- Email: [email protected]
One approach to the control of intramolecular hydrogen transfer
21 Janurary 2000
Chemical Physics Letters 316 Ž2000. 551–557 www.elsevier.nlrlocatercplett
One approach to the control of intramolecular hydrogen transfer Y. Ohta ) , T. Yoshimoto, K. Nishikawa Department of Computational Science, Faculty of Science, Kanazawa UniÕersity, Kanazawa 920-1192, Japan Received 21 September 1999; in final form 8 November 1999
Abstract One approach using the counterintuitive pulse sequence ŽCPS. is presented to control the intramolecular hydrogen transfer. We have applied the asymmetric double-well model to the substituted malonaldehyde and have shown the complete population transfer from one local minimum state to another one through the barrier using CPS. We also have found in our simulation that the high product yield is robust with respect to the changes of the laser parameters. The mechanism of the population transfer is elucidated with our approach with the Huckel theory in a systematic and pictorial way. q 2000 ¨ Elsevier Science B.V. All rights reserved.
1. Introduction The control of chemical reaction using strong coherent light has been studied by many researchers. Tannor and Rice have presented the pump–dump scheme w1,2x where the wavepacket is first created on the excited potential energy surface by the pump pulse. This wavepacket is then dropped to the desired state by the dump pulse in an appropriate time delay for the pump pulse. Rabitz et al. have proposed an optimal control theory w3,4x in which the laser pulse shape is appropriately tailored so that the molecule is led to a desired state. Paramonov et al. have presented a method in which a pulse sequence is made to induce state-selective transition by generalized p pulses w5,6x. These methods have been theoretically applied to the control of the isomerization reaction, which has
) Corresponding author. Fax: q81-762-64-5742; e-mail: [email protected]
not been experimentally demonstrated as far as we know. The isomerization reaction has been well treated with the double minimum potential model. Paramonov et al. have simulated the selective transition from the reactant to the product via some intermediate states using a pulse sequence in the infrared domain w7x. Fujimura et al. have designed a pulse by means of the optimal control theory, which leads the initial wavepacket to the product well over the potential barrier in the electronic ground state potential energy surface w8x. Quack et al. have applied the pump–dump scheme to the stereomutation of chiral molecule which has a double minimum potential in the electronic ground state and a harmonic oscillator potential in the achiral excited electronic state w9x. Sundermann et al. have presented one approach w10x to the hydrogen transfer in the substituted malonaldehyde, in which a laser pulse designed by the optimal control theory induces the direct transition from the reactant to the product. Therefore, they called this situation the hydrogen subway. In our analysis, however, the pulse applied is essentially
0009-2614r00r$ - see front matter q 2000 Elsevier Science B.V. All rights reserved. PII: S 0 0 0 9 - 2 6 1 4 Ž 9 9 . 0 1 3 1 7 - 2
552
Y. Ohta et al.r Chemical Physics Letters 316 (2000) 551–557
equal to a p-pulse which directly connects the reactant and the product. The p-pulse method practically needs some strong conditions on the laser parameters to make the required pulse so that we apply the more effective CPS to this system in this work. The CPS method which is often referred to the delayed pulse method w11–15x, has a weak restriction on the laser parameters, i.e., the temporal behavior of the system is insensitive to some changes of the laser parameters. The CPS method is well known as being an effective method for population transfer in threelevel systems. The features of the CPS method are that Ži. there is an appropriate delay time between two laser pulses, Žii. two pulses are in a counterintuitive order and Žiii. each pulse has a larger pulse area than the ordinary p-pulse. For example, let us consider the population transfer from an initial state <1: to a final state <3: in a three-level system with eigenstates <1:, <2: and <3:. In the conventional p-pulse method, two p-pulses are used where laser 1 connecting <1: and <2: is first applied to induce the complete transition from <1: to <2:; then laser 2 connecting <2: and <3: is applied to transfer completely from <2: to the final state <3:. On the other hand, in the CPS method, laser 2 precedes laser 1, contrary to the p-pulse method. As long as above three flexible conditions are satisfied, this method allows efficient population transfer from <1: to <3: without appreciable population in the intermediate state <2:. For this reason, the efficiency of the population transfer is not affected by decay resulting from the lifetime of the intermediate state. Experimentally, the vibrationally selective population transfer in the ground electronic state of some simple molecules has been demonstrated by taking a vibrational level of the excited electronic state as an intermediate state w16–23x. In such experiments, this method is called STIRAP Žstimulated Raman adiabatic passage.. However, to the best of our knowledge, it has not been applied to the isomerization reaction system. In this work, we apply the CPS method to the system with an intramolecular hydrogen bond. This reaction system could be described well by the onedimensional asymmetry double-well model, the parameters of which are chosen to describe the motion of the hydrogen in the substituted malonaldehyde w10x. Using this model, we show that the CPS method
allows the complete population transfer from the reactant to the product without appreciable population of an intermediate state above the potential barrier. Although the result is similar to that of Sundermann et al., that is, the reactant state moves to the product state through the barrier, the mechanism of the transition is substantially different, since our approach is based on the adiabatic process of the molecular system interacting with light. Thus, we show that the result with the CPS method is robust with respect to the variation of the laser parameters. In our previous paper, we have presented a pictorial and easy way using the Huckel model to under¨ stand the adiabatic process w24x. Here, we describe briefly the mechanism of complete hydrogen transfer by the CPS method using our approach. 2. Model system We are here interested in the control of the hydrogen movement in the hydrogen bond. In general, the motion of the hydrogen bond could be well described by a double-well potential. A molecule in the electromagnetic field is semiclassically described by H s H0 q Hint Ž t . Ž 1. where H0 is the molecular Hamiltonian and satisfies the following time-independent Schrodinger equa¨ tion: H 0 < f n : s En < f n : Ž 2. < : where fn and En are the eigenstate and eigenvalue of the molecular system, respectively. Hint is the interaction Hamiltonian which describes the interaction of molecular system with the classical electromagnetic field: Hint Ž t . s ym ´ Ž t . Ž 3. where m is the molecular dipole moment operator and is assumed to be linear for the reaction coordinate q, i.e., m Ž q . s eq. ´ Ž t . is the oscillating electric field, which is assumed to be linearly polarized along the same direction of the molecular dipole moment. The model Hamiltonian of the hydrogen bond is given by H0 s y
1
E2
2 m E2 q
qV Ž q. ,
Ž 4.
Y. Ohta et al.r Chemical Physics Letters 316 (2000) 551–557
where m is the hydrogen mass. Potential V Ž q . is given by an asymmetric double-well type: D V Ž q. s Ž q y q0 . 2 q0 V X y Dr2 2 2 q Ž q y q0 . Ž q q q0 . Ž 5. q04 with an asymmetry parameter D s 0.000257, the barrier height V X s 0.000625, and potential minima at q s "q0 , Ž q0 s 1 .. These values are taken to describe the motion of the hydrogen in the substituted malonaldehyde and in the atomic unit. The eigenvalues En and associated eigenfunctions fnŽ q . of the model Hamiltonian are computed by the Fourier grid Hamiltonian ŽFGH. method w25x. The V Ž q . and the wavefunctions are represented on the 101-point spatial grid with qmin s y2.0 a.u., D q s 0.04 a.u., and qmax s 2.0 a.u. Fig. 1 shows explicitly the shape of V Ž q . and some wavefunctions Ž f 0 , f 1 , and f 3 .. It shows that the ground state < f 0 : and first excited state < f 1 : are localized in the left and right minimum of this potential. In the CPS method, the two pulses are applied so that the electric field is given by ´ Ž t . s ´ 1 Ž t . cos v 1 t q ´ 2 Ž t . cos v 2 t where v i is the carrier frequency of laser pulse i. Lasers 1 and 2 induce the dipole transitions between
553
< f 0 : and < f 3 :, and < f 1 : and < f 3 :, respectively. The pulse shapes are assumed to be the Gaussian curve:
´ i Ž t . s ´ i0 g i Ž t .
Ž 6.
with 2
g i Ž t . s exp y Ž t y t i . rs 2
Ž 7.
where ´ i0 and t i are the maximum amplitude and the center time of the Gaussian pulse i, respectively. For simplicity, we also assumed that the two pulses have same width s . For convenience, we define t s Ž t 1 y t 2 .rs which measures the delay time of two pulses in unit of s . If t ) 0, pulse 2 is followed by pulse 1, namely, this pulse sequence is called the counterintuitive pulses. The dynamics of the hydrogen bond system is described by the time-dependent Schrodinger equa¨ tion i"
d
s H
Ž 8.
An arbitrary state,
Ž 9.
n
Substituting Eq. Ž9. into Eq. Ž8., we can obtain the following differential equation for the expansion coefficients. i"
dCi Ž t . dt
s y´ Ž t . Ý ² f i < m < f j : exp i v i j t C j Ž t . j
Ž 10 . with
vi j s
Fig. 1. The plot of the double-well potential energy surface for the transfer of hydrogen in substituted malonaldehyde; V Ž q . s D Ž V X y D r2. Ž q y q0 . q Ž q y q0 . 2 Ž q q q0 . 2 , where an 2 q0 q04 X asymmetry parameter D s 0.000257 a.u., the barrier height V s 0.000625 a.u., and q0 s1 a.u. The lowest eigenenergies E0 s 0.00218 a.u., E1 s 0.00248 a.u., E2 s 0.00583 a.u., E3 s 0.00691 a.u. and eigenfunctions f 0 , f 1 , f 3 are also depicted.
ž /
Ei y E j "
Ž 11 .
where m i j is the value of the transition dipole moment between states < f i : and < f j :. These values were calculated by the wavefunction fnŽ q . obtained by the FGH method. This equation is numerically solved by the Runge–Kutta method with the initial condition C0 Ž0. s 1.0, CnŽ0. s 0.0 Ž n / 0. and the time step D t s 0.1 a.u. The population of the eigenstates f i at time t is evaluated by PnŽ t . s <² fn
Y. Ohta et al.r Chemical Physics Letters 316 (2000) 551–557
554
For the simulation of the adiabatic time evolution, it is convenient to introduce the pulse area A i :
transitions are hardly induced due to large detuning. Therefore, V 10 and V 20 are given by
Ai s Vi Ž t . d t
V 10 s
Ž 12 .
H
where V i Ž t . is the time-dependent Rabi frequency related to the laser field i and means the coupling strength of the induced dipole transitions:
V 1 Ž t . s V 10 g 1 Ž t . ,
V 2 Ž t . s V 20 g 2 Ž t . .
m 03 ´ 10 "
,
V 20 s
m 13 ´ 20 "
where m 03 s ² f 0 < m < f 3 : s 0.159 , m 13 s ² f 1 < m < f 3 : s 0.148 .
Ž 14 .
Ž 15 .
Ž 13 .
In our simulations, we evaluate the only Rabi frequencies which related to the transitions between f 0 and f 1 , and f 1 and f 3 since the other dipole
3. Results and discussion In this section, we discuss the results obtained by solving Eq. Ž10. with the three eigenstates < f 0 :,
Fig. 2. Ža. Time variation of the Rabi frequencies. s s 6.3 ps, t s 1, V 10 s V 20 s 4.8 = 10y5 a.u., and A1 s A 2 s 7p. Žb. Time evolution of the populations. Final product yield P1 s 99.8%.
Y. Ohta et al.r Chemical Physics Letters 316 (2000) 551–557
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Now we will show briefly the mechanism of the adiabatic population transfer w24x. We expand an arbitrary state
Fig. 3. Ža. Time variation of the Rabi frequencies. s s6.3 ps, t s1.4, V 10 s V 20 s 4.8=10y5 a.u., and A1 s A1 s 7p. Žb. Time evolution of the populations. Final product yield P1 s96.9%.
< f 1 :, and < f 3 :. Note that we had confirmated on the simulation with the lowest 10 eigenstates that the amplitudes of the other states are hardly participated for the relevant transitions. In our simulations, the laser frequencies resonant to the corresponding transitions, i.e., v 1 s 1038 cmy1 , v 2 s 986 cmy1 are used. Here we simulated three cases with the different laser parameters in order to show explicitly the effectiveness of the CPS method. Figs. 2–4 show the time variation of the Rabi frequency and the resultant populations. The laser parameters in Fig. 2a are s s 6.3 ps, t s 1.0, V 10 s V 20 s 4.8 = 10y5 a.u., and A i s 7p. The final product yield P1 is 99.8%. In Fig. 3a, where we changed only t to be 1.4, we obtained P1 s 96.9%. In Fig. 4a, the V 20 is 1.5 times larger than the case in Fig. 2a; therefore, A 2 s 10.5p. Then, the final product yield P1 is 99.7%. In all cases, we realized the almost complete population transfer, so we could easily understand that the counterintuitive method is very flexible with respect to the shape of the laser pulse contrary to the p-pulse method.
2
0
V 1Ž t .
V 1Ž t .
0
0
V2Ž t .
0
0
V2Ž t . . 0
Ž 17 .
If the time variation of V 1Ž t . and V 2 Ž t . is slow enough, it is possible to carry out the instantaneous diagonalization of the Hamiltonian to yield three eigenstates Ždressed states., < q Ž t .:, < y Ž t .: and <0Ž t .: which can be expressed in terms of the bare states < f 0 :, < f 1 :, and < f 3 :. However, it is difficult to understand the mechanism of the adiabatic process because the Hamiltonian depends on time explicitly and the order of the laser pulses is counterintuitive. In our previous pa-
Fig. 4. Ža. Time variation of the Rabi frequencies. s s6.3 ps, t s1, V 10 s 4.8=10y5 a.u., V 20 s 7.2=10y5 a.u., A1 s 7p and A 2 s10.5p. Žb. Time evolution of the populations. Final product yield P1 s99.7%.
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Y. Ohta et al.r Chemical Physics Letters 316 (2000) 551–557
Fig. 5. The adiabatic time evolution of three dressed states, < q Ž t .:, < y Ž t .: and <0Ž t .:. Time regions I, II, III, IV and V as in Fig. 2a. The circles in < q :, < y : and <0: refer to the amplitudes of f 0 , f 1 and f 3 . The relative phase between the white and black circles is p.
per, we have presented an easy and pictorial way based upon the Huckel model, which has facilitated ¨ understanding of the adiabatic mechanism without diagonalizing the Hamiltonian. The diagonalization method gives the quantitative values of the eigenvalues and eigenvectors of the Hamiltonian. However, we usually lose the physical picture on the propagation of the interaction, which is fundamental to generate the molecular orbital from the related atomic orbitals ŽAO.. In our approach, we have utilized the fundamental concept of how to make a molecular orbital ŽMO. from the AO. Namely, the bonding and anti-bonding MOs are produced as a result of the constructive and destructive interference of two interacting AOs, respectively. Here, we briefly explain the adiabatic time evolution of each dressed state using our approach. We divide the whole time domain into five time regions as shown in Fig. 2a. Fig. 5 shows the adiabatic time evolution of three dressed states. The radius of circle means the magnitude of the contribution of the bare states to each dressed state and the white and black circles mean that the relative phase of them is p. In our approach, the dressed and bare states correspond to the MO and AO in the Huckel ¨ theory, respectively. In region I Ž V 1Ž t . s 0., laser 2 couples the state < f 1 : to < f 3 : to produce the two dressed states < y Ž t .: and < q Ž t .:, which correspond to the bonding and anti-orbitals in the Huckel molec¨ ular orbitals, respectively. Here the phase of < f 1 : and < f 3 : in the bonding orbital is the same, and their phase in the anti-bonding orbital is opposite. We
note that the initial state < f 0 : does not change by laser 1 in this region. In region II Ž V 2 Ž t . 4 V 1Ž t .. where laser 1 is applied gradually, < f 0 : begins to couple with the < f 3 : component in < q Ž t .: and < y Ž t .: in region I, and then new dressed states < q Ž t .:, <0Ž t .:, < y Ž t .: are created. By means of the Huckel method, the ¨ middle state <0Ž t .: is generated by the superposition of following two virtual states. One is the bonding orbital resulting from the coupling of < f 0 : to < q Ž t .:, in which the signs of coefficients of < f 0 :, < f 3 : and < f 1 : are q, q, and y, respectively. We call this orbital VDBO Žvirtual dressed bonding orbital.. Other orbital is anti-bonding resulting from the coupling of < f 0 : to < y Ž t .:, in which the signs of coefficients of < f 0 :, < f 3 : and < f 1 : are q, y, and y, respectively. We call this VDAO Žvirtual dressed anti-bonding orbital.. Therefore, the contribution of < f 3 : to <0Ž t .: is completely cancelled out due to the interference between the VDBO and VDAO, and <0Ž t .: has a large amplitude of < f 0 : rather than that of < f 1 : due to the stronger coupling of V 2 than that of V 1. In region III Ž V 2 Ž t . f V 1Ž t .., where V 1Ž t . is comparable to V 2 Ž t ., the contribution of < f 1 : to <0Ž t .: is also becoming comparable to that of < f 0 :, while the vanishment of the amplitude of < f 3 : is maintained. When V 2 Ž t . s V 1Ž t ., the Hamiltonian corresponds to the allyl radical in the Huckel method, ¨ and the middle state <0Ž t .: is the non-bonding orbital which has only the component of < f 0 : and < f 1 :. In region IV Ž V 1Ž t . 4 V 2 Ž t .., the strong coupling of V 1Ž t . produces < q Ž t .: and < y Ž t .: which
Y. Ohta et al.r Chemical Physics Letters 316 (2000) 551–557
are composed of < f 0 : and < f 3 : and then the week coupling of V 2 Ž t . produces three dressed states. Note that the contribution of < f 3 : to <0Ž t .: is canceled out as well as that in region II and that < f 1 : component becomes predominant in <0Ž t .:. In region V Ž V 2 Ž t . s 0., <0Ž t .: has only < f 1 : component because of the vanishment of V 2 Ž t ., while < q Ž t .: and < y Ž t .: turn into the two dressed states composed only of < f 0 : and < f 3 : by V 1Ž t .. Generally, the condition for the time evolution of <0Ž t .: to be adiabatic is given by
Ž d²0 Ž t .
<1
Ž 18 .
where the left-hand means the degree of the diabaticity and EnŽ t .Ž n s ". is the eigenvalue of the dressed state w26x. Using Eq. Ž13. and associated eigenvalues with the condition t , 1 and V 10 s V 20 s V 0 , inequality Ž18. reduces to w24x
V 0 s 4 1 or
A4p .
Ž 19 .
It is important that the adiabatic condition depends not only on the pulse width s but also coupling strength V 0 . 4. Conclusion We proposed a new approach with the CPS method to control the hydrogen motion in molecule. Our numerical simulation has shown that the CPS method allows complete population transfer from the reactant to the product without an appreciable intermediate population above potential barrier. We also have shown that the CPS method gives rise to a high product yield under the very flexible conditions of laser parameters. This could facilitate an experimental demonstration. We have also explained the mechanism of the population transfer qualitatively in a simple way based upon the Huckel model without diagonaliza¨ tion of the Hamiltonian. Our approach clearly showed that the non-population of an intermediate state strongly coupled to the initial and final states through two laser fields is due to the interference between the VDBO and VDAO. The present strategy could be easily applied to the other isomerization reaction if the potential could be
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described by the asymmetric double well. We could expect a broad application of this method not only for selective isomerizations but also for other chemical systems.
References w1x D.J. Tannor, S.A. Rice, J. Chem. Phys. 83 Ž1985. 5013. w2x D.J. Tannor, R. Kosloff, S.A. Rice, J. Chem. Phys. 85 Ž1985. 5805. w3x S. Shi, A. Woody, H. Rabitz, J. Chem. Phys. 88 Ž1988. 6870. w4x W. Zhu, J. Botina, H. Rabitz, J. Chem. Phys. 108 Ž1998. 1953. w5x G.K. Paramonov, in: J. Manz, L. Woste ¨ ŽEds.., Femtosecond Chemistry, vol. 2, Verlag-Chemie, Weinheim, 1995, pp. 671–712. w6x M.V. Korolkov, G.K. Paramonov, B. Schmidt, J. Chem. Phys. 105 Ž1996. 1862. w7x J.E. Combariza, B. Just, J. Manz, G.K. Paramonov, J. Phys. Chem. 95 Ž1991. 10351. w8x M. Sugawara, Y. Fujimura, J. Chem. Phys. 100 Ž1994. 5646. w9x R. Marquardt, M. Quack, Z. Physik D 36 Ž1996. 229. w10x N. Doslic, J. Manz, K. Sundermann, J. Phys. ˘ ´ O. Kuhn, ¨ Chem. A 102 Ž1998. 9645. w11x J. Oreg, F.T. Hioe, J.H. Eberly, Phys. Rev. A 29 Ž1984. 690. w12x A.V. Smith, J. Opt. Soc. Am. B 9 Ž1992. 1543. w13x C.E. Carroll, F.T. Hioe, Phys. Rev. A 42 Ž1990. 1522. w14x J.R. Kuklinski, U. Gaubatz, F.T. Hioe, K. Bergmann, Phys. Rev. 40 Ž1989. 6741. w15x B.W. Shore, K. Bergmann, J. Oreg, S. Rosenwaks, Phys. Rev. A 44 Ž1991. 7442. w16x U. Gaubatz, P. Rudecki, S. Schiemann, K. Bergmann, J. Chem. Phys. 92 Ž1990. 5363. w17x G. He, A. Kuhn, S. Schiemann, K. Bergmann, J. Opt. Soc. Am. B 7 Ž1990. 1960. w18x H.-G. Rubahn, E. Konz, S. Schiemann, K. Bergmann, Z. Physik D 22 Ž1991. 401. w19x P. Dittmann, F.P. Pesl, J. Martin, G.W. Coulston, G.Z. He, K. Bergmann, J. Chem. Phys. 97 Ž1992. 9472. w20x S. Schiemann, A. Kuhn, S. Steuerwald, K. Bergmann, Phys. Rev. Lett. 71 Ž1993. 3637. w21x J. Lawall, M. Prentiss, Phys. Rev. Lett. 72 Ž1994. 993. w22x T. Halfmann, K. Bergmann, J. Chem. Phys. 104 Ž1996. 7068. w23x A. Lindinger, M. Verbeek, H.-G. Rubahn, Z. Physik D 39 Ž1997. 93. w24x Y. Ohta, H. Kizu, K. Yoshimoto, A. Yamada, K. Nishikawa, Int. J. Quantum Chem. Symp. 1999, to be published. w25x C.C. Marston, G.G. Balint-Kurti, J. Chem. Phys. 91 Ž1989. 3571. w26x A. Messiah, Quantum Mechanics, vol. 2, North Holland, Amsterdam, 1980, p. 754.
Chemical Physics Letters 316 Ž2000. 551–557 www.elsevier.nlrlocatercplett
One approach to the control of intramolecular hydrogen transfer Y. Ohta ) , T. Yoshimoto, K. Nishikawa Department of Computational Science, Faculty of Science, Kanazawa UniÕersity, Kanazawa 920-1192, Japan Received 21 September 1999; in final form 8 November 1999
Abstract One approach using the counterintuitive pulse sequence ŽCPS. is presented to control the intramolecular hydrogen transfer. We have applied the asymmetric double-well model to the substituted malonaldehyde and have shown the complete population transfer from one local minimum state to another one through the barrier using CPS. We also have found in our simulation that the high product yield is robust with respect to the changes of the laser parameters. The mechanism of the population transfer is elucidated with our approach with the Huckel theory in a systematic and pictorial way. q 2000 ¨ Elsevier Science B.V. All rights reserved.
1. Introduction The control of chemical reaction using strong coherent light has been studied by many researchers. Tannor and Rice have presented the pump–dump scheme w1,2x where the wavepacket is first created on the excited potential energy surface by the pump pulse. This wavepacket is then dropped to the desired state by the dump pulse in an appropriate time delay for the pump pulse. Rabitz et al. have proposed an optimal control theory w3,4x in which the laser pulse shape is appropriately tailored so that the molecule is led to a desired state. Paramonov et al. have presented a method in which a pulse sequence is made to induce state-selective transition by generalized p pulses w5,6x. These methods have been theoretically applied to the control of the isomerization reaction, which has
) Corresponding author. Fax: q81-762-64-5742; e-mail: [email protected]
not been experimentally demonstrated as far as we know. The isomerization reaction has been well treated with the double minimum potential model. Paramonov et al. have simulated the selective transition from the reactant to the product via some intermediate states using a pulse sequence in the infrared domain w7x. Fujimura et al. have designed a pulse by means of the optimal control theory, which leads the initial wavepacket to the product well over the potential barrier in the electronic ground state potential energy surface w8x. Quack et al. have applied the pump–dump scheme to the stereomutation of chiral molecule which has a double minimum potential in the electronic ground state and a harmonic oscillator potential in the achiral excited electronic state w9x. Sundermann et al. have presented one approach w10x to the hydrogen transfer in the substituted malonaldehyde, in which a laser pulse designed by the optimal control theory induces the direct transition from the reactant to the product. Therefore, they called this situation the hydrogen subway. In our analysis, however, the pulse applied is essentially
0009-2614r00r$ - see front matter q 2000 Elsevier Science B.V. All rights reserved. PII: S 0 0 0 9 - 2 6 1 4 Ž 9 9 . 0 1 3 1 7 - 2
552
Y. Ohta et al.r Chemical Physics Letters 316 (2000) 551–557
equal to a p-pulse which directly connects the reactant and the product. The p-pulse method practically needs some strong conditions on the laser parameters to make the required pulse so that we apply the more effective CPS to this system in this work. The CPS method which is often referred to the delayed pulse method w11–15x, has a weak restriction on the laser parameters, i.e., the temporal behavior of the system is insensitive to some changes of the laser parameters. The CPS method is well known as being an effective method for population transfer in threelevel systems. The features of the CPS method are that Ži. there is an appropriate delay time between two laser pulses, Žii. two pulses are in a counterintuitive order and Žiii. each pulse has a larger pulse area than the ordinary p-pulse. For example, let us consider the population transfer from an initial state <1: to a final state <3: in a three-level system with eigenstates <1:, <2: and <3:. In the conventional p-pulse method, two p-pulses are used where laser 1 connecting <1: and <2: is first applied to induce the complete transition from <1: to <2:; then laser 2 connecting <2: and <3: is applied to transfer completely from <2: to the final state <3:. On the other hand, in the CPS method, laser 2 precedes laser 1, contrary to the p-pulse method. As long as above three flexible conditions are satisfied, this method allows efficient population transfer from <1: to <3: without appreciable population in the intermediate state <2:. For this reason, the efficiency of the population transfer is not affected by decay resulting from the lifetime of the intermediate state. Experimentally, the vibrationally selective population transfer in the ground electronic state of some simple molecules has been demonstrated by taking a vibrational level of the excited electronic state as an intermediate state w16–23x. In such experiments, this method is called STIRAP Žstimulated Raman adiabatic passage.. However, to the best of our knowledge, it has not been applied to the isomerization reaction system. In this work, we apply the CPS method to the system with an intramolecular hydrogen bond. This reaction system could be described well by the onedimensional asymmetry double-well model, the parameters of which are chosen to describe the motion of the hydrogen in the substituted malonaldehyde w10x. Using this model, we show that the CPS method
allows the complete population transfer from the reactant to the product without appreciable population of an intermediate state above the potential barrier. Although the result is similar to that of Sundermann et al., that is, the reactant state moves to the product state through the barrier, the mechanism of the transition is substantially different, since our approach is based on the adiabatic process of the molecular system interacting with light. Thus, we show that the result with the CPS method is robust with respect to the variation of the laser parameters. In our previous paper, we have presented a pictorial and easy way using the Huckel model to under¨ stand the adiabatic process w24x. Here, we describe briefly the mechanism of complete hydrogen transfer by the CPS method using our approach. 2. Model system We are here interested in the control of the hydrogen movement in the hydrogen bond. In general, the motion of the hydrogen bond could be well described by a double-well potential. A molecule in the electromagnetic field is semiclassically described by H s H0 q Hint Ž t . Ž 1. where H0 is the molecular Hamiltonian and satisfies the following time-independent Schrodinger equa¨ tion: H 0 < f n : s En < f n : Ž 2. < : where fn and En are the eigenstate and eigenvalue of the molecular system, respectively. Hint is the interaction Hamiltonian which describes the interaction of molecular system with the classical electromagnetic field: Hint Ž t . s ym ´ Ž t . Ž 3. where m is the molecular dipole moment operator and is assumed to be linear for the reaction coordinate q, i.e., m Ž q . s eq. ´ Ž t . is the oscillating electric field, which is assumed to be linearly polarized along the same direction of the molecular dipole moment. The model Hamiltonian of the hydrogen bond is given by H0 s y
1
E2
2 m E2 q
qV Ž q. ,
Ž 4.
Y. Ohta et al.r Chemical Physics Letters 316 (2000) 551–557
where m is the hydrogen mass. Potential V Ž q . is given by an asymmetric double-well type: D V Ž q. s Ž q y q0 . 2 q0 V X y Dr2 2 2 q Ž q y q0 . Ž q q q0 . Ž 5. q04 with an asymmetry parameter D s 0.000257, the barrier height V X s 0.000625, and potential minima at q s "q0 , Ž q0 s 1 .. These values are taken to describe the motion of the hydrogen in the substituted malonaldehyde and in the atomic unit. The eigenvalues En and associated eigenfunctions fnŽ q . of the model Hamiltonian are computed by the Fourier grid Hamiltonian ŽFGH. method w25x. The V Ž q . and the wavefunctions are represented on the 101-point spatial grid with qmin s y2.0 a.u., D q s 0.04 a.u., and qmax s 2.0 a.u. Fig. 1 shows explicitly the shape of V Ž q . and some wavefunctions Ž f 0 , f 1 , and f 3 .. It shows that the ground state < f 0 : and first excited state < f 1 : are localized in the left and right minimum of this potential. In the CPS method, the two pulses are applied so that the electric field is given by ´ Ž t . s ´ 1 Ž t . cos v 1 t q ´ 2 Ž t . cos v 2 t where v i is the carrier frequency of laser pulse i. Lasers 1 and 2 induce the dipole transitions between
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< f 0 : and < f 3 :, and < f 1 : and < f 3 :, respectively. The pulse shapes are assumed to be the Gaussian curve:
´ i Ž t . s ´ i0 g i Ž t .
Ž 6.
with 2
g i Ž t . s exp y Ž t y t i . rs 2
Ž 7.
where ´ i0 and t i are the maximum amplitude and the center time of the Gaussian pulse i, respectively. For simplicity, we also assumed that the two pulses have same width s . For convenience, we define t s Ž t 1 y t 2 .rs which measures the delay time of two pulses in unit of s . If t ) 0, pulse 2 is followed by pulse 1, namely, this pulse sequence is called the counterintuitive pulses. The dynamics of the hydrogen bond system is described by the time-dependent Schrodinger equa¨ tion i"
d
s H
Ž 8.
An arbitrary state,
Ž 9.
n
Substituting Eq. Ž9. into Eq. Ž8., we can obtain the following differential equation for the expansion coefficients. i"
dCi Ž t . dt
s y´ Ž t . Ý ² f i < m < f j : exp i v i j t C j Ž t . j
Ž 10 . with
vi j s
Fig. 1. The plot of the double-well potential energy surface for the transfer of hydrogen in substituted malonaldehyde; V Ž q . s D Ž V X y D r2. Ž q y q0 . q Ž q y q0 . 2 Ž q q q0 . 2 , where an 2 q0 q04 X asymmetry parameter D s 0.000257 a.u., the barrier height V s 0.000625 a.u., and q0 s1 a.u. The lowest eigenenergies E0 s 0.00218 a.u., E1 s 0.00248 a.u., E2 s 0.00583 a.u., E3 s 0.00691 a.u. and eigenfunctions f 0 , f 1 , f 3 are also depicted.
ž /
Ei y E j "
Ž 11 .
where m i j is the value of the transition dipole moment between states < f i : and < f j :. These values were calculated by the wavefunction fnŽ q . obtained by the FGH method. This equation is numerically solved by the Runge–Kutta method with the initial condition C0 Ž0. s 1.0, CnŽ0. s 0.0 Ž n / 0. and the time step D t s 0.1 a.u. The population of the eigenstates f i at time t is evaluated by PnŽ t . s <² fn
Y. Ohta et al.r Chemical Physics Letters 316 (2000) 551–557
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For the simulation of the adiabatic time evolution, it is convenient to introduce the pulse area A i :
transitions are hardly induced due to large detuning. Therefore, V 10 and V 20 are given by
Ai s Vi Ž t . d t
V 10 s
Ž 12 .
H
where V i Ž t . is the time-dependent Rabi frequency related to the laser field i and means the coupling strength of the induced dipole transitions:
V 1 Ž t . s V 10 g 1 Ž t . ,
V 2 Ž t . s V 20 g 2 Ž t . .
m 03 ´ 10 "
,
V 20 s
m 13 ´ 20 "
where m 03 s ² f 0 < m < f 3 : s 0.159 , m 13 s ² f 1 < m < f 3 : s 0.148 .
Ž 14 .
Ž 15 .
Ž 13 .
In our simulations, we evaluate the only Rabi frequencies which related to the transitions between f 0 and f 1 , and f 1 and f 3 since the other dipole
3. Results and discussion In this section, we discuss the results obtained by solving Eq. Ž10. with the three eigenstates < f 0 :,
Fig. 2. Ža. Time variation of the Rabi frequencies. s s 6.3 ps, t s 1, V 10 s V 20 s 4.8 = 10y5 a.u., and A1 s A 2 s 7p. Žb. Time evolution of the populations. Final product yield P1 s 99.8%.
Y. Ohta et al.r Chemical Physics Letters 316 (2000) 551–557
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Now we will show briefly the mechanism of the adiabatic population transfer w24x. We expand an arbitrary state
Fig. 3. Ža. Time variation of the Rabi frequencies. s s6.3 ps, t s1.4, V 10 s V 20 s 4.8=10y5 a.u., and A1 s A1 s 7p. Žb. Time evolution of the populations. Final product yield P1 s96.9%.
< f 1 :, and < f 3 :. Note that we had confirmated on the simulation with the lowest 10 eigenstates that the amplitudes of the other states are hardly participated for the relevant transitions. In our simulations, the laser frequencies resonant to the corresponding transitions, i.e., v 1 s 1038 cmy1 , v 2 s 986 cmy1 are used. Here we simulated three cases with the different laser parameters in order to show explicitly the effectiveness of the CPS method. Figs. 2–4 show the time variation of the Rabi frequency and the resultant populations. The laser parameters in Fig. 2a are s s 6.3 ps, t s 1.0, V 10 s V 20 s 4.8 = 10y5 a.u., and A i s 7p. The final product yield P1 is 99.8%. In Fig. 3a, where we changed only t to be 1.4, we obtained P1 s 96.9%. In Fig. 4a, the V 20 is 1.5 times larger than the case in Fig. 2a; therefore, A 2 s 10.5p. Then, the final product yield P1 is 99.7%. In all cases, we realized the almost complete population transfer, so we could easily understand that the counterintuitive method is very flexible with respect to the shape of the laser pulse contrary to the p-pulse method.
2
0
V 1Ž t .
V 1Ž t .
0
0
V2Ž t .
0
0
V2Ž t . . 0
Ž 17 .
If the time variation of V 1Ž t . and V 2 Ž t . is slow enough, it is possible to carry out the instantaneous diagonalization of the Hamiltonian to yield three eigenstates Ždressed states., < q Ž t .:, < y Ž t .: and <0Ž t .: which can be expressed in terms of the bare states < f 0 :, < f 1 :, and < f 3 :. However, it is difficult to understand the mechanism of the adiabatic process because the Hamiltonian depends on time explicitly and the order of the laser pulses is counterintuitive. In our previous pa-
Fig. 4. Ža. Time variation of the Rabi frequencies. s s6.3 ps, t s1, V 10 s 4.8=10y5 a.u., V 20 s 7.2=10y5 a.u., A1 s 7p and A 2 s10.5p. Žb. Time evolution of the populations. Final product yield P1 s99.7%.
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Y. Ohta et al.r Chemical Physics Letters 316 (2000) 551–557
Fig. 5. The adiabatic time evolution of three dressed states, < q Ž t .:, < y Ž t .: and <0Ž t .:. Time regions I, II, III, IV and V as in Fig. 2a. The circles in < q :, < y : and <0: refer to the amplitudes of f 0 , f 1 and f 3 . The relative phase between the white and black circles is p.
per, we have presented an easy and pictorial way based upon the Huckel model, which has facilitated ¨ understanding of the adiabatic mechanism without diagonalizing the Hamiltonian. The diagonalization method gives the quantitative values of the eigenvalues and eigenvectors of the Hamiltonian. However, we usually lose the physical picture on the propagation of the interaction, which is fundamental to generate the molecular orbital from the related atomic orbitals ŽAO.. In our approach, we have utilized the fundamental concept of how to make a molecular orbital ŽMO. from the AO. Namely, the bonding and anti-bonding MOs are produced as a result of the constructive and destructive interference of two interacting AOs, respectively. Here, we briefly explain the adiabatic time evolution of each dressed state using our approach. We divide the whole time domain into five time regions as shown in Fig. 2a. Fig. 5 shows the adiabatic time evolution of three dressed states. The radius of circle means the magnitude of the contribution of the bare states to each dressed state and the white and black circles mean that the relative phase of them is p. In our approach, the dressed and bare states correspond to the MO and AO in the Huckel ¨ theory, respectively. In region I Ž V 1Ž t . s 0., laser 2 couples the state < f 1 : to < f 3 : to produce the two dressed states < y Ž t .: and < q Ž t .:, which correspond to the bonding and anti-orbitals in the Huckel molec¨ ular orbitals, respectively. Here the phase of < f 1 : and < f 3 : in the bonding orbital is the same, and their phase in the anti-bonding orbital is opposite. We
note that the initial state < f 0 : does not change by laser 1 in this region. In region II Ž V 2 Ž t . 4 V 1Ž t .. where laser 1 is applied gradually, < f 0 : begins to couple with the < f 3 : component in < q Ž t .: and < y Ž t .: in region I, and then new dressed states < q Ž t .:, <0Ž t .:, < y Ž t .: are created. By means of the Huckel method, the ¨ middle state <0Ž t .: is generated by the superposition of following two virtual states. One is the bonding orbital resulting from the coupling of < f 0 : to < q Ž t .:, in which the signs of coefficients of < f 0 :, < f 3 : and < f 1 : are q, q, and y, respectively. We call this orbital VDBO Žvirtual dressed bonding orbital.. Other orbital is anti-bonding resulting from the coupling of < f 0 : to < y Ž t .:, in which the signs of coefficients of < f 0 :, < f 3 : and < f 1 : are q, y, and y, respectively. We call this VDAO Žvirtual dressed anti-bonding orbital.. Therefore, the contribution of < f 3 : to <0Ž t .: is completely cancelled out due to the interference between the VDBO and VDAO, and <0Ž t .: has a large amplitude of < f 0 : rather than that of < f 1 : due to the stronger coupling of V 2 than that of V 1. In region III Ž V 2 Ž t . f V 1Ž t .., where V 1Ž t . is comparable to V 2 Ž t ., the contribution of < f 1 : to <0Ž t .: is also becoming comparable to that of < f 0 :, while the vanishment of the amplitude of < f 3 : is maintained. When V 2 Ž t . s V 1Ž t ., the Hamiltonian corresponds to the allyl radical in the Huckel method, ¨ and the middle state <0Ž t .: is the non-bonding orbital which has only the component of < f 0 : and < f 1 :. In region IV Ž V 1Ž t . 4 V 2 Ž t .., the strong coupling of V 1Ž t . produces < q Ž t .: and < y Ž t .: which
Y. Ohta et al.r Chemical Physics Letters 316 (2000) 551–557
are composed of < f 0 : and < f 3 : and then the week coupling of V 2 Ž t . produces three dressed states. Note that the contribution of < f 3 : to <0Ž t .: is canceled out as well as that in region II and that < f 1 : component becomes predominant in <0Ž t .:. In region V Ž V 2 Ž t . s 0., <0Ž t .: has only < f 1 : component because of the vanishment of V 2 Ž t ., while < q Ž t .: and < y Ž t .: turn into the two dressed states composed only of < f 0 : and < f 3 : by V 1Ž t .. Generally, the condition for the time evolution of <0Ž t .: to be adiabatic is given by
Ž d²0 Ž t .
<1
Ž 18 .
where the left-hand means the degree of the diabaticity and EnŽ t .Ž n s ". is the eigenvalue of the dressed state w26x. Using Eq. Ž13. and associated eigenvalues with the condition t , 1 and V 10 s V 20 s V 0 , inequality Ž18. reduces to w24x
V 0 s 4 1 or
A4p .
Ž 19 .
It is important that the adiabatic condition depends not only on the pulse width s but also coupling strength V 0 . 4. Conclusion We proposed a new approach with the CPS method to control the hydrogen motion in molecule. Our numerical simulation has shown that the CPS method allows complete population transfer from the reactant to the product without an appreciable intermediate population above potential barrier. We also have shown that the CPS method gives rise to a high product yield under the very flexible conditions of laser parameters. This could facilitate an experimental demonstration. We have also explained the mechanism of the population transfer qualitatively in a simple way based upon the Huckel model without diagonaliza¨ tion of the Hamiltonian. Our approach clearly showed that the non-population of an intermediate state strongly coupled to the initial and final states through two laser fields is due to the interference between the VDBO and VDAO. The present strategy could be easily applied to the other isomerization reaction if the potential could be
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described by the asymmetric double well. We could expect a broad application of this method not only for selective isomerizations but also for other chemical systems.
References w1x D.J. Tannor, S.A. Rice, J. Chem. Phys. 83 Ž1985. 5013. w2x D.J. Tannor, R. Kosloff, S.A. Rice, J. Chem. Phys. 85 Ž1985. 5805. w3x S. Shi, A. Woody, H. Rabitz, J. Chem. Phys. 88 Ž1988. 6870. w4x W. Zhu, J. Botina, H. Rabitz, J. Chem. Phys. 108 Ž1998. 1953. w5x G.K. Paramonov, in: J. Manz, L. Woste ¨ ŽEds.., Femtosecond Chemistry, vol. 2, Verlag-Chemie, Weinheim, 1995, pp. 671–712. w6x M.V. Korolkov, G.K. Paramonov, B. Schmidt, J. Chem. Phys. 105 Ž1996. 1862. w7x J.E. Combariza, B. Just, J. Manz, G.K. Paramonov, J. Phys. Chem. 95 Ž1991. 10351. w8x M. Sugawara, Y. Fujimura, J. Chem. Phys. 100 Ž1994. 5646. w9x R. Marquardt, M. Quack, Z. Physik D 36 Ž1996. 229. w10x N. Doslic, J. Manz, K. Sundermann, J. Phys. ˘ ´ O. Kuhn, ¨ Chem. A 102 Ž1998. 9645. w11x J. Oreg, F.T. Hioe, J.H. Eberly, Phys. Rev. A 29 Ž1984. 690. w12x A.V. Smith, J. Opt. Soc. Am. B 9 Ž1992. 1543. w13x C.E. Carroll, F.T. Hioe, Phys. Rev. A 42 Ž1990. 1522. w14x J.R. Kuklinski, U. Gaubatz, F.T. Hioe, K. Bergmann, Phys. Rev. 40 Ž1989. 6741. w15x B.W. Shore, K. Bergmann, J. Oreg, S. Rosenwaks, Phys. Rev. A 44 Ž1991. 7442. w16x U. Gaubatz, P. Rudecki, S. Schiemann, K. Bergmann, J. Chem. Phys. 92 Ž1990. 5363. w17x G. He, A. Kuhn, S. Schiemann, K. Bergmann, J. Opt. Soc. Am. B 7 Ž1990. 1960. w18x H.-G. Rubahn, E. Konz, S. Schiemann, K. Bergmann, Z. Physik D 22 Ž1991. 401. w19x P. Dittmann, F.P. Pesl, J. Martin, G.W. Coulston, G.Z. He, K. Bergmann, J. Chem. Phys. 97 Ž1992. 9472. w20x S. Schiemann, A. Kuhn, S. Steuerwald, K. Bergmann, Phys. Rev. Lett. 71 Ž1993. 3637. w21x J. Lawall, M. Prentiss, Phys. Rev. Lett. 72 Ž1994. 993. w22x T. Halfmann, K. Bergmann, J. Chem. Phys. 104 Ž1996. 7068. w23x A. Lindinger, M. Verbeek, H.-G. Rubahn, Z. Physik D 39 Ž1997. 93. w24x Y. Ohta, H. Kizu, K. Yoshimoto, A. Yamada, K. Nishikawa, Int. J. Quantum Chem. Symp. 1999, to be published. w25x C.C. Marston, G.G. Balint-Kurti, J. Chem. Phys. 91 Ž1989. 3571. w26x A. Messiah, Quantum Mechanics, vol. 2, North Holland, Amsterdam, 1980, p. 754.