Appendix: Overview of our methodology in"Regularity and Non-recrossing Paths in Transition States of Chemical Reactions"Tamiki Komatsuzaki and R. Stephen Berry

Department of Chemistry, The University of Chicago, Chicago, IL 60637(February 8, 1999)

The crux of our method, Lie canonical perturbation
theory, abbreviated as LCPT, is a classical version of
Van Vleck perturbation theory. These theories a priori assume existence of M invariants of motion for theM-dimensional Hamiltonian systems in question. It is
well known that with such methods, the new, transformed Hamiltonian diverges if the system encounters(near-)resonances, and becomes meaningless. However,
we clarify in this work that LCPT is still powerful enoughto reveal an invariant manifold even if almost action variables fail to maintain constants of motion. Rather, themethod provides us with a new phase-space dividing surface free from recrossing up to moderately high energy.
We briey describe our method to construct a model
Hamiltonian from any arbitrary autonomous Hamiltonian to which LCPT applies. We use an ecient technique, so-called algebraic quantization, to carry out the
LCPT without any special mathematical manipulations
and achieve a reformulated (classical) transition state
theory by using the hyperbolic phase space orbit in thesea of chaotic motions of the other stable modes. At theend, we describe shortly three distinct levels of local dynamics in the transition state in terms of regularity ofthe dynamics, which have not been addressed previously.

the zeroth-order Hamiltonian H0 is assumed to be integrable, e.g., a system of M harmonic oscillators. Such azero-order system is a function of action variables J only,and does not depend on the conjugate angle variables ,soX nH(p q) = Hn(p q)
(A3)
n=0
X
= H0(J) + n Hn(J )
(A4)
=

M
X
k=1

n=1

!k Jk +X n
 Hn(J )

n=1

(A5)

where !k is the fundamental frequency of the kth modeof H0. Assuming the new Hamiltonian H and the generating function w are also expandable in , we substitute and w to Eq. A2 and determine the new Hamilall H, H,
tonian, at each order in , to be as a simple a form as
possible by eliminating its dependencies on the new angle variables  . (1  3  7 ) If the H is obtained free fromthe angle  (at the order of the perturbative calculation
performed): p q) = H( J  ) = H( J) = X nH n(J) (A6)H(
n=0

APPENDIX A: LIE CANONICALPERTURBATION THEORY (LCPT)

the new action and angle variables for the kth mode areexpressed as J)dJk = J_ = ; @ H(
(A7)
k
dt
@  k = 0Jk = constant (k = 1 2 3 : : : M)
(A8)
and
 J)

_ k = @ H(
(A9)
@ Jk  !k (J) = constant k = ! k (J)t + k 
(A10)
where k is the arbitrary initial phase factor of the kthmode. From there, the equations of motion with respect
to the new coordinates q and momenta p are obtained
from the Hamiltonian equations of motion obeying H 
d2qk (p q) + ! 2 q (p q) = 0
(A11)
k k
dt2
and
(A12)
pk (p q) = !! k dqk (dtp q)

Canonical perturbation theories (1 ) transform (p q)
to a new (p q) coordinate system so as to make the p q) as close to integrable as possinew Hamiltonian H(
ble. Lie canonical perturbation theory (LCPT) (1 {7 ) is
the most sophisticated theory among them, applicable toimplementing higher-order perturbations and to treating
systems with many degrees of freedom. The LCPT isbased on Lie transforms, that is, the exponential of a Lieoperator induces a canonical transformation: let Lw be
the Lie operator associated with a generating function w,
Lw  fw g(A1)
where fg denotes the Poisson bracket. Then the transformation of an autonomous Hamiltonian H to a newHamiltonian H, p  q) = exp(Lw )H(p  q)
H(p q) ;! H((A2)
is canonical. We let H be an M-dimensional Hamiltonian
expandable in  (=strength of the perturbation) wherek1where !k (= !k (J) = !k (p q)) is independent of timet because the J are constant through all t Eq. (A7)].
The equations (A11) and (A12) tell us that even though
the motions look quite complicated in the old coordinate
system, they could be reformulated as simple decoupledperiodic orbits in the phase space.
If one tries to regularize a general Hamiltonian globally, one almost surely encounters the problem that the(near-)commensurable conditions that an integer linear
combination of fundamental frequencies vanishes identically at a given order n ,MX
nk !k  O(n)(A13)multidimensional Hamiltonian systems are strongly nonuniform at low and moderate energies, depending on thelocal topography of their potential energy surfaces. (8  9 )We rst expand the full 3N-DOF potential energy surface about a chosen stationary point, i.e., minimum, saddle, or higher rank saddle. By taking the zeroth order Hamiltonian as a harmonic oscillator system, whichmight include some negatively curved modes, i.e., reactive modes, we establish the higher-order perturbation terms to consist of nonlinear, anharmonic couplings
which we may express in arbitrary combinations of coordinates. Thus,
1X
H = H0 + n Hn(B1)(nk is arbitrary integer), makes the corresponding
new Hamiltonian diverge and destroys invariants ofmotion. (1 ) It is quite likely in many M-dimensional
systems that the near-commensurable conditions densely
distribute in the phase space, i.e., the occupation ratio of
the M-dimensional tori in the phase space is negligibly
small, and hence Eqs. (A11)-(A12) become meaningless.
Here, however, we consider only approximately regularbehavior in local regions, specically saddles, where wend that such diculties of fully-developed chaos for thefull set of degrees of freedom only at quite high energies.
We rst showed that, by monitoring the new action of
the kth mode Jk (p q) along molecular dynamics (MD)trajectories obeying equations of motion of the original Hamiltonian H(p q), one can detect whether theqk mode tends to maintain an invariant of motion. If
the Jk (p q) and its associated ! k (p q) exhibit nearconstants of motion through a certain range in time andlocality, it implies that pk and qk are approximately decoupled from the other modes, and represent the local
dynamics analytically. Those quantities associated witha reactive mode exhibit such near-constants of motion
up to a moderately high energy in the region of a saddle. Furthermore the dividing hypersurface, dened by
the condition that the reactive coordinate in the transformed coordinates is zero, is almost free from recrossing
problems.

whereX
H0 = 12 (p2j + !j2 qj2 )

k=1

n=1

j

1
X
X
n Hn =  Cjklqj qk ql

n=1

jkl

+ 2

X

jklm(B2)(B3)

Cjklmqj qk ql qm + : : : (B4)Here, qj and pj are the jth normal coordinate and its conjugate momentum, respectively !j and Cjkl, Cjklm ,...
are, respectively, the frequency of the jth mode, the coupling coecient among qj , qk , and ql and that among
qj , qk , ql , and qm and so forth. The frequency associated with an unstable reactive mode F and those of
the other stable modes B are pure-imaginary and real,respectively. At any stationary point there are six zerofrequency modes corresponding to the total translational
and innitesimal rotational motions, and the normal coordinates of the innitesimal rotational motions appear
in the perturbation terms Hn(q) (n > 0). The contribution of the total translational motion is simply separated. We make no more mention of this. If one deals
with a system whose total angular momentum is zero,one could eliminate the contributions of the total rotational motions from Hn(q) (n > 0) by operating witha suitable projection operator (10 ) at the stationary
point it corresponds to putting to zero each normal coordinate and corresponding conjugate momentum representing the innitesimal total rotational motion. For thesake of simplicity we focus on a (3N-6)-DOF Hamiltoniansystem with total linear and angular momenta of zero, sothat the kinetic and potential energies are purely vibrational. For such a zeroth-order Hamiltonian !k 6= 0 for
all k (= 1 2 3 : : : 3N ; 6( M)), the associated actionangle variables of the stable modes B (!B 2 < : real)
and the unstable mode F (!F 2 = : imaginary) are
expressed as
 p2

I
1
1
B
2
JB = 2 pB dqB = 2 ! + !B qB 
(B5)B

APPENDIX B: THE REGIONAL HAMILTONIAN
Despite its versatility, LCPT has not been applied to many-degrees of freedom (DOF) realistic
atomic/molecular systems. One of the two main diculties is how one could handle the analytical derivativeand integral calculations that appear successively in the
LCPT procedure. The other is the near-impossibilityof obtaining even moderately simple analytical expressions to describe the accurate (e.g., ab initio) potential energy surfaces in full. As shown for a variety ofatomic/molecular clusters, the dynamical properties of2B = tan;1 !pBq 
B B

In the present work, we analyze A up to a (nite) i order(i=0,1,2):i
XAith = Aith (p q) = nAn (p q)
(C6)

(B6)

and

1 Im Z
JF = 2
pF dqF (B7)barrier 2
= 2i j!pF j ; j!F jqF2 
(B8)
F


!F  ;j!F ji: (B9)F = i tanh;1 j!pFjq 
F F
Here the action associated with the reaction mode F,
having rst been postulated in the semiclassical transition state theory, (11 {13 ) is purely imaginary and isconnected with the barrier penetration integral in thesemi-classical theory. It is easily veried (14 ) that any
such set of variables J and  is canonical, including those
associated with the unbound mode F.

n=0

where no (near-) commensurable conditions were encountered at these orders during our LCPT procedure. The
details of the computational recipe are given in (7 ).
To indicate the complexity of the transform, we show
the expressions for p1(p q) and q1(p q) up to rst orderat saddle I of Ar6 (Here mode 1 is reactive and the others
are non-reactive). The contributions of the original p1
and q1 in pi1th (p q) and q1ith (p q) are not necessarily large
and almost all modes contribute to pi1th and q1ith (p q) fori 1.
p01th = p1 (C7)
p11st = p1 + 0:103353p1q1 + 0:097266q7p8+ 0:084426q1p10 + 0:079721q4p11 ; 0:075768p1q10; 0:070628p4q11 + 0:067259q1p12 ; 0:061966p1q12+ 0:057190q5p10 + 0:056576q6p11 ; 0:056558p7q8+ 0:051195q10p12 ; 0:048788p6q11 ; 0:044909p5q10; 0:044499q5p12 + 0:039476q2p12 + 0:038653p5q12; 0:038584q1p2 ; 0:036376p10q12 ; 0:035646p2q12+ 0:035423p2q2 ; 0:032806p6q6 + 0:031813p3q3; 0:031358q1p5 + 0:026383p7q7 + 0:025753p1q5+ 0:025277p1q2 ; 0:024068p11q11 + 0:019803p8q8+ 0:017695q4p6 ; 0:017171p9q9 + 0:016610q2p10; 0:014351p2q10 + 0:010428p5q5 ; 0:008864p4q6+ 0:005637q2p5 ; 0:003872p2q5 ; 0:003792p12q12+ 0:003657p4q4 ; 0:002603p10q10 + 0:001701q3p9; 0:001423p3q9
(C8)
q10th = q1
(C9)

APPENDIX C: THE ALGEBRAIC
QUANTIZATION METHOD
For practical LCPT calculations of the above Hamiltonians, a quite ecient method, called \algebraic quantization," has been developed. (6  7 ) This method rst
transforms (p q) in Eqs. B2-B4 to (a  a) by the customary means:ak = p1 (pk + i!k qk ) ak = p1 (pk ; i!k qk)(C1)2
2
which are expressed in terms of the old action variable,
the associated frequency Jk , !k , and time obeying
Hamiltonian H0 aspp(C2)ak ( ) = !k Jk eik = !k Jk ei(!k  +k ) ppak ( ) = !k Jk e;ik = !k Jk e;i(!k  +k ) : (C3)
The cumbersome analytical calculations of Eqs. B4-B8
that appear in the LCPT calculations are then replacedby simultaneous algebraic, symbolic operations, thanks
to the simple Poisson bracket rules for the (a  a),faj  akg = faj  akg = 0 faj  ak g = i!k jk (C4)
where fg and denote Poisson bracket and Kronecker
delta, respectively. One simply substitutes (C1)-(C3)into (B1), to set up the simultaneous algebraic equations,which readily yield the desired dynamical quantities withthe help of (C4). (7 )Finally, we obtain new transformed physical quantities
 i.e., the new Hamiltonian H,
 and new action Jk , freA,
quency !k , momentum pk , and coordinate qk of the kthmode, in terms of the original p and q as
 p q) = X nAn (p q):(C5)
A = A(2q11st = q1 + 0:599212q7q8 ; 0:559687q11
22+ 0:412029p1 + 0:338508q7 + 0:328906q82; 0:319091q62 + 0:310549q10q12 ; 0:291435q92+ 0:183270q32 + 0:170034q5q10 + 0:162288p7p8+ 0:152283q22 + 0:141216p22 + 0:136269q4q11; 0:130785p26 + 0:128438q6q11 + 0:126825p23+ 0:105180p27 + 0:101029q52 ; 0:098218q5q12
2 + 0:078946p2; 0:095949p211 ; 0:090680q12
8+ 0:075768q1q10 + 0:072618q4q6 ; 0:068455p29+ 0:061966q1q12 + 0:059076p10p12 + 0:054028q2q12; 0:053049p1p2 ; 0:051677q12 + 0:048960p5p10

n=0

3

2
; 0:046850q10+ 0:041571p25 + 0:036248p4p11(D1)+ 0:035207p4p6 + 0:034517p1p10 + 0:031050p6p11 where h(x) and (x), respectively, denote the Heaviside; 0:025753q1q5 ; 0:025277q1q2 + 0:025194q2q10 function and Dirac's delta function of x. The canonical+ 0:024569q42 ; 0:023307p5p12 ; 0:022347p1p5 form is also formulated straightforwardly.
no approximate invariant of motion exists in the sad+ 0:021104p1p12 + 0:015270p2p12 ; 0:015119p212 dleIfregion,
kGTST (E) deviates from the (classically) ex+ 0:014577p24 + 0:012340q2q5 ; 0:010375p210
act reaction rate constant k(E). Therefore one can in+ 0:009008p2p10 + 0:007034p2p5 + 0:003167q3q9 troduce a new transmission coecient c , the deviationof the kGTST (E) from the k(E):+ 0:001111p3p9 :k = c kGTST :
(D2)
(C10)
The new momenta and coordinates pk (p q) and We may use c to estimate the barrier recrossing e ect,as a measure of the extent to which the quasi-invariantsqk (p q) have the following forms respectively,of motion associated with the reactive mode, i.e., theX 2n;1 mpk (p q) = cj p q (C11) local action and its local frequency, cease to be approximate invariants. Their nonconstancy reects the degreejof fully-developed chaos in which no invariant of motion
X 2n mqk (p q) = dj p q (C12) exists, if the vibrational energy relaxation is fast enoughto let us assume quasi-equilibration in the reactant's poj
tential well.
In order to focus on how the recrossings over a given
where cj and dj denote the coecient of the jthterm, n, m( 0) P
are arbitrary integers, and qm = dividing surface contribute to c , we estimate the timeithmM ( M mj = m) etc. The newdependent quantities MD
q1m1 q2m2 q3m3 qMc (t S(qF = 0)) in terms of ourj =1pk (p q) and qk (p q) maintain time reversibility. The MD trajectories dened byh ith
iunits of pk and qk are 1=2 and m1=2 , respectively. As
_
h
j(t
=
0)h
q

(
p

q
)
iE
F
we increase the order of LCPT to be evaluated, the total
MD (t S(qith = 0)) =(D3)cF
number of the terms rapidly increases. For example,
hj+ (t = 0)iE
2
nd
2
nd
450 for p1 and q1 .
where j(t = 0), and j+ (t = 0), respectively, denote theinitial total, and initial positive uxes crossing the ith order LCPT dividing surface S(qFith (p q) = 0). The origin
APPENDIX D: REFORMULATION OF
of time t is taken to be zero when the system trajectory
TRANSITION STATE THEORYrst crosses the given dividing surface. This equation
should also tell us how the vanishing of the approximate
In the cases that saddle crossing dynamics has ap- invariants of motion of the reactive mode reects on thec.
proximate invariants of motion associated locally withthe reactive mode F in a short time interval but longenough to determine the nal state of the saddle crossAPPENDIX E: DISTINCT CLASSES OF
ings, the qF can be identied as a \good" reaction coordiDYNAMICS
IN THE TRANSITION STATEnate. This is because there is no means or force returning
the system to the new dividing surface S(qF = 0) even
We found, with the analysis a orded by the minimalthough the system may recross the original naive surrecrossing
trajectories provided by the LCPT analysis,
face S(qF = 0). The reformulated microcanonical (clasthat
there
are
at least three distinct energy regions abovesical) transition state theory(TST) rate constant kGTST
is obtained (7 ) as a thermal average of the one-way uxes the saddle point energy that can be classied in terms ofj+ (= q_ F (p q)h(q_ F (p q))) across S(qF = 0) over micro- the regularity of saddle-crossing dynamics. Let us now
canonical ensembles constructed over a range of energies articulate the distinctions among them.
Quasiregular region All or almost all the degrees ofE.
freedom of the system locally maintain approximate constants of motion in the region of the transition state. The
kGTST (E) = hj+ iE
saddle crossing dynamics from well to well is fully deterministic, obeying M-analytical solutions (see Eqs. A11__
= hZqF (p q) qFZ(p q)] h qF (p q)]iE A12) for systems of M degrees of freedom. The dynami= dq1dp1 : : : dqN dpN
cal correlation between incoming and outgoing trajecto1
N
ries from and to the transition state region is quite strong,E ; H(p q)]
and the dimensionality of saddle crossings is essentiallyq_ F (p q) qF (p q)]h q_ F (p q)] one, corresponding to the reactive mode qF in the (p q)4

space. Barrier recrossing motions observed over a naive
dividing surface dened in the conguration space are
all rotated away to no-return single crossing motions
across a phase space dividing surface S(qF (p q) = 0).
If the vibrational energy relaxes fast enough to let us
assume quasi-equilibration in the wells, the initial conditions (p(0) q (0)) of the system as it enters the transition state from either of the stable states can be simply sampled from microcanonical ensembles. One maythen evaluate the (classical) exact rate constant, freefrom the recrossing problem. The staircase energy dependence observed by Lovejoy et al. (15 ) for highly vibrationally excited ketene indicates that the transverse
vibrational modes might indeed be approximately invariants of motion. (16 ) We classify such a range of energy,
in which the rate coecient shows staircase structure, as
corresponding to this quasiregular region.
Intermediate, semi-chaotic region Due both to significant (near-)resonances and to strong anharmonic modemode couplings emerging at these intermediate energies,
almost all the approximate invariants of motion disappear, consequently inducing a topological change in dynamics from quasiregular to chaotic in the regions of saddle crossings. However at least one approximate invariantof motion survives during the saddle crossings, associatedwith the reactive coordinate qF (p q). This is due to thefact that an arbitrary combination of modes cannot satisfy the resonance conditions of Eq. A13 if one mode has
an imaginary frequency, the reactive mode in this case, is
included in the combination. The other frequencies associated with bath modes fall on the real axis, orthogonal
to the imaginary axis in the complex !-plane. That is, MX ynk !k  j!F j > O(n)
(E1)
k=1


the convergence radius becomes negligibly small for the
LCPT near the xed (saddle) point for the invariant of
motion associated with the reactive coordinate qF . In
this energy region, no approximate invariant of motion
can be expected to exist, even in the passage over the saddle between wells. The saddle-crossing dynamics is entirely stochastic, with dimensionality essentially equal to
the number of degrees of freedom of the system. Here it isprobably not be possible to extract a dividing surface freefrom barrier recrossings. At these high energies above thelowest, presumably (but not necessarily) rst-rank saddle, the system trajectories may pass over higher-ranksaddles of the potential energy surface. These provides
us with a new, untouched problem, i.e., what is the role ofresonance in the imaginary !-plane for the bifurcation?
(This even arises in the degenerate bending modes fora linear transition state of a triatomic molecule.) With
this, we encounter one of the related open subjects instatistical theories of many-DOF systems.

1: A. J. Lichtenberg and M. A. Lieberman, Regular andChaotic Dynamics 2nd Ed. (Springer-Verlag, New York,
1992).
2: G. Hori, Pub. Astro. Soc. Japan 18, 287 (1966) ibid.
19, 229 (1967).
3: A. Deprit, Celest. Mech., 1,12 (1969).
4: A. J. Dragt and J. M. Finn, J. Math. Phys. 17, 2215
(1976) ibid. 20, 2649 (1979).
5: J. R. Cary, Phys. Rep. 79,130 (1981).
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(1988).
7: T. Komatsuzaki and M. Nagaoka, J. Chem. Phys. 105,
10838 (1996).
8: C. Amitrano and R. S. Berry, Phys. Rev. E 47, 3158
(1993).
9: R. J. Hinde and R. S. Berry, J. Chem. Phys. 99, 2942
(1993).
10: M. Page and J. W. McIver. Jr. ibid. 88, 922 (1988).
11: W. H. Miller, Faraday Discussions Chem. Soc. 62,40(1977).
12: T. Seideman and W. H. Miller, J. Chem. Phys. 95, 1768
(1991).
13: R. Hernandez and W. H. Miller, Chem. Phys. Lett. 214,129 (1993).
14: H. Goldstein, Classical Mechanics, (Addison-Wesley,
Massachusetts, 1950).
15: E. R. Lovejoy, S. K. Kim and C. B. Moore, Science 256,
1541 (1992).
16: R. A. Marcus, ibid. 256, 1523 (1992).

for arbitrary integers nk with nF 6= 0, where !y denotes
the combination including the reactive mode. This wasrst pointed out by Hernandez and Miller. (13 ) In thisregion the dynamical correlation between incoming andoutgoing trajectories to and from the transition state becomes weak (but non-zero!), and the saddle crossings'
dimensionality is ' M ; 1, excluding the one dimension of qF , in this region. If the associated imaginary
frequency ! F (p q) is approximately constant during asaddle crossing as the action JF (p q) is, the reaction coordinate qF decouples from the Z-subspace composed of
the other bath-DOF, in which the system dynamics is
manifestly chaotic. The qF dynamics is then represented
analytically during saddle crossings, and a dividing surface S(qF (p q) = 0) can still be extracted free from thebarrier recrossings, even for saddle crossings chaotic in
the bath modes. This class does not exist near potential
minima, but is inherent associated with the transition
state.
Stochastic (=fully-developed chaotic) region The system becomes subject to considerable nonlinearities of thepotential energy surface at much higher energies, and5