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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

Distortion symmetry of the PF5 pseudorotation.The PF5 molecule undergoes pseudorotation from the λ=−1 state (left inset in a, blue atom is P and yellow atoms are F), through a transition state at λ=0 (middle inset in a), to the λ=+1 state (right inset in a). Pairs of static images at λ and −λ are related by a fourfold rotation along the PF1 bond (see inset in b for atom labels). The orange (light blue) arrows represent portions of the pathway going in the direction of increasing (decreasing) λ. The path segment over an infinitesimal path segment, Δλ to the left of λ=0 is transformed under 4 into the path segment −Δλ to the right of λ=0. The displacements of atoms between consecutive images on the left can be related to the displacements of atoms between consecutive images on the right. Thus 4 transforms images between λ and −λ and also the atomic displacements between consecutive images in such a way that the overall distortion path remains invariant. The set of all such operations that leave this pathway invariant form the complete distortion-symmetry group of this pathway, 4mm, where starred symmetries are distortion reversing and are highlighted by blue colour. The blue circles in a are energies from NEB calculations and the black line is the symmetrized fit as guaranteed by the 4mm symmetry group. The PF1 bond length (labelled in the inset in b, which shows the superimposed images of the molecule along the distortion path) as a function of λ for the NEB calculated path is plotted in b as blue circles; it is also guaranteed to be an even function with respect to λ, as is consistent with the symmetrized fit (blue line). Similarly, the PF2 (green circles and line) and PF3 (red circles and line) bond lengths are required by the 4mm symmetry to be mirror images of each other; this is consistent with the plot in b.
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f2: Distortion symmetry of the PF5 pseudorotation.The PF5 molecule undergoes pseudorotation from the λ=−1 state (left inset in a, blue atom is P and yellow atoms are F), through a transition state at λ=0 (middle inset in a), to the λ=+1 state (right inset in a). Pairs of static images at λ and −λ are related by a fourfold rotation along the PF1 bond (see inset in b for atom labels). The orange (light blue) arrows represent portions of the pathway going in the direction of increasing (decreasing) λ. The path segment over an infinitesimal path segment, Δλ to the left of λ=0 is transformed under 4 into the path segment −Δλ to the right of λ=0. The displacements of atoms between consecutive images on the left can be related to the displacements of atoms between consecutive images on the right. Thus 4 transforms images between λ and −λ and also the atomic displacements between consecutive images in such a way that the overall distortion path remains invariant. The set of all such operations that leave this pathway invariant form the complete distortion-symmetry group of this pathway, 4mm, where starred symmetries are distortion reversing and are highlighted by blue colour. The blue circles in a are energies from NEB calculations and the black line is the symmetrized fit as guaranteed by the 4mm symmetry group. The PF1 bond length (labelled in the inset in b, which shows the superimposed images of the molecule along the distortion path) as a function of λ for the NEB calculated path is plotted in b as blue circles; it is also guaranteed to be an even function with respect to λ, as is consistent with the symmetrized fit (blue line). Similarly, the PF2 (green circles and line) and PF3 (red circles and line) bond lengths are required by the 4mm symmetry to be mirror images of each other; this is consistent with the plot in b.

Mentions: We first demonstrate distortion symmetry in a distortion of a simple molecule and show how it can predict relevant property changes. Figure 2 shows the pseudorotation distortion of phosphorus pentafluoride, PF5, a well-known fluxional molecule. The ground-state geometry of PF5 has symmetry. The distortion proceeds by the Berry mechanism2 where the pair of fluorine atoms on the high-symmetry axis move down as another pair of fluorine atoms move up. The structure goes through an intermediate transition state with 4 mm symmetry to a final state with symmetry. Although this distortion is not a rotation, the final state is equivalent to the original structure rotated by 90°, hence the term ‘pseudorotation'. We calculated the MEP using the NEB method32. The MEP represents the set of most likely trajectories that atoms will follow when physically transitioning between these states, and NEB calculations discretize the distortion pathway into a sequence of ‘images'. The highest-energy point on the MEP is known as the transition state and corresponds to λ=0 in Fig. 2. The energy of the transition state corresponds to the activation energy.


The antisymmetry of distortions.

VanLeeuwen BK, Gopalan V - Nat Commun (2015)

Distortion symmetry of the PF5 pseudorotation.The PF5 molecule undergoes pseudorotation from the λ=−1 state (left inset in a, blue atom is P and yellow atoms are F), through a transition state at λ=0 (middle inset in a), to the λ=+1 state (right inset in a). Pairs of static images at λ and −λ are related by a fourfold rotation along the PF1 bond (see inset in b for atom labels). The orange (light blue) arrows represent portions of the pathway going in the direction of increasing (decreasing) λ. The path segment over an infinitesimal path segment, Δλ to the left of λ=0 is transformed under 4 into the path segment −Δλ to the right of λ=0. The displacements of atoms between consecutive images on the left can be related to the displacements of atoms between consecutive images on the right. Thus 4 transforms images between λ and −λ and also the atomic displacements between consecutive images in such a way that the overall distortion path remains invariant. The set of all such operations that leave this pathway invariant form the complete distortion-symmetry group of this pathway, 4mm, where starred symmetries are distortion reversing and are highlighted by blue colour. The blue circles in a are energies from NEB calculations and the black line is the symmetrized fit as guaranteed by the 4mm symmetry group. The PF1 bond length (labelled in the inset in b, which shows the superimposed images of the molecule along the distortion path) as a function of λ for the NEB calculated path is plotted in b as blue circles; it is also guaranteed to be an even function with respect to λ, as is consistent with the symmetrized fit (blue line). Similarly, the PF2 (green circles and line) and PF3 (red circles and line) bond lengths are required by the 4mm symmetry to be mirror images of each other; this is consistent with the plot in b.
© Copyright Policy - open-access
Related In: Results  -  Collection

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Show All Figures
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f2: Distortion symmetry of the PF5 pseudorotation.The PF5 molecule undergoes pseudorotation from the λ=−1 state (left inset in a, blue atom is P and yellow atoms are F), through a transition state at λ=0 (middle inset in a), to the λ=+1 state (right inset in a). Pairs of static images at λ and −λ are related by a fourfold rotation along the PF1 bond (see inset in b for atom labels). The orange (light blue) arrows represent portions of the pathway going in the direction of increasing (decreasing) λ. The path segment over an infinitesimal path segment, Δλ to the left of λ=0 is transformed under 4 into the path segment −Δλ to the right of λ=0. The displacements of atoms between consecutive images on the left can be related to the displacements of atoms between consecutive images on the right. Thus 4 transforms images between λ and −λ and also the atomic displacements between consecutive images in such a way that the overall distortion path remains invariant. The set of all such operations that leave this pathway invariant form the complete distortion-symmetry group of this pathway, 4mm, where starred symmetries are distortion reversing and are highlighted by blue colour. The blue circles in a are energies from NEB calculations and the black line is the symmetrized fit as guaranteed by the 4mm symmetry group. The PF1 bond length (labelled in the inset in b, which shows the superimposed images of the molecule along the distortion path) as a function of λ for the NEB calculated path is plotted in b as blue circles; it is also guaranteed to be an even function with respect to λ, as is consistent with the symmetrized fit (blue line). Similarly, the PF2 (green circles and line) and PF3 (red circles and line) bond lengths are required by the 4mm symmetry to be mirror images of each other; this is consistent with the plot in b.
Mentions: We first demonstrate distortion symmetry in a distortion of a simple molecule and show how it can predict relevant property changes. Figure 2 shows the pseudorotation distortion of phosphorus pentafluoride, PF5, a well-known fluxional molecule. The ground-state geometry of PF5 has symmetry. The distortion proceeds by the Berry mechanism2 where the pair of fluorine atoms on the high-symmetry axis move down as another pair of fluorine atoms move up. The structure goes through an intermediate transition state with 4 mm symmetry to a final state with symmetry. Although this distortion is not a rotation, the final state is equivalent to the original structure rotated by 90°, hence the term ‘pseudorotation'. We calculated the MEP using the NEB method32. The MEP represents the set of most likely trajectories that atoms will follow when physically transitioning between these states, and NEB calculations discretize the distortion pathway into a sequence of ‘images'. The highest-energy point on the MEP is known as the transition state and corresponds to λ=0 in Fig. 2. The energy of the transition state corresponds to the activation energy.

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