A contribution to a theory of mechanochemical pathways by means of Newton trajectories

Abstract

The reaction path is a central subject in theoretical chemistry. It is a pathway imagined on the potential energy surface (PES). It provides a one-dimensional description of a chemical reaction in an N-dimensional configuration space. Additionally, one can apply mechanical stress in a defined direction to the molecule and generate an effective PES. Changes for minima and saddle points by the stress are described by Newton trajectories on the original PES. The barrier of a reaction fully breaks down for the maximal value of the norm of the gradient of the PES along a pulling Newton trajectory. This point is named barrier breakdown point. We discuss topologically different, two-dimensional examples for this model to understand and classify the mechanochemistry of molecules.

Appendix

We report the used formulas of the examples.

• Exam 1 The surface is

  $$V(x,y)=4.5\,(1 - Exp[-x + 1])^2 + (1.75 y^2 - 0.1 y^4).$$

  (12)

  The minimum left below is at zero level, the last point on the x-axis is at level 4.34, where the SP on the y-axis is at level 7.65, and the maximum is at level 12. $D_e=4.5$ is the final dissociation energy, $x-1$ is the bond length displacement.

• Exam 2 The surface is

  $$\begin{aligned} V(x,y)=10\,(1 - Exp[-x + 1])^2 + (1.75 y^2 - 0.1 y^4). \end{aligned}$$

  (13)

  The minimum left below is at zero level, and the SP on the x-axis is at level 9.64, where the SP on the y-axis is at a lower level of 7.65.

• Exam 3 The surface is an uncoupled combination of a Morse- and a quartic/sixtic potential

  $$\begin{aligned} V(x,y)=10\,(1 - Exp[-y + 1])^2 + ( 0.1 x^2 + 0.75 x^4 - 0.125 x^6) . \end{aligned}$$

  (14)

  The minimum left below is at zero level, the SP on the x-axis is at level 4.4 U, where the SP on the y-axis is at level 9.64, and the maximum is at 14.04 .

• Exam 4 Put $h1=1,\ h2=-1.11,\ h3=3,\ h4=1,\ h5=0,\ h6=12.5$; see Ref. [8]. Form the symmetric matrices $H1=((h1,h2)^T,(h2,h3)^T)$ and $H2=((h4,h5)^T,(h5,h6)^T)$ and put

  $$V(x,y)=[(x-1,y-1)\, H1\, (x-1,y-1)^T ]\ [(x+1,y+1)\, H2\, (x+1,y+1)^T].$$

  (15)

  The minimums lie at zero level, the SP is at level 24, and the left VRI is 145 levels high, where the right VRI is at level 400.

• Exam 5 The PES of Konda et al. [40] is

  $$V(x,y)= 0.5\,x^2-x^3/3+0.5\,(y^4/4+y^2\,(0.75-x)/2)+0.2\,x\,y^3/3.$$

  (16)

• Exam 6 It is a PES with two bound states and two SPs, corresponding to Fig. 3b of Ref. [32]

  $$V(x,y)= (3.5 + y)\ (1 - Exp[-x + 1 + 0.2 y])^2 + (-3 y^2 + y^4 + y^3).$$

  (17)

• Exam 7 The BQC surface [41] is given by

  $$\begin{aligned} V(x,y)= 1/3 (x^3 - 3 x y^2) - \pi (x - y) + 1/40 ((x + 7/4)^4 + y^4) . \end{aligned}$$

  (18)

• Exam 8 The modified NFK-PES [44, 52] is

  $$V(x,y)= 0.03\,(x^2+y^2)^2+ x\,y-9\,Exp[-(x-3)^2-y^2]-9\,Exp[-(x+3)^2-y^2].$$

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