A Generalized Force-Modified Potential Energy Surface (G-FMPES) for Mechanochemical Simulations

Abstract

We describe the modifications that a spatially varying external force produces on a Born-Oppenheimer potential energy surface (PES), and in this chapter, we present a formulation for describing a Generalized Force-Modified Potential Energy Surface (G-FMPES). Our formulation shows that the spatially varying force resembling hydrostatic pressure results in the G-FMPES having curvature different from that of the unmodified PES. Using electronic structure methods, the effect of pseudo-hydrostatic pressure on the PES is exemplified by calculating atomistic quantities (including transition states) for (i) conformational transitions in ethane ($\text {C}_{2}\text {H}_{6}$) and RDX (hexahydro-1,3,5-trinitro-s-triazine) molecules, (ii) the decomposition of RDX, and (iii) a Diels-Alder reaction between 1,3-butadiene and ethylene. The calculated transition states and Hessian matrices of stationary points of ethane and RDX molecules show that spatially varying external forces shift the stationary points and modify the curvature of the PES, thereby affecting the harmonic transition rates by altering both the energy barrier as well as the prefactor. The harmonic spectra of both molecules are blue-shifted with increasing compressive “pressure.” Some stationary points on the RDX-PES disappear under the application of the external force, indicating the merging of an energy minimum with a saddle point. This change in the topology of the PES demonstrates that new reaction pathways may be introduced by the application of mechanical forces. Part of this chapter is reproduced with permission from Refs. (J Chem Phys 143(13):134109 [1]) Copyright 2015 AIP Publishing, (J Chem Phys 145(7):074307 [2]) Copyright 2016 AIP Publishing, and  (Int J Quantum Chem 117(20):e25426 [3]) Copyright 2017 John Wiley & Sons.

Appendices

Appendix 1: Proof that the external force field is conservative

For a 3-dimensional, N-atom system, the position vector of the $j\textrm{th}$ atom, and the external force vector on it are given in component form as

$$\begin{aligned} \textbf{r}^{(j)} &= \left( { x^{(j)},y^{(j)},z^{(j)} }\right) \end{aligned}$$

(2.14)

$$\begin{aligned} \textbf{f}_\textrm{ext}^{(j)} &= \left( { f_{\textrm{ext},x}^{(j)},f_{\textrm{ext},y}^{(j)},f_{\textrm{ext},z}^{(j)} }\right) \end{aligned}$$

(2.15)

where various $\left\{ { \textbf{f}_\textrm{ext}^{(j)} \ \forall \ j = 1,2,3 \ldots N }\right\}$ make up the external force field $\textbf{F}_\textrm{ext}(\textbf{R})$.

Geometric centroid of the configuration is given by the mean position of all the atoms in the configuration as

$$\begin{aligned} \textbf{c} = \left( { c_{x}, c_{y}, c_{z} }\right) = \left( { \left\langle {x^{(j)}}\right\rangle , \left\langle {y^{(j)}}\right\rangle , \left\langle {z^{(j)}}\right\rangle }\right) \end{aligned}$$

(2.16)

where $\left\langle {\cdot }\right\rangle$ denotes an average taken over all N atoms.

According to our prescription of pseudo-hydrostatic pressure, external force on the $j\textrm{th}$ atom is given as

$$\begin{aligned} \textbf{f}_\textrm{ext}^{(j)} = P_{HP} \left[ { \textbf{r}^{(j)} - \textbf{c} }\right] ; \quad \forall \quad j = 1,2,3 \ldots N \end{aligned}$$

(2.17)

where $P_{HP}$ is a user-defined “pressure.”

First derivatives of the external force vector are given by

$$\begin{aligned} \left\{ {\frac{\partial \textbf{f}_\textrm{ext}^{(j)}}{\partial x^{(k)}}, \frac{\partial \textbf{f}_\textrm{ext}^{(j)}}{\partial y^{(k)}}, \frac{\partial \textbf{f}_\textrm{ext}^{(j)}}{\partial z^{(k)}} ; \quad \forall \quad j,k = 1,2,3 \ldots N}\right\} \end{aligned}$$

(2.18)

The first of this set is

$$\begin{aligned} \frac{\partial \textbf{f}_\textrm{ext}^{(j)}}{\partial x^{(k)}} = P_{HP}\left( { \frac{\partial \textbf{r}^{(j)}}{\partial x^{(k)}} - \frac{\partial \textbf{c}}{\partial x^{(k)}}}\right) \end{aligned}$$

(2.19)

with

$$\begin{aligned} \frac{\partial \textbf{r}^{(j)}}{\partial x^{(k)}} &= \left( { \frac{\partial x^{(j)}}{\partial x^{(k)}}, \frac{\partial y^{(j)}}{\partial x^{(k)}}, \frac{\partial z^{(j)}}{\partial x^{(k)}} }\right) = \left( {\delta _{jk},0,0}\right) \end{aligned}$$

(2.20)

$$\begin{aligned} \frac{\partial \textbf{c}}{\partial x^{(k)}} &= \left( { \frac{\partial \left\langle {x}\right\rangle }{\partial x^{(k)}}, \frac{\partial \left\langle {y}\right\rangle }{\partial x^{(k)}}, \frac{\partial \left\langle {z}\right\rangle }{\partial x^{(k)}} }\right) = \left( {\frac{1}{N},0,0}\right) \end{aligned}$$

(2.21)

where $\delta _{jk}$ is the Kronecker delta. Using this, and similar results for the other derivatives, first derivatives of the external force vector are

$$\begin{aligned} \frac{\partial \textbf{f}_\textrm{ext}^{(j)}}{\partial x^{(k)}} &= P_{HP}\left( {\delta _{jk} - \frac{1}{N},0,0 }\right) \end{aligned}$$

(2.22)

$$\begin{aligned} \frac{\partial \textbf{f}_\textrm{ext}^{(j)}}{\partial y^{(k)}} &= P_{HP}\left( {0,\delta _{jk} - \frac{1}{N},0 }\right) \end{aligned}$$

(2.23)

$$\begin{aligned} \frac{\partial \textbf{f}_\textrm{ext}^{(j)}}{\partial z^{(k)}} &= P_{HP}\left( {0,0,\delta _{jk} - \frac{1}{N} }\right) \end{aligned}$$

(2.24)

which are all constant for all values of j and k and therefore exist and are continuous everywhere, proving that the external force field is conservative.

Appendix 2: The NEB method on a G-FMPES

With the understanding of how to compute energies, forces, and curvatures on a G-FMPES, the NEB implementation for finding MEPs on a G-FMPES closely follows the original implementation, but with a few crucial modifications. For clarity, our implementation of a G-FMPES is outlined below.

The start and end point structures are re-optimized with the external force and a set of images is initialized between them. Consecutive images are connected by harmonic springs with an equilibrium length of zero and a user-specified spring stiffness $k_\textrm{spring}$ (the actual value of which is not particularly important as long as it is greater than zero, but needs to be on the order of the system forces for efficient convergence). The band is iteratively optimized until the net force on each image is minimized to within a user-specified tolerance. At each iteration, the net force on the $i^\textrm{th}$ image located at $\textbf{R}^{(i)}$ on the G-FMPES is given by the projected (nudged) forces as:

$$\begin{aligned} \overline{\textbf{F}}_\textrm{net,nudged}^{(i)} = \left[ {\textbf{F}_\textrm{grad}^{(i)}} + \textbf{F}_\textrm{ext}^{(i)}\right] _\perp + \left[ { \textbf{F}_\textrm{spring}^{(i)} }\right] _\parallel \end{aligned}$$

(2.25)

where the subscript $\perp$ (or $\parallel$) on a vector indicates its component perpendicular (or parallel) to the unit tangent $\hat{\boldsymbol{\tau }}^{(i)}$. The un-normalized tangent $\boldsymbol{\tau }^{(i)}$ is computed using Henkelman and Jónsson’s improved tangent estimate [50]. Defining

$$\begin{aligned} \boldsymbol{\tau }^{(i)+} &= \textbf{R}^{(i+1)} - \textbf{R}^{(i)}\end{aligned}$$

(2.26)

$$\begin{aligned} \boldsymbol{\tau }^{(i)-} &= \textbf{R}^{(i)} - \textbf{R}^{(i-1)} \end{aligned}$$

(2.27)

the tangent is estimated as

$$\begin{aligned} \boldsymbol{\tau }^{(i)} = {\left\{ \begin{array}{ll} \boldsymbol{\tau }^{(i)+} &{} \textrm{if } \quad \overline{V}^{(i+1)}> \overline{V}^{(i)} > \overline{V}^{(i-1)}\\ \boldsymbol{\tau }^{(i)-} &{} \textrm{if } \quad \overline{V}^{(i+1)}< \overline{V}^{(i)} < \overline{V}^{(i-1)}\\ \end{array}\right. } \end{aligned}$$

(2.28)

where the different $\overline{V}$ are computed using the line integral, as defined in the main article. In the event that the three consecutive values of $\overline{V}$ are neither strictly increasing nor strictly decreasing, i.e., if $\overline{V}^{(i+1)} \le \overline{V}^{(i)} \ge \overline{V}^{(i-1)}$ or $\overline{V}^{(i+1)} \ge \overline{V}^{(i)} \le \overline{V}^{(i-1)}$, in order to prevent abrupt switching between two possible tangents, the tangent is taken to be a weighted average as

$$\begin{aligned} \boldsymbol{\tau }^{(i)} = {\left\{ \begin{array}{ll} \boldsymbol{\tau }^{(i)+}\Delta \overline{V}^{(i)+} + \boldsymbol{\tau }^{(i)-}\Delta \overline{V}^{(i)-}&{} \textrm{if } \quad \overline{V}^{(i+1)}> \overline{V}^{(i-1)}\\ \boldsymbol{\tau }^{(i)+}\Delta \overline{V}^{(i)-} + \boldsymbol{\tau }^{(i)-}\Delta \overline{V}^{(i)+}&{} \textrm{if } \quad \overline{V}^{(i+1)}< \overline{V}^{(i-1)}\\ \end{array}\right. } \end{aligned}$$

(2.29)

with

$$\begin{aligned} \Delta \overline{V}^{(i)+} &= \max \left( \;{ \left| { \overline{V}^{(i+1)} - \overline{V}^{(i)}}\right| ,\left| { \overline{V}^{(i-1)} - \overline{V}^{(i)}}\right| \;}\right) \end{aligned}$$

(2.30)

$$\begin{aligned} \Delta \overline{V}^{(i)-} &= \min \left( \;{ \left| { \overline{V}^{(i+1)} - \overline{V}^{(i)}}\right| ,\left| { \overline{V}^{(i-1)} - \overline{V}^{(i)}}\right| \;}\right) \end{aligned}$$

(2.31)

The component of the spring force parallel to the tangent is computed as

$$\begin{aligned} \left[ { \textbf{F}_\textrm{spring}^{(i)} }\right] _\parallel = k_\textrm{spring} \left( \;{ \left\Vert { \boldsymbol{\tau }^{(i)+} } \right\Vert - \left\Vert { \boldsymbol{\tau }^{(i)-} } \right\Vert }\;\right) \hat{\boldsymbol{\tau }}^{(i)} \end{aligned}$$

(2.32)

At each iteration, the climbing-image method [51] is applied to the highest energy image along the band. This image, identified by $i=h$, is free from all spring forces and is assigned a climbing force computed by inverting the component of the gradient plus external forces along the tangent to this image which is expressed as

$$\begin{aligned} \overline{\textbf{F}}_\textrm{net,climb}^{(i)} = \left( {\textbf{F}_\textrm{grad}^{(i)}} + \textbf{F}_\textrm{ext}^{(i)}\right) - 2 \left[ { \left( {\textbf{F}_\textrm{grad}^{(i)}} + \textbf{F}_\textrm{ext}^{(i)}\right) \cdot \hat{\boldsymbol{\tau }}^{(i)} }\right] \hat{\boldsymbol{\tau }}^{(i)} \end{aligned}$$

(2.33)

and makes the highest image move uphill along the direction of the first eigenvector, and downhill along all other directions.

In the more recent variation with two climbing images [52], the highest image (with $i=h$) is not allowed to climb but is nudged in the usual manner by assigning its net force to be equal to $\overline{\textbf{F}}_\textrm{net,nudged}$ as in Eq. (2.25). However, its two nearest neighbors (one from each side of the band, i.e., images with $i=h\pm 1$) are assigned climbing forces as in Eq. (2.33). This prescription results in a higher density of images near the saddle point and is particularly useful for MEPs with unusually high curvatures near the saddle point which would cause the NEB tangent direction to be different from the MEP tangent direction.

Having computed the net force on each image, the band is iteratively optimized until it is well-converged by moving each image toward the point of minimum net force using a suitable optimizer.