Questions tagged [quantum-mechanics]

Refers to the branch of mechanics that deals with the mathematical description of the motion and interaction of subatomic particles, in particular electrons.

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How do I get the wavelength of light from a diffraction experiment using Python? [closed]

What is the procedure after using a diffraction grating (device used to see the spectrum) using python? We would see the entire electromagnetic spectrum emitted by an object but what then? how do I ...
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Equation (4) in Hohenberg-Kohn Paper

In their landmark paper of 1964, Inhomogeneous Electron Gas, Hohenberg and Kohn wrote down: We shall be considering a collection of an arbitrary number of electrons, enclosed in a large box and ...
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How do scalar states with different multiplities make spin-forbidden reactions possible?

This question is related to How spin-orbit coupling makes spin-forbidden reactions possible? - Matter Modeling Stack Exchange Even more importantly, [H,S2]=0 fails for multi-electron systems even ...
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How spin-orbit coupling makes spin-forbidden reactions possible?

My question is related to the Wikipedia page of spin-forbidden reactions https://en.wikipedia.org/wiki/Spin-forbidden_reactions When a reaction converts a metal from a singlet to triplet state (or ...
4 votes
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What is a "transient" state?

I was analyzing this source code of the Ising model. I found the term "transient state". I also found the term in this text: There are two absorbing states in this Markov chain because once ...
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What is the best algorithm for ERIs of contracted gaussian atomic orbitals

I am currently working on python script that does Hartree-Fock-Calculations. I try to avoid packages as good as possible. I've figured out how to calculate the one electron integrals when it comes to ...
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What occupation should be used for non metallic energetic materials in Quantum espresso?

I am trying to understand the Quantum espresso input file and I intend to do DFT calculation on an Energetic material (TKX-50) consisting of C, H, O, AND N. I have seen that "smearing" ...
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Cross-post: Matrix elements <n,k|x|n',k'> for Bloch states

Cross posted at Physics.SE I believe this is just elementary QM, but I'm getting awfully confused. The question is drawn from this paper on Wannier-Stark localization (but is self-contained). Let: \...
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Green function KMS boundary condition

How to obtain the relation between $G_{ab}(\tau,0,0,0)$ and $G_{ab}(\beta-\tau,0,0,0)$ for two-particle fermion Green function $$G_{ab}(\tau_1,\tau_2,\tau_3,\tau_4)=\langle \mathcal{T}a^\dagger(\tau_1)...
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How are the exponents in a STO-nG basis set obtained?

STO-3G has exponent parameters which can be found in a reference book. Can these parameters be calculated using, for instance, a Hartree–Fock equation for one atom? Or using another way?
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What does it mean when the first order correction energy is 0?

Suppose I have the following Hamiltonian to start $$ H_0 = \begin{pmatrix} 0 & 0 & 0 & 0\\ 0 & 0 & 2 & 0\\ 0 & 2 & 0 & 0\\ 0 & 0 & 0 & 0 \end{pmatrix} $$...
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Do we know for sure that all atomic and molecular wavefunctions decay exponentially as r goes to infinity?

Slater type orbitals (STO) are considered to be more accurate than gaussian type orbitals (GTO) for atomic and molecular QM calculations because - among other reasons - they decay with $e^{-\alpha r}$ ...
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How to numerically calculate quantum state distance using quantum metric?

In Ran Cheng's review of the quantum geometric tensor, eq. (11) gives the tensor as: $$ Q_{\mu\nu}=\sum_{n\neq 0}\frac{\langle\phi_0|\partial_\mu H|\phi_n\rangle\langle\phi_n|\partial_\nu H|\phi_0\...
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How to solve time-dependent Schrodinger equation and plot trajectory on Bloch sphere?

Let's say I have a $2 \times 2$ Hamiltonian that I am solving using the time-dependent Schrodinger equation: $$ i \frac{d}{dt} |{\Phi}\rangle=H|{\Phi}\rangle.\tag{1} $$ Consider a generic Hamiltonian $...
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Calculating the excited state dipole moment

I am trying to calculate the dipole moment of an excited state, using "Gaussian" software. My input commands were: ...
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Electric quadrupole operator for periodic systems

Considering one component of the electric quadrupole operator $\hat{r}_x \hat{r}_y$, I'm wondering if the following equalities hold: $$ \langle u_{n\mathbf{k}} | \hat{r}_x \hat{r}_y |u_{m\mathbf{k}} \...
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Quantum mechanical antireflection coating; what quantity would be analogous to index of refraction?

Background Quasiparticle interference (QPI) is a technique that can be used to study 2D surface state band structure; carrier reflections at boundaries or impurity sites will interfere producing ...
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How to get energy eigenstates using plane wave basis wavefunction .hdf5 output from Quantum ESPRESSO? [closed]

When compiled with the HDF5 flag on, Quantum ESPRESSO (QE) saves wavefunctions in the .save folder. At each k-point, we have an $m\times n$ matrix, where $m$ are the complex number entries of ...
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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}...
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How to formulate the second quantization of Dzyaloshinskii-Moriya interaction?

The Dzyaloshinskii-Moriya interaction (DMI) existing in the interface of the ferromagnetic insulator and the metal with strong spin-orbit coupling (SOC) is shown below. Mathematically, it can be ...
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Tools for symbolic calculations of quantum transport using the Keldysh NEGF formalism [closed]

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 <...
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Confusions in interpreting Dirac strings (wormholes) in Haldane's Hilbert space picture ft. two tori joined by strings at gapless points

I had a question about Haldane's wormhole interpretation (picture below). I believe he first proposed it in his paper Berry Curvature on the Fermi Surface: Anomalous Hall Effect as a Topological Fermi-...
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Help with definitions in k-space twisted bilayer graphene model

I am trying to numerically do calculations using Eq. 8 of MacDonald's simple model for twisted Bilayer graphene. I only want to calculate the Berry phase. However, I don't think I have my definitions ...
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Valleys and time reversal symmetry (Zeeman effect)

Right now I'm focused on transition metal dichalcogenides. These compounds in the Brillouin zone have valleys in the valence band and in the conduction band at points K and -K. From what I've seen of ...
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Example of 2D k-space models allowing interband transitions after a closed-loop trajectory? [closed]

I have been mainly exposed to 2D momentum-space condensed matter models in the context of Berry-related topology. I now want to study models where, if I take a closed loop in momentum space, I will ...
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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 ...
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Help with Definitions in Numerical Calculation of Multiband Berry Phase

In the third chapter of Vanderbilt's book, they discuss the so-called multiband parallel transport and provide a scheme for numerical calculations that is similar to the single band case (where the ...
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How to numerically solve real-space 1D time-independent Schrodinger equation using 2D momentum-space Hamiltonian?

Consider the usual simple 2-level graphene Hamiltonian with mass in momentum-space where: $$ H(k,V)=-t \sum_{\delta} [\cos(k\cdot\delta)\sigma_x-\sin(k\cdot\delta)\sigma_y+V\sigma_z], $$ where $t$ is ...
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Help with understanding topologically-protected edge states in domain wall systems

Let's say that I have a simple domain wall system for the following Hamiltonian with added on-site potential $M(x)$: $$\tag{1} H(k,M)=-t \sum_{\delta} [\cos(k\cdot\delta)\sigma_x-\sin(k\cdot\delta)\...
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How to determine which levels correspond to each other in LS coupling and jj-coupling?

In the LS coupling scheme, L, S and J are good quantum numbers; whereas in jj-coupling scheme, j1, j2 and J are good quantum numbers. I have learnt how to get the term symbols for LS coupling, and for ...
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How are molecular rotational states modeled within the Born Oppenheimer Approximation for polyatomic molecules?

Molecular vibrational states associated with an adiabatic electronic state within the Born-Oppenheimer Approximation are typically defined by doing the harmonic approximation for the potential at an ...
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What does “strongly correlated” mean?

For quantum many body problem, there is a common terminology “strong correlated systems” that appears in different context. However, it seems that the definition of it is ambiguous and sometimes ...
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How can we say that the KS equation is describing a noninteracting many-electron system?

Based on HK's two theorems, the density functional theory was built. Because one can't find the universal energy functional $F_{HK}[n(r)]$, Kohn and Sham further proposed the Kohn-Sham ansatz: mapping ...
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Example of a standard/archetypal/simple 4-band un-gapped condensed matter model with analytic results?

I am looking to study Berry phase-like phenomena in an un-gapped material model. However, I am having trouble finding a widely-used 4-band model with analytic expressions for wavefunctions and ...
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How is the error in a wavefunction related to the error in energy?

While studying the variational method in McQuarrie's Quantum Chemistry, I came across the following problem: to relate the difference between an approximation $\phi$ and the exact ground-state wave ...
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Why does numerical computation of Berry curvature give me a correct Berry phase when it is supposed to diverge? [closed]

I implemented the standard numerical algorithm for calculating the Berry curvature in MATLAB. For a given 2D system, I can visualize the Berry curvature over parameter space. If I sum the Berry ...
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What is the best program to manipulate numerical DFT wavefunctions to calculate custom matrix elements?

For pedagogical reasons, I am looking for ways to calculate quantum-mechanical quantities such as $\langle m | \dot{m}\rangle, \langle m | \dot{n}\rangle, \langle m | \ddot{n}\rangle$ using ...
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What will break the time-reversal symmetry?

Specifically, I am interested in the time-reversal symmetry $(\mathcal{T})$ in quantum mechanics. The $\mathcal{T}$ is considered as an anti-unitary operator, namely: $$\mathcal{T}\psi(\vec{r},t)=\psi^...
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How do I figure out how many monomer configurations I need for my simulation?

I am running a quantum mechanical simulation on Psi4 for a certain number of monomers to generate data for a database. Some of these monomers are short, while some are long. At the moment, I am ...
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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_{...
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How to understand the formation of covalent bond from the viewpoint of wavefunction interaction?

The formation of the covalent bond is usually understood from the electron sharing, like explained by the following figure: However, this simple physical picture seemingly hinders many things. In ...
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How exact is DFT, really?

It is often claimed (e.g. here), that Density Functional Theory is in principle exact. This seems to be a very strong statement to me. Are all current limitations only of a technical nature rather ...
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Does the Schrödinger equation have unique solutions?

I am learning DFT and the Hohenberg Kohn Theorem of Existence. It says that there is a one-to-one correspondence between the external potential and the density. However the proofs that I have seen ...
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Evaluating $C_v$ for one mole $H_2$ molecules in a quantum simulation [closed]

Given an ensemble of $N$ diatomic molecules, we know that the rotational partition function is given by $$Z_r = z_r^N$$ where $$z_r = \sum_{l} (2l+1)e^{-Kl(l+1)}$$ where $K = \beta \hbar ^2 /2I$. I ...
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How to explain Density Functional Theory results to an experimentalist?

When we present our Density Functional Theory simulation results e.g. lattice parameters, stacking fault energies, band gaps, etc. to people who are experimentalists then the very first question ...
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Velocity operator for a Hamiltonian that satisfies the generalized Schrodinger equation

The velocity operator is defined as $\mathbf{v}=i[H,\mathbf{r}]$ for the Hamiltonian $H$ satisfying $H\psi=\epsilon \psi$. This can be obtained from the Ehrenfest theorem. I'm wondering if $\mathbf{v}=...
11 votes
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Which method gives the most accurate electron density, and how can it be verified experimentally?

I read a paper from Science (DOI https://doi.org/10.1126/science.aah5975), “Density functional theory is straying from the path toward the exact functional”. They were trying to make a point that ...
3 votes
1 answer
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Interpretation of electronic band structure diagram

I want to understand the electronic band structure diagram of the following image, corresponding to $\text{MoS}_2$ (TMD): I read about DFT (density functional theory). DFT is based on solving the ...
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How to derive the effective Hamiltonian of two-dimensional TMDCs monolayers?

TMDs are transition metal dichalcogenides and have the chemical formula MX$_2$ where M is the transition metal and X is the chalcogen. An example of a TMD is MoSe$_2$. I would like to demonstrate that ...
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Anomalous Quantum Hall Effect

I am studying the transition metal dichalcogenides (TMDs) and I have seen webinars and articles that said that these materials exhibited the anomalous quantum Hall effect related to the curvature of ...