22

To complement mykd's already excellent answer, I will just add that the approximant we all learn in school (the Taylor approximant) is nice to teach and easy to help students learn the concept of approximations, but in practice it's one of the worst options in terms of it's accuracy-to-complexity ratio. The Taylor approximant matches the $n^{\textrm{th}}$ ...


16

PADE refers to the mathematician Henri Padé, who developed the Padé approximation, which can approximate a function using rational polynomial functions. For ex: $$ \sin(x)\approx {\frac {(12671/4363920)x^{5}-(2363/18183)x^{3}+x}{1+(445/12122)x^{2}+(601/872784)x^{4}+(121/16662240)x^{6}}} $$ This feature is used to approximate the total correlational ...


16

Great question! Protein folding has been in open question for decades. Just recently, there's been a lot of discussion regarding DeepMind's AlphaFold project, which was discussed at length on our very own site here. My answer will be complementary to the one above, but the references I will provide will be closer to the physics side of the problem. First ...


10

Someone more familiar with the problem might have a better suggestion, but I recently came across Daniel B. Dix' notes on Mathematical Models of Protein Folding. This is not my field, so I won't guarantee correctness. However, to a layman at least, these notes seem well suited for someone with your background. The abstract reads We present an elementary ...


10

Plasticity is still an actively researched area. The Ludwik-Hollomon equation is one model that is used for the strain-hardening region since it captures the convex shape of the curve using a power law: $$\sigma = K \epsilon^{n} \tag{1}$$ Here, $n$ is known as the strain hardening coefficient or strain hardening exponent. This equation only captures the ...


9

You are developing the right idea, but your Eq. 1 is a bit awkward. To understand why, let me say a bit about functionals and functional derivatives. A functional acts on a function in its entirety, for example we need to know the full input function $n(r)$ in order to obtain the corresponding value for the functional: $$\tag{1} E[n(r)] = \int_0^1n(r)\textrm ...


9

By asking about the discretization of $\mathbf{r}$, I think you may be trying to visualize/intuitively understand the spatial variation of the energy and density, $\mathrm{d}E/\mathrm{d}\mathbf{r}$ and $\mathrm{d}n(\mathbf{r})/\mathrm{d}\mathbf{r}$, and you are conflating that with the meaning of the functional derivative $δE[n]/δn(\mathbf{r})$. If you have ...


9

The Davidson-Method is the best algorithm when you want a comparatively small number of eigenvectors/eigenvalues from a sparse, diagonally dominant matrix. MKL doesn’t have a sparse matrix diagonalization method, but it’s a good idea to use it for the Linear Algebra operations used when implementing Davidson. If you want ALL the eigenvectors then you ...


9

Following Nike Dattani's suggestion, I now add some comparisons of the Davidson method with methods that are more "modern" than it, to supplement the existing answers which compared the Davidson method with methods that predates it. However my answer will be more or less off-topic because I will mainly talk about disadvantages of the Davidson ...


9

NetworkX This is a widely-used Python library for graph theory stuff. Open source (https://github.com/networkx/networkx) under a 3-clause BSD license. Caveat: since this is a tool that knows nothing about chemistry, the "chemical" nature of the structure might get lost in the projection of the network (e.g., some bond lengths might not look ...


8

It looks Doi makes some extra simplifications (beyond expanding the exponential) that are valid when the external field is weak. Let's start with what you wrote and make one simplification, $$ \begin{eqnarray} \overline{\delta \phi _a} &= \frac{\langle\delta \phi_a \rangle -\langle \delta \phi _a \beta U_{ext}\rangle}{\langle 1-\beta U_{ext}\rangle} \tag{...


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

In the Huckel method, you are just generating a very simplified version of the molecular Hamiltonian and determining it's eigenvalues. The molecular Hamiltonian will always be a Hermitian matrix (for the Huckel method, we can be more specific and say it's a real symmetric matrix by construction). Hermitian matrices are guaranteed to have all real eigenvalues ...


7

There are two related structures. Dodecahedrane is known - $\ce{C20H20}$ in which the carbon atoms form the lattice you described, with hydrogen atoms on the outside. Beyond that, $\ce{C20}$, a fullerene is known, and you can find a great variety of similar structures, from $\ce{C20}$ to $\ce{C720}$. As for your discussion of quasi-crystals, I think it's ...


7

Preamble Since I don't know your specific background, this is a generic answer for any applied mathematician wishing to enter the field of protein folding. Not everything will apply specifically to you, and please don't feel offended if there's something I assume you don't already know or do! First of all, as a fellow mathematician (I was trained in ...


7

To supplement Cody's excellent answer, Davidson diagonalization originates from quantum chemistry, where solving configuration interaction problems requires diagonalizing the Hamiltonian in the space of electron configurations. The Davidson algorithm and refinements thereof enabled full configuration interaction calculations with matrix sizes of $10^9 \times ...


7

The angles $\phi$ and $\theta$ never contain a non-zero imaginary part (they are always purely real numbers). The general solution to Eq. 1 is: $$ \tag{1} |\Phi(t)\rangle = e^{-\textrm{i}H t} | \Phi(0)\rangle. $$ This can be written as: $$ \tag{2} \begin{bmatrix} a(t) \\ b(t) \end{bmatrix} = U \begin{bmatrix} a(0) \\ b(0) \end{bmatrix} . $$ where the ...


6

Finally, I figure out this question on my own. Put Eq.$(2)$ and $(3)$ into Eq.$(1)$: \begin{equation} -e \Re\{\mathcal{E}_a e^{i\omega t}\} (\partial_a f_0+\partial_a f_1)+(\partial_tf_1+\partial_tf_2)=\dfrac{-f_1-f_2}{\tau} \tag{7} \end{equation} in which $\Re\{\mathcal{E}_a e^{i\omega t}\}f_2\propto |\mathcal{E}|^3$ is droped and $\partial_t f_0=0$. ...


6

There are a few issues with the derivation: (not necessarily a mistake, but) please note that if you mean the Coulomb Hamiltonian, the RHS of (1) should involve the reciprocals of $|r_i|$ etc., not $r_i^2$ etc. The Coulomb interaction enters the Hamiltonian in the form of potential energy, rather than in the form of force. Please ignore this comment if you ...


6

The question is not very well-posed. However, the question appears to be about whether rotational invariance is honored by LCAO calculations. The old-time approach is to use Cartesian basis functions. Translating our origin to the nucleus, we can write the Cartesian basis functions in the form $\psi({\bf r}) = N x^k y^l z^m \exp(-\alpha r^2)$, where $\lambda:...


6

Higher-order integrators are used, but usually the way they perform the calculation is not through directly calculating higher order derivatives, but essentially through multiple force calculations. The Wikipedia page on symplectic integrators gives some information on this, including 3rd order and 4th order examples. I've personally used the 4th order ...


5

Here is a short video tutorial to explain the following points about FEM: Why study FEM Engineering systems and FEM What is FEM? Layman's explanation Mathematical treatment The power of FEM Hope it helps.


5

I recommend you to visit and go through the COMSOL Multiphysics site. COMSOL software is based on finite element method (FEM) with applications in several areas of Physics. In the site, you can request a demonstration of the software (so you will have the experience to use FEM in practice) and also can visit the WEBMINARS section that is full of applications ...


5

Fullerene graphs One example of a category of graphs that is used in chemistry is the set of Fullerene graphs. One Fullerene graph that is especially interesting to some chemists, is the 26-Fullerene graph. You may also be interested in this paper about Fullerene graphs, or the book An Atlas of Fullerenes by theoretical chemists Patrick Fowler and David ...


5

The other answer is great and comes from a viewpoint of macroscopic plasticity. I'd just like to note that another perspective on plasticity exists, a multiscale view based on following atomistic mechanisms up through the length scales to aim for an understanding of plasticity that is increasingly based on physical mechanisms. The enormous range of length ...


5

I searched for the word "minimum" on the Wikipedia page for Lennard-Jones potential, and the 2nd paragraph says: "The potential minimum is at $r=r_m = 2^{1/6}\sigma$." You can obtain this by calculating the derivative of the Lennard-Jones potential with respect to $r$ and finding which $r$ value makes that derivative equal to 0.


5

For context to future readers, Doi starts with a model of an $N$ unit polymer formed by a random walk along a uniform grid of lattice length $b$. It seems to be implicitly assumed in the described derivation that the lattice used is $b\mathbb{Z}^3$, that is the uniform 3D grid. Note there are other types of lattices for which Eq.$\,(\ref{2})$ is not true. ...


5

The derivatives of a molecular Hamiltonian with respect to nuclear coordinates can be performed analytically (as you correctly pointed out), and therefore does not need PySCF or Psi4, which are programs for doing numerical calculations for things that cannot be done analytically. This is how it's done: \begin{align} H &= -\sum_{i}\frac{1}{2}\nabla_{i}^{2}...


5

Yes, the sum of these potentials is zero everywhere with the true ground-state density. But the true ground-state density is generally only expressible when the basis set is complete (if you know the true ground-state density in advance, you can of course construct a finite basis set that reproduces it exactly, but if you don't know the exact density, you ...


3

The Hubbard Model is already manifestly SU(2) spin-rotation invariant because there are no terms that connect the $\uparrow$ and $\downarrow$ sectors (here by connect, I mean change a $\uparrow$ to a $\downarrow$ or vice versa. The full Hamiltonian in terms of fermion annihilation/creation operators is: $$ H = - t \sum \limits_{t,\sigma} \left( c^\dagger_{i,\...


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