2156
J . Phys. Chem. 1991, 95, 2156-2166
only in both phases. The TX' of this carbon has the same value in each phase. The decays of the /3 and 7 carbohs have two components in both phases, with the T2G' values of Gaussian component larger than that of the 6 carbon. These results confirm that the C, rotation of the adamantyl group persists in both high-
and low-temperature phases.
Acknowledgment. This research was supported by grants from the NSERC (Canada) and FCAR (Quebec). Y.H.acknowledges an award of fellowship from McGill University.
An Operator-Algebraic Method for Propagation of a Wave Packet: An Application to Photodissociation Dynamics Kiyohiko Someda; Tamotsu Kondow,* and Kozo Kuchitsut Department of Chemistry, Faculty of Science, The University of Tokyo, Bunkyo-ku, Tokyo 113. Japan (Received: April 24, 1990)
A theory of wave packet dynamics based on the operator-algebraic method is developed and applied to photodissociation. A wave packet is propagated by an approximate time-evolution operator, which involves several variational parameters. The time evolution of these variational parameters is determined from the timedependent variational principle; it can be obtained
practically from numerical integration of the initial-value problem, which requires a calculation comparable with integration of several classical trajectories. The approximate time-evolution operator is shown to be connected to an affine transformation of the Wigner function in the phase space. In the limit where the Wigner function is localized in the phase space, the time evolution described by this approximate time-evolution operator results in the classical mechanics, and the variational parameters are shown to Correlate with the classical dynamical variables. A formula for the vibrational-state distribution of a diatomic fragment produced in the collinear dissociation of a triatomic molecule is derived; this formula is expressed explicitly by the variational parameters. As an example the vibrational distribution of the symmetric deformation of the CF3 radical produced in the photodissociation of CF31 is calculated by regarding CF31 as a pseudotriatomic molecule, (Fd-C-1. The general trend of this distribution agrees with that calculated by Henning et al. by the close-coupling method.
1. Introduction The time-dependent treatment of the quantum dynamics such as the wave packet method sometimes provides a clear picture on certain problems of molecular processes.Id In particular, photodissociation has been studied extensively by using the wave packet and the dissociation dynamics, e.g., the vibrational-state distribution of the photofragment, has been explained clearly by the picture that the wave packet, which is the vibrational eigenfunction of the bound state, m o w on the potential energy surface of the dissociative state. In this connection, there have been reported many variants of the wave packet theory,'9296312-15 and considerable progress has been made in the accuracy of the propagation by superposing many wave packets or by performing heavy numerical calculation. In return, a simple and clear picture for the molecular processes of interest is inclined to be lost in these elaborate treatments. From the viewpoint of a formal theory, the wave packet method based on the time-dependent variational principle is clasely rdated to the theory of quantum mechanics of parametrized states (or generalized coherent a representative example is the time-dependent Hartree-Fock (TDHF) method originally developed in nuclear physics for the purpose of solving the manyfermion problems.' In this theory, the time evolution of a state vector is approximately described by a finite number of timedependent parameters, which are determined by the time-dependent variational principle; namely, the time evolution of a quantum state is represented by a trajectory in the phase space of the parameters as dynamical variables. The way of parametrization of the quantum states, which determines the nature of the approximation, has been studied from the standpoint of the Prcsent address: Division of nKoretical Studies, Institute for Molecular Science, Myodaiji, Okazaki 444, Japan. *Present address: Department of Chemistry, Nagaoka University of Technology, Nagaoka 940-21, Japan.
0022-3654/91/2095-2156$02.50/0
"geometry"* in the Hilbert space of the state vectors by using the languages of differential geometry and the Lie algebra of the quantum mechanical operators. In the present study the wave packet dynamics was formulated by manipulating the quantum mechanical in the framework of the quantum mechanics of parametrized states. This approach serves to clarify the relation between the wave packet and the generalized coherent states and provides a clue to develop a theory of wave packet in a certain abstract coordinate-momentum space. The formalism was applied to the photodissociation, and an explicit formula for the vibrational-state distribution of the photofragment was derived based on the picture of the oscillating wave packet. Such a model calculation helps to provide a certain intuitive understanding of the dissociation dynamics. In order to retain the relation to the TDHF method, the time-dependent variational principle of a Dirac-Frenkel type was employed, although the Dirac-Frenkel-McLachlan principle,laJ2 or ( I ) (a) Heller, E. J . J . Chem. Phys. 1975, 62, 1544. (b) Heller, E. J. J . Chem. Phys. 1915,62,63. (c) Heller, E. J. J . Chem. Phys. 1981, 75, 2923. (2) Lee, S. Y.; Heller, E. J. J . Chem. Phys. 1982, 76, 3035. (3) Shapiro, M.; Bcrsohn, R. Annu. Rev. Phys. Chem. 1982, 33, 409. (4) Williams. S. 0.;h e , D. 0.J . Phys. Chem. 1988, 92, 3363. (5) Zhang, 2.H.; Kouri, D. J. Phys. Reu. A 1986, 34, 2687. (6) (a) Alvarollos, J.; Metiu, H.J . Chem. Phys. 1988, 88, 4957. (b) Sawada, S.; Heather, R.; Jackson, B.; Metiu, H. J . Chem. Phys. 1985, 83, 3009. (c) Heather, R.; Metiu, H. J . Chem. Phys. 1986.84, 3250. (7) Blaizot, J.-P.; Ripka, G. Quantum Theory ojNnire Systems; MIT Press: Cambridge, MA, 1986. (8) Kramer, P.; Saraceno, M. Geomerry ofrhe Time-Dependcnr Variational Principle in Quantum Mechanics; Springer: Berlin, 1981. (9) Tishby. N. 2.;Levine, R. D. Phys. Reu. A 1984, 30, 1477. (10) Levine, R. D. J . Phys. Chem. 1985, 89, 2122. ( I 1) Alhassid, Y.;Levine, R. D. Phys. Reo. A 1978, 18, 89. (12) (a) Meyer, H.-D.; Kucar, J.; Ccderbaum. L. S.J. Marh. Phys. 1988, 29, 1417. (b) Coalson, R. P.; Karplus, M. J . Chem. Phys. 1983, 79, 6150. (13) Shi, S.; Rabitz, H. J . Chem. Phys. 1988,88, 7508. (14) Shi, S.J . Chem. Phys. 1984,81, 1794. (15) Shi, S. J . Chem. Phys. 1983, 79, 1343.
0 1991 American Chemical Society
The Journal of Physical Chemistry, Vol. 95, No. 6. 1991 2157
Propagation of a Wave Packet a minimum error method,6 is used in general because it is believed to give more accurate results in practical calculations than the original Dirac-Frenkel principle does. When the latter principle is applied, the time evolution of the time-dependent parameters obeys the equations of motion having a structure similar to the Hamilton equation, and therefore, correspondence to classical mechanics can be made clear. In the present formulation, the time evolution of the wave packet is described by an approximate time-evolution operator, which is a product of component unitary operators?.I0 Each component unitary operator has a form e"C'w, where v(r) is a time-dependent parameter and X is a Hermitian operator. The wave packet dynamics is derived purely mathematically from the time-dependent variational principle of a Dirac-Frenkel type. The equations of motion for the parameters ~ ( t have ) a structure similar to those of the classical mechanics. The motion and deformation of the wave packet are determined by the choice of the operator X. The effect of each component unitary operator on the position and the shape of the wave packet can be examined in the phase space by utilizing the Wigner function.I6 As a consequence, one can construct the approximate time-evolution operator with desired properties. For example, the wave packet that exactly describes the motion of a free harmonic oscillator with a given frequency can be constructed by using only two parameters. This wave packet is considered as "frozen", because only the mean position and the momentum are taken into consideration by employing two parameters, but its shape is changed automatically so that its time evolution is exact for a harmonic oscillator. This wave packet can be regarded as a generalization of the frozen wave packet and has the advantage of obtaining a simple analytical formula of the vibrational-state distribution of a fragment produced in photodissociation, since the projection of this wave packet to each eigenstate is independent of time. On the other hand, the projection of the conventional frozen Gaussian wave packet oscillates with time, and hence, the final-state distribution varies with time even in the dissociation limit. In addition, this generalized frozen wave packet can easily be defrosted by adding further component unitary operators, for example, to be the wave packet, the shape of which changes like the Gaussian wave packet with variable width. The present paper is organized as follows: The time-dependent treatment of photodissociation is reviewed briefly in section 2, and the basic theory of quantum mechanics of parametrized states is surveyed in section 3. The approximate time-evolution operators employed in the present work are presented in section 4, and the nature of the approximation is discussed in section 5 by using the Wigner function. In section 6, the classical limit is discussed. The collinear dissociation of a triatomic molecule ABC into A BC is formulated in section 7, and the results of a numerical calculation are presented in section 8.
+
2. Time-Dependent Treatment of Photodissociation In the present paper, attention is focused only on direct photodissociation, which can be regarded as one of the simplest cases in the dynamical processes of molecules. In the time-dependent picture, the cross section for photodissociation is given by (in atomic ~ n i t s ) ~ J
represents the stationary state of the bound state and is spatially localized in ordinary photodissociation; the wave function of the vibrationally ground state corresponds to the minimum-uncertainty wave packet. Furthermore, in direct photodissociation the potential energy surface of the dissociative state is sometimes so smooth that the wave packet is hardly deformed. In such a case, it is expected to be effective to make some approximation in estimating deformation of the wave packet. As an example, a collinear photodissociation of a triatomic molecule ABC* A + BC (2.3) +
in which an electronically excited molecule ABC dissociates into an atom A and a diatomic fragment BC maintaining a linear configuration is considered. The motions of the system are described by the vibration of BC and the relative translation of A and BC. The Hamiltonian of the dissociative state is expressed as
H =
+
f/2p~~ f/2p?
1/2JQ,2
+ V(QR.Q,)
where Q,and QR represent the mass-weighted coordinates of the vibration of BC and the relative translation of A and BC, respectively, P, and PR are the momentum operators conjugate to Q,and QR, respectively, and w is the harmonic vibrational frequency of the fragment BC. The interaction, V(QR,Q,), between the translation and vibration vanishes in the dissociation limit, QR a. The final state is described by the vibrational quantum number, n, and the momentum of the relative translation, p , and the final-state distribution is given by (nplUpoV'(np), or I(np1vl'kO)l2.
-
3. Basic Theory A heuristic introduction of the approximate time-evolution operator is as follows: The exact time-evolution operator, U, is given by
u(t)= (3.1) The Hamiltonian, H, can be expressed generally by a sum of the operators, X , as
+ a2X2 + ... + aJ,,
H = aJ1
where H is the Hamiltonian for the nuclear motion on the potential energy surface of the dissociative state. One can obtain the cross section by propagating the initial wave function (or the initial density operator po) on the potential energy surface of the dissociative state and then taking the overlap with the eigenstates of the separated system, l ' k ~ ~ The ) . initial wave function, (16) Wigner,
E. Phys. Rev. 1932, 40, 749.
(3.2)
The U operator in eq 3.1 can be written as U ( t ) = exp[-i(alXl a2X2+ ... a,,X,,)t]
+
+
= exP[h(t)XIl exp[i72(t)X21 '.' exp[bf(t)X,l (3.3) The last expression is called the "product representation", where { X I ,...,XN]is the smallest Lie algebra which involves {XI,..., X,,], and q,(t) is determined from the structure of the Lie algebra.9J0 However, N becomes infinity unless the Hamiltonian is very simple. In the present approximation, the product representation of the U operator in eq 3.3 is approximated by a finite number of the component operators, exp[iq,(r)X,] as U(t) =
li exp[irl,(r)Xl
(3.4)
v-l
by which the state vector, l'ko),is approximately propagated as
I W ) = U(t)l*,) or
(2.4)
(3.5)
The product representation of the U operator (eq 3.4) indicates that the time evolution described by U ( t ) can be expressed as a sequence of the transformations represented by exp[iq,(r)X,]. The approximate U operator in 3.4 can be expressed in 'the sum representation" by using the time-ordered exponential as
fs exp[iq,(r)X,]
v=
I
= T exp[ilfdt'
?qu(r')&] u=
I
(3.6)
where the operator 2"is defined by (3.7)
2158 The Journal of Physical Chemistry, Vol. 95, No. 6 , 1991
Someda et ai.
Equation 3.6 can be verified by differentiating both sides with respect to t . The operator X,can be expressed by a linear combination of operators X, as N
2,= CD,&,
( v = 1,
..., m)
(3.8)
Ir- 1
where X, ( u = 1, ...,N(Lm)) forms a Lie algebra, and the matrix D is a function of the parameters vu( v = 1 , ..., m)?,Io From eq 3.6 one can see that the time evolution of the system described by the approximate U operator in eq 3.4 is equivalent to the time evolution by a time-dependent effective Hamiltonian, %dt)
= -Cv,(t)X ( t )
where the operators X,(lIv)r - Z((QR)r)2W2p1/2U{1 z((QR)t)IQr + ~ / ~ Z ( ( Q R ) I ) ~ W - ~Z M ((Q ~R ~ {) J~I ~(F.4)
(E.15)
Registry No. CFII, 2314-97-8; CFI, 2264-21-3.
