Questions tagged [mathematical-modeling]

Questions related to mathematical modeling of materials systems.

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How to find magnetic materials in terms of magnetic point group?

Are there any methods to find materials in terms of the assigned magnetic point group? For example, I know the magnetic point groups $6'mm'$ and $6'22'$, and I want to find the corresponding ...
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What does it mean when the first order correction energy is 0?

Suppose I have the following Hamiltonian to start $$ H_0 = \begin{pmatrix} 0 & 0 & 0 & 0\\ 0 & 0 & 2 & 0\\ 0 & 2 & 0 & 0\\ 0 & 0 & 0 & 0 \end{pmatrix} $$...
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What are the precise forms for the SU(2) and rotation matrices in VASP?

Some time ago, I conducted a discussion with Dr. Gui-Bin Liu on topological materials here. One relevant thing he said was: "The format of trace.txt file is only a necessary condition to be used ...
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Understanding derivation of discretized FIRE algorithm

Introduction/Preamble @SusiLehtola's answer to Basics of numerical energy minimization techniques used in molecular dynamics? mentions conjugate gradients, BFGS for energy minimization, Metropolis ...
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Basics of configuration initialization strategies used in molecular dynamics? [duplicate]

This answer to Basics of numerical energy minimization techniques used in molecular dynamics? mentions conjugate gradients, BFGS for energy minimization and Metropolis Monte Carlo (and the) the ...
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Basics of numerical energy minimization techniques used in molecular dynamics?

The question below describes my plan to make a basic molecular dynamics calculation using a Python script rather than a canned, self-contained program. There seems to be three parts: a model of the ...
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How to get a Wannier function for a tight-binding model numerically?

I have a question about construction of a Wannier function for a tight-binding model. Let's say we consider the tight-binding model of a 1D chain with two atoms (site A and B in a unit cell). In k-...
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Do we know for sure that all atomic and molecular wavefunctions decay exponentially as r goes to infinity?

Slater type orbitals (STO) are considered to be more accurate than gaussian type orbitals (GTO) for atomic and molecular QM calculations because - among other reasons - they decay with $e^{-\alpha r}$ ...
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Can superconductivity be faked by using precise mathematics in a simulated circuit & specific materials in an actual circuit? [closed]

A type of electrical amplifier is proposed in which only magnetism and it’s analogous current exists for the most part. There is no voltage and no wattage (to speak of) in which electricity has to ...
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How to rotate polarizability tensor depending upon the molecular coordinates?

My question is somewhat related to molecular rotation. I have calculated the polarizability tensor of $\ce{HCHO}$ molecule in PSI4. The output tensor is this, ...
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Which specific perturbation do we consider applying to the potential when defining the energy derivative of the log derivative?

Norm-conserving pseudopotentials are defined such that the energy derivative of the log derivative of the real and pseudo wavefunctions agree at $r>r_c$. I understand that although valence ...
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How to solve time-dependent Schrodinger equation and plot trajectory on Bloch sphere?

Let's say I have a $2 \times 2$ Hamiltonian that I am solving using the time-dependent Schrodinger equation: $$ i \frac{d}{dt} |{\Phi}\rangle=H|{\Phi}\rangle.\tag{1} $$ Consider a generic Hamiltonian $...
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Cutoff length for vdW correction in DFT calculation

In classical MD simulations of lattice structures, whenever we are trying to incorporate vdW corrections to the atomic force calculation, we need to set up cutoff length beyond which the vdW ...
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Use of higher-order integrators in molecular dynamics?

The usual verlet integration scheme for propagating molecular dynamics according to Newton's equations of motion looks like the following: $$ x(t+\Delta t)=2x(t)-x(t-\Delta t)+a(t)\Delta t^2 $$ This ...
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Spacetime translation of Noether's current density [closed]

I've heard so many times that spacetime translation invariance is assumed when we speak about Noether's current density. Could someone explain to me, why is that so? What if we don't assume it? To be ...
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8 votes
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Given the adjacency matrix of a molecule, how can I get a graphical representation using only open source software?

In Huckel Method, by numbering the sp2 carbons in a molecule with conjugated double bonds, we can assemble its secular determinant in a form similar to the adjacency matrix of a graph. Taking trans-...
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How do you Select a Time Step for Molecular Dynamics Simulations?

It seems there is a general agreement among the practitioners of Molecular Dynamics that 1fs is a fairly reasonable time step, with shorter time steps being required for materials with higher ...
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Relationship between functional derivative and potential value at a position

Thanks to very helpful and detailed answers for my previous question, I understand the functional derivative of the energy with respect to the density. In addition, this functional derivative is equal ...
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9 votes
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Functional derivative of energy with respect to density

I have read the paper of "A bird's-eye view of density-functional theory [PDF]" and I have a question about the functional derivative of the energy by the density. I have found the following ...
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1 answer
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What is the equilibrium bond length for a Lennard-Jones potential?

If I have $\epsilon$ and $\sigma$ can I calculate equilibrium distance $r_e$ in one run? What I have tried is to put $V = \epsilon$ and bring out the $r$ from the formula, but it seems not solvable. ...
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Ground State energy trick for many-body electronic structure calculations?

I am an outsider to this field, so I am not sure about the validity of my work below. Let us define the following Hamiltonian from DFT: $$ \tag{1}H_{ij} \psi_{ij} \equiv (-\frac{\hbar^2 \nabla_i^2}{2m}...
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What are the advantages of the Davidson diagonalization method over other sparse matrix diagonalization methods?

I am interested to understand the advantages of the Davidson diagonalization method over other sparse matrix diagonalization routines. For instance, Intel MKL.
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21 votes
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What is a Padé approximant?

I have been looking at using Goedecker-Teter-Hutter (GTH) pseudo-potentials and I came across the abbreviation PADE. I was wondering what this abbreviation actually stood for and how it is related to ...
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Analytic solution of Boltzmann equation

This question is related to nonlinear Hall effect proposed in this paper. The Boltzmann equation in the electric field under relaxation time approximation is: $$-e E_a \partial_a f+\partial_t f=\dfrac{...
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4 votes
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Hubbard model SU(2) symmetry: manifest invariance

Could someone explain, is it possible to make Hubbard Hamiltonian manifestly SU(2) invariant? I know about the interaction term, but how would kinetic (hopping effect) term have to look like? Here I'm ...
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8 votes
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Tools for symbolic calculations of quantum transport using the Keldysh NEGF formalism [closed]

I am looking for tools for symbolic calculation of quantum transport or quantum mechanics that involves Keldysh NEGF (non-equilibrium Green's function) formalism. The closest tool I heard/seen is <...
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What is special about valley-focused Hamiltonians that make them give quantized/rational (valley) Chern numbers?

I have been thinking about so-called valley Chern numbers $C_v$ and associated topological phenomena. To my knowledge, they are usually applicable when inter-valley scattering is suppressed, leaving ...
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1 answer
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Derivation on correlation function and response functions in polymer physics

I am reading Introduction to Polymer Physics by Doi and I am having trouble understanding a derivation by him on the concentration fluctuations in polymer solutions. I have outlined his method and ...
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5 votes
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Derivation of probability density of isolated polymers

(Crossposted on physics SE) I am reading Introduction to Polymer Physics by Doi, and in his proof for the probability distribution for ideal polymers of length $N$ and end-to-end vector $\mathbf{R}$, ...
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Calculating first and second derivatives of a molecular Hamiltonian?

I'm interested in computing first and second derivatives of molecular Hamiltonians with respect to nuclear coordinates. I've been using Psi4 and PySCF to perform Hartree-Fock calculations, and I was ...
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9 votes
1 answer
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Help with translating Hamiltonian into matrix

Eq. 19 in this paper gives the following Hamiltonian: $\sigma_a, \tau_a, \eta_a$ are respectively the spin, sublattice pseudospin and valley pseudospin respectively. Normally, I would have chosen a ...
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7 votes
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Molecular Structure Vectorization for Computational Quantum Mechanics [closed]

What are the different ways that a molecule can be encoded into a vector? My answer to that question would employ machine learning but maybe there are other approaches, perhaps analytical, e.g., graph ...
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References related to the molecular distance geometry problem (estimating true distances based on noisy distances)

One aspect of the molecular distance geometry problem (MDGP) described in this PDF, can be written as follows: "Given observations of noisy distances between atoms in a molecule, estimate the ...
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9 votes
2 answers
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Classes of Graphs used in Chemistry

I am looking for common categories of graphs used in chemistry, for math research I am doing in graph theory. When I write categories, the meaning is not for graphs design or types of design, but for ...
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9 votes
2 answers
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Mathematical models for the plastic region in the tensile test

The tensile test of a material consists of subjecting a standardized specimen to an increasing axial tensile stress until it breaks. During its performance in the laboratory, we can plot a stress-...
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Construct a parity operator at a TRIM point? [closed]

I want to calculate the band parity at some TRIM (time-reversal invariant momentum) point in Brillouin zone. Parity was defined as the eigenvalue of the inversion operator. My question is how to ...
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11 votes
0 answers
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How to solve general wave equation and dispersion relation using Fourier series? [closed]

In this paper (open access), the authors used Fourier series with most general wave equation to find the dispersion relation. I am presenting some main equations as snippets to depict their solution. ...
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6 votes
2 answers
91 views

Books and online resources to get me started in the finite element method (FEM)

In a course on the mechanical properties of materials, I have been asked to do a paper on the "Finite Element Method" to improve my current grade. However, most of the resources I find on ...
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5 votes
0 answers
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What's the information contained in a character table of a group? [closed]

This question has an answer on the Chemistry Stack Exchange: Understanding group theory easily and quickly. Anyone wishing to add alternative or additional perspectives that aren't already covered in ...
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9 votes
1 answer
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How can I implement the Cartesian harmonics?

I'm studying the quantum chemistry calculation and now trying to implement the basic methods. For example, given a water molecule, $M = {(\ce{O}, R_\ce{O}), (\ce{H}, R_\ce{H}), (\ce{H}, R_\ce{H})}$, ...
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1 answer
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Is it possible to get complex numbers as solutions for a secular determinant in simple Hückel method (SHM)?

Past semester we revisited the Hückel molecular orbital theory at class. One day I was trying to solve some problems with SymPy, a Python module that is a computer algebra system (CAS), and noticed ...
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3 votes
1 answer
198 views

Can corundum be considered a covalent network?

I know that I learned in Chemistry class that a bond is defined to be ionic if the electronegativity difference between the atoms is more then 1.7, half the electronegativity difference between the ...
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7 votes
1 answer
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Can replacing some of the atoms in a covalent network with that of the element one atomic number higher make it nonstick? [closed]

I originally made the title the question I had that is a more suitable question but then it ended up too long. I had the idea that if you replace some of the atoms in a covalent network with the atom ...
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12 votes
1 answer
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Has a form of diamond with certain properties ever been theorized?

The following is what a Penrose tiling looks like: I know the verticies of a dodecahedron can be grouped into 5 groups each of which are the verticies of a tetrahedron. So I thought of the idea that ...
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24 votes
4 answers
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Introduction to protein folding for mathematicians

My background is mostly in (applied) math with healthy doses of physics and computer science. Are there any good introductions to protein folding and its challenges for someone with that kind of ...
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12 votes
1 answer
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Why linear response is absent in a non-centrosymmetric system with time reversal symmetry?

In this paper, it is mentioned that a non-centrosymmetric system with time-reversal symmetry doesn't have a linear response. It is actually a consequence of the Onsager reciprocal theorem. But I didn'...
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15 votes
1 answer
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First order variation of the wave function of conduction states

The first order variation of the wave function $\Delta \psi_n$ is obtained by standard perturbation theory (Eq. 25 of ref 1): \begin{equation} (H_{SCF}-\epsilon_n)|\Delta \psi_n \rangle = -(\Delta V_{...
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7 votes
0 answers
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Deriving relations for a hard sphere phase diagram [closed]

In Torquato's book "Random Heterogeneous Materials", he has written: $$\frac{p}{\rho kT} = 1+2^{d-1}\eta g_2 (D^{+})\tag{1}$$ where $g_2(D^+)$ is the contact value from the right-side of the ...
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8 votes
1 answer
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What software can perform block analysis on time sequence data?

I have a .xyz file for a trajectory and I want to perform block analysis on it, and to find the decorrelation time and to perform analysis on that as well. What ...
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12 votes
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A viscoelastic material with nonconvex memory kernel?

The title is basically my question. Viscoelastic materials are characterized by a constitutive equation between stress and strain involving a convolution integral. This integral is weighted with a ...
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