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Quantum optimal control strategies for photoisomerization via electronically excited states
3 July 1998
Chemical Physics Letters 290 Ž1998. 415–422
Quantum optimal control strategies for photoisomerization via electronically excited states Jorn ¨ Manz a, Karsten Sundermann a
a,1
, Regina de Vivie-Riedle
b,2
Institut fur ¨ Physikalische und Theoretische Chemie, Freie UniÕersitat ¨ Berlin, Takustraße 3, D-14195 Berlin, Germany b Max-Planck-Institut fur ¨ Quantenoptik, Hans-Kopfermann-Str. 1,D-85748 Garching, Germany Received 9 March 1998; in final form 14 April 1998
Abstract Optimal control of the photoisomerization of Li 2 Na from the stable acute to the near-degenerate obtuse triangular configuration is simulated by means of representative wave packet dynamics on two ab initio potential energy surfaces for the electronic ground ŽX 2AX . and excited Ž4 2AX . states. Product state specifity is achieved by means of new iteration methods wW. Zhu, J. Botina and H. Rabitz, J. Chem. Phys. 108 Ž1998. 1953x which incorporate feedback from the control field. An additional restriction yields a smooth switch on and off behaviour of the optimized pulses. A windowed Fourier transform decomposes the optimized laser field into efficient pump–dump pulses q 1998 Elsevier Science B.V. All rights reserved.
1. Introduction Control of chemical reactions by ultrashort laser pulses is a hot topic in current femtosecond chemistry w1–5x.Various theoretical approaches include the methods of Ži. infrared ultrashort laser pulses for selective transitions w6–9x, Žii. control by visible or ultraviolet pump–dump laser pulses w10–12x, Žiii. coherent control by ultrashort laser pulses w13x and Živ. optimal control w12,14–18x. Several of these concepts have been verified by experimental applications, see, e.g., Refs. Žii. w1,19,20x and Živ. w21x, based on ultrafast pulse shaping techniques w22x, providing a flexible tool for feedback strategies w15x. The experimental demonstrations have been restricted, however, to photodissociation and photoion1 2
E-mail: [email protected] E-mail: [email protected]
izations versus non-reactive processes of diatomic or triatomic molecules. To the best of our knowledge, control of photoisomerizations by ultrashort laser pulses has not yet been achieved – these reactions provide, therefore, an outstanding challenge. Three theoretical concepts towards this goal have been published, based on method Ži. w8x, Žii. w23x and Živ. w24x. The purpose of this paper is to suggest a new approach towards laser control of photoisomerizations, based on optimal control Živ. w15x. For reference we shall also consider the pump–dump Žii. w10,23x approach. The relation of the two methods Žii. and Žiii. has been recognized first in Ref. w12x, with application to laser control of photodissociations. In order to increase the experimental feasibility of the resulting optimal laser pulses we shall also employ two extensions of a new family of iteration methods which incorporate feedback from the con-
0009-2614r98r$19.00 q 1998 Elsevier Science B.V. All rights reserved. PII: S 0 0 0 9 - 2 6 1 4 Ž 9 8 . 0 0 4 7 2 - 2
J. Manz et al.r Chemical Physics Letters 290 (1998) 415–422
416
trol field w15x with Ža. smooth switch on and off behaviours w25x and Žb. only two frequencies, corresponding to the pump–dump frequencies of the alternative strategy Žii.. Moreover, we shall go beyond the simple one dimensional Ž1D. model of Refs. w15,23x towards a more demanding 2D system. Specificly, our new method will be applied to the realistic model system, Li 2 Na, which has been suggested to us by our colleagues, V. Bonacic-Koutecky ˇ´ ´ and L. Woste. ¨ According to the theoretical prediction w26x, this mixed alkali cluster has two different isomers of acute and obtuse triangular shapes, with rather small potential energy difference V Žobtuse. y V Žacute. s 0.07 eV and separated by a small potential barrier DV /s 0.03 eV. The two isomers could not be separated, up to now, using traditional nanosecond or cw-spectroscopy. The selective preparation of, say, the obtuse isomer from the acute one, by means of femtosecond laser pulses is, therefore, a demanding challenge, both to theory and experiment. In order to achieve this goal, we shall design optimal laser pulses for molecular wavepackets which are propagated on ab initio potential energy surfaces of Li 2 Na. For simplicity, we shall consider the case of C 2 Õ symmetry along the reaction path, as suggested by the ab initio configurations of the two isomers w26x. The corresponding 2D model and theory, the results and the conclusions are presented in Sections 2–4.
For the pump–dump scheme with two PES, coupled by the transition dipole moment meg , the system Hamiltonian is HŽ t. s
ž
T q Vg
0
0
T q Ve
/
According to the C 2 Õ symmetry of the two isomer states, the system is treated in a two dimensional model using Jacobi coordinates. With R denoting the Li 2 –Na distance,
m1 s
m Li 2 P m Na m Li 2 q m Na
the reduced mass of Li 2 –Na, r the Li–Li distance and m 2 s 12 m Li the reduced mass of Li–Li, the kinetic energy operator is given by "2 Tsy
E2
2 m1 E R 2
"2 y
E2
2 m2 E r 2
.
Ž 1.
ž
0
mge
meg
0
s H0 y e Ž t . P m
/ Ž 2.
The PES Vg and Ve for the electronic ground Žg. and excited Že. states have been calculated using the quantum chemistry program MOLPRO w27x, starting with a restricted Hartee–Fock calculation and refining the molecular orbitals first with a MCSCF calculation and correlating the three valence electrons with a MRCI calculation. Taking into account that the three normal modes in Li 2 Na break the C 2 Õ symmetry, the overall PES for our 2D modell have been calculated in C s symmetry. To check the feasibility of the photoisomerization process, with separable initial and target states of the reactants and products, the vibrational eigenfunctions of the electronic ground state had to be computed. For this purpose we have employed a combination of two techniques, i.e. filter diagonalization w28x with propagation in imaginary time w29x as filtering operation. Explicitly the diagonalization works as follows: Ø select an arbitrary starting function c 0 Ø compute n low energy wave functions C i by propagation in imaginary time H F 0 s exp yi P Ž yi P dt . c 0 Ž 3. " Fk Ck s Ž 4. 5F k 5
ž
2. Model and theory
ye Ž t.
/
iy1
ž
Ci s 1 y
Ý Pk ks0
/
exp Ž yi P H Ž yi P dt . .
iy1
ž
= 1y
Ý Pk ks0
/
c0
Ž 5.
with Pk being the projector on C k Ø setup the orthonormal basis B s C i < 0 F i - n4
Ž 6.
Ø construct the Hamilton matrix Hi , j s ²C i < H0
Ž 7.
in the basis B of Eq. Ž6. and diagonalize it by means of conventional methods.
J. Manz et al.r Chemical Physics Letters 290 (1998) 415–422
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The quality of the computed eigenstates can be evaluated by
Ž13. with the given boundary conditions C i Ž0. being the initial state and Cf ŽT . the desired target state.
f Ž C . s H0C y ²C < H0
i"
Ž 8.
Conventionally, a computed wave function C is considered a good approximation to the true eigenstate if f ŽC . - 10y6 Eh . The time evolution of the wave functions, both in real and in imaginary time, has been performed by using the split operator technique w30x as a short time propagator. For the analytical determination of the optimal laser field, which will drive the system from an initial state C i Ž0. at t s 0 to a desired target state Cf ŽT . at time t s T the following set of coupled differential equations has been solved.
a0 e Ž t . s ys Ž t . P Im Ž ²C i Ž t .
E i"
Et
C i Ž t . s H0 y m P e Ž t . C i Ž t .
Ž 10 .
C f Ž t . s H0 y m P e Ž t . C f Ž t .
Ž 11 .
E i"
Et
Eq. Ž9. describes the optimal electric field in terms of the evolving wave packets C i and Cf , whereas Eqs. Ž10. and Ž11. guarantee the compliance of the Schrodinger equation for C i and Cf under the ¨ influence of the driving field e Ž t .. Compared to the original equations of Zhu et al. w15x, the equations above are modified, with a shape function sŽ t . being added in the differential equation ŽEq. Ž9.. for the electric field e Ž t .. This additional shape function sŽ t . may be used to satisfy the experimental requirements that the electric field has to have a smooth start and end. For example a function satisfying this demand is sŽ t . s sin2 Žp Tt . w6x where T denotes the overall pulse duration. For the special case sŽ t . s 1 one recovers the original equations of Ref. w15x. The explicit derivation for the set of differential equations including sŽt. and their numerical implementation will be given elsewhere w25x. The parameter a 0 limits the time averaged laser intensity. The equations are solved by inserting Eq. Ž9. into Eqs. Ž10. and Ž11. and then solving iteratively Eqs. Ž12. and
E Et
C i Ž t . s H0 y m P s Ž t . P Im Ž ²C i Ž t .
Ž 12 .
E i"
Et
Cf Ž t . s H0 y m P s Ž t . P Im Ž ²C i Ž t .
Ž 13 .
3. Results 3.1. Quantum chemistry The intermediate PES involved in the pump–dump scheme has been preselected according to the selection rules for optical transitions. Considering the 2D model system in C 2 Õ symmetry, the electronic ground state wave function of the acute isomer transforms according to A 1 symmetry and for the obtuse isomer according to B 1 symmetry. Therefore, transitions into the electronic states of A 2 symmetry are forbidden in the Franck–Condon ŽFC. region of the acute form. Likewise transitions from electronic states of B 2 symmetry are forbidden in the FC region of the obtuse form. Test calculations for the first excited state of AXX symmetry revealed that these selection rules are also conserved for the molecule in C s symmetry and therefore the AXX state is not appropriate for the pump–dump process. Except for the first AXX state all the other excited states exhibit couplings and avoided crossings. We restricted our calculations to the set of the electronic ground state Ž12AX , 22AX . and the lowest lying set of excited states Ž3–52AX . in AX symmetry. The PES for the electronic ground state X2AX and the excited 42AX state are represented in Fig. 1. The two isomers can easily be distin˚ guished. The acute isomer is located at r s 2.75 A ˚ ˚ and R s 3.10 A and the obtuse one at r s 3.35 A ˚ A detailed analysis of the PES and R s 2.57 A. shows that the energy difference between the isomers is DV s 0.07 eV with an energy barrier of only DV /s 0.03 eV. For the selection of the pump–dump intermediate state a one dimensional cut has been extracted from the ab initio data for all the five computed PES, leading through the minima of the
418
J. Manz et al.r Chemical Physics Letters 290 (1998) 415–422
X X Fig. 1. Ab initio ground ŽX2A . and excited Ž42A . state potential energy surface for the photoisomerization of Li 2 Na. The intuitive pump and dump scheme for the laser control is indicated by the arrows. The equipotential lines differ by D Es 0.033 eV.
isomers. In Fig. 2 the one dimensional cuts are presented along the reaction coordinate, which is projected onto the Li–Li bond length. Also indicated in Figs. 1 and 2 is an intuitive pump–dump cycle which transfers population from the acute to the obtuse conformation and which should serve as a reference for the presented control. For this purpose
Fig. 3. Vibrational eigenstates of Li 2 Na shown by equidistant contours superimposed on equidistant contours Ž D Es 0.033 eV. for the potential energy surface of the electronic ground state.
we have chosen the third electronically excited state as the intermediate one, because it is the only one which permits a dynamical evolution of the pumped wave packet towards the target isomer. The fourth excited state would stop the wave packets motion too early, whereas the second excited state is too similar to the ground state, so that the relevant dynamics towards the desired isomer would not develop. As the pump–dump process occurs on a femtosecond time scale, the couplings to the 32AX and 52AX state were neglected in the 2D model of our simulation. In Condon approximation we chose a constant transition dipole moment m which has been set to m s 1ea0 reflecting typical values at some ab initio points. Table 1 Fundamental frequencies of the two isomers with the corresponding vibrational modes in C2 Õ symmetry
Fig. 2. One-dimensional cuts through the ab initio potential energy surfaces along the reaction coordinate for the photoisomerization of Li 2 Na, projected on the Li–Li bond length. Laser control by the intuitive pump dump scheme is indicated by arrows.
Isomer acute obtuse
n1
n2 y1
151 cm ŽLi 2 –Na. 143 cmy1 Ž Q x .
335 cmy1 ŽLi–Li. 234 cmy1 Ž Q s .
J. Manz et al.r Chemical Physics Letters 290 (1998) 415–422
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Table 2 Populations of the product vibrational eigenstates achieved by hand-optimized pump–dump laser pulses for the case when Õ s 0,0 is the target state Target state
PŽ Õ s 0,0.
PŽ Õ s 1,0.
PŽ Õ s 0,1.
PŽ Õ s 2,0.
P s Õ Ž1,1.
X PŽX2A .
Population
0.26
0.003
0.11
0.001
0.004
0.62
3.2. Vibrational eigenfunctions In the upper left panel of Fig. 3 the vibrational ground state of the acute isomer is shown and in the upper right the vibrational ground state of the obtuse isomer. These eigenfunctions clearly show, that the isomers can be separated. At the bottom of Fig. 3, from left to right, the first excited vibrational eigenfunctions of the obtuse isomer representing the Q x vibration Ž Õ s 0,1. and the Q s vibration Ž Õ s 1,0. are presented. Evidently, a small part of these eigenfunctions already tunnels towards the lower potential minimum, but a detailed investigation shows that in both cases more than 90% of the wave function remains in the product isomer. This tunnelling already indicates, that it is not a trivial task to selectively populate the product isomer. By analyzing higher excited eigenfunctions, one can assign six of them to the higher energy isomer. For the subsequent detection of the isomers their different vibrational spectra are of importance and given in Table 1. For the acute isomer the vibrations can be assigned approximately as Li 2 –Na or Li–Li vibrations, whereas for the obtuse isomer the eigenfunctions correspond to vibrations along the normal modes Q x and Q s . 3.3. Pump–dump photoisomerization by ‘hand’ optimized laser pulses In a first step we tried to achieve the photoisomerization of Li 2 Na by the traditional pump–dump ap-
proach of Tannor and Rice w10,11x, i.e. we assumed a sequence of two Gaussian type pulses with frequencies n 1 , n 2 , pulse width ŽFWHM. s 1 , s 2 and a delay time DT. With an initial guess for the frequencies according to the diagram in Fig. 2, subsequent optimization ‘by hand’ yields the best parameters n 1 s 10643 cmy1 , n 2 s 10029 cmy1 , s 1 s 51 fs, s 2 s 34 fs and DT s 92 fs. Table 2 shows the resulting populations of the eigenstates in the target isomer together with the total population of the electronic ground state. A large portion of the wave packet remains on the excited surface. By our experience it is impossible to design ‘hand’ optimized pump–dump pulses for this twofold electronic inversion process in order to achieve a high transfer yield to the obtuse isomer with simultaneous selectivity for the vibrational target states. This deficiency was overcome by means of optimal control methods. 3.4. Optimal control Using Eqs. Ž12. and Ž13. for the field optimization, and the vibrational ground state of the acute isomer as initial wave function C i Ž0., we computed optimal laser fields to drive the system towards selective eigenstates of the obtuse isomer as target states Cf ŽT .. The time T for the overall process has been set to T s 35000 E"h f 850 fs. The differential equations ŽEqs. Ž12. and Ž13.. have been solved first for the original approach of Zhu et al. w15x i.e. sŽ t . s 1, then for our new extended method includ-
Table 3 Populations for the product vibrational eigenstates achieved by optimal laser pulses for three different target states, using the shape function sŽ t . s sin2 Žp 85 0t fs . Target state
PŽ Õ s 0,0.
PŽ Õ s 1,0.
PŽ Õ s 0,1.
PŽ Õ s 2,0.
PŽ Õ s 1,1.
X PŽX2A .
Õ s 0,0 Õ s 1,0 Õ s 0,1
0.65 0.02 0.00
0.02 0.57 0.00
0.01 0.00 0.65
0.00 0.00 0.00
0.00 0.00 0.00
0.84 0.78 0.86
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Table 4 Populations for the product vibrational eigenstates achieved by optimal laser pulses for three different target, using the shape function sŽ t . s 1 Žyielding the original equations of Ref. w15x. Target state
PŽ Õ s 0,0.
PŽ Õ s 1,0.
PŽ Õ s 0,1.
PŽ Õ s 2,0.
PŽ Õ s 1,1.
X PŽX2A .
Õ s 0,0 Õ s 1,0 Õ s 0,1
0.50 0.01 0.03
0.01 0.36 0.03
0.04 0.03 0.43
0.00 0.00 0.00
0.00 0.00 0.00
0.79 0.71 0.77
ing the shape function sŽ t . s sin2 Žp 850t fs .. For the optimization procedure it was necessary to provide an initial guess for the laser field. According to Ref. w15x a constant initial field is satisfactory when optimizing transfer yields for one-dimensional systems on one potential surface, but this proved to be insufficient in our simulation of the pump–dump process. The initial guess had to include frequency components close to the potential energy gaps for vertical FC transitions in the domain of the reactants and products. Nevertheless, an initial guess with arbitrary envelope and a rough estimate of the pump–dump frequencies proved to be good enough. The resulting populations of the different vibrational states for the shape functions sŽ t . s sin2 Žp 850t fs . and sŽ t . s 1 are given in Tables 3 and 4. For both cases of sŽ t . the optimization has been performed with a 0 s 4.0 and using six iteration steps. The total populations of the electronic ground state PŽX2AX . are given in both tables in the last column, i.e. the loss of selectivity due to transfer of the system to the electronically excited state in the photoisomerization process is given by P Ž42AX . s 1 y P ŽX2AX .. Complementary populations of rather marginal highly excited delocalized vibrational states in the electronic ground state are omitted in Tables 3 and 4. Fig. 4 shows the optimized laser fields, the analysis in time and frequency of the electric field and population dynamics for our modified algorithm. For comparison, the optimal laser pulse for sŽ t . s 1 with high field intensities in the beginning is placed on top, followed by the optimized field with sŽ t . s sin2 Žp 850t fs ., which smoothly increases in intensity in the beginning and decreases at the end. The corresponding dynamics of the population transfer between the electronic ground state X2AX and excited state 42AX as well as the population dynamics of the target state Õ s 0,0 are shown at the bottom. The
final population of the target state is 0.65 of the total wave function, or, since 0.16 remained in the 42AX state, 0.77 of the electronic ground state population. The temporal evolution of the population dynamics suggests that the optimized laser field is composed mainly of two pulses similar to the intuitive pump– dump process. A frequency analysis revealed the two major frequencies, which when traced in time yield two pulses with their envelopes shown in the third panel. Superposition of two laser pulses with these
Fig. 4. Optimized laser pulses and population dynamics for the photoisomerization of Li 2 Na, starting from the acute isomer and targeting the vibrational ground state Õ s 0,0 of the obtuse one.
J. Manz et al.r Chemical Physics Letters 290 (1998) 415–422
envelopes and corresponding frequencies n 1 s 10600 cmy1 , n 2 s 10030 cmy1 approximates very closely the optimal field. The reconstructed laser pulse transfers 0.38 of the population to the desired target state, which is 0.53 of the ground state population. This is less than achieved by the approach of Zhu, Botina and Rabitz w15x but much more than the transfer rate achieved by ‘hand optimized’ laser pulses. Additionally this reconstructed laser pulse is much more likely to be realized experimentally than the one of the original optimal control scheme.
4. Conclusions The presented model simulations of the photoisomerization of Li 2 Na from the stable acute to the near-degenerate obtuse triangular configuration suggest an approach to optimal laser control with increasing experimental feasibility of the resulting laser fields. The approach is based on the new iteration method of Zhu, Botina and Rabitz w15x and it invokes the following steps. First Ži. one should determine the reactant and product configurations, together with the corresponding two laser frequencies of vertical FC transitions between the electronic ground and suitable excited states. Second Žii. it is helpful but not necessary to design two pump–dump laser pulses for laser control of the photoisomerization by means of the pump–dump scheme of Tannor and Rice w10x, as advocated for photoisomerization by Marquardt and Quack w23x. If the pump–dump scheme yields sufficient product selectivity one may omit any further optimization steps. Else one should make the third step Žiii. and use the resulting laser fields of steps Ži., Žii., i.e. either Ži. a superposition of two rectangular pulses with appropriate FC transition frequencies, or Žii. the hand optimized fields of the pump–dump-approach, as a start for the iterative scheme of the method of optimal control w15x, together with our modification Ža. of smooth switch on an off behaviours w25x. For an experimental realization of the optimized laser pulse, one may add our second modification Žb., i.e. to determine first the dominant frequencies by an ordinary Fourier transform and subsequently their corresponding envelopes by a windowed Fourier transform. In the present case only two relevant frequencies occur in accord with
421
an intuitive pump–dump cycle. The time-dependent superposition of the two or more dominant frequency components weighted with their corresponding envelopes should provide an efficient driving field which, according to our experience, should combine the virtues of optimal control and a step towards experimental feasibility. Beyond these general conclusions, we also note a number of special items based on the results shown in Section 3.4. ŽA. Our results show that a rather optimistic conclusion of Ref. w15x has to be modified when one goes from simple 1D-models to more complex and demanding systems such as the present one, Li 2 Na. Specifically a simple laser field like a constant one as in w15x could not be used for starting the field optimization. But already laser fields with rather arbitrary none zero envelopes comprising the FC frequencies for the pump–dump scheme proved to be a good starting point for the optimization process. ŽB. It is both amazing and rewarding that the state specifity achieved with our new extension is even higher than the result of the original method Ref. w15x, even though our extension implies a restriction of the original method since we impose the constraint of smooth switch on and off behaviours of the resulting optimal field, via the additional shape function sŽ t . w25x. The apparent paradox of increasing selectivity with restricted fields may be due to the fact that the method of optimal control w14–18x yields local optima of the driving laser pulse, not global ones. As a consequence, the iteration method of the unbiased technique w15x may approach a local optimum which is different from the global one, whereas our extension yields another local optimum which is even closer to the global one. This item suggests further investigations of suitable conditions for global optimization in laser control of chemical reactions. ŽC. Close analysis of the unbiased w15x and the present extended optimal laser fields indicates similar time evolutions of the dominant frequency components, except for some time delay. In the unbiased approach, the dominant components appear immediately, causing the abrupt increase of the optimal field which is likely to be not experimentally feasible. In our extended approach, the switch on of those components is delayed such as to achieve the smooth
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J. Manz et al.r Chemical Physics Letters 290 (1998) 415–422
overall behaviour of the optimal pulse. Except for this different switch-on behaviour, both optimal pulses obtained by the methods of w15x and w25x have similar switch off behaviours, i.e. they yield rather long tails with very weak intensities causing only marginal population dynamics at the end of the pulses. It is suggestive that these tails are not decisive for obtaining optimal population transfer and that they can therefore be omitted. This would allow to design even shorter optimal laser pulses – an important goal for photoisomerization of more complex systems where selective reactivity has to compete against additional ultrashort dissipative processes such as intra-molecular vibrational redistribution. Work along this line is in progress.
Acknowledgements We would like to acknowledge fruitful discussions with the groups of Prof. V. Bonacic-Koutecky ˇ´ ´ ŽHU-Berlin. and Prof. L. Woste ŽFU-Berlin. and ¨ financial support of the DFG through project No. Sfb 337. J. Manz also thanks Fonds der chemischen Industrie for continuous support.
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Chemical Physics Letters 290 Ž1998. 415–422
Quantum optimal control strategies for photoisomerization via electronically excited states Jorn ¨ Manz a, Karsten Sundermann a
a,1
, Regina de Vivie-Riedle
b,2
Institut fur ¨ Physikalische und Theoretische Chemie, Freie UniÕersitat ¨ Berlin, Takustraße 3, D-14195 Berlin, Germany b Max-Planck-Institut fur ¨ Quantenoptik, Hans-Kopfermann-Str. 1,D-85748 Garching, Germany Received 9 March 1998; in final form 14 April 1998
Abstract Optimal control of the photoisomerization of Li 2 Na from the stable acute to the near-degenerate obtuse triangular configuration is simulated by means of representative wave packet dynamics on two ab initio potential energy surfaces for the electronic ground ŽX 2AX . and excited Ž4 2AX . states. Product state specifity is achieved by means of new iteration methods wW. Zhu, J. Botina and H. Rabitz, J. Chem. Phys. 108 Ž1998. 1953x which incorporate feedback from the control field. An additional restriction yields a smooth switch on and off behaviour of the optimized pulses. A windowed Fourier transform decomposes the optimized laser field into efficient pump–dump pulses q 1998 Elsevier Science B.V. All rights reserved.
1. Introduction Control of chemical reactions by ultrashort laser pulses is a hot topic in current femtosecond chemistry w1–5x.Various theoretical approaches include the methods of Ži. infrared ultrashort laser pulses for selective transitions w6–9x, Žii. control by visible or ultraviolet pump–dump laser pulses w10–12x, Žiii. coherent control by ultrashort laser pulses w13x and Živ. optimal control w12,14–18x. Several of these concepts have been verified by experimental applications, see, e.g., Refs. Žii. w1,19,20x and Živ. w21x, based on ultrafast pulse shaping techniques w22x, providing a flexible tool for feedback strategies w15x. The experimental demonstrations have been restricted, however, to photodissociation and photoion1 2
E-mail: [email protected] E-mail: [email protected]
izations versus non-reactive processes of diatomic or triatomic molecules. To the best of our knowledge, control of photoisomerizations by ultrashort laser pulses has not yet been achieved – these reactions provide, therefore, an outstanding challenge. Three theoretical concepts towards this goal have been published, based on method Ži. w8x, Žii. w23x and Živ. w24x. The purpose of this paper is to suggest a new approach towards laser control of photoisomerizations, based on optimal control Živ. w15x. For reference we shall also consider the pump–dump Žii. w10,23x approach. The relation of the two methods Žii. and Žiii. has been recognized first in Ref. w12x, with application to laser control of photodissociations. In order to increase the experimental feasibility of the resulting optimal laser pulses we shall also employ two extensions of a new family of iteration methods which incorporate feedback from the con-
0009-2614r98r$19.00 q 1998 Elsevier Science B.V. All rights reserved. PII: S 0 0 0 9 - 2 6 1 4 Ž 9 8 . 0 0 4 7 2 - 2
J. Manz et al.r Chemical Physics Letters 290 (1998) 415–422
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trol field w15x with Ža. smooth switch on and off behaviours w25x and Žb. only two frequencies, corresponding to the pump–dump frequencies of the alternative strategy Žii.. Moreover, we shall go beyond the simple one dimensional Ž1D. model of Refs. w15,23x towards a more demanding 2D system. Specificly, our new method will be applied to the realistic model system, Li 2 Na, which has been suggested to us by our colleagues, V. Bonacic-Koutecky ˇ´ ´ and L. Woste. ¨ According to the theoretical prediction w26x, this mixed alkali cluster has two different isomers of acute and obtuse triangular shapes, with rather small potential energy difference V Žobtuse. y V Žacute. s 0.07 eV and separated by a small potential barrier DV /s 0.03 eV. The two isomers could not be separated, up to now, using traditional nanosecond or cw-spectroscopy. The selective preparation of, say, the obtuse isomer from the acute one, by means of femtosecond laser pulses is, therefore, a demanding challenge, both to theory and experiment. In order to achieve this goal, we shall design optimal laser pulses for molecular wavepackets which are propagated on ab initio potential energy surfaces of Li 2 Na. For simplicity, we shall consider the case of C 2 Õ symmetry along the reaction path, as suggested by the ab initio configurations of the two isomers w26x. The corresponding 2D model and theory, the results and the conclusions are presented in Sections 2–4.
For the pump–dump scheme with two PES, coupled by the transition dipole moment meg , the system Hamiltonian is HŽ t. s
ž
T q Vg
0
0
T q Ve
/
According to the C 2 Õ symmetry of the two isomer states, the system is treated in a two dimensional model using Jacobi coordinates. With R denoting the Li 2 –Na distance,
m1 s
m Li 2 P m Na m Li 2 q m Na
the reduced mass of Li 2 –Na, r the Li–Li distance and m 2 s 12 m Li the reduced mass of Li–Li, the kinetic energy operator is given by "2 Tsy
E2
2 m1 E R 2
"2 y
E2
2 m2 E r 2
.
Ž 1.
ž
0
mge
meg
0
s H0 y e Ž t . P m
/ Ž 2.
The PES Vg and Ve for the electronic ground Žg. and excited Že. states have been calculated using the quantum chemistry program MOLPRO w27x, starting with a restricted Hartee–Fock calculation and refining the molecular orbitals first with a MCSCF calculation and correlating the three valence electrons with a MRCI calculation. Taking into account that the three normal modes in Li 2 Na break the C 2 Õ symmetry, the overall PES for our 2D modell have been calculated in C s symmetry. To check the feasibility of the photoisomerization process, with separable initial and target states of the reactants and products, the vibrational eigenfunctions of the electronic ground state had to be computed. For this purpose we have employed a combination of two techniques, i.e. filter diagonalization w28x with propagation in imaginary time w29x as filtering operation. Explicitly the diagonalization works as follows: Ø select an arbitrary starting function c 0 Ø compute n low energy wave functions C i by propagation in imaginary time H F 0 s exp yi P Ž yi P dt . c 0 Ž 3. " Fk Ck s Ž 4. 5F k 5
ž
2. Model and theory
ye Ž t.
/
iy1
ž
Ci s 1 y
Ý Pk ks0
/
exp Ž yi P H Ž yi P dt . .
iy1
ž
= 1y
Ý Pk ks0
/
c0
Ž 5.
with Pk being the projector on C k Ø setup the orthonormal basis B s C i < 0 F i - n4
Ž 6.
Ø construct the Hamilton matrix Hi , j s ²C i < H0
Ž 7.
in the basis B of Eq. Ž6. and diagonalize it by means of conventional methods.
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The quality of the computed eigenstates can be evaluated by
Ž13. with the given boundary conditions C i Ž0. being the initial state and Cf ŽT . the desired target state.
f Ž C . s H0C y ²C < H0
i"
Ž 8.
Conventionally, a computed wave function C is considered a good approximation to the true eigenstate if f ŽC . - 10y6 Eh . The time evolution of the wave functions, both in real and in imaginary time, has been performed by using the split operator technique w30x as a short time propagator. For the analytical determination of the optimal laser field, which will drive the system from an initial state C i Ž0. at t s 0 to a desired target state Cf ŽT . at time t s T the following set of coupled differential equations has been solved.
a0 e Ž t . s ys Ž t . P Im Ž ²C i Ž t .
E i"
Et
C i Ž t . s H0 y m P e Ž t . C i Ž t .
Ž 10 .
C f Ž t . s H0 y m P e Ž t . C f Ž t .
Ž 11 .
E i"
Et
Eq. Ž9. describes the optimal electric field in terms of the evolving wave packets C i and Cf , whereas Eqs. Ž10. and Ž11. guarantee the compliance of the Schrodinger equation for C i and Cf under the ¨ influence of the driving field e Ž t .. Compared to the original equations of Zhu et al. w15x, the equations above are modified, with a shape function sŽ t . being added in the differential equation ŽEq. Ž9.. for the electric field e Ž t .. This additional shape function sŽ t . may be used to satisfy the experimental requirements that the electric field has to have a smooth start and end. For example a function satisfying this demand is sŽ t . s sin2 Žp Tt . w6x where T denotes the overall pulse duration. For the special case sŽ t . s 1 one recovers the original equations of Ref. w15x. The explicit derivation for the set of differential equations including sŽt. and their numerical implementation will be given elsewhere w25x. The parameter a 0 limits the time averaged laser intensity. The equations are solved by inserting Eq. Ž9. into Eqs. Ž10. and Ž11. and then solving iteratively Eqs. Ž12. and
E Et
C i Ž t . s H0 y m P s Ž t . P Im Ž ²C i Ž t .
Ž 12 .
E i"
Et
Cf Ž t . s H0 y m P s Ž t . P Im Ž ²C i Ž t .
Ž 13 .
3. Results 3.1. Quantum chemistry The intermediate PES involved in the pump–dump scheme has been preselected according to the selection rules for optical transitions. Considering the 2D model system in C 2 Õ symmetry, the electronic ground state wave function of the acute isomer transforms according to A 1 symmetry and for the obtuse isomer according to B 1 symmetry. Therefore, transitions into the electronic states of A 2 symmetry are forbidden in the Franck–Condon ŽFC. region of the acute form. Likewise transitions from electronic states of B 2 symmetry are forbidden in the FC region of the obtuse form. Test calculations for the first excited state of AXX symmetry revealed that these selection rules are also conserved for the molecule in C s symmetry and therefore the AXX state is not appropriate for the pump–dump process. Except for the first AXX state all the other excited states exhibit couplings and avoided crossings. We restricted our calculations to the set of the electronic ground state Ž12AX , 22AX . and the lowest lying set of excited states Ž3–52AX . in AX symmetry. The PES for the electronic ground state X2AX and the excited 42AX state are represented in Fig. 1. The two isomers can easily be distin˚ guished. The acute isomer is located at r s 2.75 A ˚ ˚ and R s 3.10 A and the obtuse one at r s 3.35 A ˚ A detailed analysis of the PES and R s 2.57 A. shows that the energy difference between the isomers is DV s 0.07 eV with an energy barrier of only DV /s 0.03 eV. For the selection of the pump–dump intermediate state a one dimensional cut has been extracted from the ab initio data for all the five computed PES, leading through the minima of the
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J. Manz et al.r Chemical Physics Letters 290 (1998) 415–422
X X Fig. 1. Ab initio ground ŽX2A . and excited Ž42A . state potential energy surface for the photoisomerization of Li 2 Na. The intuitive pump and dump scheme for the laser control is indicated by the arrows. The equipotential lines differ by D Es 0.033 eV.
isomers. In Fig. 2 the one dimensional cuts are presented along the reaction coordinate, which is projected onto the Li–Li bond length. Also indicated in Figs. 1 and 2 is an intuitive pump–dump cycle which transfers population from the acute to the obtuse conformation and which should serve as a reference for the presented control. For this purpose
Fig. 3. Vibrational eigenstates of Li 2 Na shown by equidistant contours superimposed on equidistant contours Ž D Es 0.033 eV. for the potential energy surface of the electronic ground state.
we have chosen the third electronically excited state as the intermediate one, because it is the only one which permits a dynamical evolution of the pumped wave packet towards the target isomer. The fourth excited state would stop the wave packets motion too early, whereas the second excited state is too similar to the ground state, so that the relevant dynamics towards the desired isomer would not develop. As the pump–dump process occurs on a femtosecond time scale, the couplings to the 32AX and 52AX state were neglected in the 2D model of our simulation. In Condon approximation we chose a constant transition dipole moment m which has been set to m s 1ea0 reflecting typical values at some ab initio points. Table 1 Fundamental frequencies of the two isomers with the corresponding vibrational modes in C2 Õ symmetry
Fig. 2. One-dimensional cuts through the ab initio potential energy surfaces along the reaction coordinate for the photoisomerization of Li 2 Na, projected on the Li–Li bond length. Laser control by the intuitive pump dump scheme is indicated by arrows.
Isomer acute obtuse
n1
n2 y1
151 cm ŽLi 2 –Na. 143 cmy1 Ž Q x .
335 cmy1 ŽLi–Li. 234 cmy1 Ž Q s .
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Table 2 Populations of the product vibrational eigenstates achieved by hand-optimized pump–dump laser pulses for the case when Õ s 0,0 is the target state Target state
PŽ Õ s 0,0.
PŽ Õ s 1,0.
PŽ Õ s 0,1.
PŽ Õ s 2,0.
P s Õ Ž1,1.
X PŽX2A .
Population
0.26
0.003
0.11
0.001
0.004
0.62
3.2. Vibrational eigenfunctions In the upper left panel of Fig. 3 the vibrational ground state of the acute isomer is shown and in the upper right the vibrational ground state of the obtuse isomer. These eigenfunctions clearly show, that the isomers can be separated. At the bottom of Fig. 3, from left to right, the first excited vibrational eigenfunctions of the obtuse isomer representing the Q x vibration Ž Õ s 0,1. and the Q s vibration Ž Õ s 1,0. are presented. Evidently, a small part of these eigenfunctions already tunnels towards the lower potential minimum, but a detailed investigation shows that in both cases more than 90% of the wave function remains in the product isomer. This tunnelling already indicates, that it is not a trivial task to selectively populate the product isomer. By analyzing higher excited eigenfunctions, one can assign six of them to the higher energy isomer. For the subsequent detection of the isomers their different vibrational spectra are of importance and given in Table 1. For the acute isomer the vibrations can be assigned approximately as Li 2 –Na or Li–Li vibrations, whereas for the obtuse isomer the eigenfunctions correspond to vibrations along the normal modes Q x and Q s . 3.3. Pump–dump photoisomerization by ‘hand’ optimized laser pulses In a first step we tried to achieve the photoisomerization of Li 2 Na by the traditional pump–dump ap-
proach of Tannor and Rice w10,11x, i.e. we assumed a sequence of two Gaussian type pulses with frequencies n 1 , n 2 , pulse width ŽFWHM. s 1 , s 2 and a delay time DT. With an initial guess for the frequencies according to the diagram in Fig. 2, subsequent optimization ‘by hand’ yields the best parameters n 1 s 10643 cmy1 , n 2 s 10029 cmy1 , s 1 s 51 fs, s 2 s 34 fs and DT s 92 fs. Table 2 shows the resulting populations of the eigenstates in the target isomer together with the total population of the electronic ground state. A large portion of the wave packet remains on the excited surface. By our experience it is impossible to design ‘hand’ optimized pump–dump pulses for this twofold electronic inversion process in order to achieve a high transfer yield to the obtuse isomer with simultaneous selectivity for the vibrational target states. This deficiency was overcome by means of optimal control methods. 3.4. Optimal control Using Eqs. Ž12. and Ž13. for the field optimization, and the vibrational ground state of the acute isomer as initial wave function C i Ž0., we computed optimal laser fields to drive the system towards selective eigenstates of the obtuse isomer as target states Cf ŽT .. The time T for the overall process has been set to T s 35000 E"h f 850 fs. The differential equations ŽEqs. Ž12. and Ž13.. have been solved first for the original approach of Zhu et al. w15x i.e. sŽ t . s 1, then for our new extended method includ-
Table 3 Populations for the product vibrational eigenstates achieved by optimal laser pulses for three different target states, using the shape function sŽ t . s sin2 Žp 85 0t fs . Target state
PŽ Õ s 0,0.
PŽ Õ s 1,0.
PŽ Õ s 0,1.
PŽ Õ s 2,0.
PŽ Õ s 1,1.
X PŽX2A .
Õ s 0,0 Õ s 1,0 Õ s 0,1
0.65 0.02 0.00
0.02 0.57 0.00
0.01 0.00 0.65
0.00 0.00 0.00
0.00 0.00 0.00
0.84 0.78 0.86
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Table 4 Populations for the product vibrational eigenstates achieved by optimal laser pulses for three different target, using the shape function sŽ t . s 1 Žyielding the original equations of Ref. w15x. Target state
PŽ Õ s 0,0.
PŽ Õ s 1,0.
PŽ Õ s 0,1.
PŽ Õ s 2,0.
PŽ Õ s 1,1.
X PŽX2A .
Õ s 0,0 Õ s 1,0 Õ s 0,1
0.50 0.01 0.03
0.01 0.36 0.03
0.04 0.03 0.43
0.00 0.00 0.00
0.00 0.00 0.00
0.79 0.71 0.77
ing the shape function sŽ t . s sin2 Žp 850t fs .. For the optimization procedure it was necessary to provide an initial guess for the laser field. According to Ref. w15x a constant initial field is satisfactory when optimizing transfer yields for one-dimensional systems on one potential surface, but this proved to be insufficient in our simulation of the pump–dump process. The initial guess had to include frequency components close to the potential energy gaps for vertical FC transitions in the domain of the reactants and products. Nevertheless, an initial guess with arbitrary envelope and a rough estimate of the pump–dump frequencies proved to be good enough. The resulting populations of the different vibrational states for the shape functions sŽ t . s sin2 Žp 850t fs . and sŽ t . s 1 are given in Tables 3 and 4. For both cases of sŽ t . the optimization has been performed with a 0 s 4.0 and using six iteration steps. The total populations of the electronic ground state PŽX2AX . are given in both tables in the last column, i.e. the loss of selectivity due to transfer of the system to the electronically excited state in the photoisomerization process is given by P Ž42AX . s 1 y P ŽX2AX .. Complementary populations of rather marginal highly excited delocalized vibrational states in the electronic ground state are omitted in Tables 3 and 4. Fig. 4 shows the optimized laser fields, the analysis in time and frequency of the electric field and population dynamics for our modified algorithm. For comparison, the optimal laser pulse for sŽ t . s 1 with high field intensities in the beginning is placed on top, followed by the optimized field with sŽ t . s sin2 Žp 850t fs ., which smoothly increases in intensity in the beginning and decreases at the end. The corresponding dynamics of the population transfer between the electronic ground state X2AX and excited state 42AX as well as the population dynamics of the target state Õ s 0,0 are shown at the bottom. The
final population of the target state is 0.65 of the total wave function, or, since 0.16 remained in the 42AX state, 0.77 of the electronic ground state population. The temporal evolution of the population dynamics suggests that the optimized laser field is composed mainly of two pulses similar to the intuitive pump– dump process. A frequency analysis revealed the two major frequencies, which when traced in time yield two pulses with their envelopes shown in the third panel. Superposition of two laser pulses with these
Fig. 4. Optimized laser pulses and population dynamics for the photoisomerization of Li 2 Na, starting from the acute isomer and targeting the vibrational ground state Õ s 0,0 of the obtuse one.
J. Manz et al.r Chemical Physics Letters 290 (1998) 415–422
envelopes and corresponding frequencies n 1 s 10600 cmy1 , n 2 s 10030 cmy1 approximates very closely the optimal field. The reconstructed laser pulse transfers 0.38 of the population to the desired target state, which is 0.53 of the ground state population. This is less than achieved by the approach of Zhu, Botina and Rabitz w15x but much more than the transfer rate achieved by ‘hand optimized’ laser pulses. Additionally this reconstructed laser pulse is much more likely to be realized experimentally than the one of the original optimal control scheme.
4. Conclusions The presented model simulations of the photoisomerization of Li 2 Na from the stable acute to the near-degenerate obtuse triangular configuration suggest an approach to optimal laser control with increasing experimental feasibility of the resulting laser fields. The approach is based on the new iteration method of Zhu, Botina and Rabitz w15x and it invokes the following steps. First Ži. one should determine the reactant and product configurations, together with the corresponding two laser frequencies of vertical FC transitions between the electronic ground and suitable excited states. Second Žii. it is helpful but not necessary to design two pump–dump laser pulses for laser control of the photoisomerization by means of the pump–dump scheme of Tannor and Rice w10x, as advocated for photoisomerization by Marquardt and Quack w23x. If the pump–dump scheme yields sufficient product selectivity one may omit any further optimization steps. Else one should make the third step Žiii. and use the resulting laser fields of steps Ži., Žii., i.e. either Ži. a superposition of two rectangular pulses with appropriate FC transition frequencies, or Žii. the hand optimized fields of the pump–dump-approach, as a start for the iterative scheme of the method of optimal control w15x, together with our modification Ža. of smooth switch on an off behaviours w25x. For an experimental realization of the optimized laser pulse, one may add our second modification Žb., i.e. to determine first the dominant frequencies by an ordinary Fourier transform and subsequently their corresponding envelopes by a windowed Fourier transform. In the present case only two relevant frequencies occur in accord with
421
an intuitive pump–dump cycle. The time-dependent superposition of the two or more dominant frequency components weighted with their corresponding envelopes should provide an efficient driving field which, according to our experience, should combine the virtues of optimal control and a step towards experimental feasibility. Beyond these general conclusions, we also note a number of special items based on the results shown in Section 3.4. ŽA. Our results show that a rather optimistic conclusion of Ref. w15x has to be modified when one goes from simple 1D-models to more complex and demanding systems such as the present one, Li 2 Na. Specifically a simple laser field like a constant one as in w15x could not be used for starting the field optimization. But already laser fields with rather arbitrary none zero envelopes comprising the FC frequencies for the pump–dump scheme proved to be a good starting point for the optimization process. ŽB. It is both amazing and rewarding that the state specifity achieved with our new extension is even higher than the result of the original method Ref. w15x, even though our extension implies a restriction of the original method since we impose the constraint of smooth switch on and off behaviours of the resulting optimal field, via the additional shape function sŽ t . w25x. The apparent paradox of increasing selectivity with restricted fields may be due to the fact that the method of optimal control w14–18x yields local optima of the driving laser pulse, not global ones. As a consequence, the iteration method of the unbiased technique w15x may approach a local optimum which is different from the global one, whereas our extension yields another local optimum which is even closer to the global one. This item suggests further investigations of suitable conditions for global optimization in laser control of chemical reactions. ŽC. Close analysis of the unbiased w15x and the present extended optimal laser fields indicates similar time evolutions of the dominant frequency components, except for some time delay. In the unbiased approach, the dominant components appear immediately, causing the abrupt increase of the optimal field which is likely to be not experimentally feasible. In our extended approach, the switch on of those components is delayed such as to achieve the smooth
422
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overall behaviour of the optimal pulse. Except for this different switch-on behaviour, both optimal pulses obtained by the methods of w15x and w25x have similar switch off behaviours, i.e. they yield rather long tails with very weak intensities causing only marginal population dynamics at the end of the pulses. It is suggestive that these tails are not decisive for obtaining optimal population transfer and that they can therefore be omitted. This would allow to design even shorter optimal laser pulses – an important goal for photoisomerization of more complex systems where selective reactivity has to compete against additional ultrashort dissipative processes such as intra-molecular vibrational redistribution. Work along this line is in progress.
Acknowledgements We would like to acknowledge fruitful discussions with the groups of Prof. V. Bonacic-Koutecky ˇ´ ´ ŽHU-Berlin. and Prof. L. Woste ŽFU-Berlin. and ¨ financial support of the DFG through project No. Sfb 337. J. Manz also thanks Fonds der chemischen Industrie for continuous support.
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