Attosecond electronic and nuclear quantum photodynamics of ozone monitored with time and angle resolved photoelectron spectra
P. Decleva, N. Quadri, A. Perveaux, D. Lauvergnat, F. Gatti, B. Lasorne, G. J. Halász, Á. Vibók
aa r X i v : . [ phy s i c s . a t m - c l u s ] O c t Attosecond electronic and nuclear quantumphotodynamics of ozone monitored with time andangle resolved photoelectron spectra
Piero Decleva, † Nicola Quadri, † Aurelie Perveaux, ‡ David Lauvergnat, ‡ FabienGatti, ¶ Benjamin Lasorne, ¶ Gábor J. Halász, § and Ágnes Vibók ∗ , k , ⊥ Dipartimento di Scienze Chimiche, Universita’ di Trieste, Via L. Giorgieri 1I - 34127 Trieste,Italy, Laboratoire de Chimie Physique, CNRS, Université Paris-Sud, F-91405 Orsay, France,Institut Charles Gerhardt, CNRS, Université de Montpellier, F-34095 Montpellier, France,Department of Information Technology, University of Debrecen, H-4002 Debrecen, PO Box 400,Hungary, Department of Theoretical Physics, University of Debrecen, H-4002 Debrecen, PO Box400, Hungary, and ELI-ALPS, ELI-HU Non-Profit Ltd, Dugonics tér 13, H-6720 Szeged, Hungary
E-mail: [email protected]
Abstract
Recently we reported a series of numerical simulations proving that it is possible in princi-ple to create an electronic wave packet and subsequent electronic motion in a neutral moleculephotoexcited by a UV pump pulse within a few femtoseconds. ∗ To whom correspondence should be addressed † Dipartimento di Scienze Chimiche, Universita’ di Trieste, Via L. Giorgieri 1I - 34127 Trieste, Italy ‡ Laboratoire de Chimie Physique, CNRS, Université Paris-Sud, F-91405 Orsay, France ¶ Institut Charles Gerhardt, CNRS, Université de Montpellier, F-34095 Montpellier, France § Department of Information Technology, University of Debrecen, H-4002 Debrecen, PO Box 400, Hungary k Department of Theoretical Physics, University of Debrecen, H-4002 Debrecen, PO Box 400, Hungary ⊥ ELI-ALPS, ELI-HU Non-Profit Ltd, Dugonics tér 13, H-6720 Szeged, Hungary e considered the ozone molecule: for this system the electronic wave packet leads to adissociation process. In the present work, we investigate more specifically the time-resolvedphotoelectron angular distribution of the ozone molecule that provides a much more detaileddescription of the evolution of the electronic wave packet. We thus show that this experimentaltechnique should be able to give access to observing in real time the creation of an electronicwave packet in a neutral molecule and its impact on a chemical process. ntroduction Since the advent of femtochemistry remarkable and decisive progress has been achieved on theexperimental front and it is now possible to monitor electronic motion in the context of atto-physics . In other words, electronic wave packets can be created and observed in real time,which will improve our understanding of fundamental quantum concepts such as coherence andcoherent light-matter interaction on the time scale of the electrons in a molecule.Exciting molecules with attosecond XUV light pulses may populate several electronic statescoherently, thus creating an electronic molecular wave packet. Its evolution will eventually triggernuclear motion on a longer timescale via the effective potential created by the electrons and gov-erning nuclear dynamics. In this context, a crucial challenge for attosecond sciences is to createspecific electronic wave packets able to induce nuclear motion, e.g. a chemical process, selec-tively and efficiently. This should lead, on the long term, to what some already call attochemistry,where, at each step of a molecular process, the coupled motions of electrons and nuclei could becontrolled on their natural time scales . For example, if the attosecond pulse ionizes the molecule,the hole thus created will move, a process which is termed charge migration . This may yield, in asecond step, to selective bond dissociation . Another possibility is to populate a limited numberof electronic states in the neutral molecule by means of UV subfemtosecond pulses in order totrigger a selective chemical process. Experimentally, attosecond pulses are already available in theXUV spectral domain but few-cycle UV subfemtosecond pulses are expected to emerge in a nearfuture.A complete theoretical description of such processes is not a trivial task: it requires a quantummechanical description of both the motion of the electrons and the nuclei in interaction with theexternal ultrafast field. In previous studies, we presented a full quantum mechanical simulation ofthe excitation of the ozone (neutral) molecule after excitation by a 3 fs UV pump pulse . Thecentral wavelength of the pulse at 260 nm was selected so as to create a coherent superpositionof only two electronic states: the ground state, X ( A ) , and the excited B ( B ) state . Theozone molecule was chosen since, for obvious environmental reasons, its electronic excited states3re well-known and understood . In addition the B state is rather well isolated and, moreimportantly, the transition dipole between the X and B state is very large, leading to the so-calledHartley band in the UV domain that is responsible for the properties of the ozone layer. As aconsequence, exciting the molecule to the B state does not require very high intensity (we useda value of 10 W/cm ), and we can assume that only this state is populated by the laser pulse.However, it is worth noting that obtaining such intensities for very short UV pulses remains anexperimental challenge at the moment.In Ref. , we investigated the creation of an electronic wave packet (see Fig. 6 in Ref. ) leadingto an oscillation of the electronic charge density from one O-O bond to the other on the subfem-tosecond time scale (with a period of 0 . and O + O.Upon propagating nuclear wave packets with the Heidelberg Multi-Configuration Time-DependentHartree (MCTDH) package , we showed that, at the end of the laser pulse, the molecule startedto vibrate (see Fig. 4 in Ref. ). The quantum coherence between the two electronic states couldthus be expected to be destroyed rapidly due to vibrations, even more so because of the dissocia-tion outcome making this process irreversible. However, we observed a revival of coherence afterthe external field was off, with a time delay corresponding to a single vibrational period in the B state. This was attributed to a portion of the wave packet being trapped in the B state around ashallow potential energy well. Obviously, electronic coherence would have been preserved longerif the potential energy well of the B state had been deeper. In any case, this revival of quantumcoherence is the signature that the coherent superposition of the two electronic states is not de-stroyed as soon as the nuclear motions starts. To conclude, we showed that it was possible to firstcreate an electronic wave packet in the bound molecule, which would lead, in a second step, tothe dissociation of the molecule and monitor the whole process with time-resolved spectroscopy.In principle, one could also expect to control this process upon manipulating the initial electronicwave packet via modulating the pump pulse.From the experimental point of view, a wave packet cannot be observed as such, or at least not4directly” but rather from its consequences on the photodynamics of the system, via time-resolvedobservables obtained from pump-probe spectroscopy techniques. Attosecond XUV probe pulsescan be used to ionize the molecule during the whole process with a time resolution compatible withthe electronic motion . The resulting time-resolved spectra from both electronic states, X and B , will provide precious information about the detailed dynamics of the system. Our probe pulse iscentered around 95 eV. This high value generates electrons that are ejected with high velocities. Asudden approximation can thus be invoked to describe one-photon XUV ionization . In addition,it is desirable that the ionization process is as instantaneous as possible so that it does not perturbthe electronic motion induced by the pump pulse. In Ref. , we calculated the relative ionizationprobabilities based on an approach exploiting Dyson orbitals (see Ref. for the calculation ofthese). Within the sudden approximation regime one can estimate relative cross sections as thesquare norms of the Dyson orbitals. Then, after convolution of the stick photoelectron spectrafrom X and B , we could calculate the time-resolved photoelectron spectrum (TRPES) as a functionof time and photoelectron kinetic energy. This spectrum clearly exhibited depletion of X andproduction of B .Now, in order to analyze the wave packet created by the pump pulse in more detail, it is usefulto consider a more accurate and complete description of the time-resolved photoelectron spectrum,including both realistic cross sections and angular distributions, and their photon energy depen-dence. For instance, molecular frame photoelectron angular distributions (MFPAD) give accessto the shape of the electronic wave packet . Even photoionization from molecules that are ran-domly distributed in terms of their orientation in space show important dependence on the anglebetween the polarization axis of the pump pulse and the direction of the ejected electron. The aimof the present work is precisely to provide such a time-resolved photoelectron angular distributionfor the dissociation of ozone with the aforementioned pump pulse. This completes an ab-initiotheoretical framework for the accurate description of pump-probe experiments in small molecules,represented here by O , able to deal with electronic and nuclear motion on equal footing, describ-ing the combined electron-nuclear wave packet. 5he outline of the paper is as follows: in the next section we describe briefly the methodsused for quantum chemistry calculations and quantum dynamics simulations. In the third section,the resulting photoelectron spectra are presented and discussed. Finally, conclusions provide anoutlook for the future of molecular attophysics. Theoretical background
A molecule such as ozone can be viewed as a collection of N nuclei and n electrons. Let ~ R = ( ~ R , . . . , ~ R N ) and ~ r = ( ~ r , . . . ,~ r n ) denote the position vectors of the nuclei and the electrons,respectively. Using a semi-classical approach with respect to the external electromagnetic field andthe so-called dipole approximation, the non-relativistic Coulomb molecular Hamiltonian operatorfor the system interacting with a time-dependent external electric field, ~ E ( t ) , reads H ( ~ r , ~ R , t ) = T nu ( ~ R ) + H el ( ~ r ; ~ R ) − ~ m ( ~ r , ~ R ) · ~ E ( t ) , (1)where T nu ( ~ R ) is the kinetic energy operator of the nuclei, H el ( ~ r ; ~ R ) the electronic Hamiltonianoperator (the sum of the latter two terms being the field-free molecular Hamiltonian), and ~ m ( ~ r , ~ R ) the electric dipole moment of the molecule.The time-dependent Schrödinger equation reads H ( ~ r , ~ R , t ) Y ( ~ r , ~ R , t ) = i ¯ h ¶ Y ( ~ r , ~ R , t ) ¶ t , (2)with Y ( ~ r , ~ R , t ) the wave packet of the molecule.The adiabatic electronic basis functions, F i ( ~ r ; ~ R ) , satisfy for each ~ RH el ( ~ r ; ~ R ) F i ( ~ r ; ~ R ) = E eli ( ~ R ) F i ( ~ r ; ~ R ) , (3)where ~ R are to be viewed as parameters and E eli ( ~ R ) play the role of potential energy surfaces forthe nuclei. 6ere, we consider only a pair of adiabatic electronic states for ozone: X ( A ), the ground state,and B ( B ), the Hartley excited state. The total wave function of the molecule can be expanded as Y ( ~ r , ~ R , t ) = (cid:229) i = X , B Y i ( ~ R , t ) F i ( ~ r ; ~ R ) . (4)In the following, we assume the Born-Oppenheimer approximation to be valid and thus neglect thenon-adiabatic couplings between the two electronic states stemming from the nuclear kinetic en-ergy operator. The only coupling between X and B is induced by the external field through the term − ~ m XB ( ~ R ) · ~ E ( t ) , where the transition dipole is defined as ~ m XB ( ~ R ) = ´ F ⋆ B ( ~ r ; ~ R ) ~ m ( ~ r , ~ R ) F X ( ~ r ; ~ R ) d ~ r .We also neglect the diagonal terms involving ~ m XX ( ~ R ) and ~ m BB ( ~ R ) since ~ E ( t ) is an external fieldresonant between X and B with respect to the central wavelength of the spectrum of the pulse.Thus, the evolution of Y X ( ~ R , t ) and Y B ( ~ R , t ) is governed by a set of two coupled equationsinvolving only E elX ( ~ R ) , E elB ( ~ R ) , − ~ m XB ( ~ R ) · ~ E ( t ) , and T nu ( ~ R ) . To solve this set of equations, i.e. tosolve the Schrödinger equation for the nuclei, we use the MCTDH method . The nuclearwave functions are expanded in a basis set of time–dependent functions, the so–called single–particle functions (SPFs), Y ( Q , · · · , Q f , t ) = n (cid:229) j · · · n f (cid:229) j f A j , ··· , j f ( t ) f (cid:213) k = j ( k ) j k ( Q k , t ) , (5)where f denotes the number of nuclear degrees of freedom ( Q k are single coordinates or groups ofcoordinates involved in ~ R ). There are n k SPFs for the k th nuclear degree of freedom. The equationsof motion for the A -coefficients and the SPFs are derived from a variational principle thatensures optimal convergence.In this work, Q , · · · , Q are (polyspherical) valence coordinates ( R and R , the two bondlengths, and a , the angle between the two bonds). The corresponding expression of the ki-netic energy operator, T nu ( R , R , a ) , with zero total angular momentum can be found in Ref. .The potential energy surfaces, E elX ( R , R , a ) and E elB ( R , R , a ) , and the transition dipole surface, ~ m XB ( R , R , a ) , are those from Schinke and coworkers . They are implemented in MCTDH7nd have already been tested on accurate applications in spectroscopy .The parameters defining ~ E ( t ) , the laser pump pulse (see Fig. 1) are: central wavelength at 260nm, intensity of 10 W/cm , Gaussian envelope with a full duration at half maximum (FDHM)equal to 3 fs. Note that, due to the C v symmetry of the ozone molecule at the Franck-Condon (FC)point ( R = R = .
275 Å; a = . ◦ ), the y -component ( B ) of the transition dipole between X and B is the only non-vanisihing one at the FC point and is thus primarily responsible for the light-induced electronic transitions. Consequently, the effective polarization axis of the electric field is y . Further details regarding our calculations – the (time-independent) primitive basis sets, theparameters for the complex absorbing potentials, the refitting of the potential energy and transitiondipole surfaces in a form adapted to MCTDH, and the number of SPFs – can be found in previouswork, for instance in Sec. 3 of Ref. .Starting from the vibrational ground state in the electronic ground state X , MCTDH calcula-tions will generate Y X ( ~ R , t ) and Y B ( ~ R , t ) at any subsequent time. Assuming that only the B elec-tronic state is populated by the laser pulse (see Fig. 1), the total molecular wave packet (see Eq. 4)can be constructed provided the corresponding adiabatic electronic wave functions are known.Thus, with this approach, we can obtain in principle the full electronic and vibrational wavepacket (note again that we only consider the case where the total angular momentum is equal to0). However, this quantity cannot be observed directly in actual experiments and we need a time-resolved property that will characterize the time evolution of the system: the TRPES for instance,which can be measured and compared to calculations. The procedure that we used to compute thisquantity is explained below.As a first approximation, we can consider that the early stages of the process will be dominatedby the behavior of the wave packet at the FC point, ~ R FC . The corresponding renormalized densitymatrix of the molecule at the FC point (see Sec. II B of Ref. for further details) reads, for8 , i ′ = X , B , r ii ′ ( t ) = Y ⋆ i ( ~ R FC , t ) Y i ’ ( ~ R FC , t ) (cid:229) l = X , B (cid:12)(cid:12)(cid:12) Y ( l ) ( ~ R FC , t ) (cid:12)(cid:12)(cid:12) (6)Note that such local populations of X and B are not classical quantities but extracted from theactual quantum wave packets.Assuming a “stationary” picture, the approximate photoelectron spectra from either X or B at the FC point appear as stick spectra, I k ( e ) = (cid:229) i I ik d ( e − e ik ) , (7)where e is the kinetic energy (KE) of the ejected electron, i = X or B , and k is used to label thevarious cation states. e ik are the corresponding peaks appearing in the spectra. They satisfy e ik = E photon − IP ik IP ik = E k − E i , (8)where E photon denotes the energy of the probe photon, 95 eV here. E i are the energies of the X and B states at the FC geometry, E k the energies of the cation that can be populated by the photon atthe same geometry, and IP ik are the relative ionization potentials. Our calculations show that 19cation states can be populated (up to about 20 eV above the X state) . For the calculation of thepeak intensities, I ik , we adopt an approach based on Dyson orbitals . The latter are defined as f Dysoni , k ( ~ r ; ~ R ) = √ n ˆ d ~ r . . . d ~ r n F eli ( ~ r = ~ r ,~ r , . . . ,~ r n ; ~ R ) × F cat ⋆ k ( ~ r , . . . ,~ r n ; ~ R ) , (9)where F eli are the electronic functions of the neutral molecule as defined above and F catk the elec-tronic functions of the cation. We calculated Dyson norms at the FC point (see Ref. ) at theCASSCF(17,12)/aug-cc-pVQZ (no state average) level of theory for the cation wave functions and9ASSCF(18,12)/aug-cc-pVQZ (no state average) for the neutral wave functions with the MOL-PRO quantum chemistry package . The energies of the neutral and the cation were further re-fined with MRCI-SD(Q) calculations, including Davidson corrections, and based on the previousCASSCF references.If a sudden approximation is assumed, the squares of the Dyson norms, h f Dysoni , k | f Dysoni , k i , areproportional to the relative ionization probabilities I ik . Ionization potentials and I ik = h f Dysoni , k | f Dysoni , k i are reported in 1. The corresponding stick spectrum is displayed in Figure 2. To obtain the energyresolved spectra we convoluted the stick spectra with a Gaussian envelope function G ( e ) to mimicthe bandwidth of the XUV probe pulse, I k ( e ) = (cid:229) j G jk ( e ) I jk G jk ( e ) = s √ p e − ( e − e jk ) s . (10)Here s is the standard deviation of the intensity: s = . d s jk ( e jk ) d W = s jk ( e jk ) p [ + b jk ( e jk ) P ( cos q )] (11)where P ( cos q ) = ( q − ) is the second order Legendre polynomials and q is the anglebetween the direction of the electron momentum and the polarization of the electric field. W isthe angle relative to electron emission momentum in the LF system and the two energy depen-dent parameters are s jk (partial cross section) and b jk (asymmetry parameter). (The LF systemdefines the experiment i.e. the direction of the polarization and propagation of light as well as thedirection of electron detection. The reference system is the molecular frame (MF) system in whichthe molecule is fixed and the electronic structure, transition dipole moment etc. calculations areperformed.) 10alculation of s and b parameters require an explicit description of the continuum wave func-tion for the final state. Neglecting interchannel coupling effects, generally very small far fromthresholds, a single channel approximation of the form Y ( − ) k ,~ k = A F catk j ( − ) ~ k (12)is generally quite accurate. Here j ( − ) ~ k describes an electron with asymptotic momentum ~ k (andincoming wave boundary conditions, appropriate for photoionization), and A describes antisym-metrization and proper symmetry couplings. Actually it is computationally easier to work in anangular momentum basis, employing eigenstates Y k , e lm = A F catk ( ~ r , . . . ,~ r N − ) j e lm ( ~ r N ) (13)where the continuum wavefunctions j e lm are characterized by suitable asymptotic conditions, inour case K-matrix boundary conditions, defined as j e lm ( ~ r ) → (cid:229) l ′ m ′ ( f l ′ ( k r ) d l ′ l d m ′ m + g l ′ ( k r ) K l ′ m ′ , lm ) Y l ′ m ′ (14)which has the advantage of working with real wave functions. Here f l and g l are regular andirregular coulomb functions. The j e lm so obtained can be transformed to incoming wave boundaryconditions and then to linear asymptotic momentum by standard transformation j ( − ) e lm = (cid:229) l ′ m ′ j e l ′ m ′ ( + iK ) − l ′ m ′ , lm (15) j ( − ) k ,~ k = √ m (cid:229) l ′ m ′ i l e − i s l Y lm ( ˆ k ) j ( − ) e lm (16)The same transformation can be directly applied to the transition dipole moments. The many-11article transition dipole moment D ik ; lm g ( e ) = h A F catk j e lm | D g | F eli i (17)reduces to the single particle moment involving the Dyson orbital (9) D ik ; lm g ( e ) = h j e lm | d g | f Dysoni , k i (18)plus an additional term (conjugate term) which is generally small and is usually neglected . Here g is the Cartesian component of the dipole, D and d are the many-particle and the single particledipole operators.From dipole moments (and the K -matrix) s jk ( e ) and b jk ( e ) , as well as any angular distributionfrom oriented molecules, can be computed according to well known formulas .In our formulation, the continuum wave function ( ?? ) is computed as an eigenfunction of theKohn-Sham Hamiltonian defined by the initial state electron density r h KS j e lm = ej e lm (19) h KS = − D + V eN + V C ( r ) + V XC ( r ) (20)where V eN is the nuclear attraction potential, V C the coulomb potential and V XC the exchange-correlation potential defined in terms of the ground state density r . The latter is obtained from aconventional LCAO SCF calculation, employing the ADF program with a DZP basis. A specialbasis is employed for the continuum solutions of ( ?? ). Primitive basis functions are products of a B -spline radial function times a real spherical harmonic c ilm ( r , q , f ) = r B i ( r ) Y lm ( q , f ) (21)The full basis comprises a large one-center expansion on a common origin, with long range R max ,12nd large maximum angular momentum, L max . This is supplemented by additional functionscentered on the nuclei, of very short range, R maxp , and small angular momenta L maxp . A short rangeis necessary to avoid almost linear dependence of the basis, which spoils the numerical stability ofthe approach. Despite the very limited number of LCAO functions these choices ensure a very fastconvergence of the calculated quantities. The basis is then fully symmetry adapted.The calculation of continuum eigenvectors is performed at any selected electron kinetic energyby the Galerkin approach originally proposed in ref. and the generalized to the multichannelcase . From the energy independent Hamiltonian H and overlap S matrices continuum vectorsare obtained as eigenvectors of the energy dependent matrix A ( E ) = H − ES with eigenvaluesvery close to zero. These give the correct number of independent open channel solutions, and areefficiently obtained by block inverse iteration, since they are separated by large gaps from the restof the spectrum. Actually the more stable form A + A is currently employed . Final normalizationto K -matrix boundary conditions is obtained by fitting the solutions to the analytical asymptoticform at the outer boundary R max .In the present calculation the LB94 V XC potential was employed, due to the correct asymp-totic behavior, important in photoionization. Parameters were L max = R max = . L maxp = R maxp = .
50 a.u. for the O atoms, for a total of 23013 basisfunctions.Such an approach, called static-DFT proves in general remarkably accurate for the descriptionof cross sections and angular distributions . In conjunction with the Dyson orbital formulationit is able to describe ionization involving multiconfigurational initial and final cationic states .We refer to previous work for details of the implementation . s jk and b jk are obtained on adense electron KE e jk grid, so that the value at any KE dictated by the given photon energy canbe accurately obtained by interpolation. With these the angularly resolved photoelectron intensitybecomes: I k ( e , q ) = (cid:229) j G jk ( e jk ) s jk ( e jk ) p [ + b jk ( e jk ) P ( cos q )] . (22)13pplying the same convolution procedure as in Eq. 9 of Ref. we arrive to the appropriate formulaof the angle resolved photoelectron spectrum: I ( e , q , t ) = (cid:229) k r kk ( t ) I k ( e , q ) . (23)Here the r kk ( t ) comes from eq. 6 and from now on the above expression (eq. ?? ) will serve as ourworking formula in the forthcoming part of the paper. E x c it a ti on e n e r gy / e V R / a.u. Figure 1: Potential energy cut of the ozone molecule as a function of the dissociation coordinate, R : ground state ( X , solid line) and Hartley state ( B , dashed line), the arrow denotes the excitationof the B state. The other bond is fixed at R = .
43 a.u. and the bond angle a = ◦ . R e l a ti v e i n t e n s it y
75 76 77 78 79 80 81 82 83 84 85 86 87 88 89
Electron kinetic energy / eV X - B - X - B - X - B - X - B - B - X - B - X - B - X - B - X - B - X - B - X - B - X - B - X - B - X - B - Figure 2: Stick photoelectron spectra from X (blue) or B (red) as functions of the energy of theejected electron for a probe photon at 95 eV. Cation states (see Table 1) are labeled according tothe order given in Ref. ; our calculations give E < E and E < E , which is why B −
18 isbefore B −
17. 14 E n e r gy / e V -4 -2 0 2 4 6 Time delay / fs = 0 o I n t e n s it y -4 -2 0 2 4 6 8 Time delay / fs = 80eV= 81eV= 82eV= 83eV= 84eV= 85eV= 86eV= 87eV= 88eV= 89eV= 90eV = 0 o ǫǫǫǫǫǫǫǫǫǫǫ I n t e n s it y
70 75 80 85 90 95
Energy / eV t delay = -4fst delay = -2fst delay = -1fst delay = 0fst delay = 1fst delay = 2fst delay = 4fs = 0 o E n e r gy / e V -4 -2 0 2 4 6 Time delay / fs = 45 o I n t e n s it y -4 -2 0 2 4 6 8 Time delay / fs = 80eV= 81eV= 82eV= 83eV= 84eV= 85eV= 86eV= 87eV= 88eV= 89eV= 90eV = 45 o ǫǫǫǫǫǫǫǫǫǫǫ I n t e n s it y
70 75 80 85 90 95
Energy / eV t delay = -4fst delay = -2fst delay = -1fst delay = 0fst delay = 1fst delay = 2fst delay = 4fs = 45 o E n e r gy / e V -4 -2 0 2 4 6 Time delay / fs = 90 o I n t e n s it y -4 -2 0 2 4 6 8 Time delay / fs = 80eV= 81eV= 82eV= 83eV= 84eV= 85eV= 86eV= 87eV= 88eV= 89eV= 90eV = 90 o ǫǫǫǫǫǫǫǫǫǫǫ I n t e n s it y
70 75 80 85 90 95
Energy / eV t delay = -4fst delay = -2fst delay = -1fst delay = 0fst delay = 1fst delay = 2fst delay = 4fs = 90 o Figure 3: Angle resolved photoelectron spectrum (ARPES). First column: ARPES (logarithmicscale) as a function of the time delay (horizontal axis) and energy of the ejected electrons (verticalaxis). The different panels correspond to different q orientation angle ( q is the angle between thedirection of the electron momentum and the polarization of the electric field). The intensity of theejected electrons are coded by colors according to the scale on the right side. Second column: Onedimensional cuts for the intensity of the ejected electrons via time delay with fixed q and e . Thirdcolumn: One dimensional cuts for the intensity of the ejected electrons via energy with fixed q and t delay . 15 O r i e n t a ti on () / d e g r ee
76 78 80 82 84 86 88 90 92 94
Energy / eV t delay = -4fs I n t e n s it y
70 75 80 85 90 95
Energy / eV = 0 o = 15 o = 30 o = 45 o = 60 o = 75 o = 90 o t delay = -4fs I n t e n s it y Orientation ( ) / degree = 80eV= 81eV= 82eV= 83eV= 84eV= 85eV= 86eV= 87eV= 88eV= 89eV= 90eV t delay = -4fs ǫǫǫǫǫǫǫǫǫǫǫ O r i e n t a ti on () / d e g r ee
76 78 80 82 84 86 88 90 92 94
Energy / eV t delay = -1fs I n t e n s it y
70 75 80 85 90 95
Energy / eV = 0 o = 15 o = 30 o = 45 o = 60 o = 75 o = 90 o t delay = -1fs I n t e n s it y Orientation ( ) / degree = 80eV= 81eV= 82eV= 83eV= 84eV= 85eV= 86eV= 87eV= 88eV= 89eV= 90eV t delay = -1fs ǫǫǫǫǫǫǫǫǫǫǫ O r i e n t a ti on () / d e g r ee
76 78 80 82 84 86 88 90 92 94
Energy / eV t delay = 2fs I n t e n s it y
70 75 80 85 90 95
Energy / eV = 0 o = 15 o = 30 o = 45 o = 60 o = 75 o = 90 o t delay = 2fs I n t e n s it y Orientation ( ) / degree = 80eV= 81eV= 82eV= 83eV= 84eV= 85eV= 86eV= 87eV= 88eV= 89eV= 90eV t delay = 2fs ǫǫǫǫǫǫǫǫǫǫǫ O r i e n t a ti on () / d e g r ee
76 78 80 82 84 86 88 90 92 94
Energy / eV t delay = 5fs I n t e n s it y
70 75 80 85 90 95
Energy / eV = 0 o = 15 o = 30 o = 45 o = 60 o = 75 o = 90 o t delay = 5fs I n t e n s it y Orientation ( ) / degree = 80eV= 81eV= 82eV= 83eV= 84eV= 85eV= 86eV= 87eV= 88eV= 89eV= 90eV t delay = 5fs ǫǫǫǫǫǫǫǫǫǫǫ Figure 4: Angle resolved photoelectron spectrum (ARPES). First column: ARPES (logarithmicscale) as a function of the energy of the ejected electrons (horizontal axis) and orientation angle q ( q is the angle between the direction of the electron momentum and the polarization of the electricfield) (vertical axis). The different panels correspond to different time delays between the pumpand probe pulses. The intensity of the ejected electrons are coded by colors according to the scaleon the right side. Second column: One dimensional cuts for the intensity of the ejected electronsvia energy with fixed t delay and q . Third column: One dimensional cuts for the intensity of theejected electrons via electron emission orientation with fixed t delay and e .16 O r i e n t a ti on () / d e g r ee -4 -2 0 2 4 6 Time delay / fs = 76eV ǫ I n t e n s it y -4 -2 0 2 4 6 8 Time delay / fs = 0 o = 15 o = 30 o = 45 o = 60 o = 75 o = 90 o = 76eV ǫ I n t e n s it y Orientation ( ) / deg t delay = -4fst delay = -2fst delay = -1fst delay = 0fst delay = 1fst delay = 2fst delay = 4fs = 76eV ǫ O r i e n t a ti on () / d e g r ee -4 -2 0 2 4 6 Time delay / fs = 79eV ǫ I n t e n s it y -4 -2 0 2 4 6 8 Time delay / fs = 0 o = 15 o = 30 o = 45 o = 60 o = 75 o = 90 o = 79eV ǫ I n t e n s it y Orientation ( ) / deg t delay = -4fst delay = -2fst delay = -1fst delay = 0fst delay = 1fst delay = 2fst delay = 4fs = 79eV ǫ O r i e n t a ti on () / d e g r ee -4 -2 0 2 4 6 Time delay / fs = 82eV ǫ I n t e n s it y -4 -2 0 2 4 6 8 Time delay / fs = 0 o = 15 o = 30 o = 45 o = 60 o = 75 o = 90 o = 82eV ǫ I n t e n s it y Orientation ( ) / deg t delay = -4fst delay = -2fst delay = -1fst delay = 0fst delay = 1fst delay = 2fst delay = 4fs = 82eV ǫ O r i e n t a ti on () / d e g r ee -4 -2 0 2 4 6 Time delay / fs = 85eV ǫ I n t e n s it y -4 -2 0 2 4 6 8 Time delay / fs = 0 o = 15 o = 30 o = 45 o = 60 o = 75 o = 90 o = 85eV ǫ I n t e n s it y Orientation ( ) / deg t delay = -4fst delay = -2fst delay = -1fst delay = 0fst delay = 1fst delay = 2fst delay = 4fs = 85eV ǫ O r i e n t a ti on () / d e g r ee -4 -2 0 2 4 6 Time delay / fs = 88eV ǫ I n t e n s it y -4 -2 0 2 4 6 8 Time delay / fs = 0 o = 15 o = 30 o = 45 o = 60 o = 75 o = 90 o = 88eV ǫ I n t e n s it y Orientation ( ) / deg t delay = -4fst delay = -2fst delay = -1fst delay = 0fst delay = 1fst delay = 2fst delay = 4fs = 88eV ǫ Figure 5: Angle resolved photoelectron spectrum (ARPES). First column: ARPES (logarithmicscale) as a function of the time delay t delay (horizontal axis) and orientation angle q ( q is the anglebetween the direction of the electron momentum and the polarization of the electric field) (verticalaxis). The different panels correspond to different energies of the ejected electrons. The intensityof the ejected electrons are coded by colors according to the scale on the right side. Second column:One dimensional cuts for the intensity of the ejected electrons via time delay with fixed e and q .Third column: One dimensional cuts for the intensity of the ejected electrons via electron emissionorientation with fixed E and t delay . 17able 1: Ab initio ionization potentials (MRCI-SD(Q) level of theory) and I ik , the squares of theDyson norms (CASSCF/aug-cc-pVQZ level of theory) with respect to either X or B at the FC point.The energy difference between the X and B states is 5 .
78 eV. (Experimental ionization potentialsand further theoretical values can be found for comparison in Ref. .)cation states ( j ) E j − E X / eV I ik ( X ) E j − E B / eV I ik ( B ) A ) 12.38 0.72 6.59 0.082 (1 B ) 12.51 0.69 6.72 0.093 (1 A ) 13.20 0.71 7.42 0.414 (1 B ) 14.14 0.00 8.36 0.005 (2 A ) 14.45 0.00 8.66 0.006 (2 B ) 15.18 0.01 9.40 0.017 (2 A ) 15.58 0.00 9.80 0.028 (2 B ) 16.35 0.29 10.56 0.249 (3 A ) 16.50 0.00 10.72 0.0010 (3 B ) 17.10 0.06 11.32 0.0211 (3 A ) 17.33 0.27 11.54 0.3212 (3 B ) 17.65 0.13 11.87 0.4113 (4 B ) 18.18 0.01 12.41 0.0314 (4 A ) 18.64 0.00 12.85 0.0015 (4 B ) 18.61 0.00 12.83 0.0016 (4 A ) 19.07 0.01 13.29 0.0117 (5 B ) 19.61 0.04 13.83 0.0218 (5 A ) 19.48 0.26 13.70 0.1119 (6 B ) 19.94 0.42 14.16 0.0418 esults and Discussion Figure 3 displays the intensity of the ejected electrons as a function of energy and time delaybetween the pump and probe pulses for three different fixed values of the orientation angle, q . Itcan be seen that the ionization probability is larger for smaller angles. For q > ◦ it is drasticallyreduced. At early times, when t delay < − X . Here, two clearly distinct high intensity bands are observed within the 75 −
78 eV and the80 −
85 eV energy intervals. These are consistent with the large Dyson norms calculated betweenthe X state of the neutral and some of the states of the cation (see Table 1). In particular, largeDyson norms are found between X and the 1st (0 . . . . . . .
42) cationic states. The corresponding ionization potential valuesfor these lie within (12 . − .
2) eV and (16 . − .
94) eV, thus resulting in two well separatedenergy regions, ∼ ( − ) eV and ∼ ( − ) eV. However, from t delay = − B as well. The explicit consequence of this is anew band that appears around 88 eV in the t delay = − B state starts to be populated, owing to the large value of the Dyson norm between B and the 3rdcationic state (0 . B to the 8th (0 . . .
41) cationic states, which corresponds to the energy band around (80 −
85) eVin the t delay = − t delay > X electronic state slowlydepletes, thus providing fewer electrons ejected from the ground state, which results in smallerintensity values (see the color in the 75 −
78 eV energy region). The structure of the figures atlarger angles ( q > ◦ ) are quite similar to the former ones, but the colors are much lighter dueto lower intensities, reflecting that large orientation angles are much less likely to be involvedefficiently in the ionization.The above findings are confirmed on Figure 4 and Figure 5, where the same results are pre-sented differently. On Figure 4, the electron emission orientation is given against the energy ofthe ejected electrons at several consecutive times. We observe that, up to t delay = − ( − ) eV and ( − ) eV, exhibit significant intensity. They correspond to19onization taking place from X only. Ionization occurring from B , once t delay > − t delay > t delay = − ( − ) eVenergy region and t delay < B state resulting in ionization taking place only from the X state. For t delay > X state. For e >
80 eV, a jointeffect of ionizations from X and B is observed, more substantially from X . Again, the shape andthe structure of the band for e >
85 eV and t delay = ( − ) − B .From Figure 5 it also appears that the angular distribution is strongly peaked along the probefield polarization, which is consistent with a high b value, close to two, for all ionizations. This isnot surprising because of the high photon energy of the probe, 95 eV, which implies high kineticenergy of the outer valence ionized electrons, typically characterized by high b values, similar forall ionizations.Finally the oscillatory patterns appearing in Figures 3 and 5 are clear fingerprints of the timedependence of the external electric field. Specifically, the pump pulse is a few-cycle pulse of width3 fs and period 0.87 fs, centered around 260 nm (4.8 eV) in the deep UV (UV-C) domain andtherefore its oscillation is faster than the nuclear motion.In summary, the most representative signal is perhaps the upper-right panel in Figure 3 (inten-sity against electron kinetic energy at different time delays for q = ◦ ). It is clear that the largesttemporal change in the spectrum is associated with the highest kinetic energies, from 86 to 89 eV,which are exclusively emitted from the B state, where the intensity increases significantly just after20he pump pulse. Correspondingly, the decrease of the intensity after the pump is most evident inthe low kinetic energy region, from 75 to 78 eV, due to the depleting of the X state, which is thedominant contribution in this energy window. Conclusions
A numerical simulation protocol has been developed for describing the electron dynamics of theozone molecule in the Franck-Condon region involving only the ground ( X ) and Hartley ( B ) elec-tronic states in the dynamics. Assuming isotropic initial distribution for the molecular ensemble,angle resolved photoelectron spectra have been calculated for various time delays between thepump that creates the wave packet (coherent superposition of X and B ) and the probe that ionizesfrom either X or B . This physical quantity can be measured in actual experiments and comparedto our calculations.The present results are very encouraging and call for further improvements concerning theaccuracy of the dynamics simulations. Therefore, our future aim is to perform more realistic simu-lations upon going beyond the presently assumed limiting hypotheses: isotropic initial distributionand populations extracted at the FC geometry only. This will be manifested by two significantchanges in the numerical protocol: i) after the pump pulse is off alignment of the molecular en-semble will be assumed; ii) instead of performing calculations at a single FC geometry, severalother nuclear geometries will be involved in the FC region where the nuclear density has signifi-cant value too.We stress again that given the dipole matrix elements and K-matrix, all photoionization observ-ables can be computed, like photoionization from fixed-in-space molecules (MFPADS) or partiallyoriented molecules, as well as suitable averages over final detector energy and angle resolution ,to accurately describe any specific experimental setup. Actually the 95 eV pulse employed in thepresent study was suggested by an experimental colleague. With hindsight angular distributionfrom unoriented molecules turn out not to be very informative, given the b values close to 2 for all21nal states at this relatively large photon energy. Working at lower energies would produce largeranisotropies. Moreover working with oriented molecules, which is a goal actively pursued in suchstudies, would further much enhance anisotropies, different for each initial and final state.The present numerical simulations clearly indicate that angle and time resolved photoelectronspectra can be used in molecular attophysics to characterize the creation of an electronic wavepacket in a neutral molecule on the subfemtosecond time scale. We expect our computationalstudy to be followed by experiments showing similar results.As the number of experimental choices is quite large, we found it important to set up a fullyab-initio general formulation that will accommodate any specific experimental setup. We lookforward to upcoming experiments to validate the theoretical framework provided here. References (1) M. Drescher, et al.,: Time-resolved atomic inner-shell spectroscopy. Nature,
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The authors thank H.-D. Meyer for very helpful discussions about the MCTDH calculations. P.D.and Á.V. acknowledge the supports from the CORINF and from the COST action CM1204 XLIC.
Author contributions statement
P.D., B.L., and A.V. initiated the concept of the calculations. P.D., G.J.H. and D.L. conducted thecalculations. G.J.H. prepared the figures. F.G., B.L., P.D. and A.V. wrote the manuscript. Allauthors analyzed the results and reviewed the manuscript.
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