The Influence of a Pitchfork Bifurcation of the Critical Points of a Symmetric Caldera Potential Energy Surface on Dynamical Matching
TThe Influence of a Pitchfork Bifurcation of the CriticalPoints of a Symmetric Caldera Potential EnergySurface on Dynamical Matching
Y. Geng a , M. Katsanikas a , M. Agaoglou a , S. Wiggins a, a School of Mathematics, University of Bristol,Fry Building, Woodland Road, Bristol, BS8 1UG, United Kingdom.
Abstract
Many organic chemical reactions are governed by potential energy surfaces thathave a region with the topographical features of a caldera. If the caldera has asymmetry then trajectories transiting the caldera region are observed to exhibita phenomenon that is referred to as dynamical matching. Dynamical matchingis a constraint that restricts the way in which a trajectory can exit the calderabased solely on how it enters the caldera. In this paper we show that bifurca-tions of the critical points of the caldera potential energy surface can destroydynamical matching even when the symmetry of the caldera is not affected bythe bifurcation.
Keywords:
Phase space structure, Chemical reaction dynamics, Calderapotential, Chemical Physics, Pitchfork bifurcation.
1. Introduction
In this paper we study a symmetric caldera potential energy surface (PES)and the effect of bifurcation of critical points. The geometry of this PES re-sembles that of the collapsed region of an erupted volcano. This feature wasthe main reason Doering [1] and co-workers used the word ‘’caldera” in refer-ence to this type of PES. Features of the caldera PES occur in many organicchemical reactions, such as the vinylcyclopropane-cyclopentene rearrangement[2, 3], the stereomutation of cyclopropane [4], the degenerate rearrangement ofbicyclo[3.1.0]hex-2-ene [5, 6] or that of 5-methylenebicyclo[2.1.0]pentane [7].The topography of the caldera PES is characterized by a shallow minimumand four index-1 saddles that surround this region. Two of the index-1 saddleshave higher energy (upper index-1 saddles) than the other two (lower index-1saddles). The four index-1 saddles control the entrance into and exit from the
Email address:
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Preprint submitted to Chemical Physics Letters February 5, 2021 a r X i v : . [ n li n . C D ] F e b entral area of the caldera. Chemical reaction occurs when trajectories fromthe region of one of the upper index-1 saddles (reactants) cross the caldera andapproach the region of one of the two lower index-1 saddles (products).Detailed studies of trajectories in a two dimensional caldera PES have beencarried out in [8–12]). Broadly speaking, there are two distinct situations: thecase when the PES is symmetric and the case when the PES is asymmetric (to bemore precisely defined below). The symmetric case was studied in [8, 9] whereit was found that trajectories entering the caldera from the region of an upperindex one saddle exited the caldera from the opposite lower index one saddle.Hence 100% of the products exit from the region of one lower index-1 saddleand not 50%, as would be predicted by statistical theories. This phenomenonis referred to as dynamical matching and was first reported in ([13, 14]).A phase space analysis of dynamical matching was carried out in [9]) whereit was shown that the controlling mechanism is the existence of heteroclinicorbits between the unstable manifolds of the unstable periodic orbits associatedwith the upper saddles and the stable manifolds of invariant sets in the centralregion. In particular, the non-existence of such heteroclinic orbits implies theexistence of dynamical matching and the existence of such heteroclinic orbitspromotes trapping in the central region and inhibits dynamical matching. Inall cases of the symmetric caldera PES studied thus far the non-existence ofsuch heteroclinic trajectories was found and, hence, dynamical matching wasalways observed. In [8, 10] it was shown that if the symmetry of the calderaPES was broken by ‘’stretching” the PES then dynamical matching could bebroken. Indeed, [10–12]) showed that a rich variety of dynamical behavior couldbe created as a result.In this paper we show that there is a different mechanism for breaking dy-namical matching in a symmetric caldera PES that does not require symmetrybreaking. Rather, it involves a pitchfork bifurcation of the critical points of thePES.The structure of the paper is as follows: In section 2 we describe the struc-ture of the caldera PES. In section 3 we describe the pitchfork bifurcation of thecritical points of the PES and compare the topography of the PES before andafter the bifurcation. In section 4 we show how the bifurcation influences tra-jectories crossing the caldera and demonstrate its effect on dynamical matching.In section 5 we summarize our conclusions.
2. Model
In this section we describe the Caldera potential energy surface (PES) andits geometrical features that will be the focus of our dynamical studies. Thecaldera PES that we use is taken from [8] and has the form: V ( x, y ) = c r + c y − c r cos(4 θ )= c ( x + y ) + c y − c ( x + y − x y ) , (1)2here ( x, y ) are cartesian coordinates, ( r, θ ) are standard polar coordinates, and c , c , c are parameters. We make the important observation that the PES issymmetric with respect to the y-axis.Hamilton’s equations are straightforward. We let p x and p y denote themomentum of the particle in x − and y − direction respectively, the 2 degree-of-freedom (DoF) Hamiltonian is given by: H ( x, y, p x , p y ) = p x m + p y m + V ( x, y ) , (2)where we consider m to be a constant equal to 1. The Hamiltonian equationsof motion are therefore:˙ x = ∂H∂p x = p x m ˙ y = ∂H∂p y = p y m ˙ p x = − ∂V∂x ( x, y ) = − (2 c x − c x + 12 c xy )˙ p y = − ∂V∂y ( x, y ) = − (2 c y − c y + 12 c x y + c ) (3)
3. Bifurcation of the Critical Points of the Caldera PES
In ([8–12]) the parameters of the caldera PES were fixed at c = 5 , c = 3and c = − .
3. In this case the PES has one minimum and with four index-1saddles, two for low values of energy and other two for high values of energy,as listed in table 2. The index-1 saddles control the exit from and the entranceinto the caldera.We now consider how the critical points of the PES change as we vary c butleave the other two parameters fixed at c = 3 and c = − .
3. It is importantto note that the PES remains symmetric, in the sense described above, for allrange of values of c that we consider.We consider c in the interval 0 to 5 and find that there is a critical valueof c = 1 .
32 for which the number of critical points of the PES change from3 to 5. This is a result of a pitchfork bifurcation involving the minimum andthe two lower energy index-1 saddles. Hence, for 0 ≤ c ≤ .
32 the two higherenergy index-1 saddles exist and one lower energy index-1 saddle on the y -axis.In other words, there is no longer a minimum which is indicative of the existenceof a well. This is a regime for the symmetric caldera which has not receivedattention from the point of view of trajectory analysis.In table 1 we give the location of the critical points and their energies for c = 0 . c = 5. 3 able 1: Stationary points of the caldera potential for c = 0 . , c = 3 and c = − . Critical point x y ELower saddle 0.000 -1.194 -2.402Upper LH saddle -1.204 0.840 2.321Upper RH saddle 1.204 0.840 2.321
Table 2: Stationary points of the caldera potential for c = 5 , c = 3 and c = − . Critical point x y ECentral minimum 0.000 -0.297 -0.448Upper LH saddle -2.149 2.0778 27.0123Upper RH saddle 2.149 2.0778 27.0123Lower LH saddle -1.923 -2.003 14.767Lower RH saddle 1.923 -2.003 14.767In Fig. 1 we show contours of the PES for two sets of parameter valuesshowing the two different configurations of critical points, i.e. 3 critical pointsin panel A and 5 critical points in panel B.In Fig. 2 we show 3D views of the PES for the same set of parameter values(panels A and B show different views for the same set of parameter values).
4. Results
In this section we investigate the behavior of trajectories that enter thecaldera from the region of the upper index-1 saddles for 0 ≤ c ≤
5. To achievethis we will analyze the fate of the trajectories that are initialized in the regionof the right upper index-1 saddle. The results will be similar if we choose theother upper index-1 saddle due to the symmetry of the potential.First, we specify the choice of initial conditions. In particular, we choose aline in configuration space that passes through the upper right index-1 saddleand is perpendicular to the line that connects the upper right index-1 saddlewith the lower index-1 saddle on the y-axis for 0 ≤ c ≤ .
32 or the lower leftindex-1 saddle for 1 . ≤ c ≤
5. The momentum for these initial conditions ischosen to be in the direction of the line that connects the two index-1 saddlesand to satisfy the constant energy condition. This choice guarantees that theinitial conditions correspond to trajectories that enter into the caldera from theregion of the upper right index-1 saddle.We choose 1000 initial conditions in this way (equally spaced along thechosen line in configuration space) and for each initial conditions c is chosenin increments of 0 .
01 in the appropriate interval. These initial conditions areintegrated for a fixed time interval t = 3 time units.4)B) Figure 1: The contours of the caldera potential: A) for c = 0 . , c = 3 and c = − .
3. B)for c = 5 , c = 3 and c = − .
3. We indicate the location of the index-1 saddles and of thecenters using blue and red points, respectively.
Figure 2: The 3D PES of the caldera potential A) and B) for c = 0 . , c = 3 and c = − . c = 5 , c = 3 and c = − .
3. We indicate the location of the index-1 saddles and ofthe center using black and red points respectively.
6e consider that a trajectory has exited through the lower left exit regionor the lower right exit region, if the y component of the trajectories are belowthe line y = − . x -coordinate or a positive x -coordinates,respectively. We note that the lower left exit region and the lower right exitregion are the regions on the left, or on the right, of the index-1 saddle of the y -axis, respectively, before the bifurcation, i.e. for 0 ≤ c ≤ . . ≤ c ≤ y = 2 . x -coordinates orpositive x -coordinates, respectively. If a trajectory did not exit from any regionin the fixed integration time we consider that it is trapped in the central areaof the caldera.The results of our simulations showed us that we have three types of behaviorof the trajectories that come from the region of the upper right index-1 saddlefor 0 ≤ c ≤ .
32. The first and second type are represented by the red andgreen curves, respectively in the panel A of Fig. 3. The red and green curvescorrespond to trajectories that exit through the lower left exit region or thelower right exit region, respectively. The third type is represented by the blueline (see the panel A of Fig. 3) that corresponds to a trajectory that is trappedin the central region of the caldera.We computed the ratio of the trajectories that exit through the lower left exitregion and the ratio of the trajectories that are trapped or they exit throughthe lower right exit region as the parameter c increases. We see in Fig. 4that initially the ratio of the trajectories that exit through the lower left exitregion (red line) is lower than the ratio of the trajectories that are trappedor exit through the lower right exit region (black line). Then the ratio of thetrajectories that exit through the lower left exit region increases and it exceedsthe ratio of the trajectories that are trapped or they exit through the lower rightexit region. This ratio increases until it converges (see the first plateau of thered line in Fig. 4) to a high value (approximately to 0.974). The bifurcationpoint ( c = 1 .
32) corresponds to the end of this plateau which is the point wherewe have the final increase of the ratio of the trajectories that exit through theregion of the lower left exit region until it converges to one. We observe theexact opposite behavior for the ratio of the trajectories that are trapped or theyexit through the region of the lower right exit region (see Fig.4).Furthermore, we see that as c increases from the bifurcation value of 1 . Figure 3: The contours of the caldera potential and trajectories in the configuration spacethat begin from the region of the upper right saddle (for a value of energy 0.5 units abovethe energy of the higher index-1 saddles): A) for c = 0 . , c = 3 and c = − .
3. B) for c = 5 , c = 3 and c = − .
3. The trajectories that are trapped or they exit through theregion of the lower left saddles or they exit through the region of the lower right saddle aredepicted by blue, red and green color respectively. igure 4: The ratio of the trajectories that exit through the lower left exit region (red line)and the ratio of the trajectories that are trapped or they exit through the lower right exitregion (black line) versus parameter c . the upper left index-1 saddle as a result of the symmetry of the potential. Thistype of behavior is that all the trajectories that come from the region of theupper index-1 saddles cross the caldera and they exit though the opposite lowerexit region (see for example the panel B of Fig. 3). This is the phenomenon ofdynamical matching that is found in all previous studies of trajectories in thesymmetric caldera (see [8–12]).
5. Conclusions
In this paper we investigated the behavior of the trajectories that come fromthe region of the high energy saddles in caldera potential energy surfaces. Wefound that the family of the equilibrium points of the y-axis (the symmetryaxis of the caldera) undergoes a pitchfork bifurcation by increasing a parameterof the potential c . We observed that for 0 ≤ c ≤ .
32 the trajectories thatenter into the central area of the caldera from the region of the upper index-1saddle can be trapped in the central area or to follow two different paths to theexit. The one path is to exit through the lower left exit region and the otheris where the trajectories exit through the lower right exit region. The situationfor these trajectories is different for 1 . ≤ c ≤
5. In this case the trajectoriesthat enter into the central area of the caldera from the region of the high energyindex-1 saddles are not trapped, and they have only one choice for their exit.This choice is to cross the caldera and to exit through the opposite lower exitregion. This behavior is the dynamical matching that has been studied in manypapers ([8–12]). Hence, we have shown that a pitchfork bifurcation of criticalpoints of the PES can destroy dynamical matching, even in the case where thePES is symmetric. 9 cknowledgments
The authors would like to acknowledge the financial support provided by theEPSRC Grant No. EP/P021123/1.
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