21

Quasicrystals were known well before 2015. The recent developments is in showing that such crystals exist naturally, rather than just in synthesized samples. The first (nowadays generally accepted) clear demonstration of an aperiodic crystal was published in 1984. Examples of such crystals remain very rare compared to periodic crystals, however. So rare in ...


16

Two-band model for graphene. To simplify the discussion of Dirac points, it is sufficient to consider a nearest-neighbor tight-binding 2-band model for graphene. This is a spinless model because spin-orbit is negligible in graphene. The Bloch Hamiltonian is: $$ \hat{H}(\mathbf{k})= \begin{pmatrix} 0 & h(\mathbf{k}) \\ h^{\dagger}(\mathbf{k}) & 0 \end{...


15

Background theory. In the harmonic approximation, the potential energy surface (PES) is expanded about an equilibrium point to second order, to obtain the Hamiltonian: $$ \hat{H}=\sum_{p,\alpha}-\frac{1}{2m_{\alpha}}\nabla_{p\alpha}^2+\frac{1}{2}\sum_{p,\alpha,i}\sum_{p^{\prime},\alpha^{\prime},i^{\prime}}D_{i\alpha;i^{\prime}\alpha^{\prime}}(\mathbf{R}_p,\...


14

The question is too broad to be answered directly so I will provide a somewhat general scheme. Basically in an integral like $$ \int d\mu A B C $$ one would seek to expand each part in irreducible representations of a given group, say for instance \begin{align} B=\frac{1}{\vert \mathbf{r}-\mathbf{r}^\prime\vert} =\frac{1}{r} \sum_{\ell} \left(\frac{r'}{r}\...


12

The answer of @ProfM is already very complete, but I wanted to tackle your question from a more practical point of view. The presence of imaginary frequencies indicate that there are atomic positions which are more energetically favorable at the ground state. So, the concept of "following" a mode means condensing it onto the reference structure, ...


11

I will try to outline this in simple terms. There are certainly more rigorous explanations. The high-symmetry points in the Brillouin zone are those that remain physically identical when certain symmetry operations of the point-group are applied. Therefore, we notice that the first and higher derivatives of the dispersion relation have the same magnitude in ...


11

Jmol/JSmol Did you consider Jmol/JSmol? It is freely available for Windows / Mac / Linux, scriptable, may export what is being displayed in formats relevant to chemistry (e.g., .pdb, .sdf), as image (e.g., .png, .pngj, [animated] .gif) and already is in use to teach symmetry in molecules and crystals. As an example, the interactive compilations by Symmetry@...


10

A Weyl point is a crossing of two bands. A two-band crossing can be described using the general Hamiltonian: $$ \hat{H}(\mathbf{k})=d_0(\mathbf{k})+d_1(\mathbf{k})\sigma_1+d_2(\mathbf{k})\sigma_2+d_3(\mathbf{k})\sigma_3 $$ where $\sigma_i$ are the Pauli matrices. The eigenvalues are given by $E_{\pm}=d_0\pm\sqrt{d_1^2+d_2^2+d_3^2}$, so that a band crossing ...


10

Yes, broken symmetry solutions do break time-reversal symmetry, and that's one of the reasons why they are unphysical when the total magnetic moment of the system is 0 (although they are physical when the total magnetic moment is not 0, due to spin polarization). The reason is that, suppose you have a spin polarized $M_S=0$ state, then if you flip the spin ...


9

From that reference, what we got is the bond length from the atomic positions. One easy way is to look for a CIF file with the crystal information. From the CIF49801 bellow, we get the following info: (label type_symbol symmetry_multiplicity Wyckoff_symbol fract_x fract_y fract_z B_iso_or_equiv occupancy) Mo1 Mo4+ 2 c 0.3333 0.6667 0.25 . 1. S1 ...


9

First, you have to transform all the basis functions into the irreducible representations (irreps) of the point group of the molecule. You can do this with standard projection formulas. Once, you know the irreps of the basis functions, you have to look at the product table of the point group to find out whether the product of those four basis functions ...


9

A possible tool is the ASE python package (open-source): https://wiki.fysik.dtu.dk/ase/ase/cluster/cluster.html Another tool is materials studio software. Unfortunately, it is not free.


9

Yes, nematic order is an alignment of rotational degrees of freedom without spatial structure. One good example of a nematic phase occurs in a liquid crystal composed of elongated molecules. The molecules can align their long axes (breaking the rotational symmetry) without forming any lattice or long-range correlations in their spatial positions. If you were ...


8

Nanocut From its site: Nanocut is a program designed to cut out various objects from three dimensional crystal structures. It is aimed to be helpful when creating geometry input for atomistic simulations. Currently it can create following objects: Spherical cluster Polyhedral cluster Cylindrical cluster Spherical wire (1D periodic) Polyhedral wire (1D ...


8

If you know which atoms correspond to each other in the two structures, you can use a structural superposition method. Least-squares superposition methods find the rotation matrix and translation that minimizes the RMSD between given points. There are a few well-established methods. Recently, I had to use one and I picked QCP (because it comes with BSD-...


8

Another way to explicitly break time-reversal symmetry is by applying circularly polarized light. Under time reversal, left circularly polarized light transforms to right circularly polarized light, and vice versa. One way to see this is to note that circularly polarized light is an eigenstate of the spin angular momentum operator, which is odd under time ...


8

I will first take a generic view-point and then quote some examples in condensed matter & materials modeling. Time-reversal symmetry is one of two discrete symmetries usually discussed in the context of condensed-matter, the other being Parity(Inversion). The simplest way in which this concept is presented is a transformation : $ t \rightarrow -t $ . ...


8

Let's give it another go: In the monolayer you would place the inversion centre on the green atom. But this would reverse the direction of the trigonal prism formed by the yellow atoms. Hence, there can't be an inversion centre. In the bilayer the inversion centre is between the layers. Then the direction of the trigonal prism is reversed but it is also ...


8

The 1-dimensional unitary group U(1) simply corresponds to all complex numbers with a modulus of 1. This is isomorphic to the special orthogonal group SO(2), which corresponds to all real 2x2 rotation matrices. This is the case because because any element of U(1) is uniquely defined by its complex phase (going from 0 to 2pi) and this can be mapped uniquely ...


8

The typical calculation flow for the density of states is: Geometric relaxation to obtain the lowest-energy structure (CONTCAR) [1relax] (For your system, you should do the spin-polarized calculation by setting ISPIN=2). Using the relaxed structure to perform the electronic self-consistent calculation to obtain converged charge density [2scf]. Using the ...


7

Based on the comments I shared above, it seems that the quality of the charge density is fine. It is just the projection onto atoms that may not be done correctly. If this is indeed the case, then you don't need to re-relax the structure (Step #1). You should be able to read in the charge density and set ISYM=0 (symmetry disabled) and be okay. Disclaimer: ...


6

This is standard output of bands.x which is a post-processing code to plot band structures used in QE under the PP folder.


6

There is no guaranteed increase in performance if you specify a non-zero value for ibrav. There is an additional layer to this - Whether the atomic positions are specified in cartesian or crystal coordinates (sometimes some symmetries can be missed out on QE). On both points, you would need to run a test calculation and see how many symmetries QE detects. ...


6

When your lattice is primitive you have only the (0,0,0)+ set; when your lattice has some kind of centering (body- or face-centering) other sets are present, such as (1/2, 1/2, 1/2)+ or (1/2, 1/2, 0)+ It's not clear to me what you write. In the first page of the Internationl Tables you find all the symmetry operations that are listed with Roman numerals (1),(...


6

It seems you are looking for a package to compute the irreducible representation of electronic states computed by VASP. This has been recently developed: J. Gao J, Wu Q, Persson C, Z. Wang. Irvsp: to obtain irreducible representations of electronic states in the VASP. Comput. Phys. Comm. 261, 107760 (2021). https://doi.org/10.1016/j.cpc.2020.107760. And the ...


5

To start with, while it's not as necessary, you can apply symmetry arguments to one-electron integrals. Consider $\langle\mu|O_1|\nu\rangle$, where $O_1$ is some one-electron operator. If the molecule has some point group symmetry, we can form basis functions/operators that are irreducible representations of the group. Once we have the functions expressed in ...


5

CCDC Mercury The Cambridge Crystallographic Data Centre (CCDC) provides a free-to-use visualization program Mercury that offers a wide variety of features for crystal structures including, perhaps not surprisingly crystallographic symmetry. The trick, of course, is to have appropriate crystal structures to illustrate points. CCDC does provide a free teaching ...


4

Boy, you're not starting off easy. Proper implementation of symmetry is quite a job, especially since most systems of interest nowadays have no symmetry. For a reference, you can look at e.g. Dovesi's work on the use of symmetry in CRYSTAL, which is a periodic Hartree-Fock code using Gaussian orbitals. Symmetry is much more important in the periodic case, ...


4

Assuming that the structure starts from one symmetry and you distort it into another, VASP will want to find the symmetry it expects and use it. One straightforward way to fix this problem while also increasing the chance of finding the actual minima is to break the symmetry yourself. One way to do this is to randomly displace atoms by a small amount. This ...


4

These are some general comments about modelling disorder which I think address your question, but note I cannot provide specifics about the codes you are using. Imagine a simple configuration with only two sites for the discussion: Both atoms are of the same type (blue) so that this configuration has a mirror plane down the middle of the two atoms (dashed ...


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