Second-order Saddle Dynamics in Isomerization Reaction

Abstract

The role of second-order saddle in the isomerization dynamics was investigated by considering the $E-Z$ isomerization of guanidine. The potential energy profile for the reaction was mapped using the ab initio wavefunction method. The isomerization path involved a torsional motion about the imine (C-N) bond in a clockwise or an anticlockwise fashion resulting in two degenerate transition states corresponding to a barrier of 23.67 kcal/mol. An alternative energetically favorable path ($\sim$1 kcal/mol higher than the transition states) by an in-plane motion of the imine (N-H) bond via a second-order saddle point on the potential energy surface was also obtained. The dynamics of the isomerization was investigated by ab initio classical trajectory simulations. The trajectories reveal that isomerization happens via the transition states as well as the second-order saddle. The dynamics was found to be nonstatistical with trajectories exhibiting recrossing and the higher energy second-order saddle pathway preferred over the traditional transition state pathway. Wavelet based time-frequency analysis of internal coordinates indicate regular dynamics and existence of long-lived quasi-periodic trajectories in the phase space.

APPENDIX

The sampling procedure for trajectories initiated at the reactant and transition state regions are discussed elsewhere [28]. Here, the sampling procedure for trajectories initiated at the SOS region from a microcanonical ensemble is presented. The procedure involved the following steps:

• 1) The classical harmonic vibrational frequencies, $\omega_{i}=2\pi\nu_{i}$ ($i=1,\ldots,3N-6$) and their corresponding eigenvectors (normal modes, $Q_{i}$) are first obtained by diagonalizing the mass-weighted force constant ($\mathbf{F}$) matrix given by $\mathbf{\Lambda}=\mathbf{L^{-1}FL}$. At the SOS there are two imaginary frequencies corresponding to two normal mode vectors $Q_{1}$ and $Q_{2}$.

• 2) The energy $E$ available to the SOS is distributed randomly among the $m=3N-8$ normal modes with positive frequencies such that the $i^{\text{th}}$ mode has an energy

  $$E_{i}=\left(E-\sum_{j=1}^{i-1}\right)(1-R_{i}^{1/(m-i)}),$$

  (A.1)

  where $R_{i}$ is a random number between 0 and 1.

• 3) The remaining energy $E_{2}=E-\sum_{j=1}^{m}E_{j}$ is then calculated and part of it is assigned randomly to the mode $Q_{1}$ and the rest of the energy is then assigned to the mode $Q_{2}$.

• 4) Once the energy for each normal mode is assigned, the normal mode coordinates and their conjugate momenta are chosen randomly using a random phase for each mode given by

  $$Q_{i}=\left(\frac{\sqrt{2E_{i}}}{\omega_{i}}\right)\cos(2\pi R_{i})\text{ and }P_{i}=-(\sqrt{2E_{i}})\sin(2\pi R_{i}),$$

  (A.2)

  where $R_{i}$ is a new random number between 0 and 1 different from the one given in Eq. (A.1).

• 5) The normal mode coordinates and momenta $\mathbf{Q}$ and $\mathbf{P}$ are then transformed to Cartesian coordinates $\mathbf{q}$ and momenta $\mathbf{p}$ according to

  $$\mathbf{q}=\mathbf{q_{0}}+\mathbf{m}^{-1/2}\mathbf{L}\mathbf{Q}\text{ and }\mathbf{p}=\mathbf{m}^{-1/2}\mathbf{L}\mathbf{P},$$

  (A.3)

  where $\mathbf{m}$ is a diagonal matrix with atomic masses as its elements and the vector $\mathbf{q_{0}}$ corresponds to the equilibrium (SOS) coordinates of the molecule.

• 6) Any spurious angular momentum that arises from the linear transformation is subtracted. Then the energy $E_{\text{chosen}}$ for the chosen Cartesian coordinates and momenta is calculated. If $E_{\text{chosen}}$ and the desired energy $E$ do not agree, the Cartesian coordinates and momenta are rescaled by a factor of $(E/E_{\text{chosen}})$ and any spurious center-of-mass translation energy arising from the rescaling is then subtracted. This step is repeated until the agreement between $E_{\text{chosen}}$ and $E$ is within 0.1 %.