Questions tagged [mathematical-modeling]
Questions related to mathematical modeling of materials systems.
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Difference between Phase Diagram and Convex Hull
in literature I sometimes see a mixed use of the words Phase Diagram and Convex Hull. However from my current knowledge, there are some differences between them.
A phase diagram views the difference ...
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What is the real form of p-type gaussian basis functions?
I am attempting to write my own Hartree-Fock program, and I am currently working on implementing the STO-3G basis set for a few atoms. For C, it is of the form
...
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Are there any software packages to find the minimum number of constraints to fix a molecule structure?
I wanted to know if there are any modules or packages which help in identifying the bonds which should be fixed to make the molecule rigid. For example in $\ce{H2O}$ there are just three (trivial), $\...
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What is the computational scaling of DFT energy vs gradient vs Hessian?
I am trying to understand the computational expense of calculating only the energy, versus the gradient/Hessian of the DFT energy with respect to nuclear coordinates. How do these scale with the size ...
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Proof that equilibrium phase fraction(s) is a continuous function of composition
It seems that the equilibrium phase fractions (i.e., relative amounts of phase A1, A2, ..., Aq for a system in which there are q total phases) as a function of composition (for a constant pressure and ...
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A problem to simulate the kinetic energy operator using some field going through a crystal
We know about a differentiation and integration operations and also other ones can be simulated by an analog way using electric circuits with operational amplifiers (here are articles of the ...
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Are there non-antisymmetric solutions to the electronic Hamiltonian?
In electronic structure theory, one seeks solutions of the molecular electronic Hamiltonian
$$
H_e(\vec{R}) = T_n(\vec{r}) + V_{ne}(\vec{r};\vec{R}) + V_{ee}(\vec{r}) + const,
$$
where $\vec{r}$ are ...
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For one atom, what is the correct point group?
I have a concern related to the follwoing question When orbitals are labeled based on their irreps in D2h, how are the orbitals ordered for an N atom? In which the following Answer clearly states that ...
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Is the electron-electron interaction to blame, for the added complexity of using "orbitals" for an N-electron system?
In a one-electron system such as the H atom, we are able to say that the orbitals are valid wavefunctions. But for complex systems with N-electrons, I have heard people say that they are just ...
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How can I compute the longest relaxation time?
Cross-posted on Stats.SE and on SciComp.
In the case of Monte Carlo simulations:
Autocorrelation Time ($\tau_{\text{int}}$): A measure of how many steps are needed for the correlations in the chain ...
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How do I get the energy, if I have the wavefunction?
I recently wrote a simple HF SCF program for my understanding and realized that I do not fully understand such QM calculations.
Suppose I have the ground state gaussian wave function of a system,
$$
\...
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Speed of atom from kinetic energy of molecule
What is the velocity of each hydrogen atom in an H2 molecule, given a kinetic energy of hydrogen molecule is 0.12 eV, when expressed in angstroms per femtosecond and assuming equal velocities for both ...
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Why does the C6z operator have 6 eigenvectors in this paper?
In this paper, I couldn't understand this line in the 3rd section (i.e. III. SYMMETRY ARGUMENTS):
In the absence of SOC, the eigenvalues of $C_{6z}$ rotation operator are $e^{i2\pi n/6}$ where $n = 0$...
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Correct expressions for one-center Gaussian radial two-electron integrals
I am working on a new atomic solver library, for which I need expressions for the one-center two-electron integrals. Pitzer has furnished the necessary expressions for Slater-type and Gaussian-type ...
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Evaluating analytic gradients for overlap integrals
I have been working on a toy HF code for some time. From this reference I am trying to work out an analytic expression for the first derivatives of overlap integrals.
The original expression for ...
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Convex hull calculations with WIEN2k
How does one do the calculation of a convex hull with the WIEN2k code for the formation energy of alloys?
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Gauge constraint in the definition of the Z2 invariant
Cross-posted at Physics.SE.
In Fu and Kane's paper from 2006, the authors define the $\mathbb{Z}_2$ invariant for time-reversal invariant topological insulators as an obstruction to Stoke's theorem,
$$...
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How is the equation for the position of a virtual site derived?
Cross posted on Math SE
I am trying to understand virtual sites in MD simulations, and I came across this configuration:
Here, coordinate $\mathbf{s}$ represents the virtual site, which is formed by ...
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Inconsistencies in a famous point group table
The original resource
"Tables for Group Theory" By Peter Atkins, Mark Child and Courtenay Phillips (1970). I have been using this 2008 version, in which Pg. 37 has the following table:
Some ...
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Is the following description talking about the simulation box or the boundary condition?
Is the following description talking about the simulation box or the boundary condition?
To prevent the argon atoms in the gas phase from running off to
infinity, we enclose the system in a "...
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How can I properly plot this .cube file in Python?
Motivation for Post: I saw a question about how to plot a .cube file recently on the Mathematica & Wolfram Language Stack Exchange. It was marked "off-...
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Cubic spline interpolation through a series of images
Cubic spline interpolation is used in a lot of methods like Nudged Elastic Band (NEB), or String or Growing String Method (GSM) either to reparameterize the path, or to estimate the position of the ...
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What is the meaning of a "free energy per unit surface area" vs "pressure" plot?
I don't understand this graph. From the graph, does it mean that at high pressures the oxide layer is stable and this stability reduces at low pressures and temperatures?
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Why are some point groups for single atoms listed as Oh in the CCCBDB?
Does anyone know why the CCCBDB database lists the tin cation and antimony atom as having the Oh point group (please see the screenshot below from CCCBDB when I look at "All molecules sorted by ...
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Why, if the potential is different from the Coulomb one, but has spherical symmetry, the eigenvalues of the system are non-degenerate? [closed]
Question duplicated from here https://mathematica.stackexchange.com/questions/287433/why-if-the-potential-is-different-from-the-coulomb-one-but-has-spherical-symme and here https://physics....
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Help with Mathematica for optimizing a Gaussian basis set for a hydrogen atom in a magnetic field
This is a follow-up to my previous question: Optimal Gaussian basis set for hydrogen atom in magnetic field
Brief description of the problem
I would like to find the ground and excited states of the ...
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Why is strain rate used instead of absolute strain for modelling stress relaxation of a viscoelastic material using the Maxwell model?
I don't quite fully understand the derivation of the Maxwell model for prediction of stress relaxation for viscoelstic materials. usually the governing equation can be described in terms of strain ...
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Predict if a group will be acceptor/donor of H bonds [closed]
Context:
I am a student of cheminformatics, and I am trying to solve a problem in predicting whether atom can be an acceptor/donor of hydrogen bonds, using Python programming language and RDKit ...
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Molecular integrals in spherical harmonics?
I have been trying to write a basic HF code that can compute the MOs and total energy of a molecule. I followed the following reference for overlap and kinetic integrals. During a conversation with a ...
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How can one determine the time-step for imaginary time evolution?
The real-time evolution of a quantum system described by the state $\psi(t)$ is formally given by,
$$ \psi(t+\Delta t) = \exp^{- i \hat{H} \Delta t / \hbar} \psi(t)\tag{1}$$
For dynamical simulations ...
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Inverse Matrix Calculation
I am working on the charge transport calculation, based on the non-equilibrium green function method, for the Hamiltonian with the non-orthogonal basis. Suppose $H$ and $S$ are the Kohn-Sham ...
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Estimate the time between different residues?
Imagine now, I have 5 structures, and I want to estimate the time when the 5 structures are closer to each other at the same time. I thought I could extract the distance between the different ...
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Database (or table) of functions obeying symmetry properties of space groups
I'm looking for a table or database containing mathematical functions that belong to crystallographic space groups by having the same symmetry elements.
Imagine a 2D square planar lattice. It has a 4-...
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What is the X in Almlöf and Taylor's "Unified treatment of energy derivatives?"
I have been studying the possible methods for basis set optimizations. One notable paper is "Energy-optimized GTO basis sets for LCAO Calculations. A Gradient Approach" by Knut Faegri Jr. ...
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Constrained optimisation on a hypersphere
I am currently trying to implement a GS2 (Gonzalez-Schlegel second order) IRC algorithm in a python code. I am following the original paper ref(1).
The main problem is in the constrained optimisation ...
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Understanding the complexity of geminal-based wavefunctions
Cross-posted on Reddit.
I have been reading through [1] to get a better understanding of geminal-based methods. Some short passages are included below:
The occupation of each orbital in the expansion ...
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Understanding derivation of geminal based methods [closed]
Cross-posted on Reddit.
I have been reading through [1] to get a better understanding of geminal based methods. This is one of two questions that I'm asking about that paper on this site, the other ...
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Numerical Evaluation of Hessian?
I am trying to understand the construction of Hessian matrix using numerical differentiation (NUMFREQ option in ORCA for example). I understand that Hessian is a square matrix containing second ...
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How to Derive the Kong-Chakrabarty Mixing Rules
Background
In the world of atomistic modeling with classical force fields, one is often given a force field defined by like interactions (e.g. argon-argon interactions). If one is working with a ...
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Decay rate of DSD-PBEPBE-D3BJ
I gave up on the wB97X-2(TQZ) functional, which would have an exact decay rate of -1/r, re: my previous question and decided to use DSD-PBEPBE-D3BJ, which is available on vanilla Psi4. I then looked ...
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how to incorporate size polydispersity in colloidal suspensions [closed]
Basically, I'm working on spherical colloids and investigating the phase behavior of these monodisperse colloids using Monte Carlo simulations (Fortran 77 coding). Now I want to introduce size ...
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Question on possible cancellations in the computation of Dyson orbitals from FCC/FCI wavefunctions [closed]
Dyson orbitals are mathematically meaningful, in the sense that they, unlike the DFT orbitals for instance, are actual observables of the wavefunction, instead of being "the other way around"...
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Question on known expression (see post) of exact XC functional in analytic (but not closed) form [closed]
This paper analytically solves the "inverse DFT problem" of mapping the ground-state density to the exact XC functional. And the exact density, i.e. the FCC/FCI density, can be expressed as ...
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Is there an analogue of the K-S theorem for double hybrids
In my previous question, I was basically asking whether the results of the double hybrid using the exact XC functional are the same as those of just the exact XC functional.
Even that sentence is hard ...
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Is this method of obtaining only the exact "first excited state" density correct, under select conditions that makes it mathematically non-ambiguous? [closed]
(For context, see my other question here; this has been disproven (albeit contestedly), and is not even well-defined in the first place, so I'm going to ask a slightly different (and well-defined) ...
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Is this method of obtaining only the exact "other" densities, and no other properties w.l.o.g, correct?
In Kohn-Sham DFT, the exact density of the ground-state wavefunction is given by summing the squares of the filled K-S orbitals.
Suppose now that one only needs the exact density (and nothing else w.l....
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Is an "exact" double hybrid density the same as the "exact" DFT density?
Double hybrid approximate functionals have "unoccupied" Kohn-Sham orbitals in their formulations due to their MP2 component; however, the "exact functional" depends only on the &...
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Why is the Alphafold PAE (predicted aligned error) not symmetric?
Cross-posted on AI Stack Exchange.
We are running alphafold2 multimer on Google Colab to predict the association between two proteins.
It generally works fine, but we get an asymmetric PAE plot for a ...
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How to apply FIRE to many atoms where P=F·v seems to be a vector rather than a scalar?
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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Do the cc/pc/def2 basis sets mathematically converge to the CBS limit, assuming exact CI/DFT?
The cc-, pc- and def2- basis sets are often described as "systematic", in the sense that the results of these basis sets at different cardinal numbers (i.e. the number of ζ's) can be ...
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