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18

Adding to the answer given by Cody Aldaz, there are many situations in chemistry when the Born-Oppenheimer approximation (BOA) breaks down, conical intersection is just one of them! In fact, the electronic states do not necessarily have to cross/intersect each other for BOA to be invalid. A classic example of this is given by what is famously known as "...


17

TL;DR conical intersections, and polarons. Or any other case when the velocity of the nuclei is faster than the electrons can respond nearly instantaneous The long answer requires a lot of mathematics! The Mathematical derivation The total nuclear and electronic Hamiltonian can be written as $\hat{H} = \hat{T}_N + \hat{T}_e +\hat{V}_{ee} + \hat{V}_{NN} ...


15

In the paper "A Continued-Fraction Representation of the Time-Correlation Functions", generalized susceptibilities and transport coefficients for materials are obtained using a continued-fraction expansion of the Laplace transform of the time-correlation functions. This was the precursor to what is now called the "hierarchical equations of ...


15

The solutions to the Schrödinger equation are not unique in general, and uniqueness depends on several things such as the form of the potential and boundary conditions. Many papers have discussed uniqueness of solutions to the Schrödinger equation for specific classes of potentials and boundary conditions, but in general it is possible to come up with cases ...


14

The coupled-cluster hierarchy is a systematic approach to the exact many-body solution to the electronic Schrödinger equation, which yields size extensive energies and often converges extremely rapidly with respect to the maximum rank of excitations included in the model. CCSD(T) is widely known as the "golden standard of quantum chemistry", since ...


14

The specific questions: Only $l = 0$ overlaps are (currently) considered, in the spirit of atoms being represented by (only) a spherical charge distribution (point charges). The overlap/attenuation function for the bond capacities is very (!) preliminary. It should be designed to reflect the actual physics, and this is work in progress. However, any ...


13

You may take a look at the Method of Continued Fractions used in quantum scattering theory—this was only formed in 19831 so is rather recent. Related is the PhD thesis by Kónya (2000)2; §3.3 onwards. Reference [1] Horáček, J., Sasakawa, T. (1983). Method of continued fractions with application to atomic physics. Physical Review A. 28(4):2151–2156....


13

In the jj-representation, each electron from $i$ to $N$ will have: $\vec{l}_i$ (orbital angular momentum), $\vec{s}_i$ (spin angular momentum), and $\vec{j}_i=\vec{l}_i + \vec{s}_i$ (total angular momentum). In your example we have $N=2$ and both electrons are $p$-type so we have: $$\tag{1} {l}_1=1,~~~~~~~{l}_2=1. $$ Let's also assign the spins for each ...


12

tldr; it depends on flexibility / number of rotatable bonds A while ago, I answered a related question - in general, molecules with fewer "rotatable bonds" need fewer conformers geometries generated to sample properly. Based on that, I would normally have said "50 is more than enough for up to 3-4 rotatable bonds" and beyond that, I'd ...


11

First of all, let me emphasize that it is more appropriate to speak of KS equations (plural), which you correctly denoted by an index $i$ in your post. This index goes over all KS orbitals (i.e. single-particle wavefunctions) of the system. Additionally, as you mentioned, these equations have the same form as the single-particle Schrödinger equation. And ...


11

Here I assume that we are focus on the condensed matter system which is composed of nuclei and electrons with the fundamental force: Coulomb Force. Furthermore, I just discuss the electron degree of freedom. A naive (wrong) answer: The interaction gets stronger when electron density is higher. The Coulomb interaction is proportional to $\dfrac{1}{r}$, ...


10

Spherical harmonics are not themselves full atomic orbitals. Consider the Hydrogen wave function, which separates into a radial part and an angular part. The latter is a spherical harmonic, but the former is some other function (in the case of Hydrogen it's a Laguerre polynomial). In general, we can approximate the angular part for other atoms with the same ...


10

Here is my summary of definition of strong correlated systems in different context. For ab initio electronic structure Hamiltonian The eigen value of the system can not be well approximated by single slater determinate approach (Hartree Fock or DFT). These system normally has large coefficient in there configurational interaction (CI) expansion. For some ...


9

What is the quantum anomalous Hall effect? Figure from C-X. Liu, S-C. Zhang, and X-L. Qi. "The Quantum Anomalous Hall Effect: Theory and Experiment," Annual Review of Condensed Matter Physics 7, 301-321 (2016) (arXiv link). In short, the quantum anomalous Hall effect (QAHE) is the quantized version of the regular anomalous Hall effect (AHE). That ...


9

I'll try to give a short but reasonably rigorous way of thinking about the exactness of density functional theory (DFT). Consider $N$ electrons under the influence of a fixed external potential $v(\mathbf{r})$ for which the ground state electron density is $n(\mathbf{r})$. The external potential might be a sum of individual potentials from atomic nuclei, but ...


9

The exponential comes from solving a linear differential equation: \begin{align} \frac{\textrm{d}|\psi(t)\rangle}{\textrm{d}t} &= -\frac{\textrm{i}}{\hbar}H|\psi(t)\rangle\tag{1}\\ |\psi(t)\rangle &=e^{-\frac{\rm{i}}{\hbar}Ht}|\psi(t=0)\rangle\tag{2}\label{eq:matrixDynamics}. \end{align} Now if you diagonalize $H$ then instead of $H$ you have a ...


9

The Born-Oppenheimer approximation comprises two different approximations: Adiabatic separation of electron and nuclear coordinates Semi-classical approximation for nuclei Previous answers have already addressed (1). I would also like to add that although it neglects electron-phonon coupling, this can be added back in using density functional perturbation ...


9

In addition to the classic examples of where non-adiabatic effects are important, the Born-Oppenheimer approximation cannot be taken for granted in the electronic structure computations of small chemical systems, where high-accuracy is needed (e.g. isotope dependence of molecular properties, high-resolution rovibrational spectroscopy, and quantum nuclear ...


9

As you note, the interacting electrons and the Kohn-Sham non-interacting electrons have the same density. How is this possible when the Hamiltonians for the two systems are so different? The answer is that the Kohn-Sham potential—the potential felt by the non-interacting electrons—is constructed very carefully. Namely, if we want to remove electron-electron ...


8

The first answer is most relevant to materials modeling, but I also want to chip in that Born-Oppenheimer is more than just a kinetic assumption, it also assumes nuclei are point charges. This is especially important for actinides and super heavy elements where nuclei become not only much larger relative to the electron cloud but also more ovoid with unequal ...


8

It is certainly possible to develop ML models that yield more accurate results than would be possible without ML. One route to do this is through so-called "Δ-learning" where you use ML to learn a correction to a less expensive, often less accurate level of theory. An example can be found here for thermochemical properties of organic molecules. ...


8

I will first take a generic view-point and then quote some examples in condensed matter & materials modeling. Time-reversal symmetry is one of two discrete symmetries usually discussed in the context of condensed-matter, the other being Parity(Inversion). The simplest way in which this concept is presented is a transformation : $ t \rightarrow -t $ . ...


8

Another way to explicitly break time-reversal symmetry is by applying circularly polarized light. Under time reversal, left circularly polarized light transforms to right circularly polarized light, and vice versa. One way to see this is to note that circularly polarized light is an eigenstate of the spin angular momentum operator, which is odd under time ...


8

You got it completely correct when you said: "it seems that the definition of it is ambiguous and sometimes inconsistent in different field of study." As the other answers show, it's a bit of a "loose" term that can be used in slightly different ways depending on the sub-field of Matter Modeling in which the term is being used. This ...


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 ...


7

There is more than one way to answer this question well. I'll give one answer here. Let's treat the two atoms (that will form a covalent bond) as two localized election states (orbitals). The detailed shape isn't important. The Hamiltonian of this system could be written: $$ H = V (n_{\uparrow,1} n_{\downarrow,1}+ n_{\uparrow,2} n_{\downarrow,2}) - t \sum \...


7

Here are some basic comments: I don't think you should call Eq(1) as the Schrödinger equation. It's just defining the energy eigen states/values, or in other words, establishing the relation of those with the Hamiltonian. The Schrödinger equation is $i\hbar\frac{d}{d t} |n(\lambda)\rangle = \hat{H}(\lambda)|n(\lambda)\rangle$, telling you how the quantum ...


7

How do you get the Hamiltonian formula for transition metal dichalcogenides? How to derivate the expression above (equation (1))? You can use the $ k \cdot p$ method to derive this effective Hamiltonian. Please take a look at this paper for the details. $k \cdot p$ theory for two-dimensional transition metal dichalcogenide semiconductors. What does the ...


7

You can just exchange the $\mu,\nu$ indices to verify the antisymmetry: $$ \Omega_{n,\mu\nu}(\mathbf{k})=\partial_{\mu}A_{n\nu}(\mathbf{k})-\partial_{\nu}A_{n\mu}(\mathbf{k})\\ \Rightarrow \Omega_{n,\nu\mu}(\mathbf{k})=\partial_{\nu}A_{n\mu}(\mathbf{k})-\partial_{\mu}A_{n\nu}(\mathbf{k}) = - \left( \partial_{\mu}A_{n\nu}(\mathbf{k})-\partial_{\nu}A_{n\mu}(\...


7

You can use QE.6xxx with the support of the hdf5 library. To realize that purpose, you should add the following command when you compile QE: --with-hdf5=yes or take a look at the official guide. Then the saved wavefunction can be manipulated with the python package h5py. import h5py read_wf=h5py.File("wfc1.hdf5")


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