J . Phys. Chem. 1984,88, 4829-4839 printed page statistic, is not entirely without merit. The formal structure of quantum mechanics is much simpler than that of classical mechanics and does not support the rich variety of distinctions in dynamical behavior that one has with the latter. Even von Neumann had to strain to find chaos in the orderly abstract dynamics of quantum systems, and as we have seen it was not his most successful effort. I have tried to make two points in this paper: first, it is the correspondence limit h 0 in which one should look for solid evidence of statistical behavior in quantum systems; and second, it is finitetime behavior that is of real interest in quantum mechanics, not the infinite-time limit used in classical ergodic theory. Perhaps there is even a challenge here to classical
-
4829
theory: given only a finite segment of a classical trajectory, is there any meaningful measure of how ”chaotic” that trajectory is? Certainly the surfaces of section that a theoretical chemist uses to illustrate classical chaos are generated by finite calculation! Finally, it is encouraging that experimentalists are now interested in this subject, and wonderful that some of the predictions for the irregular spectrum of highly vibrationally excited molecules have now been confirmed by stimulated emission pumping experiments.*’ (27) E. Abramson, R. W. Field, D. Imre, K. K. Innes, and J. L. Kinsey, J . Chem. Phys., 80, 2298 (1984).
Classical Liouville Mechanics and Intramolecular Relaxation Dynamics Charles Jaff$ and Paul Brumer*t Department of Chemistry, University of Toronto, Toronto, Ontario, Canada M5S 1A1 (Received: February 21, 1984)
The formal Hilbert space structure of classical mechanics is reviewed with emphasis on the relationship of the spectrum of the Liouville operator to regular vs. irregular motion. Two aspects of this approach are then described. First, eigendistributions of the Liouville operator for the harmonic oscillator and pendulum are displayed, providing insight into this alternate view of classical mechanics. Second, the dynamics of selected classical distributions in regular systems are discussed, with emphasis on the broad range of possible behavior, including periodicity, dephasing, and relaxation.
I. Introduction Classical mechanics has, for several decades, been useful in studies of molecular dynamics.’ Interest in conditions for the applicability of classical mechanics, particularly in intramolecular dynamics,2 has grown in recent years as a consequence of the recognition that classical conservative Hamiltonian systems undergo a transition, with increasing energy, from regular quasiperiodic motion to irregular or chaotic dynamic^.^ Although this change in character justifies historical concepts of statistical relaxation in highly excited isolated molecules, firm connections with quantum mechanics have yet to be e~tablished.~ At least some of the difficulty which arises in analyzing relationships between classical and quantum dynamics stems from the distinctly different approaches advocated to study them. In the former case dynamics is usually described via Hamiltonian mechanics, with trajectories as the focus. In the latter case, however, the quantum uncertainty principle insists, a priori, on the study of distributions with inherent position-momentum widths. The purpose of this paper is (a) to advocate the distribution dynamics viewpoint in the purely classical framework and (b) to emphasize the conceptual utility of the Liouville eigenfunction formulation which provides an alternate picture of classical mechanics and a useful view of relaxation phenomena. The specific technical emphasis is on the nature of the spectrum of the Liouville operator and its relationship to regular vs. irregular motion, as well as on the nature of relaxation in classically regular systems. As such, our goal is distinct from previous sporadic effortsS to utilize the Liouville equation as a computational tool or formulations in statistical mechanics which focus on the thermodynamic limit.6 The basic approach of interest below is the Liouville formulation of classical mechanics in which a distribution p is defined on phase space (p,q) and evolves in time in accord with the Liouville equation: ‘Current address: Department of Chemistry, Columbia University, New York, New York 10027. *I. W. Killam Research Fellow (1981-1983).
0022-3654/84/2088-4829$01 .50/0
Here H is the (conservative) Hamiltonian, { , is tHz: Poisson bracket, and p,q are any set of generalized canonical momenta and coordinates. Trajectories, which arise by choosing p(p,q;O) = 6(p-po) 6(q-qo),are the characteristics of Liouville’s equation and, as such, provide an equivalent dynamics. However, fundamental arguments have repeatedly been advanced in favor of p(p,q;t)dynamics over trajectory dynamics. These include those due to the Ehrenfests7 and to Born* in which recognition is made of in-practice limitations of classical measurement devices which prevent the preparation of an initial phase space point. More recently, Brillouin9 and Prigogine” have formulated such arguments with explicit reference to the rapid growth of errors associated with the incomplete specification of the initial state in the regime of irregular motion. Prigogine, in partittdar, argues for (1) E.g., M. Karplus, R. N. Porter, and R. D. Sharma, J . Chem. Phys., 43, 3259 (1965); D. L. Bunker and M. Pattengill, ibid., 48, 772 (1968); L. Verlet, Phys. Reu., 165, 201 (1968). (2) See, e&: (a) Discuss. Faraday SOC.,75 (1983). (b) Adu. Chem. Phys.,
47 (1981)-both containing several relevant articles. (3) For reviews, see: M. V. Berry in “Topics in Nonlinear Dynamics”, S. Jorna, Ed., American Institute of Physics, New York, 1978; P. Brumer, Adu. Chem. Phys., 47, 201 (1981); S . A. Rice, ibid., 47, 117 (1981). (4) E&: (a) D. W. Noid, M. L. Koszykowski, and R. A. Marcub, Ann. Rev. Phys. Chem., 32, 267 (1981); (b) K. Kay, J . Chem. Phys., 72, 5955 (1980). . ( 5 ) W. H. Miller and B. M. Skuse, J. Chem. Phys., 68, 295 (1978); B. C. Eu, ibid., 54, 559 (1971) are but a few examples. (6) I. Prigogine, “Non-Equilibrium Statistical Mechanics”, Wiley, New York, 1962; R. S. Zwanzig in “Lectures in Theoretical Physics”, Vol. 3, Wiley, New York, 1961, p 135. (7) P. Ehrenfest and T. Ehrenfest, “Encyklopadia der Matimatischen Wissenschaften IV”, 32, Leipzig, 1911. ( 8 ) M. Born and D. J. Hooten, Proc. Cambridge Philos. SOC.,52, 287 (1956), and references therein. (9) L. Brillouin, “Scientific Uncertainty and Information”, Academic Press, New York, 1d64. (10) I. Prigogine, “From Being to Becoming”, W. H. Freeman, San Francisco, 1980.
0 1984 American Chemical Society
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The Journal of Physical Chemistry, Vol, 88, No. 21, 1984
the outright rejection of trajectory mechanics as fundamental to dynamics in the irregular regime. While these arguments do not provide compelling evidence for the in-principle rejection of classical trajectory dynamics, numerical studies clearly display the difficulty associated with obtaining accurate long-time trajectory dynamics for irregular motion.” The extent to which isolated trajectory dynamics can be conceptually misleading is now also being recognized.12 Other arguments in favor of classical distribution dynamics, which also emphasize the role of the Liouville eigenspectrum, are readily discernible, specifically in the important area of relaxation in small systems. The ideal cases of relaxation, e.g., mixing and K-flow behavior, are defined” in terms of distribution dynamics with the measure-zero behavior of single trajectories explicitly excluded. Formally, such behavior has its expression in terms of the spectral properties of the Liouville operator. Thus, seeking characteristic features of the Liouville spectrum associated with the transition from regula! to irregular motion is easily motivated. This issue is discussed below, with additional motivation gained from misconceptions in the literature about behavior of the spectrum on passage through the transition. As a specific application of the classical distribution dynamics approach we discuss relaxation in reg%lar quasiperiodic systems. Recent enthusiasm regarding the existence of irregular motion has obkured the fact that particular characteristics of relaxation occur in quasiperioeic systems which, in some sense, is closer to quantum expectation than is irregular motion. Recognition of the advantages afforded in classical mechanical thinking by the Liouville eigenfunction viewpoint is the principal emphasis of the work described below. While the approach permits a clearer uGdersta?ding of the similarities and differences betwen classical and quantum mechanics, provides conceptually pleasing results regarding the classical limit, and allows a useful extension of classical mechanics, such results are relegated to a companion paper.14 Organization of this paper is as folIows: Section II contains a description of the Hilbert space structure of classical mecfianics and several simple examples of eigenfunctions and eigenvalues of the Liouville operator. The qualitative properties of the eigenspectrum are emphasized in a discussion. Section I11 contains a treatment of relaxation and dephasing in regular systems with various types of behavior demonstrated by specific example. Section IV contains a brief discussion and summary. 11. Classical Mechanics in Hilbert Space A . Formulation. Koopman,15 in a classic note, formally established classical Liouville mechanics as a well-defined theory on Hilbert space. Subsequently, Prigogine and Zwanzig6 took advantage of some aspects of this approach in applications to statistical mechanics. Appreciation of this Hilbert space structure is necessary for this paper as well as its compani~n.’~ Hence, the theory is summarized briefly below, with substantially less rigor than Koopman but with the goal of clarifying features of the rather terse formulation in the literature. The density function p(p,q;t) dp dq in classical mechanics is interpreted as the probability of finding the system, at time t, about the point (p,q) in 2N-dimensional phase space. The probabilistic interpretation requires that P(P,q;t) 2 0 for all ( p d
Jaff6 and Brumer where F ( f ) is the phase space average of a measurable function F(p,q). If, in addition, p(p,q;t) is required to be square integrable, then the tr (plp2) of two distributions is well-defined. In particular, we can define an inner product (,) as (Pl(t),PZ(t?) = tr [Pl*(O P2(t?l = JdP dq Pl*(PKf) Pz(P9q;t’) (3) The functions p(p,q;t) are then elements of a Hilbert space with parameter t and metric given by eq 3. More importantly, p(p,q;t) is but one representation of I p ( t ) ) , the abstract vector in Hilbert space. Transformation to alternative representations are achieved via canonical transformations introduced to take advantage of specific system features. Thus, eq 1, with the standard definition of L, should be denoted L(p,q), indicating that it is a (p,q) repThis cumbersome notation resentation of the abstract L = -i( ,H). is avoided below without loss of clarity. Classical mechanics, as quantum mechanics, allows for the classification of states p(p,q) as either pure of mixed. In accord with measurement theory,16 a pure state is one in which all theoretically measurable details of the state are determined. In principle, classical mechanics allows for the precise specification of all coordinates and momenta; Le., the pure states pp(p,q) are phase space points 6(ppo) 6(q-qo). Unlike bound-state quantum mechanics, however, tr ( p p 2 ) = gzru(O) # tr (pp) and p p are not elements of the Hilbert space, being of nonfinite norm. Other 6 function type states appear below; throughout they are regarded as improper states which could be acceptably included as limits of a well-defined sequence of ever-sharpening distributions in Hilbert space. As is familiar from quantum scattering theory, such improper distributions are associated with a continuous We note two important features of the Hilbert space formalism.16J8 First, L can be readily shown to be Hermitian and, second, any p(p,q) can be expanded in a linear combination of basis states. Of particular interest below is the complete orthonormal basis of eigendistributions of L. B. Time Evolution and L Eigendistributions. The time evolution in Hilbert space is formally given by
(4) the operator L, k i n g Hermitian, possesses a complete orthonormal set of eigendistributions /pa) with real eigenvalues A,, Le., Lip,) = AJp,). Thus, the general expression I p ( t ) ) = Ca,(t)Ip,) U
= Caa(0)e-9pa)
(5)
U
describes the evolution of Ip) in a particularly convenient (and familiar) fashion. Here aa(0) are constants determined by the initially prescribed Ip(0)); Le., a,(O) = ( p , , p ( O ) ) , and the sum in eq 5 implies an integral in the case of a continuous spectrum. The (p,q) representation of I p ( t ) ) is then, with (p,qlp,) = pa(p,q), of the form p(p,q;l) = Ca,(o)e-%,(p,q) a
(6)
with a,(O) = S d p dq p(p,q;O) p,*(p,q). Equation 6 displays an explicit separation between the time development of the system, embodied in the exponentials, and the functional dependence in phase space, contained in the eigendistributions p,(p,q). A similar
tr [p(p,q;Ol = JdP dq p(p,q;t) = 1
(! 1) Consider, e.g., the classically mixing stadium systems for which double precision IBM arithmetic limits accurate dynamics to approximately 50 collisions with the walls (C. Jaffe and P. Brumer, unpublished). (12) R. D. Taylor and P. Brumer, J . Chem. Phys., 77,854 (1982); see also the relaxation of intertrajectory distances in phase space discussed by I. Hamilton and P. Brumer, ibid., 78, 2682 (1982). (13) V. I. Arnold and A. Avez, “Ergodic Problems of Classical Mechanics”, W. A. Benjamin, New York, 1968. (14) C. Jaffe and P. Brumer, to be submitted for publication. (15) B. 0. Koopman, Proc. Nail. Acad. Sei. U.S.A., 17, 315 (1931).
(16) U. Fano, Reu. Mod. Phys., 29, 74 (1957). (17) (a) Throughout this paper our use of the terms discrete and continuous spectrum and their tacit association with L2 and singular functions respectively is in accord with Dirac’s use of improper functions and consistent with current usage in scattering theory. The language is, therefore, of physics and not of formal mathematics. (b) Attention is drawn to footnote 17a. In no sense have we demonstrated that an ergodic system has a continuous spectrum in the formal mathematical sense, but rather that all h provide solutions to the Liouville equation. Difficulties with using this approach to formally analyze the spectrum of an ergodic system are related to the unusual nature of 8/87 which generates motion along zero measure trajectories in phase space. (18) R. L. Liboff, “Introduction to the Theory of Kinetic Equations”, Krieger, New York, 1980.
The Journal of Physical Chemistry, Vol. 88, No. 21, 1984 4831
Liouville Mechanics and Intramolecular Relaxation expression holds for the time evolution of any property B, i.e.
B(t) = tr
[p(t)B] =
(9)
Ca,(O) tr [Bp,]e-iX*‘ U
If, in addition, B is expandable in p,(p,q) as B = &bsps(p,q), then the orthogonality of the basis set gives B(t) = &z,(0)6gf~. We note two well-known6 features of the p,(p,q) for an N degree-of-freedom system:
(7) CPu(P,q) PU*(P’d) = W-P’) J(q-q’)
(8)
(1
The first of these equations indicates that the nonstationary (Le., A, # 0) eigendistributions do not, by themselves, satisfy the basic conditions imposed by a probability interpretation of the phase space density; Le., they are not positive everywhere. Indeed, their contribution to the total probability tr ( p ( t ) ) is zero. Equation 8, the completeness condition on L eigendistributions, makes clear that specifying point particle initial conditions (Le., a trajectory) implies an infinite linear combination, necessary to localize the system at a point, of thesigendistributions. Finally, note that if p.,(p,q) is an eigenfunction with eigenvalue A,, then py*(p,q) is an eigenfunction with eigenvalue -A7: both are included in eq 5 without explicit recognition of this relationship. From this one can show that, for real p(t), if the coefficient of p., in the expansion of p ( t ) is a?, then the coefficient of pr* is a?*. There has been considerable emphasis in the literature of the similarity in the formal Hilbert space structure of classical and quantum mechanics. Less well appreciated is the remarkable qualitative similarity in the time evolution of classical and quantum distributions, particularly as it relates to questions of current interest in intramolecular relaxation dynamics. We comment here briefly on this matter; explicit applications to relaxation are provided later below and further issues of quantum/classical correspondence are discussed e1~ewhere.l~ Consider the evolution of p(p,q;t); then the coefficients a,(O) are determined, as noted above, from the initial p(p,q;O). Note first that the population of the stationary eigendistributions (Xu = 0) in the expansion (eq 6 ) are unchanged with time; i.e., tr [p,*p(t)] = tr [p,*p(O)] = a,(O). Changes in the probability of finding the system in particular regions of phase space arise from changes of the weighting of eigendistributions during the time evolution. Furthermore, a distribution initially localized about po,qo can only populate a distant region in phase space at later times if the initial expansion in eigendistributions contained p,(p,q) which are nonzero in these regions. The resultant qualitative picture of time evolution is far closer to the quantum picture, described via an expansion in eigenfunctions of the quantum L or H , than is the trajectory viewpoint in which dynamics unfolds under time evolution. Quite clearly the range of possible dynamics for a given system depends intimately on the nature of the p,(p,q) and 1,. Rather general statements which can be made about these quantities is the subject of the next section. C. Submanifolds. Classical systems can be classified by the number of isolating integrals of motion which they possess. With attention restricted to conservative systems energy is always an integral and we shall make the crucial assumption that the remaining integrals of motion are also known. Consider an N degree-of-freedom system with M independent isolating integrals denoted by K, = K,(p,q), i = 1, ..., M I N . They satisfy (K,,HJ = 0 and (K,,K,) = 0 for all i j . The set of K, can be chosen as M of the N generalized momenta describing the system, with ignorable conjugate coordinates v,(p,q). In many cases the physical properties of the system will be periodic in the 9,. This is not, however, guaranteed and a nonperiodic dependence on qr strongly influences the spectrum of the classical Liouville operator, as discussed later below. A consequence of the existence of M isolating integrals is that a distribution initially confined to the 2N - M dimensional metrically indecomposable submanifold in phase space labeled by K , will remain on this submanifold during its subsequent time evolution. This implies that elementary distributions of the form
are eigenfunctions of L. This form allows consideration of the L eigenvalue problem restricted to the submanifold defined by fixed K. The restricted Liouville operator on this submanifold will be denoted Lk. For two specific cases, M = 1 or M = N , the formal solution to the Lk eigenvalue problem simplifies dramatically. If M = N, i.e., the system is regular, H = H(K) and we possess a complete set of N isolating integrals K,. These may be used to construct a set of N Hermitian operators K, = -i[ ,K,) which satisfy [L,K,] = 0 and [Kj,K,] = 0, where [ , ] is the commutator. Hence, the set L,K,,i = 1, ..., N, have a common complete set of eigenfunctions, , and Lk can be expressed as a function of the K ~Le.
Lk = W(k)*K w(k) = BH(K)/dKIK=k (10) In the (K,q) representation eq 10 becomes Lk = -iw(k)d/aq with eigenfunctions f, and eigenvalues Xk,e given by
A(d = exp(i6-d
Xk,c
=t4k)
(11)
If the physical properties of the system are periodic in q, then the requirement that f;(q) be single valued restricts 6 to all integer values n. The result is a point spectrum. IfJmwever, system properties are not periodic in q, then e may assume any real value yielding a continuous spectrum.” For integrable systems K , may be conveniently chosen as the actions, I, with 7, = e,, the conjugate angles, givingf”(0) = ( 2 ~ ) ~ ’ exp(im0), Xk,, = mw(k), where a convenient (2n)-’normalization has been included. For the ergodic case, M = 1, K 1 = H , v1 = T , where T is the variable (“time”) conjugate to the Hamiltonian. The rgstricted Liouville operator LE is then given by LE = id/& with eigenSince the system is not periodic in T , functions p A ( 7 ) = e--IxT(P,q). any real X is an acceptable eigenvalue aqd the spectrum in the submanifold is c o n t i n ~ o u s . ’In~ ~no sense should pi(?) be viewed as a simple function of (p,q), T(p,q) being a highly Singular distribution. An example of the rich structure of the Liouville eigenvalue problem for an ergodic (in fact, mixing) system, and its relationship to relaxation, is discussed e1~ewhere.l~ For the case of M C N , M # 1 the general form of eq 9 still holds, with$(p,q) defined on the 2N - M dimension submanifold specified by the M values of K. The functionfk(p,q) satisfies the Liouville equation L&p,q) = Xfk(p,q) and is a function of 2N - M variables p,q, M of which may be chosen as the conjugate ignorable angles vt. No obvious simplifications to the Liouville equation on the submanifold emerge in other than special sebarable cases. However, the expectation that the system is not periodic in the p,q suggests that the nonstationary states will belong to a continuous, rather than point, spectrum. Thus, the spectrum of the Liouville operator restricted to the submanifold defined by K is readily characterized. In the event that the system is periodic on the submanifold defined by K, then the restricted Liouville operator has a pure point spectrum. Nonstationary states are nonuniform on the submanifold. Stationary states either are uniform on the submanifold or satisfy the resonance condition m w = 0. If, on the other hand, the physical properties of the systems are not periodic on the submanifold, then the point spectrum of Lk is simple, Le., has only one member, corresponding to the stationary state which is uniform on the submanifold. Nonstationary states correspond to elements in the continuous spectrum which are nonuniform on the submanifold. D. Full Phase Space Spectrum. Eigenfunctions of the Liouville operator in the full phase space, givenJk(p,q) on the submanifolds, are given by eq 9. Qualitatively, the eigenvalues are obtained as the union of the spectra of the restricted Liouville operators. Formally,20however, there are subtle issues relating to questions of which eigenvalues correspond to the point or continuous components of the spectrum. These results are consistent with the (19) R. Dumont and P. Brumer, to be submitted for publication. (20) H. Spohn, Physica A (Amsterdam), 80, 323 (1975).
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The Journal of Physical Chemistry, Vol. 88, No. 21, 1984
following qualitative discussion. Generally, the spectrum of L will be comprised of both continuous and discrete components. This is the case even for regular systems in which all Lk have point eigenspectra. The simple physical origin of this continuous component is clear, arising from the k dependence of At,". In the full phase space K is a continuous variable and arbitrarily small changes in K (Le., a change in submanifold) yields an arbitrarily close nearby eigenstate. There is, however, an important distinction between stationary and nonstationary eigenstates of L. Consider, for example, the regular system. In the former case there are an infinite number of eigenstates of the type 6(I-Io) with zero eigenvalue, corresponding to the uniform distributions on submanifolds with different I. These eigendistributions may be recombined to remove the 6 functions and to construct an L2 basis which spans the null space of the Liouville operator. As a direct consequence there are stationary states which are members of the point spectrum. Since the integration over all N 6 functions is necessary to produce L2 eigenfunctions, the degenerate nonstationary states, whose degeneracy is limited to N - 1 dimensional surfaces in I space, cannot be recombined to construct Lznonstationary distributions. Such states correspond to continuous components of the Liouville eigenspectrum. E. Regular us. Irregular Motion. The above discussion clarifies the difficulty associated with designing practical methods to identify regular vs. irregular motion3 based upon features of the L spectrum. The issue of regular vs. irregular motion in classical mechanics formally corresponds to the question of whether N global integrals of motion do or do not exist. As discussed above, the distinction in the qualitative nature of solutions to the Liouville eigenvalue problem for regular and irregular systems occurs on the level of the restricted Liouville operators. That is, a regular system will display a discrete Lk spectrum whereas an irregular system will display a continuous 4 spectrum. Unfortunately, this qualitative difference does not carry over to the full L spectrum which, in general, has continuous components. Since the ability to examine the Lk spectrum directly (e.g., via diagonalization) requires prior knowledge of the K, no useful new technique emerges. The alternative (dynamics) route is to construct a distribution on the submanifold, evolve the distribution, and Fourier transform the dynamics to obtain the spectrum. Under sufficient frequency resolution the discrete vs. continuous nature of the spectrum of Lk can be revealed. This is, in fact, what is done when the power spectrum associated with a trajectory is computed. Here one defines the initial point distribution p ( 0 ) = 6(ppo) 6(q-qo) which implicitly places it on a particular submanifold. Numerical values of all existent constants of motion are hence set implicitly. The dynamics continues on the submanifold, with peaks in the power spectrum displaying the specific Liouville operator eigenvalues contributing in the evolution of that property. Any effort to broaden the initial distribution introduces additional eigenvalues from adjacent submanifolds resulting in a broadening of the spectrum and a loss in a qualitative distinction between regular and irregular motion based on the feature of continuity of the ~pectrum.'~ Similarly, direct examination of the eigendistributions of the Liouville operator to distinguish regularity will only be useful if 4 eigenfunctionsare examined, again most easily done by allowing a trajectory to trace out the manifold. F. N = 1 Examples. In order to display various features of the eigenfunctions and eigenvalues we consider briefly below two N = 1 examples, the harmonic oscillator and the pendulum. The former is atypical in that it displays action-independent frequencies, the pendulum being, therefore, of considerable greater interest in mechanics. We emphasize below the contrast in the spectrum properties, for these two cases, due to the unique character of the harmonic system. ( i ) Harmonic Oscillator. Consider H = p2/2m
+ mw2q2/2 = w l
with well-known action-angle variables
JaffE and Brumer
In accordance with eq 11 eigenfunctions and eigenvalues of the restricted Liouville operator are given by p,'"
= e'J8/27r= (1/27r)[@
+ i m w q ) / ( p - i m ~ q ) ] J / ~ (12)
with
bo,= jw
(13)
In eq 12 and 13, p,q are understood to be confined to the curve defined by constant Z(p,q) = Zo and j is an integer. Eigendistributions in the full phase space are then given by prOJ = (2~)-'6(Z(p,q)-Zo) [ (p + imwq) /(p - imwq)]jI2
(14)
with
XI,, = j w The uniqueness of the harmonic oscillator manifests itself here by the fact that each submanifold has precisely the same discrete spectrum, each eigenvalue therefore being infinitely degenerate. As a direct consequence a complete orthonormal set of L2 functions may be constructed in the full phase space for each eigenvalue by a suitable recombination of the eigenfunctions given in eq 14. For example, the set (indexed by integer n ) pnJ(P9q) = ~ m d z o ~ J ( z o p) I o J ( P ? q )
(15)
defined through the coefficient functionsfnJ(Zo)are eigenfunctions of L with eigenvalue j w . Recognition that the Wigner representation of the quantum Liouville operator for the harmonic oscillator is precisely the same as the classical Liouville operator allows one to use the known quantum eigendistributions as a guide to choosing one possiblefnJ(Zo) set. In particular, choosing (with j 1 0, n integer, n - j even)
f n)0'( J
NnJ
= NtlJ exP(-2'~ / A)
IO / pY/2L(Pj)/z'(4r~/ A)
= ((-l)n/TA)[((n - j ) / 2 ) ! / ( ( n+ j ) / 2 ) ! 1 1 / 2 fn,-J
= fnJ*
(16)
provides a complete orthonormal L2 basis set for the classical eigendistributions of the harmonic oscillator. Here A is an arbitrary fixed parameter defining the Io width of the distributions, n labels degenerate states of eigenvalue h = j w , and L,J is a generalized Laguerre polynomial. Note that each A value yields a different set of eigendistributions, unlike quantum mechanics where A = h / 2 is fmed by explicitly constructing eigendistributions from eigenstates of the Schrodinger equation. Several sample classical eigenfunctions are shown, in the form P ~ , - ~in] , Figure 1. In general, these of the real combination [p, functions have 2j angular nodes and ( n - j ) / 2 "radial" (Le., Io) nodes. The fact that each submanifold has the same spectrum makes this case highly unusual. A more typical situation, the pendulum, is described below. ( i i ) Pendulum. Here the Hamiltonian is given by
+
H(p,q) = p 2 / 2 -
COS
q
(17)
with 0 5 q I27r. Three energy regimes are readily identified: For -e < E < E the pendulum oscillates between the turning points f q o , qo = cosw1( - E / € ) ;for E > c the pendulum rotates and E = e corresponds to the separatrix motion. In the oscillation regime the metrically indecomposable submanifold is labeled solely by energy whereas the higher energy regimes require an additional label corresponding to the direction of rotation. Each regime requires separate treatment as described below. For convenience and we introduce the parameters a = ( E + r ( a ) = [ ( E + e ) / 2 ~ ] '=/ ~[ c ~ / 2 e ' / ~=] l / ~ [p2/4c + sinZ ( q / 2 ) ] 1 / 2(18)
The Journal of Physical Chemistry, Vol. 88, No. 21, 1984 4833
Liouville Mechanics and Intramolecular Relaxation
Figure 1. [on,/+ p n , 4 superposition of eigendistributions of the classical Liouville operator for the Harmonic oscillator. Dashed contour is -0.05. Solid contours are at 0.005 and 0.05. (a) n = 10, I = 0; (b) n = 11, I = 1; (6) n = 11, I = 3; (d) n = 11, I = 5 ; (e) n = 11, I = 9; (0 n = 11, 1 = 11.
Then 0 Ir
< 1 for E < e,
> 1 for E >
r = 1 for E = e , and r
E.
In the regime 0 5 r < 1 the transformation between q,p and the elliptic action-angle variables a,P is given by eq 19. Here q(a,P) = 2 sin-' (r(a) sn(plr(a)) 25
p(a,P) = 2e'I2r(a) cn(@lr(a)) a(p,q) = p2/2c1l2
for P
E the transformation to a,/3 is given by q(a,P) = 2 sin-' (sn(r(a)PII /r(a))
25
50
+ 2e1I2sin2 (q/2)
P ( P d = f ( l /r(p,q))F(q/2;1 / r h q ) )
with eq 9 the full L eigendistributions and eigenvalues are given by P&n(P,q) = 6(ao-4P,q)) /2K(l/r(ao)) exp [ i * ~ P @ , d/ K ( 1/r(ao))
A,,,
= me'/2/K(1 /r(ao))
1
(23)
Projections of the clockwise eigendistributions onto the coordinate and momentum axes are shown in Figure 3. The y,,,Jq) are complex with symmetric real part and antisymmetric imaginary part. Both real and imaginary parts of -yao,Jq)have 2n nodes whereas y,,,(p) has n nodes. The spectra of the restricted Liouville operator in both regimes discussed above are pure discrete. The situation at E = e is quite different since motion of this separatrix is not periodic (Le., the period is infinite). Specifically, at E = e, (49) are related to (a,@) by q(a,P) = f 2 ~ ' sin-' / ~ (tanh
P)
p(a,P) = f2e'l2 sech P cu(p,q) = p2/2e'I2
+ 2c1/2sin2 (q/2)
P G n d = In tan
= 2dJ2
((* f q)/4)
(24)
where plus and minus signs correspond to clockwise and counterclockwise rotation. Eigendistributions and eigenvalues are then P&
) p(a,@) = f 2 ~ ' / ~ r ( adn(r(a)Bll/r(a)) a(p,q) = p2/2e'12
0
for p 2 0
W r h q ) ) - F(sin-' [sin (q/2)/r(p,q)l;r(p,q))
= aH/aa =
-25
Figure 2. Projections Y&Q) and for pendulum (f = 6 2 5 ) in oscillatory regime: (a and c) a. = 40, n = 1; (b and d) a0 = 40, n = 8.
P@d =
H(a) =
-50
D
+ 2e'l2 sin2 (q/2)
P(p,q) = W n - ' [sin (q/2)lr(p,q)l;r(p,q))
50
= ( N a 0 - 4 P , q ) ) / 2 ) exP[~PPo,q)l Yeo# =
(22)
where dn is a Jacobi elliptic function, F is the incomplete elliptic integral of the first kind, and the sign in eq 22 is chosen as plus for clockwise pendulum rotation and minus for counterclockwise rotation. The tranformed H a n d w are given by eq 20. In accord
Since the system is not periodic in P, p can assume any real value. Coordinate and momentum properties of P:,,~ for g = 2 are shown in Figure 4. The projections have an infinite number of nodes, displaying the improper nature of these eigendistributions associated with the continuous spectrum of the restricted Liouville operator.
JaffE and Brumer
The Journal of Physical Chemistry, Vol. 88, No. 21, 1984
4834 0.40
I
I
C 40
I
I
courage clarity of terminology we utilize the term dephasing to denote the apparent relaxation process associated with short-time dynamics in a system with a discrete frequency spectrum only and relaxation to denote true irreversible behavior associated with a continuous spectrum. As is clear from the above discussion, both regular and statistical systems (the latter term is here adopted to denote systems which are at least mixing) have Liouville operators with continuous frequency components. Hence, true relaxation is possible in both cases. The essential difference between these systems lies in the nature of the dependence of the final state upon the initial conditions, as discussed later below. Since this distinction relies on long-time dynamics, in practice experimental limitations in intramolecular studies may prevent a distinction between regular and statistical relaxation. In general, the time evolution of p ( t ) is given by
I
I
(b)
(a)
pu(t)
= Caapae-'hf
(25)
01
where the superscript a denotes the set of coefficients {aa].Relaxation is readily discussed in terms of the correlation functions tr [p'(t)pb*], where p b is an arbitrary density, pb = cabsps, or in terms of the time evolution of a property B, Le., B ( t ) = tr [ B p a ( t ) ] .In the former case Pab(t)= tr [p'((t)pb'] = Caaba*e-ixJ a
=
-I -
a
1
i
0.20-
--a 1I-
a. 0
qO L
-0.20
-0 2 o t
-.
I
uu
- 0.40 -60.25
0
P
60.25-0'%?0.25
60.25
0
P
Figure 3. Projections y20,n(q)and y;,Jp) for pendulum (e = 625) in regime of rotation: (a, c, e) a. = 60, n = 8 ; (b, d, f) a. = 60, n = 1.
Eigenvalues of the simple pendulum are shown, as a function of action in Figure 5. The clustering of eigenvalues at the action corresponding to the separatrix is evident. 111. Relaxation in Regular Systems Within classical mechanics, intramolecular dynamics focuses on the nature of the asymptotic distribution in isolated molecules and the rate at which the system approaches this limit. To en-
Figure 4. Projections Y&,Jq) and y:o,n(p) for pendulum
(E
= 625) at E =
a;X.#O
aaba*e-ixJ
(26)
with a similar form for & t ) . Two qualitatively different forms of the long-time limit of Pab(t)are readily discerned. In one case the a&* terms are such that the contributing A, are pure discrete. Then Pab(t)oscillates indefinitely. In the second case aabs leads to continuous A, # 0 contributions with a long-time limit given by ,&=O~aba, Le., a sum over stationary states. In either case short-time observations can reveal apparent relaxation behavior, dephasing in the former and true relaxation in the latter. Note further that the nature of the time evolution depends upon both the spectrum of L and the nature of p',pb reflected in aaba*. A basic distinction between regular and statistical systems lies in the types of terms which contribute to the first sum in eq 26. For statistical systems the only stationary states in the discrete spectrum are 6(E-H); hence, the sum of ha = 0 terms corresponds to a weighted sum of uniform distributions over each energy shell represented in the initial distribution. In principle this is not substantially different from the long-time limit in regular cases involving continuous A, where the long-time limit is a weighted sum over stationary Liouville eigenstates. The difference, however, is of extreme practical significance since integrals of motion other than simple symmetry-related properties are rarely known. Below we examine a sequence of successively more detailed examples which display dephasing and relaxation in regular systems. Connections with previous discussions of relaxation in classical/quantum system^^^,^^ is evident and will be emphasized
0 4 0 r -
+2
C aaba* +
a;X,=O
E.
The case of n = 2 is shown.
The Journal of Physical Chemistry, Vol. 88, No. 21, 1984 4835
Liouville Mechanics and Intramolecular Relaxation 125
75
is of considerable use in understanding the time evolution. It, as all Fourier transforms of dynamic properties, explicitly displays the contributions due to the eigenvalues A, and coefficients
25
a,*
x
A. One-Dimensional Torus. Angle Broadening. Consider a one-dimensional system with action-angle variables Z,0 and an initial distribution
C exp[-(B
p(Z,0;0) = (rA)-l/%(Z-Z0)
25
0
75
50
d
- OC
n=--m
+ 2 ~ n ) ~ / A (28) ]
Here, the distribution is localized on a specific torus ( I = Io) with a Gaussian-like distribution in 0. The particular form of the 0 dependence, chosen for mathematical convenience (in particular to allow use of the Poisson summation formula) produces a localized distribution for small A. For this, and all regular systems, expansion in the eigenfunctions of the Liouville operator is straightforward:
Figure 5. Pendulum eigenvalues as function of elliptic action a. For clarity only a few of the curves have been labeled.
where
I
I
I
I
pIt,n(z,e) = (2r)-I 6 (I’-I) eine
= no(Z’)
A#’) a,(Z’) = 6(Zo-Z’)
exp[-An2/4
- in0,]
(30)
Explicitly then m
p(Z,O;t)
= (2r)-’6(Z04)
+ in(0 - 0,) - inwot]
exp[-An2/4 n=-m
where wo = w(Zo). The normalized autocorrelation function P(t) is then given by m
P(t) =
JdZ d0 p*(Z,O;O) p(Z,0;t)
-
n=--m
exP[-An2/2
5
JdZ d0 IP(Z,~;O)~~
- inmot1
exp[-Am2/2]
m=-u
=
c
n=-”
exp[-(2rn
c
- ~ ~ t ) ~ / 2 m=-” A ] / exp[-2r2rn2/A]
(31)
The initial falloff of the P ( t ) (see Figure 6) is due to the movement of the distribution away from 0 = 0,. An estimate of this initial decay rate, valid for small A, arises from the time dependence of the n = 0 term in eq 31 which behaves as exp(-w2t2/2A) leading to a characteristic initial time t, and rate k, = l / t , given by
k, = 0 ~ / ( 2 A ) l / ~ t, = (2A)1/2/w0 t Figure 6. P(r) for 0 broadened distribution, with various A values, on a torus, over one period 27r/w0 for oo= 25: (solid curve) A = 0.2, (dashed curve) A = 1.0, (dot-dashed curve) A = 4.0, (dotted curve) A = 10. Labeled arrows provide t, with alphabetic sequence corresponding to this ordering.
elsewhere. Principle focus is upon the normalized autocorrelation function P ( t ) = tr [ p a * ( 0 ) pa(t)]/tr [pa*(0)pa(O)], ~ “ ( 0 )real, although other functions may be examined with comparable ease.
That is, t, is proportional to the 0 width at half-height of the initial p distribution. Two time scales are therefore evident in the dynamics, periodicity with frequency w,, and short-time falloff. The latter is not relaxation, which requires continuous frequency contributions, but rather dephasing. Their appearance in the Fourier-transformed picture is of interest for comparison with later examples. In particular, P(w) is readily evaluated by using eq 27 and 30 as -m
c
P ( w ) = e-w2/4k2
n=--m
(21) E. J. Heller in “Potential Energy Surfaces and Dynamics Calculations”, D. G. Truhlar, Ed., Plenum Press, New York, 1981.
(32)
m
G(w-nwo)/
m=--m
exp(-Am2/2)
(33)
That is, P ( w ) is comprised of a set of 6 functions spaced by wo
4836
The Journal of Physical Chemistry, Vol. 88, No. 21, 1984
and reflecting the long-time periodicity, with overall Gaussian weighting whose half-width, proportional to k,, determines the initial decay. Two limiting cases are of interest. For fixed wo, A = 0 corresponds to a point particle whereas A = m is a uniform distribution on the torus. These limits are reflected, in quantities in eq 29-33 in a straightforward manner. That is, as A 0 all n values contribute to eq 3 1 and 33, the spectrum being distributed over all multiples of wo. As A m only n = 0 contributes throughout, these quantities reflecting the stationary character of the uniform distribution on the torus. The transition between these two types of behavior, as indicated in Figure 6, lies in the region of A = 1. The initial distribution in this example explicitly extracts discrete frequencies from the continuous set associated with L. Behavior resulting from the inclusion of continuous contributions is considered in the next example wherein broadening over Z is introduced. B. One-Dimensional Torus. Action and Angle Broadening. Consider an initial distribution, in a one-degree-of-freedomsystem, given by
-
-
a
p(l,O;O) =
C exp[-(Z - Zc)z/S] ,=--' exp[-(0 - 8,
+ 2an)'/A] (34)
Jaff6 and Brumer Pm(t) = A-'$-dZ'exp[-2(10
+
n=-m
a,(Z? = exp[-(Z'-
Zc)2/6
- An2/4 - in8,]
(36)
pp,,(Z,8) given in eq 30. with X,(Z'), Consideration of P(t) requires integrals over action of functions necessitating the study of a particular system. To this of X,(Z'), end we choose the particle in a one-dimensional box of length n with H = p2/2. In this case the action-angle variables are related to particle coordinate q and momenta p by
(1
+ @((2/6)'/'(1 + 6t2/4A)Z,)] X + 6t2/4A]-1/2 exp[-Z2t2/(2A( 1 + 6t2/4A)-')]
>0 8 = 2 a - q for p < 0 8 = q for p
(37)
with frequency w(Z) = Z and X,(Z') = nZ'. We proceed, as above, by explicitly examining both P(t) and P(w). The quantity P ( t ) = tr ( p * ( O ) p(t))/tr ( p * ( O ) p ( 0 ) ) can be written as P(t) = C,",-mP,(t) with lafl(Z')lZe-inP'
0
+ @[(2/6)'/2Zci(6/8)'/2nt]) exp[-An2/2
= 8'(7~6/8)'/~ (1
- 6n2t2/8 - inZ,t] (38)
1 @((2/6)'/2Zc]] B = ( ~ 6 / 8 ) ' / ~ [+
5 exp(-An2/2)
,=--"
(39)
This simplest of regular systems is seen to relax in the long-time limit to m
P ( m ) = Po(-)= [
,=-a
(42)
The short-time decay is therefore seen to proceed with a rate k, given by
k, = Z,/(2A)'/2 = ~ ( 1 , ) / ( 2 A ) ' / ~
(43)
Le., inversely proportional to the coordinate space width of the initial distribution. This result, eq 43, is similar to that obtained in the single-torus case (see eq 32) but with w now evaluated at the center of the Gaussian-broadened action distribution. Once again it measures the rate at which the distribution moves away from its initial location. Prior to displaying specific sample results we evaluate the Fourier transform to unify the overall picture of dynamics. Explicitly
5 $-dt
,=-a
eiUrP,(t) =
0
P,(w) = E' exp(-An2/2)$-dl' 0
5 P,(w)
(44)
,=-a
exp(-2(Z'-
Zc)z/6)6(w-nZ') (45)
or m
Po(w) = 6 ( w ) /
n=--'
P,(w) = (nB)-' exp(-An2/2
exp(-An2/2)
- 2(w - nZ,)z/tin2)
= (nB)-' exp(-Z,Zn2/4ks2) exp(-(w - nZ,)2/4k,2n2)
n # 0 (46)
I= lpl
Pn(t) = B'$-dZ'
(41)
Po(2) = A-'(*6/8)'/'[1
P(w) = (1/2a)
-
exp(-2a2nz/A)
The short-time limit is then governed by the Po(t) term which can be explicitly evaluated to give
(35)
Here @(x) is the standard probability integral. This initial distribution, broadened in 8 about 8, and Z about I,, reduces eq 28 0. The eigenfunction expansion (eq 29) of eq 34 gives as 6
- (2am - Z't)2/2A]
m
c
A = ( ~ 6 / 8 ) ' / ~ (+1 @[(2/6)1/2Zc]]
with normalization constant C C = 2 ( ~ ~ A 6 ) - '1/ ~ [ @(Zc/6'/2)]-'
Z,)2/6
exp(-An2/2)]-'
(40)
with a rate governed by the n = f l terms in eq 38. These approach zero with a rate k , = (6/8)1/2, Le., proportional to the momentum space width of the initial distribution. This true relaxation arises through nonzero 6 which introduces continuous spectral contributions. Examination of behavior in the short-time limit is facilitated by rewriting P(t) as P(t) = C&-Pm(t) with
Thus, the Fourier transform is the sum of Gaussian in frequency, each centered at integer multiples of o(Z,) = Z, plus a 6 function at zero. The width of each Gaussian is proportional to k,n with an overall envelope proportional to ks/Zc, With appropriate 6, A parameters three time scales are evident, short-time relaxation with rate k, related to the overall frequency width, long-time relaxation related to the width of the individual Gaussians, and recurrences related to the spacing between the Gaussian peaks. It should be noted that it is possible to construct initial distributions in such a manner that the Gaussian peaks of P,(w) overlap in which case the system will relax prior to the first recurrence. The resultant autocorrelation function will, at least qualitatively, appear truly statistical, enhanced by the fact that one-dimensional systems are ergodic on the energy shell, albeit not mixing. Sample numerical results are displayed in Figure 8 for three initial distributions shown in Figure 7. They are centered at Z = Z, = 12.5 and q = qc = n/2; the first is well localized in coordinate space but broad in momentum space, the third localized in momentum space, but broad in q, and the second is of an intermediate nature. In going from large to small 6 (Le., Figure 8a-c) P(t) is seen to display increasingly large oscillations with decaying envelope, with the time to reach this asymptotic value increasing. This time is reasonably well approximated by t, = l / k m = (~5/8)'/~.This behavior is reflected in P(w) (Figure 8d-f) by a sharpening of spectral peaks. Narrowing of the overall spectral width with increasing A is also evident, reflected in the slower initial P(t) falloff in moving from panel a to c. The long-time limit which is approached in all cases is evident. Below, we consider the extension of these results to two-degree-of-freedom regular systems to consider cases where the system
The Journal of Physical Chemistry, Vol. 88, No. 21, 1984 4831
Liouville Mechanics and Intramolecular Relaxation
p(Z,0;0) = (a2AlA2)-1/26(Zl-ZlC)6(Z2-ZZc)
[exp[-(6, - BIc
+
nlm2
2 ~ n , ) ~ /-A(0,~ - 02c + 2 ~ n , ) ~ / A , ](47) p
12.5
on the torus labeled by Zlc,Z2c. Unless otherwise indicated sums extend from --m to 03. For small AI, A2 the initial distribution is well localized on the torus near 0 = 6, whereas for AI or A2 large the initial distribution is localized in one dimension as a ring on the torus. An expansion of the time development in terms of Liouville eigenstates gives
1 I
I
01
I
"2
25.0
I
ll
3v4
I
I
p(1,O;t) = l m d I ' ~an(I')e-i~(t')'p~,,n(I,O) (48) O
p
n
with
I
12.5
pr,,n(I,O)= (4a2)-'6(1-1') exp[in.O] A,(I') = nlwl(I')
1
a#) = 6(Il'-Ilc) 6(1{-12c) exp[-(Aln12
+ n2w2(I')
+ A2n22)/4 - i(nlOlC+ n202C)] (49)
A direct evaluation of P(t) and P(w) gives
+ n22A2)/2 - i(nlwlc + n 2 ~ 2 C ) t ] / M P(w) = Cexp[-(Aln12 + A 2 n 2 2 ) / 2 ] 6 [ w - ( n l w l c + n 2 w 2 c ) ] / M n P(t) = Eexp[-(nl2A1 n
1
M = xexp[-(Alrn12
+ A2mz2)/2]
m
Figure 7. Particle in one-dimensional box. Distribution parameters: Bo = 0, (a) 6 = 100, A = 0.029; (b) 6 = 5, A = 0.180; (c) 6 = 0.5, A = 1.929. Contour values are 0.2, 0.4, 0.6.
r.: 0.25
1 :
'0
Q25
4.02
801
'0
4.02
8.01
402
0
35
70
w
wf
= (wl
(51)
The A. is a measure of width of the distribution in the direction of motion on the torus. Subsequent dynamics depend intimately on the choice of parameters which are readily examined via eq 50. None are, of course, examples of true relaxation but P ( t ) displaying strong short-time dephasing features are plentiful. D. Two-Dimensional Torus. Action and Angle Broadening. Relaxation appears once again, now for the N = 2 case, through broadening of the distribution in the action variable, introducing continuous components to the spectrum. As an extension of cases B and C consider the initial distribution
1
cw
+ w2)/2 A0 = A1A2(wl2 + ~ 2 ~ ) / ( ~ 1+~ ~A22~ A 1 ) t , = (2AO)ll2/wf
8.04
4
02%
where w: = w1(Z1c,Z2c). The range of possible behaviors of P(t) is now substantial due to the host of possible values of wIc, w2c, Al, A2. For example, the extent to which each i n & ~ $ f i i Ia,w,C sequence affects the dynamics depends intimately on the AI values. Increasing A,, and hence broadening the initial distribution about e,, reduces the influence of higher order harmonics nIu: in the dynamics. Thus, by variation of A1,A2individual sequences associated with either nlwlCor n2wZccan be made predqminant. The initial P(t) falloff, once again merely describing the initial motion of the distribution away from its location at t = 0, occurs, within the accuracy limits discussed for the one-dimensional case, over a time
1
'0
(50)
U'
0
Figure 8. P(t) and P(w) for the cases shown in Figure 7. Panels a and d correspond to a in Figure 7, panels b and e to b in Figure 7, and panels c and f to c in Figure 7. P ( w ) is symmetric about w = 0; hence, only w > 0 is shown. In addition, the P ( w ) ordinate is the fraction bf the maximum P ( w ) (less the 6 function) located at w = 0. Actual P(0) values are (d) 0.068, (e) 0.17, and ( f ) 0.55. The initial rapid falloff from P ( t ) = 1 is barely visible on this scale.
is not ergodic the energy shell, as it must be for all one-degree-of-freedom cases. At this point the complete generality of the above approach should be evident and, for the sake of brevity, a minimum of algebra and detail is presented. C. Two-Dimensional Torus. Angle Broadening. Consider the initial distribution
- ~ ~ ~ - (z2 ) ~- z,.)~/S,) 6 ~ x - OlC + 2 ~ n , ) ~ / A- ,(0, - OZc + 2 a r ~ ~ ) ~ / A(52) ,]
p(1,O;O) = C exp(-(Zl
exp[-(8, nhn2
with C=
(4 / a ) ( A I A28 62)-1/2{1 + @ (II'/ 6 1/2))-1 ( 1
+ +(ZZc/ 621/2)]-1
Once again it is convenient to focus on a specific example, e.g., the particle in a two-dimensional box, of sides of length a. Here 11
=
lPll
12
=
lP21
O1 = q 1 for p1 > 0
(53)
4838 The Journal of Physical Chemistry, Vol. 88, No. 21, 1984 O1 = 2 a
JaffE and Brumer
- q1 for p1 C 0
O2 = q2 for p 2 > 0
O2 = 2 7 - q2 for p 2 < 0
hn(I) = nlZl + n2Z2 Expanding this distribution in eigenstates (see eq 48 and 49) gives an@’)= exp[-(Zl’
- Zlc)2/61 - (Z2’ - Z2c)2/62] exp[-(Aln12 + A ~ n 2 ~ ) /-4 i(nlOIC+ n2BzC)](54)
0.25
As in example B, P ( t ) can be written in two forms, one of general use and the second for explicit examination of short-time behavior. Specifically, the general form is JYt)
=
E
tu
00
25
50
t
Pnl,n2(t)
nlPZ
0.75 “ O O m
-
d0.501
(
- “)“2njt] 2
exp[-A,n:/2
- (sjnj2/8)t2- injZj”t] ( 5 5 )
r
2
with W
P, ,nz appears, therefore, as the product of terms arising in the one-dimensional box problem (eq 38), a direct consequence of the separability of the Hamiltonian in ZlJ2 and the particular choice of initial distribution. P(t) can be seen to approach the long-time limit
P(m) = Po,o(”) = [Eexp[-(Aln12 + A2n,2/2)/211-’ n
with each Pnl,,,(t)term decaying with rate
Thus, the long-time relaxation characteristics are controlled by the inverse width of the distributions in momentum space, the slowest rate being dependent on the particular choice of 61, a2. An analysis similar to that provided for the one-dimensional box allows a reexpression of P(t) to extract the rate k, of the initial fast decay, found to be
involving both the central frequencies w , ( Z ~ )= Z,C and the coordinate space widths. Finally, the Fourier transform may be obtained by extension of the one-dimensional box calculation to give P(w) =
E
Pnl,nz(w)
%.“2
for nl # 0, n2 # 0 and a 6 function at w = 0 with weighting given by eq 56 for nl = n2 = 0. The main feature of the Fourier transform is therefore a series of Gaussian peaks centered at = nlZIC+ n2Z2cplus a 6 function at zero. The extent hn(Z1c,Z2c) to which these peaks overlap or remain isolated depends upon the parameters defining the initial distribution. Two examples of the variety of possible behavior are shown in Figure 9; they are in no sense the most extreme examples. In Figure 9a rapid initial decay is observed followed by erratic oscillations prior to approaching the long-time value. In this case both a1 and 6 , are small, as reflected in the narrow spectral peaks and associated relatively slow approach to the long-time limit. The overall spectral width is also narrow, leading to a relatively slow initial falloff. In sharp contrast, Figure 9b and associated transform 9d show very broad individual peaks and a broad spectrum, associated with a rapid approach to the long-time limit after an initial rapid falloff. These are due to small A, but a substantial 61. In both cases the
30 w
Figure 9. P(t) and P ( w ) for two initial distributions in a two-dimensional box. Parameters for a and c are 6, = 0.5, a2 = 1.0, A, = 7.29, P2 = 3.80 and for b and d they are 61 = 50, 62 = 1.0, PI = 0.05, h2= 3.80. In both cases I I c = 4.5, IZc= 12.5, = 6’; = 1.0. Only w > 0 is shown for the symmetric P(w) and the ordinate is as a fraction of the 6 function weighting a t w = 0. For c this weighting is 0.84; for d it is 0.07. The initial rapid falloff from P ( t ) = 1 is barely visible on this scale.
spectrum shows low intensity at w = 0, other than the 6 function contribution. However, an alternate choice of Zlcand Z2c, such that these frequencies are nearly commensurate, leads to a peak in the w = 0 region as well, reflected in a low beat frequency observed in P(t). Substantially greater broadening of the spectrum is possible by a larger choice of A,, as it is a completely sharp spectrum associated with small 6,. There is no “generic” case. E . Comment. These examples have served to strengthen the formal results of section I11 regarding the role of the continuous spectrum in the dynamics of regular systems. Results which have emerged make clear that the p ( t ) behavior, as reflected in P ( t ) , spans a broad range, dependent primarily on the nature of the individual eigenvalues and on the coefficients a,, predetermined in establishing the initial conditions. Rapid initial falloff will be observed if the overall spectral width is large, recurrences will be observed if the spectrum is highly structured, and long-time relaxation is evident when the spectrum has no sharp structure away from w = 0. Although focus has been on P ( t ) , a quantity which is principally measured in certain emission experiments?2 a similar range of possible time-dependent behavior is expected for Pab(t) and B ( t ) since the basic picture remains the same.
IV. Summary The major results of this paper fall into three distinct categories. First, formal features of the Liouville approach have been emphasized with focus on the relationship of the spectrum to questions of regular and irregular motion. The conclusion that the major distinction in spectral properties lies at the level of the restricted Liouville operator leads naturally to the formal recognition that relaxation is natural in regular systems, albeit not generally to the statistical limit. Explicit examination of the dynamics of several sample distributions serve to demonstrate this behavior. In doing so a second aspect of this approach emerges, that is, the technical ease with which the time evolution of representative distributions can be analytically and computationally examined in a simple regular system. Third, the expansion of p in Liouville (22) K. F. Freed and A. Nitzan, J . Chem. Phys., 73, 4765 (1980).
J. Phys. Chem. 1984,88,4839-4844 eigenstates serves as a useful alternate way of thinking about classical dynamics. Classical distributions are seen to be superpositions of highly structured eigenfunctions whose weighting changes as time evolves. The underlying nature of the eigendistributions and their associated eigenvalues are at the heart of the detailed dynamics of regular and statistical systems. The close qualitative relationship between this classical approach and that
4839
of quantum dynamics is evident; the quantitative relationship is discussed e1~ewhere.l~ Acknowledgment. The partial support of NSERC Canada and the donors of the Petroleum Research Fund, administered by the American Chemical Society, is gratefully acknowledged. P.B. thanks Professor K. Kay for useful conversations.
Siegert Quantization, Complex Rotation, and Molecular Resonancest R. Lefebvre Laboratoire de Photophysique MolBculaire du CNRS,‘ Campus d’Orsay, 91 405 Orsay, France (Received: December 8, 1983)
A quantization scheme based on the applicationof the Siegert asymptotic boundary conditions to the coupled equations describing
a fragmentation process provides an accurate method for the determination of resonant energies. Complex rotation of the coordinate for relative motion leads to a variant of the method where bound and resonant states are treated in exactly the same manner. The case of numerical or piecewise potentials can be treated within the present context by using Simon’s complex exterior scaling. Such a quantization scheme can also be applied to the description of the interference which results from the existence of both direct and indirect routes to fragmentation. An example of this interference effect is worked out for illustrative purposes in the case of a van der Waals complex.
Introduction Resonant states (or compound states) are playing a very important role in many collisional or half-collisional processes. There are various ways of approaching the determination of such states and of their associated (complex) energies. A very interesting recent development is the complex scaling of the coordinate which describes the interfragment relative motion. The result of this scaling is to transform the resonant function (which normally diverges at infinity) into a localizable one. This opens up the possibility of using basis sets of integrable functions. Applications in various fields of atomic and molecular physics have now amply illustrated the power of such a technique.’ This paper is also concerned with the determination of resonant wave functions and energies. A different viewpoint is adopted however. The relative motion of the two entities which participate in a fragmentation process can very often be described with a set of coupled equations. There are various schemes by which the solutions of such coupled equations can be exploited to characterize the fragmentation or half-collision (see section V for a brief formal description of two of them). However, since a resonant state is characterized by well-defined boundary conditions (originally given by Siegert2) it is possible to draw numerically the consequences of imposing these boundary condition^.^ Since they concern the behavior of the system both close to the origin and at infinity, this situation, which is similar in some respect to that met in the definition of bound eigenstates, leads to quantized energies. These, however, are complex quantities. One hopes, then, that decay rates can be derived, as usual, from the imaginary parts of such energies. A variant of the method4 consists in introducing complex scaling into the coupled equations. Bound and resonant states are then calculated with exactly the same prescriptions. In section I1 of the paper two simple examples leading to Siegert quantization are given for pedagogical purposes. Section I11 describes a multichannel Siegert quantization procedure which can in principle be applied to any situation leading to a set of coupled equations. An example (atomic Stark effect) where the dissociation (ionization) rate is the desired quantity is given in section IV. This case shows that complex rotation may serve to
give a very simple solution of an otherwise difficult initialization problem. The determination of partial rates poses a more difficult problem. Section V gives a brief formal view of two methods in current use (artificial channel procedure and driven equation) which give a solution, which is then compared to that offered by the Siegert procedure. The combination of complex scaling with numerical or piecewise potentials leads to some difficulties in the coupled channel context. A solution is given in section VI with the so-called complex exterior scaling pr~cedure.~Finally, section VI1 gives an example related to the infrared photodissociation spectrum of van der Waals complexes, with the aim of showing how interference effects can be accounted for in a complex quantization method.
Prepared for the 8th Canadian Symposium on Theoretical Chemistry, Halifax, 1983. tAssociC 2 l’Universit6 Paris-Sud.
mentation of the method see J. Turner and C. W. McCurdy, Chem. Phys.,
0022-3654/84/2088-4839.$01.50/0
11. Siegert Solutions of the Wave Equation
Two familiar examples will serve to recall that we may obtain various types of quantized energies from the constraint that the wave function should behave asymptotically as an “outgoing” wave and that a class of these energies are complex and are associated to resonances. ( a ) Rectangular Wells6 Let us consider the radial Schrodinger equation
for a rectangular well v ( r ) = -Vo
< r < ro ro < r
0
u(r) = 0
We can define two wavenumbers k and inner (b) regions from
(2a) (2b)
for the outer (a) and
(1) For recent reviews see W. P. Reinhardt, Annu. Rev. Phys. Chem., 33, 223 (1982); B. R. Junker, Adv. At. Mol. Phys., 18, 207 (1982). (2) A. F. J. Siegert, Phys. Rev., 56, 750 (1939). (3) 0. Atabek and R. Lefebvre, Chem. Phys., 55, 395 (1981). (4) 0. Atabek and R. Lefebvre, Phys. Rev. A , 22, 1817 (1980). (5) B. Simon, Phys. Left., 71A, 211 (1979). For a one-channel imple-
71, 127 (1982). (6) D. L. Huestis, J . Math. Phys., 16, 2148 (1975).
0 1984 American Chemical Society
