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The antisymmetry of distortions.

VanLeeuwen BK, Gopalan V - Nat Commun (2015)

Bottom Line: Distortions are ubiquitous in nature.The symmetry of a distortion pathway is then uniquely defined by a distortion group; it has the same form as a magnetic group that involves time reversal.Distortion symmetry has important implications for a range of phenomena such as structural and electronic phase transitions, diffusion, molecular conformational changes, vibrations, reaction pathways and interface dynamics.

View Article: PubMed Central - PubMed

Affiliation: Materials Research Institute and Department of Materials Science and Engineering, Pennsylvania State University, University Park, Pennsylvania 16802, USA.

ABSTRACT
Distortions are ubiquitous in nature. Under perturbations such as stresses, fields or other changes, a physical system reconfigures by following a path from one state to another; this path, often a collection of atomic trajectories, describes a distortion. Here we introduce an antisymmetry operation called distortion reversal that reverses a distortion pathway. The symmetry of a distortion pathway is then uniquely defined by a distortion group; it has the same form as a magnetic group that involves time reversal. Given its isomorphism to magnetic groups, distortion groups could have a commensurate impact in the study of distortions, as the magnetic groups have had in the study of magnetic structures. Distortion symmetry has important implications for a range of phenomena such as structural and electronic phase transitions, diffusion, molecular conformational changes, vibrations, reaction pathways and interface dynamics.

No MeSH data available.


Related in: MedlinePlus

The consequences of distortion symmetry and balanced forces for NEB calculations.(a–d) Superimposed images along oxygen (red atom) diffusion paths on graphene (grey carbon atoms connected by grey bonds). In a and b, an initial linear path is assumed for the diffusion of a single oxygen atom from right (λ=−1) to left (λ=+1), across a C6 graphene ring. The symmetry of the path in a and b is mm2; the symmetry traps the path and prevents convergence to a minimum-energy pathway (MEP). To break the mm2 symmetry, we perturb this linear path as mm2→2 and mm2→1, respectively, as shown schematically exaggerated as green curves in a and b and indicated by the text in the inset. (c,d) The final paths after NEB relaxation starting from the perturbed paths of a and b, respectively, as indicated by red vertical arrows. The paths c and d have distortion symmetries of 2 and m, respectively. The 2 symmetry continues to trap the transition state (just like mm2 did for the linear path), whereas the initial path with trivial symmetry can correctly converge to a MEP with m symmetry. (e) The calculated energies of the images and the interpolation provided by QE's NEB module45. (f) Results for an example two-dimensional potential energy surface inspired by the above problem, using a simple NEB implementation. The plots are smoothed and rescaled histograms showing the frequency of NEB convergence at a given number of iterations in this example system for 100,000 randomly generated initial paths each with m or with trivial symmetry of 1. The two curves are rescaled to the same maximum height. Symmetrizing using the correct symmetry, m (red curve) reduced the number of NEB iterations needed in 98.97% of test cases. The average reduction was ∼2.3 × as compared with conventional symmetry (blue curve). This demonstrates that distortion symmetry can reduce the number of NEB iterations necessary for convergence.
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f3: The consequences of distortion symmetry and balanced forces for NEB calculations.(a–d) Superimposed images along oxygen (red atom) diffusion paths on graphene (grey carbon atoms connected by grey bonds). In a and b, an initial linear path is assumed for the diffusion of a single oxygen atom from right (λ=−1) to left (λ=+1), across a C6 graphene ring. The symmetry of the path in a and b is mm2; the symmetry traps the path and prevents convergence to a minimum-energy pathway (MEP). To break the mm2 symmetry, we perturb this linear path as mm2→2 and mm2→1, respectively, as shown schematically exaggerated as green curves in a and b and indicated by the text in the inset. (c,d) The final paths after NEB relaxation starting from the perturbed paths of a and b, respectively, as indicated by red vertical arrows. The paths c and d have distortion symmetries of 2 and m, respectively. The 2 symmetry continues to trap the transition state (just like mm2 did for the linear path), whereas the initial path with trivial symmetry can correctly converge to a MEP with m symmetry. (e) The calculated energies of the images and the interpolation provided by QE's NEB module45. (f) Results for an example two-dimensional potential energy surface inspired by the above problem, using a simple NEB implementation. The plots are smoothed and rescaled histograms showing the frequency of NEB convergence at a given number of iterations in this example system for 100,000 randomly generated initial paths each with m or with trivial symmetry of 1. The two curves are rescaled to the same maximum height. Symmetrizing using the correct symmetry, m (red curve) reduced the number of NEB iterations needed in 98.97% of test cases. The average reduction was ∼2.3 × as compared with conventional symmetry (blue curve). This demonstrates that distortion symmetry can reduce the number of NEB iterations necessary for convergence.

Mentions: Next, we demonstrate a symmetry-based approach to testing the stability of a pathway and checking the results of numerical computations for accuracy. This is demonstrated in the NEB calculation of activation energy for an oxygen atom diffusing across a C6 ring on the surface of graphene (Fig. 3a,b). Although not typically thought of as a ‘distortion', this diffusion path is a distortion within the symmetry framework presented in this article. Linear interpolation from the state with oxygen on the right (λ=−1), to the state with oxygen on the left (λ=+1) creates a path with m*m2* symmetry with a high-activation-energy barrier (Fig. 3a,b); this is not an MEP. Typically, only the first and last images are specified when setting up a NEB calculation and a linear path, such as this would be constructed by default by the NEB implementation. For example, this is the case for VTST Tools for VASP and neb.x for Quantum Espresso (QE). Relaxing this path using NEB cannot and does not change the m*m2* symmetry, because every NEB iteration must conserve distortion symmetry (Fig. 3c,d; Supplementary Note 2; Supplementary Figs 2–5), since the forces are balanced by symmetry. Without understanding that the symmetry needs to be broken, one might incorrectly conclude that the activation barrier for oxygen diffusion on graphene is several times larger than it actually is. We can now systematically explore perturbations to this path by using the irreps of m*m2* summarized by the character table given in Table 1.


The antisymmetry of distortions.

VanLeeuwen BK, Gopalan V - Nat Commun (2015)

The consequences of distortion symmetry and balanced forces for NEB calculations.(a–d) Superimposed images along oxygen (red atom) diffusion paths on graphene (grey carbon atoms connected by grey bonds). In a and b, an initial linear path is assumed for the diffusion of a single oxygen atom from right (λ=−1) to left (λ=+1), across a C6 graphene ring. The symmetry of the path in a and b is mm2; the symmetry traps the path and prevents convergence to a minimum-energy pathway (MEP). To break the mm2 symmetry, we perturb this linear path as mm2→2 and mm2→1, respectively, as shown schematically exaggerated as green curves in a and b and indicated by the text in the inset. (c,d) The final paths after NEB relaxation starting from the perturbed paths of a and b, respectively, as indicated by red vertical arrows. The paths c and d have distortion symmetries of 2 and m, respectively. The 2 symmetry continues to trap the transition state (just like mm2 did for the linear path), whereas the initial path with trivial symmetry can correctly converge to a MEP with m symmetry. (e) The calculated energies of the images and the interpolation provided by QE's NEB module45. (f) Results for an example two-dimensional potential energy surface inspired by the above problem, using a simple NEB implementation. The plots are smoothed and rescaled histograms showing the frequency of NEB convergence at a given number of iterations in this example system for 100,000 randomly generated initial paths each with m or with trivial symmetry of 1. The two curves are rescaled to the same maximum height. Symmetrizing using the correct symmetry, m (red curve) reduced the number of NEB iterations needed in 98.97% of test cases. The average reduction was ∼2.3 × as compared with conventional symmetry (blue curve). This demonstrates that distortion symmetry can reduce the number of NEB iterations necessary for convergence.
© Copyright Policy - open-access
Related In: Results  -  Collection

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getmorefigures.php?uid=PMC4660051&req=5

f3: The consequences of distortion symmetry and balanced forces for NEB calculations.(a–d) Superimposed images along oxygen (red atom) diffusion paths on graphene (grey carbon atoms connected by grey bonds). In a and b, an initial linear path is assumed for the diffusion of a single oxygen atom from right (λ=−1) to left (λ=+1), across a C6 graphene ring. The symmetry of the path in a and b is mm2; the symmetry traps the path and prevents convergence to a minimum-energy pathway (MEP). To break the mm2 symmetry, we perturb this linear path as mm2→2 and mm2→1, respectively, as shown schematically exaggerated as green curves in a and b and indicated by the text in the inset. (c,d) The final paths after NEB relaxation starting from the perturbed paths of a and b, respectively, as indicated by red vertical arrows. The paths c and d have distortion symmetries of 2 and m, respectively. The 2 symmetry continues to trap the transition state (just like mm2 did for the linear path), whereas the initial path with trivial symmetry can correctly converge to a MEP with m symmetry. (e) The calculated energies of the images and the interpolation provided by QE's NEB module45. (f) Results for an example two-dimensional potential energy surface inspired by the above problem, using a simple NEB implementation. The plots are smoothed and rescaled histograms showing the frequency of NEB convergence at a given number of iterations in this example system for 100,000 randomly generated initial paths each with m or with trivial symmetry of 1. The two curves are rescaled to the same maximum height. Symmetrizing using the correct symmetry, m (red curve) reduced the number of NEB iterations needed in 98.97% of test cases. The average reduction was ∼2.3 × as compared with conventional symmetry (blue curve). This demonstrates that distortion symmetry can reduce the number of NEB iterations necessary for convergence.
Mentions: Next, we demonstrate a symmetry-based approach to testing the stability of a pathway and checking the results of numerical computations for accuracy. This is demonstrated in the NEB calculation of activation energy for an oxygen atom diffusing across a C6 ring on the surface of graphene (Fig. 3a,b). Although not typically thought of as a ‘distortion', this diffusion path is a distortion within the symmetry framework presented in this article. Linear interpolation from the state with oxygen on the right (λ=−1), to the state with oxygen on the left (λ=+1) creates a path with m*m2* symmetry with a high-activation-energy barrier (Fig. 3a,b); this is not an MEP. Typically, only the first and last images are specified when setting up a NEB calculation and a linear path, such as this would be constructed by default by the NEB implementation. For example, this is the case for VTST Tools for VASP and neb.x for Quantum Espresso (QE). Relaxing this path using NEB cannot and does not change the m*m2* symmetry, because every NEB iteration must conserve distortion symmetry (Fig. 3c,d; Supplementary Note 2; Supplementary Figs 2–5), since the forces are balanced by symmetry. Without understanding that the symmetry needs to be broken, one might incorrectly conclude that the activation barrier for oxygen diffusion on graphene is several times larger than it actually is. We can now systematically explore perturbations to this path by using the irreps of m*m2* summarized by the character table given in Table 1.

Bottom Line: Distortions are ubiquitous in nature.The symmetry of a distortion pathway is then uniquely defined by a distortion group; it has the same form as a magnetic group that involves time reversal.Distortion symmetry has important implications for a range of phenomena such as structural and electronic phase transitions, diffusion, molecular conformational changes, vibrations, reaction pathways and interface dynamics.

View Article: PubMed Central - PubMed

Affiliation: Materials Research Institute and Department of Materials Science and Engineering, Pennsylvania State University, University Park, Pennsylvania 16802, USA.

ABSTRACT
Distortions are ubiquitous in nature. Under perturbations such as stresses, fields or other changes, a physical system reconfigures by following a path from one state to another; this path, often a collection of atomic trajectories, describes a distortion. Here we introduce an antisymmetry operation called distortion reversal that reverses a distortion pathway. The symmetry of a distortion pathway is then uniquely defined by a distortion group; it has the same form as a magnetic group that involves time reversal. Given its isomorphism to magnetic groups, distortion groups could have a commensurate impact in the study of distortions, as the magnetic groups have had in the study of magnetic structures. Distortion symmetry has important implications for a range of phenomena such as structural and electronic phase transitions, diffusion, molecular conformational changes, vibrations, reaction pathways and interface dynamics.

No MeSH data available.


Related in: MedlinePlus