The Role of Depth and Flatness of a Potential Energy Surface in Chemical Reaction Dynamics

Abstract

In this study, we analyze how changes in the geometry of a potential energy surface in terms of depth and flatness can affect the reaction dynamics. We formulate depth and flatness in the context of one- and two-degree-of-freedom (DOF) Hamiltonian normal form for the saddle-node bifurcation and quantify their influence on chemical reaction dynamics [1, 2]. In a recent work, García-Garrido et al. [2] illustrated how changing the well-depth of a potential energy surface (PES) can lead to a saddle-node bifurcation. They have shown how the geometry of cylindrical manifolds associated with the rank-1 saddle changes en route to the saddle-node bifurcation. Using the formulation presented here, we show how changes in the parameters of the potential energy control the depth and flatness and show their role in the quantitative measures of a chemical reaction. We quantify this role of the depth and flatness by calculating the ratio of the bottleneck width and well width, reaction probability (also known as transition fraction or population fraction), gap time (or first passage time) distribution, and directional flux through the dividing surface (DS) for small to high values of total energy. The results obtained for these quantitative measures are in agreement with the qualitative understanding of the reaction dynamics.

APPENDIX

Derivation of the Normal Form Hamiltonian for the Saddle-node Bifurcation

The following is concerned with the classical Hamiltonian saddle-node bifurcation and using it to model some phenomena of interest relevant to chemical reactions —bifurcations of NHIMs, depth and flatness of the potential energy surface.

The normal form for the one DOF Hamiltonian saddle node bifurcation is given by

$$H(q,p)=\frac{p^{2}}{2}-\mu q+\frac{q^{3}}{3},$$

(A.1)

with corresponding Hamilton’s equations

$$\displaystyle\dot{q}=p,$$

$$\displaystyle\dot{p}=-q^{2}+\mu,$$

(A.2)

and $\mu$ is the bifurcation parameter. The equilibria are given by $q^{2}=\mu,p=0$.

The Jacobian of the vector field is

$$\left(\begin{array}[]{cc}0&1\\ -2q&0\end{array}\right).$$

(A.3)

We evaluate the Jacobian at each equilibrium point to determine stability:

$$(-\sqrt{\mu},0):\qquad\left(\begin{array}[]{cc}0&1\\ 2\sqrt{\mu}&0\end{array}\right),\qquad\mbox{saddle},$$

(A.4)

$$(\sqrt{\mu},0):\qquad\left(\begin{array}[]{cc}0&1\\ -2\sqrt{\mu}&0\end{array}\right),\qquad\mbox{center}.$$

(A.5)

In the Hamiltonian saddle node described above the equilibria move as $\mu$ is varied. It will be useful to fix the saddle point at the origin. To do this we introduce the following coordinate transformation: $q=x-\sqrt{\mu},p=y$. Substituting this into (A.2) gives

$$\displaystyle\dot{q}=\dot{x}=p=y,$$

$$\displaystyle\dot{p}=\dot{y}=-(x-\sqrt{\mu})^{2}+\mu,$$

$$\displaystyle=-x^{2}+2x\sqrt{\mu}$$

(A.6)

or

$$\displaystyle\dot{x}=y,$$

$$\displaystyle\dot{y}=2\sqrt{\mu}x-x^{2},$$

(A.7)

with the corresponding Hamiltonian

$$H(x,y)=\frac{y^{2}}{2}-\sqrt{\mu}x^{2}+\frac{x^{3}}{3}.$$

(A.8)

The equilibria are given by

$$(x,y)=(0,0),(2\sqrt{\mu},0).$$

(A.9)

The Jacobian of the vector field is

$$\left(\begin{array}[]{cc}0&1\\ -2x+2\sqrt{\mu}&0\end{array}\right).$$

(A.10)

We evaluate the Jacobian at each equilibrium point to determine stability:

$$\displaystyle(0,0):\qquad\left(\begin{array}[]{cc}0&1\\ 2\sqrt{\mu}&0\end{array}\right),\qquad\mbox{saddle},$$

(A.11)

$$\displaystyle(2\sqrt{\mu},0):\qquad\left(\begin{array}[]{cc}0&1\\ -2\sqrt{\mu}&0\end{array}\right),\qquad\mbox{center}.$$

(A.12)

The depth of the potential well is controlled by the cubic term in the potential energy surface. Therefore, we introduce a parameter that allows us to vary the amplitude of this term:

$$H(x,y)=\frac{y^{2}}{2}-\sqrt{\mu}x^{2}+\frac{\alpha x^{3}}{3},\qquad\alpha>0,$$

(A.13)

$$\displaystyle\dot{x}=y,$$

$$\displaystyle\dot{y}=2\sqrt{\mu}x-\alpha x^{2}.$$

(A.14)

The equilibria are given by

$$(x,y)=(0,0),\left(\frac{2}{\alpha}\sqrt{\mu},0\right).$$

(A.15)

The Jacobian of the vector field is given by

$$\left(\begin{array}[]{cc}0&1\\ 2\sqrt{\mu}-2\alpha x&0\end{array}\right).$$

(A.16)

We evaluate the Jacobian at the equilibria to determine their stability:

$$\displaystyle(0,0):\qquad\left(\begin{array}[]{cc}0&1\\ 2\sqrt{\mu}&0\end{array}\right),\qquad\mbox{saddle}.$$

(A.17)

$$\displaystyle\left(\frac{2}{\alpha}\sqrt{\mu},0\right):\qquad\left(\begin{array}[]{cc}0&1\\ -3\sqrt{\mu}&0\end{array}\right),\qquad\mbox{center}.$$

(A.18)

The depth of the potential energy surface is determined by the difference between the potential evaluated at the saddle minus the potential evaluated at the center (minumum of the well). The potential energy function is given by

$$V(x)=-\sqrt{\mu}x^{2}+\frac{\alpha x^{3}}{3},\qquad\alpha>0,$$

(A.19)

and this difference is given by

$$\displaystyle V(0)-V\left(\frac{2}{\alpha}\sqrt{\mu}\right)=\frac{4}{\alpha^{2}}\mu\sqrt{\mu}-\frac{\alpha}{3}\frac{4}{\alpha^{2}}\mu\frac{2}{\alpha}\sqrt{\mu},$$

$$\displaystyle=\left(\frac{4}{\alpha^{2}}-\frac{8\alpha}{3\alpha^{3}}\right)\mu\sqrt{\mu},$$

$$\displaystyle=\left(\frac{12}{3\alpha^{2}}-\frac{8}{3\alpha^{2}}\right)\mu\sqrt{\mu},$$

$$\displaystyle=\frac{4}{3\alpha^{2}}\mu\sqrt{\mu}.$$

(A.20)

Hence, for a fixed $\mu$ (that is, the distance apart from the two equilibria) the potential is made less deep by taking large $\alpha$.

Derivation of the Expression for the Ratio of the Bottleneck Width and the Well Width for the Coupled two DOF System

The following expression is defined as $A$:

$$A=e-\left(\dfrac{\alpha}{3}x^{3}-\sqrt{\mu}x^{2}+\dfrac{\varepsilon}{2}x^{2}\right)+\frac{2x^{2}\varepsilon^{2}}{4(\omega^{2}+\varepsilon)}.$$

(A.21)

Then the ratio of the bottleneck width and the well width can be written as

$$\displaystyle R_{bw}=\dfrac{w_{b}}{w_{w}}=\dfrac{\text{width of the bottleneck}}{\text{width of the well}}$$

$$\displaystyle=\dfrac{\sqrt{\dfrac{2e}{\omega^{2}+\varepsilon}}}{\sqrt{A|_{x=x^{e},V=e}}\sqrt{\dfrac{2}{\omega^{2}+\varepsilon}}}$$

$$\displaystyle=\sqrt{\dfrac{e}{e-(\dfrac{\alpha}{3}(x^{e})^{3}-\sqrt{\mu}(x^{e})^{2}+\dfrac{\varepsilon}{2}(x^{e})^{2})+\dfrac{2(x^{e})^{2}\varepsilon^{2}}{4(\omega^{2}+\varepsilon)}}}$$

$$\displaystyle=\sqrt{\dfrac{e}{e-(\dfrac{\alpha}{3}x^{e}-\sqrt{\mu}+\dfrac{\varepsilon}{2}-\dfrac{2\varepsilon^{2}}{4(\omega^{2}+\varepsilon)})((x^{e})^{2})}}$$

$$\displaystyle=\sqrt{\dfrac{e}{e-(\dfrac{\alpha}{3}\dfrac{1}{\alpha}(2\sqrt{\mu}-\dfrac{\omega^{2}\varepsilon}{\omega^{2}+\varepsilon})-\sqrt{\mu}+\dfrac{\varepsilon}{2}-\dfrac{2\varepsilon^{2}}{4(\omega^{2}+\varepsilon)})\dfrac{1}{\alpha^{2}}(2\sqrt{\mu}-\dfrac{\omega^{2}\varepsilon}{\omega^{2}+\varepsilon})^{2}}}$$

$$\displaystyle=\sqrt{\dfrac{e}{e-\dfrac{1}{\alpha^{2}}(\dfrac{2\sqrt{\mu}}{3}-\dfrac{\omega^{2}\varepsilon}{3(\omega^{2}+\varepsilon)}-\sqrt{\mu}+\dfrac{\varepsilon}{2}-\dfrac{2\varepsilon^{2}}{4(\omega^{2}+\varepsilon)})(2\sqrt{\mu}-\dfrac{\omega^{2}\varepsilon}{\omega^{2}+\varepsilon})^{2}}}$$

$$\displaystyle=\sqrt{\dfrac{e}{e-\dfrac{1}{\alpha^{2}}(-\dfrac{\sqrt{\mu}}{3}+\dfrac{\omega^{2}\varepsilon}{6(\omega^{2}+\varepsilon)})(2\sqrt{\mu}-\dfrac{\omega^{2}\varepsilon}{\omega^{2}+\varepsilon})^{2}}}$$

$$\displaystyle=\sqrt{\dfrac{e}{e+\dfrac{1}{6\alpha^{2}}(2\sqrt{\mu}-\dfrac{\omega^{2}\varepsilon}{\omega^{2}+\varepsilon})^{3}}}$$

$$\displaystyle=\sqrt{\dfrac{e}{e+\mathcal{D_{\varepsilon}}}.}$$

(A.22)