# Improved Initialization of Optimal Path Calculations Using Sequential
Traversal over the Image Dependent Pair Potential Surface

Yorick L. A. Schmerwitz  [  Vilhjálmur Ásgeirsson  [  Hannes Jónsson  [
<EMAIL_ADDRESS>[

###### Abstract

In reaction path optimization, such as the calculation of a minimum energy
path (MEP) between given reactant and product configurations of atoms, it is
advantageous to start with an initial guess where close proximity of atoms is
avoided and bonds are not unnecessarily broken only to be reformed later. When
the configurations of the atoms are described with Cartesian coordinates, a
linear interpolation between the endpoints can be problematic, and a better
option is provided by the so-called image dependent pair potential (IDPP)
approach where interpolated pairwise distances are generated to form an
objective function that can be used to generate an improved initial path. When
started with a linear interpolation, this method can, however, still lead to
unnecessary bond breaking in, for example, reactions where a molecular
subgroup undergoes significant rotation. In the method presented here, this
problem is addressed by constructing the path gradually, introducing images
sequentially starting from the vicinity of the endpoints while the distance
between images in the central region is larger. The distribution of images is
controlled by systematically scaling the tightness of springs acting between
the images until the desired number of images is obtained and they are evenly
spaced. This procedure generates an initial path on the IDPP surface, a task
that requires insignificant computational effort, as no evaluation of the
energy of the system is needed. The calculation of the MEP, typically using
electronic structure calculations, is then subsequently carried out in a way
that makes efficient use of parallel computing with the nudged elastic band
method. Several examples of reactions are given where the linear interpolation
IDPP (LI-IDPP) method yields problematic paths with unnecessary bond breaking
in some of the intermediate images, while the sequential IDPP (S-IDPP) method
yields paths that are significantly closer to realistic minimum energy paths.

University of Iceland] Science Institute and Faculty of Physical Sciences,
University of Iceland VR-III, 107 Reykjavík, Iceland [Julius-Maximilians-
Universität Würzburg] Institut für Physikalische und Theoretische Chemie,
Fakultät für Chemie und Pharmazie, Julius-Maximilians-Universität Würzburg,
Emil-Fischer-Straße 42, Würzburg, D-97074, Germany University of Iceland]
Science Institute and Faculty of Physical Sciences, University of Iceland VR-
III, 107 Reykjavík, Iceland University of Iceland] Science Institute and
Faculty of Physical Sciences, University of Iceland VR-III, 107 Reykjavík,
Iceland University of Iceland] Science Institute and Faculty of Physical
Sciences, University of Iceland VR-III, 107 Reykjavík, Iceland

##  1 Introduction

Theoretical studies of the mechanism and rate of atomic rearrangements such as
chemical reactions, diffusion events, and conformational changes, often
involve finding an optimal transition path. Typically, this path is taken to
be a minimum energy path (MEP) between given configurations of the atoms
corresponding to initial and final state minima on the energy surface. The MEP
illustrates the transition mechanism, and the highest rise in the energy along
the path gives an estimate of the activation energy, an essential parameter
for estimating the transition rate within the harmonic approximation to
transition state theory.1, 2 It can also be used to parameterize a reversible
work evaluation of the free energy barrier needed for the estimation of the
transition rate in classical and quantum transition state theory.3 Paths can
be represented in a discrete way by a sequential set of atom configurations
referred to as images of the system (see ref. 4 for a review), as in the
nudged elastic band (NEB) method 5, 6 which is widely used for finding MEPs.
NEB is a local optimization method starting from some initial path that should
ideally represent a good guess. The computational efficiency of the method is
in part due to the effective use of parallel computing, as the calculation of
the energy and atomic forces on all the images representing the path can be
carried out simultaneously. The success and computational efficiency of an NEB
calculation, however, depends on the quality of the path from which the
calculation starts. The closer the initial path is to the MEP, the smaller the
computational effort will be to reach convergence. The quality of the initial
path is particularly important when the energy and atomic forces are evaluated
using an electronic structure method requiring significant computational
effort. Furthermore, sometimes more than one MEP exists connecting given
initial and final states, and since the NEB method represents a local
optimization, it will converge on the one closest to the initial path. A good
initial path is of particularly great importance in such cases.

The simplest way of generating an initial path is linear interpolation of the
Cartesian coordinates of the initial and final state configurations. This
method works well in many cases and has in particular become widely used in
computational studies of heterogeneous catalysis as well as diffusion in and
on the surfaces of solids.4 The linear interpolation can, however, lead to the
creation of unrealistic atomic configurations in the intermediate images
resulting in large atomic forces and even problems with convergence in the
electronic structure calculations. Such problems can, for example, arise when
significant rotation of a group of atoms occurs during the transition, as
shown below. One way to address such problems is to define a minimum distance
between atoms and/or construct a linear interpolation to and from an
intermediate configuration that represents a good guess for a transition
structure. This approach, however, makes the calculations more tedious and
requires significant user input, which is highly undesirable in the context of
large scale studies using MEP calculations. Internal coordinates can work
better than Cartesian coordinates in part because of the inclusion of dihedral
angles,7 but the complexity of the implementation is then greatly increased,
and there is still no guarantee that the initial path will be good since large
rotating groups can still overlap when the linear interpolation is carried
out.

An alternative to the linear interpolation and subsequent optimization of the
images in parallel is a sequential generation of the MEP starting from each of
the two endpoints until the two path segments meet. This approach has been
developed and used in electronic structure calculations of various
reactions.8, 9, 10 By growing the path sequentially, unphysical high-energy
configurations that may be generated in a linear interpolation can be avoided.
This method, however, has the disadvantage that the efficient use of parallel
computation in simultaneous calculation of all the images is no longer
possible.

A more recent approach is to generate an initial path on the so-called image
dependent pair potential (IDPP).11 There, an interpolation of the distance
between pairs of atoms in the initial and final state configurations is
carried out for each of the intermediate images. A weighted sum over squared
differences between the pairwise distances of a given path and the
interpolated values gives an objective function surface for which an NEB
calculation is carried out to get the path that matches the interpolated
pairwise distances as closely as possible. In the previous implementation, the
NEB calculation on the IDPP surface starts from a linear interpolation of the
Cartesian coordinates of the endpoints. This approach will be referred to as
the LI-IDPP method. In many cases, LI-IDPP has been shown to generate
significantly better initial paths than a linear interpolation of the
Cartesian coordinates of the endpoint configurations. However, large rotations
of groups of atoms during the transition can cause the linear interpolation in
the initial phase of the LI-IDPP method to generate intermediate
configurations where atoms come too close to each other and unnecessarily
break bonds. Several examples of such problems are given below. An NEB
calculation starting from such an initial path will converge on an unphysical
MEP with high activation energy.

In this article, we present a variation on the use of the IDPP surface where
the initial path is generated sequentially. This approach will be referred to
as the S-IDPP method. The computational effort is minimal since the IDPP
surface can be evaluated readily, as it does not involve an evaluation of the
energy of the system. In this way, a linear interpolation of Cartesian
coordinates is avoided altogether, and a better initial path is generated for
a subsequent calculation of the MEP using the NEB method taking advantage of
parallel computing. The article is structured as follows: In section 2, the
LI-IDPP method is briefly reviewed and the S-IDPP method presented with an
illustration using a two-dimensional model surface. Section 2.3 contains
information about the electronic structure calculations that are presented as
applications, and section 3 describes the results of S-IDPP calculations as
well as a comparison with LI-IDPP calculations. A discussion of the results is
given in section 4, and conclusions are presented in section 5.

##  2 Methodology

In this section, the notation is defined, the LI-IDPP reviewed briefly for
completeness, and the S-IDPP method presented. Then, the computational
methodology and parameters are given for the molecular reactions that are
presented as applications of the S-IDPP method.

The requested number of images in the discrete representation of the path is
denoted as $N_{\mathrm{im}}^{\mathrm{req}}$. As for any calculation based on
discretization of a variable, it is important to use a large enough number of
discretization points. Since NEB calculations are typically carried out in
combination with computationally intensive electronic structure calculations,
the number of images is usually kept to a minimum, typically between 5 and 10.
However, when working with the IDPP surface, the calculations are fast and it
can be advantageous to use a larger number of images when the path is
generated, for example twice the number of images than eventually will be used
in the NEB calculations. Most importantly, a larger number of images gives a
better estimate of the local tangent. In the subsequent MEP calculations, the
number of images in the converged S-IDPP path can be reduced, for example by
including only every other image.

The Cartesian coordinates of the images are
$\mathbf{r}=\left\\{r_{1},r_{2},\dots,r_{N_{\mathrm{im}}^{\mathrm{req}}-1},r_{N_{\mathrm{im}}^{\mathrm{req}}}\right\\}$
, where $r_{1}$ and $r_{N_{\mathrm{im}}^{\mathrm{req}}}$ are the fixed atom
coordinates corresponding to the initial and final configurations of the
atoms. These quantities are vectors with 3$N_{\mathrm{at}}$ coordinates where
$N_{\mathrm{at}}$ is the number of atoms in the system. The three Cartesian
coordinates of atom $j$ in image $l$ are denoted by $r_{l,j}$.

Before the calculation starts, it is important to shift and rotate the initial
and final state configurations so as to minimize the root-mean-square
deviation between the structures by using, for example, quaternion rotation12,
13 or Kabsch alignment.14. Otherwise, the path becomes unnecessarily long
because of irrelevant translation and/or rotation.

###  2.1 Image dependent pair potential

The IDPP is based on an interpolation of the distances between pairs of atoms
in the initial and final states. For a path with equal spacing of images, the
interpolated distance between atoms $i$ and $j$ corresponding to image $l$ is

$d_{ij}^{l}=d_{ij}^{R}+\frac{l-1}{N_{\mathrm{im}}^{\mathrm{req}}-1}\,\left(d_{ij}^{P}-d_{ij}^{R}\right)\,,$
(1)

where $d_{ij}^{R}=\left|{r_{1,j}-r_{1,i}}\right|$ is the distance between the
atoms in the initial state configuration, and the expression is analogous for
the final state, $d_{ij}^{P}$. For a given path, an objective function
denoting how well image $l$ matches the interpolated values is given by the
weighted sum of squared differences between the atom distances in the image
and the interpolated values

$S_{l}^{\mathrm{IDPP}}\left(r_{l}\right)=\sum_{i=1}^{N}\sum_{j>i}^{N}\omega\left(\left|r_{l,j}-r_{l,i}\right|\right)\left(\left|r_{l,j}-r_{l,i}\right|-d_{ij}^{l}\right)^{2}\,.$
(2)

A weight function $\omega\left(x\right)=x^{-4}$ is typically used to place
greater emphasis on short atom distances. This IDPP objective function can be
used instead of an energy surface in an NEB calculation to give a path that is
locally optimal in a least squares sense.

The NEB method involves defining an effective force on the atoms in such a way
as to ensure convergence to an optimal path on the objective function surface
and giving control over the location of the images along the path when the
forces are zeroed using some optimization algorithm. The effective force
consists of the component of the gradient of the objective function (usually
the energy of the system, but here the IDPP objective function) perpendicular
to the path and a spring force that controls the distribution of the images
along the path. The NEB force acting on the atoms in image $l$ is

$F_{l}^{\mathrm{NEB}}=F_{l}^{\perp}+F_{l}^{\parallel\hskip 1.0pt\mathrm{sp}}$
(3)

with

$F_{l}^{\perp}=-{\nabla S_{l}^{\mathrm{IDPP}}}+\left(\nabla
S_{l}^{\mathrm{IDPP}}\cdot\hat{\tau}_{l}\right)\hat{\tau}_{l}$ (4)

and

$F_{l}^{\parallel\hskip
1.0pt\mathrm{sp}}=\left(k_{l}\left|r_{l+1}-r_{l}\right|-k_{l-1}\left|r_{l}-r_{l-1}\right|\right)\hat{\tau}_{l}\,,$
(5)

where $\hat{\tau}_{l}$ is a normalised estimate of the tangent to the path at
image $l$. The spring constant $k_{l}$ is for line-segment
$\left[r_{l},r_{l+1}\right]$. If all the spring constants are chosen to have
the same value, equal spacing of the images will result. In the application
below, the distance between images is necessarily quite different during the
sequential construction of the path and it is, therefore, important to choose
unequal values of the spring constants. In regular NEB calculations, it can
also be advantageous to choose unequal spring constants, for example weighted
by the energy, in order to get higher density of images near the top of an
energy barrier.15, 4

The tangent to the path at image $l$ is estimated by the line segment to the
neighboring image with higher IDPP objective function value (the superscript
”IDPP” in $S^{\mathrm{IDPP}}$ is omitted for brevity)

${\tau}_{l}=\begin{cases}{\tau}_{l}^{+}=r_{l+1}-r_{l}\,,\ \ \ \mathrm{if}\
S_{l+1}>S_{l}>S_{l-1}\\\ {\tau}_{l}^{-}=r_{l}-r_{l-1}\,,\ \ \ \mathrm{if}\
S_{l+1}<S_{l}<S_{l-1}\end{cases},$ (6)

and a weighted average is used for an image that has the locally highest or
lowest value of the objective function

${\tau}_{l}=\begin{cases}\hat{\tau}_{l}^{+}\Delta
S_{l}^{\mathrm{max}}+\hat{\tau}_{l}^{-}\Delta S_{l}^{\mathrm{min}}\,,\ \ \
\mathrm{if}\ S_{l+1}>S_{l-1}\\\ \hat{\tau}_{l}^{+}\Delta
S_{l}^{\mathrm{min}}+\hat{\tau}_{l}^{-}\Delta S_{l}^{\mathrm{max}}\,,\ \ \
\mathrm{if}\ S_{l+1}<S_{l-1}\end{cases},$ (7)

where

$\Delta
S_{l}^{\mathrm{max}}=\mathrm{max}\left(\left|S_{l+1}-S_{l}\right|,\left|S_{l-1}-S_{l}\right|\right)$
(8)

and

$\Delta
S_{l}^{\mathrm{min}}=\mathrm{min}\left(\left|S_{l+1}-S_{l}\right|,\left|S_{l-1}-S_{l}\right|\right)\,.$
(9)

Note that a normalised tangent is used here, given by
$\hat{\tau}_{l}={\tau}_{l}/|{\tau}_{l}|$. The difference between eq. 7 and the
tangent definition in ref. 16 is that ${\tau}_{l}^{+}$ and ${\tau}_{l}^{-}$
are normalised before the calculation of the weighted average. This
modification is found to be necessary for the S-IDPP method.

As for any NEB calculation, an initial path is needed as a starting point for
the optimization using the IDPP objective function. In the original IDPP
method, LI-IDPP, a linear interpolation of the Cartesian coordinates of the
endpoints is used as the initial path. As will be shown in several examples
below, the local optimization starting from such a path can lead to
problematic solutions where bonds are unnecessarily broken and then reformed
along the path. The problem lies in using the linear interpolation of
Cartesian coordinates as an initial path.

The optimization of image $l$ involves shifting the location of the atoms in
the direction of the effective force, ${F_{l}}^{\mathrm{NEB}}$, until the
force has been zeroed. For this purpose, the velocity projection optimization
algorithm6, 4 is applied using a small enough time step to avoid
irregularities in the path.

###  2.2 S-IDPP

Instead of the initial linear interpolation of the reactant and product atomic
configurations, the S-IDPP method constructs the path sequentially starting
from the endpoints. The first images are added close to the endpoint
configurations, then images farther away, etc., until the path contains the
requested number of images, ${N_{\mathrm{im}}^{\mathrm{req}}}$. At each stage
in the sequential construction of the path, spring constants are chosen to
give roughly equal spacing of the images in each of the two parts of the path
near the endpoints, but larger spacing between the two parts. Finally, an NEB
calculation on the IDPP surface is carried out to refine the path, mainly to
adjust the distribution of the images along the path.

The first iteration of the path involves adding two images, one at $r_{2}$
near the reactant and another one at $r_{N_{\mathrm{im}}^{\mathrm{req}}-1}$
near the product configuration. The corresponding spring constants are chosen
to take the large difference in the distance between these images into account
as compared to their distance from the nearest endpoint. When $k$ denotes a
typical value for a spring constant when equal spacing of the images is
desired, the spring constant for these newly added images is chosen to be

$k_{2}=\frac{d_{\mathrm{id}}}{\left|r_{2}-r_{N_{\mathrm{im}}^{\mathrm{req}}-1}\right|}\,k\,,$
(10)

where $d_{\mathrm{id}}$ is the ideal distance between images if
$N_{\mathrm{im}}^{\mathrm{req}}$ images are distributed evenly along the
current path, which at this point is just a linear interpolation between the
endpoints

$d_{\mathrm{id}}=\frac{\left|r_{1}-r_{N_{\mathrm{im}}^{\mathrm{req}}}\right|}{\left(N_{\mathrm{im}}^{\mathrm{req}}-1\right)}\,.$
(11)

The positions of the images are adjusted in order to zero the NEB force to a
given tolerance. Because of the choice of spring constants, the newly added
images are close to the endpoints, but they are separated from each other by a
large distance.

As soon as the optimization of one of the images has reached convergence, a
new image is added to that part of the path, initially placed a distance
$d_{\mathrm{id}}$ from the previously placed image, in a direction given by
the tangent (always eq. 7). At each introduction of a new image, the total
length of the current path is updated, simply by summing over the length of
the linear segments, the ideal distance between images reevaluated, and values
for the spring constants chosen in order to place the image at an appropriate
distance from the previous one. The NEB force is zeroed by adjusting the
positions of all images currently placed on the path. This process is repeated
until the requested number of images has been placed.

A flowchart of the algorithm is shown in figure 1 and an example calculation
shown in figure 2 for different stages of the sequential construction of the
path. This example shows how the path has the tendency to follow the slowest
ascent direction which in this case is different from the optimal path.
However, eventually, as more images are added (beyond 9 in this case), the
path recovers and converges on the optimal path.

Figure 1: Flowchart of the S-IDPP method. The steps where images are added are
shown with a blue background, while the optimization steps are shown with a
green background. The current number of images is denoted by
$N_{\mathrm{im}}$, while $N_{\mathrm{im}}^{\mathrm{req}}$ is the requested
total number of images. The current number of images in the parts of the path
near the reactant and product configurations are denoted by $n_{R}$ and
$n_{P}$, respectively. Once the number of images reaches the requested number,
an NEB calculation on the IDPP surface is carried out with equal spring
constants to refine the path, especially the distribution of images along the
path. The resulting path is then ready for use in a CI-NEB calculation using
an electronic structure method to estimate the energy and atomic forces.

Once the requested number of images has been added, an NEB calculation is
carried out with equal spring constants to refine the path and distribute the
images evenly along the path. Then, the path is ready as an initial path for
an MEP calculation using the CI-NEB method15 in combination with some method
for estimating the energy and atomic forces, typically an electronic structure
calculation, as illustrated in the examples given below. There, it can be
advantageous to use energy weighted springs.15, 4

Figure 2: Illustration of the S-IDPP method for a 2-dimensional model IDPP
surface. The path is visualised for intermediate steps after adding 2 images
(circles), 9 images (triangles) and 14 images (squares) using the spring
constant scaling algorithm. An optimal path obtained in an NEB calculation
with 17 images is shown by red crosses. Note how the images initially line up
along the slowest ascent path from the lower right endpoint (triangles), but
later converge to the optimal path.

###  2.3 Computational Settings

The convergence thresholds used in the S-IDPP calculations are
$0.01\,\mathrm{Å}^{-3}$ for the maximum atomic force component and
$0.005\,\mathrm{Å}^{-3}$ for the RMS value. The base spring constant is 1
$\mathrm{Å}^{-2}$. The chosen number of images in the S-IDPP construction
matches the one used in the subsequent CI-NEB calculation in all cases except
the rotation of 9,9’-bianthracene where using twice as many images in the
S-IDPP calculation is found to improve the quality of the initial path.

The subsequent CI-NEB calculation using electronic structure estimates of
energy and atomic forces in the molecular reactions make use of 9 images to
describe the path. The B3LYP+D3(BJ)/def2-SVP17, 18, 19, 20, 21 and GFN2-XTB 22
calculations are performed with the ORCA software23, 24 using energy-weighted
spring constants between $0.01$ and $0.1$ a.u. and are converged to a
threshold of $2.5\cdot 10^{-3}$ a.u. for the RMS of the force orthogonal to
the image tangent and $5\cdot 10^{-3}$ a.u. for the largest perpendicular
force component for all images. The total force on the climbing image is
converged to threshold values one order of magnitude tighter than for the
other images.

##  3 Results

We present four examples of molecular reactions where the LI-IDPP method turns
out to give unphysical paths that are so different from the optimal MEP that
subsequent NEB calculations would converge on an MEP with very high activation
energy. The S-IDPP method, however, gives initial paths that are realistic,
and the subsequent NEB calculations using electronic structure calculations to
estimate energy and atomic forces converge on what appear to be optimal MEPs.
The first system is the Diels-Alder reaction between ethylene and
cyclopentadiene. The second reaction is an isomerisation of
tris[(3-methyl-1H-benzimidazol-1-yl-2(3H)-ylidene)-1,2-phenylene]iridium(III)
(TMBPI) due to the rotation of one of its
[(3-methyl-1H-benzimidazol-1-yl-2(3H)-ylidene)-1,2-phenylene]- (MBP) ligands
by 180°. The third reaction is a [3+2]-cycloaddition reaction of
9-(2-azidoethyl)-10-(but-3-yn-1-yl)anthracene, in which an azidoethyl group
rotates around a large aromatic group. The fourth reaction involves sterically
hindered rotation of an anthryl group in 9,9’-bianthracene.

####  3.0.1 Diels-Alder Reaction

Figure 3: Diels-Alder reaction of ethylene and cyclopentadiene. (a) Energy
profiles calculated using B3LYP+D3(BJ)/def2-SVP along paths obtained by LI-
IDPP (red) and S-IDPP (blue). The reactant, product and highest-energy image
configurations are shown as insets. Along the LI-IDPP path, a C-C bond is
unnecessarily broken, as highlighted by the red dashed line. In contrast, the
S-IDPP path preserves the structural integrity of the cyclopentadiene
molecule. (b) Energy along the path for various stages of the CI-NEB
calculation starting with the S-IDPP path (blue as in (a)). Intermediate
stages of the optimization (gray) and the final converged MEP (green) are
shown. The saddle point configuration corresponding to the maximum along the
MEP is included as an inset.

The reactant and product configurations of this Diels-Alder reaction system
have been taken from ref. 9. The molecules in this system are unfavorably
oriented with respect to each other. The necessary rotation of the
cyclopentadiene fragment in the initial structure of the system poses problems
for the LI-IDPP method, as is visualized in fig. 3. The bond between carbon
atoms 3 and 4 breaks and reforms over the course of the path. The S-IDPP
method produces a significantly better path preserving the structural
integrity of the cyclopentadiene molecule. The two paths represent different
locally optimal paths on the IDPP surface that differ because of the different
ways the paths are constructed.

The subsequent CI-NEB calculation using the energy and atomic forces estimated
by the B3LYP+D3(BJ)/def2-SVP electronic structure method and the S-IDPP
initial path converges to a sensible MEP that has low activation energy. The
long tail of the MEP corresponds to a 180∘ rotation of the cyclopentadiene
fragment.

####  3.0.2 Isomerisation Reaction

Figure 4: Isomerization of TMBPI where the MBP ligand rotates by 180∘. The
full Ir-complex (left) and the most relevant region of the reactant (center)
and product (right) configurations are shown in the upper panel. The lower
panel shows the image with highest energy when the LI-IDPP path (left) and
S-IDPP path (center) are used in a GFN2-XTB calculation without further
optimization, and then the saddle point configuration (right) obtained by a
CI-NEB calculation started from the S-IDPP path, using first GFN2-XTB and
subsequently a saddle point optimization at the B3LYP+D3(BJ)/def2-SVP level.
The red circle and the dashed line in the highest image of LI-IDPP highlight
the problems occurring there, where a C-C bond is broken and an H atom
dissociates from the ligand. The S-IDPP path, however, yields a highest-energy
configuration that is quite similar to the one obtained in the subsequent DFT
calculation.

The isomerisation reaction between the facial and meridional isomers of metal
carbene complexes is of interest since these complexes offer favorable
properties for diverse applications.25 The properties of the two isomers
differ significantly. Here, the isomerisation of TMBPI in the absence of an
acid is calculated. This reaction is challenging because of the large rotation
involved in the isomerisation reaction. As shown in fig. 4, a large group of
atoms constituting an MBP ligand rotates by about 180°. The LI-IDPP method
produces a path where the structural integrity of this rotating ligand is
lost, as the bond between carbon atoms 2 and 3 is broken and then later
reformed over the course of the reaction. Additionally, hydrogen atom 1 is
dissociated from the molecule. As a result, the LI-IDPP path is unsuitable as
an initial path for a CI-NEB calculation in an electronic structure
calculation of the MEP. The S-IDPP path, however, preserves the structural
integrity of the ligand and gives a path that can be used as an initial path
in a CI-NEB calculation with an electronic structure method. First, the MEP is
found on the GFN2-XTB energy surface. Then, the saddle point is determined
more accurately using the final position of the climbing image in the GFN2-XTB
calculation as input and carrying out a saddle point optimization at the
B3LYP+D3(BJ)/def2-SVP level. The resulting saddle point configuration is shown
in fig. 4. The highest-energy image on the S-IDPP path is quite similar
showing that the S-IDPP path is a good initial guess for the electronic
structure calculation, while the LI-IDPP path is not.

####  3.0.3 [3+2]-Cycloaddition Reaction

Figure 5:  [3+2]-cycloaddition reaction of
9-(2-azidoethyl)-10-(but-3-yn-1-yl)anthracene. In the upper panel, the
reactant (left) and product (right) configurations are shown. In the lower
panel, the highest-energy image configurations along the LI-IDPP (left) and
S-IDPP (center) paths are shown along with the saddle point configuration
(right) obtained by a CI-NEB calculation using B3LYP+D3(BJ)/def2-SVP starting
from the S-IDPP path. In the image from the LI-IDPP path, the azidoethyl group
is dissociated from the molecule making this path a poor initial guess for the
electronic structure calculation of the MEP.

Figure 5 shows a [3+2]-cycloaddition of
9-(2-azidoethyl)-10-(but-3-yn-1-yl)anthracene. The main challenge here for the
initial path generation is not the cycloaddition itself, but rather the
rotation of the azidoethyl group to the other side of the aromatic rings. In
the LI-IDPP path, the azidoethyl group is dissociated from the molecule, and a
dangling bond is formed on carbon atom 1. In the S-IDPP path, however, the
azidoethyl group rotates to the other side of the aromatic groups without any
bond breaking, and the highest-energy image corresponds to the reactive step
of the cycloaddition, as in the MEP on the B3LYP+D3/def2-SVP surface obtained
with the subsequent CI-NEB calculation.

####  3.0.4 Rotational Isomerization

Figure 6:  Rotation of an anthryl group in 9,9’-bianthracene. In the upper
panel, the reactant (left) and product (right) configurations of the reaction
are shown. In the lower panel, the highest-energy image along the LI-IDPP
(left) and S-IDPP (center) paths obtained using B3LYP+D3(BJ)/def2-SVP without
further relaxation are shown along with the saddle point configuration (right)
obtained from a converged CI-NEB calculation started from the S-IDPP path. The
rotating anthryl group is colored for clarity. In the LI-IDPP path, a C atom
and multiple H atoms dissociate from the molecule. The red arrow points to a C
atom that is abstracted from one of the aromatic rings.

The rotation of an anthryl group in 9,9’-bianthracene is shown in fig. 6. One
of the two anthryl groups rotates by 180°. This rotation is sterically
hindered enough that the $\sigma$-bond around which the rotation occurs breaks
and is then reformed subsequently along the MEP calculated at the
B3LYP+D3(BJ)/def2-SVP level. In the LI-IDPP path, a carbon atom and multiple
hydrogen atoms dissociate from the anthryl group that does not rotate. In
addition, the planarity of both of the anthryl groups is lost. However, the
S-IDPP path maintains structural integrity in both anthryl groups, and the
distortion of the aromatic groups is much less pronounced, leading to a
correct description of the rotation. The highest-energy image of the S-IDPP
path is quite similar to the highest-energy image along the MEP on the
B3LYP+D3(BJ)/def2-SVP surface. For this reaction, it turns out to be better to
use a larger number of images in the S-IDPP calculation, 17 rather than 9. The
subsequent MEP computation at the B3LYP+D3(BJ)/def2-SVP level is started with
only every other image of the converged S-IDPP path. Such an increase in the
number of images does not, however, improve the path obtained with the LI-IDPP
method.

##  4 Discussion

The central problem with the LI-IDPP method is the linear interpolation of
Cartesian coordinates between the endpoint structures prior to the path
optimization on the IDPP surface. For all the problematic reactions presented
here, this interpolation places some atoms too close to each other, typically
around the midpoint of the initial path. At the point along the path at which
the distances become minimal, these atoms are approximately in the same plane
which we refer to as a partitioning plane in the linear interpolation path.
The IDPP surface was formulated in order to increase too small atom distances
and thereby improve upon the linear interpolation originally used in NEB
calculations. For atoms close to a partitioning plane, the weight function
contribution, which only depends on the atom distances without taking chemical
bonding into account, dominates the IDPP force by far and points away from the
partitioning plane. As atoms move through a partitioning plane, the direction
of the full IDPP gradient, in particular its perpendicular projection on the
path tangent, can change dramatically, even up to a complete reversal. If one
part of a group of atoms has moved past a partitioning plane, while another
has not, the perpendicular IDPP gradient will point away from the partitioning
plane in opposite directions for the two parts of the group leading to
fragmentation during the IDPP path optimization, even if conservative settings
are used for the optimization. The isomerisations of TMBPI and
9,9’-bianthracene are examples of this effect.

The accumulation of atoms in a partitioning plane can also lead to atoms being
placed inside bonds between other atoms, in which case the bond is broken
during the path optimization on the IDPP. The Diels-Alder reaction is a
testament to this effect. In the [3+2]-cycloaddition, a partitioning plane is
formed by the anthracene aromatic plane and already visible in the endpoint
structures. In this case, atoms are placed inside the aromatic plane in
intermediate images, destroying the structural integrity of the plane during
the IDPP NEB optimization.

When an odd number of images is used, these problems with the linear
interpolation are more likely to be encountered since an image is then placed
at the midpoint of the path where a partitioning plane is most likely to
occur. By choosing an even number of images, the problems may be avoided, but
not in all cases, in particular not for the four systems studied in this
article. Once structural integrity is lost in one image along the path, the
parallel spring forces between the images gradually disperse the problem along
the entire path due to pairwise coupling of adjacent images. The S-IDPP method
presented here avoids these problems by not starting from a linear
interpolation of the Cartesian coordinates.

In particularly challenging cases, such as the rotation of 9,9’-biantracene,
the quality of the S-IDPP initial path can be improved by using more images
for the S-IDPP calculation than for the subsequent MEP calculation. The reason
for this improvement is the decrease in optimal spacing between the images
when more images are used. As a result, new images are placed at shorter
distance from converged images meaning they venture less into the unknown
region of the IDPP surface. The same cannot be said for LI-IDPP since there,
all images are placed on the path immediately, which leaves little opportunity
for improvement when using more images. The simplest way to reduce the number
of images between the two paths is to use twice the number of images (one less
for an odd number of images) for the initial path and then using every second
image for the MEP calculation.

It is worth noting that some of the problems with the linear interpolation
could be solved manually. The unfavorable alignment of fragments in the Diels-
Alder reaction can be resolved by changing one of the endpoint structures. A
useful initial path for the isomerisation of TMBPI can be constructed with LI-
IDPP if a good guess for the saddle point configuration is used as an
intermediate image along the path and two linear interpolations constructed,
one from the initial configuration to this guess for the saddle point, and the
other from there to the endpoint configuration. However, these strategies
require human intervention and introduce bias into the calculation, so they
are not satisfactory solutions, especially in the context of large scale
studies using NEB calculations. The S-IDPP method constitutes a way of solving
these problems automatically and without introducing bias into the
computation.

##  5 Conclusion

The S-IDPP method is an improvement on the LI-IDPP method used to generate
initial paths for MEP calculations. Problems with the LI-IDPP method derive
from the use of an initial interpolation of Cartesian coordinates at the start
of the calculation. Atoms can then move through a partitioning plane resulting
in an abrupt change of the direction of the perpendicular IDPP force acting on
the atoms. The root of the problem is not numerical instability, which could
be eliminated by reducing the step size of the optimizer, but rather the
dominance of the weight function in the IDPP surface for atoms near the
partitioning plane. S-IDPP avoids most of the linear interpolation of
Cartesian coordinates and therefore, represents a significant improvement over
LI-IDPP. A two-dimensional test surface is used to demonstrate the concept of
the S-IDPP method. It demonstrates that even when the slowest ascent direction
differs from that of the optimal path, convergence to a good initial path can
be obtained.

Y.L.A.S. thanks the Erasmus+ program of the European Union for a mobility
grant. V.Á. acknowledges a doctoral fellowship from the University of Iceland.
This work was supported by the Icelandic Research Fund and by the European
Union’s Horizon 2020 research and innovation program under grant agreement No
814416. The authors thank Prof. Bernd Engels for useful discussions. The
calculations were carried out at the Icelandic High Performance Computing
Center (IHPC).

The authors confirm that the data supporting the findings of this study are
available within the article and/or its supplementary materials.

Data related to the results presented in this article is available at
Zenodo.26 Atomic configurations of all systems: Reactant and product
configurations for Diels-Alder reaction of ethylene and cyclopentadiene;
Reactant and product configurations for isomerisation of TMBPI; Reactant and
product configurations for [3+2]-cycloaddition reaction of
9-(2-azidoethyl)-10-(but-3-yn-1-yl)anthracene; Reactant and product
configurations for rotational isomerisation of 9,9’-bianthracene.

The S-IDPP method has been implemented in a branch of the EON software27 that
is part of the ReaxPro multiscale heterogeneous catalysis software suite.

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##  6 Atomic configurations of all systems

All geometry parameters are given in Å.

###  6.1 Diels-Alder reaction of ethylene and cyclopentadiene

####  6.1.1 Reactant configuration

H | -1.07081 | -2.21835 | 0.68124
---|---|---|---
H | 1.74670 | 1.26854 | -0.43957
H | 3.40352 | -1.29647 | 0.09967
H | -1.46930 | 1.36543 | -1.77800
H | -0.93779 | 2.21928 | 0.67592
H | 3.51034 | 1.18505 | 0.16951
H | 1.63908 | -1.19534 | -0.50456
H | -1.54919 | -1.34021 | -1.77498
H | 0.13419 | -0.03191 | 1.71674
H | -1.58776 | 0.02003 | 2.11400
C | 2.54849 | -0.67576 | -0.18815
C | -1.10667 | -1.17743 | 0.35768
C | -1.35015 | -0.72623 | -0.89481
C | -1.30651 | 0.74268 | -0.89659
C | -1.03610 | 1.18164 | 0.35475
C | 2.60638 | 0.65647 | -0.15090
C | -0.87608 | -0.00256 | 1.26677

####  6.1.2 Product configuration

H | -0.11792 | 2.16141 | 0.69058
---|---|---|---
H | 2.08994 | -1.18017 | -0.02456
H | 1.14290 | 1.20885 | -1.52535
H | -1.92208 | -1.33305 | -1.08980
H | -0.11797 | -2.16141 | 0.69058
H | 1.14287 | -1.20887 | -1.52535
H | 2.08996 | 1.18012 | -0.02456
H | -1.92205 | 1.33310 | -1.08979
H | -0.91830 | 0.00001 | 2.04334
H | 0.88169 | -0.00001 | 1.98245
C | 1.18922 | 0.77880 | -0.51486
C | -0.08626 | 1.12671 | 0.32357
C | -1.27598 | 0.67156 | -0.50884
C | -1.27600 | -0.67153 | -0.50884
C | -0.08628 | -1.12671 | 0.32357
C | 1.18920 | -0.77883 | -0.51486
C | -0.04120 | 0.00000 | 1.37936

###  6.2 Isomerisation of TMBPI

####  6.2.1 Reactant configuration

C | -3.28867 | 0.80367 | 1.53343
---|---|---|---
N | -3.07193 | -0.40854 | 0.77019
C | -1.92858 | -0.74766 | 0.11805
N | -2.17065 | -1.96912 | -0.47579
C | -3.47606 | -2.39512 | -0.21355
C | -4.04664 | -1.38657 | 0.59613
C | -4.24172 | -3.50778 | -0.57868
C | -5.55642 | -3.58147 | -0.10718
C | -6.10807 | -2.57908 | 0.70606
C | -5.35555 | -1.45890 | 1.06878
C | -1.10936 | -2.53778 | -1.24221
C | 0.03914 | -1.71319 | -1.31043
C | 1.11553 | -2.21951 | -2.05710
C | 1.06931 | -3.47598 | -2.67448
C | -0.07378 | -4.27000 | -2.55951
C | -1.17682 | -3.79942 | -1.83837
Ir | -0.11451 | 0.12020 | -0.27475
C | -0.25840 | 1.94027 | 0.82260
C | 0.27510 | 1.88920 | 2.13149
C | 0.39597 | 3.01746 | 2.94893
C | -0.05833 | 4.25076 | 2.47102
C | -0.60974 | 4.33957 | 1.19056
C | -0.69421 | 3.20013 | 0.38108
N | 0.69592 | 0.58251 | 2.52990
C | 0.64234 | -0.41574 | 1.57665
N | 1.06163 | -1.55175 | 2.20254
C | 1.37240 | -1.29940 | 3.53531
C | 1.13095 | 0.07562 | 3.75794
C | 1.32122 | 0.61656 | 5.03470
C | 1.77221 | -0.23692 | 6.04723
C | 2.02467 | -1.59705 | 5.80996
C | 1.82148 | -2.15128 | 4.54298
C | 1.15115 | -2.86790 | 1.60232
C | 1.57277 | 0.86613 | -1.12299
N | 2.90211 | 0.85694 | -0.83676
C | 3.62273 | 1.51774 | -1.83219
C | 2.67917 | 1.94389 | -2.79719
N | 1.43470 | 1.52825 | -2.31803
C | 3.10883 | 2.62045 | -3.94376
C | 4.47751 | 2.86692 | -4.08565
C | 5.40427 | 2.45280 | -3.11659
C | 4.98683 | 1.76830 | -1.97193
C | 3.47643 | 0.25190 | 0.34849
C | 0.10778 | 1.64036 | -2.83188
C | -0.86007 | 0.99858 | -2.02310
C | -2.18738 | 1.08116 | -2.46920
---|---|---|---
C | -2.53593 | 1.75933 | -3.64389
C | -1.55142 | 2.38449 | -4.41062
C | -0.21469 | 2.32886 | -4.00287
H | 5.70542 | 1.44575 | -1.21734
H | 6.46567 | 2.66809 | -3.25677
H | 4.83003 | 3.39480 | -4.97434
H | 2.41744 | 2.94786 | -4.71485
H | -2.98249 | 0.60681 | -1.88828
H | -3.58182 | 1.79880 | -3.96125
H | -1.81480 | 2.91604 | -5.32777
H | 0.54353 | 2.82693 | -4.60155
H | 4.56244 | 0.16425 | 0.22218
H | 3.05526 | -0.75016 | 0.48901
H | 2.00197 | -3.20986 | 4.35035
H | 2.37655 | -2.23156 | 6.62596
H | 1.92727 | 0.16996 | 7.04875
H | 1.11601 | 1.65961 | 5.25875
H | -1.11394 | 3.28957 | -0.62375
H | -0.96586 | 5.30355 | 0.81663
H | 0.02850 | 5.13848 | 3.10205
H | 0.85115 | 2.96645 | 3.93492
H | 2.16891 | -3.27153 | 1.72372
H | 0.91581 | -2.78568 | 0.53725
H | -5.77514 | -0.66905 | 1.69364
H | -7.13794 | -2.67275 | 1.05614
H | -6.16764 | -4.44300 | -0.38468
H | -3.85340 | -4.29802 | -1.21524
H | 1.92977 | -3.83701 | -3.24496
H | -0.11652 | -5.25503 | -3.02922
H | -2.05851 | -4.42903 | -1.74828
H | -4.12756 | 1.37436 | 1.10575
H | -2.38106 | 1.41272 | 1.48641
H | 2.02600 | -1.62084 | -2.15665
H | 3.26375 | 0.85636 | 1.24229
H | 0.43810 | -3.55693 | 2.08159
H | -3.52010 | 0.55815 | 2.58191

####  6.2.2 Product configuration

C | -3.21387 | 0.70955 | -1.04295
---|---|---|---
N | -2.96372 | -0.42896 | -0.18535
C | -1.74006 | -0.78335 | 0.29812
N | -1.95467 | -1.85903 | 1.13096
C | -3.31533 | -2.16538 | 1.20528
C | -3.96132 | -1.23958 | 0.35299
C | -4.06894 | -3.10429 | 1.91637
C | -5.45667 | -3.09955 | 1.73895
C | -6.08716 | -2.18388 | 0.88249
C | -5.34371 | -1.23170 | 0.17871
C | -0.81179 | -2.37556 | 1.81287
C | 0.35525 | -1.59902 | 1.62618
C | 1.51125 | -2.04461 | 2.28181
C | 1.51133 | -3.20017 | 3.07309
C | 0.34491 | -3.95548 | 3.21365
C | -0.83080 | -3.54609 | 2.57600
Ir | 0.14237 | 0.03340 | 0.31124
C | 2.11810 | 0.71875 | 0.58133
C | 3.08923 | 0.07874 | -0.22228
C | 4.42866 | 0.47691 | -0.25852
C | 4.84143 | 1.53084 | 0.56280
C | 3.91665 | 2.16808 | 1.39284
C | 2.57443 | 1.76833 | 1.39071
N | 2.56385 | -0.99186 | -1.00667
C | 1.19659 | -1.15556 | -0.99044
N | 0.94004 | -2.26437 | -1.74020
C | 2.12485 | -2.83426 | -2.20133
C | 3.17842 | -2.01701 | -1.72972
C | 4.50563 | -2.35716 | -2.00825
C | 4.73939 | -3.50663 | -2.77074
C | 3.68680 | -4.30548 | -3.24268
C | 2.35693 | -3.98096 | -2.95794
C | -0.36867 | -2.85249 | -1.92750
C | -0.09478 | 1.79392 | -0.71698
N | 0.27873 | 2.28141 | -1.93301
C | 0.09142 | 3.66100 | -1.99088
C | -0.45255 | 4.04897 | -0.74398
N | -0.56232 | 2.87202 | 0.00066
C | -0.72931 | 5.39593 | -0.49128
C | -0.46412 | 6.32017 | -1.50749
C | 0.06875 | 5.92447 | -2.74379
C | 0.36016 | 4.58159 | -3.00157
C | 0.90915 | 1.50087 | -2.97579
---|---|---|---
C | -0.94800 | 2.62035 | 1.35207
C | -0.65823 | 1.30623 | 1.78642
C | -0.99342 | 1.00041 | 3.11274
C | -1.58889 | 1.94430 | 3.95905
C | -1.88428 | 3.22499 | 3.48614
C | -1.56801 | 3.57072 | 2.16819
H | 0.79322 | 4.26962 | -3.95290
H | 0.26869 | 6.67472 | -3.51167
H | -0.67368 | 7.37674 | -1.32694
H | -1.12261 | 5.73508 | 0.46331
H | -0.78144 | -0.00122 | 3.49366
H | -1.83183 | 1.67533 | 4.99051
H | -2.36299 | 3.96108 | 4.13632
H | -1.82547 | 4.56084 | 1.79873
H | 0.54197 | 1.82531 | -3.96036
H | 0.65639 | 0.44681 | -2.82149
H | 1.53486 | -4.60906 | -3.30346
H | 3.90907 | -5.19933 | -3.82919
H | 5.76978 | -3.78996 | -2.99557
H | 5.34093 | -1.77045 | -1.63593
H | 1.86186 | 2.28830 | 2.03581
H | 4.24013 | 2.98954 | 2.03841
H | 5.88569 | 1.85173 | 0.54464
H | 5.14856 | 0.00353 | -0.92322
H | -1.12394 | -2.09399 | -1.69889
H | -0.51181 | -3.71089 | -1.25215
H | -5.82960 | -0.50386 | -0.47283
H | -7.17347 | -2.20863 | 0.76990
H | -6.06110 | -3.82481 | 2.28802
H | -3.61016 | -3.81204 | 2.60214
H | 2.42954 | -3.51716 | 3.57502
H | 0.34258 | -4.86838 | 3.81384
H | -1.72892 | -4.15299 | 2.66989
H | -3.99228 | 0.45969 | -1.77811
H | -2.28588 | 0.96075 | -1.56664
H | 2.43595 | -1.47305 | 2.17019
H | 2.00507 | 1.61029 | -2.94039
H | -0.48605 | -3.18586 | -2.96903
H | -3.53674 | 1.58277 | -0.45393

###  6.3 [3+2]-Cycloaddition reaction of
9-(2-azidoethyl)-10-(but-3-yn-1-yl)anthracene

####  6.3.1 Reactant configuration

C | -5.15792 | -0.78667 | 0.43715
---|---|---|---
C | -5.58535 | 0.41936 | -0.05368
C | -4.68283 | 1.51356 | -0.13724
C | -3.38146 | 1.36207 | 0.26598
C | -2.88544 | 0.11903 | 0.78531
C | -3.81072 | -0.99189 | 0.88586
H | -6.61800 | 0.53761 | -0.38928
H | -5.02502 | 2.47369 | -0.53045
H | -2.70893 | 2.21445 | 0.18151
H | -5.86096 | -1.61582 | 0.46799
C | -1.53805 | -0.03325 | 1.18976
C | -1.10224 | -1.27363 | 1.71342
C | -2.02964 | -2.38124 | 1.82087
C | -3.37675 | -2.23019 | 1.41413
C | 0.24198 | -1.47493 | 2.17379
C | 0.66147 | -2.67494 | 2.68652
C | -0.24278 | -3.76703 | 2.77558
C | -1.53990 | -3.61854 | 2.3582
H | 0.94619 | -0.64647 | 2.14206
H | 1.69048 | -2.78982 | 3.03420
H | 0.09416 | -4.72178 | 3.18567
H | -2.21510 | -4.46807 | 2.44797
C | -0.59579 | 1.14756 | 1.12523
C | -0.69534 | 2.07222 | 2.35323
H | -0.79866 | 1.75646 | 0.23237
H | 0.44382 | 0.81463 | 1.02053
C | -4.32566 | -3.40220 | 1.52896
C | -4.22165 | -4.41216 | 0.35883
H | -5.36347 | -3.05749 | 1.60806
H | -4.13080 | -3.95074 | 2.46119
C | -4.59694 | -3.86199 | -0.94128
H | -4.86367 | -5.28249 | 0.58045
H | -3.18899 | -4.79486 | 0.29831
C | -4.91430 | -3.38871 | -2.00989
N | -0.33104 | 1.40568 | 3.60938
H | -1.71276 | 2.49497 | 2.42485
H | 0.00842 | 2.90907 | 2.23957
N | -1.19984 | 0.75690 | 4.18546
H | -5.18981 | -2.96479 | -2.95639
---|---|---|---
N | -1.91308 | 0.14166 | 4.82329

####  6.3.2 Product configuration

C | -5.43629 | -0.87306 | 0.97441
---|---|---|---
C | -5.90975 | 0.36890 | 0.63406
C | -5.11516 | 1.52447 | 0.86837
C | -3.87870 | 1.40722 | 1.45151
C | -3.37119 | 0.14170 | 1.87997
C | -4.17104 | -1.03761 | 1.61798
H | -6.89533 | 0.47342 | 0.17558
H | -5.49074 | 2.50773 | 0.57815
H | -3.27550 | 2.29938 | 1.62590
H | -6.04610 | -1.75899 | 0.78808
C | -2.05694 | -0.02688 | 2.37326
C | -1.73941 | -1.23250 | 3.04931
C | -2.52195 | -2.41180 | 2.75963
C | -3.58603 | -2.30701 | 1.82995
C | -0.55868 | -1.38139 | 3.83908
C | -0.18309 | -2.60216 | 4.34924
C | -0.94871 | -3.76237 | 4.05784
C | -2.06993 | -3.66747 | 3.26743
H | 0.03211 | -0.49336 | 4.07253
H | 0.70283 | -2.68160 | 4.98366
H | -0.64813 | -4.72860 | 4.46890
H | -2.65787 | -4.56120 | 3.05058
C | -0.92821 | 0.68660 | 1.67646
C | -0.51730 | -0.23557 | 0.48406
H | -0.03500 | 0.82506 | 2.30364
H | -1.21818 | 1.66553 | 1.27490
C | -3.61976 | -3.30175 | 0.69962
C | -2.38349 | -2.94374 | -0.20858
H | -4.54162 | -3.23153 | 0.10597
H | -3.50854 | -4.34769 | 1.02199
C | -2.39198 | -1.54981 | -0.77853
H | -2.35456 | -3.65206 | -1.05288
H | -1.47058 | -3.11354 | 0.37945
C | -3.23440 | -1.04374 | -1.75549
N | -1.59855 | -0.46479 | -0.47233
H | -0.18303 | -1.20839 | 0.86529
H | 0.31015 | 0.22583 | -0.07449
N | -1.95302 | 0.60391 | -1.21819
---|---|---|---
H | -4.03310 | -1.55641 | -2.28769
N | -2.93724 | 0.25965 | -1.98581

###  6.4 Rotational isomerisation of 9,9’-bianthracene

####  6.4.1 Reactant configuration

C | -2.68605 | 0.78807 | 1.16242
---|---|---|---
C | -2.58108 | 2.15674 | 0.70917
C | -1.59264 | 2.49714 | -0.22287
C | -0.69933 | 1.54314 | -0.72566
C | -0.79626 | 0.17340 | -0.27339
C | -1.79094 | -0.18482 | 0.66377
C | -3.70802 | 0.46604 | 2.11302
C | 0.31054 | 1.88567 | -1.68040
C | 1.17700 | 0.93884 | -2.16327
C | 1.08442 | -0.41231 | -1.71624
C | 0.13041 | -0.78215 | -0.80193
C | -1.90326 | -1.60015 | 1.12782
C | -2.70297 | -2.51371 | 0.40531
C | -2.82164 | -3.87946 | 0.86431
C | -2.14101 | -4.27632 | 2.02181
C | -1.34284 | -3.38216 | 2.74596
C | -1.21708 | -2.01582 | 2.29055
C | -3.40904 | -2.13457 | -0.78164
C | -4.17698 | -3.04042 | -1.46871
C | -4.29435 | -4.38666 | -1.01335
C | -3.6359 | -4.79079 | 0.11938
C | -0.39281 | -1.12244 | 3.04810
C | -4.56433 | 1.42831 | 2.58574
C | -4.45807 | 2.77831 | 2.13866
C | -3.49514 | 3.12862 | 1.22714
C | 0.25793 | -1.54676 | 4.17891
C | 0.12865 | -2.89361 | 4.62937
C | -0.64755 | -3.78248 | 3.93098
H | -1.51725 | 3.53216 | -0.56697
H | 0.37418 | 2.92307 | -2.01879
H | 1.94236 | 1.21470 | -2.89288
H | 0.0631 | -1.81642 | -0.46263
H | -2.23451 | -5.30907 | 2.36836
H | -3.32164 | -1.10653 | -1.13524
H | -4.91037 | -5.09401 | -1.57366
H | -3.71983 | -5.82090 | 0.47488
---|---|---|---
H | -4.70260 | -2.73148 | -2.37518
H | 1.78083 | -1.15724 | -2.10829
H | -5.14817 | 3.53118 | 2.52668
H | -3.40761 | 4.16069 | 0.87815
H | -3.79291 | -0.56486 | 2.45865
H | -5.33604 | 1.16114 | 3.31185
H | -0.75211 | -4.81665 | 4.26933
H | 0.65005 | -3.21226 | 5.53493
H | -0.29049 | -0.09139 | 2.70747
H | 0.88091 | -0.84874 | 4.74324

####  6.4.2 Product configuration

C | -0.79626 | 0.17340 | -0.27339
---|---|---|---
C | -0.69933 | 1.54314 | -0.72566
C | -1.59264 | 2.49714 | -0.22287
C | -2.58108 | 2.15674 | 0.70917
C | -2.68605 | 0.78807 | 1.16242
C | -1.79094 | -0.18482 | 0.66377
C | 0.13041 | -0.78215 | -0.80193
C | -3.49514 | 3.12862 | 1.22714
C | -4.45807 | 2.77831 | 2.13866
C | -4.56433 | 1.42831 | 2.58574
C | -3.70802 | 0.46604 | 2.11302
C | -1.90326 | -1.60015 | 1.12782
C | -2.70297 | -2.51371 | 0.40531
C | -2.82164 | -3.87946 | 0.86431
C | -2.14101 | -4.27632 | 2.02181
C | -1.34284 | -3.38216 | 2.74596
C | -1.21708 | -2.01582 | 2.29055
C | -3.40904 | -2.13457 | -0.78164
C | -4.17698 | -3.04042 | -1.46871
C | -4.29435 | -4.38666 | -1.01335
C | -3.63590 | -4.79079 | 0.11938
C | -0.39281 | -1.12244 | 3.04810
C | 1.08442 | -0.41231 | -1.71624
C | 1.17700 | 0.93884 | -2.16327
C | 0.31054 | 1.88567 | -1.68040
C | 0.25793 | -1.54676 | 4.17891
C | 0.12865 | -2.89361 | 4.62937
C | -0.64755 | -3.78248 | 3.93098
H | -1.51725 | 3.53216 | -0.56697
H | -3.40761 | 4.16069 | 0.87815
---|---|---|---
H | -5.14817 | 3.53118 | 2.52668
H | -3.79291 | -0.56486 | 2.45865
H | -2.23451 | -5.30907 | 2.36836
H | -3.32164 | -1.10653 | -1.13524
H | -4.91037 | -5.09401 | -1.57366
H | -3.71983 | -5.82090 | 0.47488
H | -4.7026 | -2.73148 | -2.37518
H | -5.33604 | 1.16114 | 3.31185
H | 1.94236 | 1.21470 | -2.89288
H | 0.37418 | 2.92307 | -2.01879
H | 0.06310 | -1.81642 | -0.46263
H | 1.78083 | -1.15724 | -2.10829
H | -0.75211 | -4.81665 | 4.26933
H | 0.65005 | -3.21226 | 5.53493
H | -0.29049 | -0.09139 | 2.70747
H | 0.88091 | -0.84874 | 4.74324