TRANSITION STATE GEOMETRY NEAR HIGHER-RANK SADDLES … · TRANSITION STATE GEOMETRY NEAR...

TRANSITION STATE GEOMETRY NEARHIGHER-RANK SADDLES IN PHASE SPACE

G. Haller(1), C. Jaffe(2), J. Palacian(3), T. Uzer(4), P. Yanguas(3)

(1)Department of Mechanical and Process Engineering, ETH Zurich, Switzerland,

(2)Department of Chemistry, West Virginia University, Morgantown, WV, USA,

(3)Departamento de Ingenierıa Matematica e Informatica, Universidad Publica de Navarra, Spain,

(4)Center for Nonlinear Sciences, School of Physics, Georgia Institute of Technology, Atlanta, GA, USA

Workshop on Dynamical Systems and Applications

BCAM, Bilbao (Spain), December 10th–11th (2013)

J. Palacian et al. (UPNa) TST FOR m-SADDLES (m > 1) 10/12/13 1 / 35

Contents

Contents

1 Rank-1 Saddles

2 Rank-m (m > 1) SaddlesGeneralisation of the Transition StateExample: Ionization of the Helium Atom


The Key

Transition State Theory (TST) provides at the same time a simple way tocompute reaction rates, and an intuitive understanding of the implieddynamical mechanism. It was started by E.P. Wigner around 1930.

The key idea is to define the TS in phase space, where (locally) a surface ofno return can be defined.

The hypervolumes of initial conditions leading to products are identified, thusallowing a correct and explicit calculation of the corresponding reactive flux.


Need of Algorithms

Very precise determination of the geometrical structures that play a rolein the reaction.

Calculation of reaction rates related to the flux through the TransitionState.

Visualization strategies for trajectories, periodic solutions, invariant tori,manifolds, TS, . . . .


Rank-1 Saddles

Transition State Theory #1

We start with a Hamiltonian system of the type:

H =∑n−1

i=1

ωi

2(p2

i + q2i)

+ λ qn pn

+ f1(q1, . . . , qn−1, p1, . . . , pn−1, I)

+ f2(q1, . . . , qn−1, p1 . . . , pn−1)

where

(q1, . . . , qn, p1, . . . , pn) denote canonical coordinates;

I ≡ pn qn;

f1, f2 are functions of order three, at least;

f1(q1, . . . , qn−1, p1, . . . , pn−1, 0) = 0.


Rank-1 Saddles


The normally hyperbolic invariant manifold (NHIM) is the intersection of thecentral manifold of the origin with the energy surface given through:

E =∑n−1

i=1

ωi

2(p2

i + q2i)

+ λ qn pn

+ f1(q1, . . . , qn−1, p1, . . . , pn−1, I)

+ f2(q1, . . . , qn−1, p1 . . . , pn−1) = h = C > 0.


Rank-1 Saddles


The dynamics associated withH takes place on the energy surface ofdimension 2 n− 1.

The normally hyperbolic invariant manifold associated withH is giventhrough:

M2 n−3h =

{(q1, . . . , qn, p1, . . . , pn) | qn = pn = 0,∑n−1

i=1

ωi

2(p2

i + q2i)

+ f2(q1, . . . , qn−1, p1, . . . , pn−1)

= h = C > 0}.

The NHIM acts as a multidimensional saddle and its dimension is 2 n− 3.


Rank-1 Saddles


The stable manifold of the NHIM is:

Ws(M2 n−3

h

)=

{(q1, . . . , qn, p1, . . . , pn) | qn = 0,∑n−1

i=1

ωi

2(p2

i + q2i)

+ f2(q1, . . . , qn−1, p1, . . . , pn−1)

= h = C > 0}.

The unstable manifold of the NHIM is:

Wu(M2 n−3

h

)=

{(q1, . . . , qn, p1, . . . , pn) | pn = 0,∑n−1

i=1

ωi

2(p2

i + q2i)

+ f2(q1, . . . , qn−1, p1, . . . , pn−1)

= h = C > 0}.

They are (2 n− 2)–dimensional objects which act asmultidimensional separatrices.


Rank-1 Saddles


The transition state for this system is obtained by taking qn = pn:

T S2 n−2h =

{(q1, . . . , qn, p1, . . . , pn) | qn = pn,∑n−1

i=1

ωi

2(p2

i + q2i)

+ f1(q1, . . . , qn−1, p1, . . . , pn−1, p2n)

+ f2(q1, . . . , qn−1, p1, . . . , pn−1) = h = C > 0}.

It has the following properties:1 The trajectories crossing the transition state correspond to reactive

trajectories.2 The transition state is a “surface of no return” existing in the original

Hamiltonian in fact.3 The transition state is a sphere of dimension 2 n− 2.4 All reactive particles must pass through the transition state.


Rank-1 Saddles


Projection of a forward and backward reacting trajectory into the qn-pn plane

NHIM

Stable manifold of the NHIM

Unstable manifold of the NHIM


Rank-1 Saddles

Normal Forms and Transition State Theory

Once the normal form is obtained we can compute analytically theexpressions of the NHIM, its stable and unstable manifolds and the transitionstate in the original coordinates, let us say,

(Q,P) = (Q1, . . . ,Qn,P1, . . . ,Pn)

in terms of the so-called “normal form” coordinates, let us say,

(q, p) = (q1, . . . , qn, p1, . . . , pn).

First we compute the direct change of coordinates, relating (Q,P) with (q, p).

The NHIM is obtained by setting qn = pn = 0;the stable manifold of the NHIM is obtained by setting qn = 0;the unstable manifold of the NHIM is obtained by doing pn = 0;the transition state results after making qn = pn.


Rank-m Saddles Generalisation of the Transition State

Contents

1 Rank-1 Saddles




Transition State for Rank-m (m > 1) Saddles

Equilibria with linearisation:

centre× . . .× centre× saddle × . . .× saddle

Is it interesting to apply the transition state theory to rank-m saddles?

1 The transition state as formulated with the standard theory does notseparate the phase space conveniently.

2 It is not possible to apply the standard theory.

3 There are interesting examples in several fields.



Applications

Celestial Mechanics:

1 Three-body problems where one of the bodies is a gyrostat.

2 Restricted full three-body problems (one of the primaries is a sphere andthe other one is a triaxial ellipsoid).

3 Elliptic Restricted Three Body Problem, around the periodic solutions L1,L2 and L3 in the case that the linearisation is centre × saddle × saddle.

Molecular Dynamics and Atomic Physics:

1 Lennard-Jones potentials used to model molecular reactions with a highnumber of atoms.

2 An atom subject to an electric field.



Elliptic Restricted Three Body Problem #1

(With Shinnosuke Kawai, Hokkaido University)

Elliptic restricted three-body problem in 3D:

After linearising around L1, L2 and L3 and using symplectic Floquet theory totransform the periodic linear vector field into an autonomous linear vector

field, the characteristic multiplier related to the periodic solutions are:

centre × centre × saddlecentre × saddle × saddlesaddle × saddle × saddle

It appears when handling cases where the eccentricity of the primaries ismoderately high: useful for exoplanets, science missions around cometsand asteroids, etc.



Elliptic Restricted Three Body Problem #2

For each Li we have calculated 1600 monodromy matricesL1 L2 L3

1 Evolution of the NHIM according to the parameters of the problem: theeccentricity of the primaries ep and the mass ratio µ.

2 How does the transition state and the related invariant manifolds behavewhen the saddles are of rank two?



Linear Theory

The quadratic part of the Hamilton function is:

H2 =∑m

i=1

ωi

2(p2

i + q2i)

+∑n

i=m+1λi qi pi

and the eigenvalues satisfy

0 ≤ λm+1 ≤ λm+2 ≤ . . . ≤ λn−1 < λn.



Linear Centre, Stable and Unstable Manifolds

The spaces Ec, Es and Eu are the usual linear invariant manifolds of Rn whoserespective dimensions are 2m, n− m and n− m.

We define two linear spaces:

Es is the linear stable invariant manifold of dimension n−m− 1 spannedby the eigenvectors related to the eigenvalues

−λm+1, . . . ,−λn−1.

Eu is the linear unstable invariant manifold of dimension n− m− 1spanned by the eigenvectors related to the eigenvalues

λm+1, . . . , λn−1.



Linear Pseudo-Stable and Pseudo-Unstable Manifolds

The pseudo-unstable invariant manifold of the origin related to λn−1:maximal invariant subspace in which the norm of solutions grows, staysconstant, or decays no faster than e−λn−1 t:

Euλn−1

= Ec × Es × Eu.

The pseudo-stable invariant manifold of the origin related to λn−1:maximal invariant subspace in which the norm of solutions decays, staysconstant, or grows no faster than eλn−1 t:

Esλn−1

= Ec × Es × Eu.

Both spaces are (2 n− 1)–dimensional.



Linear Pseudo-NHIM

TakingC = Es

λn−1∩ Eu

λn−1= Ec × Es × Eu

and

Eh ={

(q1, . . . , qn, p1, . . . , pn) |H2(q, p) = h > 0}

we define the linear pseudo normally hyperbolic invariant manifold(pseudo-NHIM) as:

M2n−3h = Eh ∩ C={

(q1, . . . , qn, p1, . . . , pn) |∑m

i=1

ωi

2(p2

i + q2i)

+∑n−1

i=m+1λi qi pi = h, qn = pn = 0

}.

It is unbounded.



Linear Pseudo-Stable and Pseudo-Unstable Manifolds ofthe NHIM

Its stable and unstable manifolds are (2n− 2)-dimensional and separatethe phase space:

Ws(M2n−3

h

)={

(q1, . . . , qn, p1, . . . , pn) |∑m

i=1

ωi

2(p2

i + q2i)

+∑n−1

i=m+1λi qi pi = h, pn = 0

},

Wu(M2n−3

h

)={

(q1, . . . , qn, p1, . . . , pn) |∑m

i=1

ωi

2(p2

i + q2i)

+∑n−1

i=m+1λi qi pi = h, qn = 0

}.



Linear Transition State

We define it as:

T S2n−2(M2n−3

h

)={

(q1, . . . , qn, p1, . . . , pn) |∑m

i=1

ωi

2(p2

i + q2i)

+∑n−1

i=m+1λi qi pi + λnq2

n = h, pn = qn

}.

Its dimension is 2 n− 2.

It is not an invariant manifold.

It is unbounded and contains the NHIMM2n−3h .



Full Hamiltonian

Problem

The nonlinear continuation of Ws(M2n−3h ) and Wu(M2n−3

h ) cannot bedone using the standard theory for NHIMs.

The reason is thatM2n−3h is not bounded.

Solution

We make a nonlinear continuation of Esλn−1

(resp. Euλn−1

) in order toobtain the pseudo-stable (resp. pseudo-unstable) Wps(0) (resp. Wpu(0))tangent to Es

λn−1(resp. Eu

λn−1) of dimension 2 n− 1 and of class Cr,

where r = min {[λn/λn−1] , r} and the Hamiltonian H is of class Cr.

Both Wps(0) and Wpu(0) are invariant manifolds.

They are not unique but are tangent to Esλn−1

and Euλn−1

, that are unique.



Nonlinear Pseudo-NHIM

If Eh is the nonlinear energy surface (for fixed h > 0), we define:

Wpsh = Wps(0) ∩ Eh and Wpu

h = Wpu(0) ∩ Eh,

which are of codimension 1 in Eh.

ThenM2n−3

h = Wpsh ∩Wpu

h

is a smooth continuation of the surfaceM2n−3h for h small enough.

M2n−3h is the nonlinear pseudo-NHIM of dimension 2 n− 3.

It is not compact as it has n− m− 1 hyperbolic directions.

The nonlinear TS is built in a similar way to the linear case.


Rank-m Saddles Example: Double Ionization of the Helium Atom

Contents

1 Rank-1 Saddles




Ionization of the Helium Atom with an Electric Field

Initial Hamiltonian (three-body problem: the nucleus and two electrons):

H =12

(P2ρ1

+P2φ1

ρ21

+ P2z1

)+

12

(P2ρ2

+P2φ2

ρ22

+ P2z2

)− E(t)(z1 + z2)

− 2√ρ2

1 + z21

− 2√ρ2

2 + z22

+1√

ρ21 + ρ2

2 − 2ρ1 ρ2 cos(φ1 − φ2) + (z1 − z2)2.

The problem is of five degrees of freedom, and is autonomous if the electricfield is chosen to be conservative.



Physical CoordinatesTransition state geometry near higher-rank saddles in phase space 543

Figure 8. The physical coordinates. The physical variables are chosen to be the cylindricalcoordinates and corresponding momenta of each of the electrons.

of the electric field. Each of the vectors describing the positions of the two electrons arerepresented in cylindrical coordinates, (!1, "1, z1) and (!2, "2, z2). An additional integral ofthe motion exists, namely the angular momentum about the z-axis defined by the electric field.Consequently, the problem can be reduced to one of five degrees of freedom, which is standardfor the n-body problem in a static electric field, see [37, 38].

The presence of the electric field gives rise to a Stark barrier. To ionize classicallythe electrons must escape over this barrier (quantum-mechanically they can also ionize bytunnelling under the barrier). If the external electric field is taken to be constant the Hamiltoniansystem defining the problem is autonomous. The Stark barrier gives rise to a fixed point in theten-dimensional phase space associated with the five degrees of freedom. This fixed point is arank-two saddle. In other words, it has three stable or elliptic directions (central directions) andtwo unstable or hyperbolic directions. One of the unstable directions is associated with the non-sequential (or correlated) ionization (He ! He2+ + 2e") while the second unstable directionis associated with the non-sequential (or correlated) exchange (He+ + e"(A) ! He+ + e"(B))of electrons.

The initial Hamiltonian is given by

H = 12

!

P 2!1

+P 2

"1

!21

+ P 2z1

"

+12

!

P 2!2

+P 2

"2

!22

+ P 2z2

"

" F(z1 + z2) " 2#

!21 + z2

1

" 2#

!22 + z2

2

+1

#!2

1 + !22 " 2!1!2 cos("1 " "2) + (z1 " z2)2

. (16)

The first two terms are the kinetic energy of the two electrons. The third term representsthe interaction of the two electrons with the electric field. Here the scalar F represents thestrength of the electric field. The fourth and fifth terms are the attractive interactions betweenthe nucleus and the two electrons and the last term is the repulsive interaction between thetwo electrons. This Hamiltonian accounts for the motions of the electrons with respect to thenucleus.

The Hamiltonian (16) has six degrees of freedom. One of these can be eliminated bymaking use of the existence of the additional constant of the motion: the projection of theangular momentum on the z-axis. To achieve this we introduce a new set of coordinates



Results

1 We have computed the normal forms to degree 12. Computing the localintegrals, the errors are of the order of 10−10 within balls of radii 1/5.550 G Haller et al

Figure 9. Logarithm of the estimate of the error in the integrals of the motion versus the degree towhich the calculations were carried. The line is a linear fit to the data.

C1 one might have expected difficulties to occur. Nevertheless, the estimations of the tail ofthe normal-form transformations made in the previous subsection suggest the good behaviourof the approximations to the degree we have carried out the computations. The errors inthe approximate integrals of motion and in the estimates based on the composition of theHamiltonians decrease as we increase N . In figure 9 we plot the logarithm of the estimateof the error in the integrals versus the degree to which the calculations were carried. Herewe have fixed a ball in phase space of size 1/10 centred at the rank-two saddle and take themaximum values of |{I !

k , H }|/|H | for k = 1, . . . , 5 for the different approximations I !k for

different N ’s. Note that I !k are polynomials of degree N . We see that this measure of the error

appears to vary linearly with the order of the calculation.As a conclusion, even when a high-order of smoothness of the manifolds cannot be

guaranteed, the simulations show that the approximations of the geometrical structures throughhigh-order normal forms are well behaved in the neighbourhood of the rank-two saddle.

4.4. The dynamical implications

In addition to obtaining the transformed or normal-form Hamiltonian (23), the normalizationalgorithm also yields the transformation equations between the normal mode variables asdefined in (19) (also see table 1) and the normal-form action-angle variables, that is,

U = U(I1, . . . , I5, !1, . . . , !5) PU = PU(I1, . . . , I5, !1, . . . , !5),

V = V (I1, . . . , I5, !1, . . . , !5) PV = PV (I1, . . . , I5, !1, . . . , !5),

u = u(I1, . . . , I5, !1, . . . , !5) Pu = Pu(I1, . . . , I5, !1, . . . , !5),

v = v(I1, . . . , I5, !1, . . . , !5) Pv = Pv(I1, . . . , I5, !1, . . . , !5),

" = "(I1, . . . , I5, !1, . . . , !5) P" = P"(I1, . . . , I5, !1, . . . , !5).

(24)

The normal-form Hamiltonian is integrable and we can immediately write down the timedevelopment of action-angle variables. The actions are constants of the motion and the anglesdevelop linearly in time,

Ii(t) = Ii(0),

!i (t) = #i t + !i (0)(25)

for i = 1, . . . , 5 and where the angular frequency is

#i (I1, . . . , I5) = $K(I1, . . . , I5)

$Ii

. (26)

2 We have found the nonlinear transition state, the NHIM and the rest ofstructures (stable and unstable manifolds of the NHIM, periodicsolutions and hyperbolic tori of dimensions 2 and 3).



Quasiperiodic Solutions and 2-Tori



Manifolds and Trajectories #1

Main idea for the projections: collect the unessential elliptic directions.Green (NHIM), yellow (TS), red/blue (stab./unst. manifold of NHIM).



Manifolds and Trajectories #2



Reactive and Nonreactive Trajectories

Two very close trajectories: the yellow solution reacts and the red one doesnot.



Normal Modes in Phase Space

Almost identical initial conditions but yielding to different situations:(a) The two electrons ionise; (b) no electron ionises.

0.3 0.3 u

0.1

0.1

Pu

0.3 0.3 U

0.1

0.1

PU

0.3 0.3 V

0.1

0.1

PV

0.3 0.3 v

0.1

0.1

Pv

0.3 0.3 Φ0.1

0.1

PΦ

0.3 0.3 u

0.1

0.1

Pu

0.3 0.3 U

0.1

0.1

PU

0.3 0.3 V

0.1

0.1

PV

0.3 0.3 v

0.1

0.1

Pv

0.3 0.3 Φ0.1

0.1

PΦ

(a)

(b)



Electrons that Ionise and Electrons that Do Not Ionise

Seen in the physical space:

e1e2

He 2

2 1 1 2 y1

123

z

e1e2

He 2

2 1 1 2 y1

123

z

e1e2

He 2

2 1 1 2 y1

123

z(a) (b) (c)

+ + +

-- - ---

- --

- --

- --



Another Example(a) No ionisation. (b) Double ionisation.556 G Haller et al

Figure 18. Two trajectories embedded in the the energy shell. The initial conditions of these twotrajectories are chosen to be near identical. The difference being that they are on opposite sidesof the unstable manifold of the pseudo-NHIM. Trajectory (a) has sufficient energy in the B2 tosurmount the barrier, while that seen in (b) does not have sufficient energy and is reflected.

Figure 19. The two trajectories seen in figure 18 shown in the physical variables. Trajectory (a)has sufficient energy in the B2 to surmount the barrier, while that seen in (b) does not have sufficientenergy and is reflected, (c) a superposition of the two trajectories illustrating that they can only bedistinguished after approaching the transition state.

are such surfaces. They are also codimension one surfaces and partition the energy shell intoreactant and product volumes. They are not invariant, and the reaction is said to occur whenthe system crosses these surfaces.

The relationships between these three codimension one surfaces are illustrated in figures 18and 19. In figure 18 we see the normal mode representation. These three surfaces have thepseudo-NHIM in common, that is, all possible partitions of the energy among the three ellipticmodes and the slow hyperbolic mode are allowed. The difference between these three surfacesoccurs in the phase space associated with the fast (B2) hyperbolic mode. In this figure thestable manifold is shown in red, the unstable is blue. The transition state bisects the stableand unstable manifolds of the fast hyperbolic direction. Also shown in this figure are twotrajectories whose initial conditions are chosen to be on opposite sides of the stable manifold(red). We see that as the orbits approach the saddle point the one is reflected and the otherpasses through the transition state as it passes above the saddle point.

Finally, these two trajectories are also shown in the physical coordinate space in figure 19.Again we see that as the two electrons approach the transition state (one from one side and theother from the other side) that in the first case they pass over the saddle point and in the otherare reflected by the barrier.

556 G Haller et al

Figure 18. Two trajectories embedded in the the energy shell. The initial conditions of these twotrajectories are chosen to be near identical. The difference being that they are on opposite sidesof the unstable manifold of the pseudo-NHIM. Trajectory (a) has sufficient energy in the B2 tosurmount the barrier, while that seen in (b) does not have sufficient energy and is reflected.

Figure 19. The two trajectories seen in figure 18 shown in the physical variables. Trajectory (a)has sufficient energy in the B2 to surmount the barrier, while that seen in (b) does not have sufficientenergy and is reflected, (c) a superposition of the two trajectories illustrating that they can only bedistinguished after approaching the transition state.

are such surfaces. They are also codimension one surfaces and partition the energy shell intoreactant and product volumes. They are not invariant, and the reaction is said to occur whenthe system crosses these surfaces.

The relationships between these three codimension one surfaces are illustrated in figures 18and 19. In figure 18 we see the normal mode representation. These three surfaces have thepseudo-NHIM in common, that is, all possible partitions of the energy among the three ellipticmodes and the slow hyperbolic mode are allowed. The difference between these three surfacesoccurs in the phase space associated with the fast (B2) hyperbolic mode. In this figure thestable manifold is shown in red, the unstable is blue. The transition state bisects the stableand unstable manifolds of the fast hyperbolic direction. Also shown in this figure are twotrajectories whose initial conditions are chosen to be on opposite sides of the stable manifold(red). We see that as the orbits approach the saddle point the one is reflected and the otherpasses through the transition state as it passes above the saddle point.

Finally, these two trajectories are also shown in the physical coordinate space in figure 19.Again we see that as the two electrons approach the transition state (one from one side and theother from the other side) that in the first case they pass over the saddle point and in the otherare reflected by the barrier.


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