24
votes
Accepted
Do we know for sure that all atomic and molecular wavefunctions decay exponentially as r goes to infinity?
I'll answer this question from the theoretical side. The exponential behavior follows simply from the Schrödinger equation. Consider the one-electron Schrödinger equation:
$$
(-\frac{1}{2}\nabla^2 + V(...
- 9,076
18
votes
What are the situations/problems where Born-Oppenheimer approximation is invalid?
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 ...
- 701
18
votes
What are the situations/problems where Born-Oppenheimer approximation is invalid?
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 ...
- 7,957
15
votes
Which method gives the most accurate electron density, and how can it be verified experimentally?
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 ...
- 18k
15
votes
Accepted
Does the Schrödinger equation have unique solutions?
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 ...
- 33.6k
15
votes
How is Poier and Jensen's Bond Capacity Model Parameterized and Optimized?
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 ...
- 1,311
15
votes
Accepted
How are continued fractions related to quantum materials?
In the paper "A Continued-Fraction Representation of the Time-Correlation Functions", generalized susceptibilities and transport coefficients for materials are obtained using a continued-...
- 33.6k
13
votes
How are continued fractions related to quantum materials?
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 ...
- 1,607
12
votes
Derivation of Slater-Koster equations
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 ...
- 4,696
12
votes
Accepted
How do I figure out how many monomer configurations I need for my simulation?
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 ...
- 9,045
12
votes
How can we say that the KS equation is describing a noninteracting many-electron system?
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. ...
- 376
12
votes
Accepted
How to determine which levels correspond to each other in LS coupling and jj-coupling?
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 ...
- 33.6k
11
votes
Accepted
How exact is DFT, really?
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(\...
- 1,883
11
votes
What does “strongly correlated” mean?
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 ...
- 14.9k
11
votes
Accepted
What does it mean when the first order correction energy is 0?
The first-order correction $\langle \psi_0|H|\psi_0\rangle$ vanishes since $H|\psi_0 \rangle$ is orthogonal to $|\psi_0\rangle$. You need to go higher in perturbation theory to get a correction that ...
- 18k
10
votes
Accepted
Anomalous Quantum Hall Effect
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 ...
- 4,696
10
votes
What are the situations/problems where Born-Oppenheimer approximation is invalid?
The Born-Oppenheimer approximation comprises two different approximations:
Adiabatic separation of electron and nuclear coordinates
Semi-classical approximation for nuclei
Previous answers have ...
- 6,582
9
votes
Can Machine Learning lead to the more accurate theories and methods for matter modeling?
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 ...
- 7,281
9
votes
Accepted
Dynamic phase in an adiabatic system
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 ...
- 33.6k
9
votes
What are the situations/problems where Born-Oppenheimer approximation is invalid?
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 ...
- 1,073
9
votes
Accepted
How can we say that the KS equation is describing a noninteracting many-electron system?
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 ...
- 1,883
9
votes
What does “strongly correlated” mean?
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 ...
- 3,813
9
votes
What does “strongly correlated” mean?
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 ...
- 33.6k
9
votes
How spin-orbit coupling makes spin-forbidden reactions possible?
TL;DR: The statement $[H,S^2]=0$ indeed holds for no-pair (i.e. without considering the production of electron-positron pairs) calculations of hydrogen-like ions, but not for general atoms and ...
- 9,076
8
votes
What are the situations/problems where Born-Oppenheimer approximation is invalid?
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 ...
- 852
8
votes
What will break the time-reversal symmetry?
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 ...
- 2,332
8
votes
What will break the time-reversal symmetry?
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, ...
- 4,696
8
votes
Help with translating Hamiltonian into matrix
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 ...
- 33.6k
8
votes
Accepted
Ground State energy trick for many-body electronic structure calculations?
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 $...
- 9,076
8
votes
Accepted
Why is chemical bond dissociation difficult to simulate?
Some hints : all methods that attempt to describe the ground state of a many-body electronic system are often reduced to one electron in a mean-field or effective potential that includes the effects ...
- 896
Only top scored, non community-wiki answers of a minimum length are eligible