christian buth and robin santra- rotational molecular dynamics of laser-manipulated...

8/3/2019 Christian Buth and Robin Santra- Rotational molecular dynamics of laser-manipulated bromotrifluoromethane studie…

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Rotational molecular dynamics of laser-manipulated bromotrifluoromethane studied

by x-ray absorption∗

Christian Buth1, † and Robin Santra1, 2

1Argonne National Laboratory, Argonne, Illinois 60439, USA2 Department of Physics, University of Chicago, Chicago, Illinois 60637, USA

(Dated: October 14, 2008)

We present a computational study of the rotational molecular dynamics of bromotrifluo-romethane (CF3Br) molecules in gas phase. The rotation is manipulated with an offresonant,800 nm laser. The molecules are treated as rigid rotors. Frequently, we use a computationally effi-cient linear rotor model for CF3Br which we compare with selected results for full symmetric-rotorcomputations. The expectation value

˙cos2 ϑ

¸(t) is discussed. Especially, the transition from im-

pulsive to adiabatic alignment, the temperature dependence of the maximally achievable alignmentand its intensity dependence are investigated. In a next step, we examine resonant x-ray absorptionas an accurate tool to study laser manipulation of molecular rotation. Specifically, we investigate theimpact of the x-ray pulse duration on the signal (particularly its temporal resolution), and studythe temperature dependence of the achievable absorption. Most importantly, we demonstratedthat using picosecond x-ray pulses, one can accurately measure the expectation value

˙cos2 ϑ

¸(t)

for impulsively aligned CF3Br molecules. We point out that a control of the rotational dynamicsopens up a novel way to imprint shapes onto long x-ray pulses on a picosecond time scale. Forour computations, we determine the dynamic polarizability tensor of CF3Br using ab initio molec-ular linear-response theory in conjunction with wave function models of increasing sophistication:

coupled-cluster singles (CCS), second-order approximate coupled-cluster singles and doubles (CC2),and coupled-cluster singles and doubles (CCSD).

PACS numbers: 33.20.Sn, 33.55.-b, 31.15.Qg, 33.20.Rm

I. INTRODUCTION

We consider an ensemble of bromotrifluo-romethane (CF3Br) molecules subjected to the light of an optical, nonresonant laser. Due to a dynamic secondorder Stark effect, the electric field of the laser exertsa net torque on the molecular axis with the largest

polarizability. It is directed in such a way that it alignsthe molecular axis with the linear polarization axis of thelaser and thus aligns the molecules. The laser intensity istoo low to excite or ionize the molecules.1,2,3 In combina-tion with x rays, we have a so-called two-color problem;the laser can be used in a pump-probe-way precedingthe x rays or simultaneously with them. The former caseleads to impulsive (transient) alignment where a shortpulse creates a rotational wavepacket which evolves freelyafterwards and periodically goes through brief periods of alignment; the latter case typically is used in conjunctionwith long laser pulses leading to adiabatic alignmentwhere the molecular rotation follows the laser pulseenvelope. Laser alignment has been extensively studiedexperimentally [see, e.g., Ref. 1]. It was explained inthe adiabatic case by Friedrich and Herschbach4 and inthe case of pulsed lasers inducing rotational wavepacketdynamics leading to transient alignment by Seideman.5

The laser alignment of symmetric rotors was studied byHamilton et al.6 Recently, we formulated a theory forlaser-aligned symmetric-top molecules which are probedin situ by x rays; it was applied to study brominemolecules.7 In a joint experimental and theoreticalinvestigation, the alignment of CF3Br was studied for

various conditions.3,8

In this work, we would like to deepen the theoreti-cal understanding of the x-ray probe of molecules. Es-pecially, we are interested in trends. Resonant x-rayabsorption is sensitive to molecular alignment, if theresonance has certain symmetry properties. Then se-lection rules can be established which lead to a differ-ent absorption for perpendicular and parallel x-ray po-larization vectors with respect to the polarization axisof the laser. However, the absorption of x rays abovethe edge is not sensitive to molecular alignment. Rota-tional dynamics is conventionally specified in terms of the expectation value

cos2 ϑ

(t). This value is a the-

oretical quantity. Experimentally, one typically has re-sorted to the Coulomb explosion method to study align-ment.1,2 This method relies on the intricate nature of molecular strong-field ionization.9,10 The x-ray absorp-tion technique uses the simpler one-photon interactionand thus is physically simpler. We address the followingquestions in this paper. In what way is

cos2 ϑ

(t) ac-

cessible from x-ray absorption measurements in a cross-

correlation experiment? What is the achievable temporalresolution? We see our work in conjunction with the fas-cinating possibilities offered by upcoming ultrafast x-raysources.11,12,13,14 They will provide intriguing direct in-sights into the molecular rotational dynamics in the timedomain. Additionally, our research lays the foundationfor molecular imaging of laser-aligned molecules using x-ray diffraction.15,16

In addition to the analytical value of the x-ray absorp-tion probe, laser-induced molecular alignment opens up a

http://arxiv.org/abs/0809.0146v2














































3

(a,b,c)T in the molecule-fixed reference frame. Due tothe vanishing integrals rule,7,28,29 the direct product of representations Γ∗Br 1s⊗Γi⊗Γσ∗ with i ∈ {a,b,c} needsto contain the totally symmetric representation such that

Br 1s | ( d)i |σ∗ = 0, i.e., the transition Br 1s → σ∗ is al-

lowed via ( d)i. Using the product table of the C3v point

group,29 we find that the c component of d is totally sym-

metric but the components a and b are not. This givesrise to a dichroism with respect to the figure axis (C–Br)of CF3Br: absorption for parallel x-ray polarization vec-tor, no absorption for perpendicular x ray polarizationvector. In the Appendix, we show that this result holdsin spite of potential symmetry-breaking effects in CF3Br.

For our two-level model, we have Br 1s | d |σ∗ =

dc ec = dc e0 ≡ d ′10 with the unit vectors ec and e0 along

the c and the m = 0 axes, respectively.7,24,25,30 We ne-glect the dependence of the denominator in Eq. (1) onrotational quantum numbers. This is justified because,in view of the large inner-shell decay widths, rotationalstates remain unresolved in x-ray absorption. Let us de-

compose the fraction into a real part and an imaginarypart as follows:

F R + i F I ≡ 1

E 1 − E 0 − ωX − i Γ/2. (2)

We tune the x rays to the resonance energy, i.e., ωX =E 1 − E 0. Thus, we find from Eq. (2) F R = 0 and F I =2Γ . The resonant cross section (1) of a two-level systembecomes:

σ2(t) = 8π α ωX|dc|2

Γ

J,J ′,K,M

Re(KM )JJ ′ (t)

× J ′′,K′′,M ′′

s′c,J ′KM,J ′′K′′M ′′ s

′c,J ′′K′′M ′′,JKM ,

(3)with s ′c,J ′KM,J ′′K′′M ′′ ≡ (s ′J ′KM,J ′′K′′M ′′)c ands ′c,J ′′K′′M ′′,JKM ≡ (s ′

J ′′K′′M ′′,JKM )c.The initial, thermal distribution over rotational states

is sensitive to the nuclear spin statistics.30 The three flu-orine atoms in CF3Br, are 19

9F, the only stable fluorineisotope, with nuclear spin quantum number I = 1

2 .31 Inthis case, the statistical weights of the symmetric rotorare given by30

gI (J, K ) = g 1

2

(K ) = 4 ; K mod 3 = 0

2 ; K mod3 = 0 .

(4)

In most of the computations of this paper, we use a linearmodel for CF3Br. In this case, the CF3 portion of CF3Bris assumed to be a pseudo-atom which is distinct from abromine atom. Thus the nuclear statistical weight is inthis case:

gI (J ) = 1 . (5)

There is no quantum number K for a linear rotor.30

B. Experimentally accessible quantities

The instantaneous cross section (1) is not easily ex-perimentally accessible. However, we can use it to de-vise quantities which can be measured more readily. Thelaser pulse is imprinted on the cross section of the two-level model σ2,i(t) [Eq. (3)]. We determine it for parallel(i =

) and perpendicular (i =

⊥) x-ray and laser polar-

ization vectors. Additionally, we consider the cross sec-tion without laser σ2,th(t), i.e., a thermal ensemble. Thex rays are characterized by the photon flux J X(t). Thecross correlation of the cross section and the x-ray pulseis defined by

P i(τ ) =

∞ −∞

σ2,i(t) J X(t − τ ) dt , i =, ⊥, th . (6)

It represents the total probability of x-ray absorption fora time delay of τ between the laser and the x-ray pulse.We define following ratios of cross correlations:

R/⊥(τ ) =P (τ )P ⊥(τ )

, (7a)

R/th(τ ) =P (τ )

P th(τ ), (7b)

R⊥/th(τ ) =P ⊥(τ )

P th(τ ). (7c)

The ratios were studied experimentally in Ref. 3,8.Let us assume ultrashort x-ray pulses which can be

modeled by a δ distribution as follows:

J X(t − τ ) = J X,0 δ(t − τ ) . (8)

Inserting this flux into Eq. (6) yields P i(τ ) = J X,0 σ2,i(τ ).When the ratios (7) are formed, the constant J X,0 cancelsand the ratios are given by the ratios of the participatingcross sections. Hence, in the limit of ultrashort x-raypulse duration, Eq. (7), goes over into the new ratios

r/⊥(τ ) =σ2,(τ )

σ2,⊥(τ ), (9a)

r/th(τ ) =σ2,(τ )

σ2,th(τ ), (9b)

r⊥/th(τ ) =σ2,⊥(τ )

σ2,th(τ ). (9c)

C. Relation between the cross section

and˙

cos2 ϑ¸

(t)

We set out from the coupling matrix elements inEq. (1) reading7,23

d ′ ∗0i′′ ·s ′J ′KM,J ′′K′′M ′′ = J ′KM , 0 | d ·eX |J ′′K ′′M ′′, i′′ .

(10)



4

The x-ray polarization vector is eX and d denotes thedipole operator. Both vectors are Cartesian vectors inthe laboratory reference frame. The scalar product onthe left hand side of Eq. (10) is formed in the spheri-cal basis.24,25 With |J ′′K ′′M ′′, i′′ we denote the directproduct of the symmetric-rotor state |J ′′K ′′M ′′ and theelectronic state | i′′ .

Let us assume the two-level model of Sec. IIA; the

only dipole vector d01 and the polarization vector eX arereal vectors. The scalar product on the right hand sideof Eq. (10) becomes

J ′KM | d∗01 · eX |J ′′K ′′M ′′ = | d01| J ′KM | cos ϑ |J ′′K ′′M ′′ .

(11)

We use |eX| = 1 and ϑ is the angle between the two

vectors d01 and eX. We choose eX = ez along laser po-

larization axis. The dipole vector d01—here given in thelaboratory frame—points along the C–Br molecular axis;in the body-fixed reference frame the C–Br axis was cho-sen to be the c axis [Sec. IIA]. Thus, ϑ is the Euler

angle between the z and c-axes of the space-fixed andmolecule-fixed frame, respectively. The dependence of the denominator in Eq. (1) on rotational quantum num-bers is disregarded as in Eq. (2). The sum over inter-mediate symmetric-rotor states can then be eliminatedtotally yielding J ′KM | cos2 ϑ |JKM in the numera-tor of Eq. (1). The remaining summation forms a tracewith these matrix elements and the density matrix:

cos2 ϑ

(t) =

J,J ′,K,M

(KM )JJ ′ (t) J ′KM | cos2 ϑ |JKM .

(12)Finally, we arrive at the cross section of the two-

level model (3) which clearly exhibits the relationto

cos2 ϑ

(t) as follows:

σ2,(t) = 8π α ωX|dc|2

Γ

cos2 ϑ

(t) . (13)

Assuming an x-ray polarization vector perpendicularto the laser-polarization axis yields

σ2,⊥(t) = 4π α ωX|dc|2

Γ[1 −

cos2 ϑ

(t)] . (14)

The prefactor of 1−cos2 ϑ

(t) in σ2,⊥(t) is only half the

prefactor of cos2 ϑ (t) in σ2,(t) because there are two

perpendicular directions to the laser-polarization axiscompared to only one for parallel polarization vectors.The cross section of a thermal ensemble can be inferredeasily from Eq. (13) by letting

cos2 ϑ

(t) = 1

3 whichleads to

σ2,th(t) =8π

3α ωX

|dc|2Γ

. (15)

Given the formulas for the cross sections (13), (14), and(15), we can write explicit expressions for the ratios in

Eq. (9) as follows:

r/⊥(τ ) =2

cos2 ϑ

(τ )

1 − cos2 ϑ (τ ), (16a)

r/th(τ ) = 3

cos2 ϑ

(τ ) , (16b)

r⊥/th(τ ) =3

2[1 −

cos2 ϑ

(τ )] . (16c)

D. Thermal average of the rotational period

The character of molecular alignment—impulsive toadiabatic see, e.g., Fig. 3 and the ensuing discussion—crucially depends on the rotational temperature T .1,2

Conventionally, the rotational period of a linear rotoris defined by T RP = 1

2B with the rotational constant B.The regimes of molecular alignment from impulsive toadiabatic are distinguished by comparing T RP with theduration of the aligning laser pulse τ L. Yet, as T RP isindependent of T , the classification thus does not varywith T , despite the changed alignment characteristics.

We derive a thermally averaged rotational period fora linear molecule which does not exhibit this shortcom-ing. Therefore, we equate the energy expression of theclassical linear rotor of an angular frequency ωJ with thequantum mechanical rotational energy corresponding tothe angular momentum J as follows:

E J =1

2I B ω2

J = BJ (J + 1) . (17)

The moment of inertia for the rotation around axis Bis I B = 1

2B .30 This leads to the rotational period

T J =π√

B E J . (18)

A thermal ensemble of rotational states of tempera-ture T = 1

β consequently has the average rotational pe-

riod:

T th =1

Z

∞J =1

T J gI (J ) (2J + 1) e−βEJ . (19)

The nuclear statistical weight is given by gI (J ) [Sec. IIA].The partition function reads

Z =

∞J =1

gI (J ) (2J + 1) e−βEJ . (20)

The term for the rotational ground state J = 0 is ex-plicitly excluded in the sums of Eqs. (19) and (20) be-cause a molecule in the J = 0 state does not rotate andthus T 0 = ∞.

III. COMPUTATIONAL DETAILS

Bromotrifluoromethane (CF3Br) [Fig. 1] is a prolatesymmetric-top molecule of C3v symmetry. We use the



5

CCS CC2 CCSD

Basis set ωL [ Eh ] α [a.u.] ∆α [a.u.] α [a.u.] ∆α [a.u.] α [a.u.] ∆α [a.u.]

cc-pVDZ 0 25.54 14.81 26.11 15.97 25.05 14.80

0.057 25.71 15.09 26.32 16.32 25.24 15.12

aug-cc-pVDZ 0 36.60 11.67 38.83 13.57 37.20 12.16

0.057 36.93 11.89 39.27 13.90 37.60 12.43

aug-cc-pVTZ 0 37.71 10.91 40.05 12.78

0.057 38.07 11.10 40.52 13.08

TABLE I: Dynamic average dipole polarizability α(ωL) and dynamic dipole polarizability anisotropy ∆α(ωL) o f aCF3Br molecule in the field of a laser with photon energy ωL determined with coupled-cluster linear response methods forseveral basis sets.7 Goebel and Hohm32 obtained the experimental values α(0 Eh ) = 41.23 a.u. and ∆α(0 Eh ) = 11.00 a.u.

structural data from Taylor33 where the bond lengthsare specified as |C F| = 1.328 A and |C Br| = 1.918 Aand the bond angles are specified as ∠(FC F) = 108.42◦

and ∠(FCBr) = 110.52◦. The rotational constants A =0.191205 cm−1 and B = 0.069834 cm−1 were obtained

with theab initio

quantum chemistry program pack-age dalton.20,34 In the linear rotor approximation, weset A = 0 which implies no rotations around the fig-ure axis.30 For CF3Br, the rotational times are T A =1/(2A) = 87.2 ps and T B = 1/(2B) = 238.8 ps. Here, Bis the crucial constant because it corresponds to a rota-tion about an axis perpendicular to the figure axis whichis the only one influenced by the laser.36 The other con-stant A creates only a substructure of the energy lev-els of the thermal density matrix of the symmetric-topmolecule.7

We compute the average dynamical dipole polarizabil-ity α(ωL) of the molecule and its anisotropy ∆α(ωL)in the field of the aligning laser.37 The polarizabilityanisotropy is needed to describe the impact of the laseron the alignment later on; the average polarizabilitymerely causes a constant energy shift of the Hamiltonianwhich can be neglected for our purposes.7 Both quantitiesare determined in fixed-nuclei approximation using dal-

ton.20 We harness molecular linear-response theory inconjunction with wave function models of increasing so-phistication: coupled-cluster singles (CCS), second-orderapproximate coupled-cluster singles and doubles (CC2),and coupled-cluster singles and doubles (CCSD).38 Thecc-pVDZ [correlation consistent polarized valence dou-ble zeta] and aug-cc-pVX Z [augmented cc-pVX Z] (forX = D, T [D: double, T: triple]) basis sets21,22 are used.

We examine the static case, ωL = 0 Eh , and the dy-namic case, ωL = 0.057Eh , for the photon energy of an800 nm Ti:sapphire laser. Our data are compiled in Ta-ble I. A moderately rapid convergence with respect tothe basis set quality is observed for the individual meth-ods. We were not able to determine data for the com-bination of CCSD with aug-cc-pVTZ due to insufficientcomputational resources. Our best results for ωL = 0 Eh

are determined using the CC2 method with the aug-cc-pVTZ basis set. We note a satisfactory agreement

of our values to those of Goebel and Hohm,32 who ob-tain an average static polarizability of 41.23 a.u. and ananisotropy of 11.00 a.u.. In our computations, we use thevalue of ∆α(0.057Eh ) = 12.00 a.u. for the polarizabilityanisotropy.

The computations in this paper were carried out withthe program alignmol of the fella package.39 The ini-tial density matrix of a thermal ensemble of rotationalstates is represented using symmetric-rotor functions upto an angular momentum denoted by J max which waschosen high enough for the respective contribution tothe partition function of the thermal ensemble to be lessthan7 10−6. For temperatures lower than 6 K, N eq = 25equations of motions are used for each angular momen-tum to describe the time-evolution of the rotational den-sity matrix of the molecules interacting with the laser;for higher temperatures, N eq = 50 is used. Unless other-wise stated, we assume Gaussian laser and x-ray pulses;their durations (FWHM) are denoted by τ L and τ X, re-

spectively. The wave packet propagation takes place overthe time interval which is subdivided into N t time steps.The number was always chosen sufficiently high with re-spect to τ L and τ X for the numerical integration of thedifferential equations to be converged. The number of x-ray pulses which are used for cross correlation calcula-tions N X is sufficiently high to ensure smooth curves inthe respective graphs of this paper.

IV. RESULTS AND DISCUSSION

A. Laser alignment

Let us examine

cos2 ϑ

(t) which is typically used to

quantify molecular alignment.1,2 Here, ϑ represents thepolar Euler angle which describes the tilt of the c axis—identical to the figure axis in our case—of the molecule-fixed coordinate system with respect to the laser polar-ization axis, which is the z axis of the space-fixed coor-dinate system.24,30 The quantity

cos2 ϑ

(t) can be in-

ferred, e.g., with the help of the Coulomb explosion tech-nique.1,2 To this end, the molecule is highly ionized with



6

FIG. 3: (Color online) The time evolution of ˙

cos2 ϑ¸

(t) dueto molecular rotation is shown for four temporal regimes of the laser pulse: a) impulsive alignment, τ L = 50 fs; b) short-pulse intermediate regime (still almost impulsive), τ L = 10 ps;c) long-pulse intermediate regime (quasi-adiabatic), τ L =95 ps; d) adiabatic alignment, τ L = 1 ns. Other computa-tional parameters are T = 1 K, and I L,0 = 1012 W

cm2 .

an intense laser pulse. It breaks up into fragments thatsubsequently undergo Coulomb explosion. The locationwhere the fragments hit the detector plate is used to in-fer

cos2 ϑ

(t).

In Fig. 3, we investigate molecular dynamics for fourregimes of laser-pulse durations. These regimes are char-acterized by the ratio of the laser-pulse duration and thethermally averaged rotational constant [Sec. IID].

First, there is pure transient alignment [top panel of Fig. 3]; the molecules rotate freely after an initial “kick”with a laser pulse which is short compared to the ro-tational response time T th [Eq. (19)] of the molecule.The short laser pulse induces a brief period of align-ment. After well-defined periods of time, half revivalsoccur around 120 ps and around 360ps. At such oc-currences, there is a brief period of alignment followedby a brief period of antialignment. The molecule un-dergoes full revivals around 240 ps exhibiting the re-verse sequence—antialignment followed by alignment—compared with half revivals.6 Antialignment means thatthe molecules are aligned perpendicular to the laser po-larization axis. Then, the expectation value

cos2 ϑ

(t)

FIG. 4: (Color) Influence of the temperature on the rota-tional dynamics of CF3Br. The time evolution of

˙cos2 ϑ

¸(t)

is depicted for a linear rotor by the solid (black) curves fora range of temperatures given in blue. For comparison, weshow results from a symmetric-rotor computation as dashed(green) curves. The Gaussian laser pulse with τ L = 95ps andI L,0 = 1012 W

cm2 is represented by the dash-dotted (red) curve.

is suppressed with respect to the value of a thermal en-semble.

Second, the short-pulse intermediate regime is dis-played in the upper middle panel of Fig. 3. Here, themolecular dynamics is still almost impulsive. The purelyimpulsive character of the top panel of the figure, how-ever, is distorted due to a longer interaction with thelaser pulse during which rotational motion takes place.

Third, the long-pulse intermediate regime is shown inthe lower middle panel of Fig. 3; the molecular responseis almost adiabatic. Only slight nonadiabaticities occur,

e.g., the ripples on top of the peak. Moreover, the molec-ular ensemble does not fully return to a thermal distribu-tion after the laser pulse has ended.40 This causes beat-ing of excited states which leads to the oscillations be-yond 100 ps.

Fourth, in the lowest panel of Fig. 3, we display pureadiabatic alignment. The rotational response followsclosely the laser pulse shape. After the pulse, the en-semble has returned to the initial thermal distribution.We observe that the peak alignment of the molecules ismuch larger in the adiabatic case compared with the im-pulsive case in the top panel. This occurs due to the factthat a larger number of rotational states can be excitedin the wave packet during a longer laser pulse which, in

turn, facilitates a stronger localization of the molecule,i.e., alignment.1,2

In Fig. 4, we demonstrate the temperature dependenceof the molecular response to the laser in the temperaturerange from 0 K to 20 K. The transition from the interme-diate region of molecular alignment to purely adiabaticalignment is observed: at T = 0 K, the curve is muchbroader than the laser pulse and non-Gaussian which ex-hibits the nonlinear response of

cos2 ϑ

(t) with respect

to the Gaussian laser pulse. The rising temperature leads



7

FIG. 5: (Color online) Influence of the laser intensity on themaximally attainable alignment of CF3Br. We show curvesfor the linear rotor as solid (black) lines and curves for thesymmetric rotor as green (dashed) lines. The data are forvarious temperatures given by the numbers in blue. We uselaser pulses with a FWHM duration of τ L = 1 ns.

to a considerable reduction of the peak alignment. Alsothe large oscillations beyond a time of 150 ps vanish com-pletely. A broad thermal distribution is responsible fordephasing effects which diminish nonadiabaticities.7 Fora given temperature, the degree of adiabaticity can be

judged by applying our criterion from Sec. II D: we com-pare the FWHM laser-pulse duration with the thermallyaveraged rotational period [see Table II for examples].Generally, the values of T th in the table turn out to besmaller than T RP = 1/(2B) = 238.8 ps for CF3Br, as theyshould, to reflect the right trend: for small temperatures,

the period is longer than the laser pulse, thus correctlypredicting a nonadiabatic molecular response in Fig. 4;for larger temperatures an adiabatic behavior is rightlyindicated. The considerable sensitivity of the rotationalresponse of molecules to the rotational temperature sug-gests to use it to measure

cos2 ϑ

(t), i.e., the effect can

be harnessed as a thermometer to determine the rota-tional temperature of a gas sample. However, one has tokeep in mind that in experiments, the precise determina-tion of the laser intensity is often very difficult.

Overlaid on the curves for the linear rotor are datafrom a symmetric rotor computation. The deviationsbetween both descriptions of CF3Br are minute. For T =

0, there is no difference between both because only therotational ground state with J = 0 is populated. Asthe interaction with the laser does not mix states withdifferent K quantum numbers,7 no difference may arisewhen a laser is turned on. Also for higher T , the deviationremain tiny.

In Fig. 5, we investigate the influence of the laser in-tensity on the maximally attainable alignment. We takethe maximum of

cos2 ϑ

(t) over the propagation time

interval for a number of temperatures and laser inten-

sities. The computational parameters are chosen suchthat we are in the adiabatic regime. For all the curves,we have the following two limits: if the laser intensityis zero, there is no alignment, only a thermal ensembleand

cos2 ϑ

(t) = 1

3 . Conversely, if I L,0 → ∞, then

max{cos2 ϑ

(t)} → 1. Of course, the latter limit cannever be approached closely in practice because moleculesare ionized above a certain value for the intensity. With

increasing temperature, the curves in Fig. 5 change dra-matically. For 0 K, we find a strong nonlinear dependenceof max{cos2 ϑ

(t)} on the intensity. For 20 K, the de-

pendence is essentially linear. In analogy to the discus-sion of Fig. 4, we conclude that the strong temperaturedependence of the functional shape of max{cos2 ϑ

(t)}

provides a means to determine the rotational tempera-ture of a gas sample, if the laser intensity is accuratelyknown.

B. X-ray absorption

The expectation value

cos2 ϑ

(t) examined in theprevious Sec. IV A can be inferred experimentally viathe Coulomb explosion technique. However, due to thecomplicated nature of the interaction of a laser witha molecule, this experimental technique is a somewhatintricate way to study molecular alignment; e.g., re-cently, the strong-field ionization probability of alignedmolecules was measured and a nontrivial, molecule-specific angular dependence was found.10 Therefore, thistechnique is prone to systematic errors9 and it is highlydesirable to have an alternative method for the measure-ment of molecular alignment. We advocate resonant ab-sorption of polarized x rays as a novel route to study

molecular alignment.3,7,8

In Fig. 6, we study the four regimes of molecular align-ment from Fig. 3 using picosecond x-ray pulses (τ X =1 ps). The x-ray pulse duration was chosen much shorterthan the typical rotational time scale T RP. We inves-tigate the amount of information on the rotational dy-namics which is provided by the ratio R/⊥(τ ) [Eq. (7a)]because it is an experimentally accessible quantity whichhas been measured successfully before by Peterson et al.8

The dependence of R/⊥(τ ) [Eq. (7a)] on the laser pulseis shown for a large range of time delays τ between laserand x-ray pulses. With the picosecond pulses, the molec-ular dynamics is reproduced clearly in Fig. 6; even thedetailed features of Fig. 3 are resolved. This shows thatshort-pulse x-ray absorption is a promising novel avenueto study the details of molecular rotations.

In Fig. 7, we investigate the impact of the duration of the x-ray pulse on the absorption R/⊥(τ ) [Eq. (7a)]. Wefind that the curve in the top panel of Fig. 7 for τ X = 50fscan hardly be distinguished from the curve for R/⊥(τ ) inthe top panel of Fig. 6 for τ X = 1 ps. This indicates thatpicosecond x-ray pulses are sufficiently short to deliversnapshots without averaging noticeably over the rotationof the molecule. In other words, for CF3Br picosecond



8

T [K] 0 0.1 1 3 5 10 20

T th [ps] ∞ 167 99 65 52 39 28

TABLE II: Thermally averaged rotational periods T th [Eq. (19)] for a number of rotational temperatures T for the linear rotormodel of CF3Br. The periods were determined by using up to J = 1000 terms in the sum over J in Eqs. (19) and (20).

FIG. 6: (Color online) Study of the four regimes of the align-ment of CF3Br from Fig. 3 with a picosecond x-ray source(τ X = 1 ps). The dependence of R/⊥(τ ) [Eq. (7a)] on thetime delay τ between laser and x-ray pulses is shown.

x-ray pulses can be modeled as a δ(t−τ ) distribution (8)and one obtains r/⊥(τ ) [Eq. (9a)]. In the lower panel of Fig. 7, we use long x-ray pulses τ X = 100 ps, instead, todetermine R/⊥(τ ). We observe that the detailed infor-mation on the molecular dynamics revealed in the upperpanel is basically invisible in the lower panel. The longx-ray pulse averages out all fine details.

Comparing the details of the curves in Fig. 6 with thecorresponding curves from Fig. 3 reveals a vertical distor-tion. Consider, for instance, the bottom panel in Fig. 3.While the curve is basically flat around the peak, thecorresponding curve in Fig. 6 exhibits a clear peak. Thedifferences between the curves can be understood by ex-amining the relation between

cos2 ϑ

(t) and R/⊥(τ )

(we assume a δ-distribution x-ray pulse (8) and equate tand τ here). For this purposes, we carried out absorp-tion calculations for adiabatically aligned molecules for a

FIG. 7: (Color online) Impulsive alignment of CF3Br studiedby x-ray absorption. The setup is the one of the top panelin Figs. 3 and 6. The ratio R/⊥(τ ) [Eq. (7a)] is shown here.We use τ X = 50 fs (top) and τ X = 100 ps (bottom). Othercomputational parameters are T = 1 K and τ L = 50 fs.

FIG. 8: (Color online) The relation of ˙

cos2 ϑ¸

(t) to thecross correlation ratios R/⊥(τ ) (dashed red), R/th(τ ) (solidblack), and R⊥/th(τ ) (dash-dot blue) [Eq. (7)] for CF3Br. Thelong dashed (green) vertical line indicates the value of a ther-mal ensemble. Other computational parameters are T = 0K,τ L = 1ns, and τ X = 10 ps.

series of laser intensities. In Fig. 8, we plot R/⊥(τ ) ver-

sus cos2 ϑ (t). To this end, we use the maximum of thetwo quantities over the propagation interval of individualcomputations with varying laser intensity. The curve re-veals the nonlinear relationship between both quantities,see Eq. (16a). In transient alignment, e.g., the top panelof Fig. 3, there is also a suppression of alignment with re-spect to the signal of a thermal ensemble due to antialign-ment (i.e., alignment perpendicular to the laser polariza-tion axis). We can easily add this part of the curve toFig. 8 by realizing that at

cos2 ϑ

(t) = 0, the molecules

are perfectly antialigned. According to Sec. IIA, there is



9

FIG. 9: (Color online) Dependence of the x-ray absorptionsignal max{R/⊥(τ )} [Eq. (7a)] of CF3Br on the rotationaltemperature of the molecules. The solid (black) curve isfor τ L = 95ps and τ X = 100 ps whereas the dashed (red)curve is for τ L = 1ns and τ X = 10 ps. For both curves, thepeak laser intensity is I L,0 = 1012 W

cm2 .

no absorption of x rays with a polarization parallel to thelaser polarization axis. We interpolate linearly betweenthis point and the value unity for a thermal ensemble.

In addition to a curve for R/⊥(τ ), we also show re-sults for R/th(τ ) and R⊥/th(τ ) in Fig. 8. These two

mappings of x-ray absorption ratios to

cos2 ϑ

(t) arelinear, see Eqs. (16b) and (16c). They are opposite toeach other: where the one is maximal, the other is min-imal. Yet the maximum of R⊥/th(τ ) is only half themaximum of R/th(τ ). This is due to the fact that thereare two axes perpendicular to the laser polarization axisbut only one parallel to it. The curves were extendedinto the range from

cos2 ϑ

(t) = 0 to the value of a

thermal ensemble

cos2 ϑ

(t) = 13 in the same way as

for R/⊥(τ ). For R/th(τ ), we have the two values 0and 1, respectively. The curve for R⊥/th(τ ) certainly as-sumes the value unity for a thermal ensemble as well.However, in

cos2 ϑ

(t) = 0, we used the maximum

value of R/th(τ ) divided by 2. The signature of molec-ular alignment from the ratios R⊥/th(τ ) and R/th(τ )is weaker than from R/⊥(τ ). The latter ratio is thuspreferable in experiments.

In Fig. 9, we investigate the dependence of the x-rayabsorption signal R/⊥(τ ) [Eq. (7a)] on the rotationaltemperature. To this end, we computed the absorptionof adiabatically aligned molecules for a series of tempera-tures. The high sensitivity of the molecular dynamics onthe temperature revealed in Fig. 9 parallels the one foundin Fig. 4. Our prediction stresses once more how impor-tant a low temperature is to observe a substantial signal.Conversely, the maximum of the x-ray absorption ratiocan again be used to determine the rotational tempera-ture of a gas sample. For this purpose, the precise laserand x-ray pulse parameters need to be known to tailor a

FIG. 10: (Color online) Exemplary one-dimensional controlof x-ray absorption by laser-aligned CF3Br molecules. In theupper panel, we show the induced molecular dynamics subjectto the sequence of 50 laser pulses with intensity I L(t) whichis shown in the lower panel. The individual pulses have aFWHM duration of τ L = 1 ps and are spaced by 5.2ps. Thecurves in the upper panel represent instantaneous ratios of cross sections: r/th(t) [Eq. (9b)] (dashed, red) and r⊥/th(t)[Eq. (9c)] (solid, black). The rotational temperature is 1 K.

figure like Fig. 9 for a given experimental situation.

C. Control of x-ray absorption

In the previous Sec. IVB, we used x-ray absorption tostudy the rotational molecular dynamics of a gas sam-ple. In this subsection, we focus on the inverse pro-cess: the control of x-ray absorption by molecular align-ment. Controlling x-ray pulse shapes represents a power-ful tool to control inner-shell electronic processes; it has

been theoretically established in terms of electromagneti-cally induced transparency (EIT) for x rays.3,16,41,42,43,44

There we use a shaped laser pulse to cut out two shortx-ray pulses from a long Gaussian x-ray pulse. Here,we would like to investigate similar avenues using laser-aligned molecules. The control of molecular rotation withappropriately shaped laser pulses has been investigated,e.g., in Ref. 45,46. The x-ray absorption cross section of the molecules depends on the alignment [Sec. IIA]. Weassume that the gas sample absorbs most x rays for ran-domly oriented molecules. Once the molecules are suffi-ciently antialigned, a large fraction of the incident x-rayflux is transmitted through the gas sample for parallellaser and x-ray polarization vectors.

Let us examine examples of controlled x-ray absorp-tion. In the upper panel Fig. 10, we show the ratios of cross sections r/th(t) [Eq. (9b)] and r⊥/th(t) [Eq. (9c)]of CF3Br for parallel and perpendicular x-ray and laserpolarizations, respectively. The molecules are subjectto the sequence of 50 laser pulses shown in the lowerpanel. As derived in Sec. II C, σ2,(t) [Eq. (13)] is propor-

tional to

cos2 ϑ

(t) and thus is well-suited to examinethe molecular motion. The sequence imprints an oscil-latory occurrence and vanishing of enhanced alignment



10

and antialignment. This rotational motion is enforcedby the laser and only present during the pulse sequence.Afterwards, t > 260 ps, we see—by comparison with theupper panel in Fig. 3 (keep in mind that

cos2 ϑ

(t) is

plotted there)—that the free time-evolution after a singlepulse is approximately assumed. The ratios r/th(t) andr⊥/th(t) are complementary: the former peaks where thelatter is minimal.

Likewise, the ratios of cross sections corresponding tothe physical situation in the top panel of Fig. 3 can beinvestigated where a single, short laser pulse creates arotational wavepacket. About every 120 ps, the moleculeundergoes short time intervals of transient antialignment.The brief periods of antialignment then lead to brief pe-riods of transparency of the sample for x rays, i.e., x-raybursts of a few picoseconds may be cut out of a longer x-ray pulse. We have constructed a picosecond clock whichprovides us with short x-ray bursts. For very long x-raypulses, several bursts with a well-defined recurrence timecan be generated.

Control of x-ray absorption with laser-aligned

molecules has advantages and disadvantages com-pared with the competing technique of EIT forx rays.3,16,41,42,43,44 An advantage of molecular alignmentover EIT is that by the choice of the molecule, x-ray ab-sorption can be controlled for a large number of wave-lengths, i.e., by replacing CF3Br with other molecules,one can shift the absorption edge over a wide range of energies. Additionally, other edges, L, M , N , O, P , Q,can be used alternatively without any extra effort. Thisadds an additional degree of freedom of available x-raywavelengths. In the case of EIT for x rays, on the otherhand, the number of suitable atoms seems to be quitelimited by the requirements to sustain strong laser fieldsand the availability of intense laser systems with the rightwavelength to couple two empty Rydberg orbitals. Alsowith molecular alignment, absorption can be controlledfor hard x rays where K -shell core-hole decay widths arelarge and EIT for x rays is suppressed.3,16,41,42,43,44

Let us estimate the required length of CF3Br gas wherethe laser and the x rays need to overlap for the pro-posed x-ray pulse shaping to work. We approximate thecross section of CF3Br on the pre-edge resonance by the1s → 5 p cross section for krypton atoms ∼ 17 kb.41 Letthe gas jet have a number density of 5 × 1014 cm−3 as ithad in the experiment of Ref. 8. From Beer’s law,26 weestimate that the overlap region needs to be 1 km longfor the x-ray flux to drop by a factor of 1/e. Clearly, thisis an absurd requirement in practice. A number densityof 7 × 1019 cm−3 would be required to reduce the lengthof the overlap region to 0.8 cm. Yet the number densityin the gas jet cannot be increased indefinitely becausethe molecules start to interact strongly with each otherforming clusters and, eventually, a condensed phase. Thex-ray cross section assumed in this estimate holds for per-fectly aligned molecules parallel to the x-ray polarizationdirection. With laser-alignment, this cross section canbe reduced to zero for perfectly antialigned molecules.

Note that the cross section is about a hundred timeslarger for neon atoms than for krypton atoms.42 The es-timate with respect to the fluorine pre-edge resonance isthus far more favorable than for bromine. On the downside of molecular alignment over EIT is that the timescales of the former is about three orders of magnitudeslower (picoseconds) than the latter (femtoseconds). Themolecules need to have a low rotational temperatures,

too; it is not yet clear whether this is physically possiblebecause interactions between molecules have a consider-able damping influence on molecular rotations in densergases and condensed matter.46 The latter points seemto represent a big, if not insurmountable, obstacle forcontrol of x-ray absorption by laser-manipulated molec-ular rotation. However, the ideas presented in this sub-section might nevertheless prove fruitful in conjunctionwith recent work on the laser-alignment of molecules insolution.47

V. CONCLUSION

This study is devoted to laser-induced rotationaldynamics of bromotrifluoromethane (CF3Br) moleculesprobed by x rays. First, we investigated the molecularrotations in terms of the expectation value

cos2 ϑ

(t).

Second, we investigate the properties of a probe of molec-ular rotations by x rays. This represents an alternativeto the established Coulomb-explosion technique.1 Third,we explore possibilities to employ laser manipulation of molecular rotation to control x-ray pulse shapes, i.e., toimprint a certain pulse shape onto a longer x-ray pulse.

We start by investigating

cos2 ϑ

(t). Specifically,we examine its temporal evolution for varying laser-

pulse durations, its temperature dependence, and itsdependence on the peak laser intensity. Then, we re-visit these physical situations with short-pulse x rays.Additionally, we explore the dependence on the x-raypulse duration and the relation between the maximumof x-ray absorption and the maximum of

cos2 ϑ

(t).

We show that

cos2 ϑ

(t) can be directly measured byshort-pulse x-ray absorption measurements. Such shortpulses will be produced by the emerging ultrafast x-raysources.11,12,13,14 The method permits to stroboscopi-cally photograph the steps in the time-evolution of ro-tational wavepackets. We can learn about the beamsand the experimental conditions; x-ray absorption is anin situ probe.

Our investigations are based on our recent theory7 forthe laser alignment of symmetric-top molecules probedby x rays. To carry out the computations in this paper,we devise a parameter-free two-level model for the elec-tronic structure of CF3Br; it models the Br 1s → σ∗ pre-edge resonance. The x-ray absorption cross section for-mula from Ref. 7 is reduced to this model. We employcoupled-cluster linear-response techniques to determinethe average dynamic dipole polarizability and the dy-namic dipole polarizability anisotropy for CF3Br in the



12

presence of spin-orbit coupling is

| ϕ2, ms = |ϕ2, ms

+

4i=1i=2

1

2m′

s=− 1

2

ϕi, m′s | V SO |ϕ2, ms

ε2 − εi

× |ϕi, m′s .

(A.3)The notation | ϕ2, ms shall not imply that ms is a goodquantum number; it merely indicates that | ϕ2, ms be-comes |ϕ2, ms in the limit of vanishing spin-orbit inter-action.

Spin-orbit coupling is associated with the atomic po-tential near the bromine nucleus. We therefore assumethat the sp3 hybrid orbital on carbon is unaffected byspin-orbit coupling and set

χ1, ms | V SO |χi, m′s = 0 , (A.4)

with i = 1, 2, 3, 4 and ms, m′s = −1

2 , 12 . However,

V SO causes mixing between the Br 4 p orbitals

|χ2, ms

,

|χ3, ms , and |χ4, ms . For an atom, V SO is proportional

to25,50,51 l · s, where l and s are the orbital and spin an-gular momentum operators. Thus, V SO is diagonal withrespect to the spin-orbit coupled states25,50

|4 pj, m =1

ml=−1

1

2ms=−

1

2

C (1, 1/2, j; ml, ms, m)

× |4 pml ⊗ |ms .

(A.5)In this expression, C (1, 1/2, j; ml, ms, m) is a Clebsch-Gordan coefficient,25 j = 1/2 or 3/2, and m = − j , . . . , j.Upon inverting Eq. (A.5),25 we obtain for the spin-orbit

coupling matrix elements in the atomic basis

4 pml| ⊗ ms | V SO |4 pm′

l ⊗ |m′

s

= δml+ms,m′

l+m′

s

3

2j= 1

2

C (1, 1/2, j; ml, ms, ml + ms)

× C (1, 1/2, j; m′l, m′

s, ml + ms) ∆E 4 p,j,(A.6)

where51

∆E 4 p,j =∆E SO4 p

3×

1 , j = 3/2−2 , j = 1/2

, (A.7)

and ∆E SO4 p is the fine-structure splitting in the va-lence shell of atomic bromine. Expression (A.6)—in combination with Eqs. (A.1), (A.2), and (A.4)—

justifies that terms involving matrix elements of the formϕ2, ms | V SO |ϕ2, m′

s for ms = m′s were excluded from

Eq. (A.3).

Hence, upon collecting results, the spin-orbit coupledσ∗ orbital may be written in a transparent form:

| ϕ2, ±1/2 = |ϕ2, ±1/2 + c2,2

√2

3

∆E SO4 pE gap

|ϕ3, ∓1/2 ,

(A.8)

We have introduced E gap = ε3 − ε2 using Eqs. (A.1)and (A.2) to denote the HOMO-LUMO gap. Employ-ing standard angular momentum algebra24 and exploit-ing the fact that the Br 1s orbital |1s, ms has practicallyno overlap with the sp3 hybrid orbital on carbon, the x-ray absorption probability at the Br 1s → σ∗ resonanceis proportional to

1

2ms,m′

s=− 1

2

ϕ2, ms | d · eX |1s, m′s 2

=2

3|c2,2|2 | 4 p ||d||1s |2

×e2X,c + (∆E SO4 p )

2

9 E 2gap[e2X,a + e2X,b]

.

(A.9)

Here, d is the electric dipole operator, eX is the x-raypolarization vector, 4 p ||d||1s is a reduced matrix ele-ment,24 and eX,a, eX,b, and eX,c are the Cartesian com-ponents of eX in the body-fixed frame, assuming that themolecular symmetry axis is the c axis.

X-ray polarization parallel to the CF3Br symmetryaxis thus corresponds to e2X,a + e2X,b = 0 and e2X,c = 1;x-ray polarization perpendicular to the CF3Br symme-try axis corresponds to e2X,a + e2X,b = 1 and e2X,c = 0. It

follows from Eq. (A.9) that, assuming perfect alignmentof the molecule relative to the x-ray polarization axis,the ratio between x-ray absorption in the parallel con-figuration () and x-ray absorption in the perpendicularconfiguration (⊥) is given by

f /⊥ =9 E 2gap

(∆E SO4 p )2. (A.10)

The spin-orbit splitting ∆E SO4 p in a bromine atom

is 0.46 eV.52 Using dalton20 and the aug-cc-pVTZ ba-

sis set,21,22 we find that the HOMO-LUMO gap E gapin CF3Br is 14 eV. We may therefore conclude fromEq. (A.10) that for perfect alignment, f /⊥ ≈ 8000. Thisupper limit is much higher than any of the values for f /⊥calculated in Sec. IV. In comparison to the restrictionsimposed on f /⊥ by laser-induced alignment, the impactof spin-orbit coupling is negligible.

∗ The following article appeared in J. Chem. Phys. 129, 134312 (2008) (12 pages) and may be found at



13

dx.doi.org/10.1063/1.2987365. Copyright 2008 AmericanInstitute of Physics. This article may be downloaded forpersonal use only. Any other use requires prior permissionof the author and the American Institute of Physics.

† Present address: Department of Physics and Astronomy,Louisiana State University, Baton Rouge, Louisiana 70803,USA; Electronic address: [email protected]

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20

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14

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http://physics.nist.gov/asd3

http://physics.nist.gov/asd3




http://arxiv.org/abs/physics/0611122

http://www.cse.anl.gov/Fundamental_Interactions/FELLA_main.shtml

christian buth and robin santra- rotational molecular dynamics of laser-manipulated...

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