1047
J . Phys. Chem. 1991, 95, 1047-1053 highest probability via direct collisional activation at high translational energies (So> 0.01 for EN > 65 kJ/mol) rather than via trapping or precursor mediated-processes at low incident translational energies (ET< 40 kJ/mol).
Acknowledgment. We thank the Department of Energy, Chemical Sciences Division, Office of Basic Energy Sciences (Grant DE-FG03-86ER13468), for their financial support of this work.
Optimal Control of Molecular Rotation in the Sudden Limit Liyang Shen and Herschel Rabitz* Department of Chemistry, Princeton University, Princeton, New Jersey 08544 (Received: April 3, 1990)
Optimal control theory is introduced for the control of quantum molecular rotational excitations induced by electric fields. Particular emphasis is given to the case where the electric field pulse is sufficiently short (- 1 ps) so that the sudden approximation can be made. Co:isequently, the time evolution of the rotational wave functions can be obtained analytically for general molecular rotations. A hyperbolic curve is shown to explicitly describe the relationship between the rotational energy and the action integral, x ( T ) = JrdSc(t) dt where d is the molecular permanent dipole moment, S the direction cosine matrix, and c ( r ) the applied electric field over time [0, r ] . For the case that the control cost functional has the form Q = F ( x ( T ) ) + /3.frc2(t) dt, it is found that the optimal fields are constant in time. The controllabilityof both rotational energy and transition probability is investigated. A detailed discussion on properties of the optimal fields, as well as the initial and final rotational states, is presented. Numerical calculations are performed for the molecule CsF.
Introduction Molecular rotation plays an important role in various chemical phenomena. For example, collision-induced chemical reactions can be strongly dependent on the relative orientations of the colliding molecules.’ Different initial rotational states and rotational energies may result in different product-state distributions. Thus the preparation of selected rotational states or distributions is an important objective. Currently, two laboratory methods have been developed to address this goal. One is to use inhomogeneous electrostatic or magnetic fields to select out the molecules with certain quantum rotational states from a molecular beam and then use a weak homogeneous field to orient the molecules (via the first-order Stark e f f e ~ t ) . ~ The - ~ other is to use lasers to pump molecules to the new rotational state^.^^^ However, both of these methods have their limitations. For example, the first method fails to select out the molecules of km 1 0 rotational states’ and the second requires the changes of either vibrational or electronic states. Another motivation to seek control of molecular rotation in this work arises from the recent progress in the theoretical study of bond-selective excitation and dissociation.’ The results from that work thus far are based on the assumption that the molecules being vibrationally excited are already oriented before the optimized pulse is applied. Given this assumption, controlling the molecular rotation (or molecular orientation) may therefore be needed as the first step before the bond-selected excitation or dissociation pulse is applied in the laboratory. (Simultaneous control of vibration and rotation also may be possible.) The capability of generating ultrafast arbitrarily shaped microwave pulsess suggests the possibility of controlling molecular rotation by designing suitable electric field pulses. Optimal control ( I ) For example: Bernstein, R. 9. Chemical Dynamics oia Molecular Beam and Loser Techniques; Clarendon Press: New York, 1982. (2) Xu, Q.-X.; Quesada, M. A.; Jung, K.-H.; Mackay, R. S.;Bernstein, R. 9. J. Chem. Phys. 1989, 91, 3477. (3) Mosch. J. E.; Safron, S. A.; Toennies, J. P. Chem. fhys. 1975.8, 304. (4) Carman, H. S.;Harland, P. W.; Brooks, P. R. J. fhys. Chem. 1986, 90, 944. (5) Karny, Z.; Estler, R. C.; Zare, R. N. J. Chem. fhys. 1978,69, 5199. (6) Scherer, N. F.; Khundkar, L. R.; Rose, T. S.; Zewail, A. H. J. Chem. Phys. 1987, 91, 6478. Baskin, J. S.; Felker, P. M.; Zewail, A. H. J. Chem. Phys. 1987.86, 2483. (7) Shi, S.;Woody, A.; Rabitz, H. J. Chem. fhys. 1988,88, 6870. Pierce, A. P.;Dahleh, M. A.; Rabitz, H. Phys. Reo. A 1988, 37,4950. Shi, S.; Rabitz, H. J. Chem. Phys. 1990, 92. 364. (8) Haner, M.; Warren, W. S. Appl. Opt. 1987, 26, 3687.
0022-3654/91/2095-1047%02.50/0
theory provides a rational set of tools for designing such pulses and a framework to accommodate rotational state objectives and field constraints to solve the inverse problem for the desired electric field. As an initial study, this research is carried out to understand the fundamental properties of the optimal control of molecular rotation. Particular attention is given to the investigation of various controllability problems, such as how the optimal fields vary with the initial and final rotational states and what relationship exists between the rotational energy and the optimal fields, etc. To avoid computational complications and be able to clearly reveal the relationship between the optimal field and molecular rotation, we consider the case that the interaction happens fast on the rotational time scale so that the sudden approximation can be applied. Ultimately this regime may be of significance for vibration-rotation control where a short (- 1 ps) pulse is used to ensure that the rotational portion is in the sudden regime. In a second paper pure rotational control in the nonsudden regime will be treated.9 This paper is organized in the following fashion: In the next section, we review the time-dependent Schriidinger equation for molecular rotation in the sudden regime, based on the rigid-rotor model, and then we discuss the controllability problems of molecular rotation with electric field pulses. The results can be expressed generally for any molecule up to an asymmetric top. A discussion will then be presented on the optimal control problem for molecular rotation. Two applications of optimal control to rotational energy and state preparation are then demonstrated. Finally we present some discussion and conclusions. Atomic units are utilized throughout this paper except in identified places. Rotational Wave Functions in the Sudden Limit We first assume that the molecule under consideration is rigid with nonzero permanent dipole moment d. In the molecule-fixed abc frame, the dipole moment d = (d,,db,d,) is a constant vector, and the free rotational Hamiltonian operator can be written as Ho = AJ,Z + BJb2 CJ,Z (1.1)
+
where J,, Jh, and J , are the angular momentum components with respect to the a, b, and c axes, respectively. A, B, and C denote the equilibrium rotational constants defined by
(9) Judson, R. S.; Lehmann, Siruci. 1990, 223, 425.
K. K.;Rabitz, H.; Warren, W. S. J. Mol.
0 1991 American Chemical Society
1048 The Journal of Physical Chemistry, Vol. 95, No. 3, 1991
Shen and Rabitz
CHART I cos 0 cos 4 cos x - sin 4 sin x -cos 0 cos 4 sin x - sin 4 cos x sin 0 cos 4
where I,, 16, and I , are the three principal moments of inertia. One can regard the molecule (and hence the dipole) rotating in space equivalently as the abc frame rotating in the laboratory frame XYZ described by the Euler angles 8, $, and x.Io Upon application of an electric field pulse c ( t ) with three components €At).e&), and tdt) along the X Y Z directions over a time interval [ O , q , the molecule will interact with the field and may change its rotational state. The interaction Hamiltonian operator is H , ( t ) = dSt(t) (11.3) where t(r) is defined as the column vector of (cx(t),tdt),cz(t)) and S is the direction cosine matrix in Chart 1.Io The time-dependent Schriidinger equation is then i dlJ.(t,8,4,x))/dt = ( H o + Hi(t))l$(t,e,+,x)) (0 I t I 77 (IISa) subject to IJ.(o,e,$,x)) = I$o(e,$,x))
(I1.5b)
where IJ.(t,e,+,x)) is the rotational wave functioin at time t and I$o(O.$,x)) is the initial rotational state. By defining (11.6) IJ.(t7e,4,x)) = exp(-iHot)14(t,8,$,x) ) the Schrdinger equation can then be transformed into i W 4 O 9 4 , x ) ) / d t = Bi,t(t)ld(rJ,d,x))
(0 It I T )
(11.7)
with BinI(f)= exp(iHot)Hi(t) exp(-iHot) = H i ( ? ) + it[Ho,H,(t)] + ( ( i t ) 2 / 2 ! ) [ H o , [ H o , H i ( t ) ] ]... (11.8)
+
The last equation (the so called Baker-Hausdorff expansion) results from the expansion of the exponential operators. The matrix element of the nth term (n = I , 2, ...) in the basis of H
