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J. Phys. Chem. B 2004, 108, 4916-4921
Chiral Molecular Motors Ignited by Femtosecond Pump-Dump Laser Pulses Kunihito Hoki,† Makoto Sato,† Masahiro Yamaki,† Riadh Sahnoun,† Leticia Gonza´ lez,‡ Shiro Koseki,§ and Yuichi Fujimura*,† Department of Chemistry, Graduate School of Science, Tohoku UniVersity, Sendai, 980-8578, Japan, Institut fu¨r Physikalische und Theoretische Chemie, Freie UniVersita¨t Berlin, D-14195, Germany, and Department of Material Science, College of Integrated Arts and Sciences, Osaka Prefecture UniVersity, Sakai, Osaka 599-8531, Japan ReceiVed: August 14, 2003; In Final Form: February 1, 2004
The results of a theoretical study on a chiral molecular motor ignited by a femtosecond pump-dump laser excitation are presented. The rotational direction of the motor is determined by the gradient of the potential energy surface (PES) of the electronic excited state in the Franck-Condon region. The pump-dump ignition method is applied to (R)-2-methyl-cyclopenta-2,4-dienecarbaldehyde, which is one of the simplest chiral molecular motors. The aldehyde group is the engine of the chiral molecular motor. The magnitudes of the angular momentum of the rotational motion are quantum-mechanically evaluated. The motor dynamics is analyzed in terms of the rotational wave packet propagation on the ground-state PES. A time-frequencyresolved photoionization method for observing the motor dynamics in real time is also described.
I. Introduction Much interest has been shown in recent years in the fabrication of molecular motors.1-13 Such a microscopic dynamical system possessing a unidirectional motion driven by external energy is a fundamental part of nanomachines. As an external driving source, chemical, thermal, and/or photon energies have been considered. Lasers are expected to become an important external energy source because recent advances in laser technology have made it possible to manipulate molecular motions by controlling laser parameters such as pulse duration or frequency.14-16 In our recent papers,17,18 we have proposed a chiral molecular motor driven by linearly polarized laser light. Its mechanism has been clarified using both classical and quantum mechanics. An internal rotational motion in the electronic ground state is rocked by infrared (IR) laser pulses which drive the motor in a picosecond time regime. The essential point in driving a molecular motor in the ground state is that its rotation originates from its molecular chirality. Moreover, the unidirectional motion depends on the slope of the groundstate potential, that is, the direction of the rotational motion is toward the gentle slope of the potential. From the viewpoint of the existence of other relevant degrees of freedom of molecular motors, such as whole molecular rotations, it is desirable to ignite molecular motors within ultrashort time regimes to avoid such competing dynamical effects. In ordinary cases,19,20 the periods of the whole molecular rotations range from nanoseconds to hundreds of picoseconds. In the case in which femtosecond laser pulses are applied, the effects of molecular rotations during the ignition of the molecular motor can be considered negligible. In this paper, we propose a new scheme for the ignition of molecular motors using pump and dump visible (vis) or * Author to whom correspondence should be addressed. E-mail: fujimura@ mcl.chem.tohoku.ac.jp. † Tohoku University. ‡ Freie Universita ¨ t Berlin. § Osaka Prefecture University.
ultraviolet (UV) laser pulses in a femtosecond time regime. The use of pump and dump pulses is well-known in the field of quantum control of chemical reactions.21 To demonstrate the effectiveness of the pump-dump laser ignition method, we adopted a simplified model of molecular motors, namely, the chiral molecule (R)-2-methyl-cyclopenta-2,4-dienecarbaldehyde. The unidirectional motion of the motor is determined by the gradient of the potential energy surface (PES) in the electronic excited state in the Franck-Condon region. The motor dynamics was analyzed in terms of rotational wave packets. We also propose an experimental method for probing the motor dynamics in real time, based on time-frequency-resolved photoionization. In the next section, we first present an outline of the pumpdump laser ignition method. Here, the time evolution of the rotational wave packets created by the pump and dump laser pulses is treated in the Hilbert space within a two-electronicstate model. An expression of the time evolution of the rotational wave packet is derived in the frame of the Born-Oppenheimer approximation, in order to analyze dynamical effects such as dephasing or rephasing which make an essential contribution to the rotational dynamics of the motor. We also present a method for detecting the motor dynamics which is based on time-frequency-resolved photoionization. In section III, an application of the pump-dump laser ignition method to (R)2-methyl-cyclopenta-2,4-dienecarbaldehyde is described, together with the results of analyzing the motor dynamics. Finally, the summary is presented in section IV. II. Chiral Molecular Motor Driven by Femtosecond Pulses II.A. Pump-Dump Laser Ignition Method. Consider an ideal chiral molecular motor consisting of two rigid groups, A and B, as shown in Figure 1. A and B can rotate around an axis to which they are both connected. The coordinate of the internal rotation is defined by R. In this paper, the pair of chiral molecular motors will be called the (S) and (R) motors. To drive the internal rotation of the motor, we aim at manipulating wave packets by the use of linearly polarized pump
10.1021/jp036437l CCC: $27.50 © 2004 American Chemical Society Published on Web 03/23/2004
Chiral Molecular Motors
J. Phys. Chem. B, Vol. 108, No. 15, 2004 4917 chirality of the molecular motor, and its gradient can be estimated using molecular orbital calculations. The time evolution of the molecular motor expanded in terms of the ground and excited electronic states (n ) 0 and n ) 1, respectively), is determined by the time-dependent Schro¨dinger equation:
ip
(
) [
](
hˆ 0 -µ01(R)E(t) φ0(R,t) ∂ φ0(R,t) ) -µ10(R)E(t) hˆ 1 φ1(R,t) ∂t φ1(R,t)
)
(1)
where µ01(R) is the transition dipole moment, E(t) is the electric field of the laser, φn(R,t) is the nuclear wave packet in the electronic state n, and hˆ n is the nuclear Hamiltonian. hˆ n can be written as
hˆ n ) -
p2 ∂ 2 + Vn(R) 2I ∂R2
(2)
where -(p2/2I) (∂2/∂R2) is the nuclear kinetic energy operator, I is the moment of inertia, and Vn(R) is the potential energy. The electric field of the linearly polarized pump laser with central angular frequency ωp and the linearly polarized dump laser with ωd is expressed as
E(t) ) fp(t) cos(ωpt) + fd(t) cos(ωdt)
(3)
where fj(t) (j ) p and d) denotes the envelope function of the pulse Figure 1. Ideal chiral (S) and (R) molecular motors (upper figure). A denotes the body of the motor, and B denotes the engine. R denotes the rotational coordinate of the motor. A schematic illustration of ignition of a unidirectional rotation using a pump-dump laser excitation method is shown in the lower figures. V0 and V1 are the potential energy in the electronic ground state and that in an electronically excited state, respectively. Solid lines with an arrow denote the pump (1) and dump (2) processes. Dashed lines trace the rotational wave packet motions after the pump and dump processes. (S) and (R) molecular motors rotate in opposite directions. The positive or negative direction, with respect to the coordinate R, depends on the molecular structure characteristic of chirality. Here, as an example, the stable configuration of an (R) motor in the electronic ground state is assumed to be located at a positive R.
and dump laser pulses. The control scheme is shown in Figure 1. A femtosecond vis (or UV) pump pulse denoted by (1) generates a rotational wave packet on the PES V1 in an electronic excited state. When the wave packet moves along the PES and reaches an appropriate position, it is transferred back to the PES V0 of the ground electronic state by applying a dump pulse (2). In the dumping process, the kinetic energy of the internal rotation is conserved before and after the dumping process as a result of the Franck-Condon principle. Therefore, the pumpdump laser excitation can ignite a unidirectional motion of the rotational wave packet because the wave packet has sufficient kinetic energy to pass over its transition state region on the ground-state PES. The direction of the rotational motion is determined by the gradient of the PES around its Franck-Condon region in the electronic excited state as shown in Figure 1. If the molecule has a symmetric plane as in the case of an achiral molecule, its gradient should be zero from symmetry considerations. Consequently, such an achiral molecule cannot produce unidirectional motion using a linearly polarized laser. As shown in cases 1 and 2 in Figure 1, on the other hand, from symmetry considerations, the gradient has opposite signs between the (R) and (S) motors. Its rotational direction is characteristic of the
[
fj(t) ) Aj sin2
]
π(t - tj) Tj
(4)
with Aj the pulse amplitude, Tj the pulse duration, and tj the time delay. Equation 1 is solved using the split-operator method22-24 and the fast Fourier transform method.25 In order to estimate the magnitude of rotational motion of the motor, the quantum-mechanical expectation value of the angular momentum for the wave packet Ψ(R,t) is defined as
∫-ππ dRΨ*(R,t)(-ip ∂R∂ )Ψ(R,t) l(t) ) ∫-ππ dR|Ψ(R,t)|2
(5)
II.B. Time-Evolution of Rotational Wave Packets and Angular Momentum. We now derive an expression for the time evolution of the rotational wave packets created by the pump-dump laser method in order to analyze the rotational dynamics. For this purpose, we consider a case in which the wave packets created rotate almost freely above the groundstate PES. In that case, we can derive an expression for the time evolution within a free-rotor approximation. The wave packet Ψ(R,t) can be expanded in terms of eigenfunctions of a free-rotor, 1/x2π exp(imR), with eigenvalue m ) p2m2/2I as
Ψ(R,t) )
1
x2π
mb
( ) i
∑ Cm exp(imR) exp - p mt m)m a
(6)
The coefficient Cm which represents the probability amplitude of the rotational quantum number m is determined at the time when the dump pulse is applied. Let m ) m0 be the quantum number of the maximum probability. In eq 6, the range of the summation, ma e m e mb, covers the energy distribution of the wave packet. We assume that the distribution of the angular
4918 J. Phys. Chem. B, Vol. 108, No. 15, 2004
Hoki et al.
momentum is localized, that is, mb - ma , m0. By rearranging the index of summation in eq 6 as m ) m0 + m′, we obtain
Ψ(R,t) )
1
x2π
[ (
exp im0 R -
)] ∑ [ ( m∆
pm0 2I
×
t
m′)-m∆
Dm′ exp im′ R -
pm0 I
t-
pm′ 2I
t
)]
(7)
where Dm′ ) Cm0 + m′ . By substituting eq 7 into eq 5 and taking into account that m∆ , m0, we obtain the following expression for the timeindependent expectation value of the angular momentum:
l = pm0
(8)
II.C. Detection of Rotational Wave Packets by a TimeFrequency-Resolved Photoionization Method. We now consider a simple method to detect motor dynamics using a timefrequency-resolved photoionization technique. The method involves the preparation of a rotational wave packet in an ionic state using a probe pulse with time delay t1. The analysis is based on the R-dependence of the ionization potential. The probing process was assumed to be independent of the abovementioned pump-dump process. Within the Born-Oppenheimer approximation in the photoionization process, the wave packet dynamics after the dump pulse is turned off at t ) t2 + T2/2 are described as
( ) [
]( )
hˆ 0 -µ02(R)E(t) φ0(R,t) ∂ φ0(R,t) ) ip ∂t φ2(R,t) -µ20(R)E(t) hˆ 2 φ2(R,t)
(9)
where hˆ 2 is the nuclear Hamiltonian of the ionized state, φ0(R,t) is the rotational wave packet in the electronic ground state, and φ2(R,t) is the rotational wave packet in the ionized state, in which φ2(R,t ) t2 + T2/2) ) 0 is satisfied as the initial condition. The electric field with the central frequency ωi for the ionization is expressed as E(t) ) fi(t) cos(ωit), in which fi(t) denotes an envelope function of the pulse. The time-frequency-resolved ionization spectrum as a function of ωi and its time delay t3 between the dump pulse and the ionization pulse is defined as
I(ωi,t3) )
∫-ππ dR|φ2(R,t ) t3 + T3/2)|2
(10)
where t3 + T3/2 is the time at which the ionized pulse with its duration T3 is turned off. III. Results and Discussion III.A. Electronic Structure of (R)-2-Methyl-cyclopenta2,4-dienecarbaldehyde. In order to theoretically examine the femtosecond pump-dump ignition method, we selected (R)-2methyl-cyclopenta-2,4-dienecarbaldehyde (see Figure 2a) as the model system for our investigation. This is one of the simplest molecules that fulfills the criteria of chiral molecular motors.17,18 From a practical point of view, complex molecules, in which the hydrogen numbered 4 (shown in Figure 2a) is substituted by a methyl group, can be considered as a chemically stable molecular motor. One of the reasons that prompted us to choose (R)-2-methyl-cyclopenta-2,4-dienecarbaldehyde is that its relatively small size allows the application of a high level of theory. The internal rotation of the dihedral angle O1-C2-C3-H4 is taken as the rotational angle R of the motor. For simplicity, we
Figure 2. (a) (R)-2-Methyl-cyclopenta-2,4-dienecarbaldehyde as a model system. The internal rotation of the dihedral angle O1-C2-C3H4 is taken as a rotational angle R of the motor. (b) Potential energy surfaces (PES) as a function of R. The solid line shows the PES of the electronic ground-state S0 and the dashed line shows that of the first electronic excited-state S1. The Franck-Condon region for a pump process is pointed to by an arrow. The broken-dashed line indicates the PES of the electronic ground state of its cation. (c) Transition dipole moment function between S0 and S1. The solid line shows the transition dipole moment along the C2-C3 bond.
assume that the molecule is aligned and that the polarization of the electric field vector is parallel to the C2-C3 bond. The full relaxed geometrical structure of the molecule was calculated at the MP2 (fc)/6-31+G(d,p) level of theory at each point of the chosen reaction coordinate R. Along the reaction path, the main change in the geometry takes place within the C-CHO group. The torsion of the CHO group has little effect on the five-ring scaffolding. The potential energy in the electronic ground-state S0 and that in the first excited singlet state S1 were calculated by means of multiconfiguration methods. We used the state-averaged complete active space SCF (SA-CASSCF)26 method with equal weights over both roots (S0 and S1) and the 6-31+G(d,p) basis set. The geometrical structures were calculated using the Gaussian 98 package of programs,27 and the SA-CASSCF calculations were performed using the GAMESS package.28 Figure 2b shows the PESs of the molecule. The solid line indicates the S0 PES, and the dashed line shows the S1 PES. Because the S1 PES slope in the Franck-Condon region (pointed to by an arrow) is negative, a wave packet created by a pump pulse would run with a positive angular momentum. The active space is composed of eight electrons correlated in seven orbitals. Within this active space, the electronic configuration of the ground state conforms to (πCdC)2(πCdO)2(πCdC)2(nO)2(π*CdO)0(π*CdC)0(π*CdC)0. The electronic configuration of the S1 state is mainly obtained by the excitation of one electron from nO to π*CdO. The electronic configuration in
Chiral Molecular Motors
J. Phys. Chem. B, Vol. 108, No. 15, 2004 4919
Figure 3. Upper figures: contour plots of the rotational wave packet on S0,|φg(R,t)|2, as a result of the pump-dump laser excitation. Panels a, b, and c show the wave packets at different time regimes. Lower figure: expectation values of the angular momentum. Time regimes indicated by a, b, and c correspond to those in the upper figure, respectively.
S1 reflects the reverse character in the potential minima and maxima compared with those in S0. This is because a considerable change in charge of the hydrogen of the CHO group upon excitation takes place from a negative to a positive one. It should be noted that the minimum energy position in the ground state is slightly shifted in a positive direction. This is reflected in the geometry of the molecule by a slight bend of the CHO group toward the methyl group (by about 7°). This can be explained by a weak interaction that takes place between the oxygen atom and the nearest hydrogen atom of the methyl group forming a weak O‚‚‚H bond. The transition dipole moment shown in Figure 2c was estimated using SA-CASSCF wave functions. Figure 2b also shows the potential energy function in the ionic ground-state D0. The active space is composed of seven electrons correlated in seven orbitals. The electronic configuration of the ground ionized state D0 was found to be that of the parent neutral molecule except that one electron was removed from the πCdC molecular orbital. We assumed that the moment of inertia I for the -CHO group is a constant, 17.6 amu A2, as estimated with the most stable configuration of the -CHO group. III.B. Driving a Motor by a Pump-Dump Laser Ignition Method. We now apply the pump-dump laser ignition method to drive a rotation of a -CHO group of (R)-2-methyl-cyclopenta-2,4-dienecarbaldehyde. To study the rotational dynamics, we solved the time-dependent Schro¨dinger equation by means of the split-operator method with the help of an FFT algorithm using 256 grid points for R; the motor-laser interaction, -µ02(R)E(t) was treated numerically using Pauli matrices. The upper panels a, b, and c in Figure 3 show the time evolution of the rotational wave packets created on S0 by applying the pump-dump laser ignition method. We omitted wave packets trapped in the S0 rotational potential well because they do not evolve after the dump pulse is turned off. Here the parameters of pulses used are Ap ) Ad ) 1010 V/m, Tp) T d ) 100 fs, tp ) 50 fs, td )180 fs, ωp ) 34 300 cm-1, and ωd ) 32 800 cm-1. The initial state is set on the lowest rotational level in the electronic ground state. It can be seen that, in the
early stage of the time evolution, the motor takes about 550 fs for one cycle, which corresponds to the angular frequency ω of 1.1 × 1013 s-1. From eq 8 and l = ωI, the expectation value of the angular momentum of the motor is estimated to be l ≈ 31p. An interesting feature appears in panel b where the rotational wave packet is split into two wave packets. These wave packets converge to a single one again after about 15 000 fs, as shown in panel c. This feature can be explained by eq 7 derived in section 2. It should be noted, however, that the rephasing time is given by 2πI/p because
Ψ(R,t ) 2πI/p) )
m∆
1
x2π
exp[im0(R - π)]
∑ m′)-m
× ∆
Dm′ exp[im′(R - π)]
) Ψ(R - π,t ) 0)
(11)
where exp(iπm′2) ) exp(iπm′) has been used. Moreover, the wave packet at half of the rephasing time, πI/ p, can be expressed as
Ψ(R,t ) πI/p) ) 1
1
x2π
[
[(
exp i m0R -
Dm′ exp(im′R) + exp
x2
)
)] ∑ {} π 4
m∆
×
m′)-m∆
iπ 2
exp{im′(R - π)}
]
1 π exp -i Ψ(R,t ) 0) + 4 x2 1 π exp i Ψ(R - π,t ) 0) (12) 4 x2
[ ]
[ ]
This expression indicates that the wave packet splits into two parts with the same magnitudes at half of the rephasing time.
4920 J. Phys. Chem. B, Vol. 108, No. 15, 2004
Hoki et al.
Figure 4. Ionization potential of (R)-2-methyl-cyclopenta-2,4-dienecarbaldehyde as a function of R.
The rephasing time in the present motor system is roughly estimated to be 2πI/p ) 17 000 fs, which is the same order as the numerically evaluated one, as shown in panel c. This explains why the splitting of the rotational wave packet into two parts takes place in panel b. The lower panel in Figure 3, panel d, shows the timedependent expectation value of the angular momentum l(t) of the model motor. This was calculated using eq 5. The time evolution of the expectation value l(t) can be divided into three regions, a, b, and c, within the rephasing time 2πI/p ) 17 000 fs. These three regions are the same as those in the upper panels. As we can see, the expectation value of the angular momentum is about 30p, which is similar to the value of 31p estimated above. Revival structures of the angular momentum appear in Figure 3d. These are called in fractional revivals that originate from the nonlinearity of the system Hamiltonian. Revival structures have been seen in other systems such as Rydberg atoms,29-31 rotational motions,20,32,33 and vibrational motions.34,35 III.C. Determination of Motor Dynamics by a TimeFrequency-Resolved Ionization Method. We now apply a time-resolved ionization method to the model system described above in order to observe motor dynamics in real time. To calculate the time-resolved ionization spectrum, the timedependent Schro¨dinger equation, eq 11, was numerically solved in a way similar to that described in the preceding section. For simplicity, the transition dipole moment between S0 and D0 was assumed to be constant (1 D) and independent of R. The pulse duration was set to T3 ) 100 fs, and the maximum value of the pulse amplitude was set to a small value (107 V/m) in order to satisfy a one-photon ionization in a weak field limit condition. Figure 4 shows the ionization potential as a function of R. This was estimated by calculating the difference between the PES of D0 and that of S0 at each R. The ionization potential energy has a maximum value of 60 900 cm-1 around R ) π, at which the potential energy of S0 is maximum, whereas the ionization potential energy has a maximum value of 62 800 cm-1 around R ) 0, at which the potential energy of S0 is minimum. Figure 5 shows a time-frequency-resolved spectrum I(ω3,t3) of (R)-2-methyl-cyclopenta-2,4-dienecarbaldehyde in the initial time region. Here again we have omitted the stationary component of the wave packet inside the potential well. We can see an oscillatory behavior of the spectrum at ω3 ) 60 900 cm-1 corresponding to the ionization potential around R ) 0. The time for one cycle oscillation is estimated to be 540 fs since two peaks of the oscillation are located at 380 and 920 fs. Figure 6 shows a time-resolved spectrum I(t3) in which the probe pulse frequency is fixed at ω3 ) 60 900 cm-1; the other parameters are the same as those in Figure 5. Here, we can see the dephasing and rephasing effects that are characteristic of
Figure 5. Time-frequency-resolved spectrum I(ω3,t3) of (R)-2-methylcyclopenta-2,4-dienecarbaldehyde. The abscissa is the frequency of a probe pulse, and the ordinate is its intensity. t3 denotes the time delay of the probe pulse. The intense spectra around ω3 ) 62 800 cm-1 were omitted in order to clearly show the weak spectra.
Figure 6. Time-resolved spectra I(t3). A fixed probe pulse frequency of ω3 ) 60 900 cm-1 was used.
rotating wave packets, as discussed above. The rephasing time is 17 000 fs. As can be seen in Figure 6, an oscillation with half of the period of the original one occurs at about half of the rephasing time and a one-third period at around one-third and two-thirds of the rephasing time. These observations indicate that the ionization spectrum reflects the time-dependent behavior of the rotational wave packets on the PES of S0 shown in Figure 4. Finally, it should be noted that in this paper we have treated the femtosecond pump-dump laser ignition method within the Schro¨dinger approach. As a consequence, we have omitted temperature and ultrafast relaxation effects due to intramolecular or intermolecular interactions. However, these must be taken into account for quantitative discussions on molecular motor dynamics using the Liouville equation approach.36 IV. Summary In this paper, we have proposed a new method for igniting a chiral molecular motor within a femtosecond time scale. This method is based on a pump-dump vis (or UV) laser excitation
Chiral Molecular Motors scheme, which creates a rotational wave packet in the electronic ground state. The direction of the rotational motion is determined by the gradient of the PES of the resonant electronic excited state in the Franck-Condon region. This method was applied to (R)-2-methyl-cyclopenta-2,4-dienecarbaldehyde. Motor dynamics such as dephasing or rephasing was analyzed in terms of the rotational wave packet propagation on the ground-state potential surface. We have also proposed a time-frequencyresolved photoionization method for observing motor dynamics in real time. Another ignition method of a model chiral rotor by means of ultrashort IR + UV laser pulses has recently been designed to obtain a high angular momentum (about 110 p).37 Acknowledgment. This work was partly supported by Grants-in-Aid for Scientific Research from the Ministry of Education, Science, Sports, Culture and Technology, Japan (No. 1555002), a Grant-in-Aid for Scientific Research on Priority Areas, “Control of Molecules in Intense Laser Fields” (No. 419), and the DFG, Project Ma 515/18-3. R.S. acknowledges the receipt of JSPS fellowship (P02353). References and Notes (1) Balzani, V.; Gomez-Lopez, M.; Stoddart, J. F. Acc. Chem. Res. 1998, 31, 405-414. (2) Molecular Machines and Motors; Sauvage, J.-P., Ed.; Springer: Berlin, 2001. (3) Sauvage, J.-P. Acc. Chem. Res. 1998, 31, 611-619. (4) Armaroli, N.; Balzani, V.; Collin, J.-P.; Gavin˜a, P.; Sauvage, J.P.; Ventura, B. J. Am. Chem. Soc. 1999, 121, 4397-4408. (5) Davis, A. P. Nature 1999, 401, 120-121. (6) Vacek, J.; Michl, J. New J. Chem. 1997, 21, 1259-1268; Proc. Natl. Acad. Sci. U.S.A. 2001, 98, 5481-5486. (7) Kelly, T. R.; Silva, H. D.; Silva, R. A. Nature 1999, 401, 150152. (8) Koumura, N.; Zijlstra, R. W. J.; van Delden, R. A.; Harada, N.; Feringa, B. L. Nature 1999, 401, 152-155. (9) Feringa, B. L.; Koumura, N.; van Delden, R. A.; ter Wiel, M. K. J. Appl. Phys. A 2002, 75, 301-308. (10) van Delden, R. A.; Koumura, N.; Schoevaars, A.; Meetsma, A.; Feringa, B. L. Org. Biomol. Chem. 2003, 1, 33-35. (11) Tuzun, R. E.; Noid, D. W.; Sumpter, B. G. Nanotechnology 1995, 6, 52-63. (12) Space, B.; Rabitz, H.; Lo¨rincz, A.; Moore, P. J. Chem. Phys. 1996, 105, 9515-9524. (13) Brouwer, A. M.; Frochot, C.; Gatti, F. G.; Leigh, D. A.; Mottier, L.; Paolucci, F.; Roffia, S.; Wurpel, G. W. H. Science 2001, 291, 21242128.
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