Laser Control for Coupled Torsions in Chiroptical Switches - American

Article pubs.acs.org/JPCC

Laser Control for Coupled Torsions in Chiroptical Switches: A Combined Quantum and Classical Dynamics Approach Dominik Kröner,* Selina Schimka, and Tillmann Klamroth Institut für Chemie, Universität Potsdam, Karl-Liebknecht-Str. 24-25, D-14476 Potsdam, Germany ABSTRACT: We present a novel laser pulse control for the chiroptical switch 1-(2-cis-fluoroethenyl)-2-fluoro-3,5-dibromobenzene mounted on adamantane, where the latter imitates a linker group or part of a solid surface. This molecular device offers three switching states: a true achiral “off”-state and two chiral “on”-states of opposite handedness. Due to the alignment of its chiral axis along the surface normal several defined orientations of the switch have to be considered for an efficient stereocontrol strategy. In addition to these different initial conditions, coupled torsional degrees of freedom around the chiral axis make the quest for highly stereoselective laser pulses a challenge. The necessary flexibility in pulse design is accomplished by employing the iterative stochastic pulse optimization method we presented recently. Still, the size and complexity of the system dictates a combined treatment by fast molecular dynamics and computationally intensive quantum dynamics. Although quantum effects are found to be of importance, the pulses optimized within the classical treatment allow us to turn on the chirality of the switch, achieving high enantioselectivity in the quantum treatment for all orientations at the same time.

1. INTRODUCTION The fascination for molecular switches and motors continues due to their application as components of nanoscale devices. Light-driven changes in molecules are particularly attractive because the characteristics of light allow for a very flexible and efficient addressing, even within large and complex environments, as well as for fast response times. Therefore, control by light can become an advantage for the design of molecular machines in comparison to those driven by heat or chemical reactions. Light-induced molecular switches may even trigger macroscopically observable changes, as observed for instance in polymer films, which bend depending on the direction of the linear polarization of the light due to switching azobenzene moieties of compatible orientation.1 In particular, chiroptical switches are an interesting group of light-driven molecular devices because of their possibility to change their chirality by irradiation. One popular example is the light-induced cis−trans isomerizations of bulky alkene derivatives with helical structures.2 Here the reversible change between the two diastereomeric forms is achieved by irradiation with monochromatic light of proper wavelength. The process runs highly stereoselectively in both directions, even when the molecules are mounted on the surface of gold nanoparticles.3 Applied as a dopant, chiral molecules may also transfer their chirality to the nematic phase of liquid crystals, initiating the formation of a cholesteric phase. If the handedness of the chiral dopant can be switched by light, the helicity of the cholesteric phase can be defined.4 This allows us to control the optical properties of a liquid crystal using a light-controlled dopant, as shown, for instance, by Ma et al.:5 They dynamically tune the reflection color of an achiral liquid crystal host with light© 2013 American Chemical Society

induced cis−trans isomerizations of its axially chiral binaphthyl bisazobenzene type of dopant, where the spatial structure of the molecular switch determines the pitch length of the helical superstructure of the cholesteric phase. Another attractive use of a molecular switch with lightcontrolled chirality is the application in asymmetric synthesis as catalyst. Here the handedness of the products could be controlled by a switchable catalyst. This was successfully demonstrated for an asymmetric autocatalysis with very high enantiomeric excess and yield, where the light-controlled handedness of an olefin mediator determines the configuration of the stereogenic center of the product.6 However, it is a tiny imbalance of the initially racemic photoequilibrium of the axially chiral olefin, created by very long irradiation with circularly polarized light, which induces the asymmetry in the autocatalytic reaction. In general, chiral molecular switches addressable by light are potentially applicable in data storage or information processing, as their handed forms are easily distinguished by polarized light. While in most experiments the chirality of molecules is influenced by the irradiation with cw light, we propose to control chiral molecular switches using shaped laser pulses. Applying modern feedback-optimized pulse-shaping techniques,7 femtosecond laser pulses may indeed be used to steer chemical reactions, as shown in the control of the retinal isomerization in bacteriorhodopsin8 or of the wavepacket dynamics of mixed alkaline clusters.9 Apparently switchable Received: October 18, 2013 Revised: December 16, 2013 Published: December 17, 2013 1322

dx.doi.org/10.1021/jp410342a | J. Phys. Chem. C 2014, 118, 1322−1331

The Journal of Physical Chemistry C

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molecules often undergo cis−trans isomerization, or bonds are broken and formed intermediately or even permanently, for instance in photocyclization or ring-opening reactions.10 The chirality of molecules may, however, also be changed by an isomerization without altering the bond grade. In quantum simulations the handedness of the helical structure of chiral difluorobenzo[c]phenanthrene was changed using shaped laser pulses.11 Similarly, optimized laser pulses may induce a torsion of the central single bond of an axially chiral 1,1′-binaphthyl to achieve a selective P−M isomerization, as shown in quantum simulations.12 Some time ago we presented a stereoselective laser pulse control for a molecular switch supporting three stable conformations, one being achiral and two being of opposite chirality along the torsion about the chiral axis of the model system.13 The chirality of this molecule, the fluorine-substituted styrene derivative 1-(2-cis-fluoroethenyl)-2-fluorobenzene, can enantioselectively be turned on with shaped IR laser pulses. In addition, the transformation to the opposite handedness is also possible using enantioselective UV pulses. As the high stereoselectivity of the developed control strategy relies on the orientation of the molecular switch, we proposed to mount the molecules onto a surface.14 In that way a high degree of orientation, in particular, of the chiral axis with respect to the surface normal is possible. However, to achieve high stereoselectivity for several orientations of the adsorbed switch, although they are defined by the rotational symmetry of the surface, became a challenge for pulse shaping. Here our stochastic pulse optimization algorithm succeeded in generating laser pulses which accomplish a high enantioselectivity for all possible orientations at the same time.14 The model system was eventually further extended by introducing adamantane as a model for either a nonmetal surface or, if properly substituted with thiol groups,15 as a linker group for, e.g., a gold surface.16 We showed that even a well-defined elliptically polarized pump−dump pulse sequence optimized by hand allows us to control the chirality of the switch independent of its conformation with respect to adamantane. Nevertheless, so far our model simulations considered only the torsion of the ethylene group of the molecule, assuming that the much heavier phenyl group will be little influenced by the isomerization. Now, as the phenyl ring is connected to the adamantyl “surface” by a C−C single bond, a second torsional degree of freedom has to be included. This torsional degree of freedom involves a rotation about the chiral axis of the switch, too, and thus changes its chirality. In addition, it is coupled to the first torsional degree of freedom. These two coupled torsional degrees of freedom represent a new challenge for laser pulse control, which we wish to address in this report. Although the stochastic pulse optimization we recently introduced in quantum dynamical simulations14 is basically capable of handling the task, the size of the system dictates a computationally less demanding approach. Thus, a combined classical and quantum dynamical treatment becomes necessary, which we will present here. In this context it is interesting to note that the enantioselective transformation by coupled torsion and overall rotation of axially chiral H2POSH can be achieved applying circularly polarized laser pulse.17 In this simulation, the chiral axis of the molecule was also assumed to be oriented along a space-fixed axis, however, not as result of an adsorbate−surface interaction, such that the free rotation of the entire molecule around its chiral axis had to be accounted for. Moreover, only

pure state to state transitions between torsional and rotational eigenstates of the molecule were induced, in contrast to the here-proposed control scheme. In more recent combined theoretical and experimental studies elliptically polarized laser pulses were used for threedimensional alignment of an axially chiral biphenyl derivative, while a linear polarized nonresonant “kick” pulse induces torsion about the chiral axis.18 In experiment the torsion was monitored by femtosecond time-resolved Coulomb imaging, revealing that the amplitude of the light-induced torsion never exceeded a few degrees in the dihedral angle. While simulations predict a much more efficient control of the torsion of such systems,19−21 the coupling to the overall rotation of the molecule seems to reduce the success in the laboratory. In the report at hand we not only mount our chiroptical switch onto a solid surface but also try to restrict the overall rotation of the adsorbate around its chiral axis by introducing steric demanding substituents. In Section 2, the new extended model system is introduced and the calculation and construction of the potential energy surface are explained. In addition, the stochastic pulse optimization scheme is briefly reviewed, and its application to molecular dynamics is introduced. Subsequently, the obtained control pulse and its effect on the chiral switch are presented in the Results, Section 3. The paper concludes with a summary and evaluation of the findings in the final Section 4.

2. MODEL AND METHODS 2.1. Model System. We propose 1-(2-cis-fluoroethenyl)-2fluoro-3,5-dibromobenzene mounted on adamantane as a model system for a surface-mounted molecular chiroptical switch (see Figure 1). In this model, the 1-(2-cis-fluoroethenyl)2-fluoro-3,5-dibromobenzene switch has three stable conformations for the torsion around the single C−C bond connecting the ethenyl and phenyl group, that is, the rotation characterized be the angle θ in Figure 1. The global minimum at θ = 180° is achiral, while the two local minima at θ = 39° and θ = 321° are enantiomers. This is analogous to the 1-(2-cis-fluoroethenyl)-2-

Figure 1. Illustration of the three minima of 1-(2-cis-fluoroethenyl)-2fluoro-3,5-dibromobenzene mounted on adamantane for the torsion of the ethenyl group around θ, as obtained from B3LYP/6-311G(d,p). The angle Φ, which describes the rotation of the whole molecular switch, i.e., the ethenyl−phenyl unit, relative to the adamantyl group is fixed at 180°. 1323



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Figure 2. (a) Sketch of the angles φ = Φ and ω = Φ + θ used for the dynamical calculations, drawn as a projection of the F-substituted ethenyl (red inner lever) and phenyl group (green outer levers) as well as the 3-fold frame of the adamantane (blue base). (b) Two-dimensional PES in φ and ω shown as a contour plot (contours start at 250 cm−1 with an increment of 250 cm−1; 5000 cm−1 is used as a cutoff in this representation). The three minima representing the achiral conformations are indicated by the bold contours, the minima for the (aS) configurations by dashed contours, and the ones for the (aR) configurations by dotted-dashed contours. In the lower panels, one-dimensional cuts along θ for Φ = 180° (c) and along Φ for θ = 318° (d) are shown. Also indicated are the “lowest” localized one-dimensional wave functions used as a basis for the calculation of the target state wave functions.

fluorobenzene switch, which was investigated in previous studies.13,14,16 The geometries shown in Figure 1 were optimized using the B3LYP22 hybrid functional with a 6311G(d,p) basis set.23,24 All quantum chemical calculations were performed at the same level of theory with the Gaussian0325 program package. For the geometry optimizations, the x,y coordinates of all four carbon atoms on the chiral axis, i.e., those on the z-axis, were kept frozen (= 0) (cf. the coordinate system given in Figure 1). Inspecting the carbon atoms on the z-axis in Figure 1, one immediately realizes that not only the ethenyl group can rotate around this axis but also the whole ethenyl−phenyl unit relative to the adamantyl base, which is described by the angle Φ. Here, the adamantyl group corresponds either to a surface, on which the switching unit is mounted, or to a rigid molecular tripod with, e.g., thiol “legs” for an absorption on an ordered gold surface.15 Although we assumed only three stable orientations of the chiroptical switch with respect to adamantane in ref 16, the barriers between these minima were very low compared to barriers we wish to overcome in the laser control. Therefore, steric demanding bromine substituents are introduced now, which bring about three degenerate and well-separated minima for the rotation in Φ, namely, at 60°, 180°, and 300°. The potential energy surface (PES), V(θ, Φ), and the corresponding dipole function, μ⃗ (θ, Φ), were calculated starting from the achiral minimum shown in Figure 1 by varying θ and Φ, while all other internal degrees of freedom

were kept frozen. We used a tensor grid with 72 equidistant points in θ and Φ within the interval [0°, 355°] and a Δθ = ΔΦ = 5° spacing, resulting in 5184 first-principle points, Vkl = V(kΔθ, lΔΦ) with k and l ∈ {0, 1, ..., 71}, of which only the ones unique by symmetry were calculated. A continuous representation of the PES was obtained by a cosine series as nθ − 1 n Φ − 1

V (θ , Φ) =

∑ ∑ j=0

Cij cos(jθ )cos(3i Φ)

i=0

(1)

utilizing the symmetry of the PES. We chose nθ = 35 and nΦ = 11, which resulted in a root-mean-square deviation (rmsd) of 3.5 cm−1 for the continuous PES compared to the first-principle points Vkl. The coefficients Cij are determined from the firstprinciple points by 71

Cij =

71

∑ ∑ Vkl cos(jkΔθ )cos(3ilΔΦ) k=0 l=0

(2)

In the same way, a continuous representation of μ⃗ (θ, Φ) is obtained, i.e., by building a cosine series for the x and y components. Here we used nθ = nΦ = 15, which lead to a rmsd of 9.9 × 10−5 Debye for the x component and of 9.8 × 10−5 Debye for the y component. Note that μz(θ, Φ) is not considered in the following as the laser propagates in (negative) z-direction with its z-component being zero (see below). As expected, the dipole component functionssee Appendix B for 1324



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more detailsare similar to the ones constructed from the rotated 1D function (θ) used in refs 14 and 16. The coordinates θ and Φ are perfect for the potential fitting, due to the small couplings between them in the potential energy. However, they are less suitable for dynamical simulations because of the kinetic couplings in the Hamilton function and the Hamilton operator. Therefore, we use φ=Φ

(3)

ω=Φ+θ

(4)

Ω E k (t ) = s (t ) π

pω2̂ 2Iω

+

pφ2̂ 2Iφ

l=0

(7)

i.e., a truncated Fourier series. We set the number of frequency components nf = 50. The shape function ⎧ ⎛ ⎞ ⎪ sin 2⎜ πt ⎟ 0 ≤ t < ts ⎪ ⎝ 2ts ⎠ ⎪ ⎪ ts ≤ t < (t p − ts) s(t ) = ⎨1 ⎪ ⎪ 2⎛ π ( t p − t ) ⎞ ⎪ sin ⎜⎜ ⎟⎟ (t p − ts) ≤ t ≤ t p ⎪ 2ts ⎝ ⎠ ⎩

(8)

ensures a “smooth” start and end of the laser field on the time scale of ts = 500 fs. The coefficients ak,l and bk,l are initialized by random numbers, with the constraint of a fixed Euclidean norm

+ V (φ , ω)

NΩ = (2ax2,0 + 2ay2,0 + 2bx2,0 + 2by2,0

(5)

nf − 1

with moments of inertia Iφ = 1.0790 × 107mea20 and Iω = 7.5817 × 105mea20, calculated from one of the optimized global minima shown in Figure 1. The two-dimensional PES in φ and ω is shown in Figure 2(b). The three global minima corresponding to the three degenerate achiral conformations are surrounded by the bold contours. The (aS) minima are indicated by dashed contours and the (aR) minima by dotted-dashed contours. As one can see there are three valleys in φ, i.e., at 60°, 180°, and 300°. The corresponding three wells can be clearly seen in Figure 2(d), where a cut along Φ (= φ) for θ = ω − φ = 318° is shown. As such, this cut is parallel to the diagonal (ω = Φ + θ) of Figure 2(b) and connects all three minima corresponding to the (aR) configurations. These three wells are separated by high barriers (of about 4000 cm−1), and we will refer to them as different orientations j = {1, 2, 3} in what follows. Note that the barriers between the “achiral” minima (θ = 180°) are of similar height (see contours in Figure 2(b)). For each orientation j there is an achiral and two chiral (aS,aR) minima, as in the one-dimensional case investigated in refs 14 and 16. In Figure 2(c) these minima are depicted in a 1D cut along θ = ω − φ for Φ = 180°. Thus, this curve represents a horizontal cut along ω through the middle (j = 2) of the PES in Figure 2(b), followed by a shift of 180°. Analogous cuts at j = 1 and j = 3 result in the same potential curves due to the 3-fold symmetry of the adamantyl. Note that, however, the dipole functions significantly differ for the three conformations j (see Appendix B). 2.2. Stochastic Pulse Optimization. Details of the stochastic pulse optimization (SPO) procedure have been already discussed in ref 14. Here, we will only shortly recapitulate the most important points. For the optimization procedure we use g different laser pulses (p) of the following form ⎛ Ex(t )⎞ ⎟ ⎜ Ep⃗ (t ) = ⎜ Ey(t )⎟ ⎟ ⎜ ⎝ 0 ⎠

∑ [ak ,l cos(l·Ω·t ) + bk ,l sin(l·Ω·t )],

k = x, y

instead, which are schematically depicted in Figure 2(a), and describe the independent torsion of the ethenyl and phenyl group with respect to the fixed adamantane frame. These angles result in an uncoupled kinetic term in the molecular Hamiltonian Ĥ 0(ω , φ) =

nf − 1

+

∑ [ax2,l + ay2,l + bx2,l + by2,l])Ω l=1

(9)

NΩ is only an approximation for the laser fluence, as we choose π Ω= tp (10) i.e., an overcomplete set of frequencies, which turns out to be beneficial for the optimization procedure. The final time of the pulse, tp, was chosen to be 12 ps. Note that NΩ not only restricts the amplitude of the electric field but also controls the maximum amount of energy that can be transferred to the molecule. That is a benefit for highly nonlinear light−matter interactions. The stochastic optimization starts with a set of g = 60 randomly chosen laser pulses. For these pulses the laser-driven nuclear dynamics is determined within the dipole approximation. This leads to a time-dependent Hamiltonian in quantum dynamics of the form Ĥ (t ) = Ĥ 0 − μ ⃗ F (⃗ t )

(11)

or a time-dependent potential in classical dynamics. For each pulse, propagations for three different initial conditions j = {1, 2, 3}, i.e., orientations corresponding to the achiral configurations, are performed. At the final time tf the fitness F(j) p for each orientation is evaluated. Details about the initial conditions and the fitness for the quantum and the classical propagations will be given in Sections 2.3 and 2.4. The total fitness for each pulse is then calculated from a product of the fitnesses of each orientation Fp = (∏ Fp(j))1/3 j

(12)

The chosen product ansatz (eq 12) ensures comparable fitness for all orientations, preventing an optimization for one orientation at the expense of the others: if the fitness of any orientation F(j) p vanishes, the overall fitness Fp vanishes as well. Later, we will see that this is particularly important if different orientations compete for maximum enantioselectivity and efficiency at the same time.

(6)

with 1325



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From the g initial pulses (parents) a new set of g pulses (children) is created by a random walk procedure for the coefficients

ak̃ , l = ak , l + RMak , l

PT(j) =

T = R, S

ν

(15)

Here, R indicates localized states for the (aR) configurations and S for the (aS) ones. To quantify the overall fitness of a pulse p for the quantum dynamical propagations we set F(j) p = P R(j) in eq 12, if the (aR)-enantiomer is the desired configuration. For the determination of the localized target wave functions (j) ψT,v for the quantum mechanical simulations the timeindependent Schrödinger equation was solved in the basis of localized zero-order states. These zero-order states are product states of one-dimensional wave functions in θ and Φ, as shown in Figure 2(c) and (d), which were transformed into the φ, ω space. For further detail see Appendix A. 2.4. Molecular Dynamics. For the stochastic pulse optimization based on MD calculations we apply a Velocity Verlet integration.27,28 The classical equations of motions for (j) (j) (j) initial conditions φ(j) 0 , ω0 , ω̇ 0 , and φ̇ 0 (dot denotes the time derivative) are solved, where periodic boundary conditions at 0 and 2π for φ(j)(t) and ω(j)(t) apply. In the following, we use pω = ω̇ Iω, pφ = φ̇ Iφ for the momenta. The simplest way for the pulse optimization would be to calculate just the three trajectories simultaneously for a given laser field with initial positions, which correspond to the classical minima for the three orientations j and zero velocities. However, it turned out that laser pulses optimized in such a way performed rather poorly when applied to the quantum dynamics. This was found to be mainly due to the fact that the spreading of the quantum wave packet due to the finite initial width in position and momentum space is not accounted for by a single trajectory representing one orientation. Therefore, not a single trajectory per orientation is propagated, but a set of trajectories characterized by the initial (j) (j) (j) conditions {φ(j) 0,i , ω0,i , ω̇ 0,i , φ̇ 0,i }. We choose a tensor grid with K equidistant points in each dimension α ∈ {ω, φ, pω, pφ} as α(j) i (t = 0) = α(j) 0,i = α0 + iΔα with Δα as described below. Thus, we get i = 1, ..., K4, i.e., K4 trajectories per orientation. For each orientation all trajectories of one initial set are propagated, and observables, O(j)(t), are computed as a weighted average of all trajectories

(13)

and for the bk,l alike. R is a random number between −1 and 1 and M an adjustable mutation step factor, which is chosen initially rather high, i.e., 0.32. For all child pulses also the overall fitness is determined after a propagation. From the 2g (parents and children) pulses the g fittest are kept as a new parent generation. This procedure is continued a certain number of times (typically a few hundred) until no substantial improvement in the maximal fitness is observed any more. Then, M is reduced, i.e., divided by two, and the whole optimization procedure continues. After a few of such cycles, when M is in the range between 0.04 and 0.01, an acceptable converged fitness for the best pulse in the set is reached. This kind of optimization requires little precognition of the control mechanism. The pulse duration tp may be estimated from the lowest possible frequency of the involved modes, while an educated guess of the frequency range allows us to determine the number of frequency components nf. SPO can be performed for classical and quantum systems and has been proven to be very successful in laser-coupled one-dimensional quantum systems of multiple defined orientations,14,16 where a high nonlinearity of the control was found. However, SPO relies on a large number of propagations, which are prohibitively expensive with respect to the computational time for the present 2D system. Therefore, we used for the optimization classical trajectories, where some quantum effects are approximately included by initial conditions derived from a Wigner distribution (see Section 2.4). The such obtained pulses are then tested for the full quantum system, as described in the following section. 2.3. Quantum Dynamics. For the quantum dynamical simulations the time-dependent Schrödinger equation ∂ Ĥ (t )Ψ(j)(ω , φ ; t ) = iℏ Ψ(j)(ω , φ ; t ) ∂t

∑ |⟨ψT(j,)ν|Ψ(j)(tf )⟩|2

(14)

is solved, where Ψ(j)(ω, φ; t) is the two-dimensional timedependent wave function for orientation j, which is represented on an equidistant grid with 768 points in ω and 2304 points in φ. For both angles the grid is defined between 0 and 2π. We use a time step of 1 fs with the split operator propagator,26 which gave converged results for all laser fields considered here. To get localized ground-state wave functions, characterizing the achiral conformations, as initial wave functions for each orientation, Ψ(j)(ω, φ; t = 0) imaginary time propagations were performed. Starting with a two-dimensional Gaussian located at the minimum for the respective orientation j, the wave function was propagated with a time step of 0.1 fs (for 10 ps) until convergence. By this choice we were able to obtain localized ground states, which aredue to the high tunneling barriers along Φstationary at least on the time scales considered here. The probability densities of the three localized initial wave functions are shown in Figure 6(a). We used the optimized laser field from the molecular dynamics simulations (see Section 2.4) for the propagation of all three initial wave functions. At final time tf = 12 ps the total population PT of bound localized “chiral” states for each orientation j is determined by

O(j)(t ) =

∑i W i(j)Oi(j)(t ) ∑i W i(j)

(16)

The weights, W(j) i , are chosen according to a Wigner-like distribution derived from the localized 2D ground-state wave functions in harmonic approximation ) , p(j) ) W i(j) = Wi (ω0,(ji) , φ0,(ji) , pω(j,0, i φ ,0, i ) 2 ⎛ (pω(j,0, ) (φ0,(ji) − φm(j))2 (ω0,(ji) − ωm(j))2 i = exp⎜ − − − 2 2 2 ⎜ 2σω 2σφ 2σp ⎝ ω ) 2⎞ (pφ(j,0, ) i ⎟ − 2 ⎟ 2σp ⎠ φ

(17) (j) ω(j) m and φm are the coordinates corresponding to the three stable achiral conformations. Further, it is

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σp = α

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stochastic pulse optimizations, targeting the (aR) enantiomer, have been tested in quantum wavepacket propagations to determine their success, namely, their “quantum fitness” according to eq 15. The results are given in Table 1.

ℏ 2Iαωα(tor)

(18)

1 ℏIαωα(tor) 2

(19)

Table 1. Listed are the Fitnesses F(j) for Each Orientation j as Obtained from MD and QD Simulations for Seven Different Stochastic Pulse Optimization Runsa

with α = ω,φ and ⎛ ∂2 ⎞1/2 φ ω V ( , ) 2 ⎜ ⎟ ωα(tor) = ⎜ ∂α ⎟ Iα ⎝ ⎠

pulse #1

(20) #2

ω(tor) ω

being the harmonic frequency along α. Here, we obtain = −1 19.88 cm−1 and ω(tor) = 18.16 cm . φ Test calculations indicate that the use of two different initial values per dimension, i.e., K = 2, gives the best compromise between computation time (roughly 1 to 5 million trajectories are propagated for an optimized pulse) and transferability of the classically optimized pulses to the quantum mechanical (j) (j) (j) description. Here we choose α(j) 0,1 = αm − σα/2 and α0,2 = αm + σα/2 for the coordinates and ± σpα/2 for the momenta. By this choice we get a set of 16 trajectories for each initial orientation. As a last requirement for the pulse optimization using classical mechanics, one needs to define a fitness, F(j) p . For this purpose we use Gaussian functions in phase space in conjunction with the classical observables at the end tf of the pulse p Fp(j) = g(j)(ω(j)(tf ), φ(j)(tf ), pω(j) (tf ), pφ(j) (tf ))

#3 #4 #5 #6 #7

2δ p2

ω

ω(j) c

−

pφ2 ⎞ ⎟ 2δ p2 ⎟⎠ φ

F(2)

F(3)

F

0.69 0.51 0.61 0.49 0.68 0.50 0.70 0.49 0.66 0.48 0.44 0.26 0.58 0.51

0.73 0.69 0.51 0.33 0.62 0.30 0.62 0.31 0.36 0.19 0.61 0.42 0.37 0.014

0.81 0.61 0.55 0.55 0.59 0.55 0.62 0.50 0.69 0.50 0.51 0.36 0.63 0.76

0.53 0.22 0.78 0.58 0.84 0.64 0.85 0.65 0.92 0.74 0.21 2.8 × 10−4 0.75 0.76

0.68 0.45 0.60 0.47 0.67 0.47 0.69 0.47 0.61 0.41 0.40 0.035 0.56 0.20

The following observations are made: (i) There is a fair correspondence between “quantum” and “classical” fitness. For reasonable high fitnesses (F > 0.5) both values are in the same order of magnitude, and in most cases the ordering of the fitness for different j is the same in MD and QD. However, for small “classical” fitnesses the quantum result may actually become close to zero (see pulses #6 and #7). (ii) Most optimization results in fairly equal product distributions for all orientations. The reason for the differences between fitnesses in MD and QD, applying the same pulse, is attributed to the rather low number of trajectories and/or their lack of coherence. Furthermore, note that a small “quantum fitness”, even in the case of a negligible fitness for certain orientations (pulses #6 and #7), does not necessarily mean that the undesired enantiomer is formed, as the excited population could still be distributed among states below the barriers separating achiral from chiral forms or, even if energetically above those barriers, the wavepacket could be spread equally over all torsional angles. In the following we will only consider pulse #2, which is a good compromise in terms of overall yield, adequate efficiency for each orientation, and high enantioselectivity, i.e., low (aS) population (see Table 2 below), in QD.

(21)

⎛ (φ − φc(j))2 (ω − ωc(j))2 g(j)(ω , φ , pω , pφ ) = exp⎜ − − ⎜ 2δω2 2δφ2 ⎝ pω2

F(1)

a Also given is the total fitness, F, cf. eq 12, used for the optimizations and the average fitness of all orientations, denoted øF(j).

with

−

MD QD MD QD MD QD MD QD MD QD MD QD MD QD

øF(j)

(22)

φ(j) c

Here, and are the coordinates corresponding to the desired chiral conformer for the respective orientation. On the basis of the results of test calculations we used δω = δφ = 0.25 = 14.32° and δpω = δpφ = 25ℏ. The spatial localization of the fitness functions, i.e., Σjg(j)(ω, φ, 0, 0) is shown in Figure 3.

3. RESULTS The goal of the stereocontrol is to enantioselectively turn on the chirality of the molecular switch using properly shaped laser pulses. The laser fields obtained from seven converged classical

Table 2. Population of the First 188 Localized “Chiral” States and the Torsional Excitation Energy in the (aR) and (aS) Configuration after Propagation with Pulse #2a (aR) ⟨Etor⟩ [cm ]

P j=1 j=2 j=3

Figure 3. Shown is the spatial localization of the fitness functions, i.e., Σjg(j)(ω, φ, 0, 0).

a

1327

(aS) −1

(j)

0.3312 0.5466 0.5760

186.9 384.9 126.1

P

(j)

0.01792 0.008816 0.02181

⟨Etor⟩ [cm−1] 596.0 625.8 550.2

Note that the barrier height is about 750 cm−1. dx.doi.org/10.1021/jp410342a | J. Phys. Chem. C 2014, 118, 1322−1331


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The x- (black) and y-components (red) of the electric field of pulse #2 are shown in Figure 4. Apparently, the pulse is not

sense, the optimized pulse significantly differs from the comparable ones in previous studies,14,16 where we even found systematic phase-shifted oscillations between different orientations in the observables, allowing for the design of a stereoselective laser pulse sequence by hand.16 The complexity of the here obtained control scheme must be attributed to the laser-driven coupled torsions around the chiral axis of the molecular switch. In Figure 5 we can also see that the “Wigner” classical dynamics and the fully quantum dynamical calculations mainly differ in the effectiveness of the dump step. This becomes even more evident when looking at the probability densities of all orientations, which are shown in a single graph at initial time t = 0 and final time t = 12 ps in Figure 6. The final wavepackets of each orientation consist of a rather localized part of high amplitude in the desired minima as well as a low-amplitude component spreading widely over all torsional angles ω. A further analysis of the wavepackets at final time is given in Table 2. Although the fitness F of the pulse found in the QD simulations is reduced by 0.13 compared to the MD results (see Table 1), the enantioselectivity remains very high: while on average almost 50% of the population is found in the states of the (aR) enantiomer, regardless of orientation, not much more than 1% is distributed among the localized states of the (aS) form. In addition, the population in the target states is, at least for the most part, distributed equally among the orientations. Orientation j = 1 deviates most from the average value of almost 50%, which is attributed to the noticeable difference of the expectation value of ω between QD and MD after the dump set in (see j = 1 in Figure 5). These differences result from the low number of Wigner-distributed trajectories used in the MD calculations or the strong coherence of the wavepacket in the QD simulations. Still, the torsional excitation energy, ⟨Etor⟩, of the populated localized “chiral” states is quite low for all orientations compared to the height of the barrier between chiral and achiral minima (∼750 cm−1). This confirms that, at least, this part of the wavepacket is well localized deep within the target well. The torsional energy of the undesired (aS) conformer is, in contrast, rather close to the top of the barrier for any of the three orientations, suggesting that the optimized pulse does not efficiently support a population of these states. Hence, we achieved a high and well-dumped (aR) population, i.e., with low torsional excitation, while the tiny (aS) population is clearly torsionally excited. To selectively turn on the (aS) chirality of the molecular switch, the handedness of the laser pulse must be mirrored, e.g., by inverting the sign of its ycomponent.

Figure 4. x- (black) and y-component (red) of the electric field of the optimized laser pulse #2, as obtained from the stochastic pulse optimization with MD calculations.

linearly polarized. This is due to the fact that the pulse needs to act simultaneously on all three orientations. As one can imagine a single linearly polarized pulse, stereoselective for one orientation only, could not be used for all three orientations simultaneously, as the interference between the original pulse and two pulse rotated by +120° and −120° around the z-axis (to make them stereoselective for their orientations) would result in a zero field. As in previous investigations,13,14,16 we find a pump−dumplike control scenario, which can best be seen in Figure 5 from

Figure 5. Shown are the quantum mechanical expectation values (black lines) and the classical observables as a weighted average over trajectories (red lines) for φ and ω as a function of time for all three orientations j interacting with laser pulse #2. Also indicated is tp, the time after the pulse is off. For clarity the interval in which ω is defined has been shifted to [120°; 480°] for j = 1 and to [−120°; 240°] for j = 2.

4. CONCLUSIONS We presented a laser pulse controlled chiral molecular switch mounted on adamantane. The molecule, 1-(2-cis-fluoroethenyl)-2-fluoro-3,5-dibromobenzene, offers three different stable stereoisomers, one being achiral and the other two being enantiomers, along the torsion of its chiral axis, namely, the C− C single bond connecting the ethenyl and phenyl group. Properly shaped laser pulses allow us to enantioselectively “turn on” the chirality of the switch. However, via the phenyl ring the chirality-switching degree of freedom is coupled to the torsion of the whole switch with respect to its fixed adamantyl base, which in turn serves as a model of a linker group, to adsorb the switch on a surface, or of the surface itself. In particular, the torsion of the C−C single bond between molecular switch and adamantyl changes the orientation of the switch. However, as

the classical observables and quantum expectation values for φ and ω. The first part of the pulse simultaneously excites the torsions of ethenyl (ω) and phenyl (φ) groups, where in terms of the former mode the system is pumped to energies which allow us to overcome the low barrier between the “achiral” and the desired “chiral” minima. The second part of the pulse dumps the wavepacket into the desired “chiral” minimum, effectively localizing it there. At that moment, both torsional motions are damped. The moment when control switches from pump to dump is, however, different for each orientation and cannot be observed in the field components in Figure 4. In this 1328



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Figure 6. Snapshots of the quantum wavepacket dynamics of the laser pulse induced stereoselective switching of 1-(2-cis-fluoroethenyl)-2-fluoro-3,5dibromobenzene mounted on adamantane from its achiral to its (aR)-stereoisomer. Shown are the probability densities of the wave functions Ψ(j)(ω, φ; t) at initial time t = 0 (a) and final time t = 12 ps (b) for the three stable initial conformations (results given for pulse #2).

phenyl ring to permit for a higher number of stable orientations of the chiral switch. At the same time, however, the barriers between those orientations diminish, allowing for a quasi-free torsion of the switch around its chiral axis and/or the surface normal, even at moderate temperatures/excitation energies. How stereoselective control can still be maintained is yet to be shown. Here modifications to our stochastic pulse optimization approach or alternative optimization approaches might be necessary to consider. Interestingly, this rotation may effectively turn the switch into a rotor whose direction of rotation and chirality could be controlled simultaneously.

both axesthe chiral axis and the one controlling the orientation of the switch on adamantanerun parallel, torsional excitations by light affect both chirality and orientation. In other words: if the torsion of both F-substituted groups, i.e., ethenyl and phenyl, is treated independently with respect to the “surface”, the laser interacts with two coupled chirality-changing degrees of freedom. The steric demanding bromine substituents, though, help to constrain the orientation of the chiral switch to three stable conformations, eventually preventing a change of chirality by a light-induced rotation of the phenyl ring only. Still, the two torsional degrees of freedom remain coupled and pose a great challenge for a stereoselective control by laser pulses. This challenge was addressed by employing our stochastic pulse optimization (SPO) approach, a highly flexible technique, which seeks for a maximum fitness by the random mutation of laser pulse parameters. The size and complexity of the model, however, required a combined treatment of quantum and classical dynamics, to efficiently cope with the large number of optimization steps. We showed that the SPO based on molecular dynamics allows for a very efficient (and fast) construction of stereoselective pulses. However, even a high, but still affordable, number of classical trajectories were found not to perfectly mimic the results of the quantum wavepacket dynamics. The pulses optimized within the classical treatment still allow us to turn on the chirality of the switch, achieving high enantioselectivity with comparable yields for all orientations. Although the presented molecular switch may be considered a model system, the discussed results offer interesting perspectives. In contrast to the very popular molecular switches based on cis−trans isomerization, i.e., between diastereomers, the here-presented system gives the possibility to switch between enantiomers or even turn chirality off. This has implications on possible applications as dopant in achiral crystal liquids or for information processing. Moreover, autocatalytic asymmetric syntheses that rely on a tiny imbalance in an initially racemic photoequilibrium need to be irradiated for very long periods of time in comparison to fast and direct addressing by the laser pulse control approach presented here. However, the low torsion barriers in our adamantane-mounted styrene derivative as well as the optimal IR pulses of rather high intensity present an experimental challenge that should not be concealed. The requirement of the orientation of the chiral axis along the surface normal is a rather minor restriction, as laser pulse control for not perfectly oriented molecules will merely be of lower efficiency but still very high stereoselectivity. Nevertheless, it is obvious that lifting other restrictions of our model system will make laser control even more challenging. As such it is appealing to abandon the bromine substituents at the

A. DETERMINATION OF LOCALIZED TARGET STATES The chiral configurations of the molecular switch are characterized by wave functions localized in either of the two mirror-inverted wells on the left and right (in terms of ω) of each of the three global minima (see Figure 2(b)). These wave functions may be obtained by proper positive or negative superpositions of those torsional eigenfunctions which show no amplitude within the region of the global minima.13,29 Note that these wave functions must additionally be localized with regard to the three orientations of the switch on adamantane. The corresponding eigenfunctions belong to doublets of torsional eigenstates which are formed due to the mirror symmetry and the barriers of the potential energy curve along θ (see Figure 2(c)), as it were three double well potentials (the central minimum acts like a barrier). While the generation of localized “chiral” wave functions is straightforward in one dimension, see, e.g., ref 13, it becomes more cumbersome for the here-investigated two-dimensional model because of (i) a high density of torsional states, (ii) a large number of “achiral” states energetically below and among the wanted “chiral” states, and (iii) the C3-symmetry of the φ-degree of freedom which requires proper superpositions for localization within only one of the three “orientation” minima j. Moreover, the high moment of inertia for the torsion of the bromine-substituted phenyl group (>106mea20) demands a large number of grid points along φ for a converged basis. Thus, the resulting total number of grid points for a direct diagonalization of the molecular Hamiltonian in 2D, Ĥ 0 , exceeds easily one million, which tops the limit for memory available in present standard computers. Therefore, we chose a basis of 1D product functions which are determined from cuts of the PES along θ at Φ = 60°, 180°, or 300° (j = 1, 2, or 3), respectively, and from a cut along Φ at θ = 318°, i.e., at the grid point closest to the (aR) minimum configuration of each orientation (see Figure 2(c) and (d)). From 32 torsional eigenfunctions ϕn(θ) along θ (within the range of n = 43−108), which all well reside within the outer wells, two times L = 16 localized “chiral” 1D wave functions are 1329



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Figure 7. Contour plot of the (a) x- and (b) y-component of the dipole moment μ⃗ along torsion angles ω and φ. Contours start at −1.5 D with an increment of 0.2 D and end at +1.5 D, with negative value shown as dashed lines. The dotted horizontal lines mark the minima in φ according to the three different stable orientations, j, of the switch on the adamantane.

formed, ϕλT(θ), for T = aR or aS. Those wave functions which are entirely localized in only one of the three global minima along Φ, ϕ(j) μ (Φ), are calculated from the lowest 99 torsional eigenfunctions; i.e., two times M = 33 localized “oriented” wave functions are obtained for each orientation j. Two-dimensional product functions, ψ(j) T,v, are then generated from ϕλT(θ) and ϕ(j) μ (Φ) by L

ψT(j,)ν(θ , Φ) =

M

∑ ∑ cν ,λμϕλT (θ)ϕμ(j)(Φ) λ

B. COMPONENTS OF THE DIPOLE MOMENT As outlined in Section 2.1, the dipole moment was calculated as a function of θ and Φ, μ⃗ (θ, Φ), at the B3LYP/6-311G(d,p) level of theory using the optimized minimum geometry as a reference, in agreement with the calculation of the unrelaxed potential energy surface (PES). Afterward, a cosine series was fitted to the data points, and the coordinates were transformed to ω and φ, according to the procedure explained in Section 2.1. The obtained x- and y-components of the dipole function, μξ(ω, φ) (ξ ∈ {x, y}), are depicted in Figure 7 in the form of contour plots. The z-component is not shown since it does not interact with the laser. The fluorine substituents of the ethenyl and phenyl group account for most of the observed dipole moment. The independent torsion of two fluorine-substituted moieties, each around its angle (ω or φ), results in the apparently simple functional form of the dipole components. By comparison with the PES, shown in Figure 2(b), we can however get an idea of the challenge one faces when inducing an internal rotation of the ethenyl group (with respect to the phenyl group) toward the (aR) minimum for all three orientations of the switch at the same time. For instance, the gradients of μx(ω, φ) at the achiral minimum conformations significantly differ for each orientation j of the switch: Compare μx (240°, 60°), μx(0°, 180°), and μx(120°, 300°). It is these differences in the gradients that make stochastic pulse optimization such a powerful tool, as by choice of the fitness (product ansatz in eq 12) it allows us to effectively prevent a quasi-perfect stereoselective transformation for only one orientation at the cost of unselective excitations of the other two.

μ

(23)

Afterward they are converted to ω and φ according to eq 3, resulting in ψ(j) T,v(ω, φ). The molecular Hamiltonian Ĥ 0 (ω, φ) is, then, diagonalized in the basis of these 2D product functions for each T and (j) separately L

M

∑ ∑ cν ,λμ⟨ϕλ ′ T (ω)ϕμ(j′)(φ)|Ĥ 0(ω , φ)|ϕλT (ω)ϕμ(j)(φ)⟩ λ

μ

= E ν cν , λ ′ μ ′

(24)

(j) where ⟨ϕλ′Tϕ(j) μ′ |ϕλTϕμ ⟩ = δλ′μ′,λμ. Convergence was tested for one orientation j by increasing the number of 1D functions (L and M) until the eigenenergies, Eν, did not significantly alter any more, at least for energies up to the top of the low barriers along θ at approximately 85° and 275°. A basis of L = 16 and M = 33 functions was found adequate for each orientation. To ensure localized “chiral” wave functions of sufficient lifetime, only those 2D functions with energies, well below the top of the low barriers (at 1906 cm−1, vide supra), that are 188 (ν = 0−187) out of a possible 528 (= 16 × 33) for each orientation were considered as target functions to determine the population in the “chiral” states according to eq 15. Although convergence with the number of 1D wave functions was tested, we still cannot entirely rule out that even more eigenfunctions are found which qualify for the generation of localized “chiral” wave functions of sufficiently long lifetime, if the same full size grid basis was used as for the propagation. Therefore, the calculated populations in the “chiral” states may actually be considered a lower bound.

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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS We thank Christian Hensel for technical assistance as well as Peter Saalfrank for helpful discussions. Financial support by the German Research Foundation, DFG-project KR 2942/2, is 1330



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gratefully acknowledged. T.K. appreciates financial support by the German Research Foundation, Sfb 658 project C2.

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Laser Control for Coupled Torsions in Chiroptical Switches - American

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