Tag Info

28

Quick Summary: There's no way around performing a convergence test. However, it is possible to obtain convergence much faster than the Phonopy approach by using nondiagonal supercells . The basic quantity you build when performing a phonon calculation is the matrix of force constants, given by: $$D_{i\alpha,i^{\prime}\alpha^{\prime}}(\mathbf{R}_p,\... 21 Phonons are a measure of the curvature of the potential energy surface about a stationary point. In particular, the matrix of force constants is calculated as:$$ D_{i\alpha,i^{\prime}\alpha^{\prime}}(\mathbf{R}_p,\mathbf{R}_{p^{\prime}})=\frac{\partial^2 E}{\partial u_{p\alpha i}\partial u_{p^{\prime}\alpha^{\prime}i^{\prime}}}, $$where E is the ... 17 Short answer: Modern implementations of these two methods lead to similar accuracies. Longer answer: The calculation of phonons requires the calculation of the Hessian of the potential energy surface V(\mathbf{R}), also known as the matrix of force constants:$$ \frac{\partial^2 V(\mathbf{R})}{\partial \mathbf{R}_i\partial\mathbf{R}_j}=-\frac{\partial \...

16

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,\... 15 This is a difficult question without a straight-forward answer. In general you have to perform a test to decide whether the harmonic approximation is sufficient or whether you need to include higher order anharmonic terms in the potential expansion. Due to the computational cost of including anharmonic terms, very often systems are assumed to be harmonic ... 14 Ideally, a convergence test would be the best way to decide the required size of the supercell, but it can get expensive. When phonopy (or any similar calculation technique) finds displacements in the cell based on symmetry, the idea is to see how the displacement of certain ions affects the forces on every ion within the cell. We must then take care that ... 14 Phonon calculations tend to be very expensive to run. That being said, for gas phase molecules it is very common and expected that frequency calculations are run to ensure the molecule is not on a saddle point. In general, you can publish anything if it makes it past peer review. Phonon calculations are something you would do if you fear you are on the ... 13 Assuming that all calculation parameters associated with the electronic structure are properly converged, then obtaining imaginary frequencies can mean one of two things. Physical imaginary frequencies This situation corresponds to obtaining imaginary frequencies on \mathbf{q}-points that are included in the n_1\times n_2\times n_3 grid that you ... 13 In general it is not justified to published the geometry of a system without performing a phonon calculation. This is where you may end up in the potential energy surface depending on which type of calculation you perform: Geometry optimization. With a geometry optimization, you may end up at a local minimum or at a saddle point of the potential energy ... 13 This is a description of some methods available to study dynamical and chemical stability, not limited to MD: Dynamical stability. The question here is: given a compound in a particular structure, is this structure dynamically stable or is there a lower-energy structure that can be accessed without jumping over an energy barrier? From a potential energy ... 13 Considering this in terms of a 2x2x2 supercell matrix is the wrong way to think about this, as choice depends on the cell length and bonding type. Given that rigorous convergence testing is near-impossible (see ProfM's answer), what saves the method is the rapid fall-off with distance of the force constant matrix \Phi. This "nearsigtedness" ... 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, ... 12 The potential energy surface (PES) is a 3N-dimensional function for a bulk system containing N atoms (in reality 3N-3 to account for the trivial translational degrees of freedom). For a bulk structure, N typically represents the number of atoms in a simualtion cell with periodic boundary conditions, which is of the order of 10^2-10^3, so the function is ... 11 You are basically looking for finding phonon frequencies with respect to \mathbf{q} the scattering vector in reciprocal space. From fluctuation-dissipation theory, the force constants of the system in the reciprocal space is given by:$$\Phi_{k\alpha,k^{'}\beta}(\mathbf{q}) = k_{B}T \mathbf{G}^{-1}_{k\alpha,k^{'}\beta}(\mathbf{q})$$\mathbf{G} is the ... 10 Some people do: In this paper there is a system coupled to a bath of Morse oscillators rather than a bath of harmonic oscillators, but it is not exactly solvable, they used a numerical approach called mctdh. When it is said that the Morse potential is "exactly solvable", what it means is that you can solve the vibrational Schroedinger equation for ... 10 This depends on what you are studying. For molecular systems without periodicity, the simplest approach is to carry out a vibrational frequency analysis and confirm that there are no imaginary modes. It is considered standard to carry out a vibrational frequency analysis for all investigated structures, provided the number of investigated systems is not ... 10 The phonon dispersion relates the phonon frequencies \omega_{\mathbf{q}\nu} for each branch \nu with the phonon wave vector \mathbf{q}, typically along a path in the Brillouin zone joining high-symmetry points. The phonon density of states compresses this information by integrating over \mathbf{q} and summing over \nu:$$ \tag{1} g(\omega)=\sum_{\...

10

The zero-point energy is of no importance here, since you can always choose your reference energy freely you can energy-shift your hamiltonian by $\frac{1}{2}\hbar\omega$ $$H = \frac{p^2}{2m}+\frac{1}{2}m\omega^2x^2-\frac{1}{2}\hbar\omega,$$ and the physics of the system will stay the same (the wave function will be the same). Since this wavefunction is ...

9

If you index the molecule and the atoms in the crystallographic unit cell the same, you could extract the displacement from the phonon eigenvectors and the displacement from the normal modes. By projecting each normal mode displacement onto each phonon displacement, you can get a qualitative relationship between a molecular deformation and a lattice ...

8

This is actually an important question regardless of the code you're using (I use SIESTA and NWChem). To obtain a good phonon spectrum, the first thing you need is a truly relaxed/optimized structure. This requirement entails several parameter optimizations (mesh cutoff, force tolerance, basis set ..). Afterwards, other requirements are also important. I ...

7

Whatever scattering mechanism you choose must respect detailed balance in equilibrium: on average, the number of particles hitting a patch of the wall at a given angle and velocity must equal the number of particles reflected at the same angle and velocity. If this were not the case, the system would not be in equilibrium. There are numerous ways to do this ...

6

Even in treating molecular vibrations, the Morse potential is not always the best, because: There's cases where the potential is more "harmonic" than "Morse-like", for example in the asymmetric stretching of water. It is the same case in solid state physics: considering the most crude approximation that we fix all the atoms in a solid ...

6

I have never used Phonopy, so we will have to wait for an actual user for a full answer. However, I can make some general statements about phonon calculations that could plausibly explain your error. The starting point of any phonon calculation is the construction of the matrix of force constants:  D_{\alpha i;\alpha'i'}(\mathbf{R}_p,\mathbf{R}_{p'})=\frac{...

6

Charge density waves (CDW's) exist in a few circumstances that I know of. The simplest example I can think of is the CDW due to nesting of the Fermi surface. This can be discussed in the context of the 'Peierls instability.' Kittel's Introduction to Solid State Physics provides a decent description of the phonon renormalization in this case (pp 422). ...

6

The quantum number n simply represents the different energy levels given by the harmonic oscillator. $\mathbf{n=0}$ does not correspond to a given temperature, but its relative occupation to other energy levels does correspond to a given temperature. As a system rises in temperature, the higher energy levels can be occupied at greater numbers. Likewise, at ...

5

You mention phonons, so I assume you are doing periodic structures which I am not entirely familiar with since I study mostly single molecules. Although, when I get imaginary frequencies from optimized geometries of single molecules this typically implies that the isomer in question is not a stable minimum on the potential energy surface (PES). This does not ...

5

In order to check if a geometry is a local minimum, it is a necessary and sufficient condition that the Hessian is positive (semi)definite, i.e. that the lowest eigenvalue of the nuclear Hessian is non-negative. Namely, expanding the energy $E({\bf R})$ around the reference point ${\bf R}_0$ you have the Taylor expansion $E({\bf R}) = E({\bf R}_0) - {\bf g} \... 5 ProfM's answer gets the core idea perfectly right: Symmetry really is your best friend here. However, symmetry analysis is often quite involved, especially for larger unit cells. I recently discovered the hiPhive package, which uses statistical fits (forces from random displacements fit to a force-constant potential), combined with with symmetry analysis (... 5 You can (should) use symmetry to reduce the number of displacements needed to construct the matrix of force constants. A nice pratical description of how to do this can be found in the description of the PHON package by Dario Alfè. In short: if you have the force constants for displacing a given atom, and when you apply the symmetry operations of the crystal ... 5 Since no one has responded with expertise, I'll attempt a speculative answer here. To my mind, the simplest model of the atoms on the surface of the walls would be an ensemble of independent classical 3D harmonic oscillators at some temperatures$T_{\rm wall}\$. That would be pretty easy to describe from a numerical standpoint, since their velocity could be ...

Only top voted, non community-wiki answers of a minimum length are eligible