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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{... 16 There's three scenarios that come to my mind, for when ab initio methods get abandoned: The cost becomes prohibitive (e.g. too many electrons) The insight is lost It is simply not required for what we want to do Prohibitive cost: If solving the Schrödinger equation (for example) is no longer possible, we may very well still wish we could solve the ... 15 Alot Optimizations are all about finding the minimum in something. Typically in geometry optimization, it is about finding the minimum in energy. At a minimum the derivative of energy with respect to changing position should be zero, or better put, the jacobian should be positive definite. Your question is about bulk systems, however, single molecules are ... 14 I can't answer in the context of DFT/VASP, where these terms might have specific uses, but I can offer some general comments. These terms are not necessarily mutually exclusive: Total energy probably refers to the combined kinetic and interaction energy E=K+P... resulting from summing/integrating over the whole Hamiltonian. In DFT this would be the ... 12 I think the way this question is asked is a little too simplistic. In order to execute a computational project, there is always more than one calculation required. Even if you are happy with the lowest level of theory (say, B3LYP/6-31G*), it does not mean that any package that lists B3LYP in the list of the available features would be useful. Possible ... 11 First of all, I assume that you are taking VASP to perform your calculation. Secondly, I assume that your structural defect is taking a Hf atom from your structure. (You can deal with substitutional doping with similar logic.) Thirdly, for the HfS2 monolayer, there are two phases, namely T-phase and H-phase. The T-phase monolayer was fabricated in the ... 11 A locally interacting system displaying a continuous phase transition belongs to a universality class that is determined solely by the system symmetries and dimensionality. Drawing from Wikipedia's list (itself mostly based on Ódor's paper) and this answer from Physics SE, here's a partial list of universality classes and critical exponents: \begin{array}{|... 11 First of all, let me emphasize that it is more appropriate to speak of KS equations (plural), which you correctly denoted by an index i in your post. This index goes over all KS orbitals (i.e. single-particle wavefunctions) of the system. Additionally, as you mentioned, these equations have the same form as the single-particle Schrödinger equation. And ... 11 Here I assume that we are focus on the condensed matter system which is composed of nuclei and electrons with the fundamental force: Coulomb Force. Furthermore, I just discuss the electron degree of freedom. A naive (wrong) answer: The interaction gets stronger when electron density is higher. The Coulomb interaction is proportional to \dfrac{1}{r}, ... 9 Realizing this Hamiltonian in a natural material I cannot imagine a material in which all nearest-neighbor spin-spin interactions can be adjusted arbitrarily at the same time. Spin-spin couplings that are stronger when the spins are closer together, and weaker when the spins are farther apart, can be adjusted by moving the spins relative to each other; so ... 9 The bulk band structure of a topological insulator would look just like any other insulator, with the Fermi level in the gap between the valence and conduction bands. If your band structure includes the surface states, then you would see some states in the gap that cross the Fermi level. There's an example on Wikipedia: An idealized band structure for a ... 9 The review that @Anyon cited focuses on the use of fractal geometry to classify and model the structure of disordered materials, e.g. structures synthesized by the sol-gel method. The computational work is nicely summarized in the following figure taken from the paper: Here "Reaction-Limited", "Ballistic" and "Diffusion-Limited"... 9 eDMFT Link to website: eDMFT Developer: Kristjan Haule at Rutgers University Main features from their website: DFT+DMFT derived from the stationary Luttinger-Ward functional. Local correlated orbitals: real space (rather the usual choice of Wannier space). Exact double counting for DFT+DMFT from this paper. Impurity solvers: continuous time quantum Monte ... 9 Actually, if you allow for quasi-2D systems, graphene has had a recent renaissance starting with the experimental discovery of correlated states in "magic angle" twisted bilayer graphene, which was originally predicted by Bistritzer and MacDonald. By stacking two graphene layers with a relative twist, new structures with very long periodicity can ... 9 As you note, the interacting electrons and the Kohn-Sham non-interacting electrons have the same density. How is this possible when the Hamiltonians for the two systems are so different? The answer is that the Kohn-Sham potential—the potential felt by the non-interacting electrons—is constructed very carefully. Namely, if we want to remove electron-electron ... 9 Here is my summary of definition of strong correlated systems in different context. For ab initio electronic structure Hamiltonian The eigen value of the system can not be well approximated by single slater determinate approach (Hartree Fock or DFT). These system normally has large coefficient in there configurational interaction (CI) expansion. For some ... 8 DCore: integrated DMFT software for Correlated electrons DCore is aimed at model calculations and ab-initio calculations by the dynamical mean-field theory (DMFT). This package consists of programs with text-based and hdf5-based interface. These programs enable users to perform DMFT calculations and analyze results without writing computer code. Website:... 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 One can also pose the opposite question, which may be more interesting: what significant matter modelling methods are implemented in open source software, for which there is no commercial alternative? Anna's answer above had a lot of important considerations, but also this reverse question is important to keep in mind; commercial software is not always free ... 8 I won't go over what Charlie Crown already described in his answer, but if you don't have experimental data, you can try starting off with the experimental structure of a similar compound that has the same crystal structure (i.e. ZrO2 and HfO2). If there are several options, try them and choose the relaxation that gives you the lowest energy. Just to add a ... 8 You got it completely correct when you said: "it seems that the definition of it is ambiguous and sometimes inconsistent in different field of study." As the other answers show, it's a bit of a "loose" term that can be used in slightly different ways depending on the sub-field of Matter Modeling in which the term is being used. This ... 8 For this type of calculation, I find MATLAB/Octave to be easier, at least for demonstrating what to do. You can add the np. everywhere afterwards if you want to use Python. Assuming you have the following matrices already defined in your workspace: \eta_z = eta \tau_x = taux \tau_y = tauy \tau_z = tauz m_z = tauz \sigma_z = sigmaz, and the ... 8 One way of determining this is using the projected density of states (P-DOS) This resolves the DOS into specific orbitals thereby allowing you to discretize each orbitals weight for a specific energy.  \mathrm{PDOS}_\nu(E) = \sum_i \psi^*_{i,\nu} [\mathbf S | \psi_{i}\rangle]_\nu D(E-\epsilon_i)  Note here that |\psi_i\rangle is the ith eigenvector and ... 7 USPP is not a format of pseudopotential. It is a type of pseudopotential that enables you to work with lower cutoffs. On the other hand, upf is a popular file extension for pseudopotentials. If your question is regarding the availability of fully relativistic Ultrasoft PPs, they aren't very popular but PSLibrary offers that support. For Norm-conserving and ... 7 VASP can't deal with the external magnetic field, but VASP can deal with the spin-polarized and noncolinear magnetism. You should read carefully the cited paper. As far as I have known, MnBi2Te4 is AFM semiconductor. You can do the spin-polarized calculation and noncolinear calculation for it with VASP. 7 Questaal Questaal is a first-principles DFT solver based on the LMTO basis (different from Wien2k answered by @ProfM), which is interfaced with the Continuous-Time Quantum Monte Carlo solver, developed by K. Haule and coworkers, for DMFT. Website: https://www.questaal.org/about/questaal/ Description from official website: When localized electronic ... 7 You can just exchange the \mu,\nu indices to verify the antisymmetry:$$ \Omega_{n,\mu\nu}(\mathbf{k})=\partial_{\mu}A_{n\nu}(\mathbf{k})-\partial_{\nu}A_{n\mu}(\mathbf{k})\\ \Rightarrow \Omega_{n,\nu\mu}(\mathbf{k})=\partial_{\nu}A_{n\mu}(\mathbf{k})-\partial_{\mu}A_{n\nu}(\mathbf{k}) = - \left( \partial_{\mu}A_{n\nu}(\mathbf{k})-\partial_{\nu}A_{n\mu}(\...

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Question 1. eq. 3.102 defines a matrix in "band-space", e.g. it's NxN where N = number of bands being considered. The right hand side is a regular vector inner product for fixed m,n. The vectors on the right hand side are in any basis you want, so long as they span the same Kohn-Sham/band subspace. Question 2. I'm not sure where you're lost. You ...

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This sounds like a very standard problem for multiphysics applications. The oven is comprised of a heating element, insulation, and argon gas inside it. You will have both radiative and convective transfer of heat inside the oven, and you'll need to model both to find out how the temperature distribution develops. Here is a demonstration of COMSOL for a ...

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In general the best resource for details about VASP tags and their use is the VASP wiki page. For the virtual crystal approximation the relevant tag appears to be "VCA" and the corresponding wiki page is here. I have never used the virtual crystal approximation, so cannot speak about its merits. However, I would highlight the text at the bottom of ...

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