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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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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SQE (split charge equilibration) method [closed]

Is SQE (split charge equilibration) method a type of DFT method? Can someone help me in understanding this method !
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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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6 votes
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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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9 votes
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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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1 answer
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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?

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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5 votes
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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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1 answer
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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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7 votes
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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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6 votes
1 answer
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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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8 votes
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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?

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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9 votes
1 answer
217 views

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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7 votes
1 answer
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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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12 votes
1 answer
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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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8 votes
1 answer
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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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12 votes
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300 views

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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8 votes
1 answer
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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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15 votes
3 answers
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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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11 votes
2 answers
473 views

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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6 votes
1 answer
61 views

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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7 votes
1 answer
128 views

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?

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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12 votes
1 answer
220 views

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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  • 2,091
8 votes
2 answers
364 views

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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8 votes
2 answers
259 views

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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15 votes
1 answer
233 views

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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10 votes
1 answer
118 views

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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14 votes
1 answer
467 views

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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16 votes
1 answer
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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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6 votes
0 answers
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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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12 votes
1 answer
214 views

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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6 votes
1 answer
173 views

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}=...
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11 votes
1 answer
522 views

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 ...
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3 votes
1 answer
146 views

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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11 votes
2 answers
329 views

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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11 votes
1 answer
345 views

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 ...
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12 votes
1 answer
143 views

Adiabatic equation related to the Berry phase for lambda with first order terms

Consider the following derivation in David Vanderbilt's book "Berry Phases in Electronic Structure Theory - Electric Polarization, Orbital Magnetization and Topological Insulators" (2018, ...
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9 votes
2 answers
344 views

Dynamic phase in an adiabatic system

I am trying to understand the Berry phase through the evolution of a system that evolves adiabatically. Schrodinger's equation is: \begin{equation} H(\lambda)|n(\lambda)\rangle=E_n|n(\lambda)\rangle \...
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9 votes
2 answers
417 views

Can Machine Learning lead to the more accurate theories and methods for matter modeling?

There's no doubt about it. Machine Learning (ML) is one of the hottest topics out there and it plays an important role in computational science. One application I have seen is to use ML and Density ...
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10 votes
1 answer
164 views

What are the types of Quantum Molecular Dynamics (QMD)?

In similar spirit to recent questions on Quantum Monte Carlo, ab-initio Molecular Dynamics, types of SCF, and others, I would like to ask: What are the types of Quantum Molecular Dynamics (QMD)? As I ...
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5 votes
1 answer
73 views

Property related with Berry curvature: $\Omega_{n,\mu\nu}=-\Omega_{n,\nu\mu}$

I read in David Vanderbilt's book named "Berry Phases in Electronic Structure Theory - Electric Polarization, Orbital Magnetization and Topological Insulators" the definition of Berry ...
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21 votes
2 answers
759 views

Derivation of Slater-Koster equations

I am trying to derive the Slater-Koster equations (Table. 1 of Ref. 1) for the two-centre approximation of hopping integrals between atomic orbitals. I understand that Slater-Koster approximates the ...
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14 votes
1 answer
175 views

On mass polarization terms

In Jensen's Introduction to Computational Chemistry it says that the total non-relativistic Hamiltonian operator, transformed to the center of mass system, can be written, in atomic units, as $$ \hat{...
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