The antisymmetry of distortions

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.


Supplementary Note 1 -Connection between distortion symmetry and representation analysis
We will further establish the connection between distortion symmetry and representation analysis with the example of atomic displacements of a water molecule, H2O, with conventional symmetry mm2 (C2v) as depicted in Supplementary Figure 1a. Using the irreducible representations of mm2, we can construct a symmetry-adapted basis for the atomic displacements of H2O. Our chosen basis is depicted in Supplementary Fig. 1a. Each of our basis modes correspond to a distortion path that is constructed by linearly scaling the displacements by λ, as depicted in Fig. 1h An improper ferroelectric antiferromagnet, YMnO3, distorting from one ferroelectric domain,  + at =-1 to the opposite domain  -at =+1 exhibits a distortion symmetry of P63/m*cm (Fig. 5c). This is also effectively the distortion implied by Fennie and Rabe 1 in studying the P63/mmc parent structure. It is also an interesting case in terms of the relationship between distortion symmetry and representation analysis. Fennie  ). This is an advantage of distortion symmetry over conventional representation analysis. We also note that giving the primary irrep of a distortion is not equivalent to giving a distortion group, just as is the case with magnetic symmetry 2 , because K3 actually corresponds to three different types of distortion symmetry: , , and ̅ depending on the direction of the order parameter.
Finally, we note one way in which the representations of distortion groups can be applied. The path depicted in Fig. 3a and in Supplementary Fig. 2 for an oxygen atom diffusion across a C6 ring have m*m2* distortion symmetry. The character table for the m*m2* group is given in Table 1. If we consider perturbations that displace the oxygen atom and not the carbon atoms, the 21 dimensional space of perturbations of the path (7 images across the path times 3 degrees of freedom for oxygen per image) carry a representation of m*m2* with the following irreducible components: 7 1 + 3 2 + 7 3 + 4 4. Using a symmetry-adapted basis, this can be decomposed into four symmetry invariant subspaces: where x(L) is the unit displacement of the oxygen atom in the image at =L along x (or y or z).
These four subspaces correspond to symmetry breaking to m*m2*, 2*, m, and m* respectively (these are the kernels of each irrep). Just as an ordinary symmetry-adapted basis for a static structure would put the force constants matrix in a block diagonal form, this basis will do so as well for the generalization force constants matrix that includes the nudged elastic band forces on the path; this would have blocks of 7, 3, 7, and 4 rows corresponding to each of the four subspace specified above. For this particular path, the first three blocks should be positive definite (stable, similar to having positive squared frequency with phonons of static structures).
The final block, corresponding to the 4 irrep, has one or more negative eigenvalues, indicating instability, that would then lead to a minimum energy pathway.

Supplementary Note 2: Additional analysis of NEB calculations for O diffusion across C6 ring.
The initial path created by linearly interpolating between and has m*m2*, or more specifically mx*my2z* symmetry where the subscripts represent axis associated with the operation: mx* is a starred mirror whose normal is along x, my is a mirror whose normal is along y, and 2z* is a starred two-fold axis along z (see compass on lower left in Supplementary Figure   2). Applying Neumann's principle to the force on the oxygen atom gives the following results: • my Fy(λ) = -Fy(λ) = Fy(λ), hence Fy() = 0 (see red in Supplementary Fig. 2) is an even function of λ (see green) Clearly, the forces on both the initial and converged path are consistent with these symmetry predictions in Supplementary Figure 2. The forces of one iteration are used to update the positions for the next iteration, thus guaranteeing that the mx*my2z* symmetry cannot be broken in any subsequent iteration.
In Supplementary Figure 3, we deliberately break the mx*my2z* symmetry with a sinusoidal perturbation (the green curve is exaggerated; the maximum displacement was 0.1 Angstrom).
Note that the forces on the initial path are similar to the unperturbed case, because the perturbation is small, but slightly break the previous symmetry. The perturbation is such that the path retains 2z* symmetry. Again applying Neumann's principle to the force on the oxygen atom gives the following results: is an even function of λ (see green) Again, this is consistent with the forces on both the initial and converged path in Supplementary Fig. 3. The 2z* symmetry in the initial guess prevents NEB iterations from finding a significantly lower transition state (TS).
In Supplementary Figure 4, we deliberately break the mx*my2z* symmetry to trivial symmetry (the green curve is exaggerated; the maximum displacement was 0.18 Angstrom). Note that the forces on the initial path are similar to the unperturbed case, because the perturbation is small, but slightly break all previous symmetry. NEB iterations drive the path to the much lower energy m* path seen in Figure 3d of the main text.
Supplementary Figure 5 shows two other ways of breaking m*m2* symmetry. We expect that NEB would drive the m* initial path ( Supplementary Fig. 5a) to the same low energy m* path as seen in Figure 3d of the main text. The m initial path ( Supplementary Fig.   5b) should not be able to converge to the low energy m* path because m is not a subgroup of m* and NEB iterations must conserve distortion symmetry.
Supplementary Note 3: Simple example to demonstrate the effect of distortion symmetry on

NEB convergence
In the provided Mathematica Notebook file, Simple_NEB.nb, an example 2D potential energy surface (PES) is described and a simple implementation of the nudged elastic band (NEB) method is included. This PES is given as: (1) "Simple_NEB.nb" contains dynamic and interactive plots that show what happens to the initial guess path as the NEB method iterates. Supplementary Figure 6 shows an example of what it might look like starting from a path with trivial symmetry after 25 iterations.
Using this implementation and PES, we tested the idea that applying distortion symmetry should result in a more rapid convergence of the NEB code. Our implementation was based on the explanation of the Nu ge lastic an metho given y o nsson et al. 3 . Starting from a straight path, we generated 100,000 initial paths with trivial symmetry and 100,000 initial paths with m* symmetry, which are the conventional symmetry and distortion symmetry, respectively, of the MEPs in this example. The details of how these were randomly generated are in the Simple_NEB.nb file. For each initial path, we ran our NEB implementation until the forces fell below a chosen convergence threshold. The results are summarized in Fig. 3f. Note that convergence is considerably more rapid with distortion symmetry in this example. The conventional symmetry paths typically took more than twice as long; the average number of iterations was about 443.8 for conventional symmetry and 190.8 for distortion symmetry.
Symmetrizing using the correct distortion symmetry reduced the number of NEB iterations needed in 98.97% of our test cases and by a factor of 2.3 on average. Also note that if a straight path was provided as the initial guess, convergence to a MEP would not be possible in this example. Consequently, at least for this example, understanding distortion symmetry is crucial for achieving good results.

Supplementary Note 4: Examples of distortion symmetry in quartz
Supplementary Figure 7 shows the example of quartz, a common crystal that is found in left-and right-handed configurations, and is commonly used in watches and clocks as a crystal oscillator by using its piezoelectric effect. Trigonal α-quartz transforms into hexagonal βquartz at 573°C, into hexagonal β-tridymite at 870°C and to cubic β-cristobalite at 1470°C. If βquartz is considered a parent, the distortion to α-quartz has P62*22* symmetry for left-handed quartz and P64*22* symmetry for right-handed quartz (this is the distortion depicted in Fig. 5a Figure 7b shows that, as with the examples given in the main text, the energy of the P64*22* distortion is symmetric with respect to λ due to the starred symmetry.  4 3,4,5,6 Stacking faults in olivine. All figures that should be symmetric due to starred symmetry show some degree of asymmetric error, Fig. 6 in particular. In part, this is due to having an even number of image. 5 3 Both Fig. 3a and 3b should be symmetric due to starred symmetry. Fig. 3a is a particularly extreme example of asymmetry error. The path taken in Fig. 3b is unstable and is balanced by symmetry as discussed in the main text. NEB study of diffusion of tungsten adatom on tungsten cluster surface. Figure 3 shows symmetric energy due to starred symmetry. 10 1c and 3a-d Energy versus slip in magnesium alloys. Energy is symmetric for prismatic slip in Figure 1c due to starred symmetry. Symmetric energy in Figure 3a and 3b due to starred symmetry. The restoring force is antisymmetric in Figure 3c and 3d due to starred symmetry. 11 6

Supplementary
Diffusion path of MgO/O3 cluster in MgO. Figure 6 is symmetric due to starred symmetry, but has small asymmetric errors. 12 2,3,4, and 5b Constrained nudged elastic band calculation of the Peierls barrier with atomic relaxations. There are small but apparent deviations from symmetry (numerical errors). Fig. 5b shows only half of the pathway, due to symmetry. 13 7 Stacking faults in Ni. Figure 7b (Theta=0.0) and 7c(Theta=0.0) are antisymmetric due to symmetry. 14 4a N diffusion in fcc Fe. Fig. 4a is symmetric due to starred symmetry. 15 3 Carbon diffusion in supersaturated ferrite. Figure 3 (xc = 0) is symmetric due to starred symmetry. 16 2,3 Ad-atom diffusion on copper surface. Figs. 2 and 3 are symmetric due to starred symmetry. 17 N/A Cu diffusion in cristobalite. None of these figures are symmetric due to starred symmetry. Some are nearly symmetric barriers, but not exact. This is an example that shows that apparent symmetry in energy plots is not necessarily due to distortion symmetry (and therefore not exact). 18 2,3 An ab initio study of the effect of charge localization on oxygen defect formation and migration energies in magnesium oxide. Figs. 2 and 3 are symmetric due to starred symmetry. 19 3 Vacancy diffusion pathway, CI-NEB. Fig. 3 is symmetric due to starred symmetry. 20 4 H-atom relay reactions in real space. Fig. 4 is symmetric due to starred symmetry. 21 2 Stacking fault energy. Figure 2 is symmetric due to starred symmetry. 22 3,4,5 Stacking fault energy in Mg and Mg-Y alloys. Figs. 3, 4, and 5 are symmetric due to starred symmetry. 23 4 Diffusion in aluminum. Figure 4 is symmetric due to starred symmetry. 24 N/A Lithium intercalation in TiO2-B. Shows the result of reversed pathways, e.g. Fig. 5. 25 2 Arrows depicting PF5 pseudorotation in Fig. 2. As noted in the main text, these arrows are related by symmetry. 26 2a Diffusion of oxygen in La2CoO4. Figure 2b shows Initial to Saddle images but not Saddle to Final, implying the intuitive application of symmetry. 27 7 Diffusion in Li3N. Pathways depicted in Fig. 6. Figure 7 is symmetric due to starred symmetry. 28 3 Li diffusion in TiO2. Figure 3 is symmetric due to starred symmetry. 29 5 Li diffusion on B8C24 and B24C12. The path on B8C24 has starred symmetry and thus the energy plot in Fig. 5c should be symmetric but is not, presumably due to errors. The path on B24C12 does not have starred symmetry and the asymmetry of Fig. 5d is consistent with this. 30 3e O diffusion in La2CoO4. Figure 3e is symmetric due to starred symmetry. 31 5 NEB, molecular transitions. Figure 5 is symmetric due to starred symmetry. 32 N/A Oxygen diffusion in lanthanum silicate. Very interesting pathway from O5-0 to symmetry equivalent O5-0 site depicted in Fig. 9 and Fig.  10. Shows how equivalence of initial and final states (as noted in the captions) does not guarantee symmetry. This can be seen from the labels given along the O5-0 to O5-0 pathways, e.g. in Fig. 10 the pathway goes from O5-0 to O5-s2 to O5-s1 to O5-0. Because O5-s2 and O5-s1 are inequivalent, it is impossible to superimpose this pathway with its reverse and therefore there is no starred symmetry. 33 7 Migration pathway for Li in LiFePO4. Fig.5 also depicts two different kinds of hops that occur in LixCoO2, we note that the first (Fig. 5a) has starred symmetry but the other does not. 34 2 Diffusion of hydrogen atom on MoS2. Figure 2 is symmetric due to starred symmetry. 35 N/A Fig. 6 shows another example of a minimum energy pathway that is not superimposable with its reverse and therefore does not contain starred symmetry. 36 5a,b Various migration pathways in Li2MnO3. Path 3 and 4 have starred symmetry, the rest do not. 37 3,4 Mg diffusion in Mg0.5FeSO4F. L1 and L2 pathways (Fig. 3) both have starred symmetry. Small deviations from this symmetry are apparent in Fig.4, e.g. the 5th and 13th points of Fig. 4a should be at the same height. Presumably this is due to errors in the calculations. 38 4 Na migration path on MoS2. Very clear example with superimposed images showing the pathway. Both pathways in Figure 4 have starred symmetry. 39 7 Li diffusion in LiVOPO4. Figure 7 is symmetric due to starred symmetry. 40 3 Na diffusion in NaCoO2. Figure 3 is symmetric due to starred symmetry. 41 1c,d and 3 Figure 3 shows the expected diffusion path of Li in LiFePO4. Depicted as continuous motion. Figure 1c, Fig. 3a,d, and e are symmetric due to starred symmetry. 43 2,4,6 Figs. 2, 4, and 6 are symmetric due to starred symmetry. 44 4b Pathways in Fig. 4a and c have equivalent endpoints, but do not have starred symmetry (the approximate symmetry of the energy in Fig. 4c is not due to starred symmetry). The pathway in Fig. 4b should have starred symmetry but there are clear deviations in the energy (note 2nd and 2nd to last points). The starred symmetry suggests this is an error, maybe due to using an even number of images. 45 2,3,4 Adatom Diffusion on Fe surfaces. Small asymmetry in Fig 2 due to choosing an even number of points. 46 8 Diffusion in LaCoO3. Fig. 8c is interesting in the context of distortion symmetry because μA and μB are related by a starred operation and so the should be opposite with respect to the reaction coordinate and meet in the middle. This is consistent with the figure except they do not meet in the middle. This is either an error or an interesting case of magnetism breaking starred symmetry. 47 2 MedeA Transition State Search Datasheet. Shows an example of applying the MedeA Transition State Search module to the migration of a Pd4 cluster on a MgO(001) surface. Figure 2 is symmetric due to starred symmetry. 48 2d,e (red and blue),f (red and blue) Diffusion in Li4Ti5O12, Li7Ti5O12, and Na6LiTi5O12. Fig. 2d,e (red and blue),f (red and blue) are symmetric due to starred symmetry. 49 Fig. 3 is symmetric due to starred symmetry. 50

(red)
Stacking faults in Ni, Al, and Cu. Fig. 3 (red) is symmetric due to starred symmetry. 51 10 Pseudo-rotation of 1,2,3-F3C6H3 -. In Figure 10b, starred symmetry has the consequence of making F2 symmetric and F1 and F3 mirror images. This is similar to the PF5 pseudorotation and bond lengths example given in the main text of our work. 52

5
Diffusion of Na and Li ions in Na1.5VPO5F0.5 and LiNa0.5VPO5F0.5. Fig. 5b and 5d give very clear depictions of the pathways. Fig. 5c and 5e are symmetric due to starred symmetry. 53 1 Fig. 1 is symmetric due to starred symmetry. 54 5 Fig. 5 is symmetric due to starred symmetry. 55 3 Diffusion in NiAl3. Fig. 3 is a comparison of NEB and constrained atom (CA) methods. Both show small asymmetry errors. Fig. 3 is symmetric due to starred symmetry. 56 1 Diffusion in Ti. Figure 1 is symmetric due to starred symmetry. 57 1,2 Li diffusion in olivine phosphates (FePO4 and LiFePO4). All paths with plotted energies are symmetric due to starred symmetry. 1 2,4 The distortions in this paper come from following the normal modes of a parent structure. Figure 4 is antisymmetric because of the starred symmetry. 58 3a Figure 3a is symmetric due to starred symmetry.