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INSTITUTE OF PHYSICS PUBLISHING JOURNAL OF PHYSICS B: ATOMIC, MOLECULAR AND OPTICAL PHYSICS
J. Phys. B: At. Mol. Opt. Phys. 36 (2003) 1–17 PII: S0953-4075(03)56981-4
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Rotational transition states: relative equilibriumpoints in inelastic molecular collisions
L Wiesenfeld1,2, A Faure2 and T Johann1,2
1 Laboratoire de Spectrometrie Physique, Universite Joseph-Fourier-Grenoble,F-38402 Saint-Martin-d’Heres-Cedex, France2 Laboratoire d’Astrophysique de l’Observatoire de Grenoble,Universite Joseph-Fourier-Grenoble, F-38051, Grenoble Cedex 9, France
E-mail: [email protected]
Received 8 November 2002, in final form 13 February 2003PublishedOnline at stacks.iop.org/JPhysB/36/1
AbstractTransition states (TSs) are a key ingredient in the understanding of manychemical reactions. We present here a generalization of TS theory towardsrotational excitation in molecular collisions, in a multi-dimensional classicalHamiltonian framework. The treatment is based on relative equilibrium,where the two colliding molecules behave as a single rotating solid object.We illustrate the theory with the help of a simple, yet meaningful, modelpotential for collisions between H2O and H2, which is of great importancein the astrophysical context. We show that it is the occurrence of a rotationalTS that opens up the possibility of significant angular momentum transfer.
1. Introduction
The Hamiltonian classical dynamics of bound states of molecules or atoms have been widelystudied and the knowledge gained has greatly helped in understanding many aspects of boundstate spectroscopy and dynamics (Child 1991, Brack and Bhaduri 1997, Rost 1998, Nymanand Yu 2000). However, scattering systems have been less thoroughly discussed (Kovacs andWiesenfeld 1995, 2001, Toda 2002) and even fewer papers examine the dynamical importanceof angular momentum. Here we wish to begin such an undertaking, focussing on the scatteringprocesses between two molecules and allowing for sufficiently many degrees of freedom sothat realistic inelastic processes are included in the models. We shall specifically deal withnonzero total angular momentum (J � 0) in order to concentrate on the dynamical importanceof centrifugal barriers. We will also make the specific connection between the effectivenessof angular momentum transfers and the occurrence of transition states (TSs) in the full phasespace.
Quantum mechanically, it has been known for a long time that calculating inelastic orreactive collisions between isolated molecules entails large relative angular momentum. Partial
0953-4075/03/000001+17$30.00 © 2003 IOP Publishing Ltd Printed in the UK 1
2 L Wiesenfeld et al
waves with � � 15 are of importance even at low collision energies. These partial waves are,for example, known to contribute significantly to inelastic cross sections in molecular collisionsof astrophysical interest. In particular, calculations of rotational excitation of polar molecules(such as CO, H2O, NH3) by H2 appeared recently (Flower 1990, Rist et al 1993, Phillips et al1996, Dubernet and Grosjean 2002). They showed firstly the appearance of resonances in thecross section at low energy (E � 100 K) and secondly, the absolute necessity of includinghigh � partial waves in order to obtain convergence of the quantum close-coupling calculation.
On the experimental side, the importance of high J states is known, for example, in thecontext of collision-induced absorption, for so-called van der Waals molecules (Vigasin andSlanina 1998, Borysow and Frommhold 1986, Chambers and Child 1988, Miller and Wales1996). These complexes are known, in particular, for molecules of atmospheric interest, suchas N2, O2, CO2. Complexes of these partners last long enough to exhibit radiative transitions(broad absorptions are observed). Rotational barriers are thought to determine the actualproperties of some classes of those long-lasting complexes. They are also known to be ofimportance in some collisions described in astrophysical contexts, such as the CO–H2 collision(Jankowski and Szalewicz 1998). Also, angular momentum may orient the outcome of somephoto-dissociation reactions, even if much more modest J values are usually involved (Schinke1993).
The statistical analysis of rotational transfers in chemical reactions used to be called‘orbital transition state theory’ and was studied in detail some years ago (Markus 1975, Gruceet al 1986), and also more recently (Larregaray et al 2001). Even so, despite the considerableimportance of angular momentum in elastic, inelastic or reactive collisions, very few detailednonlinear dynamical descriptions have been proposed (Kozin et al 1999, 2000, Littlejohn1997, Faure et al 2000). Here we wish to fill this gap. It seems that the paucity of detailedinvestigations stems mainly from the dimensionality of the system at hand. In order to includesubsystems with internal degree(s) of freedom, the minimally meaningful system must includethe following ingredients:
(i) A two-dimensional physical space, in order to provide the possibility to define the totalangular momentum J and relative angular momentum � both perpendicular to this physicalplane.
(ii) Two molecules a and b, each having an angular momentum by themselves ( ja, jb), inorder to allow the exchange between internal and external degrees of freedom.
(iii) An anisotropic inter-molecular potential, in order to actually couple the angular andtranslational degrees of freedom.
These requirements amount to the following scalar coordinates and momenta:
Intermolecular distance : R, pR
Intermolecular axis orientation : θ , �
Internal orientation a : φa, jaInternal orientation b : φb, jb,
summing up to four degrees of freedom and an eight-dimensional phase space. This highdimensionality had precluded until recently any detailed attempt to describe angular momentumtransfer mechanisms.
The only theory devised to cope with multi-dimensional dynamics in reactive scattering isevidently transition state theory (TST), in its many varieties (Truhlar et al 1996, Miller 1998,Pollak and Liao 1998). However, the usual TS approach does not seem to apply here. Thepotential function may have some extrema in configuration space, some of them indeed havingthe correct eigenfrequencies to be considered as TST in the usual sense (three real frequencies
Rotational transition states 3
and one imaginary frequency, in the parlance of chemical physics). But for J > 0 dynamics, itmay be that the centrifugal energy is of the same order as the usual potential energy, especiallyfor the weak intermolecular potentials we are considering here. It becomes meaningless tospeak about TST in configuration space. Thanks to the advent of a well founded TST in phasespace (Wiggins et al 2001), it is now possible to take full advantage of the detailed knowledgeof the various equilibrium points in order to describe some broad features of the inelasticprocess.
This paper is organized as follows. In section 2 we describe the relative equilibria ingeneral. Then we introduce the specific Hamiltonian for H2O–H2 (section 3); we continue byexamining the physical relevance of the relative equilibria (section 4). We end with a shortconclusion.
2. Dynamical theory
Classical models of dynamics in molecules have always proved to be useful whether modellinga reaction or an inelastic scattering process. Some equilibrium points of the dynamics(sometimes called critical points in the mathematical literature) are of special importance.Their stability patterns determine whether or not they support a TS, as has been shown inconfiguration space (for a full account, see the review paper (Truhlar et al 1996)) and alsoshown in phase space (Wiggins et al 2001, Uzer et al 2002, Komatsuzaki and Berry 2001).
2.1. Properties of equilibria
Let us recall that (absolute) equilibrium points are defined as points where x = 0, where vectorx denotes the 2N dynamical variables characterizing the system (N is the number of degreesof freedom). In other words, from Hamilton’s equations, ∇H = 0 at the equilibrium points.Equilibrium points are characterized by their linear stability. Let x0 be an equilibrium pointand let δx = x − x0 be a small increment near to this equilibrium value. At linear order, wehave
dδ x
dt= J
∂2 H
∂xi∂x jδx,
J =(
0 I−I 0
),
(1)
where I is the N ×N unit matrix. These linearized equations of motion (equation (1)) define thelinear stability of the equilibrium point by means of the eigenvalues of the linearized motionmatrix. The eigenvalues appear in pairs,λ±
k , one pair for each degree of freedom k = 1, . . . , N .The system being Hamiltonian, we have λ+
k + λ−k = 0 (Wiggins 1990, Ozorio et al 1990). If
we suppose that no degeneracy occurs (λ±k �= λ±
k′ ), then each nonzero eigenvalue pair definesa plane spanned by the corresponding two eigenvectors (there are N planes for N degrees offreedom). If we have λ±
k real, then motion in the plane is hyperbolic, if the eigenvalues λ±k are
pure imaginary, motion is elliptic. In the hyperbolic case, the equilibrium point shows stableand unstable manifolds, which are at a tangent, in the nonlinear case, to the stable and unstableeigenvectors of the linear case.
It may also occur that: (i) four complex eigenvalues appear, two by two conjugated and ofzero sum λ = ±κk ± iωk (ii) two pairs of eigenvalues are degenerate (λ±
k = λ±k′ ), and (iii) two
are zero (λ±k = 0). Of these three cases, only the latter will be taken into account here, for the
sake of simplicity of the arguments.
4 L Wiesenfeld et al
ξ
pξ
Figure 1. A schematic view of the linearized dynamics in the ξ, pξ plane. The equilibrium pointis the filled circle at the 0, 0 coordinate. The reactive trajectories are represented by full curves,and the nonreactive ones by dotted (ξ > 0 side) or dashed curves (ξ < 0 side). Arrows denote thesense of the motion.
As alluded to in the introduction, it has been known for a long time that TSs betweenreactants and products occur near to some particular equilibrium points, those for which wehave the following eigenvalue structure, for N degrees of freedom:
λ±k , k = 1, . . . , N − 1 pure imaginary; λ±
N real. (2)
This has been recently put on firm mathematical grounds (Wiggins et al 2001). Let ξ, pξ
be the position and momentum coordinates that span the N th hyperbolic plane and letqk, pk, k = 1, . . . , N − 1 be the position and momentum coordinates that span the other,N − 1 elliptic planes. A linearized Hamiltonian that supports such a decoupling between onereaction mode (the ξ sector) and N − 1 bath modes can be readily written as follows(Miller1998, Wiggins et al 2001, Uzer et al 2002):
H = 12
N−1∑k=1
(p2k + ω2
k q2k ) + 1
2 (p2ξ − κ2ξ2). (3)
The structure of equation (3) is understood as N − 1 nonresonant harmonic oscillators—thebath—and one anti-oscillator—the reaction. Figure 1 shows the reaction sector of the linearizeddynamics.
We wish to show now that this TS structure exists not only around an absolute equilibriumbut also near to a relative equilibrium (RE), as defined in the next section.
2.2. Relative equilibria
Relative equilibria are situations where the shape of the object under scrutiny does not changein time while the object as a whole is rotating. Since our system here comprises of twomolecules, a RE point means that:
(i) There is no relative radial speed nor acceleration,(ii) The composed body rotates as a whole, at uniform angular speed and zero angular
acceleration.
Rotational transition states 5
There has been various studies concerning relative equilibria (Kozin et al 1999, 2000,Faure et al 2000, Roberts and de Sousa Dias 1997, van Hecke et al 2000, Uzer et al 1998),especially for bound systems. Rotating scattering systems have been much less studied exceptwhen the rotating frequency is imposed and constant. Many gravitational N-body problemsbelong to this class. In particular, some versions of the restricted three-body problem aretreated in the rotating frame, where equilibria are actually only relative equilibria. The drift ofobjects in the solar system have been studied in a well chosen rotating reference frame (Jaffeet al 2002, Henrard and Navaro 2002, Koon et al 2000). This approach is similar to ours butis restricted to a few degrees of freedom. It also uses the determination of a TS in order totransport in phase space rates, in a manner similar to chemical theory. It must be emphasizedhere that the idea of a RE is simple only if the two objects are not deformable. Otherwise,the full rotation/vibration/deformation Hamiltonian is much more involved (Littlejohn 1997,Kozin and Pavlichenko 1996).
Let us first consider a radial Hamiltonian, with angular momentum J , reduced mass mand radial potential V :
H = p2r
2m+
J 2
2mr2+ V (r). (4)
A RE occurs when
dV (r)
dr− J 2
mr3= 0.
This RE is radially unstable if J 2/(2mr2) + V (r) is a maximum, and radially stable if it isa minimum. If an unstable RE occurs, the deflection function f = f (bi), (Kovacs andWiesenfeld 1995, Jung and Scholtz 1987), displays rainbows ( f is the final angle of exit ofthe particle in the inertial frame, bi is the initial impact parameter). The structure of theserainbows is well known in the classical or quantum cases (De Micheli and Viano 2002). Forsuch an integrable Hamiltonian like (4), there are as many singularities (rainbows) of thedeflection function as integer numbers: each singularity is characterized by an increase by 1of k = mod( f , 2π). There is one impact parameter b∗ such that k → ∞. It correspondsto the intersection of the asymptotic condition r → ∞ with the stable manifold of the uniqueRE point.
A general Hamiltonian is more complicated:
H = p2r
2m+
J 2
2mr2+ V (r, θJ ) (5)
or
H = p2r
2m+
J 2
2mr2+ V (r, θJ , t) (6)
with θJ conjugated to the total angular momentum J , and t , the time. Motion is vastly morecomplicated, exhibiting generically regular and chaotic regions in phase space as well as one orseveral RE points. The dynamically invariant objects in the relative motion frame (RE, periodicorbits, and their high-dimensional analogues, (Ozorio et al 1990, Wiggins et al 2001)) havestable/unstable manifolds. These manifolds create heteroclinic and homoclinic tangles, in away totally analogous to inertial periodic orbits (Kovacs and Wiesenfeld 1995, Meyer et al1995, Jung and Seligman 1997).
If one considers the whole Hamiltonian H , equation (5), including the rotation, the REpoint becomes a periodic orbit (PO). This PO has a trivial equation
x = x0, x �= θJ
θJ = θ0J + ωt
(7)
6 L Wiesenfeld et al
where the superscript 0 denotes the initial conditions for the orbit. In the language of linearizedmotion, the following eigenvalue structure appears for the PO (Wiggins 1990) (recall ourrestriction given in section 2.1):
(i) One eigenvalue pair λ±1 = 0, in the J, θJ plane, corresponding to the conservation of total
angular momentum, along the PO,(ii) K pairs of imaginary eigenvalues λ±
k = ±iωk .(iii) K ′ pairs of real eigenvalues λ′±
k = ±κk .
We have 1 + K + K ′ = N .Returning to the relative frame, let H ′ be the Hamiltonian restricted to the relative motion.
A RE will occur at points where ∇H ′ = 0. The linear stability is now defined with respectto the Hessian of H ′ and no more of H as in section 2.1. For general relative Hamiltonians,with several rotation-like motions (Littlejohn 1997), there are in general m � 1 degrees offreedom that are frozen in the rotating frame. For Hamiltonian (5) as well as for the planar,nondeformable bodies problem that is dealt with in this study, we have m = 1: the totalangular momentum J and its conjugated angle θJ . Then, if we have the following structure ofeigenvalues for the RE point P (see equation (2)):
λ±k , k = 2, . . . , N − 1 pure imaginary; λ±
N real, (8)
we define in a neighbourhood of P a relative TS. The first eigenvalue pair does not appearin the relative frame. In a manner totally comparable to the usual TS, we have in relativecoordinates and in the linear approximation (see equation (3))
H ′ = 12
N−1∑k=2
(p2k + ω2
k q2k ) + 1
2 (p2ξ − κ2ξ2). (9)
The dynamics across this linearized relative TS is exactly comparable to the dynamics acrossthe linearized usual TS. As with the usual TS, a relative TS defines two regions in phase space:an outer region and an inner region with ξ > 0 and ξ < 0, for Hamiltonian (9). However,the full dynamics may be qualitatively different, precisely because of the relative nature of theequilibrium and the occurrence of Coriolis terms in the relative frame. These Coriolis termsbreak the time-reversal symmetry, as will be made apparent in section 4.
3. Application to the H2O–H2 system
3.1. A model Hamiltonian
A potential energy surface for the H2O–H2 system has been calculated by Phillips et al (1995)for astrophysical applications. This potential enabled several authors to calculate rotationalexcitation rates, as a function of temperature, in the conditions prevailing in interstellar media.While this potential is rather precise, its analytical form is very intricate and does not yieldeasily written and understood equilibrium conditions. A detailed examination of a very goodquality potential will be in order once the function of the relative equilibria is fully understood.
In order to keep things as simple as possible while remaining physically relevant, wemake use here of a model potential, that includes the terms in R−4 and R−6 that dominate thelong-range H2O–H2 interaction: the dipole–quadrupole potential, the anisotropic dispersionpotential (van der Waals potential) and the anisotropic dipole/induced dipole force. The termsinvolving induced quadrupoles,as well as the quadrupole/quadrupole interactions are neglectedbecause of their R−n, n � 8 behaviours. For the sake of simplicity, we also neglect theweak, R−5 quadrupole/induced dipole potential. As stated in the preceding section, we restrict
Rotational transition states 7
x
y
y
yb
y’xb
O
H
H
H
xa
a
x’
R
χb
H
χa
θ
Figure 2. Two-dimensional scheme of the a ≡ H2O and b ≡ H2 coordinates. x, x ′, y are lab-fixedframes and the dashed line is the intermolecular axis, with R, the intermolecular distance. Thedipole of H2O is along xa and the H–H internuclear axis is oriented along xb . Angles as shown.
ourselves to a planar configuration, neglecting two more angular degrees of freedom. The threeforces (dipole/quadrupole, dispersion and induction) are derived from the following potentials(Hirshfelder et al 1964):
VµQ = +3µQ
4R4{cos χH2O(3 cos2 χH2 − 1) − sin χH2O sin 2χH2}, (10)
Vdisp = − 1
R6{3(A − B) cos2 χH2 + (A + 5B)}, (11)
Vind = −µ2α(3 cos2 χH2O + 1)
2R6− µ2δα
4R6[cos 2χH2(3 cos2 χH2O + cos 2χH2O)
+ 2 sin 2χH2O sin 2χH2 ]. (12)
The angles χH2O, χH2 are defined in figure 2 and R is the distance between the H2O centre-of-mass and the H2 centre-of-mass. The first term in equation (12) is the usual dipole/induceddipole term, for isotropic polarizabilities. A and B are molecular constants defined by meansof parallel and perpendicular polarizabilities (α⊥ and α‖, mean α, anisotropy δα = α‖ − α⊥)as well as by means of the average ionization potential I = I (H2)I (H2O)/(I (H2) + I (H2O)):
A = α(H2O)α‖(H2) I/4
B = α(H2O)α⊥(H2) I/4.(13)
The values of the physical constants for the hydrogen and water molecules are given in table 1.An order of magnitude calculation shows that we have, for angle averages,
〈Vdisp〉/〈Vind〉 � 7.
The physical problem we address here is essentially long range. However, in order tohave physically meaningful trajectories, a hard-core repulsive potential has to be added. Forthe sake of simplicity, only a hard-core, Lennard-Jones type potential is added to the precedingterms:
VLJ = +CLJ
R12. (14)
8 L Wiesenfeld et al
Table 1. Values of the physical constants (all in atomic units) involved in the H2O/H2 potential.α, polarizabilities, µ, permanent dipole, Q, permanent quadrupole, I , mean ionization potential.
H2O α 9.83a
I 0.464a
µ 0.730a
H2 α‖ 6.803b
α⊥ 4.845b
Q 0.71c
I 0.567b
a From http://www.iapws.orgb From http://webbook.nist.gov/cgi/c See Hirshfelder et al (1964).
A value of CLJ ∼ 100 au is convenient. No RE result depends on this value for more than a See endnote 1relative precision of 10−6.
The full Hamiltonian is thus written in atomic units as
H = p2R
2m+
�2
2m R2+
j 2H2O
2IH2O+
j 2H2
2IH2
+ V (R, θ, χH2O, χH2) (15)
where Ia, a = H2O, H2 denote constants of inertia (IH2O � 2π1207 au, IH2 � 2π288 au). Itmust be noted that H is written in the centre-of-mass, laboratory frame. Consequently, theangles conjugated to ja are φa = θ + χa.
3.2. Relative equilibria
The conditions for RE are given as follows:
dR
dt= pR = 0 (16a)
d pR
dt= �2
m R3− ∂V
∂ R= 0 (16b)
d jH2O
dt= ∂V
∂χH2O= 0 (16c)
d jH2
dt= ∂V
∂χH2
= 0 (16d)
dφH2O
dt= dφH2
dt= �. (16e)
It follows from the definition of total angular momentum, J = � + jH2O + jH2, and from Jconservation that equation (16e) implies θ = � (recall that θ is the angle conjugated to J ).
In order to find the possible occurrence of RE, we proceed in two stages, which we nowpresent.
(i) We try to find, in the angle sector, values of χH2O, χH2 that would satisfy the twoequations (16c) and (16d). Examining the actual form of the potentials, we find thattwo sets of values of the angles make equations (16c) and (16d) vanish identically:
χH2O = Kπ
χH2 = K ′π/2,(17)
where K , K ′ ∈ Z. Two relative orientations of H2O and H2 may lead to RE: the H2O dipolemay be parallel or perpendicular to the H2 molecular axis. Let us call the K = 0, K ′ = 0
Rotational transition states 9
case the parallel case (‖) and the K = 0, K ′ = 1, the perpendicular case (⊥). (Note,figure 2, that θ is not defined in the usual way with respect to the dipole of water.)
(ii) The radial equation, equation (16b), and the conservation of energy (H = E0) lead to twoother equations that completely determine the problem. These are
�2
m R3= 6
Dλ
R7− 4
Cλ
R4(18)
E0 = �2
2m R2+
j 2H2O
2IH2O+
j 2H2
2IH2
+Cλ
R4+
Dλ
R6(19)
where λ = ‖,⊥ and the Cλ and Dλ constants have the following values: C‖ = 3µQ/2,C⊥ = −3µQ/4, D‖ = 4A + 2B + 2µ2α‖, D⊥ = A + 5B + 2µ2α⊥.
Using both preceding points we have the following determination of the RE radial coordinatevalue, Req, and RE angular speed �eq:
�2eq = 1
m
(6Dλ
Req8 − 4Cλ
Req6
)(20)
and
E0 Req8 + Cλ Req
4 − 2Dλ Req2 − IH2O + IH2
m(3Dλ − 2Cλ Req
2) = 0. (21)
This completes the problem of the determination of RE in the model Hamiltonian chosen, ifwe recall that, in equations (18)–(21), we neglect the short-range, repulsive, R−12 term. Thisterm has no angular dependence and its numerical importance was checked to be marginal atmost.
3.3. Stability
The next point to be checked is the linear stability of the two kinds of RE just found, aswell as the possible dependence on the total energy. As was described by equation (1),one needs to examine the eigenvalues of the Hessian matrix of the Hamiltonian, taken atthe equilibrium value, R = Req, � = �eq, . . .. The linearized equations of motion arestraightforwardly derived. Figure 3 shows the real and imaginary parts of the parallel andperpendicular eigenvalues.
It may be seen that for all the energies considered here, with the physically meaningfulparameters used in the Hamiltonian, two types of RE occur. The parallel case RE is unstablein all directions, except for a small energy domain, where four eigenvalues collapse into oneset λ = ±κ ± iω. In phase space, its dynamics resembles the dynamics of a potential summit,with a locally linearized Hamiltonian behaving as H = 1/2
(∑i=1,3 p2
i − κ2i x2
i
). This type of
RE may be of importance for the overall appearance of chaotic motion. However, it does notlook like a relative TS at any energy.
The perpendicular RE has the eigenvalue structure that we are looking for. Two purelyimaginary pairs of eigenvalues, at all energies (in atomic units) 10−4 � E � 3 × 10−3,and one real pair. The linearized dynamics around this RE has thus the relevant structure ofequation (3). Hence, the relative TS defines two regions in phase space. The outer region,R > Req, is directly connected to the asymptotic regions, R → ∞ and an inner region,R < Req, where the two molecules, H2O and H2 behave as a single van der Waals complex.
In order to confirm our analysis and also to have a glimpse of the full dynamics nearbythe perpendicular RE, we present in figures 4 and 5 some trajectories integrated numericallyon the basis of the full Hamiltonian. In the projection onto the R, pR plane (figure 4), the
10 L Wiesenfeld et al
0 0.5 1 1.5 2 2.5 3x 10–3
–1.5
–1
–0.5
0
0.5
1x 10 –3
E (at. units)
Eig
enva
lues
(at
. un
its)
perpendicular
0 0.5 1 1.5 2 2.5 3x 10–3
–5
0
5
10
15
20x 10 –4
E (at. units)
Eig
enva
lues
(at
. un
its)
parallel
Figure 3. Real and imaginary parts of the eigenvalues of the linearized motion, for the two kindsof equilibrium points, as a function of energy. The real parts of the eigenvalues are depicted bycircles, the imaginary parts by crosses. Only the strictly positive real eigenvalues and strictlynegative imaginary eigenvalues are shown. Note that in the parallel case, between approximately0.6 × 10−3 and 1 × 10−3 au, we have a quartet of eigenvalues, λ = ±a ± ib.
occurrence of a TS is obvious, because of the hyperbolic character of the motion in R, pR
(Uzer et al 2002). One sees readily the curvature of the stable and unstable manifolds ofthe RE in the centre. These manifolds do not exhibit the usual time-reversal symmetry (herepR ↔ −pR) we are accustomed to, for a time-reversal symmetric Hamiltonian. The same istrue for trajectories entering or not entering the inner region (such as T2 or T1, respectively).Evidently, this asymmetry is the consequence of the relative character of the equilibriumpoint and the appearance of Coriolis terms in the relative motion equations. We also showtrajectories in ordinary space (figure 5), which corresponds to equation (15), expressed inCartesian coordinates (x = R cos θ, y = R sin θ). See endnote 2
Rotational transition states 11
–0.8 –0.6 –0.4 –0.2 0 0.2 0.4 0.6–0.8
–0.6
–0.4
–0.2
0
0.2
0.4
0.6
R (at. units)
p R (
at.u
nits
)
T2
T1
Figure 4. The numerically integrated dynamics around the perpendicular RE at E = 0.001 au.R = pR = 0 is the RE. The dynamics is projected onto the R, pR plane. The thick curvesrepresent the stable and unstable manifolds of the RE, the thin curves several trajectories; theT1, T2, nonreactive and reactive trajectories, respectively.
–10 –5 0 5 10 15
–5
0
5
10
z
x Requil
T1T2
Figure 5. Numerically integrated dynamics around the perpendicular RE at impact parameterb = 8.2 au, E = 0.001 au, this time in scattering coordinates and in ordinary space. The scatteringcentre is at x = y = 0. The RE is represented by the dashed circle. Both trajectories, T1, T2,originate from the right. T1 enters the inner region through RE, T2 not.
4. Physical relevance
4.1. The energy dependence of the RE point
It has been clear from the start of this paper that the Hamiltonian used to model the H2O–H2
interaction is just a crude model, and far from an exact description. In particular, all short-rangeinteractions are very crudely represented and the long-range interactions are just multipolar.
12 L Wiesenfeld et al
0 0.5 1 1.5 2 2.5 3x 10–3
5
5.5
6
6.5
7
7.5
8
8.5
9
9.5
10
E (at. units)
R (
at. u
nits
)
Figure 6. Numerical solutions for Req for the perpendicular RE, as a function of energy.
Also, both dihedral angles are absent. Nevertheless, because of the experimental values takenfor the multipoles and polarizability, it is hoped that some of the main features of the relativedynamics are retained in the model Hamiltonian equation (15). In this context, it seemsmeaningful to look for the actual values of the parameters determining the perpendicularRE, in support of the relative TS. Solutions of equations (16a)–(21) have been numericallycalculated, for the same energy range as before (see figures 6 and 7). The corresponding shapeof the rotating T-shaped complex is shown in figure 8. We have examined in the precedingfigures the energy range relevant to molecular astrophysical contexts, where those types ofcollision (molecular hydrogen—polar molecule) are of paramount importance (Herbst 1995,Flower 1990).
4.2. Transition state and angular momentum transfer
As a next illustration, let us try to depict more globally the importance of the TS. While in someanalogous works, it was possible to devise surfaces of section or even full representations ofphase space, this is barely imaginable here. Let us recall that an on-shell (or constant energy)Poincare section would be of dimension D� = D(phase space) − 1 − 1 = 6. Instead wedepict the asymptotic conditions, on-shell, for which trajectories enter or do not enter into theinner region. We set the following conditions (at a total rigid-body energy E = 0.001 au):
(i) The initial conditions for a trajectory are R → ∞; jH2O = jH2 = 0, in accordance withthe RE conditions; 0 � � = pR ·b � 25, where b is the impact parameter and pR is foundby energy conservation.
(ii) 0 � χH2O < 2π , 0.5 � χH2 < 1.5π : the χH2O, χH2 angles are scanned, since they haveno physical meaning. We thus have a two-dimensional set of initial conditions.
(iii) Because the inner part of the potential (R � Req) is not very meaningful and because thedepth of the potential well in the inner part is not controlled, we monitor only whether ornot the trajectory enters the inner region.
With help of all those conditions, we have the results presented in figure 9. We clearly seethree regimes.
Rotational transition states 13
0 0.5 1 1.5 2 2.5 3x 10–3
10
12
14
16
18
20
22
24
26
28
30
E (at. units)
l (at
. uni
ts)
Figure 7. Numerical solutions for �eq for the perpendicular RE, as a function of energy.
J
H
O
H
H H
Figure 8. An almost to scale image of the perpendicular RE, for an energy of 300 K. The T-shapeof the rotating complex is apparent. Angular momentum J is perpendicular to the plane of thefigure.
(i) For high-�, (� > �eq), the centrifugal barrier is too high and all collisions are quasi-elastic.This has been verified by monitoring the jH2O values before and after collisions and byfounding near-zero changes. In chemical reactions, we would say that we are belowthreshold.
14 L Wiesenfeld et al
(ii) Around � � �eq ∼ 24, we open a valley into the inner region. This valley has an image inthe asymptotic conditions described above. The transport in the outer region is regular andthe valley has an image backwards in time like a simply connected region, centred aroundthe perpendicular RE (χH2O = 0, χH2 = π/2). This also means that during the approachpart of the trajectory (Rasymptotic → Req), the relative position of the two molecules doesnot change much and almost no angular momentum is transferred. This is in agreementwith the preceding case. In chemical reactions, we would say that we are at and justabove threshold. After entering the inner region (crossing the TS) very effective angularmomentum transfer occurs.
(iii) As � diminishes, the valley opens and covers the whole χH2O domain. For low values of� (0 � � � 17) the RE that lies very distant in phase space seems to have no influenceanymore. The TS is absent and the dynamics is dominated by a complex (maybe chaotic)interplay between short- and long-range potentials. This is analogous to reaction pathsfar above threshold, where details other than TS may determine the output.
4.3. The qualitative influence of short-range dynamics
As explained above, the present short-range description of the H2O–H2 potential energy surfaceis not realistic. A quantitative description of this region and its influence on the dynamicswould require a careful statistical analysis of an accurate multi-dimensional potential surface(see section 5). Nevertheless, in order to probe qualitatively the short-range dynamics of theH2O–H2 system, a crude but illustrative approach is to consider only the isotropic component,or spherical average, of a realistic potential energy surface. As already mentioned, a relativelyaccurate potential surface was computed by Phillips et al (1995) for astrophysical applications.In figure 10, the H2O–H2 isotropic ‘effective’ potential, which is the sum of the isotropic‘electrostatic’ potential computed by Phillips et al (1995) and the centrifugal potential, isplotted for different values of the orbital angular momentum �. It can be observed that for� = 0, the isotropic potential presents a well depth of about 2.1 ×10−4 au at an intermolecularseparation of 6.7 au. When � increases, the potential well is reduced and the effective potentialbecomes less and less attractive, as expected. In particular, the isotropic effective potentialbecomes totally repulsive for angular momenta greater than about � = 12. In terms of REtheory, �eq = 12 corresponds to an energy of about 2.0 × 10−4 au, as deduced from figure 7.This result means that for energies below 2.0 × 10−4 au, angular momentum transfer willoccur within the low-� regime described above, for which the RE influence is small. In thisregime, cross sections for rotational excitation should show a marked resonance structure,presumably of ‘shape’ or ‘dynamical’ type (Abrol et al 2001, Liu 2001). Around � = 12,the influence of the TS is crucial and a time-delayed inelastic scattering mechanism is tobe expected (Faure et al 2000, Harich et al 2002). On the other hand, for energies above2.0 ×10−4 au, angular momentum transfer will occur up to intermediate-� values (� > 10) forwhich rotational excitation will arise mainly from the anisotropy of the short-range potential.In this regime, a considerable smoothing of the resonance structure should be observed as thehigh-� contribution becomes dominant.
The above predictions were indeed observed in the close-coupling calculations byDubernet and Grosjean (2002) for the rotational excitation of H2O by para-H2: a significantresonance structure was found in the cross sections for collision energies below about3.0×10−4 au. Similar results were obtained by Flower (2001) for the CO–H2 system. Finally,note that a fine description of these resonances is important for computing rate coefficients atlow temperature.
Rotational transition states 15
l = 0
χ H2O
/ π
0.5 1 1.5
0
0.5
1
1.5
2
l = 5
0.5 1 1.5
0
0.5
1
1.5
2
l = 19
0.5 1 1.5
0
0.5
1
1.5
2
l = 21
χ H2O
/ π
χH2
/ π0.5 1 1.5
0
0.5
1
1.5
2
l = 22
χH2
/ π0.5 1 1.5
0
0.5
1
1.5
2 0
2
4
6
χH2
/ π
l = 23
0.5 1 1.5
0
0.5
1
1.5
2
Figure 9. The first six panels. With varying �, at E = 0.001 au, sets of trajectories entering (blackdots) or not entering (white dots) the inner region. The last six panels. The same as previously, butjH2O final. The grey defines the final angular momentum, with black corresponding to jH2O � 6.
5. Concluding remarks
Several extensions and calculations are now in order. The obvious next step will be to usea precisely calculated intermolecular potential (for example, H2O–H2 or CO–H2) in order to
16 L Wiesenfeld et al
5 6 7 8 9 10 11
–2
0
2
4
6
8
10
12
14
x 10–4
r (atomic units)
V (
atom
ic u
nits
)
l=0l=6l=10l=12l=14
Figure 10. Radial isotropic potential of Phillips et al (1995), including the centrifugal potential,for various � values, as a function of H2O–H2 distance r .
compare diverse quantum calculations and classical or semi-classical calculations based onthe occurrence of a RE, possibly with the help of statistical theories (Gruce et al 1986, Miller1998, Truhlar et al 1996, Jaffe et al 2002). Also, while a full dynamical study of the systemis not possible, because of the high dimensionality, some insights must be gained into thedynamically invariant objects that constitute the TS: are they ‘normally invariant hyperbolicmanifolds’, (Wiggins et al 2001)? What are the multi-dimensional tubes, (Ozorio et al 1990),extending from the TS towards the asymptotic regions, of which we have just a glimpse infigure 9?
We have shown that equilibrium points and their associated TSs may be defined not onlyin an inertial framework, as usual, but also in a uniformly rotating framework. The H2O–H2
inelastic collision was used to illustrate in detail the dynamics associated with a RE point, withhelp of a simple model potential, retaining the main long-range interactions. We have foundnecessary conditions for RE to occur and analysed the linear stability of these equilibria. Forthe potential used and for an energy range comparable to astrophysical conditions, the RE ofthe T-shaped bimolecular complex has the correct stability for a TS to occur. For the wholerange of relevant energies, the partial wave angular momenta and the inter-nuclear distancesare typical of physical scattering and van der Waals type complexes. We have shown that therelative equilibria and the relative TSs are very effective in transferring translational momentumof the projectile, here H2, into rotational momentum of the target, here H2O. The thresholdoccurs when the partial wave angular momentum is high enough to let inelastic collision occuronly in the vicinity of the TS.
Acknowledgments
We thank P Valiron and M Roberts for useful discussions. Parts of this work were preparedwhile at the University of Warwick MASIE symposia (March–April 2002), whose support isgratefully acknowledged. The Laboratoire de Spectometrie Physique and the Laboratoire
Rotational transition states 17
d’Astrophysique are both ‘Unite Mixte de Recherches CNRS/Universite Joseph–Fourier–Grenoble’. One of us (AF) is supported by CNES (Centre National d’Etudes Spatiales).
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Queries for IOP paper 56981
Journal: JPhysBAuthor: L Wiesenfeld et alShort title: Rotational transition states
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