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Reply to Comment on "Reaction Coordinates and Pathways of Mechanochemical Transformations" Stanislav M. Avdoshenko, and Dmitrii E. Makarov J. Phys. Chem. B, Just Accepted Manuscript • DOI: 10.1021/acs.jpcb.6b00917 • Publication Date (Web): 02 Feb 2016 Downloaded from http://pubs.acs.org on February 8, 2016
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Reply to Comment on "Reaction Coordinates and Pathways of Mechanochemical Transformations" Stanislav M. Avdoshenko1,3 and Dmitrii E. Makarov*,1,2 1
Institute for Computational Engineering and Sciences, University of Texas at Austin, Austin,
Texas 78712, 2
Department of Chemistry, University of Texas at Austin, Austin, Texas 78712
3
Current address: Leibniz Institute for Solid State and Materials Research (IFW Dresden),
Helmholtzstraße 20, 01069, Dresden, Germany.
Abstract. We clarify the relation between mechanochemical reaction paths discussed in our work and the Branin equation approach discussed in the Comment by Quapp and Bofill.
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In our paper1, we discussed calculation of the force-dependent activation barrier in the case where two atoms of a molecule are subjected to a pulling force of magnitude F. In one of the common scenarios arising in such calculations, the force deforms both the reactant and the transition-state configurations until a mechanical instability occurs at a critical value of the force, Fc , where the activation barrier vanishes via a fold “catastrophe”; the force-dependent reactant
and transition-state configurations then form a continuous line in the conformational space, which was named FDSP-force displaced stationary points. FDSP can be regarded as the proper reaction path of the mechaniochemical transformation in that the one-dimensional potential along this path yields the correct force dependence of the activation barrier. In their comment on our work2, Quapp and Bofill point out that the same path can be obtained in one sweep using an equation due to Branin. In what follows we clarify the connection between the two approaches. Assuming constant direction of the applied force f specified by a unit vector l, f = Fl , our equation describing the force dependence of the FDSP points reads (cf. Eq. 6 in ref.1 or Eq. 1 in ref.3): dr / dF = H −1l ,
(1)
where H is the Hessian of the underlying potential energy surface U(r) . We note that FDSP does not imply a second order expansion and that the Quapp & Bofill’s Eq. 52 is not the exact FDSP equation unless the force is infinitely small. Now, the Branin equation is2 dr / dt = det H H −1∇U .
(2)
Since every point on this line satisfies the mechanical equilibrium condition, f = Fl = ∇U , Eq. 2 can also be rewritten as
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dr / dt = F det H H −1l .
(3)
It follows that Eq. 3 can be obtained from Eq. 1 using a transformation of variables dt = dF / (det HF) .
(4)
The map F(t ) is, however, not a single-valued function because det H changes its sign when the catastrophe point is crossed: while t varies monotonically along the reaction path, the force F first increases from 0 to Fc and then decreases back to 03. If the Branin equation is integrated numerically using a finite-difference scheme with a uniform grid along t and a step δ t then it is equivalent to a non-uniform grid along F, with a step size δ F decreasing as Fc is approached; thus the Branin equation provides a justification for the use of such adaptive grids proposed in ref.4 It is instructive to consider the behavior of such a finite-difference scheme in the vicinity of Fc . Assuming just two degrees of freedom and λx
introducing a local coordinate system (x,y) that diagonalizes the Hessian, H =
0
0 , λ y
such that λ x → 0 at F → Fc , one obtains a finite difference approximation to Eqs. 1 and 3: fx λ y fx λ y fx δx δr = ≈ δ F Fλx = δt λ f ≈ δt δ y 0 x y 0
,
(5)
where f x is the component of the force along x. This shows that the reaction path becomes aligned with the zero-mode of H. Moreover, the procedure proposed in ref.3 to bypass the catastrophe point (Eqs. 3-4 in ref.3; note that Eq. 4 in there contains a typo – the factor 12 should
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be in the numerator) involves a single finite-difference step along x that becomes equivalent to Eq. 5 if the step size is small. The Branin equation thus offers an elegant solution to sidestepping the catastrophe point that avoids its special treatment. In conclusion, the Qapp and Bofill comment offers useful insight that bridges two distinct fields: mechanochemistry offers physical motivation to reaction-path and transition-state search approaches while the latter have the potential to increase the efficiency of mechanochemical calculations. We are grateful to Graeme Henkelman and Gregory J. Rodin for helpful comments. Financial support from the Robert A. Welch Foundation (Grant No. F-1514) and the National Science Foundation (Grant No. CHE 1266380) is gratefully acknowledged. References 1. Avdoshenko, S. M.; Makarov, D. E., Reaction Coordinates and Pathways of Mechanochemical Transformations. The Journal of Physical Chemistry. B 2015, 10.1021/acs.jpcb.5b07613. 2. Quapp, W.; Bofill, J. M., Comment on “Reaction Coordinates and Pathways of Mechanochemical Transformations". 2016. 3. Avdoshenko, S. M.; Makarov, D. E., Finding mechanochemical pathways and barriers without transition state search. The Journal of Chemical physics 2015, 142, 174106. 4. Konda, S. S. M.; Avdoshenko, S. M.; Makarov, D. E., Exploring the topography of the stress-modified energy landscapes of mechanosensitive molecules. The Journal of Chemical Physics 2014, 140, 104114.
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