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# The quantum chemistry of attosecond molecular science

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Abstract With the advent of attosecond light pulses at the beginning of this century, the possibility to perform real‐time observations of electron motion in molecules has spurred impressive theoretical developments aimed at providing support and guidance to numerous, but still incipient, experimental efforts devoted to understand chemistry at its ultimate temporal frontier: the attosecond. The first real‐time observation of electron dynamics in a relatively large molecule, phenylalanine, was reported in 2014. This would have been difficult without the help of theory, since observations in this emerging field, recently coined attochemistry, are still indirect. While standard Quantum Chemistry methods can describe excited bound‐state dynamics, new approaches incorporating scattering theory formalisms are needed to understand the interaction with attosecond pulses. Indeed, due to their short wavelengths, lying in the XUV and X‐ray spectral regions, the interaction of such pulses with any molecule inevitably leads to ionization, which requires describing the molecular ionization continuum. Also, because of their short duration (i.e., large bandwidth), ionization is accompanied by the formation of a molecular electronic wave packet (i.e., a coherent superposition of electronic states), which evolves in time and dictates the fate of the molecule at the longer time scales where chemistry shows up. Although much has already been done up to date, current bottlenecks in the field are to account for electron correlations during the ionization process and for the coupled electron and nuclear dynamics that follows. Past and ongoing theoretical efforts are described here along with experimental work towards the solid establishment of attochemistry. This article is categorized under: Electronic Structure Theory > Ab Initio Electronic Structure Methods Software > Quantum Chemistry Theoretical and Physical Chemistry > Spectroscopy
(a) Black full lines: Ground state of the hydrogen molecule and its first three ionization thresholds (i.e., 1sσg, 2pσu, and 2pπu states of ). Orange vertical shadow is the Franck–Condon region. Blue and red shapes illustrate the evolution and decay (illustrated with arrows) of nuclear wave packets pumped into the doubly excited states (Q1 series ‐red‐ and Q2 series ‐blue‐). The longer the lifetime of these states, the farther the nuclear wave packets travel accumulating momentum, and thus the higher the vibrational states that significant overlap with the nuclear wave packet upon autoionization. (b) Vibrationally resolved photoionization cross sections normalized to the absolute cross section obtained for v = 2 (vertical transition from H2 to (1sσg)) as a function of the photon energy. Circles with bars are the experimental data. Full lines: ab initio simulations. (Adapted from Reference )
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(a) Diagram depicting the three different regions in space for the standard Gaussian and the hybrid Gaussian/B‐spline basis sets (GABS). Innermost region (r ∈ [0, R0] in red) combines Gaussian functions, centered at each individual atom (polycentric Gaussians, PC) and those at the center of mass of the molecule (monocentric, MC). Outermost region (r ∈ [R1, Rbox] in yellow) only contains B‐spline functions. In the intermediate region (r ∈ [R0, R1] in blue), both the monocentric Gaussian and B‐spline functions contribute. (b) Example taken from Reference of the radial basis set composed of Gaussian (thick purple lines) and B‐spline (thin green lines) functions composing the monocentric GABS basis set. The first B‐spline node is located at R0 = 10 a.u. (c) Radial part of the He+ continuum state with ℓ = 1 and E = 0.2 a.u., computed analytically (dots) and numerically using GABS (magenta dashed line). The Gaussian (purple solid line) and the B‐spline (green solid line) components of the numerical function are also shown
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(a) Sketch of the incoming boundary conditions that must fulfill the scattering wave function (see text for notations). (b) A basis of B‐spline functions of order 7 defined in the radial interval between 0 and 60 a.u. As an illustration, one of the B‐spline functions is indicated in red. (c) Radial part of the continuum state of the H atom with the appropriate boundary conditions for ℓ = 1 and E = 1.45 a.u. Dotted line: exact result. Black full line: Results obtained with a basis of 22 even‐tempered Gaussian functions Gi(r) ∝ rexp(−αir2), with αi = α0βi (α0 = 0.001, β = 1.46, i = 0, …, 21). (d) Results obtained with the B‐spline basis shown in (c)
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(a) Energy diagram of the 20 occupied molecular orbitals of glycine. (b) Illustration of the interaction of glycine with almost pure monochromatic light. Electrons ejected from different molecular orbitals have different kinetic energies (the kinetic energy is the difference between the energy of the absorbed photon and the binding energy of the molecular orbital from which the electron is ejected). (c) Illustration of the interaction with an ultrashort pulse (the energy bandwidth of ~400 as at full width half maximum is plotted), showing that electrons lying in different molecular orbitals can now be ejected with the same kinetic energy
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Snapshots of the charge density evolution occurring in tryptophan after ionization with an attosecond XUV pulse. These simulations employed a TDDFT approach as described in the main text. Upper row: Results obtained by fixing the positions of the nuclei. Lower row: Results obtained by allowing the nuclei to move as described by the Ehrenfest formalism. For a better visualization, at all times the hole density is referred to its average value over the complete time interval. Regions with an excess or a deficit of charge with respect to this average value are represented in purple and yellow, respectively. Only the equilibrium geometry has been considered. (Adapted from Reference )
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(a–c) Schematic representation of the different ionization processes that take place in an XUV‐pump/VIS‐probe experiment in CF4 (see text for details). When the pulses overlap (a), an XUV and a VIS photon can be simultaneously absorbed from different molecular orbitals (for the sake of clarity, only two of these processes are shown). When the XUV pulse arrives first (b), only an XUV photon is absorbed. Due to the large bandwidth of this pulse, the energy of the electron emitted from two different orbitals can be the same. When the VIS pulse arrives (c), resonant one‐ and two‐photon transitions can be induced between an occupied molecular orbital and one of the orbitals from which the electron was ejected in step (b). This path is undistinguishable from that in which the electron was directly removed from the former molecular orbital by the XUV pulse, leading to a quantum mechanical two‐way interference. (d) Photoelectron spectra of CF4 leaving a () ion as a function of the pump–probe delay. (e) Same as (d) for leaving an excited () ion. Both spectra have been normalized to the corresponding pump‐only spectra. Dotted lines show the VIS pulse. (Adapted from Reference )
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The first row shows a set of diagrams with the relevant potential energy curves of H2 and the wave packet dynamics induced by two identical time‐delayed 2 fs pulses of ~14 eV central frequency and an intensity I = 1012 W/cm2. The potential energy curves shown in these diagrams are the two lowest excited states of H2 (lowest black lines), the two lowest ionization thresholds (upper black lines) and the Q1 series of doubly excited states of symmetry (blue lines in between the upper two black lines). Shadowed areas indicate the electronic continua (light orange: only the first ionization continuum is open; yellow: both thresholds are open). The plots illustrate the interference (orange curve) of two identical but delayed direct two‐photon ionization events (left orange arrows), as well as the sequential ionization path in which a photon from the pump pulse first excites the molecule and a second photon from the probe pulse ionizes it (right orange arrow), further interfering with the direct paths and mapping the evolution of the excited molecular wave packet (blue shaded areas) at different time delays. (a) Dissociative ionization probability as a function of time delay. (b) Short‐time Fourier transform of the probability plotted in (a), obtained by employing a 2‐fs Gaussian‐shaped time window. (Adapted from Reference )
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Measured and calculated photoelectron spectrum of tryptophan at 100 eV. Calculations were performed by using the static exchange DFT approach described in the text. For a realistic comparison, the infinitely resolved spectral lines obtained in the simulations are convoluted with a Lorentzian function of 0.4 eV full width half maximum that reproduces the experimental broadening of the peaks. (Adapted from Reference )
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(a) Ionization continua between the second and third ionization thresholds of N2. White columns indicate the autoionizing states of Πu (yellow) and ∑u (red) symmetries. Gray columns show the actual ionization continua in which these resonances are embedded. Each continua is denoted by the symmetry of the ejected electron and a superscript to denote the state of the remaining cation, that is, (I), 2Πu(II), or (III). The colored shaded areas indicate autoionization widths. (b) Photoionization cross sections. Continuous lines correspond to length gauge and dashed‐dotted lines to velocity gauge. The total cross section is shown in black line, while the contributions leaving the system in a or Πu state are shown in red and yellow, respectively, following the color code in (a). The former exhibits resonances corresponding to nsσg and ndσg series of autoionizing states, while the latter exhibits the (much less pronounced) resonance features of the npπu series. Circles correspond to the experimental results of Dehmer et al. and Huber et al. (Adapted from Reference )
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Schematic diagram of an attosecond XUV‐pump/IR‐probe experiment performed in phenylalanine. The energy distribution of the XUV pulse is plotted in a blue shadow area. The amino acid is singly ionized by the attosecond XUV pulse (left column), creating a superposition of a manifold of ionic states, which results in a highly delocalized hole density (second column). The initial hole density calculated with the static exchange DFT approach for phenylalanine is also shown. The experiment captures this dynamics by using a delayed IR pulse that ejects a second electron (third column). The resulting dication follows different fragmentation paths (right column). In the experiment, only the yields corresponding to the path leading to the immonium fragment and a neutral COOH (plotted in the figure) reveals the sub‐femtosecond fluctuations
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(a) Relevant potential energy curves of . Red transparent area: Franck–Condon region. Black striped area: Region with significant population transfer between the F2g and 32g states for time delays where interferences are observed. Magenta single‐headed arrows: Two‐photon process populating the 32g state via the 52u state as a virtual intermediate state. Black double‐headed arrows: Internuclear distances where resonant single‐photon transitions occur. Insets in (a): Components of the nuclear wave packets associated to each individual electronic state as a function of time; that is, spectra resulting from the individual contributions of each ionic state to the total spectrum shown in (b). (b) N+ kinetic energy spectra calculated by including all states within the pump–probe delay interval of 5–16 fs. (Adapted from Reference )
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Electric field in the (a) time domain and (b) frequency domain of a Gaussian‐shaped pulse of 400 as of full width half maximum in time, a carrier frequency ω = 30 eV and intensity of 1012 W/cm2. The carrier envelope and frequency phases are set to zero
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