Quantum mechanical study of the attosecond nonlinear Fourier transform spectroscopy of carbon dioxide
aa r X i v : . [ phy s i c s . c h e m - ph ] F e b Quantum mechanical study of the attosecondnonlinear Fourier transform spectroscopy of carbon dioxide
Sergy Yu. Grebenshchikov ∗ R¨usterstr. 24, 60325 Frankfurt am Main, Germany
Sergio Carbajo
SLAC National Accelerator Laboratory and Stanford University,2575 Sand Hill Rd, Menlo Park, CA 94025, USA
Attosecond nonlinear Fourier transform (NFT) pump probe spectroscopy is an exper-imental technique which allows investigation of the electronic excitation, ionization,and unimolecular dissociation processes. The NFT spectroscopy utilizes ultrafastmultiphoton ionization in the extreme ultraviolet spectral range and detects thedissociation products of the unstable ionized species. In this paper, a quantum me-chanical description of NFT spectra is suggested, which is based on the second orderperturbation theory in molecule-light interaction and the high level ab initio calcu-lations of CO and CO +2 in the Franck-Condon zone. The calculations capture thecharacteristic features of the available experimental NFT spectra of CO . Approx-imate analytic expressions are derived and used to assign the calculated spectra interms of participating electronic states and harmonic photon frequencies. The de-veloped approach provides a convenient framework within which the origin and thesignificance of near harmonic and non-harmonic NFT spectral lines can be analyzed.The framework is scalable and the spectra of di- and triatomic species as well as thedependences on the control parameters can by predicted semi-quantitatively. I. INTRODUCTION
Chemical transformations, induced by the ultraviolet (UV), vacuum UV (VUV), andextreme UV (XUV) light in carbon dioxide CO and carbon dioxide cation CO +2 , are ofconsiderable importance for atmospheric, planetary, and interstellar chemistry. Spectral ∗ Email: [email protected] signatures of the cation CO +2 were detected in the Martian atmosphere and the cometcomae and tails. CO is the second common trace gas in the Earth atmosphere. It is oneof the main products of the fossil fuel burning and its photoabsorption is used in the UVdiagnostics of high-temperature and high-pressure flames. Accurate knowledge of its low-temperature UV absorption properties would improve the existing photochemical models ofthe atmospheres of Mars, which is to a 95% CO -based, and of Titan, in which CO is aminor constituent.Structure, properties, and photodynamics of carbon dioxide and carbon dioxide cation arethoroughly studied. Nevertheless, their photoreactivity in the gas phase and at catalyticinterfaces remains an area of active research mainly due to its environmental and technolog-ical relevance. For example, UV light ultimately destroys CO with a unit quantum yield.This reaction provides a one-step route towards CO reduction to carbon monoxide and isin scope of studies on the negative emission technologies. From a broader perspective,carbon dioxide excited with energetic (UV/VUV/XUV) photons undergoes a series of fun-damental photochemical processes typical of highly energy loaded molecular systems. Theseinclude non-adiabatic interactions between Rydberg and valence electronic states, roamingdissociation pathways leading to unusual photochemical products, or dissociative VUVphotoionization. Understanding the atomistic and electronic mechanisms of these processesenhances our ability to address the current major technological and climate challenges atthe molecular level.The rapidly expanding field of attochemistry, utilizing the generation of ultrashortattosecond-scale XUV pulses, interrogates the photochemical processes on the time scaletypical for the motion of valence electrons. Various spectroscopic and time-resolved pump-probe techniques have emerged, including attosecond spectroscopy and streaking,
2D correlation spectroscopy, and Raman scattering spectroscopy. The choice of theparticular method is primarily dictated by the nature of the attosecond dynamics one in-tends to study. Attosecond nonlinear Fourier Transform (NFT) spectroscopy in XUV isan experimental attosecond method which can be directly applied to the investigation ofsuper-excited ionizing and dissociating molecular systems.
With this technique, onedetects ions resulting from the coherent interaction between two attosecond pulse trains(APTs) and a molecule excited and ionized via multi-photon transitions. The spectral con-tent of the APTs produced through high-harmonic generation is commonly limited to oddharmonics of the fundamental driving frequency ω (usually in the infrared). Figure 1(a)provides examples of the intensity distributions of the harmonics in the APTs used in thiswork. The two APTs propagate along two interferometric arms with variable delay τ andare focused on the interaction region containing only the molecular species of interest. TheNFT spectroscopy is an attosecond pump-probe detection scheme, with identical pump andprobe APT pulses. The time delay increments in this technique, determined by the repro-ducibility with which one can move the interferometric arms and control the relative delay, isin the latest experiments of the order or better than 10 as. The frequency resolution in theNFT spectroscopy is limited in principle by the maximum delay τ max of the interferometer,typically of a few femtoseconds, which corresponds to ∼ and CO +2 . The nonlinear response of CO toAPT fields is encoded in the interferometric autocorrelator employing velocity map imaging(VMI), which detects the signals corresponding to the fragment ions, such as C + , O + , CO + ,as functions of the time delay τ between the two APTs. For each fragment ion type, thesignal of an electron/ion time-of-flight spectrometer such as VMI is rendered as a two-dimensional (2D) autocorrelation map giving the fragment kinetic energy distribution foreach time delay τ . The 2D NFT spectrum, showing the intensity as a function of the kineticenergy and the NFT frequency ω NFT , is obtained by a Fourier transform from the τ domainto the ω NFT domain. Integration over the kinetic energy of the ionic fragment gives a one-dimensional (1D) NFT spectrum representing the total ionic yield as a function of ω NFT and proportional to the total population of the electronic states of the ionized moleculedissociating into the channels containing detected fragments.The goal of this work is to explore the quantum mechanical aspects of the experimentalNFT spectroscopic technique, to construct a theoretical ab initio model for calculation of theNFT spectra of carbon dioxide, and to identify the NFT spectral features which can carryinformation on the chemical rearrangements within the molecule or the cation. This studyprovides transparent theoretical means to predict ab initio NFT spectra of small polyatomicmolecules with a modest computational effort. We hope that these results can be used tosupport and to inspire new experimental campaigns.
FIG. 1: (a) Amplitudes a n of the harmonics making up the APTs used in the calculations [cf. Eq.(7)]. Red rectangles: Amplitude Set 1 with only harmonics between n = 9 and n = 19 included.Grey rectangles: Amplitude Set 2 modeling the experiment of Ref. 46, but with the harmonicamplitude for n = 1 strongly amplified to the value of 2. (b) The frequency spectrum of one ofthe APTs in the calculations. The fundamental frequency of ω = 1 .
55 eV and the amplitude Set1 [red rectangles in panel (a)] are used. The temporal width of the APT envelope is 5 fs, givingthe spectral width of the harmonic lines of about 0.80 eV.
In what follows, we concentrate on the simplest type of the pump-probe signal — thetotal ionic yield after two APTs and the associated 1D NFT spectra. The extension to 2Dspectra is considered in a separate publication, although we briefly summarize our approachin appendix A. The quantum mechanical description of NFT spectra can be convenientlybased on the general framework, developed by Seel and Domcke for two-pulse timeresolved ultrafast ionization spectroscopy of polyatomic molecules. The laser radiation fieldis treated classically. The interaction between molecule or cation and light is accounted forusing the time dependent perturbation theory. The ladder of electronic states of CO +2 iscomputed using high level electronic structure theory including electron correlation.The analysis of the theoretical NFT spectra of carbon dioxide enables one to evaluateNFT spectroscopy as a research tool and to discuss the following questions: (1) How toassign spectral peaks in 1D NFT spectra? (2) Under which circumstances spectral peaks atnon-harmonic NFT frequencies can develop and what is their significance? (3) How sensitiveare NFT spectra to variations in the fundamental laser frequency ω ?The paper is organized as follows: The quantum mechanical approach to NFT spec-tra is outlined in Sect. II. This section describes the physical and chemical aspects of theAPT-induced photoionization of CO , summarizes the main simplifying assumptions, intro-duces the Hamiltonian for the system consisting of the molecule and cation interacting withthe laser light, and relates the 1D NFT spectrum to the solution of the time dependentSchr¨odinger equation. The ab initio quantum chemical calculations of the molecular andionic electronic states are discussed in Sect. III A. The NFT spectra of CO are presentedin Sect. III B and compared with the available experimental data. The calculated spectraare assigned in terms of the participating electronic states and the harmonic orders involvedin electronic transitions in Sect. III C. Section IV concludes and provides an outlook on theapplications of the developed theory, illustrating how the dependence of the NFT spectraon the control parameters can be visualized. Two appendices provide additional informa-tion. In Appendix A, the main equations extending the developed theory to 2D spectra arederived. In Appendix B, approximate analytical expressions which support and guide theassignment of the NFT spectra are presented. II. QUANTUM MECHANICAL APPROACH TO NFT SIGNALSA. The photochemical model
The APT pulses interact with the parent molecule and trigger photoionization and pho-todissociation reactions. One can broadly distinguish two major photoreaction pathways.In the first one, the parent molecule (CO ) X in the ground electronic state ˜ X Σ + g absorbsa photon with frequency ω i and becomes ionized to form cation (cid:0) CO +2 (cid:1) ⋆ , which is furtherexcited with a photon ω f into dissociative state(s) (cid:0) CO +2 (cid:1) ⋆⋆ :(CO ) X + ~ ω i → (cid:0) CO +2 (cid:1) ⋆ + e − ; (1a) (cid:0) CO +2 (cid:1) ⋆ + ~ ω f → (cid:0) CO +2 (cid:1) ⋆⋆ (1b)In the second pathway, the parent molecule is promoted into an electronically excited neutralstate before ionization: (CO ) X + ~ ω i → (CO ) ⋆ ; (2a)(CO ) ⋆ + ~ ω f → (cid:0) CO +2 (cid:1) ⋆⋆ + e − ; (2b)This pathway is akin to the one explored recently by Adachi et al. in the experimentalstudy of the ultrafast ionization spectroscopy of CO . An overview of the neutral andionic electronic states mediating different photoreaction pathways is given in Fig. 2. Thephotoexcitations via Eq. (1) are shown with brown arrows and via Eq. (2) — with dark bluearrows. In either pathway, the unstable cation (cid:0) CO +2 (cid:1) ⋆⋆ dissociates into the arrangementchannels containing fragment ions, for example O + , CO + , or C + : (cid:0) CO +2 (cid:1) ⋆⋆ → O + + CO + e − ;CO + + O + e − ;C + + OO + e − . (3)The fragment ions are ultimately detected using VMI. The dissociation threshold relevantfor the production of O + /CO + fragment ions is close to 19.0 eV (see Ref. 53 and Fig. 2 inwhich thresholds for various O + /CO and O/CO + channels are marked on the energy scaleabove the ground state of the neutral CO ). The appearance of C + ions is established tooccur between 25.0 eV and 30.0 eV and is attributed to the 3-body dissociation; in Fig. 2,the C + /O/O channel is located close to 25 eV. As an aside, we note that recent experimentsof Lu et al. demonstrated that CO can decompose into the 2-body channel C and O . This might suggest that a similar 2-body arrangement channel C + /O , lying upwards of17.0 eV, could be detected, too. However, we found no published experimental result so far.The two-photon excitations in these reactions can be either concerted or sequential. Gen-eral expressions for the NFT signals, discussed in Sects. II C and II D, account for both (X Σ g+ ;000)CO (triplets)CO (1 Σ u- ;1 Π g ;1 ∆ u )Ryd( Σ u+ ; Π u ) O( P)/CO(X Σ + )O( D)/CO(X Σ + )O( S)/CO(X Σ + )O( P)/CO(a Π )O( D)/CO(a Π )C( D)/O (X Σ g- )C( P)/O (b Σ g+ )C( P)/O ( ∆ g )C( P)/O (X Σ g- )C( P)/O (X Σ g- ) CO (X Π g ;000)CO (a Π u )CO (b Σ u+ )CO (c Σ g+ )CO (5 Π u )C + ( P)/O (X Σ g- )C + ( P)/O ( ∆ g )C+( P)/O (b Σ g+ )C+( P)/O( P)/O( P)C+( P)/O (X Σ g- )C+( P)/O ( ∆ g ) O+( S)/CO(X Σ + )O( P)/CO + (X Σ + )O( D)/CO + (X Σ + )O( P)/CO + (a Π )O( S)/CO + (X Σ + )O( S)/CO + (a Π ) Ionization Energy 13.78 eVC/C + containing channels CO/CO + containing channelsCO /CO parent excitations − h ω i − h ω f pathway (1)pathway (2) Blue/green: CO doublet states P o t en t i a l E ne r g y [ e V ] FIG. 2: An overview of ab initio electronic spectra of CO (mainly singlet states) and CO +2 (doubletstates). The ground electronic state of CO and the electronic states of CO +2 are calculated in thiswork and used in the quantum mechanical NFT spectral calculations. The excited electronic statesof CO , shown with gray color, are taken from Refs. 28 and 33. The known two-body dissociationchannels of CO and CO +2 , with dissociation products and their electronic states. Brown arrowsillustrate photochemical pathway of Eq. (1); dark blue arrows — photochemical pathway of Eq.(2). Black arrows indicate dissociations in the electronic states of CO +2 . pathways. The ab initio model in Sect. III A is constructed for the reactions of Eqs. (1) and(3) which constitute the relevant pathway under the experimental conditions of Ref. 46.The above reaction schemes illustrate the main assumptions made in this work: A1. Excitations in the Franck-Condon zone only . Equations (1) and (2) imply thatall interactions with photons take place in the Franck-Condon zone, before either the neutralCO or the cation CO +2 start to decompose. This assumption is justified: Both the APTdurations and the pump probe delay times in the NFT experiments are smaller than 10 fs and are therefore substantially shorter than the characteristic times of vibrational motionin CO or CO +2 . Judging by the resonance lifetimes of ∼ −
100 fs calculated for CO , the dissociation reactions are expected to unfold on a much longer time scale. This hastwo implications: First, the ionization of the neutral dissociation fragments, such as CO,O, or C, makes no contribution to the observed NFT signal and can be ignored. Second,the electronic states contributing to the total ion yield (or the 1D NFT spectrum), as wellas the transition matrix elements between them, can be found from the quantum chemicalcalculations limited to the Franck-Condon zone. This simplification is used in Sect. III A toset up the ab initio model. A2. Single ionizations only . The reaction schemes in Eqs. (1) and (2) involves only singleionizations. Double ionization of carbon dioxide is also possible and has been extensivelyinvestigated.
The cross section grows with energy in excess of threshold located 37.3 eVabove minimum of CO . However, the ratio of CO to CO +2 does not exceed 2% even30 eV above threshold, and can be neglected in experiments operating APTs with sumfrequencies below 70 eV or so. This assumption is also consistent with the mass spectroscopicmeasurements of Ref. 46 which indicate very low intensities for peaks corresponding to thedoubly ionized species. A3. Two-photon processes only . The reactions in Eqs. (1) and (2) consume only twophotons. This is in line with the second order perturbation theory in the molecule-lightinteraction which we use to find the time dependent excitation and ionization amplitudes.The perturbation theory is known to be reliable for ultrashort laser pulses. The role ofhigher order processes in the NFT spectroscopy of CO was discussed, but no conclusiveevidence was found so far. An extension to nonperturbative treatment of excitations andionizations can be made (see, for example, Ref. 51), but is outside the scope of this work.With these assumptions, we seek to develop a minimum theoretical description adequatefor a quantitative analysis of experimental 1D NFT spectra of photoionizing small polyatomicmolecules — taking CO as an example. B. The Hamiltonian
Quantum mechanical theory of NFT spectra in the setup involving ultrafast time resolvedionization of polyatomic molecules is based on the approach developed in the seminal papersby Seel and Domcke.
The electronic basis includes the ground electronic state | φ i ofCO ; a set of excited electronic states of neutral CO , {| φ α i} ; the one-electron continuumstates | ψ ek i , corresponding to the photoelectron kinetic energy E k ; and a set of ion core states { (cid:12)(cid:12) φ + j i} of CO +2 . Direct products (cid:12)(cid:12) φ + j ψ ek i define the ionization continua in the model. Thechoice of the electronic basis is governed by the assumption A2 of Sect. II A — only singleionizations are considered.The molecular Hamiltonian in this basis has the following form: H M = | φ i H h φ | + X α | φ α i H α h φ α | + X j Z ∞ d E k (cid:12)(cid:12) φ + j ψ ek i ( H j + E k ) h φ + j ψ ek (cid:12)(cid:12) . (4)The electronic basis states, both neutral and ionic, are treated as diabatic. Possible non-adiabatic off-diagonal interactions between them are suppressed and will be explicitly con-sidered in a separate publication on 2D NFT spectra.The Hamiltonians H , H α , and H j describe the vibrational dynamics in the electronicground state of CO , in the excited electronic states of CO , and in the electronic states ofCO +2 , respectively. Vibrational eigenstates in each electronic state are given by: H | i = ǫ | i ; H α | v α i = ǫ vα | v α i ; H j | v j i = ǫ vj | v j i ;The vibrational energies ǫ vα and ǫ vj are measured with respect to the energy ǫ of the groundvibrational state in the state ˜ X Σ + g of CO . For example, the energy ǫ vj =1 of the groundvibrational state in the ground electronic state ˜ X Π g of CO +2 is approximately equal to13.8 eV, the ionization energy of CO [cf. Fig. 2].The external electric field E ( t ) is comprised of two APT fields, one of which is delayedby a time τ : E ( t ) = E APT ( t,
0) + E APT ( t, τ ) . (5)The time profile of each APT field is determined by the envelope function L n ( t, τ ): E APT ( t, τ ) = n X n = n ′ a n L n ( t, τ ) e − iω n ( t − τ ) . (6)Here n is the harmonic order (the primed sum runs only over odd orders); ω n are theharmonic frequencies; a n is the amplitude of the n -th harmonic in the APT. Gaussian timeenvelope is often considered, L n ( t, τ ) = (cid:18) Pπ (cid:19) / T e − P ( t − τ ) /T , (7)0and will be used in the numerical calculations in this work. In this expression, T is theFWHM of the APT Gaussian envelope and P = 4 ln 2. The pump-probe time delay τ is thetime interval between the centers of the Gaussian envelopes of the two APTs. The APTspectral shape for the pulse of an experimentally realistic duration of T = 5 fs is shown inFig. 1(b). The widths of the individual harmonic peaks are slightly below 1 eV and thereforeare comparable to the fundamental frequency ω (which is 1.55 eV in the figure).The interaction between the molecule and the external electric field E ( t ) includes the fol-lowing components capable of describing the pump-probe dynamics along the photoreactionpathways of Eqs. (1) and (2): Photoreaction pathway of Eq. (1).
The interaction term describing this pathwaycomprises two components: W ( t ) = W A ( t ) + W B ( t ) . (8)with one of them mediating ionization out of the ground state of CO , W A ( t ) = − X i Z ∞ d E k (cid:12)(cid:12) φ + i ψ ek i µ i ( E k ) E ( t ) h φ | + h.c. , (9)and the other allowing optical excitations between states of the free ionic core: W B ( t ) = − X i,j = i (cid:12)(cid:12) φ + j i µ ji E ( t ) h φ + i (cid:12)(cid:12) Z ∞ d E k | ψ ek ih ψ ek | + h.c. (10)In these expressions, µ i ( E k ) is the ionization dipole moment, and µ ji is a transitiondipole moment (TDM) between ionic states j and i .2. Photoreaction pathway of Eq. (2).
The interaction term is structurally similar tothe above: W ( t ) = W A ( t ) + W B ( t ) , (11)with one term describing optical excitations in the neutral molecule, W A ( t ) = − X α | φ α i µ α E ( t ) h φ | + h.c. , (12)and the other giving rise to the ionization from the state | φ α i of CO : W B ( t ) = − X jα Z ∞ d E k (cid:12)(cid:12) φ + j ψ ek i µ jα ( E k ) E ( t ) h φ α | + h.c. . (13)1Here µ α are the TDMs for optical excitations in the neutral CO from the groundelectronic state ˜ X Σ + g , and the ionization dipole moments µ jα ( E k ) are now defined fora given pair of the ionic state j and the neutral state α .The total Hamiltonian governing the dynamics in the time dependent electric field E ( t ) isgiven by the sum of the molecular Hamiltonian [Eq. (4)] and the interaction with laser field: H T OT ( t ) = H M + W ( t ) + W ( t ) . (14)The TDMs between diabatic states are treated in the Condon approximation consistentwith the diabatic representation. For the ionization step, the dipole moments µ i ( E k ) and µ jα ( E k ) depend on the photoelectron kinetic energy E k and are subject to the boundarycondition µ ( E k ) → E k → ∞ . Following Seel and Domcke, we approximate thisdependence by a simple step function, e.g. µ i ( E k ) = µ i , if 0 ≤ E k ≤ E max k , if E k > E max k (15)The cutoff energy E max k is a parameter of the calculation controlling the width of the pho-toelectron spectrum in a given ionic state. C. NFT signals
The NFT signal calculated in this work is the total yield I ion ( τ ) of the detected fragmention [for example, C + , O + , or CO + , cf. Eq. (3)]. This 1D signal is proportional to the totalpopulation of the ionic electronic states dissociating to produce the detected ion. The laddermodel in Fig. 2, approximating the electronic spectrum of CO +2 with ab initio energies atthe Franck-Condon point, can be taken as a starting point for a calculation of I ion ( τ ). Afterthe pump and probe pulses, the system, consisting of CO +2 and a photoelectron, is left inthe state Ψ I ( t | E k , τ ). The population p f of a dissociative final state | f i of CO +2 is given by p f ( E k , τ ) = (cid:12)(cid:12) h φ + f ψ ek | Ψ I ( t → ∞| E k , τ ) i (cid:12)(cid:12) . (16)The total ion yield I ion ( τ ) is proportional to this population integrated over the photoelectronkinetic energy E k and summed over all final states having energies ǫ f above the appearance2threshold A ion of the detected ion: I ion ( τ ) = X f : ǫ f >A ion Z ∞ d E k p f ( E k , τ ) . (17)Fourier transform of I ion ( τ ) gives the frequency domain 1D NFT spectrum: I ion ( ω NFT ) = Z τ max − τ max dτ I ion ( τ ) e iω NFT τ (18)The maximum time delay τ max defines the spectral resolution in the NFT frequency ω NFT .NFT signals of CO are also reported as 2D maps I ion ( ǫ kin , τ ), in which the kinetic energydistribution of the recoiling fragment ions is measured for different time delays. The NFTobservable in this case is directly related to the dissociation dynamics in the ionic states,and the description of the signal requires basic elements of scattering theory. The scatteringapproach to the 2D NFT signals is outlined in Appendix A. Numerical applications areconsidered in a separate publication. D. Total ion yield via time dependent perturbation theory
Evaluation of the NFT signals described with Eqs. (16) and (17), as well as with Eqs.(A3) and (A5), requires the time dependent molecular wave function | Ψ I ( t ; E k , τ ) i in theinteraction representation. We expand it in the electronic basis introduced in Sect. II B: | Ψ I ( t ; E k , τ ) i = χ ( t ) | φ i + X α χ α ( t ) | φ α i + X j Z ∞ d E k χ j ( t ; E k , τ ) (cid:12)(cid:12) φ + j ψ ek i , (19)Here the coefficients χ ( t ) and χ α ( t ) are the nuclear wave functions in the ground (index 0)and excited (indices α ) electronic states of the neutral molecule; χ j ( t ; E k , τ ) are the nuclearwave functions of the molecular ion depending on the photoelectron kinetic energy E k and,through the APT fields, on the pump probe time delay τ . The initial state of the moleculeis assumed to be the vibrational ground state in the electronic ground state, | φ i | i .The coefficient χ (2) j ( t ; E k , τ ) for the CO +2 ion in one of the final dissociative states is eval-uated using the second order time dependent perturbation theory in the interactions W ( t )and W ( t ). This is in line with the assumption A3 of Sect. II A, so that the theoretical de-scription focuses on the two-photon processes. Perturbation theory allows us to concentrateon the observable ion signal and bypass a rigorous description of the ejected electron which3has not been detected in the NFT spectroscopic experiments on CO . In fact, the photo-electron dynamics is very rich in carbon dioxide, and was a subject of detailed theoreticaland experimental studies on the attosecond time scale. The contributions of the interactions with laser fields to the wave functions (and to thepump probe amplitudes) are additive. Considering the interaction W ( t ), the second ordernuclear wave function is given by: χ (2) j ( t ; E k , τ ) = 1 i X i = j Z t −∞ dt Z t −∞ dt e iH j t [ µ ji E ( t )] e iE k t e − iH i ( t − t ) [ µ i ( E k ) E ( t )] e iH t | i . (20)We use ~ = 1 throughout the paper. The amplitude χ (2) j describes ionization of the moleculein the vibrationless ground state | i into an ionic state i at time t , and a subsequent opticalexcitation of the state i into the final state j at time t [photoreaction via the Eq. (1)]. Thepump probe experiment is represented as a linear combination of photoionization/excitationevents separated by the temporal delay of length τ . Projecting this function onto the finalvibrational state h v j | and taking the limit t → ∞ gives the asymptotic second order pumpprobe amplitude due to the interaction W ( t ): c j ( v j ; E k , τ ) = 1 i X i = j X vi µ ji h v j | v i i µ i ( E k ) h v i | i× Z ∞−∞ dt e i ( ǫ vj − ǫ + E k ) t E ( t ) Z t −∞ dt e − i ( ǫ vi − ǫ + E k )( t − t ) E ( t ) . (21)The pump probe amplitudes c j ( v j ; E k , τ ) depend on the time delay τ via the time dependenceof the laser field E ( t ), cf. Eqs. (5) and Eqs. (6) and (7). The coefficients h v i | i in Eq. (21) arethe expansion coefficients of the initial vibrational state | i in the vibrational states of theintermediate ionic states, | v i i ; the coefficients h v j | v i i are the projections of the intermediateionic vibrational states onto the final state | v j i .Repeating the same steps for the interaction W ( t ), one finds another contribution tothe pump probe amplitude due to an optical excitation of the molecule in the vibrationlessground state | i into a neutral state α at time t , and a subsequent ionization of the state α into the final state j at time t [photoreaction via the Eq. (2)]: d j ( E k , v j , τ ) = 1 i X α> X vα µ jα ( E k ) h v j | v α i µ α h v α | i× Z ∞−∞ dt e i ( ǫ vj − ǫ + E k ) t E ( t ) Z t −∞ dt e − i ( ǫ vα − ǫ )( t − t ) E ( t ) . (22)4The coefficients h v α | i and h v j | v α i in this equation are similar to the respective coefficientsin Eq. (21), but refer to the projections of the vibrational states in the excited electronicstates of the neutral molecule. The total pump probe amplitude b j ( E k , v j , τ ) is given by thesum of the amplitudes for the two pathways: b j ( E k , v j , τ ) = c j ( E k , v j , τ ) + d j ( E k , v j , τ ) (23)The ionic signal I ion ( τ ), defined in the Eq. (17), is proportional to the population p f in thefinal dissociative states | f i capable of producing the detected ion, i.e. to the square of thepump probe amplitudes | b f | collected over vibronic levels ǫ vf lying above the appearancethreshold A ion : I ion ( τ ) = X f X ǫ vf >A ion Z ∞ d E k | b f ( E k , v f , τ ) | . (24)This is a working expression in the calculations discussed in Sect. III. It is valid even ifdifferent electronic states interact non-adibatically — the vibronic energies ǫ vf in this caseshould refer to adiabatic rovibronic eigenstates. III. RESULTS AND DISCUSSIONA. Ab initio calculations of CO and CO +2 The challenge in modeling NFT spectra for polyatomics is to construct an adequate(preferably ab initio) model of electronic states of the neutral molecule and the molecularion. Strictly speaking, potential energy surfaces depending on all three internal coordinatesare needed to find the eigenstates ǫ vj and to calculate even the total ion yield. Due toa multiphoton nature of the NFT technique and the XUV harmonics used for excitationand ionization, this can easily become a formidable task because tens or even hundreds(interacting) electronic states might be needed. In this work, we use the assumption A1 of Sect. II A and calculate the electronic energylevels, as well as the dipole moments for the ionization and optical excitation steps, at theequilibrium geometry of the ground electronic state of the neutral molecule. The symmetrygroup of this Franck-Condon point is D ∞ h . In the electronic structure calculations, it isrendered as D h .5The ab initio calculations are further simplified by concentrating the analysis on thephotoreaction pathway of Eq. (1), so that ionizations take place directly from the groundelectronic state of CO . This relieves us of having to calculate the densely spaced mixedRydberg-valence states of the neutral molecule close to the ionization threshold. The pho-toreaction pathway of Eq. (1) is indeed likely to make the main contribution to the observedNFT signals of CO . Carbon dioxide is transparent up to about 6.20 eV (where the absorp-tion still remains extremely weak), and the first strong absorption band is observed near11.08 eV.
The molecule ionizes at 13.8 eV, while the harmonics which substantially con-tribute to the APTs considered in the present calculations carry excitation energies of morethan 14.0 eV [cf. Fig. 1(b)]. Possible spectral signatures of the complementary pathway ofEq. (2) will be indicated in the discussion of the results.The augmented correlation consistent polarized valence quadrupole zeta (aug-cc-pVQZ)basis set due to Dunning is used for all atoms. Energies of CO and CO +2 are calculated atthe internally-contracted multireference configuration interaction singles and doubles (MRD-CI) level, based on state-averaged full-valence complete active space self-consistent field(CASSCF) calculations with 16 electrons in 12 active orbitals and 6 electrons in three fullyoptimized closed-shell inner orbitals. Active space in CASSCF comprises orbitals 2 σ u − σ u ,3 σ g − σ g ,1 π u − π u , and 1 π g . In the MRD-CI step, all 16 valence electrons are correlated.The Davidson correction is applied in order to account for higher-level excitations and sizeextensivity. The singlet ground electronic state of CO and a series of doublet states of CO +2 are calculated with this setup. The MRD-CI calculations for the ion are performed using themolecular orbitals of the neutral molecule in order to simplify the evaluation of ionizationmatrix elements. All ab initio calculations are carried out with the MOLPRO package. More than 50 doublet electronic states of CO +2 are calculated and assigned. The full listof converged states includes 1 − Σ + g ; 1 − Σ + u ; 1 − Σ − g ; 1 − Σ − u ; 1 − Π g ; 1 − Π u ;1 − ∆ g ; and 1 − ∆ u . These states span the energy range from 13.8 eV to 32.5 eV abovethe minimum of the ground electronic state of the neutral CO molecule.The ionization dipole moments µ i between the states of CO +2 and the ground electronicstate of CO are evaluated for the CI vectors calculated in the common basis of the molecularorbitals of the neutral CO . For a given ionic state i , the photoionization dipole moment6 µ i reads as µ i = X s d ks ( E k ) x si . (25)It is a sum of products of bound-free one-electron dipole integrals d ks ( E k ) which depend onthe photoelectron kinetic energy E k , and the ‘spectroscopic factors’ x si defined as x si = h φ i | ˆ c s | ˜ X Σ + g i . (26)where ˆ c s stands for the annihilation operator for the orbital s , and φ i is the electronic wavefunction of the cation state i . The spectroscopic factors are calculated for the CI vectors ofCO and CO +2 using the algorithm proposed by W. Eisfeld. Note that spin selection rulesare relaxed in photoionization, and higher multiplicity spin states of CO +2 (e.g. quartets) canin principle be ionized, too. Test calculations performed for the four lowest quartet states1 , A ′ and 1 , A ′′ demonstrate, however, that the spectroscopic factors for these ionizationsare small.Optical transitions between the states of the ion require interstate TDMs µ ji . Compo-nents ( µ x , µ y , µ z ) of the TDM vector are calculated at the high symmetry Franck-Condonpoint. The coordinate axes in these calculations are chosen such that z runs along the molec-ular figure axis, while x and y are orthogonal to z . The ab initio TDMs are computed at theCASSCF level of theory. Previous calculations demonstrate that the difference betweenthe TDMs calculated using CASSCF and MRD-CI methods is less than 10%.At the high symmetry Franck-Condon geometry, the selection rules effectively reduce thenumber of accessible optically bright states; for many pairs of symmetry species, the x , y ,and z components of the TDMs µ ji vanish. For example, for the five states 1 Π g , 1 Π u ,1 Σ + u , 1 Σ + g , and 5 Π u (they have the largest photoionization probabilities), the opticaltransitions are driven by very few non-zero TDM components: Π g µ z µ x,y − µ z Π u − − − Σ + u µ z − Σ + g − Π u (27)In this symbolic representation, the TDM components mediating transitions between stateson the main diagonal of the matrix are enumerated as off-diagonal ‘matrix elements’. Theresulting ‘matrix’ is sparse.7Consequently, only a subset of all calculated states of CO +2 are included in the quantummechanical calculations of the NFT spectra. This subset is shown in Table I and in Fig.2. The accuracy of the ab initio calculations can be judged (for the first several states) bycomparison with the known experimental energies, also shown in Table I where available.The states of CO +2 in Table I are selected as follows: (1) Five states with the largest pho-toionization probabilities | µ i | referenced in Eq. (27). They are shown above the upperhorizontal line in Table I and as magenta lines in Fig. 2. (2) Thirteen states having strong( > . n Σ + g , n Π u and n Σ + u and shown below the lower horizontal line in Table I, convergence atthe MRD-CI level was not achieved. Their energies were shifted in the quantum mechanicalcalculations to the values shown in parenthesis. These states are shown as green lines in Fig.2. These states collectively represent the electronic states lying outside the energy rangecovered by the present ab initio calculations. B. Calculated and experimental NFT spectra
The NFT signals are calculated using Eqs. (18) and (24), and only the pump probeamplitude c j ( E k , v f , τ ) of Eq. (21) is included. The vibrational excitations are suppressedfor all states, so that each electronic state contributes one energy ǫ j coinciding with the abinitio energy for this state in Table I.It is assumed that ions in Eqs. (3) are produced with the appearance thresholds of A C+ = 30 . A O+ = A CO+ = 19 . A ion contribute to the fragment ion generation. As explainedin Sect. II A, the appearance thresholds for O + and CO + are close to the known channeldissociation thresholds shown in Fig. 2, while A C+ is taken substantially higher than thelowest dissociation energy of 23 eV, at which the production of C + ions becomes energeticallyallowed.Two sets of APTs, shown in Fig. 1(a), are used in the calculations. Set 1 (red boxes) spansthe harmonic orders from n = 9 to n = 19, with a standard choice of harmonic amplitudesof the components. Set 2 covers harmonic orders from n = 1 to n = 21, as reported forthe APTs in Ref. 46. Compared to Set 1, it incorporates contributions from low harmonic8orders n ≤
7. We also artificially amplified the amplitude for n = 1. Such an amplitudedistribution is valuable because it describes realistic experimental conditions under whichthe fundamental ω or the adjacent odd harmonics are not fully suppressed e.g. by dichroicmirrors, filters, or harmonic separators. The additional change in the amplitude for n = 1on top of the reported amplitudes, helps to rationalize the intensity of this harmonic peakin the measured NFT spectrum.The ionization transition matrix elements µ i ( E k ) between the ground electronic state ofCO and the first five ionic states in Table I are assumed to be constant up to the maximumcutoff energy E max k [cf. Eq. (15)], and negligible beyond this energy; E max k determines theupper integration limit over the electron kinetic energy in Eq. (24). Most calculations inthis work are carried out with E max k = 3 . µ i ( E k ) on E k stems fromthe bound-free dipole integrals d ks ( E k ) in Eq. (25) and is essentially governed by the overlapof a valence orbital of CO and a distorted plane wave corresponding to the photoelectronejected with the kinetic energy E k . This overlap drops with increasing E k leading to narrowphotoelectron spectra of these states, with the measured and calculated electron bind-ing energy widths below 2.0 eV. Broad photoelectron kinetic energy distributions exceeding7.0 eV, were observed for CO initially pre-excited into the Rydberg states near ionizationthreshold. Additionally, electron kinetic energies up to and exceeding 10 eV were previ-ously measured in the ionization of CO with XUV APTs. In order to assess the role of E max k , some calculations for Set 1 were repeated with E max k = 10 . ω NFT . The impact of E max k on the peak intensities wasvisible only at high NFT frequencies above 13 ω .An example of the calculated NFT signal I C+ ( τ ) in the time domain is shown in Fig. 3for the pump-probe APTs with the amplitude Set 1. The fundamental harmonic frequencyis ω = 1 .
55 eV, close to the experimental value used in Ref. 46. The pump probe timedelay τ in the calculations varies between ± τ max = ±
20 fs. This guarantees a tidy renderingof the NFT spectra in the frequency domain via Eq. (18). The chosen τ max is about threetimes larger than the experimental value.In the NFT signal I C+ ( τ ) in Fig. 3, most of the visible dynamical effects are concentratedwithin | τ | ≤ ω , the signal amplitudes are observed to decrease substantially(not shown in Fig. 3). The dependence of the NFT signal intensity on the fundamental laser9frequency is considered in more detail in Sect. IV. FIG. 3: Ab initio pump probe NFT signal I C+ ( τ ) in a simulation with the fundamental frequencyof ω = 1.55 eV and the amplitude Set 1. The optical cycle of the fundamental laser pulse is T = 2 .
67 fs.
Fourier transform of the signal in Fig. 3 gives the frequency dependent spectrum I C+ ( ω NFT ) depicted in Fig. 4(a). The calculated spectrum consists of a series of non-overlapping peaks with FWHM of about 0.9 eV, similar to the width of the harmonicsmaking up the pump and probe APTs. In order to simplify visual comparison with thespectral content of the incident APTs, the NFT frequency axis is shown in the units of thefundamental frequency ω . Two groups of spectral peaks can be identified. The first groupcomprises peaks located sufficiently close to the harmonic orders, either odd, i.e. present inthe incident APTs, or even. Example is the strong triad at ω NFT ≈ ω , 11 ω , and 13 ω inpanel (a), although the peaks 9 ω and 11 ω slightly but visibly deviate from the harmonicposition. The assignment of the excitation patterns, discussed in Sect. III C, suggests thatthis triad is due to (a) ionizations into the the ground electronic state ˜ X Π g of CO +2 , as wellas the first excited doublet state A Π u , and (b) dissociations in the high lying electronicstates belonging to the series n Π g and n Π u . Previous analyses of the NFT spectra concen-trated mainly on the peaks belonging to this group. Gradually attenuating contributionsof 15 ω , 17 ω , and 19 ω can also be recognized. However, they lie beyond the detection0range of the experimental setup limited to less than 25 . ω NFT < ω . Note thatthe relative intensities of the harmonic peaks in the NFT spectrum are quite different fromthose in the incident APT [cf. Set 1 in Fig. 1(a)].
FIG. 4: Ab initio pump probe NFT spectra I C+ ( ω NFT ) of CO calculated using APTs with theamplitude Set 1 (a) and Set 2 (c). The fundamental frequency in the calculations is ω = 1.55 eV.Shown in (b) is the 1D NFT spectrum of Ref. 46 obtained from the experimental 2D NFT spectrumfor the C + fragment in their Fig. 12(c) by integrating over the fragment kinetic energy. All spectraare normalized to the area within the region of ω NFT ≤ . ω NFT /ω . Another example of peaks in this group are spectral lines at even harmonic orders 2 ω and ∼ ω . They are absent in the APTs and have been previously attributed to the1frequency beats between the incident odd harmonics. In other words, they are seen asdifference frequency peaks stemming from the ‘parent’ triad e.g. (13 ω − ω ) for thesecond harmonic peak and (13 ω − ω ) for the forth harmonic peak.The second group of spectral peaks is formed by the peaks located at non-harmonicfrequencies. Examples include the peak between 6 ω and 7 ω and, to a lesser extent, thepeaks near ∼ ω and ∼ ω , as well as the high frequency peaks above 20 ω .The experimental NFT spectrum I C+ ( ω NFT ) is shown in Fig. 4(b). It is obtained fromthe full 2D spectrum published in Ref. 46 by integrating over the kinetic energy axis. Onefinds peaks belonging to the same two groups discussed above. The triad near ω NFT ≈ ω ,11 ω , and 13 ω is again well recognizable. The difference frequency peaks at 2 ω and 4 ω are very strong. This observation is unexpected because the difference peaks turn out tobe much stronger than any ‘parent’ harmonic peak in the spectrum. Their intensities inthe normalized spectrum are six times stronger than in the calculations. Finally, severalnon-harmonic peaks are seen in the experimental spectrum, for example between 4 ω and6 ω . Note however that the experimental spectrum is quite congested with a low intensity‘noisy’ contribution covering the whole experimental NFT frequency range.The amplitude Set 1 does not properly describe the experimental APTs of Ref. 46, becausemany low frequency harmonic components are missing. The amplitude Set 2, depicted inFig. 1 with gray boxes, is a better model of this experiment. The artificially amplifiedamplitude of the fundamental n = 1 is immaterial for the case of C + ion signal and does notaffect the spectrum I C+ ( ω NFT ) at all. The spectrum calculated with Set 2 is shown in Fig.4(c). The low frequency harmonics have no impact on the spectral peaks of the strong triad9 ω − ω − ω , which are still well recognizable, similarly to the experimental spectrum.The Set 2 adds a substantial low intensity component to the spectrum producing a smallspectral peak at every harmonic order n , both even and odd. This feature is also in line withthe experiment. Note that the low intensity component is entirely absent in panel (a), sothat the ‘noise’ in the calculation stems from the harmonic orders 3 ≤ n ≤
7. Nevertheless,the agreement between the calculated and the measured spectra is at best qualitative, andthe relative intensities for many peaks, for example 2 ω and 4 ω , disagree substantially.Lowering the appearance threshold from 30.0 eV to 19.0 eV, one can use the same ladderab initio electronic states to evaluate the NFT spectrum I O+ ( ω NFT ) for the fragment ionO + . The results are summarized in Fig. 5 for the amplitude Set 1 in panel (a) and Set 2 in2panel (c). The experimental spectrum is shown in panel (b). FIG. 5: As in Fig. 4, but for the spectrum I O+ ( ω NFT ). The 1D NFT spectrum in panel (b) isobtained from the experimental 2D NFT spectrum of Ref. 46 for the O + fragment in their Fig.13(c) by integrating over the fragment kinetic energy. The experimental spectra recorded for O + and C + are similar in several respects. Forexample, the triad near 9 ω − ω − ω is still visible in I O+ ( ω NFT ) (although the peaks areshifted more clearly to non-harmonic positions), and the low energy peaks at 2 ω and 4 ω are strong. However, the experimental spectrum I O+ ( ω NFT ) is dominated by the very intensepeak n = 1 of the fundamental frequency ω . Additionally, there are more non-harmonicpeaks in the O + spectrum than in the C + spectrum, and their intensity is higher.The amplitude Set 1 is clearly incompatible with experiment: The spectrum in Fig. 5(a)3is almost the same as in Fig. 4(a) and is insensitive to a shift in the appearance threshold.Indeed, the amplitude Set 1 comprises harmonics with energies above 14 eV, so that thehigh lying states of CO +2 are primarily populated along the photochemical pathway of Eq.(1). In the adopted model, these states contribute to both O + and C + signals.The O + spectrum calculated with Set 2 and shown Fig. 5(c) is different and has a pro-nounced peak at ω NFT /ω = 1. In the C + spectrum, calculated with the same Set 2, thispeak is much weaker. Low order harmonics in Set 2 induce excitations between neighboringelectronic states of CO +2 and this directly influences the NFT spectrum. For example, thespacing between the states 1 Σ + u and 1 Σ + g is 1.26 eV (see Table I) and is nearly resonantwith ω . Still, a single resonant electronic transition in CO +2 alone is not sufficient to makethe peak n = 1 intense: In the calculations, the large amplitude a of the fundamental inthe incident APT is also a pre-requisite; the chosen amplitude a exceeds the value inferredfrom Ref. 46, by about a factor of 20. It is therefore likely that the very strong peak at ω in the experimental spectrum has a different origin missing in the present calculations. Onepossibility are higher order multiphoton excitations. The other are contributions from thephotochemical pathway of Eq. (2), in which CO is photoexcited into the vicinity of theionization threshold and then ionized with an infrared photon. Note that the reversed orderof absorptions would be unfeasible because CO is essentially transparent around 1.55 eV.Finally, there could be yet another explanation for the large amplitude a of the fundamentalin the incident APT needed to reconcile the experiment and theory. In the traditional highharmonic generation setups, the spectral filters and the reflectiveness of XUV mirrors areexpected to attenuate the fundamental frequency by a factor of 10 − to 10 − . One can there-fore reasonably expect that sufficient amount of the fundamental may reach the molecularsystem to affect its time-domain response and NFT spectra. C. Assignments of the calculated spectra: Near-harmonic and non-harmonic peaks
The comparison between the experimental and the ab initio spectra, discussed in theprevious section, offers several insights into the relation between the NFT spectrum and thecomposition of the incident APTs. We turn now to the assignment of the spectral peaks,which allows one to relate the NFT spectra to the properties of the electronic states excitedin the pump probe experiment and to expose the origin of near-harmonic and non-harmonic4spectral peaks. Our discussion is based on the calculations using the amplitude Set 1 and ω = 1 .
55 eV: The spectrum I C+ ( ω NFT ), calculated with these settings, shares many principalfeatures with the experimental spectrum.Appendix B sets the stage for the discussion and summarizes the derivations of analyticexpressions for I ion ( ω NFT ) which can be used to guide the spectral assignment. Specifically,Eqs. (B6) and (B7) provide explicit dependence of the spectral intensity on the NFT fre-quency ω NFT , on the transition frequencies between electronic states, ∆ fi = ǫ f − ǫ i > ω n of the APT components. They are derived using theapproximation akin to the temporarily non-overlapping pump and probe pulses, familiar inthe context of ultrafast pump probe spectroscopies. The accuracy of the approxi-mation of Eq. (B6) is illustrated in Fig. 6. Most of the approximate spectral lines (black)are located at the positions of the numerically exact peaks (red). The intensities, which aremore sensitive to the coherent two photon effects, are less accurate but the overall spectrumis reasonably reproduced.The NFT spectrum derived in Appendix B is a superposition of the ionizations of theparent molecule and subsequent excitations into the final dissociative state. It is thereforenatural to label the NFT peaks using electronic pairs ( i, f ) comprising the ionized state(s) i of CO +2 , reached in the reaction CO → CO +2 ( i ), and the final dissociative state(s) f ofCO +2 ( f ) contributing to the observed C + or O + signals. The second set of labels are theharmonic assignments indicating the photon frequencies ω n promoting specific transitionsleading to the ionization and dissociation. In practice, we scan the sums in Eq. (B6) for theterms making the largest contribution at a specified ω NFT .The assignments of the calculated spectrum in terms of these attributes are summarizedin Fig. 6. The intermediate states in the ionization step include the ground electronic stateof CO +2 , ˜ X Π g as well as the first excited doublet state ˜ A Π u . The final electronic statescontributing to the fragment ion signal belong to the series n Π g and n Π u .In fact, two distinct assignment schemes emerge for the spectral peaks — and they arebest illustrated using Eq. (B7) which is valid for the specific case of only one intermediateionized state i contributing to the dissociation via the final state f . The spectrum I C+ ( ω NFT )5 FIG. 6: Comparison of the numerically exact quantum mechanical NFT spectrum (red line) withthe approximation of Eq. (B6) (black line). All calculations are performed for I C+ ( ω NFT ) usingAPTs with the amplitude Set 1. The fundamental frequency is ω = 1.55 eV. Combs show theassignments in terms of the harmonic frequencies ω n and the energy differences ∆ fi between theionization and the final dissociation states. States contributing to the assigned peaks are numberedas in Table I: 1 = 1 Π g , 2 = 1 Π u , 15 = 8 Π u , 16 = 10 Π g , and 21 = n Π u . Thick (thin) verticallines indicate odd (even) integer harmonic orders ω NFT /ω . in this case consists of two additive contributions. The first is given by( µ fi µ i ) X n ,n ,n ,n a n a n a n a n A n ,n L n (∆ fi ) L n (∆ fi ) × n δ τ ( ω NFT ) + 2 δ τ ( ω NFT − ∆ fi ) + 2 δ τ ( ω NFT + ∆ fi ) o . (28)Here L n i (Ω) = e − (Ω − ω ni ) T / P are the Gaussian Fourier images of the time envelope function L n i ( t, τ ) of the APTs defined in Eqs. (5)—(7). The transition is mediated by four photonswith harmonic frequencies n , n , n , and n . The resulting NFT spectral peaks are locatedat ω NFT = 0 (the zero peak which we do not consider) and at the energy differences betweenthe intermediate ionized and the final dissociative states, ω NFT = ± ∆ fi . They are denoted δ τ because their width is only controlled by the maximum delay time in the time signal, andcan potentially be made narrow (delta-function like) by increasing τ max . Note that these6spectral peaks can appear at non-harmonic frequencies if the energy ∆ fi is off resonance.The calculation with the Eq. (B6) recognizes two such spectral lines in Fig. 6: One near ω NFT ≈ ω and the other at 11 ω . They belong to the strong central triad discussedin the previous section. We also use this assignment for the third member of the triad, ω NFT ≈ ω . The peaks of the triad exemplify the appearance of near harmonic and non-harmonic peaks in the spectrum. The pair of electronic states ( i = ˜ X Π g , f = 8 Π u ) has,according to Table I, ∆ fi = 17 .
08 eV which is nearly resonant with the harmonic frequencyfor n = 11 (17.05 eV). It gives rise to an intense line at this harmonic order. The pair ofstates ( i = ˜ A Π u , f = 10 Π g ) is spaced by 13.49 eV, which is off resonance with respectto the harmonic frequency ω = 13 .
95 eV. The detuning affects the intensity of this off-resonance non-harmonic line. Indeed, the Gaussian prefactors L n (∆ fi ) L n (∆ fi ) suppresslarge deviations of ∆ fi from integers n = n = 9. However, the Gaussians are almost 1.0 eVbroad and — additionally — τ max is large (20 fs) in this calculation, so that the intensityof this non-harmonic line becomes appreciable. Finally, the peak near ω NFT ≈ ω canbe plausibly assigned to the pair of states ( i = ˜ X Π g , f = n Π u ) which are in resonancewith the harmonic frequency for n = 13 (20.15 eV); the offset is merely 0.1 eV, and the thirdmember of the triad is perceived as a harmonic peak.The second contribution to the NFT spectrum in Eq. (B7) has a different form:( µ fi µ i ) X n ,n ,n ,n a n a n a n a n L n (∆ fi ) L n (∆ fi ) × n L n ( ω NFT ) L n ( ω NFT ) + L n ( ω NFT − ∆ fi ) L n ( ω NFT − ∆ fi )+ L n ( ω NFT + ∆ fi ) L n ( ω NFT + ∆ fi ) o (29)There are three sets of peaks associated with this contribution: One set is located at theoriginal harmonic frequencies, ω NF T ≈ ω n or ω n . In Fig. 6, they are shown with the greencomb. These peaks are locked on the spectral composition of the incident APTs and havelittle to no dependence on the ionic energy levels. The third member of the strong triad at13 ω contains a low intensity contribution of this type. In the second set, these harmonicpeaks are shifted to lower energies by the energy difference ∆ fi . In Fig. 6, they are shownwith the red comb, and the electronic states are again i = ˜ X Π g and f = 8 Π u . Becausethe energy spacing ∆ fi is in resonance with ω , the shifted peaks appear at even harmonicsbetween n = 2 ω and n = 8 ω and are interpreted as difference peaks in the NFT spectra.7According to Eq. (29), their intensity is one half the intensity of the unshifted harmonicpeaks. In the third set, the harmonic peaks are shifted to higher energies by the sameamount ∆ fi . They are located at high NFT frequencies and marked with the blue comb.In the simple approximation which we are using, their information content is the same asin spectral peaks shifted to low ω NFT . In the present calculation, the peaks of all threesets, even the shifted ones, appear near the integer harmonic orders, either even or odd.This is because the electronic states involved in the optical transitions within the ion are inresonance with one harmonic frequency. However, each NFT peak in Eq. (29) is a product offour Gaussian factors L n i , and each Gaussian has a width of the order of ω /
2. Thus, peakslocated at non-harmonic frequencies can in principle be expected for the shifted lines. Infact, deviations from integer harmonic orders can be seen for the peaks at ∼ ω and ∼ ω ,as well as ∼ ω . These deviations can be related to the specific electronic transitions inthe molecular ion.Spectral peaks at different NFT frequencies can be further characterized by the rangesof pump probe delay times, at which they are formed. The relation between ω NFT and τ isvisualized using the so-called spectrogram S C+ ( ω NFT , t ) = Z I C+ ( τ ) h ( τ − t ) e iω NFT τ dτ , (30)which is a moving window Fourier transform of the pump probe signal. Here h ( τ ) =exp( − τ / τ ) is the Gaussian window function. In the spectrogram S C+ ( ω NFT , t ), shown inFig. 7, the delay time and the NFT frequency domains are represented in the same plot atthe expense of the resolution which is smeared along both axes:
The time resolution τ in the plot is 0.9 fs and corresponds to the frequency resolution of τ − = 0 .
80 eV, about halfthe fundamental frequency ω .All major spectral peaks can be clearly recognized in the spectrogram, and they acquirean additional — time delay — dimension. The spectrogram provides a different perspectiveon the assignment and helps to identify spectral features observed over the same rangesof the delay times. For example, the peaks in the strong central triad 9 ω − ω − ω have similar spectrograms and collect their intensities over a broad time delay range from-5.0 fs to +5.0 fs. This is an additional reason to assign them using one common assignmentscheme as done in Fig. 6. The peaks at 2 ω and at 22 ω extend over a similarly broad timedelay range — they both belong follow the assignment in terms of ω n ± ∆ fi and share the8 FIG. 7: The spectrogram S C+ ( ω NFT , t ) as defined in Eq. (30), with ω NFT plotted in the units of ω . Red contours correspond to the maxima. The calculations are performed using APTs withthe amplitude Set 1. The fundamental frequency in the calculations is ω = 1.55 eV. Vertical linesmark even and odd harmonic orders. electronic states f and i with the central peak of the triad. For the weaker peaks in thespectrum, the relationships within spectral groups are less straightforward. For example, thepeaks at ∼ ω , 15 ω , and ∼ ω are due to short delay times | τ | ≤ . , shows their relation to the experiment, and rationalizes the spectral assignmentsdemonstrating the origin of near harmonic and non-harmonic peaks in the spectra. IV. CONCLUSIONS AND OUTLOOK
In this paper, we outlined a quantum mechanical approach to modeling attosecond NFTspectra of CO ionizing to CO +2 . The approach combines perturbation theory for themolecule-light interaction with ab initio calculations of the electronic energy levels of CO +2 .9 FIG. 8: Ab initio pump probe NFT spectra I C+ ( ω NFT ) of CO for a set of fundamental harmonicfrequencies ω . The incident APTs are constructed using the amplitude Set 1. All spectra arenormalized to the area within the region of ω NFT ≤ . ω NFT /ω . The ab initio calculations are performed using accurate MRCI method accounting for theelectron correlation, but in this work they are limited to the Franck-Condon zone — so thatthe resulting ladder of electronic levels of CO +2 represents a ‘toy model’ of the molecular ion.The main results can be summarized as follows:1. One-dimensional NFT spectra of CO are calculated for two different incident APTsand compared with the available experimental spectra. Several features of the exper-iment, including the positions and intensity patterns of the main harmonic spectralpeaks, the presence of the difference frequency peaks, and the origin of the low intensitycontribution at all harmonic orders, are reproduced and/or explained.2. The calculations give an overview of the NFT spectrum outside the experimental0frequency window, for ω NFT ≥ . ω NFT , the harmonicfrequencies ω n of the APT components, and the transition frequencies between elec-tronic states. These expressions guide the assignment of the spectral peaks.4. The calculated NFT spectra are assigned in terms of the participating electronic statesand the harmonic photon frequencies. The assignment demonstrates which details ofthe electronic structure of the CO +2 are captured in the NFT experiments. Usingspectrograms, the NFT peaks can be additionally attributed to specific ranges of thepump probe delay times.5. It is shown that spectral peaks at non-harmonic frequencies can be expected, espe-cially if APTs with spectral bandwidths of about 1 eV are used and photoreactionsare limited to single ionizations. Non-harmonic NFT spectral peaks carry additionalinformation on the electronic states mediating ionization of the parent molecule anddissociation of the molecular ion.The main goal of this work was to set up a framework, within which NFT spectra canbe calculated and analyzed, and to test this framework for CO . The tests demonstratethat the framework works with reasonable precision and can be used for semi-quantitativepredictions, even though CO +2 is described using an ab initio toy model. The simplicity ofthe ab initio calculations limited to a single Franck-Condon geometry makes this frameworkhighly scalable: It can be applied to polyatomic molecules and ions and one can easily scanthrough various control parameters of the experiment, such as fundamental laser frequency ω , spectral composition of the incident APTs, or the maximum time delay τ max .An example is provided in Fig. 8 which shows a series of NFT spectra of CO calculatedusing ω varying from 1.49 eV (bottom spectrum) to 1.73 eV (top spectrum). The maintriad 9 ω − ω − ω is present in all spectra, the intensity distribution between thepeaks changes slightly, and the peak at 13 ω slowly attenuates as ω grows. The peaksat higher integer harmonic orders, present in the incident APTs, clearly become strongerwith increasing ω , as more directly ionizing states become energetically accessible fromthe ground state of the parent CO . The high frequency end of the NFT spectrum above119 ω , which stems from the short time delays | τ | ≤ ω , with multiple non-harmonic contributions. In contrast, the low frequency region of the difference frequencypeaks is stable, and the peaks 2 ω and ∼ ω change neither position nor intensity. Exceptionis the weaker non-harmonic difference peak which moves between 6 ω and 7 ω . Figure 8,together with the approximate assignment schemes illustrated in Fig. 6, could in principlebe used as a starting point for spectral inversion analysis, in which the electronic states ofthe dissociating ion are reconstructed from the NFT spectra.It is highly desirable to extend the developed framework and to replace the toy modelbased on the electronic state ladder with realistic and interacting potential energy surfaces.This extension, which will be presented in a separate publication, removes two main lim-itations of the discussed approach. First, it allows calculations of two-dimensional NFTspectra as functions of both fragment kinetic energy and ω NFT . Appendix A provides thenecessary formal expressions. Second, vibronic interactions between electronic states be-come naturally incorporated into the model so that one can analyze the sensitivity of theNFT spectra to non-adiabatic interactions in the parent molecule and in the molecular ion.The time scales and the atomistic mechanisms of ultrafast photoreactions mediated by thevibronic interactions are actively explored across a wide range of applications from ad-vanced energy materials to photoprotective mechanisms in biochromophores. In CO and CO +2 , non-adiabatic interactions affect not only the absorption and ionization profilesbut also the lifetimes of electronically excited species, and the dissociation mechanisms canbe controlled by vibronic as well as spin-orbit (i.e. relativistic) interactions. Appendix A: Two-dimensional NFT signals
After the interaction with two APTs, the excited ion (cid:0) CO +2 (cid:1) ⋆⋆ and the photoelectron arein the state Ψ I ( t | E k , τ ). We first consider one electronic state | f i of CO +2 which dissociatesinto an arrangement channel with the detected fragment, and restrict the description to2-body arrangement channels, as indicated in Eq. (3). In this case, one of the dissocia-tion fragments is diatomic and its internal state is characterized by rovibrational quantumnumbers, which we collectively denote n f , and by the internal energy ǫ int ( n f ). Quantumnumbers n f label individual dissociation channels of (cid:0) CO +2 (cid:1) ⋆⋆ in the state | f i in the con-2sidered arrangement channel. The corresponding dissociation threshold is D f . Thresholdsrelevant for the NFT experiments on CO are illustrated in Fig. 2.Suppose that the final energy of (cid:0) CO +2 (cid:1) ⋆⋆ after the absorption of two photons has a valueof ǫ f lying above D f . In the dissociation channel, this energy is shared between the (centerof mass) recoil kinetic energy ǫ kin and the internal excitation: ǫ f = ǫ kin + ǫ int ( n f ) > D f . (A1)The final rovibrational distribution and the final kinetic energy distribution are complemen-tary and can be recalculated from one another using the energy conservation.The state of the dissociating fragments is a linear combination of the scattering states ψ − f,k f ,n f , corresponding to the total energy ǫ f . The probability amplitude for the fragmentto be in a state with energy ǫ f and wave vector k f (and the kinetic energy ǫ kin ( k f ) = k f / µ ),while the diatomic fragment is in the internal state n f , is given by the matrix element γ ( ǫ, k f , n f | E k , τ ) = h ψ − f,k f ,n f |h φ + f ψ ek | Ψ I ( t → ∞| E k , τ ) i . (A2)This expression is akin to the photodissociation matrix element which contains the dy-namical information on the dissociation process. The above expression is applied tothe ionizing system, and the projection on h φ + f ψ ek | additionally specifies that the ejectedphotoelectron leaves with the energy E k .The partial cross section to produce the detected fragment ion in a given dissociationchannel n f describes the rovibrational state distribution — and equivalently the kinetic en-ergy distribution — for the fixed energy ǫ f . It is given by the square of the matrix element R d E k | γ ( ǫ, k f , n f | E k , τ ) | integrated over all possible photoelectron energies. However, thiskinetic energy distribution is not accessible in the multiphoton NFT spectroscopic experi-ment. The 2D NFT signal as measured, for example, in Ref. 46, is additionally summed overall possible final energies ǫ f and all final electronic states | f i contributing to the detecteddissociation fragment: I D ion ( ǫ kin , τ ) = X f Z ∞ D f d ǫ f Z ∞ dE k | γ ( ǫ, k f , n f | E k , τ ) | . (A3)Evaluation of the photodissociation matrix element γ ( ǫ, k f , n f | E k , τ ) requires numerical so-lution of the Schr¨odinger equation with the (generally three-dimensional) potential energysurfaces of the dissociative electronic states of CO +2 . While the nuclear dynamics for the3photofragment distributions can be efficiently calculated using iterative methods, con-struction of the multidimensional potential energy surfaces of many densely spaced (andpossibly interacting) electronic states in the energy range illustrated in Fig. 2 is a truechallenge. Sum over all channel quantum numbers n f effectively brings about summation over allfragment kinetic energies and gives the 1D NFT signal, i.e. the total ion yield: I D ion ( τ ) = X f X n f Z ∞ D f d ǫ Z ∞ dE k | γ ( ǫ, k f , n f | E k , τ ) | . (A4)The scattering basis states are complete, and this expression can be rewritten in terms of thepopulations of the final electronic states integrated over the photoelectron kinetic energy: I D ion ( τ ) = X f Z ∞ dE k h Ψ cI ( t → ∞| E k , τ ) | φ + f ψ ek ih φ + f ψ ek | Ψ cI ( t → ∞| E k , τ ) i . (A5)This expression is practically identical to the total ion yield defined in Eqs. (16) and (17)of the main text. The appearance threshold A ion for the detected ion fragment correspondsto the lowest energy ǫ f (and the lowest state | f i ) for which the amplitude γ ( ǫ, k f , n f | E k , τ )does not vanish. The subscript c on the wave function in Eq. (A5) is a remainder that onlyprojections on the continuum states ψ − f,k f ,n f are considered in each state | f i . Appendix B: Analytic expressions for 1D NFT signals
The quantum mechanical expressions for the NFT signals I ion ( τ ) and I ion ( ω NFT ), as de-scribed in Sect. II D [see Eqs. (17), (18), and (21)—(24)], are well suited for numericalcalculations. One can cast them into a different form more appropriate for the analysisand assignment of the spectral peaks. This appendix summarizes analytic expressions for I ion ( ω NFT ) which help to rationalize the NFT spectra. The implications for the spectralassignments are discussed in Sect. III C.For simplicity, we consider only the photoreaction pathway in which the parent molecule isionized already with the first photon. The second order pump probe amplitude c j ( v j ; E k , τ )is given by the Eq. (21). Changing the variables to y = t − t in the inner integral with y running from 0 to ∞ , using the Fourier transform of the APT electric field, E (Ω; τ ), and4invoking the convolution theorem, the amplitude c j ( v j ; E k , τ ) can be re-written as c j ( v j ; E k , τ ) = 1 i X i = j X vi µ ji µ i ( E k ) × Z ∞−∞ d Ω2 π E ( ǫ vj − Ω; τ ) i h v j | v i ih v i | i Ω − ǫ vi + i E (Ω + E k − ǫ ; τ ) . (B1)This (still exact) pump probe amplitude is given by the convolution of two Fourier imagesof the laser electric field and the pole factor i (Ω − ǫ vi + i − corresponding to the Fouriertransform of e − iǫ vi y times the Heaviside step function Θ( y ). The Fourier transform of thelaser field is given by E (Ω; τ ) = X n ′ a n L n (Ω) (cid:0) e i Ω τ (cid:1) , (B2)with L n (Ω) being the Fourier image of the time envelope function L n ( t, τ ) defined throughthe Eqs. (5), (6), and (7).The integral over Ω has a suggestive form for the use of contour integration in the complexplane and the residue theorem. The success of this approach depends on the actual shapeof the function L n (Ω). In the main text, computations are performed using the Gaussiantime envelope L n ( t, τ ). The function L n (Ω) is then also a Gaussian, L n (Ω) = e − (Ω − ω n ) T / P , (B3)and the integral in Eq. (B1) leads to the Hilbert transform of a Gaussian function, whichdoes not have a simple analytic representation. Other envelope functions can be easier tohandle. Example is the exponential decay envelope L n ( t, τ ) = e − P | t − τ | /T . Its Fourier image L n (Ω) = P/T (Ω − ω n ) + ( P/T ) has a Lorentzian lineshape and therefore a simple residue structure which makes the in-tegration of Eq. (B1) in the complex plane straightforward. The subsequent integrationsin Z ∞ dE k Z dτ e iω NF T τ | c j | . (B4)are also analytical. The resulting expressions, however, are awkward and tedious to analyze.In fact, the nature of spectral peaks in the quantum mechanical NFT spectra can beexposed using an approximation. We replace the pole factor in Eq. (B1) with − iπδ (Ω − ǫ vi ) and ignore the principal value contribution. This is similar to the approximation oftemporarily non-overlapping pump and probe pulses, familiar in the context of ultrafastpump-probe spectroscopies. With this approximation, the pump probe amplitude c j ( v j ; E k , τ ) can be easily evaluated, and its square is written as | c j ( v j ; E k , τ ) | = X i ,i = j X vi ,vi X n ,n ,n ,n µ ji µ i ( E k ) µ ji µ i ( E k ) × h v j | v i ih v i | v j ih v i | ih | v i i a n a n a n a n × L n ( ǫ vj − ǫ vi ) L n ( ǫ vi − ǫ + E k ) L n ( ǫ vj − ǫ vi ) L n ( ǫ vi − ǫ + E k ) × h (1 + e i ( ǫ vj − ǫ vi ) τ )(1 + e i ( ǫ vi − ǫ + E k ) τ )(1 + e − i ( ǫ vj − ǫ vi ) τ )(1 + e − i ( ǫ vi − ǫ + E k ) τ ) i . (B5)In this expression, we expand the electric fields as sums over the odd harmonics, as in Eq.(B2), and assume that the coefficients { a n } and the TDMs are real. The dependence of theprobability | c j | on the delay time τ and the photoelectron kinetic energy E k is now explicit,and the integrals in Eq. (B4) can be performed directly. Let us define the energy differencesbetween the molecular or ionic states as ∆ ji = ǫ vj − ǫ vi and ∆ i = ǫ vi − ǫ . Then the NFTspectrum, obtained after the integrations, reads as I ion ( ω NFT ) = X j X ǫ vj >A ion X i ,i = j X vi ,vi X n ,n ,n ,n µ ji µ i µ ji µ i ×h v j | v i ih v i | v j ih v i | ih | v i i a n a n a n a n × (cid:20) L n (∆ ji ) L n (∆ ji ) 12 r πPT e − T ( s n − s n ) / P (cid:18) (cid:18) T √ P s n ,n (cid:19)(cid:19) × n δ τ ( ω NFT ) + δ τ ( ω NFT ± ∆ ji ) + δ τ ( ω NFT ± ∆ ji ) + δ τ ( ω NFT ± ∆ i i ) o + L n (∆ ji ) L n (∆ ji ) × n L n ( ω NFT ) L n ( ω NFT − ∆ i i ) + L n ( ω NFT + ∆ i i ) L n ( ω NFT )+ L n ( ω NFT − ∆ ji ) L n ( ω NFT − ∆ ji ) + L n ( ω NFT + ∆ ji ) L n ( ω NFT + ∆ ji ) o (cid:21) ;(B6) s n = ω n − ∆ i ; s n = ω n − ∆ i ; s n ,n = 12 ( s n + s n ) . x ) is the standard error integral. The expression holds for any lineshape function L n (Ω), and we shall use Gaussians as in Eq. (B3). The NFT spectrum arises as a superpo-sition of ionizations of the parent molecule into the intermediate ionic states states i and i which are then further excited into a dissociative state j . There are two groups of NFTspectral peaks in this expression, corresponding to the two types of τ -dependent terms inEq. (B5). The first group — placed inside the curly brackets {· · · } — includes spectralpeaks which are located at the energy differences between the intermediate ionized and thefinal dissociative states, ω NFT = ± ∆ ji . Their width is controlled by the maximum delaytime in Eq. (18), and can potentially be made narrow (delta-function like) by increasing τ max . These NFT peaks are denoted as δ τ . They stem from the terms in | c j ( v j ; E k , τ ) | withthe exponential phase factors independent of E k , such as e − i ∆ ji τ for example. The positionsof these peaks need not coincide with the multiples of the fundamental harmonic frequency ω . The second group includes terms — placed inside the curly brackets {· · · } — whichare products of two Gaussian functions L n . Each Gaussian factor peaks at ω NFT = ω n ± ∆ ji ;the spectral width is determined by the reciprocal of the APT temporal width T [see Eq.(B3)]. These spectral lines stem from the τ -dependent terms with the exponential phasefactors explicitly depending on E k . The positions of these peaks tend to cluster aroundthe harmonic frequencies nω . Note that the widths of different lines in the actual NFTspectrum are expected to be sensitive to either τ max or T .The accuracy of the approximation of Eq. (B6) is illustrated in Fig. 6 in the main text:Most of the approximate spectral lines (black) accurately reproduce the positions of thenumerically exact peaks (red). The agreement is worse for the intensities, which are moresensitive to the coherent two photon effects, but the overall spectrum is quite well recogniz-able. This makes the approximation useful in the analysis of the origin of near harmonic aswell as non-harmonic spectral peaks. The peaks assigned ∆ fi stem from the terms in {· · · } ;the peaks assigned ω n and ω n ± ∆ fi are due to the terms in {· · · } . In particular, peaksaround ω NFT /ω = 9 , ,
24 deviating from the integer harmonic orders are reproduced.Although the NFT spectrum in the above equation is already an approximation, it is stillrather bulky and awkward to use in spectral assignments. In order to simplify the discussionof the assignment in Sect. III C, we introduce yet another approximation and assume thatthe ionizations in Eq. (B6) terminate in the same ionic state i = i = i . This is a realisticscenario, at least for the amplitude Set 1 and ω = 1 .
55 eV. We also consider only the ground7vibrational states in all electronic states, drop sums over vi , and rename the final dissociativestates ǫ j > A ion as f . The NFT spectrum is then given by I ion ( ω NFT ) = X f X i = f X n ,n ,n ,n ( µ fi µ i ) a n a n a n a n × L n (∆ fi ) L n (∆ fi ) (cid:20) r πPT e − T ( s n − s n ) / P (cid:18) (cid:18) T √ P s n ,n (cid:19)(cid:19) × n δ τ ( ω NFT ) + 2 δ τ ( ω NFT − ∆ fi ) + 2 δ τ ( ω NFT + ∆ fi ) o + n L n ( ω NFT ) L n ( ω NFT ) + L n ( ω NFT − ∆ fi ) L n ( ω NFT − ∆ fi ) L n ( ω NFT + ∆ fi ) L n ( ω NFT + ∆ fi ) o (cid:21) . (B7)The simplified spectrum again consists of two groups of spectral peaks, those located atthe electronic energy differences ± ∆ fi , and those located at the unshifted ( ω n ) and shifted,( ω n ± ∆ fi ), harmonic frequencies. There can therefore be a substantial number of spectralpeaks located at non-harmonic frequencies if ∆ fi is not exactly equal to a multiple of ω .Moreover, the Gaussian factors indicate that off-resonance excitations within the spectralwidth of L n can be encountered in an NFT spectrum. This width is of the order of 1 eV (i.e.of the order of ω ) for the typical experimental APTs, so that deviations from the harmonicfrequencies within ± ω are not unexpected.Note that the factors L n , allowing non-resonant excitations within their spectral widths,carry as arguments only the energy differences ∆ fi between the ionic states involved in theoptical excitation of the ion; the energy differences between the initial state of the parentmolecule and the ionized state, ∆ i = ǫ i − ǫ , are not involved in the final expressions. This isan intrinsic feature of the photochemical pathway of Eq. (1): The mismatch between ∆ i andthe harmonic frequency ω n > ∆ i can be compensated by the kinetic energy E k = ω n − ∆ i of the photoelectron ejected in the first (ionization) step.The approximate expressions for the NFT spectra, Eqs. (B6) and (B7), are further dis-cussed in Sect. III C and used to explain the assignment scheme for NFT spectra. Acknowledgments
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