. Indeed, excitation–ionization involves transitions to 2h1p (2 hole 1 particle) states and is thus entirely due to electron correlation (i.e. two electrons must be simultaneously promoted by the absorption of a single photon), while direct ionization involves transitions to 1h (1hole) states in which only one electron is promoted. Secondly, for large molecules, the nuclear motion is neglected due to the impossibility to evaluate potential energy surfaces in a wide range of molecular geometries. Recent theoretical work incorporating nuclear motion through a classical description20,43–47 has shown that in the absence of conical intersections, ultrafast charge fluctuations are barely affected by this motion during the first 10–15 fs following the ionization event, thus validating the use of fixed-nuclei approximation in the early states of the electron dynamics.

In the following sections, we describe our theoretical method, with emphasis on the evaluation of the ionization amplitudes and the extraction of the time-varying observables, namely the time evolution of electron density after interaction with the attosecond UV pulse.

2.1 Static-exchange DFT

A thorough description of the static-exchange DFT approach to compute the electronic structure, including implementation and computational details can be found in ref. 1, 23, 31 and 32. The method uses single Slater determinants to describe bound and excited (continuum) electronic states, ensuring that the Pauli exclusion principle is fulfilled. Continuum states are constructed by promoting one electron from a bound spin orbital ϕα to a continuum spin orbital ϕεαlh with kinetic energy εα and angular quantum numbers l and h, and can be written as
 
Ψαεαlh(x1, x2,…,xn) = |ϕ1ϕ2ϕα−1ϕεαlhϕα+1ϕN|(1)

Bound and continuum orbitals are written in a basis of multicentric B-spline functions, using symmetry-adapted48 linear combinations of real spherical harmonics with origin over different positions in the molecule:

• A large one-center expansion (OCE) over the center of mass provides an accurate description of the long-range behavior of the continuum states.

• Small expansions, called off-centers (OCs), located over the non-equivalent nuclei, complement the OCE. These basis expansions improve dramatically the convergence of the calculation, allowing one to reduce the angular expansion in the OCE, since it can effectively describe the Kato cusps49 at the nuclear positions.

Each element of the basis set is then written as

 
image file: c7cp01856h-t1.tif(2)
where Λp represents a shell of equivalent centers (p = 0 refers to the OCE), q runs over the centers in the shell, n is an index over the B-spline functions Bκn, whose order is κ = 10, λμ are the indexes of the irreducible representation (see ref. 31), h runs over the linearly independent angular functions, which are constructed as linear combinations of real spherical harmonics associated with a fixed angular quantum number l, and the coefficients bqmlhλμ are determined by symmetry,48 defining the so-called symmetry-adapted spherical harmonics Xplhλμ, which are invariant under the symmetry operations of a given point group. In each center q, the B-spline expansion reaches a maximum value Rpmax, which can be different for non-equivalent centers (different values of p, see eqn (2)). A large value of R0max is required in the OCE in order to provide a good description of the oscillatory behavior of the continuum states. One can control the overlap between the basis elements, avoiding running into linear dependence, by keeping small OC expansions (Rp>0max ≃ 1 a.u.) since the Kato cusps are usually well localized at the atomic positions. Angular expansions are truncated so that l takes values up to a maximum lpmax, which can also be different for the non-equivalent centers. In general, one can keep small values of lmax in the OCs to complement the OCE in the description of the bound states, but a large angular expansion is usually required in the OCE, especially in the case of complex molecules and for the evaluation of continuum states with high kinetic energy.

For the results presented in this work on glycine, we employed a large one-center expansion (OCE) of 118 B-splines enclosed in a sphere of 30 a.u. with origin in the center of mass, using spherical harmonics of angular momentum of lmax = 20 (see eqn (2)) to account for the long-range behavior of the continuum states. The sizes of the OCs varied from 0.2 to 1.6 a.u., larger for the heavier nuclei since they accumulate more electron density, and the angular expansions were limited to lmax = 2. Electronic exchange and correlation effects have been accounted for by the LB9450 functional, which can provide a reliable description of bound and continuum states. The ground-state electron density was generated using the Amsterdam Density Functional (ADF) package51–53 using a double ζ-polarization plus basis set. The molecular geometries for the most stable conformers of glycine where previously optimized at the DFT/B3LYP54,55 level in a 6-311+g(3df,2p) basis set using Gaussian09,56 starting from the approximate optimized geometries reported in ref. 57–59. By using the ground-state electron density, continuum Kohn–Sham orbitals were evaluated in a grid of 200 photoelectron energies by means of the Galerkin approach,30 which can yield the photoelectron wave function at a given kinetic energy without changing the basis set. Since it is well known that the LB94 functional overestimates the molecular orbital eigenvalues, the first ionization potential of glycine was calculated using the outer-valence Green's function (OVGF)60–62 method implemented in Gaussian09,56 which can provide accurate values of the ionization potentials of the outer-valence shells. Then, the DFT/LB94 eigenvalues were shifted according to the energy difference between the first IP provided by the OVGF method and the DFT/LB94 calculations.

2.2 Ionization step

The electronic wave packet created by the attosecond pulse can be written as
 
image file: c7cp01856h-t2.tif(3)
where [x with combining macron] = (x1,…,xN) are the spatial and spin coordinates of the N electrons in the molecule, Ψ0([x with combining macron]) is the electronic ground state, Ψαεlh([x with combining macron]) represents a continuum state in which an electron has been promoted from the α orbital to a continuum orbital with kinetic energy ε and angular quantum numbers l and m (see eqn (1)) and the time-dependence of the wave function is included in the spectral coefficients c0 and cαlh, which satisfies the normalization condition:
 
image file: c7cp01856h-t3.tif(4)
At t = 0, the system is assumed to be in the ground state, i.e., |c0(0)|2 = 1 and cαlh(ε,0) = 0. If the attosecond pulse is weak, most of the population will remain in the ground state, i.e., |c0(t)|2 ≃ 1, and the time-dependent coefficients can be evaluated using first-order perturbation theory:63
 
image file: c7cp01856h-t4.tif(5)
where ε is the polarization direction of the electric field E, με is the dipole operator, and Eα is the energy of an ion with a hole in the α molecular orbital. After interaction with the pulse (t > T), the integral in eqn (5) can be substituted by the Fourier transform of the electric field image file: c7cp01856h-t5.tif:
 
image file: c7cp01856h-t6.tif(6)
where the dependence on time is that of the stationary phases image file: c7cp01856h-t7.tif as the wave packet evolves freely. The ionization probability is non-zero for the states reached by photon energies within the pulse bandwidth, ℏω = Eα + εE0. The interferences between the components of the wave packet will be imprinted in different observables.

The total photoionization cross sections can be simply evaluated by summing over the squares of the cαεlh(T) amplitudes evaluated for pulses of infinite duration. i.e.,

 
image file: c7cp01856h-t8.tif(7)
As in most experiments, the molecules are randomly oriented, we have computed σεα for different directions of the polarization vector.

2.3 Time-evolution of the ionic subsystem

We seek to analyze the evolution of the hole generated in the molecular target upon ionization by the attosecond pulse. This has been done in terms of the reduced density matrix, whose elements can be constructed from the spectral coefficients (eqn (5) and (6)):
 
image file: c7cp01856h-t9.tif(8)
where the double sum runs over all open ionic states. The trace of the reduced density matrix contains the population of each ionic state and the off-diagonal terms provide the coherence between pairs of states. In the case of ionization with monochromatic light, all off-diagonal terms would be zero (unless those involving degenerate states) since the parent ion would be in an incoherent superposition of states. This is the situation that one encounters in experiments performed with synchrotron radiation, where the energy of the incident photons is well defined.64 In contrast, broadband attosecond UV pulses can generate coherent superpositions of electronic states, allowing for the investigation of ultrafast dynamics with the required time resolution.10

The reduced density matrix (eqn (8)) contains all the information about the ionic subsystem and therefore can be used to retrieve any observable associated with the molecular cation. In particular, the time-dependent electron density is given by:

 
image file: c7cp01856h-t10.tif(9)
where the first term is time invariant since the trace of the reduced density matrix is constant.

Another interesting observable is the density of the hole generated upon ionization, defined by Cederbaum and coworkers6 as the difference between the electron density of the ion, ρ(ion)(r,t), and that of the (initial) neutral molecule, ρ0(r), which does not depend on time:

 
image file: c7cp01856h-t11.tif(10)
where the ground state density is given by
 
image file: c7cp01856h-t12.tif(11)
and we have assumed that the reduced density matrix of the ionic subsystem is normalized to unity, i.e.,
image file: c7cp01856h-t13.tif

Fluctuations in the hole density can arise when several ionic states are coherently populated, that is, if there are non-zero off-diagonal elements in the reduced density matrix. Although, up to now, no experiment has been able to measure the hole density of an isolated molecule directly, the ultrafast charge redistribution accompanying sudden ionization can be imprinted in observables that are experimentally accessible. For instance, it has been recently shown15,19 that fragmentation of the molecular cation can be very sensitive to the hole dynamics induced by the attosecond pulse.

3 Benchmark results

3.1 Photoionization cross sections

Neutral glycine has 40 electrons and therefore the electronic ground state constitutes a closed-shell system that can be accurately described using 20 molecular orbitals. Fig. 1 shows the occupied Kohn–Sham orbitals in the ground state of glycine. Since the molecule belongs to the Cs point group, its molecular orbitals have either a′ or a′′ symmetry. As can be seen in Fig. 1, a′ orbitals are symmetric with respect to reflection through the mirror plane and a′′ orbitals are antisymmetric and thus contain a nodal plane. Core orbitals are those constituted by the 1s orbitals of the “heavy” atoms (C, N and O), and therefore are highly localized around each atomic center (see the first row of Fig. 1). The energy required to remove an electron from a core orbital in glycine ranges from around 290 eV in the case of C 1s to 400 eV for N 1s and 535 eV for O 1s. They are thus not accessible by attosecond UV pulses as those used in the experiments of Calegari et al.15 and, in general, by most pulses generated via HHG techniques.1 Valence orbitals, with ionization potentials ranging from around 10 eV to 20 eV, are highly delocalized as shown in the figure. Inner-valence orbitals present an intermediate situation, with ionization energies from 20 to around 35 eV.
image file: c7cp01856h-f1.tif
Fig. 1 Occupied Kohn–Sham orbitals of the glycine molecule. They have been calculated using the LB9450 functional in a basis set of B-spline functions as explained in the text.

In order to provide a reliable description of the interaction with an attosecond UV pulse as that employed in ref. 15, one needs to evaluate the corresponding ionization amplitudes. We are considering an attosecond pulse with photon energies ranging from 17 to 35 eV, thus electrons can be ejected from all valence and inner-valence shells of the molecule. The spectral shape of such a pulse is shown in Fig. 2 as a thick orange curve lying over a shaded area. We have evaluated photoionization cross sections from these outer shells for the most abundant conformer of glycine in the framework of the fixed-nuclei approximation. The results are shown in Fig. 2 in the energy range accessible by the experimental attosecond UV pulse. As expected, the cross sections decrease with the photon energy. In some ionic channels, we can see pronounced peaks near the threshold that are due to electronic shape resonances associated with either centrifugal potential barriers or unoccupied molecular orbitals embedded in the electronic continuum.65–67 We must emphasize that the use of the static-exchange approximation prevents us from observing resonances associated with multiply excited states embedded in the ionization continuum.


image file: c7cp01856h-f2.tif
Fig. 2 Photoionization cross sections (eqn (7)) of glycine from different molecular orbitals calculated using the static-exchange DFT method. The numbers and colors denote the molecular orbitals from where the electron is emitted in each case (see Fig. 1). The energy spectrum of the attosecond pulse considered in this work to trigger ultrafast electron dynamics is represented by a thick orange curve lying over a shaded area.

Fig. 2 shows that all cross sections lie within a factor of three in most of the photon energy range, thus indicating that the molecule can be efficiently ionized from any valence and inner-valence shell upon interaction with the attosecond pulse. Only core electrons will remain unaffected. In fact, for every photon energy within the bandwidth, we find a similar contribution from a significant number of ionization thresholds. At 21.2 eV (He I radiation), for instance, the largest ionization probability is associated with the 14a′ orbital, but all accessible channels significantly contribute to the total cross section (see Fig. 2). Looking at the shape of these orbitals, one can already anticipate that photoionization should lead to a very delocalized hole in the parent ion, in contrast with previous theoretical simulations of charge migration,2,6,16,17,21,26 in which the hole is suddenly created in a single, rather localized molecular orbital. In the present case, however, the initial hole density will be the result of the coherent superpositions of many ionic states, most of them containing a delocalized hole.

Fig. 3 (lower panel) shows the calculated photoelectron spectrum of glycine for a well-defined photon energy, namely 40.8 eV (He II radiation), for which experimental measurements are available in the range of 8–25 eV of binding energy (upper panel in Fig. 3). For a realistic comparison, our infinitely resolved spectral lines have been convoluted with a Lorentzian function of 0.3 eV width at half maximum that reproduces the experimental broadening of the peaks. As can be seen, the agreement between theory and experiment is good, thus validating our calculations of ionization amplitudes and cross sections in this energy range.


image file: c7cp01856h-f3.tif
Fig. 3 Comparison between the calculated and experimental25 photoelectron spectra of glycine at 40.8 eV of photon energy (He II radiation).

3.2 Charge migration

In order to check the validity of our time-propagation method, which is based on a DFT-like description of the cationic states, we have performed two sets of calculations that can be directly compared with the existing ab initio results of Kuleff, Breidbach and Cederbaum,16 although for a process that is completely different from that generated by attosecond pulses. In that work, the electronic wave packet was generated by sudden electron removal from a specific Hartree–Fock (HF) molecular orbital. The hole generated in this way then moves through the molecular skeleton because the prepared state is not an eigenstate of the ionic Hamiltonian but a linear combination of several cationic states. To perform a meaningful comparison, we have started from the same initial wave function as in ref. 16. The corresponding HF orbitals have been evaluated using Gaussian 0956 employing a DZP basis set.

Then, to study the time evolution of the hole density, we have proceeded as in ref. 16 and projected the initial state onto the Slater determinants built from the KS orbitals (shown in Fig. 1) that we have used to represent the ionic stationary states. The projection leads to a coherent superposition of ionic states that freely evolves in time as dictated by the relative eigenphases of the states that conforms the wave packet. The evolution of the hole wave packet is shown in Fig. 4 and 5 for the cases of sudden ionization from the 11a′ and the 14a′ HF orbitals, respectively. As can be seen, the agreement between our results and those previously reported16 is quite satisfactory. The slight differences between both methodologies are due the contribution of two-hole one-particle (2h1p) configurations, which were explicitly included in ref. 16, but neglected in our present approach. Nevertheless, these configurations are expected to play a minor role in the hole dynamics generated by the attosecond pulse, since a transition from the ground state to a shake-up state is a two-electron process, which is less likely to occur via one-photon absorption. Note however that if we considered sudden ionization from the inner-valence HF orbitals of glycine, as in ref. 17, we would expect a poorer agreement, since, in that case, more and more 2h1p configurations are accessible.


image file: c7cp01856h-f4.tif
Fig. 4 Time evolution of the hole generated in the glycine molecule upon sudden ionization from the 11a′ HF orbital (at t = 0 the hole is completely localized in the 11a′ HF orbital). Left figure: Natural charge orbitals, as a function of time (the square of the natural orbitals provides the hole density). Reprinted with permission from ref. 16. Copyright 2005, AIP Publishing LLC. Right figure: Hole density, evaluated using the present approach.

image file: c7cp01856h-f5.tif
Fig. 5 Time evolution of the hole generated in the glycine molecule upon sudden ionization from the 14a′ HF orbital (at t = 0 the hole is completely localized in the 14a′ HF orbital). Left figure reprinted with permission from ref. 16. Copyright 2005, AIP Publishing LLC.

4 Ultrafast electron dynamics induced by attosecond pulses

4.1 Coherent charge dynamics

We consider now the evolution of the electron hole created by the 400-as UV pulse. The experimental broadband pulse covers a photon energy region from 17 to 35 eV, plotted as an orange shaded area in Fig. 2. We thus have enough energy to eject electrons from every inner and outer valence shell in the molecule leading to a coherent superposition of the corresponding 1h states. Charge fluctuations will result from the coherent superposition of states in which the photoelectron is ejected with the same energy leaving behind a different ionic state (second term in eqn (9)). Because the energy difference between the highest occupied molecular orbital (HOMO) and the innermost shell in glycine is around 23 eV (larger than the pulse bandwidth of around 18 eV), we do not expect to observe fluctuations involving all ionic states, but only among those whose energy difference is smaller than the pulse bandwidth. Using the static-exchange DFT method, we have evaluated the ionization amplitudes for all open channels, associated with a photoelectron ejected from the 15 outer shells. We have then computed the reduced density matrix of the ionic subsystem using eqn (8). The hole density was calculated as the difference between the electronic density of the neutral molecule and the time-varying electronic density of the cation (eqn (10)). Because in the experimental setup the molecules are not aligned, we have performed calculations including an average over the three orthogonal orientations with respect to the polarization vector of the attosecond pulse. Fig. 6 displays the snapshots of the relative variation of the hole density with respect to the time-averaged values. Despite the delocalized nature of the hole generated by the attosecond UV pulse, an electronic redistribution is observed in the sub-femtosecond time scale. The charge dynamics cannot be explained as a simple charge migration from one side of the molecule to the other, in contrast with the results shown in Fig. 4 and 5 where the hole was initially localized in a specific molecular orbital. Instead, sizeable charge fluctuations are seen at the different molecular centers, thus showing that the concept of ultrafast charge migration remains valid in this context. Moreover, note that these charge fluctuations occur in a time scale much faster than nuclear vibrations (the fastest vibrational period in glycine is 8.9 fs), which again justifies the use of the fixed-nuclei approximation at least during approximately the first 10 fs. This has been confirmed by recent theoretical simulations in which the effect of nuclear motion has been taken into account in this same molecule by means of an Ehrenfest semiclassical treatment.20
image file: c7cp01856h-f6.tif
Fig. 6 Relative variation of the hole density on glycine with respect to its time-averaged value as a function of time. Isosurfaces of the relative hole density are shown at cutoff values of 10−4 a.u. (yellow) and −10−4 (purple). Time is with reference to the end of the XUV pulse (first snapshot).

4.2 Fourier analysis

A more quantitative analysis of the dynamics can be performed by looking at the charge fluctuations at specific molecular sites, e.g., by integrating the time-dependent hole density around different functional groups or atomic centers. Fig. 7 shows the hole density integrated around the C, N and O atoms as a function of time (left panels) and the corresponding Fourier power spectra (right panels). In order to obtain well-resolved peaks in frequency, the hole density has been evaluated up to 500 fs. The Fourier analysis uncovers the dominant frequencies responsible for the charge fluctuations. The fact that different frequencies appear in the analysis around each atomic center reveals the complexity of the electronic redistribution over the whole molecular skeleton. Each frequency can be associated with a pair of 1h cationic states produced by the attosecond UV pulse. We have indicated them in Fig. 7 for the most prominent frequencies. The intensity of each frequency signal in the Fourier spectra is roughly proportional to the spacial overlap between the corresponding molecular orbitals and to the ionization amplitudes associated with them. From an inspection of the Kohn–Sham orbitals where the holes are created (see Fig. 2), one can see that the dominant beatings always involve two states that have holes in orbitals with significant density on, at least, two common parts of the molecule, which allows for charge redistributions.
image file: c7cp01856h-f7.tif
Fig. 7 Fourier power spectra of the calculated hole density integrated over various atoms of glycine for the case of randomly oriented molecules. In order to obtain well-resolved peaks in frequency, the hole density has been evaluated up to 500 fs.

As expected, the analysis of the hole density reveals that the dominant beatings involve cationic states that are close in energy, namely, whose energy difference, ΔEα,α = |EαEα|, is smaller than the energy bandwidth of the attosecond pulse. Consequently, the corresponding element in the reduced density matrix γαα will be small. This is why no pronounced beatings at frequencies higher than 1 PHz are observed. For the same reason, Fig. 8 only reflects interferences between states with the same final symmetry, otherwise they would be incoherently populated.


image file: c7cp01856h-f8.tif
Fig. 8 Fourier power spectra of the hole density integrated over the amino group of glycine. The results are shown for three orthogonal orientations of the molecule with respect to the polarization vector of the electric field associated with the attosecond XUV pulse and for the case of randomly oriented molecules. In order to obtain well-resolved peaks in frequency, the hole density has been evaluated up to 500 fs. The states that give rise to the dominant peaks are indicated in the spectra by labels that denote the molecular orbitals where the holes have been created (see molecular orbitals in Fig. 1).

Previous simulations on phenylalanine have revealed that the beating frequencies around the amino group were in good agreement with those observed in a UV-pump/IR-probe experiment in which the yield of a doubly-charged immonium fragment was measured as a function of the time delay.15,19 For this reason, we choose the amino group to illustrate the dependence of the characteristic frequencies on molecular orientation. Fig. 8 shows the Fourier power spectra of the hole density around the amino group for three orthogonal orientations of the molecule with respect to the polarization vector of the field (indicated in the figure) and for the case of randomly oriented molecules. The peaks appearing in the Fourier spectra provide information about the frequency and the intensity of the charge fluctuations observed in Fig. 6. In agreement with the previous discussion, one can see that the dominant beatings always involve two states that have holes in orbitals with significant density around the amino group and another part of the molecule. The orbitals associated with the largest peaks are plotted in the right-hand side of Fig. 8. It can be seen that for every molecular orientation, in spite of the large number of frequencies that could potentially be observed (for 15 active orbitals, the number of possible frequencies is 225), the Fourier spectra only show a few prominent ones. A similar behavior has been observed in phenylalanine,15,19 thus suggesting that this might be the general behavior of other complex molecules.

4.3 What about molecular conformation?

It is well known that amino acids exist in many conformations as a result of their structural flexibility. Typically, the energy barrier for interconversion between different conformers is small, of the order of a few kcal mol−1, so that, even at room temperature, thermal energy is sufficient to induce conformational changes. Theoretical investigations have shown that such changes can affect the charge migration process.17

We have performed calculations for the 4 most stable conformers of glycine considering a temperature of 430 K and assuming a Boltzmann distribution57,68 (this is approximately the temperature at which the experiment reported in ref. 15 took place). The geometries of the most stable conformers and corresponding Fourier power spectra are shown in Fig. 9, together with the results for the thermal average obtained by taking into account the estimated populations (given in Fig. 9). Although the frequencies of the dominant peaks depend on the particular conformer, we do not find substantial differences and the spectrum of the most abundant conformer is very similar to that of the thermal average.


image file: c7cp01856h-f9.tif
Fig. 9 Fourier power spectra of the hole density on the amino group of the most abundant conformers of glycine at 430 K. The relative populations have been calculated assuming a Boltzmann distribution. The lower panel shows the averaged results and the corresponding geometries are shown in the right.

5 Conclusions

Ultrashort light sources are ready to monitor with attosecond resolution electron dynamics occurring in the sub-femtosecond time scale. Processes such as ultrafast charge migration are thus expected to be captured and eventually controlled. In this context, theoretical tools that are able to provide a realistic description of ultrafast molecular dynamics occurring upon photoionization are required. Significant theoretical efforts have been made to address this problem, from the accurate evaluation of the ionization step,15,19,26–28 as discussed in the present manuscript, to the inclusion of the coupled electron–nuclear in the subsequent dynamics.20,44–47 In this context, the static-exchange DFT method is a very valuable tool because of its excellent compromise between accuracy and computational effort. Here, we have shown that the static-exchange DFT method can accurately describe the ultrafast electronic response of glycine to ionization induced by a realistic attosecond UV pulse. The electronic wave packet created in this way leads to an initially delocalized hole density, which results in a charge dynamics that is significantly different from that initiated by removing an electron from a specific molecular orbital. Given the highly delocalized nature of the initial hole, the observed dynamics does not correspond to charge migration from one molecular site to another, but rather to charge fluctuations around the different atomic centers or functional groups of the molecule. These conclusions remain valid for different molecular conformers and different molecular orientations with respect to the light polarization direction.

Acknowledgements

We acknowledge computer time from the CCC-UAM and Marenostrum Supercomputer Centers and financial support from the European Research Council under the European Union's Seventh Framework Programme (FP7/2007-2013)/ERC grant agreement 290853 XCHEM, the MINECO projects FIS2013-42002-R and FIS2016-77889-R, and the European COST Action XLIC CM1204. We thank O. Brea, L. Martínez-Fernández, I. Corral and J. González-Vázquez for their technical assistance with standard Quantum Chemistry packages. AP acknowleges a Ramón y Cajal contract from the Ministerio de Economía y Competitividad in Spain.

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Footnote

Present address: Max Born Institute, Max Born Strasse 2a, 12489 Berlin, Germany.

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