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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What are the situations/problems where Born-Oppenheimer approximation is invalid?

We use the Born-Oppenheimer approximation in both Hartree-Fock method and DFT. What are the problems where we cannot use this approximations.
Thomas's user avatar
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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 ...
kat's user avatar
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How are continued fractions related to quantum materials?

In my spare time, I have been studying and analysing continued fractions. I was having a conversation with someone on Discord in a Mathematics server and he was telling me that continued fractions ...
Mr Pie's user avatar
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19 votes
1 answer
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How is Poier and Jensen's Bond Capacity Model Parameterized and Optimized?

I am working on adding a charge polarization model into my own research and have been exploring a few approaches. One of the most attractive options is the Bond Capacity (BC herein) model of Paolo and ...
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1 answer
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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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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 ...
GalliumBeryllium's user avatar
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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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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3 answers
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What is the best program to manipulate plane-wave 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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15 votes
1 answer
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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_{...
Xiaoming Wang's user avatar
14 votes
1 answer
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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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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 ...
Jack's user avatar
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12 votes
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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 ...
S R Maiti's user avatar
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12 votes
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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 ...
TribalChief's user avatar
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12 votes
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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, ...
Carmen González's user avatar
12 votes
1 answer
282 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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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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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 ...
Quantum Bee's user avatar
11 votes
2 answers
613 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 ...
Carmen González's user avatar
11 votes
1 answer
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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 ...
Carmen González's user avatar
11 votes
1 answer
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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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10 votes
2 answers
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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 \...
Carmen González's user avatar
10 votes
1 answer
247 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 ...
Jack's user avatar
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10 votes
1 answer
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Time Evolution of the Hartree-Fock Wave Function

The Hartree-Fock Hamiltonian is given by $$ \hat{H} = \sum_i \hat{h}_i + \frac{1}{2}\sum_i\sum_j \left[\hat{J}_{ij} - \hat{K}_{ij}\right] $$ and by choosing a basis, the Hamiltonian can be written in ...
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2 answers
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Acceleration of 8110823001207866000 m/s^2 by using Coulomb's law: did I do it right?

I have this very simple function: ...
John T's user avatar
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9 votes
2 answers
961 views

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: ...
aerospace's user avatar
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9 votes
1 answer
794 views

Why is chemical bond dissociation difficult to simulate?

I'm a student doing quantum computing research trying to understand better what kinds of chemical systems are difficult to simulate on a classical computer and for which quantum simulation could be ...
Solarflare0's user avatar
9 votes
1 answer
335 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 ...
TribalChief's user avatar
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8 votes
2 answers
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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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What does it mean when the first order correction energy is 0?

Suppose I have the following Hamiltonian to start $$ H_0 = \begin{pmatrix} 0 & 0 & 0 & 0\\ 0 & 0 & 2 & 0\\ 0 & 2 & 0 & 0\\ 0 & 0 & 0 & 0 \end{pmatrix} $$...
KAJ226's user avatar
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8 votes
2 answers
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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^...
Jack's user avatar
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8 votes
2 answers
298 views

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 ...
Sha's user avatar
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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 ...
Hans Wurst's user avatar
8 votes
1 answer
138 views

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)\...
TribalChief's user avatar
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8 votes
1 answer
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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\...
TribalChief's user avatar
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8 votes
1 answer
82 views

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 ...
uhoh's user avatar
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8 votes
1 answer
357 views

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 ...
Carmen González's user avatar
7 votes
1 answer
310 views

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 ...
TribalChief's user avatar
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7 votes
1 answer
276 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 ...
1__'s user avatar
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7 votes
1 answer
353 views

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 $...
TribalChief's user avatar
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7 votes
0 answers
139 views

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 <...
Aschoolar's user avatar
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6 votes
2 answers
357 views

Density Functional Perturbation Theory

What is the difference between density functional perturbation theory (DFPT) and many-body perturbation theory (MBPT)? Also, please help me understand why approximations like PBEsol are considered as ...
AbPhys's user avatar
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6 votes
2 answers
145 views

How do I get the energy, if I have the wavefunction?

I recently wrote a simple HF SCF program for my understanding and realized that I do not fully understand such QM calculations. Suppose I have the ground state gaussian wave function of a system, $$ \...
ipcamit's user avatar
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6 votes
1 answer
190 views

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}...
More Anonymous's user avatar
6 votes
1 answer
111 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 ...
Carmen González's user avatar
6 votes
1 answer
343 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}=...
Xiaoming Wang's user avatar
6 votes
1 answer
390 views

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 ...
Jack's user avatar
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6 votes
1 answer
72 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 ...
TribalChief's user avatar
  • 2,351
6 votes
1 answer
130 views

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}} \...
Xiaoming Wang's user avatar
6 votes
1 answer
175 views

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 ...
TribalChief's user avatar
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