Alignment and orientation of absorbed dipole molecules
Y. Y. Liao, Y. N. Chen, and D. S. Chuu*
Department of Electrophysics, National Chiao-Tung University, Hsinchu 300, Taiwan
(Received 18 April 2004; revised manuscript received 27 July 2004; published 29 December 2004) Half-cycle laser pulse is applied on an absorbed molecule to investigate its alignment and orientation behavior. Crossover from field-free to hindered rotation motion is observed by varying the angel of hindrance of potential well. At small hindered angle, both alignment and orientation show sinusoidal-like behavior because of the suppression of higher excited states. However, mean orientation decreases monotonically as the hindered angle is increased, while mean alignment displays a minimum point at certain hindered angle. The reason is attributed to the symmetry of wave function and can be explained well by analyzing the coefficients of eigenstates.
DOI: 10.1103/PhysRevB.70.233410 PACS number(s): 73.20.Hb, 33.55.Be, 33.20.Sn
Alignment and orientation of molecules are important in the investigations of stereodynamics,1surface catalysis,2
mo-lecular focusing,3 and nanoscale design.4 The alignment
scheme has been demonstrated both in adiabatic and nona-diabatic regimes. A strong laser pulse can anona-diabatically create pendular states, and the molecular axis is aligned in parallel to the direction of field polarization. The molecule goes back to its initial condition after the laser pulse is switched off, and the alignment can no longer be observed again.5 To
achieve adiabatic alignment, the duration of laser pulse must be longer than the rotational period. However, an ultrashort laser pulse with several cycles is also observed to induce a field-free alignment providing the duration of laser pulse is smaller than the rotational period. In this limit, the alignment occurs periodically in time as long as the coherence of the process is preserved.6On the other hand, a femtosecond laser
pulse is found to be able to generate field-free orientations.7
The dipole molecule, kicked by an impulsive pulse, will tend to orient in the direction of laser polarization. Laser-indued molecular orientations have been demonstrated in several experiments.8–10
Recently, the rotational motion of a molecule interacting with a solid surface has attracted increasing interest. It is known that molecules can be desorbed by applying UV laser beam along the surface direction, and the quadrupole is a measure of the rotational alignment.11 To understand molecular-surface interaction, Gadzuk and his co-workers12
proposed an infinite-conical-well model, in which the ad-sorbed molecule is only allowed to rotate within the well region. Shih et al. further proposed a finite-conical-well model to generalize the study of a finite hindrance.13 Their results showed that the rotational states of an adsorbed dipole molecule in an external electric field exhibit interesting be-haviors, and theoretical calculation of the quadrupole mo-ment based on finite-conical-well model is in agreemo-ment with the experimental data.14 These findings may be very useful
for understanding the surface reaction.
In the present work we present a detailed investigation on the rotational motion of a polar diatomic molecule, which is confined by a hindering conical well. Different well-dependent signatures between the alignment and orientation of the hindered molecule under an ultrashort laser pulse are pointed out for the first time. Crossover from field-free to
hindered rotation is also observed by varying the hindered angle of potential well. These make our results promising and may be useful in understanding the molecule-surface in-teractions.
Consider now a laser pulse polarizing in the z direction interacts with the hindered molecule as shown in Fig. 1. The model Hamiltonian can be written as
H =ប
2
2IJ
2+ V
hin共,兲 + HI, 共1兲 where J2 and I are angular momentum and momentum of
inertia of the molecule. The rotational constant B is set equal toប2/ 2I. It is reasonable to assume that the surface potential
Vhin共,兲 is independent of since previous calculations have shown that its dependence onis weaker than that on
.15–17 Therefore, in the vertical absorbed configuration, the
surface potential can be written as12
Vhin共兲 =
再
0, 0艋艋␣
⬁, ␣⬍艋,
冎
共2兲where␣is the hindered angle of the conical well. In Eq.(1),
HI describes the interaction between the dipole moments (permanent and induced) and laser field:
HI= −E共t兲cos−
1 2E
2共t兲关共␣储−␣
⬜兲cos2+␣⬜兴, 共3兲
where is the dipole moment. The components of the po-larizability␣储 and␣⬜are parallel and perpendicular to the
FIG. 1. Schematic view of the hindered rotor. PHYSICAL REVIEW B 70, 233410(2004)
molecular axis, respectively. The laser field in our consider-ation is a Gaussian shape: E共t兲=Ee−共t − to兲2/2cos共t兲, where
E is the field strength andis the laser frequency. To solve the time-dependent Schrödinger equation, the wave function is expressed in terms of a series of eignfunctions as
⌿l,m=
兺
cl,m共t兲l,m共,兲, 共4兲 where cl,m共t兲 are time-dependent coefficients corresponding to the quantum numbers 共l,m兲. For the vertical adsorbed configuration, the wave function can be written asl,m共,兲 =
冦
Al,mP
l,m
兩m兩共cos兲exp共im兲
冑
2 , 0艋艋␣0, ␣⬍艋,
冧
共5兲 where Al,mis the normalization constant and Pv
l,m
兩m兩 is the
as-sociated Legendre function of arbitrary order. In the above equations, the molecular rotational energy can be expressed as
⑀l,m=l,m共l,m+ 1兲B. 共6兲 In order to determine l,m, one has to match the boundary condition
P
l,m
兩m兩共cos␣兲 = 0. 共7兲
After determining the coefficients cl,m共t兲, the orientation 具cos典 and alignment 具cos2典 can be carried out
immedi-ately.
We choose ICl as our model molecule, with dipole mo-ment= 1.24 Debye, rotational constant B = 0.114 cm−1,
po-larizability components␣储⬇18 Å3and␣
⬜⬇9 Å3. The peak
intensity and frequency of laser pulse are about 5 ⫻1011W / cm2 and 210 cm−1, respectively. For simplicity
(zero-temperature case), the rotor is assumed in ground state initially, i.e., c0,0共t=0兲=1. Besides, in order to keep the
simulations promising, the highest quantum number for nu-merical calculations is l = 15, such that the results are conver-gent and the precision is to the order of 10−7.
The solid lines in the insets of Fig. 2 show the dependence of the alignment on hindered angle ␣. For ␣= 60°, sinusoidal-like behavior is presented, and the alignment ranges from 0.63 to 0.91. As the hindered angle increases, the curves become more and more complicated and gradu-ally approach the free rotor limit as shown in the insets of Fig. 2(b)共␣= 120°兲 and 2(c) 共␣= 180°兲. This can be under-stood well by studying the populations 兩cl,m兩2 of low-lying states. In the regime of the small hindered angle, there is little chance for electrons to populate in higher excited states since the shrinking of the conical-well angle causes the in-creasing of energy spacings.
One also notes that the populations of a hindered mol-ecule for␣= 60° and 120°, shown in Figs. 2(a) and 2(b), are mainly composed of l = 0, 1 and 2 states, while the popula-tion of a free rotor is composed of l = 0, 2, and 4 states. The underlying physics comes from the reason that 具l⬘,m⬘兩cos2兩l,m典 is nonzero for all l and l
⬘
values in thecase of hindered rotation. But it is zero in free rotor limit except for l = l
⬘
or l = l⬘
± 2. The dotted lines in the insets represent the first two main contributions of the factors 兺l⫽l⬘具l⬘,m⬘兩cos2兩l,m典 summed from low-lying states, i.e., the sum of the largest two values of the off-diagonal term 具l⬘,m⬘兩cos2兩l,m典. As can be seen, the populations for the small hindered angle are mainly distributed on lower states since the main oscillation feature(e.g., the frequency) of the curve(dotted lines) is quite similar to that from whole con-tributions(solid lines).Let us now turn our attention to the case of orientation. After applying a short pulse laser, the orientation具cos典 of a hindered molecule共␣= 60°兲 oscillates sinusoidally with time as shown in Fig. 3(a). The value of具cos典 is always positive because the rotational wave function is compressed heavily. As the hindered angle␣ becomes larger, the oscillation fre-FIG. 2. The populations of the states共l,m=0兲 for different hin-dered angles:(a)␣=60°, (b) ␣=120°, (c) ␣=180°. The insets show
the corresponding alignments(solid lines) and the first two main contributions of the factors兺l⫽l⬘具l⬘,m⬘兩cos2兩
l,m典 (dotted lines).
FIG. 3. The orientations具cos典 (solid lines) of a hindered mol-ecule confined by an infinite conical-well for different hindered angles:(a)␣=60°, (b) ␣=120°, and (c) ␣=175°. The dashed and dotted lines in(c) correspond to different potential barrier height, i.e., V0=⬁ and 100, respectively.
BRIEF REPORTS PHYSICAL REVIEW B 70, 233410(2004)
quency also decreases as shown in Fig. 3(b). These signa-tures are quite close to that of the alignment. We then con-clude that even at larger hindered angle共␣= 120°兲 the role of hindered potential still overwhelms the laser pulse, other-wise, the value of具cos典 should not always be positive.
Figure 3(c) represents results of orientations in infinite 共V0=⬁兲 or finite 共V0= 100兲 conical-well potential for ␣
= 175°. Dashed and dotted lines correspond to V0=⬁ and
100, respectively. For the case of finite conical-well poten-tial, the wave function is expressed in terms of a series of the basis wave functions obtained in Refs. 13 and 14. As can be seen, the effect of the laser pulse is obvious because a nega-tive value appears. Comparing the results with the free orientation,7 the angular distributions for the finite well are
more isotropic since the wave functions can penetrate into the conical barrier.
Further analysis shows that components of orientation 具cos典 or alignment 具cos2典 can be divided into two parts:
diagonal and nondiagonal terms. The nondiagonal term rep-resents the variations of these curves such as those in the insets of Fig. 2. These variations with time are determined by the phase difference coming from various energy levels. To see the contributions from diagonal terms, we evaluate the time-averaged orientation and alignment. In this case, the nondiagonal values will be averaged out, and only contribu-tions from diagonal terms exit. Figure 4 shows the mean orientation and alignment as a function of hindered angle. As
␣ increases, the mean orientation decreases monotonically from 1 to 0. This is because the mean orientation is deter-mined by 兩cl,m兩2 and 具
l,m兩cos兩l,m典. For a larger angle ␣, the populations兩cl,m兩2is mainly composed of l = 0 , 2 , 4 states. But the value 具l,m兩cos兩l,m典 is governed by the selection rule: l = l
⬘
+ 1. Thus the net effect is the shrinking of the mean orientation in the large angle limit.Contrary to orientation, the mean alignment shows a quite different feature. The value of 具cos2典 first decreases as ␣ increases. However, it reaches a minimum point about ␣ = 140°. From the insets of Fig. 4, we know that the values of 共具l,m兩cos2兩l,m典兲 do not depend significantly on ␣. There-fore, the decrease of具cos2典 comes from the decreasing ten-dency of the population兩cl=1,m兩2, while its increasing behav-ior is caused by two other populations兩cl=0,m兩2and兩cl=2,m兩2. Competition between these two effects results in a minimum point.
A few remarks about experimental verifications should be mentioned here. The degree of alignment can be measured
with the techniques of the femtosecond photodissociation spectroscopy and the ion imaging.8The alignment is probed
by breaking the molecular bond and subsequently measuring the direction of the photofragments by a mass selective po-sition sensitive ion detector. In contrast to alignment, the orientation is probed by Coulomb exploding the molecules with a femtosecond laser pulse.9By detecting the fragment
ions with the time-of-flight mass spectrometer, a significant asymmetry should be observed in the signal magnitudes of the forward and the backward fragments. Under proper ar-rangements, orientation and alignment of an absorbed mol-ecule may be examined by these spectroscopic technologies. In conclusion, we have shown that a short laser pulse can induce alignment and orientation of a hindered molecule. The hindered angle of the hindered potential well plays a key role on the molecular alignment and orientation. Crossover from field-free rotation to a hindered one can be observed by varying the hindered angle of the potential well. Time-averaged alignment and orientation are investigated thor-oughly to understand the difference between these two quan-tities.
This work is supported partially by the National Science Council, Taiwan under Grant No. NSC 92-2120-M-009-010.
*Corresponding author. Email address: dschuu@mail.nctu.edu.tw
1Special issue on Stereodynamics of Chemical Reaction[J. Phys.
Chem. A 101, 7461(1997)].
2V. A. Cho and R. B. Bernstein, J. Phys. Chem. 95, 8129(1991). 3H. Stapelfeldt, H. Sakai, E. Constant, and P. B. Corkum, Phys.
Rev. Lett. 79, 2787(1997).
4T. Seideman, Phys. Rev. A 56, R17(1997); R. J. Gordon, L. Zhu,
W. A. Schroeder, and T. Seideman, J. Appl. Phys. 94, 669
(2003).
5B. Friedrich and D. Herschbach, Phys. Rev. Lett. 74, 4623
(1995); L. Cai, J. Marango, and B. Friedrich, Phys. Rev. Lett.
86, 775(2001).
6J. Ortigoso, M. Rodriguez, M. Gupta, and B. Friedrich, J. Chem.
Phys. 110, 3870 (1999); M. Machholm, ibid. 115, 10724
(2001).
7M. Machholm and N. E. Henriksen, Phys. Rev. Lett. 87, 193001
FIG. 4. The mean orientation 具cos典mean and alignment
具cos2典
meanin an infinite conical well. The insets show the
popula-tions 兩cl,m兩2 (solid bar) and factors 具l,m兩cos2兩l,m典 (striped bar).
Insets(a) and (b) correspond to␣=110° and ␣=170°, respectively.
BRIEF REPORTS PHYSICAL REVIEW B 70, 233410(2004)
(2001).
8W. Kim and P. M. Felker, J. Chem. Phys. 104, 1147(1996); 108,
6763 (1998); H. Sakai et al., ibid. 110, 10235 (1999); J. J. Larsen et al., ibid. 111, 7774(1999).
9H. Sakai, S. Minemoto, H. Nanjo, H. Tanji, and T. Suzuki, Phys.
Rev. Lett. 90, 083001(2003); S. Minemoto, H. Nanjo, H. Tanji, T. Suzuki, and H. Sakai, J. Chem. Phys. 118, 4052(2003).
10V. Renard, M. Renard, S. Guérin, Y. T. Pashayan, B. Lavorel, O.
Faucher, and H. R. Jauslin, Phys. Rev. Lett. 90, 153601(2003).
11I. Beauport, K. Al-Shamery, and H.-J. Freund, Chem. Phys. Lett.
256, 641(1996); S. Thiel, M. Pykavy, T. Klüner, H.-J. Freund, R. Kosloff, and V. Staemmler, Phys. Rev. Lett. 87, 077601
(2001); J. Chem. Phys. 116, 762 (2002).
12J. W. Gadzuk, U. Landman, E. J. Kuster, C. L. Cleveland, and R.
N. Barnett, Phys. Rev. Lett. 49, 426(1982).
13Y. T. Shih, D. S. Chuu, and W. N. Mei, Phys. Rev. B 51, 14 626
(1995); 54, 10 938 (1996).
14Y. T. Shih, Y. Y. Liao, and D. S. Chuu, Phys. Rev. B 68, 075402
(2003).
15R. P. Pan, R. D. Etters, K. Kobashi, and V. Chandrasekharan, J.
Chem. Phys. 77, 1035(1982).
16J. W. Riehl and C. J. Fisher, J. Chem. Phys. 59, 4336(1973). 17V. M. Allen and P. D. Pacey, Surf. Sci. 177, 36(1986).
BRIEF REPORTS PHYSICAL REVIEW B 70, 233410(2004)