Questions tagged [mathematical-modeling]
Questions related to mathematical modeling of materials systems.
52
questions
7
votes
1answer
304 views
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-...
5
votes
0answers
54 views
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 ...
5
votes
2answers
152 views
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 ...
9
votes
2answers
479 views
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 ...
5
votes
1answer
53 views
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. ...
5
votes
1answer
130 views
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:
$$ \tag{1}H_{ij} \psi_{ij} \equiv (-\frac{\hbar^2 \nabla_i^2}{2m}...
8
votes
3answers
129 views
What are the advantages of the Davidson diagonalization method over other sparse matrix diagonalization methods?
I am interested to understand the advantages of the Davidson diagonalization method over other sparse matrix diagonalization routines. For instance, Intel MKL.
17
votes
2answers
1k views
What is a Padé approximant?
I have been looking at using Goedecker-Teter-Hutter (GTH) pseudo-potentials and I came across the abbreviation PADE.
I was wondering what this abbreviation actually stood for and how it is related to ...
8
votes
1answer
98 views
Analytic solution of Boltzmann equation
This question is related to nonlinear Hall effect proposed in this paper. The Boltzmann equation in the electric field under relaxation time approximation is:
$$-e E_a \partial_a f+\partial_t f=\dfrac{...
4
votes
1answer
54 views
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 ...
8
votes
0answers
71 views
Tools for symbolic calculations of quantum transport using the Keldysh NEGF formalism
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 <...
8
votes
0answers
46 views
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 ...
9
votes
1answer
122 views
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 ...
5
votes
1answer
63 views
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 $\mathbf{R}$, ...
5
votes
1answer
93 views
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 ...
9
votes
1answer
208 views
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 ...
7
votes
0answers
63 views
Molecular Structure Vectorization for Computational Quantum Mechanics
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 ...
8
votes
0answers
73 views
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 ...
9
votes
2answers
91 views
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 ...
9
votes
2answers
76 views
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-...
7
votes
0answers
59 views
Construct a parity operator at a TRIM point?
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 ...
9
votes
0answers
68 views
How to solve general wave equation and dispersion relation using Fourier series?
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. ...
6
votes
2answers
85 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 ...
5
votes
0answers
118 views
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 ...
9
votes
1answer
262 views
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,
$M = {(\ce{O}, R_\ce{O}), (\ce{H}, R_\ce{H}), (\ce{H}, R_\ce{H})}$,
...
7
votes
1answer
108 views
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 ...
3
votes
1answer
156 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 ...
6
votes
1answer
153 views
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 ...
12
votes
1answer
150 views
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 ...
20
votes
3answers
1k views
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 ...
12
votes
1answer
148 views
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'...
12
votes
0answers
166 views
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_{...
6
votes
0answers
67 views
Deriving relations for a hard sphere phase diagram
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 ...
8
votes
1answer
77 views
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 ...
11
votes
0answers
60 views
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 ...
5
votes
0answers
66 views
Parallel algorithms on unstructured closed graphs [closed]
Some of the work I do in the study of discrete dislocation dynamics (a branch of materials modeling) involves topological changes on a graph: changes in the connectivity of nodes, as well as deletion ...
12
votes
1answer
139 views
Adiabatic equation related to the Berry phase for lambda with first order terms
Consider the following derivation in David Vanderbilt's book "Berry Phases in Electronic Structure Theory - Electric Polarization, Orbital Magnetization and Topological Insulators" (2018, ...
19
votes
4answers
374 views
How is group theory used to deduce which of these integrals are equal to 0?
The number of all two-electron integrals:
$$
\tag{1}
\langle \phi_1 \phi_2|\phi_3\phi_4 \rangle = \int d^3\mathbf r' \int d^3\mathbf r'' \, \phi_1(\mathbf r'') \, \phi_2(\mathbf r') \frac{1}{|\mathbf ...
11
votes
1answer
98 views
Regarding oscillatory strength theoretical units to experimental ones
The output of Gaussian rotatory and oscillatory strength intensities, plus a gaussian/lorentzian fit, translates to a theoretical CD/UV-vis spectra.
In order to try and compare with experimental ...
2
votes
0answers
115 views
Modeling the fluid dynamics of semen [closed]
Is there a good model for the fluid dynamics of human ejaculation of semen?
I'm looking to do some sperm motility research in the near future and would like to tinker with a model from a good paper.
19
votes
1answer
155 views
Quadrature over three Euler Angles for orientation averaging
Does anybody know about an accurate quadrature rule over three Euler angles $\theta, \phi, \chi$?
I am trying to calculate the average value of an arbitrary function $f(\theta, \phi, \chi)$ for a ...
2
votes
0answers
30 views
What is the best way to compare an actual value from an expected value on a linear model? [closed]
I'm comparing the cost of dialysis in each country. The x axis is health expenditure per capita. The y axis is cost of dialysis per patient. As you will see, this incredibly produces a near liner ...
17
votes
6answers
338 views
What are the applications of chemical graph theory?
Graph theory was originally introduced in computer science to study data structure. Chemists also introduced graph theory to study the relation between structure and properties for molecular compounds....
70
votes
6answers
9k views
Did the 2019 discovery of O(N log(N)) multiplication have a practical outcome?
Some time ago I've read this news article, Mathematicians Discover the Perfect Way to Multiply, reporting a discovery published in 2019, where Harvey and Hoeven[1] found a algorithm able to execute ...
24
votes
1answer
204 views
How do you calculate the "true" chemical potential in classical density functional theory?
In classical density functional theory, one traditionally calculates the chemical potential by taking the variational derivative,
\begin{equation}
\mu_{i} = \frac{\delta F}{\delta \rho_{i}}\tag{1}
\...
15
votes
1answer
103 views
Types of Discrete molecular models
It is often assumed in cheminformatics that molecules with similar physical structure tend to have similar chemical properties [1].
Based on this, our group has used discrete graphs as a model for the ...
15
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0answers
129 views
Is there a software that can do derivatives with respect to user defined vibrational modes? [closed]
I'm doing some property calculations that depend on a sum of derivatives of some quantity with respect to normal vibrational modes. I was hoping to find some physical intuition relating the type of ...
9
votes
1answer
73 views
What mathematics/computation heavy resources are available for understanding materials behaviour?
Since materials modelling is made possible primarily because of the availability of mathematical models, are there any resources that provide a mathematics heavy picture of materials behaviour?
I am ...
13
votes
1answer
65 views
Modelling ion Drift-Diffusion under an external electric potential, convergency in PDE?
Ion drift-diffusion under the effect of an external electric field is a phenomenon with a huge relevancy in light emitting devices, neuromorphic architectures and in general molecular electronics
Up ...
13
votes
1answer
82 views
What are the modelling techniques that can be used for simulating microstructure evolution in materials?
I am aware that the Potts Model can be used to simulate grain growth, and that Phase Field Models have also been very successful. What are the advantages and limitations of these models? What are the ...