Coupled nuclear and electron dynamics in the vicinity of a conical intersection
aa r X i v : . [ phy s i c s . c o m p - ph ] F e b Coupled nuclear and electron dynamics in the vicinity of a conicalintersection
Thomas Schnappinger and Regina de Vivie-Riedle
Department of Chemistry, LMU Munich, Germany, D-81377 Munich, Germany (Dated: 1 March 2021)
Ultrafast optical techniques allow to study ultrafast molecular dynamics involving both nuclear and electronic motion.To support interpretation, theoretical approaches are needed that can describe both the nuclear and electron dynamics.Hence, we revisit and expand our ansatz for the coupled description of the nuclear and electron dynamics in molecularsystems (NEMol). In this purely quantum mechanical ansatz the quantum-dynamical description of the nuclear motionis combined with the calculation of the electron dynamics in the eigenfunction basis. The NEMol ansatz is appliedto simulate the coupled dynamics of the molecule NO in the vicinity of a conical intersection (CoIn) with a specialfocus on the coherent electron dynamics induced by the non-adiabatic coupling. Furthermore, we aim to control thedynamics of the system when passing the CoIn. The control scheme relies on the carrier envelope phase (CEP) of afew-cycle IR pulse. The laser pulse influences both the movement of the nuclei and the electrons during the populationtransfer through the CoIn. I. INTRODUCTION
The continuous development of attosecond laser pulsesenables spectroscopic techniques which allow the time re-solved investigations of ultrafast photo-initiated processes inatoms, molecules and solids. Nowadays it is possible tostudy electronic correlation and ultrafast molecular dynam-ics through pump-probe experiments . Within these exper-iments attosecond, broad-band pulses are used to generateelectron wavepackets in highly excited states of molecules,leading to the discovery of effects such as electron local-ization in diatomic molecules and, later, of purely elec-tronic charge migration in biological relevant molecules .To explain and interpret the observations of these experi-ments, theoretical approaches are needed that can describethe dynamics of electrons in molecules. Most approachesuse time-dependent analogs of well-established quantum-chemical methods like time-dependent Hartree-Fock the-ory (TD-HF) or time-dependent density-functional theory(TD-DFT) . Furthermore, time-dependent post-Hartree-Fock methods like time-dependent configuration-interaction(TD-CI) , time-dependent coupled-cluster (TD-CC) and multi-configuration time-dependent Hartree-Fock areavailable for the correlated description of electron dynamicsin molecular systems. In other theoretical approaches theelectronic wavefunction is propagated directly in time, withthe help of Green’s function or in the basis of molecularorbitals . All these theories focus on the evolution of theelectronic subsystem, driven by electronic correlation andpredict long-lived coherences. The neglect of the nuclearmotion is justified by the assumption that the dynamics ofthe electrons is much faster than the one of the heavier nu-clei. This results in charge migration, an oscillatory motion ofelectron density with frequencies defined by the energy gapsamong the states populated with the initial laser pulse. If thestates of the superposition are close together, the electron dy-namics becomes slow and therefor the nuclear motion can nolonger be neglected. But as shown in numerous theoreticalworks , nuclear motion in general causes decoherence in molecular systems and should not be neglected in no cases.This decoherence causes the electronic wavepackets to existonly for short time scales . For small systems like H or D a full quantum treatment of the coupled electron and nucleardynamics is possible . Beyond these three particle problemsthere are computationally very demanding methods availablebased on a multi-configurational ansatz or on the coupleddescription of nuclear and electronic flux . Further tech-niques are based on the coupled propagation of the nuclearand electronic wavefunction on a single time-dependent po-tential energy surface . But for larger molecular sys-tems the main techniques used are mixed quantum classicalrepresentations . For example, the electron dynamics isdescribed using TD-DFT and the nuclear motion is consid-ered using an Ehrenfest approach . But these methods donot reflect the quantum nature of the nuclei which, however,becomes important for ultrashort pulse excitation and non-adiabatic transitions.In this paper we want to revisit and expand an ansatz forthe coupled description of the nuclear and electron dynam-ics in molecular systems (shortened NEMol) developedin our group. It is based on electronic structure calculationsas well as nuclear quantum dynamics. In its initial formu-lation the electronic wavefunctions are represented as Slaterdeterminants and propagated in the eigenstate basis. The cou-pling of the nuclear motion to the electron motion is incor-porated explicitly through the nuclear wavepacket motion aswell as through a coherence term with contributions from thenuclear and electronic wavefunctions. Compared to the simi-lar approaches , the feedback of the electron motion to thenuclear dynamics is less directly introduced by simulating thenuclear dynamics on coupled potential energy surfaces (PES).The central equation of the original NEMol ansatz re-lates the dynamics of the coupled one-electron density to thetemporal evolution of the expected value of the nuclear po-sitions. In the first part of this work we want to generalizethe NEMol ansatz by extending beyond this single geometryapproximation. Therefore, we introduce the NEMol-grid inorder to represent the electron dynamics at multiple points onthe grid used for the nuclear wavepacket propagation. In thelimit the NEMol-grid is equal to the grid representing the nu-clear wavepacket, but in practice we choose a coarser one.By means of a simple approximation it is possible to obtain acondensed representation of time-dependent electron densityin the one-electron-two-orbital (1e-2o) picture.In the second part we want to explore the potential of ourNEMol ansatz. For this purpose, we consider a situation thatcan generate coherent electron dynamics in excited states ofmolecules even without a laser pulse present. Such a scenariooccurs in the vicinity of a conical intersection (CoIn) .For this ubiquitous but nevertheless extraordinary points ina molecular system the adiabatic separation between nuclearand electronic motion breaks down and the electronicstates involved become degenerate. Beside the creation offunnels for radiationless electronic transitions a coherent elec-tron wavepacket is created whose dynamics approaches thetime scale of the nuclear dynamics. All these properties ofCoIn’s are determined by the shape and size of the non-adiabatic coupling elements (NAC’s) and the topography ofthe vicinity. As a realistic molecular system which providessuch a situation we have chosen the NO molecule. Afterexcitation into the first excited state a CoIn enables an ultra-fast non-adiabatic transition back to the ground state withinless than 100 fs. This fast relaxation as well as the photo-physics of NO in general have been widely explored boththeoretically and experimentally . Beside the freerelaxation of NO we also studied the influence on the coupledelectron dynamics when applying a few-cycle IR laser pulse inthe vicinity of the CoIn. The variation of the carrier envelopephase φ (CEP) of such a few-cycle pulse offers the possibil-ity to steer electrons and nuclei . Similar to pre-vious studies we apply this CEP-control-scheme toNO and evaluate the CEP-dependence of the resulting cou-pled nuclear and electron dynamics. II. COUPLED NUCLEAR AND ELECTRON DYNAMICS(NEMOL)
In the original NEMol ansatz the coupled one-electron density ρ ( r , t ; h R i ( t )) is defined according to equa-tion 1. For convenience the detailed derivation of this equationcan be found in the appendix adapted to the current notation. ρ ( r , t ; h R i ( t )) = ∑ j A j j ( t ) ρ j j ( r ; h R i ( t ))+ ∑ k = j Re (cid:8) A jk ( t ) ρ jk ( r ; h R i ( t )) e − i ξ jk ( t ) (cid:9) , (1)with ξ jk ( t ) = ∆ E jk ( h R i ( t )) ∆ t + ξ jk ( t − ∆ t ) . (2)The first summation consists of the state specific electronicdensity ρ j j ( r , t ; h R i ( t )) weighted with the corresponding time-dependent population A j j ( t ) . The second summation de-fines the coherent contribution to the coupled electron densityand consists of the time-dependent overlap A jk ( t ) , the one-electron transition density ρ jk ( r , t ; h R i ( t )) and its pure elec-tronic phase defined by the energy difference ∆ E jk betweenthe electronic states involved. All quantities related to the electronic wavefunction are calculated for one nuclear geom-etry per time step which is defined by the time-dependent ex-pected value of the position h R i ( t ) (for definition see the ap-pendix). As long as we are focusing on situations with quitelocalized wavepackets and/or one-dimensional systems this approximation works quite well. But in order to treathigher dimensional systems and more complex processes wewant to generalize the NEMol ansatz in this work. To ex-tend the ansatz the integration over the full nuclear coordinatespace is split up in segments to improve the resolution of thespatial dependence of the electronic phase term. For this pur-pose a second grid, the NEMol-grid, is introduced. The re-sulting modified NEMol ansatz is described in the followingsection using exemplary a system with two nuclear coordi-nates c and c . The complete two-dimensional coordinatespace is split up into M × L segments defined by their bound-aries m min , m max and l min , l max . For each of these segments ml the population terms α mlj j ( t ) and the overlap terms α mljk ( t ) arecalculated. α mljk ( t ) = Z m max m min Z l max l min χ ∗ j ( R , t ) χ k ( R , t ) dc dc . (3)The sum of these segment terms results in the correspondingtotal population and overlap. M ∑ m = L ∑ l = α mljk ( t ) = (cid:10) χ j ( R , t ) (cid:12)(cid:12) χ k ( R , t ) (cid:11) R = A jk ( t ) . (4)At the center R ml of each segment the state specific elec-tronic densities, the one-electron transition densities and theeigenenergies are determined and with these values the cou-pled one-electron density for each segment ρ ml ( r , t ; R ml ) iscalculated. ρ ml ( r , t ; R ml ) = ∑ j α mlj j ( t ) ρ j j ( r ; R ml )+ ∑ k = j Re (cid:8) α mljk ( t ) ρ jk ( r ; R ml ) e − i ξ mljk ( t ) (cid:9) , (5)with ξ mljk ( t ) = ∆ E jk ( R ml ) ∆ t + ξ mljk ( t − ∆ t ) . (6)It should be noted that for each segment the ∆ E jk values andthe electron densities are no longer dependent on h R i ( t ) . Incontrast to the original NEMol ansatz, now many ∆ E valuesare simultaneously contributing to the overall electron dynam-ics. They are addressed, whenever the nuclear wavepacket islocated there. To obtain the total coupled electron density theindividual contributions of each segment are summed up. ρ ( r , t ; R ) = M ∑ m = L ∑ l = ρ ml ( r , t ; R ml ) . (7)This total coupled electron density ρ ( r , t ; R ) describes theelectron dynamics coupled to multiple grid points on whichthe nuclear wavepacket is represented.A second aspect that we would like to introduce is a fur-ther simplification. For clarity reasons it is here formulatedin terms of the original NEMol ansatz. We now consider asystem of two electronic states described by their electronicwavefunctions ϕ and ϕ . In the simplest case the wavefunc-tions of both states are described by two Slater determinantswhich only differ in the occupation of one spin orbital θ . Now the coupled total electron density can be simplified by ex-pressing the densities and transition densities using the spinorbitals. ρ ( r , t ; h R i ( t )) = N − ∑ j = | θ j ( r ; h R i ( t )) | + ∑ k = A kk ( t ) | θ k ( r ; h R i ( t )) | + Re (cid:8) A ( t ) θ ( r ; h R i ( t )) θ ( r ; h R i ( t )) e − i ξ ( t ) (cid:9) . (8)The summation at the beginning includes the densities of allequally occupied orbitals and is followed by the densities ofthe remaining two orbitals θ and θ weighted with the popu-lations A ( t ) and A ( t ) The coherent part contains the prod-uct of the orbitals θ and θ . Within this simplification it isnow possible to neglect the contributions of the equally oc-cupied orbitals in order to study the coupled electron dynam-ics in an one-electron-two-orbital (1e-2o) picture. Under theabove mentioned approximation this 1e-2o picture is a possi-bility to examine the coherent part of the electron dynamics ina very condensed way. This simplification can also be madein combination with the NEMol-gird. III. NO COUPLED DYNAMICS
We apply our extended NEMol approach to the non-adiabatic dynamics of NO . In this molecule, a CoIn (de-picted in FIG. 1(b)) between the D and the D state enablesa radiationless relaxation. The ultrafast non-adiabatic transi-tion takes less than 100 fs and has been widely explored boththeoretically and experimentally . First we analyzethe relaxation itself and next we apply a few-cycle IR laserpulse to control the dynamics in the vicinity of the CoIn, sim-ilar to previous studies . With our NEMol ansatz wecan study its influence on the motion of the nuclei and theelectrons.The nuclear dynamics is performed on the two-dimensionaladiabatic potential energy surfaces of the D and the D stateshown in FIG. 1. The coordinates spanning the PES’s are thegradient difference and derivative coupling vectors definingthe branching space of the D / D -CoIn depicted in FIG. 1(b).These two vectors correspond to the bending angle α and theasymmetric stretching coordinate b , defined as half the differ-ence between the two NO distances. The last internal degreeof freedom, the symmetric stretch coordinate, is kept con-stant at the value of the optimized D / D -CoIn (1 .
267 Å). Asshown by Richter et al. the population dynamics obtainedwithin this two-dimensional coordinate space is in very goodagreement with the full dimensional simulations by Arasakiet al. . We performed our dynamics simulations in the adi-abatic representation and the corresponding NAC’s between D ans D are shown in FIG. 1(c). It should be mentioned thatin previous studies the simulations were performed inthe diabatic representation and therefore small deviations mayoccur due to the limitation of the grid spacing. Further infor- mation about the simulation setup can be found in section IIof the SI.In order to calculated the coupled electron density accord-ing to equation 7 we define a NEMol-grid of 15 ×
13 pointswhich are equally distributed between 1 .
34 rad to 2 .
86 radin the α -coordinate and between − .
33 Å to 0 .
33 Å in the b -coordinate. The necessary population- and overlap-termsare calculated for equal-spaced segments around these gridpoints. To cover the entire PES the segments for the bound-ary grid points are larger. The transformation of the fullwavepacket onto the NEMol-grid, the overlap terms and theresulting coherence terms are visualized in FIG. S6 (free prop-agation) and FIG. S10 (propagation with laser pulse) in theSI. The two active orbitals which are required to describe theNEMol-dynamics in the one-electron-two-orbital (1e-2o) pic-ture are shown in FIG. 2 at the optimized CoIn. The non-binding orbital n N with contributions at the nitrogen atom isassociated with the D state and the non-binding orbital n O located only at the oxygen atoms is attributed to the D state.The energy difference ∆ E between the D and D state foreach grid point is shown in FIG. S4 of the SI. A. Free dynamics of NO To initiate the dynamics simulation in the D state we as-sumed a delta pulse excitation. The temporal evolution of thepopulation of both states is shown in the upper panel of Fig. 3and the dynamics of the nuclear wavepackets integrated overthe α -coordinate, respectively, the b -coordinate are depictedin FIG. S5 for both surfaces. The nuclear wavepacket startedin D reaches the vicinity of the CoIn after approximately7 fs for the first time. While passing the coupling region inthe time interval from 7 fs to 15 fs the population of the elec-tronic ground state increases to over 60 %. The part of thenuclear wavepacket remaining in the D state reaches its turn-ing point around 15 fs and then propagates backwards. Thisleads to a second passage through the CoIn area and an in-crease of the population of the D state around 22 fs. The nu-clear wavepacket evolving on the lower adiabatic surface, re-encounters the CoIn region later at around 30 fs. During thisthird passage, a substantial part of the population is transferredback into the excited state. After 35 fs the wavepacket is delo-calized on both surfaces and the population is nearly equal inboth states. Towards the end of the simulation at around 50 fsa fourth passage occurs. The wavepacket remains symmetri- bending angle [rad] asy m . c oo r d b [ Å ] -0.30.00.3 D D (a) D D asy m . c oo r d b [ Å ] e n e r g y [ e V ] b e n d i n g a n g l e [ r a d ] (b) a(cid:0)(cid:1)(cid:2)(cid:3)(cid:4)(cid:5)(cid:6)(cid:7)(cid:8) b [ Å ] -0.10.00.11.80 1.85 1.90bending angle [rad]1.80 1.85 1.90 (cid:9) / (cid:10) (cid:11) / (cid:12) b (c) FIG. 1. (a) Adiabatic potential energy surfaces of the D state (left)and D state (right) of NO . The CoIn is marked in white and the po-sitions of the relevant minima in the two-dimensional subspace aredisplayed in black.The two marked minima are only slightly higherin energy than the fully optimized minimum structures shown inthe SI. (b) The vicinity of the D / D -CoIn. (c) Non-adiabatic cou-pling elements between D ans D at the CoIn, α -element left and b -element right. cal with respect to the b -coordinate for the whole simulation n N n O n N n O D D n N n O FIG. 2. Molecular orbital schema with the active electron indicatedin green and corresponding orbitals at the optimized CoIn. Orbitalsare shown with an isovalue of 0 . time. For the wavepacket on the lower PES (see right sightof FIG. S5(b) in the SI) the formation of a nodal structure for b = . .In the lower part of FIG. 3 snapshots of the electron den-sity in the 1e-2o picture are shown. For a better visualiza-tion also the difference in density with respect to t = yz -plane and the centerof mass defines the origin of the laboratory frame. Thereforethe internal α -coordinate points to the same direction as the y -coordinate and the internal b -coordinate is associated withthe z -coordinate. The orientation of the molecule is shownin the upper right corner of FIG. 4. In correspondence to thenon-adiabatic transition from the D state to the D state, themain feature of the electron dynamics is the loss of density atthe nitrogen and the corresponding gain of density at the oxy-gen atoms. In addition, the change in the electron density at-tributed to the motion of the nuclei (Born-Oppenheimer part)is present. Due to the high symmetry of NO , the electrondensity is mirror-symmetrical with respect to the xy -plane,which is equivalent to the symmetric behaviour of the nuclearwavepacket with respect to the b -coordinate.To analyze the electron dynamics we calculated the dipolemoment of the electron density within the 1e-2o picture. In theupper panel of FIG. 4 the temporal evolution of its three com-ponents is shown; for the molecular orientation see the upperright corner of FIG. 4. To distinguish the Born-Oppenheimerpart of the dynamics from the coherent electron dynamicsthe density was calculated once with the coherent part in-cluded and once without. For both quantities the respectivedipole moments were determined as well as their difference,hereinafter labeled as ∆ x -coordinate and thus the 1e-2o- x -component of the dipole mo-ment stays zero and is excluded from further discussions. The1e-2o- y -component shows the largest values and the strongestchanges over time. Its evolution follows the dynamics ofthe population. In the initial 20 fs the first passage throughthe CoIn region occurs and simultaneously the value of the1e-2o- y -component changes from 0 . . u . to − . . u . . Thezero crossing occurs at 10 fs. For later times when dephasingand partial recurrence of the nuclear wavepackets become im-portant the y -component approaches zero at about 40 fs andbecomes negative thereafter again. These main features dis- ! " $%&’() p*+,-./i35 D t789 :;<= FIG. 3. Free dynamics of NO . Upper panel: Populations of the D and D state as a function of simulation time. Lower panel: Snapshots ofthe electron density in the 1e-2o picture and the difference in density relative to the initial density (green electron-loss, orange electron-gain).The isovalues used are 0 .
006 respectively ± . . appear for the ∆ y -component (lower panel FIG. 4) andonly fast oscillations with one order of magnitude smaller am-plitudes are left. The largest amplitudes are observed around10 fs, 30 fs and 50 fs. These amplitudes coincide with the pas-sages of the wavepacket through the CoIn region. The largedifference between the 1e-2o and the ∆ y -component is dominated by the nu-clear motion. That is understandable, since the y -coordinate isaligned along the main direction of dynamics ( α -coordinate),which mediates the non-adiabatic transition. The temporalevolution of the 1e-2o- z -component is an order of magnitudesmaller and almost identical to its ∆ value. The dynamics ofthe z -component is not dominated by the nuclear motion butsolely induced by the coherent electron dynamics. Therefore,we can use the y and the z component to distinguish betweenthe two contributions of the coupled electron dynamics. Asthe ∆ values of both components lie amplitude wise in thesame region and show a similar pattern they are suitable tomonitor the coherent electron dynamics in the system. Over-all the nuclear motion has a much larger impact on the dipolemoment than the coherent electron dynamics.By applying the Fourier transform to the temporal evolutionof the dipole moments the corresponding frequencies are de-termined. Beside the ∆ y - and the z -component for allthree cases are shown in FIG. 5. The spectra are all normal- ized to one individually. The relative magnitude between allquantities can be estimated from figure FIG. 4. All frequen-cies with an intensity larger than 0.1 are listed in TABLE S3and TABLE S4 of the SI.The ∆ . . ∆ E (0 . . ∆ E values. The phase of the over-lap term relates to the difference in momentum of the nuclearwavepackets involved. In our test system NO the wavepacketon D approaches the CoIn with a high momentum, largerthan the ∆ E gaps near the CoIn. In other words the coher-ent dynamics of the electronic wavepacket is in the NO casealso significantly influenced by the phase-differences of thenuclear wavepackets moving on different potentials. This cor-relation is illustrated in in FIG. S7 in the SI for two indi-vidual NEMol-grid points. The frequencies for the 1e-2o-components (FIG. 5 red) are dominated by the slower nu-clear dynamics (Born-Oppenheimer part) giving rise to thestrong peaks below 0 . y -component, >?
30 40 50 @ABC EFGHdIJKLMNOPQRSTU VWX
YZ[\]^_‘ bcefg
FIG. 4. Field-free temporal evolution of the dipole moment compo-nents based on the electron density in the 1e-2o picture. Upper panel:total value of all three components. The 1e-2o- z -component is en-hanced by a factor of five. The orientation of the molecule is shownas inlay in the upper right corner. Lower panel: Difference betweenthe dipole moment components ( ∆ y - and z -components. whereas for the z -component the initial pattern is still recog-nizable. This behaviour is further increased for the full density(FIG. 5 green). For both components some peaks appear in allthree cases, especially in the energy range between 0 . .
75 eV. They can be attributed to the coherent electronic dy-namics and may also be experimentally observable.Further information can be gained by extracting the timewhen these frequencies occur. This allows us to connect themto a specific movement in the system. Therefore, we per-formed short-time Fourier transform spectra for the ∆ y and the ∆ z components using a Gaussian window-ing function with a width of 180 data points correspondingto a time of 18 .
14 fs. The resulting two spectrograms areshown in FIG. 6. The ∆ y spectrogram (left) showstwo main pairs of signals around 10 fs (0 . . . . ∆ z spectrogram (right) shows two main signals. Thefirst one appears around 10 fs (first passage through CoIn)and covers a frequency range from 0 . . nhjklmoqrsuvwxyz{| }~(cid:127)(cid:128)(cid:129)(cid:130)(cid:131)(cid:132)(cid:133) (cid:134)(cid:135)(cid:136) ]1.0 (cid:137)(cid:138)(cid:139) (cid:140)(cid:141)(cid:142)(cid:143)(cid:144)(cid:145)(cid:146)(cid:147)(cid:148) (cid:149)(cid:150)(cid:151)(cid:152)(cid:153) (cid:154)(cid:155)(cid:156)(cid:157)(cid:158)(cid:159) (cid:160)¡¢£⁄ ¥ (a) ƒ§¤'“«‹›fifl(cid:176)–†‡·(cid:181)¶• ‚„”»…‰(cid:190)¿(cid:192) `´ˆ ]1.0 ˜¯˘ ˙¨(cid:201)˚¸(cid:204)˝˛ˇ —(cid:209)(cid:210)(cid:211)(cid:212) z (cid:213)(cid:214)(cid:215)(cid:216)(cid:217) z (cid:218)(cid:219)(cid:220)(cid:221) (cid:222) (b) FIG. 5. The Fourier spectra of the y -component (a) and z -component(b) of the dipole moment obtained using the ∆ nal which extends over low-frequency components (0 . . , since here the largest electronic coherence in the field-free case exists. f r e q u e n c y [ e V ] f r e q u e n c y [ e V ] normalized intensity time [fs] time [fs] FIG. 6. Short-time Fourier transform of the ∆ y dipole moment component (left) and ∆ z dipole moment component (right). TheFourier spectrograms are normalized and a Gaussian windowing function with a width of 180 data points corresponding to a time of 18 .
14 fsis used.
B. Dynamics in the presence of a few-cycle IR pulse
Again a delta pulse excitation is used to initiate the dy-namics. With the appropriate time delay, a few-cycle IR laserpulse is applied to influence the first passage through the CoInand thereby the subsequent coupled dynamics. The used few-cycle pulse has a Gaussian shape and is defined as: E ( t ) = E max · e − (cid:16) t − t σ (cid:17) · cos ( ω ( t − t ) + φ ) , (9)with σ = FWHM p ( ) . with the central frequency ω , the time zero t , the maximalfield amplitude E max , the full width half maximum (FWHM)and the carrier envelope phase φ (CEP). The time zero t ofthe pulse, defining the position of its maximum, was chosen tomatch the time window when the wavepacket is located nearthe CoIn ( t =
10 fs). For this time the nuclear wavepacketis still very localized and the electronic coherence maximal.The central frequency ω is chosen to be resonant with the ac-tual energy gap ∆ E = 0 .
76 eV between the electronic states.The remaining three pulse parameters, the field amplitude E max , the full width half maximum (FWHM) and the CEP φ , are set to E max = .
103 GV cm − (which corresponds toa maximum intensity of 1 . × W cm − ), FWHM = φ = π . In comparison with the pulse parameters usedby Richter et al. all values are quite similar. Only ourintensity is lower to stay in the range where the influenceof the CEP pulse is mainly determined by the interplay ofthe non-adiabatic transition and the light induced electroniccoherence . By this we also ensure to stay below or at thethreshold of ionization. The light-matter interaction is treated within the dipole approximation, for details see section I in theSI. We assume, that the electric component of the pulse is opti-mally aligned with the transition dipole moment. The absolutevalue of the TDM is used, which is shown in FIG. S3(a) of theSI. As stated by Richter et al. already a moderate molecu-lar alignment distribution is sufficient to observe the effect ofsuch a control pulse.The evolution of the adiabatic populations influenced by thefew-cycle IR-field is shown in the upper panel of FIG. 7. Therelated nuclear wavepacket dynamics on both surfaces inte-grated over the α -coordinate, respectively, the b -coordinateare depicted in FIG. S8 of the SI. During the first transitionthrough the CoIn region (7 fs to 15 fs) a 50:50 population ofboth states is created. The interaction with the light pulse is re-flected in the small wriggles around 10 fs. The subsequent dy-namics is comparable to the field-free case up to 30 fs. There-after no clear passage through the CoIn region is observable.Thus the IR pulse induces a change in the nuclear dynamicswhich persists beyond the pulse duration. As an importantconsequence, the nuclear motion becomes asymmetric withrespect to the b -coordinate and the nuclear wavepacket evenloses its nodal structure (compare FIG. S5(b) and FIG. S8(b)both in the SI), which was also observed by Richter et al. .This asymmetry leads to the partly deviations from of the CoInregion after 30 fs. On the lower panel of FIG. 7 snapshots ofthe electron density in the 1e-2o picture are shown. Again thedifference in density with respect to t = xy -plane i.e. the b -coordinate. This asymmetry persists after thelaser pulse is no longer active (for example see the snapshotsat 30 . (cid:223)(cid:224)Æ (cid:226)ª(cid:228)(cid:229)(cid:230) (cid:231)Ł ØŒº (cid:236)(cid:237) (cid:238)(cid:239)(cid:240)æ (cid:242)(cid:243) (cid:244)ı
40 500.0 (cid:246)(cid:247)ł øœß(cid:252)(cid:253)(cid:254)(cid:255)i(cid:0)(cid:1) D (cid:2) t(cid:3)(cid:4)(cid:5) (cid:6)(cid:7)(cid:8)(cid:9) FIG. 7. Dynamics of NO in the presents of a few-cycle IR laser pulse. Upper panel: Populations of the D and D state as a function ofsimulation time. Lower panel: Snapshots of the electron density in the 1e-2o picture and the density difference relative to the initial density(green electron-loss, orange electron-gain). The isovalues used are 0 .
006 respectively ± . . right to the left oxygen is most prominently observable for thesnapshots at 7 . . ∆ y - and z -coordinate in the lower panel. Again the 1e-2o- x -component stays zero for the whole simulation time. As thefew-cycle IR pulse induces the asymmetry mainly along the b -coordinate, the overall temporal evolution of the 1e-2o- y -component and the ∆ y is similar to the field-free case.The 1e-2o- z -component experiences the main changes. Dur-ing the pulse strong and fast oscillations are observed withan amplitude nearly thirty times larger than for the field-free case. The oscillations stay up to ten times larger af-ter the pulse. The superimposed slow oscillation with a pe-riod of about 20 fs can be assigned to the asymmetry in thenuclear motion. It does not appear for the ∆ z com-ponent reflecting solely the coherent electron dynamics. Bybreaking the symmetry of the nuclear motion with the laserpulse the electronic coherence induced in the NO moleculeis significantly larger. Again it is observable mainly in the z -component, respectively, in the b -coordinate. During the lightpulse it is now the coherent electron dynamics which is re-sponsible for the largest changes in the dipole moment.The corresponding frequencies for the ∆ ∆ . ∆ y spectra(FIG. 9(a) blue) are in the same energy region as in the field-free case and only the ∆ z -spectrum (FIG. 9(b) bluedotted line) shows differences. Its main peaks are shiftedto higher energies by roughly 0 . .
76 eV) into the system, which influences themomentum of the nuclear wavepacket and thereby the phaseof the overlap term (equation 1) which subsequently leadsto higher frequencies observed in the coherent electron dy-namics. The correlation between the phase of the overlapterm, the electronic phase and the laser pulse is illustratedin FIG. S11 of the SI for two individual grid points. Thefrequencies for the y -component determined with the 1e-2o-density (FIG. 9(a) red) and the full-density (FIG. 9(a) green)exhibit the same behaviour as in the field-free case. The highenergy parts lose significantly intensity since the slower nu-clear dynamics (Born-Oppenheimer part) dominates this sig-nal. The dominance of the oscillating dipole moment origi-nating from the coherent electron dynamics shows up in thenearly identical spectra for the 1e-2o- z (FIG. 9(b) red) and FIG. 8. Temporal evolution of the dipole moment components (DMcomp) based on the electron density in the 1e-2o picture in thepresents of a few-cycle IR pulse. Upper panel: total value of allthree components. The orientation of the molecule is shown as inlayin the middle. Lower panel: Difference between the dipole momentcomponents one time calculated with the coherence term includedand once without it. Differences only shown for the for y - and z -DMcomp. ∆ z (FIG. 9(b) blue). For the z -spectra of the full-density (FIG. 9(b) green) the high energy parts lose some in-tensity but still more high energy contributions survive com-pared to the field-free case.The results of the short-time Fourier transform for the ∆ y and the ∆ z dipole moment component us-ing a Gaussian windowing function with a width of 180data points corresponding to a time of 18 .
14 fs are shown inFIG. 10. Both spectrograms show a dominant signal whichis attributed to the first passage through the CoIn region.The observable electron dynamics is significantly strength-ened by the simultaneous light pulse interaction. In case of the ∆ y spectrogram (left) some new features between 10 fsto 30 fs appear. Due to the symmetry breaking of the nuclearmotion by the laser pulse, signals with very low frequenciesas well as an extended signal around 1 . ∆ z component only one dominant peakis observed. In summary, the presence of a few-cycle IR pulsemodifies the coupled dynamics by breaking the symmetry ofthe nuclear motion and changing the temporal evolution ofthe population. Both factors lead to a significant increase ofelectronic coherence in the molecule especially along the z -coordinate (laboratory frame), respectively, the b -coordinate(internal frame). n(cid:10)(cid:11)(cid:12)(cid:13)(cid:14)(cid:15)(cid:16)(cid:17)(cid:18)(cid:19)(cid:20)(cid:21)(cid:22)(cid:23)(cid:24)(cid:25)(cid:26) f(cid:27)(cid:28)(cid:29)(cid:30)(cid:31) c! " ]1.0 (a) BCEFGHIJKLMNOPQRST
UVWXYZ[\] ^_‘ ]1.0 abd eghjklmop qrsuv z wxyz{ z |}~(cid:127) (cid:128) (b) FIG. 9. The Fourier spectra of the y -component (a) and z -component(b) of the dipole moment in the presents of a CEP-pulse (0 . π ) ob-tained using the ∆ IV. WAVEFORM CONTROL OF MOLECULAR DYNAMICS
In the last part we investigate the controllability of thenuclear and electron dynamics by the variation of theCEP φ of a few-cycle IR laser pulse. As shown in theliterature the CEP control scheme offers the pos-sibility to steer electrons and nuclei in the ionization processbut also during the passage through a CoIn. The few-cycle IRpulse builds up a coherent electronic and nuclear wavepacketwith a well-defined phase-relationship controllable by theCEP. In the vicinity of a CoIn also the non-trivial geomet-ric phase (Pancharatnam–Berry phase) is introduced .The interplay of both phase-terms lead to an interference pro-cess when the CoIn is passed. The interference (constructiveor destructive) can be manipulated by the CEP. A. Control of the nuclear dynamics
As a first step we focus on the controllability of the nu-clear dynamics. Therefore, we define control objectives whichare directly accessible via the nuclear wavepacket and use the0 time [fs] time [fs] f r e q u e n c y [ e V ] normalized intensity f r e q u e n c y [ e V ] FIG. 10. Short-time Fourier transform of the ∆ y dipole moment component (left) and ∆ z dipole moment component (right) witha few-cycle IR pulse included in the simulation. The Fourier spectrogramms are normalized and a Gaussian windowing function with a widthof 180 data points corresponding to a time of 18 .
14 fs is used. population P D ( t , φ ) of the D ground state as reference. P D ( t , φ ) = h χ D ( R , t , φ ) | χ D ( R , t , φ ) i R . (10)One objective is the CEP efficiency Γ ( t ) which is calculatedas the difference of the maximum and the minimum popula-tion P D ( t , φ ) for each time step: Γ ( t ) = max ( P D ( t , φ )) − min (cid:0) P D ( t , φ ′ ) (cid:1) . (11)For its maximum value the population of the target state showsthe highest CEP-dependence and consequently the highest de-gree of controllability with respect to the population trans-fer. The light pulse amplifies the coherent electron dynamicsin the system by breaking the symmetry with respect to theasymmetric stretching coordinate b , as shown in section III B.Therefore, the second objective is the CEP-dependent asym-metry parameter AN ( t , φ ) quantifying the CEP induced asym-metry in the nuclear motion with respect to the coordinate b . AN ( t , φ ) = P RD ( t , φ ) − P LD ( t , φ ) P D ( t , φ ) . (12)Where P LD ( t , φ ) and P RD ( t , φ ) are defined as follows: P LD ( t , φ ) = α max Z α min d α Z b min db χ ∗ D ( R , t , φ ) χ D ( R , t , φ ) . (13) P RD ( t , φ ) = α max Z α min d α b max Z db χ ∗ D ( R , t , φ ) χ D ( R , t , φ ) . (14)In the spirit of the efficiency Γ ( t ) a maximal asymmetry AN max ( t ) is calculated as: AN max ( t ) = max ( AN ( t , φ )) + min (cid:0) AN ( t , φ ′ ) (cid:1) . (15) For its maximum the motion of the nuclear wavepacket showsthe highest asymmetry and controllability. Its CEP depen-dence is illustrated in FIG. 11. t i m e [ f s ] prob. density asymmetric coord. b [Å] -0.2 0.0 0.2 -0.2 0.0 0.20 1.4 FIG. 11. Normalized nuclear probability density evolution in thepresents of a few-cycle IR pulse withe a CEP of 0 . π (left) and 1 . π (right) on the adiabatic D -surface integrated over the α -coordinate.For the other probability densities see FIG. S8 and FIG. S9 in the SI. The temporal evolution of Γ ( t ) and the CEP dependent pop-ulation P D ( t , φ ) at three selected times are shown in FIG. 12.The CEP efficiency (blue line) reaches its global maximum(13 %) nearly simultaneously with the peak intensity ( t =
10 fs) of the laser pulse (grey area). The increase of Γ ( t ) isslightly delayed and the subsequent decrease to 3 % occursin two steps. After the laser pulse, approximately at 15 fs, Γ ( t ) has a finite oscillating value with a maximum of about5 % around 20 fs, which indicates the second passage throughthe CoIn region. The later passages through the CoIn region1 time [fs]10 20 30 40 500 C EP e ff i c i e n cy Γ (a) (cid:129)(cid:130)(cid:131)(cid:132)(cid:133)(cid:134)(cid:135)(cid:136)(cid:137)(cid:138)(cid:139)(cid:140)(cid:141)(cid:142) D (cid:143)(cid:144)(cid:145) (cid:146)(cid:147)(cid:148) [ ]0.0 0.5 1.0 1.5 (cid:149)(cid:150)(cid:151)(cid:152)(cid:153)(cid:154) (cid:155)(cid:156)(cid:157)(cid:158) (cid:159)(cid:160) ¡¢£⁄ ¥ƒ§¤ '“ (b) FIG. 12. (a) Temporal evolution of the CEP efficiency Γ ( t ) (blue).The vertical colored lines indicate the points in time that are exam-ined more closely. The violet curve indicates the deviation of themean population (averaged over all CEP’s) from the population in thefield-free case. The envelope of the IR pulse is indicated in grey. (b)Mean difference of the CEP-dependent populations P D ( t , φ ) givenin percent for different times. at 30 fs and after 40 fs can roughly be seen in the increaseof Γ ( t ) . The deviation (violet curve) of the mean population(averaged over all CEP’s) from the population in the field-free case is significant, especially during the IR pulse and af-ter 30 fs. As discussed with respect to FIG. 7, the inducedasymmetry leads to a partial missing of the CoIn region after30 fs, which is almost independent of the CEP chosen. TheCEP-dependence of the population P D ( t , φ ) (see FIG. 12(b))is recorded for three selected times marked as vertical linesin 12(a). For better visualization the mean difference is usedhere and, unless otherwise stated, in all following respectivefigures. The first line at 15 fs (green) matches the end of thelaser pulse. The second (red line) and the third point (yel-low line) correspond to the second and fourth passage throughthe CoIn region. For all three times P D ( t , φ ) shows a sinu-soidal oscillation with a periodicity of approximately π . Forinterference a periodicity of 2 π should emerge. Thus the ob-served π dependence of the population is an indication thatit is due mostly to the temporal asymmetry of the few-cyclelaser pulse. An analog analysis is performed for the asymmetry of thenuclear motion along the stretching coordinate b and shown inFIG. 13. The maximal asymmetry AN max ( t ) shows its global time [fs]10 20 30 40 500 m ax asy mm e t r y AN (a) «‹›fifl(cid:176)–†‡ AN CEP [ ]0.0 0.5 1.0 1.5 2.00.3 15 fs20 fs40 fs0.1-0.1-0.3 (b)
FIG. 13. (a) Temporal evolution of the maximal asymmetry of thenuclei AN max ( t ) after t =
10 fs. The vertical colored lines indicatethe points in time that are examined more closely. The envelope ofthe IR pulse is indicated in grey. (b) The CEP-dependent asymmetryparameter AN ( t , φ ) for different points in time. maximum around 8 fs. As it is defined with respect to the pop-ulation in D alone, the values for the early times (in the begin-ning of the laser pulse) are overestimated compared to the ac-tual population in the D O state. Nevertheless, we can deducethat AN max ( t ) follows the envelope of the laser pulse. Thesubsequent peaks between 15 fs to 20 fs, at 30 fs and between42 fs to 48 fs correspond to the passages through the CoInregion. The decreasing height of the maxima reflects againthe delocalization of the nuclear wavepacket with time. TheCEP-dependence of the asymmetry of the nuclear motion (seeFIG. 13(b)) AN ( t , φ ) is recorded for the same times as previ-ously selected for the CEP-dependent populations P D ( t , φ ) .It should be mentioned that the entire value of AN ( t , φ ) isshown here and not the mean difference. The asymmetry inthe nuclear motion along the coordinate b shows a sinusoidaloscillation, now with a periodicity of 2 π for all three times,which is typical for interference. This means that for the twoquantities P D ( t , φ ) and AN ( t , φ ) we observe a different CEP-dependence. Or in other words there are two different mecha-2nisms active in the system which can be projected out by usingdifferent observables.In addition we calculated the temporal evolution of Γ ( t ) and AN max ( t ) , as well as the CEP-dependence of P D ( t , φ ) and AN ( t , φ ) using the y -component and the z -component of theTDM. Since the results are quite similar the ones obtainedwith the absolute value of the TDM the orientation of themolecule with respect to electric field of the pulse should notplay a major role. For more details see section IV of the SI. B. Control of the electron dynamics
As shown in section III B the laser pulse is creating a co-herent electronic superposition in the vicinity of the CoIn.Therefore, we also examined the influence of the CEP vari-ation on the electron density. The first control objective is theCEP-dependent asymmetry parameter AE ( t , φ ) of the 1e-2o-density ρ ( r , t , φ ) . AE ( t , φ ) = N R ( t , φ ) − N L ( t , φ ′ ) N R ( t , φ ) + N L ( t , φ ′ ) . (16)with the probabilities N L ( t , φ ) and N R ( t , φ ) to find the electronon the left or the right side of the molecule given by N L ( t , φ ) = x max Z x min dx y max Z y min dy Z z min dz ρ ( r , t , φ ) . (17) N R ( t , φ ) = x max Z x min dx y max Z y min dy z max Z dz ρ ( r , t , φ ) . (18)The maximal asymmetry of the electron density AE max ( t ) iscalculated as follows: AE max ( t ) = max ( AE ( t , φ )) + min (cid:0) AE ( t , φ ′ ) (cid:1) . (19)For its maximum the electron dynamics shows the highestCEP-dependence and thus the highest controllability. Thetemporal evolution of AE max ( t ) and the CEP-dependent asym-metry of the electron density AE ( t , φ ) at three selected timesare shown in FIG. 14. The maximal asymmetry AE max ( t ) ishighest during the laser pulse (grey area). It decreases within8 fs and becomes smaller by a factor of ten. However duringthis time period two peaks at 12 fs and 15 fs can be recognize.Afterwards the maximal asymmetry oscillates between nearlyzero and 0.125 until the end of the simulation time. Compar-ing the maximal asymmetry of the electron density AE max ( t ) with the one of the nuclei ( AN max ( t ) ) faster oscillations are ob-served. To further analyze the response of the electron density(see FIG. 14(b)), AE ( t , φ ) is recorded for three selected pointsin time marked as vertical lines in 14(a)). The first line at 10 fs(green) corresponds to the main peak of AE max ( t ) and is takenat the maximum of the pulse. The second point (red line) istaken at 15 fs when the laser pulse is approximately over. Thelast point in time (yellow line) is at 40 fs. At all three times AE ( t , φ ) shows a sinusoidal oscillation with a periodicity ofapproximately 2 π and a decreasing amplitude with time. The time [fs]10 20 30 40 500 m ax asy mm e t r y A E (a) ·(cid:181) ¶•‚„ ”»…‰ (cid:190)¿ CEP [ ]0.0 0.5 1.0 1.5 2.0 (cid:192)`´ˆ˜¯˘˙¨(cid:201)˚¸(cid:204) A E ˝˛ˇ -20.0-10.00.020.010.0 (b) FIG. 14. (a) Temporal evolution of the maximal asymmetry of the ac-tive electron AE max ( t ) . The vertical colored lines indicate the pointsin time that are examined more closely. The envelope of the IR pulseis indicated in grey. (b) Mean difference of the CEP-dependent asym-metry parameter of the active electron AE ( t , φ ) given in percent fordifferent times. asymmetry of the electron density thus has the same periodic-ity as the nuclear asymmetry AN ( t , φ ) which is as previouslymentioned typical for an interference process.As already discussed in section III B the response of thedipole moment to the applied laser field is an observabledirectly connected to the electron motion. In the presentcase the 1e-2o- y - and the 1e-2o- z -component are of interest.Their maximal CEP-dependence γ y ( t ) and γ z ( t ) are evaluatedas the difference of the maximum and the minimum valueof 1e-2o- y -DM ( t , φ ) respectively 1e-2o- z -DM ( t , φ ) for eachtime step. The maximal CEP-dependence γ y ( t ) is depicted asfunction of time in FIG. 15(a) and its related component 1e-2o- y in FIG. 15(b) at three selected times.The maximal CEP-dependence γ y ( t ) like all other objec-tives shows its maximum simultaneously with the maximumof the IR pulse. In this period the shape of the γ y ( t ) curve issimilar to the Γ ( t ) curve (see FIG. 12(a)), only the decreasewith decaying pulse intensity is even more asymmetric. Af-ter the pulse in the time window from 20 fs to 40 fs the CEP-dependence oscillates. Again the oscillations are significantlyfaster than for the nuclear objectives. The CEP-dependence3 time [fs]10 20 30 40 500 m ax asy mm e t r y γ y [ a u ] (a) —(cid:209) (cid:210)(cid:211)(cid:212)(cid:213) (cid:214)(cid:215)(cid:216)(cid:217) (cid:218)(cid:219) CEP [ ]0.0 0.5 1.0 1.5 2.0 (cid:220)(cid:221)(cid:222)(cid:223)(cid:224)Æ(cid:226)ª(cid:228)(cid:229)(cid:230)(cid:231)Ł o - y D M ØŒº -0.50.51.5-1.5 (b)
FIG. 15. (a) Temporal evolution of the maximal asymmetry γ y ( t ) ofthe 1e-2o- y -component of the dipole moment. The vertical coloredlines indicate the points in time that are examined more closely. Theenvelope of the IR pulse is indicated in grey. (b) Mean difference ofthe CEP-dependent 1e-2o- y -component for different points in time. of the 1e-2o- y -component is recorded in FIG. 15(b) for thesame three selected times as for AE ( t , φ ) . It shows a sinu-soidal oscillation with a periodicity of approximately π and adecreasing amplitude with later times. Thus the componentshows the same periodicity as Γ ( t ) even with the same phase.The temporal evolution of the maximal CEP-dependence γ z ( t ) and its 1e-2o- z -component as function of the CEP areshown in FIG. 16. The maximal CEP-dependence γ z ( t ) issignificantly larger than γ y ( t ) in consistency with our find-ing in section III B that the z -component reacts more stronglyto the laser pulse. The overall shape of γ z ( t ) is quite simi-lar to the temporal evolution of AE max ( t ) (see FIG. 14(a)) andthe 1e-2o- z -component shows the same periodicity of 2 π as AE ( t , φ ) . The only difference is a phase shift of π .In summary, two different responses on the CEP varia-tion are present in the nuclear and electron dynamics. Bothasymmetry parameters AN ( t , φ ) and AE ( t , φ ) as well as the1e-2o- z -component of the dipole moment provide a distinc-tion between left and right within the molecular plane ( yz -plane). The associated 2 π periodicity is typical for an in-terference process. Γ ( t ) and the 1e-2o- y -component of thedipole moment are directly sensitive to the main direction of time [fs]10 20 30 40 500 m ax asy mm e t r y γ z [ a u ] (cid:236)(cid:237)(cid:238) (a) CEP [ ]0.0 0.5 1.0 1.5 2.0 (cid:239)(cid:240)æ(cid:242)(cid:243)(cid:244)ı(cid:246)(cid:247)łøœß o - z D M (cid:252)(cid:253)(cid:254)
10 fs15 fs40 fs-252550-500 (b)
FIG. 16. (a) Temporal evolution of the maximal asymmetry γ z ( t ) ofthe 1e-2o- z -component of the dipole moment. The vertical coloredlines indicate the points in time that are examined more closely. Theenvelope of the IR pulse is indicated in grey. (b) Mean difference ofthe CEP-dependent 1e-2o- z -component for different points in time. motion along the α -coordinate, respectively the y -coordinate.The motion in this direction mediates the non-adiabatic trans-fer between the D and D O state. For these cases the CEP-dependence shows a π periodicity, arising from the tempo-ral asymmetry of the few-cycle pulse itself. Both mech-anisms are present for the nuclear as well as for the electrondynamics and can be detected depending on the chosen ob-servable.
CONCLUSION
In this paper, we expand our ansatz for the descriptionof the coupled nuclear and electron dynamics in molecularsystems (NEMol). We applied our method to the pho-toinduced ultrafast dynamics in NO which is dominated bya CoIn. We observe the appearance of a coherent electronicwavepacket at each passage of the CoIn. The coherence isnot strong and only short lived due to the high symmetry ofthe molecule which cancels out the individual contributions .Beside the field-free relaxation we also studied the influenceof a few-cycle IR laser pulse applied in the vicinity of the4CoIn. The induced symmetry breaking significantly enhancesthe degree of coherence and its life time. Inspired by previ-ous works we varied the carrier envelope phase φ (CEP) of the IR pulse to control the movement of electronsand nuclei during the passage through the CoIn.In the first part we generalized our NEMol ansatz. Theprinciple advantage of this ansatz is based on the combina-tion of highly developed quantum-chemical methods with theaccurate description of the nuclear quantum dynamics. Inthe original ansatz an expression for the time-dependentelectronic wavepacket is formulated where the electronic partof the total wavefunction is propagated in the electroniceigenstate basis. Its dynamics is extracted from the nu-clear wavepacket propagation on coupled potential energysurfaces by introducing the parametric dependence on thetime-dependent expected value of position h R i ( t ) . By extend-ing the NEMol ansatz with a grid representation, it is pos-sible to couple the electron dynamics to multiple grid pointson which the nuclear wavepacket is represented. Through asimple approximation we were able to condense the coupleddynamics of the one-electron excitation process in the den-sity of one active electron (1e-2o-picture). In the second partwe compared the coupled nuclear and electron dynamics ofNO with and without an IR pulse present when the systemreaches the CoIn for the first time. Using the NEMol ansatz,we characterized the coherent electron dynamics by analyzingthe temporal evolution of the induced dipole moment. The ob-served frequencies of the coherent electron dynamics cover arange up to 2 . the phase contribution of the nuclear overlap term ishigh and therefor provides a significant contribution to theelectron dynamics. The applied few-cycle IR laser pulse gen-erated an asymmetric movement of the nuclear and electronicwavepackets, which is vital for the controllability at the CoIn.The induced oscillating dipole reflects an enhanced build up ofthe coherent electron dynamics by the laser pulse which sur-vives for several 10 fs. In the last part the CEP of the IR pulsewas varied to influence both the nuclear dynamics as well asthe electron dynamics. The CEP-dependent effect lives con-siderably longer than the pulse in all investigated observables.Depending on the chosen observable a π or 2 π periodicity canbe found indicating two mechanisms, one based on an inter-ference process (2 π ) and the other one induced by the tempo-ral asymmetry of the few-cycle pulse itself ( π ). Both period-icities are observed for the nuclear as well as for the electrondynamics. In each case they can be projected out by usingdifferent observables.We demonstrated the potential of our NEMol ansatz to de-scribe the coupled nuclear and electron dynamics in molecularsystems beyond diatomics. In NO we followed the dynamicsin the excited state dominated by fast changing wavepacketinterference effects. The ansatz is expandable to simulate theinduced coherent electron dynamics in the excitation processitself as well as higher-dimensional molecular system as longas the underlying nuclear dynamics can be treated quantummechanically. Two electron processes could be realized byusing pair densities. SUPPLEMENTARY MATERIAL
See the supplementary material for the details of thewavepacket simulation setup, the underlying quantum chem-ical data of NO and additional figures and tables for theNEMol-dynamics. A section contains the results for the CEP-control obtained with the y -component and the z -componentof the TDM. AUTHOR CONTRIBUTIONS STATEMENT
TS performed all calculations. TS and RDVR analyzedthe results and contributed equally to the final version of themanuscript.
DATA AVAILABILITY STATEMENT
The data that support the findings of this study are avail-able from the corresponding author upon reasonable request.The following article has been submitted to ’The Journal ofChemical Physics’.
CONFLICTS OF INTEREST
There are no conflicts to declare.
ACKNOWLEDGEMENTS
The authors gratefully acknowledge the DFG Normalver-fahren and the Munich Center of Advanced Photonics (MAP).
Appendix:
The following detailed formulation of the NEMolansatz is given here in the improved notation. The totalmolecular wavefunction Ψ tot ( r , R , t ) is setup as the sum overthe electronic states with χ ( R , t ) the nuclear wavefunctions, ϕ ( r , t ; R ) the electronic wavefunctions, the nuclear and elec-tronic coordinates R and r and the time t . Ψ tot ( r , R , t ) = ∑ i χ i ( R , t ) · ϕ i ( r , t ; R ) . (A.1)Applying the Born-Oppenheimer approximation the uncou-pled electronic wavefunctions ϕ i are hereby parametricallydepending on the nuclear coordinates R and define a multi-dimensional vector ϕ tot . The total nuclear wavefunction χ tot also represents a multi-dimensional vector, spanned by thecoupled wavefunctions χ i . For details how the temporal evo-lution of the nuclear wavefunctions χ i on coupled potentialenergy surfaces (PES) is determined see section I of the SI.5Multiplying Ψ tot ( r , R , t ) from the left with χ tot and the sub-sequent integration over the nuclear coordinates results in anexpression of the coupled total electronic wavefunction . Φ tot ( r , t ; h R i ( t )) = Z χ ∗ tot ( R , t ) · Ψ tot ( r , R , t ) dR = Φ ( r , t ; h R i ( t )) Φ ( r , t ; h R i ( t )) ... Φ j ( r , t ; h R i ( t )) , (A.2)with h R i ( t ) = ∑ i h χ i ( R , t ) | R | χ i ( R , t ) i R . (A.3)The coupled total electronic wavefunction is parametricallydepending on the time-dependent expected value of the po-sition h R i ( t ) . In other words Φ tot is evaluated at one singlenuclear geometry which changes with time. The individualcomponents Φ j are defined by the following equation: Φ j ( r , t ; h R i ( t )) = A j j ( t ) · ϕ j ( r , t ; h R i ( t ))+ ∑ k = j A jk ( t ) · ϕ k ( r , t ; h R i ( t )) , (A.4)with A jk ( t ) = (cid:10) χ j ( R , t ) (cid:12)(cid:12) χ k ( R , t ) (cid:11) R . (A.5)The first part depends on the population A j j of the respectivestate j , while all others summands include the nuclear over-lap term A jk which specifies the degree of coherence inducedbetween the two states j and k . The population and coher-ence of the electronic states as well as the influence of all cou- pling terms are already determined by the nuclear quantum-dynamics simulation. If the coupling between the electronicstates is weak, the nuclear wavefunctions propagate indepen-dently and the coherence term becomes zero. In this case,the coupled electronic wavefunctions Φ j in equation A.4 be-come equivalent to the uncoupled electronic wavefunction ϕ j .Standard quantum-chemical calculations at the h R i ( t ) struc-ture yield the real-valued wavefunctions ϕ j ( r ; h R i ( t )) of therelevant electronic states and their eigenenergies. The tempo-ral evolution of ϕ j ( r , t ; h R i ( t )) is determined by the deforma-tion of the electronic structure induced by the nuclear motion(Born-Oppenheimer part) and an oscillation through phasespace defined by a pure electronic phase. ϕ j ( r , t ; h R i ( t )) = ϕ j ( r ; h R i ( t )) · e − i ξ j ( t ) (A.6)The phase term ξ j ( t ) depends on the eigenenergies E j ( h R i ( t )) and has to be calculated recursively. ξ j ( t ) = E j ( h R i ( t )) ∆ t + ξ j ( t − ∆ t ) . (A.7)This recursive evaluation is necessary to retain the memoryof the progressing electronic phase. Thereby the propagationvelocity of the phase in the complex plane changes smoothlyin time while the nuclear wavepacket propagates. Using thecoupled total electronic wavefunction Φ tot ( r , t ; h R i ( t )) the as-sociated electron density ρ ( r , t ; h R i ( t )) can be determined bymultiplying Φ tot ( r , t ; h R i ( t )) from the left with ϕ tot and thesubsequent integration over N − N being the total number of electrons). ρ ( r , t ; h R i ( t )) = Z ϕ ∗ tot · Φ tot dr . . . dr N = ∑ j A j j ( t ) ρ j j ( r ; h R i ( t )) + ∑ k = j Re (cid:8) A jk ( t ) ρ jk ( r ; h R i ( t )) e − i ξ jk ( t ) (cid:9) , (A.8)with ξ jk ( t ) = ∆ E jk ( h R i ( t )) ∆ t + ξ jk ( t − ∆ t ) . (A.9)The first summation consists of the state specific electronicdensity ρ j j ( r , t ; h R i ( t )) weighted with the corresponding time-dependent population A j j ( t ) . 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