Few-cycle laser pulses to obtain spatial separation of OHF[sup −] dissociation products

THE JOURNAL OF CHEMICAL PHYSICS 130, 024310 共2009兲

Few-cycle laser pulses to obtain spatial separation of OHF− dissociation products Nadia Elghobashi-Meinhardt,1,a兲 Leticia González,2,b兲 Ingo Barth,3,c兲 and Tamar Seideman4,d兲 1

Computational Molecular Biophysics, Interdisciplinary Center for Scientific Computing (IWR), University of Heidelberg, Im Neuenheimer Feld 368, 69120 Heidelberg, Germany 2 Institut für Physikalische Chemie, Friedrich-Schiller Universität Jena, Helmholtzweg 4, 07743 Jena, Germany 3 Institut für Chemie und Biochemie, Freie Universität Berlin, Takustrasse 3, 14195 Berlin, Germany 4 Department of Chemistry, Northwestern University, 2145 Sheridan Road, Evanston, Illinois 60208-3113, USA

共Received 7 August 2008; accepted 24 November 2008; published online 14 January 2009兲 In a two-part theoretical study, field-free orientation of OHF− is achieved by means of moderately intense half-cycle, infrared laser pulses. In the first step, a short linearly polarized pulse excites a superposition of rigid rotor rotational eigenstates via interaction with the permanent dipole moment of OHF−. After the field has been switched off, pronounced molecular orientation is observed for several picoseconds. In the second step, femtosecond few-cycle laser pulses are applied to the oriented system to steer vibrational dynamics, modeled by anharmonic vibrational wave functions calculated on a potential energy surface obtained with unrestricted fourth order Møller-Plesset ab initio calculations. The result is selective bond breaking of OHF, followed by the spatial separation of dissociation products in the space-fixed frame. Due to revivals in the rotational wavepacket, product yields can be enhanced over long times. © 2009 American Institute of Physics. 关DOI: 10.1063/1.3054276兴 I. INTRODUCTION

Controlling the orientation of molecules in the laboratory frame is desirable for the optimization of bimolecular collision experiments since the position of collision partners determines the outcome of the encounter. Likewise, in unimolecular dissociation experiments, controlling the orientation of molecules is essential for the spatial separation of dissociation products. Surface reactions can likewise greatly benefit from orientation of the gas phase molecules, which allow control of the reaction, site selectivity, insights into the reaction stereospecificity, and in some cases the creation of a layer with long range orientational order. Furthermore, several coherent control approaches developed in the past few years require that the molecules should be oriented prior to the application of the control fields. Traditional approaches to controlling molecular orientation include the application of a strong, static electric field 共termed the “brute force method”兲;1,2 the technique of hexapole state selection3,4 and the method of photoselection by resonant polarized light.5 Related to these studies, and also to the topic of the present manuscript, is the method of laser alignment by means of moderately intense laser pulses.6,7 In particular, nonadiabatic alignment by means of short 共with respect to the rotational periods兲 laser pulses, introduced in Ref. 8, has been the topic of rapidly growing experimental a兲

Electronic mail: [email protected]. Electronic mail: [email protected]. c兲 Electronic mail: [email protected]. d兲 Electronic mail: [email protected]. b兲

0021-9606/2009/130共2兲/024310/9/$25.00

and theoretical interest during the past decade. Although most of this research focused on isolated linear molecules subject to linearly polarized laser fields, extensions to complex polyatomics, to solutions, to surface adsorbed molecules, and to molecular assembly have been reported, as were extensions to three-dimensional alignment with elliptically polarized fields and to torsional control with circularly polarized light. Several approaches have been developed to augment the method of nonadiabatic laser alignment so as to affect nonadiabatic orientation. One approach, introduced and examined theoretically in Ref. 9, is based on the combination of a weak static field with a short pulse laser field. Experimental realization of this approach was reported in Ref. 10. Dion et al.11 applied two-frequency IR laser pulses to orient HCN molecules. One of the more recent approaches to controlling molecular orientation involves using a half-cycle pulse 共HCP兲. Although this technique has yet only been studied numerically,12–15 high-power 共0.8 ␮J兲 500 fs HCPs in the terahertz 关far-infrared 共IR兲兴 range have been available for over ten years and are suitable for nonlinear optics and multiphoton spectroscopy experiments.16 In fact, numerical optimal control schemes that produce laser fields based on genetic algorithms have indicated that HCPs are the most effective in achieving a high degree of orientation.15 HCPs rely on a predominantly unipolar electric field that exerts a unidirectional force on the molecular axis. Classically, this force could be considered as a torque applied to a rigid body. An ensemble of randomly oriented molecules will

130, 024310-1

© 2009 American Institute of Physics

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Elghobashi-Meinhardt et al.

O

+Z

O

H

H

F

+x

F +y

+z

FIG. 1. 共Color online兲 Schematic representation of a laser scheme containing few-cycle IR+ UV pulses applied to control bond-selective dissociation of a triatomic molecule. This control scheme benefits from an optimal reverse orientation of the molecular Z axis with respect to the space-fixed 共x , y , z兲 frame of the laboratory.

slowly begin to feel the direction of the electric field, such that the librations of the molecule are restricted to a fixed angular range in ␪, the angle between the molecular axis and the field vector. In the long pulse limit where the pulse duration t p exceeds the rotational period t p ⬎ ␲ប / Be, where Be is the rotational constant of the molecule in its equilibrium geometry, the dynamics is adiabatic—each eigenstate of the field free rotational Hamiltonian evolves adiabatically into the corresponding state of the complete Hamiltonian. In the short pulse case t p Ⰶ ␲ប / Be, the ensuing time evolution is nonadiabatic and a rotational wavepacket is formed that undergoes dephasing, and rephasing occurs on the time scale of rotational periods. Our goal in the present theoretical work is to study the application of nonadiabatic laser orientation for bondselective photochemistry. Specifically, we will apply HCPs to orient OHF− and subsequently use time-delayed IR and UV pulses to control the branching ratio of the two dissociation channels. Our approach is thus related, on one hand, to the work in Ref. 17 on control of branching ratios through adiabatic alignment, and on the other hand, to the scheme in Ref. 18 that relies on pulses short with respect to vibrational periods to manipulate unimolecular reaction channels. Several recent articles illustrated the possibility of bondselective dissociation, using a series of tuned few-cycle IR and UV laser pulses, as shown schematically in Fig. 1 for the case of OHF−. These include the dissociation of isotopically substituted ozone, 16O 16O 18O,19 HOD,20,21 FHF−,22–24 and OHF−.25 The IR pulse excites a vibrational wavepacket in the ground state. The UV pulse, timed to an instant when the bond to be broken is transiently elongated, electronically excites the wavepacket leading to selective bond breakage 共see Fig. 1兲. In these studies, the molecule was assumed to be perfectly oriented with respect to the laser fields25,26 or randomly oriented.19–24 In reality neither assumption is realistic, even in a qualitative sense. Perfect orientation is never attained in practice and, moreover, although the nonadiabatic orientation approach provides sharper orientation than other methods, the orientation is transient—the larger the degree of rotational excitation and hence the sharper the orientation the

shorter its duration. The assumption of random orientation is also incorrect, as the anisotropy of free space is broken in the presence of the two laser pulses. More importantly, as with similar control schemes, at the absence of orientation one expects 共and finds兲 much less controllability. Our goal in the present theoretical work is to examine as to what extent laser nonadiabatic orientation can make a useful tool in this and other optical control schemes. We ask also in how far the conclusions in Ref. 25, in which perfect orientation was assumed, survive in the case of practically attainable orientation. We thus recalculate the branching ratio of OHF dissociation products by considering explicitly a rotational-vibrational wavepacket subject to three laser fields, the first of which orients the molecule and the subsequent two selectively break the bond. The first goal is to achieve a high degree of molecular orientation that is conserved over times much larger than the rotational period ␶rot of the molecule, which is of the order of 50 ps for OHF−. When molecular orientation is sharp, a series of optimized ultrashort femtosecond bond-breaking pulses is applied to dissociate the oriented molecular bonds. Since rotational motion is negligible on the femtosecond timescale of vibrational dissociation dynamics, molecular fragments will retain their orientation in the space-fixed frame upon dissociation. Furthermore, due to rotational revivals that exhibit a well-defined periodicity under field-free conditions, bond-breaking pulses can be applied whenever a high degree of orientation is obtained, thus leading to a high yield of dissociation products localized in the frame of the laboratory. On the time scales of relevance, rovibrational coupling has no effect and hence the field-free eigenfunctions are separable into rotational and vibrational components. Our simulation of rotational wavepacket dynamics treats the anion OHF− as a collinear rigid rotor. The model used in the simulation of vibrational wavepacket dynamics treats both OHF− and OHF as linear, two-dimensional molecules consisting of the bonds R1 = ROH and R2 = RHF. The dissociation products that can be obtained are O + HF or OH+ F. Previously,25 it was shown that the dissociation of OHF naturally heavily favors the former set of products, while the OH+ F products are disfavored due to the topology of the neutral potential energy surface 共PES兲 that has a steep gradient in the O + HF exit channel. Here, we will limit our investigation to optimizing the yields of spatially separated O + HF products. The remainder of this article addresses the problem of selective bond breaking through control of the rotational and vibrational motions of OHF−. Sections II and III discuss the theory, starting with the rotational dynamics and proceeding to the vibrational dynamics. The methods of quantifying product yields are briefly discussed in Sec. IV, and Sec. V discusses our results. Section VI contains our conclusions.

II. THEORY: ROTATIONAL WAVEPACKETS

The time-dependent Schrödinger equation for a rigid rotor in the presence of an external laser field is given as

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024310-3

iប

J. Chem. Phys. 130, 024310 共2009兲

Simulating laser control of OHF dissociation

⳵ rot ˆ + V共t兲兲兩⌿rot共t兲典, 兩⌿ 共t兲典 = 共H 0 ⳵t

共1兲

where the field free Hamiltonian is ˆ = B Jˆ 2 H 0 e

共2兲

and Jˆ 2 is the total angular momentum operator squared. The time-dependent interaction V共t兲 is given within the electric dipole approximation as V共t兲 = − ␮0 cos共␪兲⑀rot共t兲,

⑀ជ rot共t兲 = zˆ␧mrotεrot共t兲cos共␻rot共t − t0兲兲,

共4兲

rot where zˆ is a unit vector along the space-fixed z-axis, ␧m is rot the field amplitude, and ε 共t兲 is an envelope function. The central carrier frequency of the laser pulse ␻rot corresponds to an off-resonant IR frequency, such that the molecule remains in the electronic and vibrational ground state. The envelope function is a Gaussian function centered at the time t = t 0, 2/trot2 p

共5兲

.

The pulse duration trot p is defined as trot p =

␲

␻rot2冑共ln 2兲

i i

i

共6兲

,

⍀M CJJ⍀M 共t兲兩J⍀M典e−iE t/ប , 兺 J⍀M i i

i

J

共7兲

i⍀i M i共t兲 are where the eigenstates are labeled 兩J⍀M典 and CJJ⍀M time-dependent expansion coefficients depending on the initial state 兩Ji⍀iM i典. The quantum number of the total angular momentum operator Jˆ 2 is denoted as J, and Mប and ⍀ប are the eigenvalues of the projections of Jˆ onto the space-fixed z and body-fixed Z axes, respectively. EJ are the rigid rotor rotational energies,

EJ = BeJ共J + 1兲.

共8兲

The eigenstates 兩J⍀M典 of the Hamiltonian can be expressed as normalized rotation matrices 兩J⍀M典 =

冑

2J + 1 Jⴱ ˆ D 共R兲, 4␲2 M⍀

⳵ Ji⍀i M i J ⍀i M i C 共t兲 = 兺 CJi ⍀M 共t兲 ⬘ ⳵ t J⍀M J ⬘

⫻具J⍀M兩 − ␮0 cos共␪兲⑀rot共t兲兩J⬘⍀M典 ⫻e−iEJ⬘Jt/ប

共9兲

where DJM⍀共␾ , ␪兲 = 具JM兩Rˆ兩J⍀典 are primitive Wigner matrices and Rˆ ⬅ 共␾ , ␪ , ␹兲 are the Euler angles of rotation describing

共10兲

with the initial condition i⍀i M i共t = 0兲 = ␦ ␦ CJJ⍀M JJi ⍀⍀i␦ MM i .

共11兲

In Eq. 共10兲, both M and ⍀ are conserved. The use of a linearly polarized laser field implies that Mប, the projection of the total angular momentum onto the laser polarization 共space-fixed兲 z axis, is conserved, ⌬M = 0. The projection onto the body-fixed Z axis is conserved ⌬⍀ = 0 since the electronic state of the molecule does not change. The values EJ⬘J are the rotational energy level spacings, EJ⬘J = EJ⬘ − EJ = Be关J⬘共J⬘ + 1兲 − J共J + 1兲兴.

共12兲

The interaction Hamiltonian is evaluated according to 具J⍀M兩 − ␮0 cos共␪兲⑀rot共t兲兩J⬘⍀M典 = − ␮0E共t兲 ⫻ W共J⍀M兩J⬘⍀M兲,

共13兲

where W共J⍀M 兩 J⬘⍀M兲 is the integral over the Euler angles for the dipole interaction, W共J⍀M兩J⬘⍀M兲 = 具J⍀M兩cos ␪兩J⬘⍀M典.

such that the central peak includes roughly one-half cycle of the oscillating laser field. The wave function 兩⌿rot共t兲典 is expanded in a complete ˆ , set of eigenstates of the field-free Hamiltonian H 0 兩⌿Jrot⍀ M 共t兲典 =

iប

共3兲

where ␮0 is the permanent electric dipole moment in the electronic ground state of the anion 共directed from F to O兲 and ⑀rot共t兲 is the linearly z-polarized electric field designed to control rotational motion. The angle ␪ is the angle between the molecular dipole vector 共+Z兲 and the space-fixed +z axis. For ␪ = 180°, the fluorine atom points in the +z direction and the dipole vector points in the −z direction, as shown schematically in Fig. 1. The electric field is given as

εrot共t兲 = e−共t − t0兲

the position of the body-fixed 共X , Y , Z兲 frame with respect to the space-fixed 共x , y , z兲 frame.27 Substitution of Eq. 共7兲 into Eq. 共1兲 leads to a set of coupled differential equations for the expansion coefficients,

共14兲

For moderate field strengths, the electric dipole interaction with the electric field is the leading term and the molecular polarizability—or induced dipole—can be neglected.28 The total angular momentum Jជ consists of the orbital angular momentum Rជ , the electronic angular momentum Lជ , and the spin angular momentum Sជ , as shown schematically in a vector representation in Fig. 2. The electronic angular momentum Lជ makes a projection ⌳ប onto the body-fixed Z axis, and the spin angular momentum Sជ makes a projection ជ is normal to the molecular ⌺ប onto the Z axis, whereas R ជ axis. The projection of J onto the body-fixed axis is thus ⍀ = ⌳ + ⌺.

共15兲

In the zero temperature limit, at time t = 0, OHF− is the electronic ground state 2⌸ and Jជ = 0 + 1 + 21 = 23 . However, in what follows, we neglect the spin angular momentum, since it makes a very small contribution to the total angular momentum as compared to the orbital component. Therefore, at t = 0 and T = 0 K, we have Jជ = 1. As the laser field is turned on, the field begins to exchange single units of angular momentum with the molecule, i.e., ⌬Jជ = ⫾ 1, while the projections Mប and ⍀ប are conserved. Due to the electronic angular momentum Lជ = 1, the total angular momentum of OHF− never drops below Jជ = 1.

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Elghobashi-Meinhardt et al.

具J⍀iM i兩cos ␪兩J⬘⍀iM i典 = 共− 1兲⍀i+M i冑共2J + 1兲共2J⬘ + 1兲

+z

⫻

+Z

θ

⫻

O

冉 冉

J

1 J⬘

冊 冊

Mi 0 − Mi

J

1 J⬘

⍀i 0 − ⍀i

共20兲

.

Large absolute values of 具cos ␪典 correspond to a high degree of orientation in the +z 共or −z兲 direction. As the rotational temperature increases, the thermally averaged rotational population F

J

PTJ 共t兲

FIG. 2. 共Color online兲 Schematic vector representation of the components of rigid rotor angular momentum. The electronic angular momentum Lជ makes a projection ⌳ 共up to a factor of ប兲 onto the body-fixed Z axis, and the spin angular momentum Sជ makes a projection ⌺ 共up to a factor of ប兲 onto the Z ជ due to the orbiting rigid rotor is normal axis. Since the angular momentum R to the molecular axis, its projection onto the Z axis is zero.

Molecular orientation is quantified by examining both the ␪ dependence of the time-dependent thermally averaged probability density, 兩⌿

III. THEORY: VIBRATIONAL WAVEPACKETS

The ensuing control of vibrational dynamics and bondselective dissociation is modeled using a two-level system consisting of the collinear anion 共a兲 OHF− and neutral 共n兲 vib species OHF. Vibrational wavepackets 兩⌿vib a 共t兲典 and 兩⌿n 共t兲典, respectively, are obtained as solutions of the time-dependent Schrödinger equation,

1 i,max = 兺 wJ 共T兲 Qrot Ji=Ji,min i Ji

⫻

兺

M i=−Ji

冕

2␲

iប

兩⌿Jrot⍀ M 共␪, ␾,t兲兩2d␾ , i i

0

共16兲

i

J

兺 J =J i

共2Ji + 1兲exp

i,min

册

− BeJi共Ji + 1兲 , k BT

共17兲

具cos ␪典Ji,M i共t兲 = =

兺

J,J⬘=1

␪兩⌿Jroti⍀iM i共t兲典

i

⬘

i

i

⫻ 具J⍀iM i兩cos ␪兩J⬘⍀iM i典e where 具cos ␪典Ji,M i共t兲 苸 关−1 , 1兴. 具J⍀iM i兩cos ␪兩J⬘⍀iM i典 can be symbols,30

−iEJ⬘Jt/ប

Finally, evaluated

共22兲

.

共23兲

m1 =

mO · mH mO + mH

共24兲

m2 =

mF · mH . mF + mH

共25兲

The term Va/n共R1 , R2兲 is the ab initio PES along the coordinates R1 and R2. The field-matter interaction potential V共t兲 is given by

冉

冊

冉

冊

ជ a共R1,R2兲 ␮ ជ an共R1,R2兲 ␮ Va共t兲 Van共t兲 = − ⑀ជ vib共t兲 · . Vna共t兲 Vn共t兲 ជ na共R1,R2兲 ␮ ជ n共R1,R2兲 ␮ 共26兲

J⍀M

Ji⍀i M i ⴱ i i i 共CJ⍀ M 共t兲兲 CJ ⍀ M 共t兲 i

兩⌿vib n 典

where the first three terms represent the kinetic energy, with the masses m1 and m2 given by

共18兲

where the populations of initial states J ⬎ Ji,max at the given temperature are negligible 共in this work Ji,max = 3 at T = 1 K兲. The term 具cos ␪典Ji,M i共t兲 is evaluated using 具⌿Jrot⍀ M 共t兲兩cos i i i

兩⌿vib a 典

+ Va/n共R1,R2兲,

J

冋

Ji,max

冊冉 冊

2 2 2 2 2 ˆ =− ប ⳵ − ប ⳵ + ប ⳵ ⳵ H a/n 2m1 ⳵ R21 2m2 ⳵ R22 mH ⳵ R1 ⳵ R2

where wJi共T兲 = exp共−EJi / kBT兲 and kB is the Boltzmann constant. Qrot is the rotational partitional function that runs the initial states from Ji,min = 1 to Ji,max, Qrot =

冉 冊冉

vib ˆ ˆ H ⳵ 兩⌿a 典 a Han = ⳵ t 兩⌿vib ˆ ˆ n 典 H na Hn

ˆ for ˆ or H The molecular two-dimensional Hamiltonian H a n the collinear anionic or neutral molecule, respectively, is given in terms of the bond coordinates R1 and R2 by

where ⌿Jrot⍀ M 共␪ , ␾ , t兲 = 具Rˆ 兩 ⌿Jrot⍀ M 共t兲典 and the thermally avi i i i i i eraged expectation value of cos ␪,29 i 1 i,max 具cos ␪典T共t兲 = wJi共T兲 兺 具cos ␪典Ji,M i共t兲, 兺 Qrot Ji=Ji,min M i=−Ji

共21兲

contains a broader distribution of J states and the orientation diminishes.

J

共␪,t兲兩T2

rot

J

i 1 i,max Ji⍀i M i = wJi共T兲 兺 兩CJ⍀ 共t兲兩2 兺 iM i Qrot Ji=Ji,min M i=−Ji

,

the using

共19兲 term 3−J

ជ a共R1 , R2兲 and ␮ ជ n共R1 , R2兲 The permanent dipole moments ␮ are calculated at the same level of ab initio theory as the PES ˆ and H ˆ are given by Va/n共R1 , R2兲. The coupling terms H an na vib vib ជ ជ ជ ជ −⑀ 共t兲 · ␮an共R1 , R2兲 and −⑀ 共t兲 · ␮na共R1 , R2兲, respectively; the values of the anion-to-neutral transition dipole moments ជ an共R1 , R2兲 = ␮ ជ na共R1 , R2兲 can be considered constant for a ␮

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J. Chem. Phys. 130, 024310 共2009兲

Simulating laser control of OHF dissociation

given nuclear configuration within the Condon approximation.31 Here, they are set to 1ea0 for simplicity, but smaller or larger values of the transition dipoles can be compensated by stronger or weaker UV field strengths, respecvib vib 共t兲 + ⑀ជ UV 共t兲 denotes the linearly tively. The term ⑀ជ vib共t兲 = ⑀ជ IR z-polarized electric field designed to initiate the vibrational vib dynamics 共⑀ជ IR 共t兲兲 and lead to bond selective dissociation vib vib is given by 共⑀ជ UV共t兲兲. The IR field centered about t0,IR vib vib vib vib vib vib 兲 ⑀ជ IR 共t兲 = zˆ␧m,IR εIR cos共␻IR 共t − t0,IR 兲 + ␩IR

共27兲

vib vib ⱕ t ⱕ t0,IR + tvib for t0,IR p,IR 共elsewhere zero兲. The pulse envelope vib εIR is given by, vib vib vib 共t − t0,IR 兲 = sin2共␲共t − t0,IR 兲/tvib εIR p,IR兲,

共28兲 vib ␩IR

vib ␻IR

is the central pulse frequency, and is the phase of the IR field. The UV field is identical in form, vib vib vib vib vib vib ⑀ជ UV 共t兲 = zˆ␧m,UV εUV cos共␻UV 共t − t0,UV 兲 + ␩UV 兲,

共29兲

with vib vib vib 共t − t0,UV 兲 = sin2共␲共t − t0,UV 兲/tvib εUV p,UV兲

共30兲

vib vib ⱕ t ⱕ t0,UV + tvib for t0,UV p,UV 共elsewhere zero兲. The time delay vib vib − t0,IR . In Sec. V, between the IR and UV pulses is td = t0,UV we will retain the notation IR and UV when discussing the parameters of the IR and UV laser fields, respectively. For all 2 are kept under the pulses, maximum intensities I = ␧oc␧m 13 2 limit of ⬃10 W / cm to avoid undesired processes, such as double ionization or Stark shifts. The external electric field thus contains a HCP applied to orient the anion, followed by a sequence of few-cycle IR and UV pulses to initiate the bond-selective dissociation. Prior to excitation by IR and UV fields, the system is in the ground electronic and vibrational state of the anion ␾a␷=0, rot 兩⌿vib a 共t ⬎ 共t0 + t p 兲兲典 rot 兩⌿vib n 共t ⬎ 共t0 + t p 兲兲典

=

冉 冊 兩␾a␷=0典 0

共31兲

.

IV. THEORY: QUANTIFYING PRODUCT YIELDS

After the laser pulses have been applied and the propagation came to conclusion, the branching ratio of dissociation products O + HF versus OH+ F is determined by integrating the probability density of the vibrational wavepacket 2 兩⌿vib n 共t兲兩 over the triangular grid halves, ROH ⬎ RHF and RHF ⬎ ROH. The probability of dissociation to form O + HF, for instance, is calculated as branch共tend兲 =

2 兩⌿vib n 共tend兲共ROH ⬎ RHF兲兩 2 兩⌿vib n 共tend兲共total兲兩

.

共32兲

The time-dependent product P共␪ , t兲 of oriented dissociation products O + HF is given by P共␪,t兲 = 兩⌿rot共␪,t兲兩T2 ⫻ branch共tend兲, 兩⌿rot共␪ , t兲兩T2

共33兲

is the thermally averaged probability denwhere sity of the rotational wavepacket at time t, and branch共tend兲 is the branching ratio of the vibrational wavepacket densities obtained after the end of the propagation.

Integrating over ␪ values in the forward and backward hemispheres, one then obtains the space-integrated relative yields for one set of OHF dissociation products Y forward共t兲 =

冕

␲/2

P共␪,t兲 sin ␪ d␪ ,

共34兲

P共␪,t兲 sin ␪ d␪ .

共35兲

0

and Y backward共t兲 =

冕

␲

␲/2

The time-dependent relative yield Y forward共t兲 describes the yield of the products O and HF after dissociation in the forward 0 ⱕ ␪ ⱕ ␲ / 2 and backward ␲ / 2 ⱕ ␪ ⱕ ␲ hemispheres, respectively, i.e., it is favored for the molecular orientation ␪ = 0° 共FH+ O兲. For the reverse orientation 共␪ = 180° , O + HF兲, the relative yield Y backward共t兲 is preferred and describes the yield of the products HF and O in the forward 0 ⱕ ␪ ⱕ ␲ / 2 and backward ␲ / 2 ⱕ ␪ ⱕ ␲ hemispheres, respectively 共see Fig. 1兲. The same integration could, of course, be carried out to obtain the space-integrated relative yields of the products obtained by breaking the H–F bond, i.e., OH + F and F + HO. Here, we will focus on the products obtained when the O–H bond is broken since this dissociation channel is heavily favored. V. RESULTS

We first examine the simulation of rotational wavepacket dynamics of OHF− resulting from the application of a HCP with a field strength of −500 MV/ m 共intensity Irot = 66.4 ⫻ 109 W / cm2兲 to a rotationally cold sample, T = 1 K. Our choice of a negative field amplitude is in order for the molecule to orient along the negative z-axis. The rotational constant Be for OHF− is 0.336 cm−1 leading to a rotational period of ␶rot = ␲ប / Be = 49.6 ps. The ab initio permanent dipole moment in the electronic and vibrational ground state of the anion, calculated using unrestricted fourth order MøllerPlesset perturbation theory, is found to be ␮0 = 1.05 D, which is directed from F to O, and the corresponding equilibrium bond lengths are ROH = 1.12 Å and RHF = 1.33 Å.25 The pulse frequency is centered at ␻rot = 8.1共2␲c兲 cm−1 共0.243 THz兲; this IR frequency induces essentially no vibrational transitions and is much larger than the spacing between the low-lying rotational energy levels of OHF−, 4Be , 6Be , . . .. The pulse duration trot p is chosen to contain approximately one half of the optical cycle and be short compared to the rotational period ␶rot. Here we use a Gaussian-shaped pulse with duration trot p = 1.2 ps centered at t0 = 10.0 ps. The HCP is shown in panel 共a兲 of Fig. 3; the timedependent thermally averaged populations of the J states PTJ 共t兲 are plotted in panels 共b兲 and 共c兲. At time t = 0, nearly 79% of the population is in the state J = 1 and nearly 19% is in the state J = 2. Again, due to its 2⌸ groundstate symmetry, OHF− has a minimum total angular momentum of 1 and the state J = 0 is not allowed. As the laser is turned on, the system exchanges units of angular momentum with the external field, causing population in J = 1 to spike at t = 9.46 ps, i.e., shortly before the peak of the HCP, followed by a spike in

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024310-6

1

(a)

(a)

0.8

0 -100 -200 -300

0.4 0.2

-0.2

-0.6

t2

-0.8

-500

0

J=1 J=2 J=3 J=4

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|Ψrot (θ)|2 sin (θ)

J=1

T

PJ

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80

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120

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J = 13 J = 14 J = 15 J = 16 J = 17

0.15

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4

6

8

10

12

14

t [ps] FIG. 3. 共Color online兲 A simulated HCP with ⑀rot = −500 MV/ m is applied to a rotationally cold sample 共T = 1 K兲 of OHF−. The pulse 共⑀rot共t兲兲 is shown in panel 共a兲. The calculated thermally averaged population of low-lying rotational levels, shown in panel 共b兲, begins to migrate into higher-lying T states after the field has been switched on starting from PJ=1 共t = 0兲 = 0.79 T 共red兲, PJ=2 共t = 0兲 = 0.19 共green兲, and negligible contributions from T T PJ=3 共t = 0兲 = 0.01 共dark blue兲 and PJ=4 共t = 0兲 共magenta兲. After the laser pulse has been removed 共t ⬎ 12 ps兲, the wavepacket composition remains conT stant, shown in panel 共c兲, with significant contributions from PJ=13 共t兲 共red兲, T T T T PJ=14 共t兲 共green兲, PJ=15 共t兲 共dark blue兲, PJ=16 共t兲 共magenta兲, and PJ=17 共t兲 共light blue兲.

the population in J = 2 at t = 9.67 ps. The population continT ues to migrate into higher-lying states, such that PJ=3 共t兲 T reaches a value of 0.32 at t = 9.77 ps and PJ=4共t兲 reaches 0.28 shortly thereafter, at t = 9.85 ps. This sequential rotational excitation, familiar in the area of intense laser alignment,6,7 gives rise to a broad rotational wavepacket with specific phase relations among the rotational levels. After the field has been removed 关for t ⬎ 共t0 + trot p 兲兴, the wavepacket consists T T T 共t兲 ⬇ 0.16, PJ=16 共t兲 ⬇ 0.15, and PJ=14 共t兲 ⬇ 0.12, folof PJ=15 T T lowed by PJ=17共t兲 ⬇ 0.10 and PJ=13共t兲 ⬇ 0.06 with minor contributions from remaining levels included in the simulation, Jmin = 1 艋 J 艋 12 and 18艋 J 艋 30, such that the total normalized population is contained within Jmin = 1 艋 J 艋 30. The wavepacket composition remains constant, while the relative

(b)

t5

t1 = 0.0 t2 = 11.1

1.2

t4

t3 = 41.9 t4 = 59.1 t5 = 60.8

t2

0.8

t3

t1

0.4

0 0

(c)

0 0

60

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< cos θ >T

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J. Chem. Phys. 130, 024310 共2009兲

Elghobashi-Meinhardt et al.

20

40

60

80

100 120 140 160 180

θ [deg]

FIG. 4. 共Color online兲 In 共a兲, the orientation cosine 具cos ␪典T共t兲 at T = 1 K is plotted during the simulated evolution of the rotational wavepacket after application of a HCP 共Irot = 66.4⫻ 109 W / cm2兲. The oscillatory pattern represents a rich superposition of rotational states. The coherence is conserved and rotational revivals appear in time with the rotational period of OHF−, every ⬃50 ps. In 共b兲, the thermally averaged normalized rotational wavepacket probability density 兩⌿rot共␪ , t兲兩T2 sin共␪兲 at T = 1 K is shown for several times throughout the simulated evolution of the wavepacket. An isotropic distribution of the wave function is present at time t1 = 0.0 ps 共red line兲, then the wave function is positioned in the backward hemisphere at t2 = 11.1 ps 共green line兲 corresponding to the reverse orientation of the anion before becoming a mixture of distributions in backward and forward hemispheres at t3 = 41.9 ps 共dark blue line兲. The wavepacket is positioned in the forward hemisphere at t4 = 59.1 ps 共magenta line兲 before returning to the backward hemisphere at t5 = 60.8 ps 共light blue line兲.

phases of the rotational states continue to clock on the time scale of the rotational period of 49.6 ps. This dephasing and rephasing of the rotational states give rise to a rotational revival structure that persists as long as coherence is maintained. The orientation cosine corresponding to this wavepacket propagation is plotted in Fig. 4共a兲. At t = 0, the orientation cosine is 0, indicating an isotropic distribution of the wave function. As the field is turned on, the behavior of 具cos ␪典T共t兲 is coherent, but the rapid oscillations reflect the large number of angular momentum states that compose the rotational wavepacket. The rotational period of ⬃50 ps is clearly observable. The thermally averaged normalized rotational wavepacket probability density 兩⌿rot共␪ , t兲兩T2 sin共␪兲, calculated at T = 1 K, is shown for several times during the dynamical evolution in Fig. 4共b兲. At t1 = 0 ps, the wavepacket is isotropic, with probability density distributed symmetrically about ␪ = 90°. As the pulse is turned on, an oriented rotational wavepacket is created. After the laser field has been turned off 关t ⬎ 共t0 + trot p 兲兴, the composition of the wavepacket remains

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J. Chem. Phys. 130, 024310 共2009兲

Simulating laser control of OHF dissociation

[Å][A] HFHF

1.5

OH [A]>

constant but the phases of the rotational states continue to clock, leading to dynamical behavior of the wavepacket probability density as a function of the orientation angle ␪. At the orientation extrema, t2 = 11.1 ps and later instances spaced by the revival time ␶rot, 具cos ␪典T共t兲 = −0.72 and the probability density is strongly focused in the backward hemisphere 共␲ / 2 ⱕ ␪ ⱕ ␲兲, corresponding to orientation along the negative space-fixed z-axis. At t4 = 59.1 ps+ n␶rot, n = 0 , 1 , . . . , 具cos ␪典T共t兲 = 0.70, and the probability density is centered around 20°. When the orientation cosine 具cos ␪典T共t兲 passes through zero, e.g., at t3 = 41.9 ps, 兩⌿rot共␪ , t兲兩T2 sin共␪兲 contains a mixture of states distributed between both forward and backward hemispheres, such that the orientation observable averages to zero. Next, a series of optimized ultrashort IR+ UV laser pulses is applied to the oriented system. The pulses are designed so as to optimize the dissociation of OHF− into the products O + HF. The IR and UV field intensities are optivib vib = 2.7⫻ 1012 W / cm2 共⑀0,IR = 3.2 GV/ m兲 and mized to IIR vib vib 12 2 IUV = 6.6⫻ 10 W / cm 共⑀0,UV = 5 GV/ m兲, respectively. The vib = 1565 共2␲c兲 cm−1, is tuned frequency of the IR pulse, ␻IR to drive the asymmetric stretching vibration and contains the mean energy spacing between the asymmetric vibrational vib eigenfunctions. Other parameters of the IR pulse are ␩IR vib = 0 and t p,IR = 50 fs. After a time delay td = 19 fs, the rovibrational wavepacket created by the IR pulse has been strongly displaced from the equilibrium position and ROH has been extended to 1.18 Å. This time corresponds to almost the end of the first complete cycle of the IR pulse. At this vib instant an ultrashort UV pulse with tvib p,UV = 5 fs, ␩ = 0, and a carrier frequency corresponding to the vertical energy spacvib ing between the PESs V共OHF−兲 and V共OHF兲 共␻UV −1 −1 = 28 228共2␲c兲 cm = 3.5ប eV兲 photodetaches the electron. A neutral wavepacket is prepared along the O + HF dissociation channel of V共OHF兲 and nearly exclusive O + HF neutral dissociation results. It is preferable to excite the system a few femtosecond before the IR cycle is completed, so that the maximum displacement of the H atom coincides with the maximum intensity of the IR pulse. At the end of the propagation, the wavepacket remains relatively compact and vib + 70 fs is 98:2. the final calculated branching ratio at t = t0,IR The tuning of the delay time td is critical for optimized wavepacket dissociation. By varying td to shorter or longer times, the optimized dissociation behavior will be manipulated. For delay times shorter or longer than td, the wavepacket is not compactly located at the turning point of its oscillation. As a result, the frequency of the UV pulse tuned to match the energy spacing Vn − Va at the time td is no longer valid and the efficiency of the photodissociation pulse is diminished. Since the vibrational wavepacket requires of the order of 20 fs to complete one oscillation, the delay time should be tuned within ⫾2 fs of td for maximum photodissociation yield. Above we have considered the effect of varying the angle ␪ between the molecular dipole vector 共+Z axis兲 and the space-fixed +z axis. An alternative yet equivalent method of manipulating the field-matter interaction is by modulating vib of the central carrier field of the IR pulse. In the phase ␩IR vib = 0° other words, for the field given in Eq. 共27兲, phases ␩IR

1.4 1.3 1.2 1.1 1 0

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t [fs] t - t vib 0,IR [fs]

FIG. 5. Calculated expectation values of bond lengths ROH and RHF on the vib vib = 0° 共dashed兲 and ␩IR = 180° 共solid兲. The vibrations are anion PES for ␩IR out of phase by a factor of ␲, leading to vibrational motion toward the vib transition state in the case of ␩IR = 0° and away from the transition state for vib ␩IR = 180°. vib and ␩IR = 180° will cause the wavepacket to oscillate in opposite directions, similar to opposite orientations of the molecular axis in the space-fixed frame. The resulting bond stretching is driven in the first case toward an extension of the RHF bond, in the second toward an extension of the ROH bond, as shown in Fig. 5. We proceed to analyze the wavepacket dynamics resultvib . For ing from pulses simulated with opposite phases ␩IR vib ␩IR = 0°, ROH has been extended on the anion PES from its equilibrium length of 1.12 to 1.18 Å, while RHF has been compressed from its equilibrium length of 1.33 to 1.27 Å, leading to a wavepacket that is compactly located beneath the transition state point of the neutral PES, as seen in Fig. 5. vib = 180°, the opposite is true, On the other hand, when ␩IR namely, ROH has been compressed to 1.06 Å and RHF has been extended to 1.38 Å. The wave functions that are exvib + 19 fs to the neutral surface V共OHF兲 for cited at t = t0,IR vib vib ␩IR = 0° and ␩IR = 180° behave, accordingly, to their dynamics on the anion PES, see Fig. 6. In the case of excitation to vib = 180°, the wave function has a slight the neutral PES for ␩IR tail in the direction of the O + HF dissociation channel. In the vib = 0°, as the compact wavepacket proceeds to evolve case ␩IR on the dissociative surface, the ensuing stretching of ROH and vib ⬎ 19 fs lead to a more effeccompression of RHF for t + t0,IR tive dissociation along the O + HF channel. Finally, the relative yields Y forward共t兲, Y backward共t兲, and Y forward共t兲 + Y backward共t兲 of dissociation products, calculated according to Eqs. 共34兲 and 共35兲, are shown in Fig. 7. The forward and backward Y共t兲 yields follow the time evolution of the rotational wavepacket shown in Fig. 4共a兲, undergoing

2

η=0o

η=180o

1.5

RHF [Å]

024310-7

1

1

1.5 ROH [Å]

2

1

1.5 ROH [Å]

2

vib FIG. 6. 共Color online兲 Wavepacket dynamics on the neutral PES for ␩IR vib vib = 0° and ␩IR = 180° at time t = t0,IR + 19 fs.

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024310-8

J. Chem. Phys. 130, 024310 共2009兲

Elghobashi-Meinhardt et al.

Yforward + Ybackward

of Y forward共t兲 + Y backward共t兲 is small. Systems such as HOD, for which dissociation channels are nearly equivalent, lead to a much sharper variation in the total integrated yield.21

1 0.995

Yforward Ybackward

0.99

1

0.985 0

40

80

120

0.8

Y

0.6 0.4 0.2 0 0

20

40

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100

120

t [ps] FIG. 7. 共Color兲 The space-integrated relative yields of dissociation products, Y forward共t兲 + Y backward共t兲 = 兰␲0 P共␪ , t兲 sin ␪ d␪ 共red兲, Y forward共t兲 = 兰␲0 /2 P共␪ , t兲 sin ␪ d␪ 共blue兲, and Y backward共t兲 = 兰␲␲/2 P共␪ , t兲 sin ␪ d␪ 共green兲, are plotted for the duration of the rotational wavepacket evolution. The yields demonstrate the same dynamic behavior with the same rotational period as the orientation cosine in Fig. 4共a兲.

the same oscillatory behavior at the rotational period time scale. At times where the wavepacket is oriented along the negative space-fixed z direction, the product yield in the backward hemisphere is maximized and vice versa. Our ability to stretch a given bond at times at which the wavepacket is well oriented with respect to the laser field and project the system onto the dissociative PES at that time allow, in principle, tight control over the total yield of dissociation products into a given arrangement channel. Consequently, the total yield Y forward共t兲 + Y backward共t兲 oscillates at the same frequency as the rotational wavepacket, exhibiting maxima at times when the wavepacket is oriented along the negative z-axis, corresponding to the dipole vector pointing along the positive z-axis along the dissociating laser field. We remark that the approach presented here is general and extends to larger molecules. The origin of controllability and the generality of the scheme owes to the time scale disparity between the rotational, vibrational, and dissociation dynamics. Since the orientation duration is much longer than the vibrational period, there is ample time for control of the vibrations, while the molecule is well-oriented in a desired sense. Since the dissociation pulse is fast compared to the vibrational period, it is possible to dissociate the molecule at a given instant in the course of the rotational period. The modulation depth in the total yield curve, however, depends on the extent to which the ratio of Eq. 共32兲 varies with the relative elongation of the two bonds whose dynamics is controlled. Thus, the degree to which the control approach alters the field-free ␪-integrated yield in a given channel would vary from molecule to molecule and would depend on the details of the neutral state PES. For the example considered here, the steep gradient of the neutral state PES favors the O + HF dissociation channel. Consequently the branching ratio of Eq. 共32兲 is close to 1 regardless of ␪, although it is maximized when the molecule is oriented in the negative z-direction. It follows that the modulation depth

VI. CONCLUSION

We have designed a laser scheme consisting of halfcycle and few-cycle laser pulses to control the molecular orientation and ensuing selective bond breaking of a triatomic system. The theoretical approach is based on nonadiabatic orientation of the dipole moment of the molecule along the electric field vector of a UV pulse tuned to resonance between the initial states with a dissociative PES. An IR pulse, timed to a period where the molecule is oriented, induces vibrational wavepacket dynamics in the course of which the two molecular bonds are elongated and contracted out of phase. The short UV pulse dissociates the molecule at instances where a desired of the two bonds is elongated, leading to selective bond breakage. The approach can be regarded as an example of a class of control and spectroscopic schemes that utilize the ability of current technology to probe or trigger events in the molecular frame by orienting the molecule with respect to the laboratory frame. It relies on the time-scale disparity between the rotational and vibrational periods and on the availability of laser pulses that are much shorter than vibrational periods. Due to the former feature, the orientation duration can always be chosen long with respect to the vibrational period, and hence the orientation is stationary on the time scales of relevance. Due to the latter feature, dissociation can be triggered on a time scale on which the vibrational motion is essentially stationary. In the future, one could envision a laser-driven “distillation” of spatially separated chemical products, i.e., for the purification of chemical mixtures or the targeted deposition of molecules or atoms onto a surface. ACKNOWLEDGMENTS

T.S. is grateful to the Alexander von Humboldt Foundation for a generous award that supported her visit to Berlin in the course of which this research was initiated. T.S. is grateful also to Professor Jörn Manz and Professor Martin Wolf for their kind hospitality during her visits to Berlin. N.E. gratefully acknowledges financial support from the “Berliner Programm zur Förderung der Chancengleichheit für Frauen in Forschung und Lehre.” Helpful discussions with B. Friedrich and E. L. Hamilton are acknowledged. This work has been done in the framework of the Graduiertenkolleg 788 “Hydrogen Bonds and Hydrogen Transfer” and was supported in part by the U.S. Department of Energy 共Grant No. DAAD19-03-R0017兲. H. J. Loesch and A. Remscheid, J. Chem. Phys. 93, 4779 共1990兲. B. Friedrich and D. R. Herschbach, Z. Phys. D: At., Mol. Clusters 18, 153 共1991兲. 3 K. H. Kramer and R. B. Bernstein, J. Chem. Phys. 42, 767 共1965兲. 4 T. D. Hain, R. M. Moision, and T. J. Curtiss, J. Chem. Phys. 111, 6797 共1999兲. 5 R. C. Estler and R. N. Zare, J. Am. Chem. Soc. 100, 1323 共1978兲. 6 T. Seideman and E. Hamilton, Adv. At. Mol. Opt. Phys. 52, 289 共2006兲. 7 H. Stapelfeldt and T. Seideman, Rev. Mod. Phys. 75, 543 共2003兲. 8 T. Seideman, J. Chem. Phys. 103, 7887 共1995兲. 1 2

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B. Friedrich and D. R. Herschbach, J. Chem. Phys. 111, 6157 共1999兲. H. Sakai, S. Minemoto, H. Nanjo, H. Tanji, and T. Suzuki, Phys. Rev. Lett. 90, 083001 共2003兲. 11 C. M. Dion, A. D. Bandrauk, O. Atabek, A. Keller, H. Umeda, and Y. Fujimura, Chem. Phys. Lett. 302, 215 共1999兲. 12 A. Ben Haj-Yedder, A. Auger, C. M. Dion, E. Cancès, A. Keller, C. Le Bris, and O. Atabek, Phys. Rev. A 66, 063401 共2002兲. 13 M. Machholm and N. E. Henriksen, Phys. Rev. Lett. 87, 193001 共2001兲. 14 C. M. Dion, A. Keller, and O. Atabek, Eur. Phys. J. D 14, 249 共2001兲. 15 C. M. Dion, A. Ben Haj-Yedder, E. Cancès, C. Le Bris, A. Keller, and O. Atabek, Phys. Rev. A 65, 063408 共2002兲. 16 D. You, R. R. Jones, P. H. Bucksbaum, and D. R. Dykaar, Opt. Lett. 18, 290 共1993兲. 17 J. J. Larsen, I. Wendt-Larsen, and H. Stapelfeldt, Phys. Rev. Lett. 83, 1123 共1999兲. 18 D. J. Tannor and S. A. Rice, J. Chem. Phys. 83, 5013 共1985兲. 19 B. Amstrup and N. E. Henriksen, J. Chem. Phys. 105, 9115 共1996兲. 20 B. Amstrup and N. E. Henriksen, J. Chem. Phys. 97, 8285 共1992兲. 9

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N. Elghobashi, P. Krause, J. Manz, and M. Oppel, Phys. Chem. Chem. Phys. 5, 4806 共2003兲. 22 N. Elghobashi, L. González, and J. Manz, J. Chem. Phys. 120, 8002 共2004兲. 23 N. Elghobashi, L. González, and J. Manz, Z. Phys. Chem. 217, 1577 共2003兲. 24 N. Elghobashi and J. Manz, Isr. J. Chem. 43, 293 共2003兲. 25 N. Elghobashi and L. González, Phys. Chem. Chem. Phys. 6, 4071 共2004兲. 26 M. Machholm and N. E. Henriksen, J. Chem. Phys. 113, 7838 共2000兲. 27 R. N. Zare, Angular Momentum 共Wiley, Chichester, 1988兲. 28 T. Seideman, J. Chem. Phys. 111, 4397 共1999兲. 29 L. Cai, J. Marango, and B. Friedrich, Phys. Rev. Lett. 86, 775 共2001兲. 30 A. R. Edmonds, Angular Momentum in Quantum Mechanics 共Princeton University Press, Princeton, 1957兲. 31 S. Mukamel, Principles of Nonlinear Optical Spectroscopy 共Oxford University Press, Oxford, 1995兲.

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