Quantum Nuclear Dynamics Pumped and Probed by Ultrafast

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Quantum Nuclear Dynamics Pumped and Probed by Ultrafast Polarization Controlled Steering of a Coherent Electronic State in LiH Astrid Nikodem, Raphael D Levine, and Francoise Remacle J. Phys. Chem. A, Just Accepted Manuscript • DOI: 10.1021/acs.jpca.6b00140 • Publication Date (Web): 01 Mar 2016 Downloaded from http://pubs.acs.org on March 1, 2016

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The Journal of Physical Chemistry

Quantum Nuclear Dynamics Pumped and Probed by Ultrafast Polarization Controlled Steering of a Coherent Electronic State in LiH Astrid Nikodema, R. D. Levineb,c* and F. Remaclea,b,* a. Département de Chimie, B6c, Université de Liège, B4000 Liège, Belgium b. The Fritz Haber Center for Molecular Dynamics and Institute of Chemistry, The Hebrew University of Jerusalem, Jerusalem 91904, Israel c. Crump Institute for Molecular Imaging and Department of Molecular and Medical Pharmacology, David Geffen School of Medicine and Department of Chemistry and Biochemistry, University of California, Los Angeles, CA 90095

Abstract The quantum wave packet dynamics following a coherent electronic excitation of LiH by an ultrashort, polarized, strong one cycle IR optical pulse is computed on several electronic states using a grid method. The coupling to the strong field of the pump and the probe pulses is included in the Hamiltonian used to solve the time-dependent Schrodinger equation. The polarization of the pump pulse allows to control the localization in time and in space of the non equilibrium coherent electronic motion and the subsequent nuclear dynamics. We show that transient absorption, resulting from the interaction of the total molecular dipole with the electric fields of the pump and the probe is a very versatile probe of the different time scales of the vibronic dynamics. It allows probing both the ultrashort, fs time scale of the electronic coherences, as well as the longer dozens of fs time scales of the nuclear motion on the excited electronic states. The ultrafast beatings of the electronic coherences in space and in time are shown to be modulated by the different periods of the nuclear motion.

*

Corresponding authors : [email protected], [email protected] 1 ACS Paragon Plus Environment


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1. Introduction Ultrashort, strong attopulses provide a sub to few fs time resolution that is short enough to directly probe in time electronic dynamics.

1-4

Such short pulses are necessarily broad in

energy, so that several electronic states lie in their bandwidth and can be coherently excited. In atoms, 5, 6 the states accessed differ in energy and the superposition of states that is excited will be non stationary in that the amplitude to be in any given electronic state will have a phase that varies with time in tune with its exact energy. In molecules the dynamics of such a non stationary state can be more complex because, for example, the electronic density of different states can be localized in different parts of the molecule.7 The polar LiH molecule was suggested as an early example where such charge redistribution could be observed in molecules.8 LiH has a high density of excited electronic states and many of the higher ones are unbound. The two lowest states that are optically accessible from the ground state are respectively of Σ and Π symmetry and in the Franck Condon region are about 1 eV apart in energy. A UV or IR pulse of about 1fs width in time is broad enough in energy to excite a coherent linear combination of these two states.8 The coherent electronic dynamics pumped by ultrashort pulses can be probed experimentally by high harmonic generation

9-11

, attoclock measurements12-13, transient absorption

spectroscopy6, 14 and time-resolved molecular-frame photoelectron-angular-distribution15. For molecules, it is also possible to monitor ionic fragmentation yields as a function of short fs pump-probe delay, in diatomic and in small polyatomic molecules. 11, 16-20 These experiments provide evidence of an ultrafast beating of the molecular charge, that can be attributed to electronic coherences. We here center attention on the LiH molecule because of our earlier studies suggested that it should be possible to control and steer the evolution of the excited states. Many excited states of LiH states can be optically accessed from the ground state.

21-24

By selecting the

polarization of the laser pulse one can achieve linear superpositions of states of Σ and Π symmetry where the electronic density rotates in different directions in space. In particular one can have states with the density rotating in a ring around the Li-H bond or in an ellipse in the plane of the atoms.21, 23 On the computer one can also steer the motion of the electronic wave packet along the bond so that the molecule is more polar, meaning more positive charge 2 ACS Paragon Plus Environment

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on the Li, or less polar. Such a steering of the charge density by ultrafast non equilibrium electronic density has been suggested by Weinkauf and Schalg 25-26 to explain their pioneering experimental studies of the selective ultrafast dissociation at the amine end of small tailored peptide cation locally ionized at the opposite chromophoric end. 7, 27-32 From a theoretical point of view, developing dynamical methods able to describe coherent electronic and nuclear dynamics on several electronic states remains a challenge. Among the fully quantum methodologies33-42 for solving the time-dependent Schrodinger equation (TDSE), wave packet propagation on coupled potential energy curves on a grid 43-47 is a very efficient method for systems with few nuclear degrees of freedom. The early computations

21, 23

on LiH sought a very accurate representation of the electronic

state but were carried out with the nuclei frozen at the ground state equilibrium position. More recent studies include nuclear dynamics.

48-50

In ref. 48-49, the multi-configuration electron-

nuclei dynamics (MCEND) methodology was used to investigate the electron nuclei dynamics following excitation by a 4fs UV pulse for the first 15fs. The authors concluded that limited nuclei motion takes place and that the frozen nuclei approximation holds during that time. In ref. 50, quantum electron-nuclei dynamics is pumped by a several femtosecond UV pulse and probed by computing time-resolved photoelectron spectra resulting from ionization by a fs pulse train. Here we discuss the quantum dynamics of a complete electron-nuclear wave function induced by a one cycle IR strong attopulse and its probing by transient absorption spectroscopy using a weaker one cycle UV atto pulse. In the case of the pumping of the low energy excited electronic states, the electronic coherences are induced by the interaction with the electric field of the pump pulse and not further modified by subsequent non adiabatic coupling driven by nuclear motion or by the probe pulse. We show by computing transient absorption heat maps

51-53

for various time windows and delays between the pump and the

probe pulses that the coherences between the different electronic states modulate the nuclear motion for several vibrational periods in the different excited electronic states.

3 ACS Paragon Plus Environment


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2. Coupled Electronic-Nuclear Dynamics The nuclear time-dependent Schrödinger equation (TDSE) was integrated on a grid 43, 45, 54 along the internuclear distance R, for the 5 state Σ manifold. The adiabatic electronic states of LiH were computed as a function of the internuclear distance R using the suite of quantum chemistry program MOLPRO. 55 First 10 Σ, 8 Πx, 8 Πy and 5 ∆ electronic states were computed at the state average CAS-SCF level using the MULTI program56-57 for the 4 electrons with 13/7/7/2 active molecular orbitals (MO) of Σ, Πx, Πy and ∆ symmetry and the Gaussian 6-31+G(df,p) basis of atomic orbitals. These states were used as reference configurations for getting the MRCI electronic states with the internally contracted MRCI method. 58-59 The energies of these states and the matrix elements of the transition dipole and of the non adiabatic couplings were computed for 36 R values between 0.94 and 34 a.u. and for 11 values between 34 and 75 a.u. Standard spline interpolation60 was used to build dense enough grids to ensure numerical precision when integrating the TDSE for the potential

( )

( )

energy, Vi R , dipole matrix elements, µij R , and the non adiabatic couplings. After numerical checks, the effect of non adiabatic couplings for the band of the 4 lowest Σ states involved in the dynamics investigated in this study was neglected. The reason is that the non adiabatic coupling are sufficiently small and the adiabatic potential energy curves of the 4 lowest sigma states sufficiently far apart in energy that at any R the effective coupling remains small.61 The Hamiltonian is expressed on a grid in the basis of the adiabatic electronic states coupled by the dipole interaction with the electric field of the optical pulse, E(t) : (1) where Vi (R) is the potential energy for the adiabatic state i. The third term in Eq. (1) is the dipole interaction, including the nuclear response to the electric field.

( )

The potential energy curves, Vi R in Eq. (1), of the lowest 5 Σ and 2 Π adiabatic states below the IP (7.8 eV) are plotted in Figure 1. Our computational results follow already reported trends. 8, 50, 62 The Π states are fully repulsive. Only the lowest excited state Σ1 of the Σ manifold is bound, with a shallow well. The higher excited Σ states are even shallower, 4 ACS Paragon Plus Environment

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except Σ4, which is not significantly accessed in our dynamical simulation. As explained above, the effect of the non adiabatic coupling between the Σ2 and Σ3 states is found to be very small and is not included in the Hamiltonian in Eq. (1).

Figure 1. Potential energy curves (in eV) computed at the MRCI level as a function of the internuclear distance R, in a.u. The Σ states are shown in blue and the Π states in red. The LiH bond length at equilibrium geometry is 3.08 a.u. (1.63 Å).

( )

( )

( ) ( R ) = ∫ drρ (r; R ) r where r

In Eq. (1), the total dipole operator, µtot R = µ elec R + µ nucl R . The electronic part of the dipole is computed in the basis of adiabatic states, with µijelec

elec ij

( )

stands for the electron coordinates and ρ ijelec r; R is the electronic transition matrix computed

( )

at the Li-H bond length R. The nuclear component of the dipole is µ nucl R = f R with

f = ( Z H M Li − Z Li M H M T ) . Note that µ nucl ( R ) is diagonal in the electronic index but not

µ elec ( R ) . The total dipole matrix elements curves between the 5 state Σ manifold in the direction parallel to the bond (z direction) are shown in Figure 2. The origin of the molecular frame is put at the center of mass. Its orientation is shown in Figure 2a. The strong value of the



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eq permanent dipole moment of the GS (Figure 2a) at Req, µGS = + 2.28 a.u., suggests that it is

experimentally possible to align the molecules.63-65 We showed in ref. 66 that for a partial

(

alignment corresponding to a value of cosθ

)

2

of 0.7, the beatings of the non stationary

electronic density in LiH could still be probed accurately by time resolved photoelectron angular distribution. The simulations reported below were therefore carried out in the molecular frame. From Figure 2b, one sees that only the transition dipoles between the GS and the two lowest excited states are large in the Franck Condon region. The two higher Σ states are essentially not accessed for the strength and wavelength of the optical pulses used, see section 3 below. Note that the permanent dipoles of the two lowest excited are of opposite sign, as is the transition dipole between them. This feature allows for polarization control by the pump and the probe pulses.


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Figure 2 : (a) Permanent dipole moments of the three lowest Σ states and (b) the transition total dipole moments between them as a function of R, computed in the z direction, parallel to the molecular axis at the MRCI level. The molecular frame orientation is shown as an inset in (a). In (b), only the Σ1−Σ2 transition dipole is shown for clarity but all the transition dipoles between excited states are included in the Hamiltonian.

Since we are using short strong pulses with few oscillations, to ensure that the frequency Fourier transform of the field is zero at zero frequency, we define the time profile of E(t) in



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()

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()

Eq. (1) from the derivative of the vector potential E t = − dA t dt . For a Gaussian envelope,

(E

A ( t ) = −∑ p

p

)

(

/ ω p exp  − t − t p 

)

2

(

σ 2p  sin ω pt + φ p 

)

(2)

where the sum is over the pump and the probe pulses, ω p is the pulse frequency, σ p is the width

of

the

Gaussian

envelope

and φ p

the

carrier

envelope

phase

(CEP).

E p = Exp e x + Ezp e z where two polarization directions of the electric field are considered in the molecular frame for aligned molecules, see inset in Figure 2a, z is parallel to the bond and x perpendicular. For a linear molecule, x and y are equivalent. The total wave function on the grid point Rg is given by

( )

( )

n

Ψ Rg ,t = ∑ cig ( t )θ i Rg

(3)

i

( )

()

where the θ Rg are othornormal grid functions46 centered on Rg and the amplitudes cig t of the wave function in the electronic state i at the grid point g are obtained by solving the nuclear

TDSE

for

the

matrix

of

the

Hamiltonian

given

in

Eq.

(1),

where the index i and j denote the electronic states and g and g’ the grid points. The advantage of the grid representation is that the operators in Eq. (1) are all local on the grid except the nuclear kinetic energy : (4) The kinetic energy operator is calculated using the Fourier method.44, 46-47, 67 Note that the transition dipole matrix elements µigelec, jg ' (R) are not diagonal in the index of the electronic states. Since we have a fast varying time dependent Hamiltonian, we use a 5th order RungeKutta integrator. 512 grid points ( ∆Rg = 0.13 a.u.) are used from 0.97 to 68.03 a.u. The integration time step is 0.001 fs. The total charge density is at given grid point, Rg, is not diagonal in the electronic states and reflects the electronic coherences resulting from the excitation by a short, few fs, optical 8 ACS Paragon Plus Environment

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pulses. From Eq. (3), where an integration over the electronic coordinates has already been performed, we get

( )

N

N

i, j

i

( )

( )

ρ Rg ,t = ∑ cig* ( t ) c jg ( t ) = ∑ ρ ii Rg ,t + 2 Re ∑ ρ ij Rg ,t i> j

( )

(5)

() ()

The off diagonal terms in the electronic density, ρ ij Rg ,t = cig* t c jg t , beats with a frequency given by the energy difference ∆Eij (R) = V j (R) −Vi (R) between the field free state

( )

electronic energies, Vi (R) and V j (R) . In addition, the time dependence of ρ Rg ,t reflects the periods of the nuclear motion over the different electronic states accessed by the dipole

( )

coupling. The mean value of R over ρ Rg ,t is diagonal in the grid points and in the different electronic states:

R (t ) = Ψ (t ) R Ψ (t ) = ∑ i

∑

( )

ρ Rg ,t Rg = ∑

g

∑

i

cig ( t ) Rg = ∑ Ri ( t ) (6) 2

g

i

( )

The non stationary inherent character of the coherent non equilibrium total density, ρ Rg ,t

()

in space and in time leads to a time dependent dipole, µ t , which is given by the mean value

( )

( )

( )

of the dipole operator, µtot R = µ elec R + µ nucl R , over the density:

( ) ( )

( )

( )

µtot ( t ) = ∑ ρ Rg ,t µtot Rg = ∑ cig* ( t ) c jg ( t ) µ i,elej Rg + ∑ cig ( t ) µ nucl Rg g

= ∑ cig ( t ) µ 2

elec ii

(R )+ ∑ g

ig

g

ig , jg

2

ig

2   2 Re  ∑ cig* ( t ) c jg ( t ) µi,elej Rg  + ∑ cig ( t ) µ nucl Rg  i> j  ig

( )

( )

(7)

Since it is a local function of R, the total dipole is diagonal in the grid points. Because of the

()

coherent non equilibrium electronic character of the density, the electronic part of µ t is not diagonal in the electronic states and sensitive to the electronic coherences built by the excitation by a short, few fs, pump pulse. The nuclear dipole term in Eq. (7) can be simply written as a sum over the electronic states

µ nucl ( t ) = ∑ µinucl ( t ) = ∑ f ∑ cig ( t ) Rg = ∑ f Ri ( t ) 2

i

i

g

(8)

i

It therefore appears that transient absorption and stimulated emission from the total time dependent dipole with a second pulse is an ideal probe of the non equilibrium character of the 9 ACS Paragon Plus Environment


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( )

total density ρ Rg ,t . The induced dipole can also be probed by high order molecular harmonics which offer a similar route for probing the effect of nuclear motion on electronic coherences.33, 68 Transient absorption has been used for probing coherent electronic excitation by an ultrashort atto second XUV pulse in rare gas atoms.6,

14, 69-70

In that case, it is the

electronic coherence between the atomic electronic states that is evidenced, as well as the effect of their coupling by spin orbit interaction. Stimulated emission and transient absorption have also been used to probe electronic and nuclear dynamics on a longer, dozens of fs time scale.71-73

( )

We compute the linear response function, S ω for positive frequencies, ω , as in 53 :

S (ω ) = −2 Im  µ (ω ) E * (ω ) 

( )

(9)

( )

()

where µ ω and E ω are the Fourier transform (FT) of the total dipole µ t (Eq. (7)) and

()

of the complete time course of the electric field, E t (see Eq. (2)) including the pump and the probe pulse. See also ref

()

Since µ t

()

and E t

52

for an expression that isolates the effect of the pump pulse.

( )

( )

are real, their FT obeys the property µ −ω = µ * ω

and

E ( −ω ) = E * (ω ) . Note also that µ ( t ) is an even function of time but not E ( t ) as defined from Eq. (2).

3 Selective Nuclear Dynamics by Polarization Controlled Steering of Coherent Electronic Motion Ultrashort and strong pump pulses polarized in different directions in the molecular frame allow to control of the localization of the electronic density in space and in time. The reason is that the short pump pulse is broad enough in energy to create a coherent superposition of electronic states that is beating in time on different moieties of the molecule after the pump is over. We have documented this effect for nuclei frozen in their equilibrium geometry in several systems, ranging from LiH8, 22-24 to modular polyatomic molecules such as ABCU,74 PENNA75 and small peptides76-77 by computing time and angular resolved molecular frame photoelectron maps.


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We also suggested early on 8, 23 that the steering of the coherent electronic motion by a short, essentially one cycle, pump pulse could be used to control the nuclear dynamics. The reason is that the localization of the electron density induced by the response to the pulse on different parts of the molecule will lead to different forces on the nuclei. The pulse needs to be essentially a one cycle pulse, otherwise this effect is averaged out over the many cycles. We here report on the effect of the pulse polarization for a full quantum dynamical simulation that includes the nuclear motion as obtained by solving the TDSE for the Hamiltonian given by Eq.(1). We use two different pulses, polarized along the molecular axis with the maximum of the electric field either toward the Li atom (+z direction) or to the H atom (-z direction). Only Σ states can be accessed by a pulse polarized along the molecular axis. The two lowest excited Σ states have opposite polarity properties in the Franck Condon region around Req. As shown in Figure 2a, Σ1 has a negative permanent dipole, of opposite sign to that of the GS, while the permanent dipole of Σ2 is parallel to that of the GS. Thereby the transition dipole from the GS to Σ1 is negative while that to Σ2 is positive, see Figure 2b. Therefore, when the maximum of electric field in the one cycle pulse is oriented in the –z direction, towards the H atom, the Σ1 state is preferentially accessed and the electronic density localized on the Li atom, while when the maximum electric field occurs in the opposite, +z, direction, the Σ2 state and states of the same polarity are preferentially accessed and the electronic density localized on the H atom,

()

()

2

as in the GS. The population in the different electronic states, Pi t = ∑ cig t , resulting g

from the excitation by one cycle 800 nm pulses centered at 2 fs with a width σ=0.68 fs, a field strength of 0.05 a.u., which corresponds to a field intensity of 8.75 1013 W/cm2 , and φ = 0 in Eq. (2) with a maximum of the electric field oriented in the +z and – z direction are plotted in Figure 3, together with the pulse time profile. After a rich transient dynamics during the pulse, the ‘+z pulse’, that is the pulse with the maximum of the electrical field oriented in the +z direction builds a complex superposition of the GS (80%) and of three excited Σ states, Σ1, Σ2 and Σ4 in equal amounts, see Figure 3a. Σ2 and Σ4 are dissociative and accessed when the electric field is oriented in the + z direction. Overall, 16 % of the population dissociate. On the other hand, the ‘–z pulse’ for which the maximum of the electric field occurs in the -z direction essentially builds a clean superposition of the GS and the Σ1 state, with only 4% in



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the Σ4 state, see Figure 3b. Therefore, for the ‘–z pulse’, essentially no dissociation should be observed. Note that the population in Σ4 is reduced to less than 1% for the same excitation pulse with a field strength of 0.02 a.u which corresponds to 1.4 1013 W/cm2. Photoionization is not included in this study. For a one cycle IR 0.68 fs pulse, no significant ionization can take place. The reason is that for a 800nm excitation wavelength, only states with energies within about 1.5 eV below the IP can photoionize during the pulse. Our simulations show that Σ states higher than Σ4 are not populated higher than 0.5% during the one cycle 0.68 fs IR pulse. The pulse is short enough that multiphotoionization cannot occur.

()

Heatmaps of the localization of the wave function on the different electronic states, Cig t

2

are plotted as a function of time in Figure 4, in panel a for Σ2 state and an excitation by the one cycle IR ‘+z pulse’ and in panel b for Σ1 state and an excitation in the –z direction. The motion on the Σ1 state is bound, with a period of ≈ 80 fs. This period of the Σ1 state is clearly seen in the heatmap plotted in Figure S1a, where the computation is carried out for a larger grid and for longer times. The period on the GS, not shown in Figure 4, is equal to 20 fs. In Figure S1b that reports the localization of the wave function on the Σ2 state, one clearly sees that a small fraction of the population trapped in the well of the Σ2 state.


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Figure 3. Population in the Σ state manifold computed for an excitation by a one cycle pulse polarized along the molecular with the maximum of the electric field in the +z direction (a) and in the –z direction (b). The time profile of the electric field of the two pulses is given in dashes and their parameters listed in the text. Note that the populations in each electronic state are stationary when the pulse is over since the adiabatic electronic states are not coupled by the nuclear motion.



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()

2

Figure 4. Heatmaps of the localization of the wave function ( Cig t ) on selected electronic states as a function of time. a) Localization of the wave function on the Σ2 state for an excitation by a one cycle pulse polarized along the molecular axis with the maximum of the electric field in the +z direction (Figure 3a). b) Localization of the wave function on the Σ1 state for an excitation by a one cycle pulse with its maximum in the -z direction (Figure 3b).

( )

A heatmap in space and in time of the total density, ρ Rg ,t , Eq. (5), is shown in Figure 5a for the one cycle IR pulse polarized in the –z direction (parameters as in Figure 3b). One can clearly discern the fast oscillations on a fs time scale of localization in space and in time of the coherence terms, and their modulation by the slower nuclear motion. Isocontours of

(

)

snapshots of the electronic density, ρ elec r; R,t , are plotted in Figure 5b for selected values of time. Before the pulse, at 1fs, the electronic density is that of the GS at Req with an excess of electron density on the H atom. In the second half of the pulse, at 3.5fs, the electron density has shifted to the Li, responding to the strong electric field in the –z direction. For longer


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times, the electronic density beats between the Li and the H atoms on a fs time scale, because of the electronic coherence between the GS and the Σ1 state.

( )

Figure 5. a) Heatmap of the total density ρ R,t , defined by Eq. (5), in the nuclear coordinate and in time computed for an excitation by a one cycle IR ‘-z pulse’ polarized (parameters of Figure 3b). The behavior of the density during the pulse is shown as an inset. Note, see Eq. (5), that there can be destructive interference and it can be large enough that the density almost vanishes. One clearly sees the fraction of the density in the GS (small oscillations as a function of time centered around 3.08 a.u.), the fraction in Σ1 that oscillates with a longer 80 fs period and the small amount in Σ4 that is not bound. One also clearly distinguishes the fast oscillations of the electronic coherences, on a 1 to 3 fs time scale. (b) Isocontours of the

(

electronic density, ρ elec r; R,t

)

computed taking the nuclear motion into account. The

changes in the Li-H bond length can be seen as well as the changes in the localization of the electronic density due to the beating of the electronic coherence.

()

As discussed in section 2, the total dipole moment µtot t , Eq. (7), is an observable over the

( )

density, ρ R,t , and therefore its time dependence should reflect both the electronic coherence and the effect of the nuclear motion. The electronic and nuclear contributions to the

()

total dipole moment µtot t are plotted in Figure 6 for the two pulses used in Figure 3. From

()

the electronic component, µ elec t , one can see that during the pulse, the localization of the



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electronic density is on the H atom for the ‘+z pulse’ (Figure 6a) and on the Li atom for the ‘– z pulse’ (Figure 6b). One also clearly sees from Figure 6 that the fast oscillations due to the electronic coherences are modulated by the nuclear motion on the different potentials. The modulation reflects the variation of the dipole curves shown in Figure 2 with the internuclear

()

distance R. We show in Figure 6 the nuclear dipole contribution of the GS, µ nucl−GS t on a separate scale on the right y axis so that the 20 fs of vibrational ground state can be seen. Since the extension of the bond length is much larger on the Σ1 state than on the GS, the modulation by the nuclear motion on Σ1 dominates the time profile of the total dipole. The

()

reason why the oscillation of µ nucl−GS t are less clearly seen in panel b than in panel a is that there is less back transition to the GS for the pulse with its maximum in +z direction (Figure

()

3a) than for the pulse with its maximum in the –z one (Figure 3b). The µ nucl−Σ1 t component beats with a period of 80 fs. The total dipole response computed for the -z pulse with a weaker strength (0.02 a.u.) up to 260 fs is shown in Figure 7. For this pulse, essentially two electronic states are populated, the GS with 78% and the Σ1 state with 21% and 1% in the Σ3 state. One can clearly see in Figure 7 the modulation of the electronic coherences by the nuclear motion on the Σ1 state with a 80fs period. Each time the vibrational wave packet revisits the Franck Condon region, there is a revival of the electronic coherence due to the increase of the transition dipole, see Figure 2. The results shown in Figures 6-7 suggest that it is possible to probe the electronic coherences by transient absorption.

()

Figure S2 of the SI shows a comparison between the value of the total dipole, µtot t , computed with and without nuclear motion. During the pulse, up to 5 fs, there is little effect of the nuclear motion on the total dipole moment, as was already reported.

49

At later time, the

fast oscillations due to the electronic coherences are modulated by the nuclear motion.


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()

Figure 6. Time dependent total dipole moment, µtot t , Eq. (7), and its electronic and nuclear components as indicated, computed for a one cycle 800 nm pump pulse with its maximum in the +z direction (a) and in the –z direction (b). The parameters of the pulses are the same as in

()

Figure 3. A separate scale for for µ nucl−GS t is shown on the y axis on the right side, so that the 20 fs vibrational period is clearly seen.



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Figure 7. Time dependence of the total dipole moment. Computed for the one cycle ‘-z pulse’ with the same parameters as the pulse used in Figure 4 but at a weaker strength of 0.02 a.u. The time dependence reflects three distinct time scales. The ultrafast, fs, time scale is primarily the coherence between the first excited, Σ1 state and the ground state, that at Req are separated in energy by 3.46 eV. The slower time scales are due to nuclear motions. The slowest, 80 fs, is the vibration on the shallow potential of the first excited state. A faster vibrational period of 20 fs on the ground state is superposed on the slower 80 fs period.


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4. Probing of the Electronic Coherences and Nuclear Dynamics by Transient Absorption. The transient absorption spectra are computed by adding an ultra short weak probe pulse to the dynamics (σ=0.24 fs, λ=300 nm, field strength of 0.005 a.u., φ = π in Eq. (2)). The probe pulse (σ=0.68 fs, field strength of 0.02 a.u. and φ = π in Eq. (2)) is applied at increasing time delays, τ, with respect to the pump pulse centered at 2fs. Since φ = π the maximum of the electric field occurs in the –z direction. To monitor the resulting dynamics we consider first the Fourier Transform (FT) of the dipole. To unravel different time scales we use different time windows in computing the FT. Shorter windows reveal the faster, electronic, dynamics. The longer time scales associated with nuclear motion are seen using wider windows.

()

The absolute value of FT of the total dipole, µtot t , Eq. (7), is plotted in Figure 8 for pumpprobe delays τ = 3 fs (panel a) and 34 fs (panel b), for increasing durations of time window. The frequency range is centered on the GS-Σ1 (3.46 eV) and Σ1-Σ3 (2.66 eV) electronic transitions at Req. The frequencies of the electronic transition are present for both time delays, indicating the existence of the electronic coherence also at τ = 34 fs. The modulation of the electronic coherence in time by the nuclear motion translates in the frequency domain as the emergence of a vibrational structure in the peak of the electronic transition as the time window increases. For time windows of 32 and 65 fs, only the vibrational states of the GS (20 fs corresponds to 0.2 eV) are resolved. Longer time windows provide the resolution of the vibrational states of the Σ1 state (80 fs corresponds to 0.05 eV). Note that the vibrational peaks are not exactly equally spaced because of anharmonicity.



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()

Figure 8. FT of the total dipole moment, µtot t , Eq. (7), for a pump-probe delay τ = 3 fs (panel a) and 34 fs (panel b), computed for increasing durations (in fs) of the time window as indicated.

( )

Heatmaps of the dipole µ ω

as a function of delay times in the 4.5 fs range (left, panels a,c)

and in the 35.5 fs range (right, panels b, d) are shown in Figure 9 for two values of time


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windows, 65 (top) and 160 (bottom) fs. The frequency range (y axis) is centered on the energy of the GS-Σ1 transition (3.46 eV). One clearly sees that the vibrational motion on the Σ1 state which corresponds to energy spacings of 0.05 eV is only resolved for the long time window (panels c and d). The spacing corresponding to the nuclear motion on the GS (0.2 eV) only appears for the short window time. Note that the beating of the electronic coherence GS-Σ1 (1fs) is resolved for all delay times, those centered on 3.5 fs (panels a and c) and on 35.5 fs (panels b and d).

( )

Figure 9. Heatmaps of the absolute value of the FT of the total dipole, µ ω

computed in

two ranges of delay times between the pump and the probe pulses, a short delay time, τ, in the left column (panels a and c), centered on 4.5 fs and a longer delay, right column (panels b and d), centered on 35.5 fs. For each delay times, two different duration of the time window were used, top row (panels a and b): 65 fs and bottom row (panels c and d) : 160 fs. The 65 fs time window is not long enough to resolve the vibrational spacings on the Σ1 state (0.05 eV) but the GS vibrational spacings of 0.2 eV are clearly seen for the two ranges of values of τ. The time window of 160 fs allows the resolution of the 0.05 eV spacings on Σ1. The fast 1 fs



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beating of the GS- Σ1 electronic coherence is seen as a function of the delay time, both for the 4.5 fs and the 35.5 fs ranges. See text for the parameters of the pump pulse.

( )

The linear coupling of the dipole to the field is characterized by the response function, S ω , Eq. (9), plotted in Figure 10 as a heat map for the same values of the delay times and time

( )

windows of the FT of the total dipole µ ω

( )

( )

as in Figure 9. Unlike µ ω , S ω can be

positive (absorption, red in Figure 10) or negative (emission, blue in Figure 10). Emission is stimulated to the vibrational states on the GS. As seen in Figure 3b there is only a small stimulation back to the GS, primarily to the ground and first excited vibrational states as seen in the two peaks in Figure 10. These peaks are particularly clear in Figures 10a and b, which are computed for the short time window. When using a long window in the FT, panels c and d of Figure 10, one resolves also the vibrational structure in the first excited state. Clearly seen in all panels of Figure 10 is the electronic coherence with a time scale of 1 fs. In summary, the vibrational structure appears along the frequency axis of the heat map while the electronic coherence is resolved along the delay time axis.


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( )

Figure 10 : Heatmaps of the response, S ω , Eq. (9), for the same values of the delay times and time windows as in Figure 9. Panels a and b : short time window of 65 fs with ranges of delay times τ centered on 4.5 fs (a) and 35.5 fs (b). Panels c and d, same range of τ as in a and b but for a longer time window of 160 fs. Negative values that corresponds to emission appear in blue and positive values (absorption) appear in red. See text for the parameters of the pump and probe pulses.

5. Conclusions Our full quantum dynamical study of the electronic and nuclear response of LiH to strong, one cycle ultrashort pump pulses shows that the beating of the electronic coherence built by the broad in energy pump pulse is modulated but not dephased by the vibrational periods of the nuclear motion that develop on longer, dozens of fs, time scales. In the particular case of LiH, when the pump pulse excites a vibronic wave packet made of the lowest excited electronic states, these coherences are also not modified by non adiabatic coupling, or by the interaction with a weak probe pulse. Transient absorption spectroscopy was previously used for probing the electronic response in rare gas atoms and the vibronic response in polyatomic molecules. Our computations of the electronic and dipole response indicate that it is an ideal tool for probing both the fast, fs, time scale of the coherent non equilibrium electronic dynamics and the longer nuclear response.

Supporting information description: Figure S1 provides more insights about the localization of the wave function and Figure S2 highlights the role of nuclear dynamics on the time dependent profile of the total dipole. This material is available free of charge via the Internet at http://pubs.acs.org. The authors declare no competing financial interest.

Acknowledgments This work is supported by the U.S. Department of Energy (DOE), Office of Science, Basic Energy Sciences (BES) under Award #DE-SC0012628 and in part by the Einstein Foundation



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Quantum Nuclear Dynamics Pumped and Probed by Ultrafast

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