Questions tagged [born-oppenheimer-approximation]
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Strange manipulation of Hamiltonian operator and gradient
I'm reading M. V. Berry's Quantal Phase Factors Accompanying Adiabatic Changes and came across an unfamiliar identity in eq. (8), namely $\langle m | \nabla _Rn \rangle = \frac{\langle m | \nabla_R \...
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Entanglement in the Born-Oppenheimer approximation
Consider a scenario where we aim to model an atom as a system consisting of two components: the nucleus and the valence electron.
In this approach, we simulate the behavior of the electron within the ...
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Born-Oppenheimer approximation in many-body perturbation theory
In order to obtain phonon spectrum, we usually do Born-Oppenheimer approximation and assume that the electrons are always at the ground state when the atoms move, and by calculating the force on each ...
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Understanding the effective potential Energy in the Born-Oppenheimer approximation
In the Born-Oppenheimer approximation, the effective potential energy, is the potential energy that an electron gains when considering all the inter-particle interactions in a molecule? Said in ...
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Is it a coincidence to calculate Rutherford scattering using the first order Born approximation?
It is well-known that we can use Born first approximation to calculate the differential cross section. To be precise, firstly we modify the potential as Yukawa potential
$$V(r)=\dfrac{\alpha e^{-r/a}}{...
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Clarification on a Detail in the Born-Oppenheimer Approximation
In treatments of the Born-Oppenheimer approximation, people take the adiabatic states to be real eigenfunctions of the electronic Hamiltonian. The Wikipedia article states "The electronic wave ...
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Born-Oppenheimer approximation: electronic wave function and nuclear kinetic energy term?
The Born-Oppenheimer approximation in a nutshell
Given a molecule's Schrödinger equation
$$
[\hat{T}_N + \hat{T}_e+\hat{V}_{NN}(\textbf{R})+\hat{V}_{eN}(\textbf{r}, \textbf{R})+\hat{V}_{ee}(\textbf{r})...
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Non-additivity of higher order terms in the intermolecular potential
The intermolecular potential energy can be written as
$$u(r_{1},...,r_{N})=\sum_{i<j}^{N} u_{2} (r_{i},r_{j}) + \sum_{i<j<k}^{N} u_{3} (r_{i},r_{j},r_{k})+...$$
where the nuclear coordinates ...
user27771
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Second born amplitude for the electron-hydrogen scattering
can someone explain how could they got the result of the second born formula written in equation 2.36 from 2.32
I failed to demonstrate it.
The image from Byron 1973 here the link
enter link ...
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Hilbert space of a diatomic molecule
In molecular quantum mechanics, it is very common to model a diatomic molecule as a two-level harmonic oscillator with vibrational levels lying within the electronic states:
In most of the textbooks ...
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Expected value of Hamiltonian with operators in atomic units [closed]
Hello I am currently working on a problem sheet for university, and am confused on how to solve it correctly.
We deal with a helium atom and are asked to express the Hamiltonian in atomic units. This ...
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What steps are exact in Born-Oppenheimer and adiabatic approximation?
I am reading about adiabatic and Born-Oppenheimer approximation and I can't understand what steps are exact and what steps are approximations.
Both these approximations are based on the idea that as ...
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Mathematical justification of the Born-Huang expansion in the derivation of the Born-Oppenheimer Approximation
In the book K. Huang and M. Born, Dynamical Theory of Crystal Lattices (1954, Appendix VIII) and also in the Wikipedia article https://en.wikipedia.org/wiki/Born%E2%80%93Oppenheimer_approximation#...
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Why do we treat molecular vibrations as linear harmonic oscillations and not 3D isotropic?
Born-Oppenheimer approximation leads to this equation for the eigenfunctions and eigenvalues describing molecular vibrations: $$\left[-\frac{\hbar^2}{2\mu}\frac{1}{R^2}\frac{\partial}{\partial R}\left(...
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Does the Born-Oppenheimer (adiabatic) approximation Ansatz allow excitations?
The way I understand the Born-Oppenheimer approximation is as follows:
In the general solid Hamiltonian for $N$ electrons at $3N$ coordinates $r$ and $M$ ions at $3M$ coordinates $R$,
$$
H = T_e + T_I ...
user301288
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Born-Oppenheimer approximation: calculating the expectation values for the molecular hamiltonian
Here's what I've understood so far.
The Hamiltonian for a molecule made of $N$ nuclei, and $n$ electrons is:
$${\cal \hat H} = \underbrace{{-\frac {\hbar^2}{2} \sum _{\alpha =1}^N } \frac {\nabla_\...
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Localized nuclei in Born Oppenheimer approximation
In the discussion of the Born Oppenheimer approximation, we think about the electron eigenstates as a function of fixed nuclear positions $R$. It is then assumed that the nuclear positions vary slowly,...
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Difference between Fixed Nuclei approximation, Born-Oppenheimer approximation, adiabatic nuclei approximation
I was reading some papers and depending on the author some would use the terms Fixed Nuclei approximation, Born Oppenheimer approximation, adiabatic nuclei approximation almost interchangeably whilst ...
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Correct indices on coupling matrix elements
Lets say i have an initial state $|i\rangle$ and a final state $|f\rangle$. A transition from $|i\rangle \rightarrow |f\rangle$ is coupled by an operator $\hat O$. Is the relevant coupling matrix ...
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Can the sign of non-relativistic potentials be determined from QFT amplitudes?
In many QFT books and courses people use a non-relativistic approximation to extract potentials from QFT. They do so by comparing the scattering in QFT with the scattering in Quantum Mechanics, in the ...
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Why non-adiabatic matrix is an antisymmetric matrix?
In deriving coupling term beyond Born-Oppenheimer approximation, there is a term non-adiabatic coupling term written as:
$$
\tau_{ij} = \langle\zeta_i| \nabla\zeta_j\rangle
$$
where $\zeta$ here is ...
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About Born-Oppenheimer approximation
I was just going through random lecture notes on Born-Oppenheimer approximation where I came across the following statement:
We first invoke the Born-Oppenheimer approximation by recognizing that, ...
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What is a nuclear wave packet?
What is the definition of a nuclear wave packet ? I often see the term used but i don't know how it is defined. It seems to be connected to the Born-Oppenheimer approximation. Is it defined there and ...
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Why first-order Born Approximation doesn't satisfy optical theorem?
First-order Born Approximation in Quantum Mechanics states that scattering amplitude is a Fourier transform of potential:
$$
f(\theta) = \int d^3 r^{\prime} e^{-i (\bf k - k_i)r^{\prime}} V(r^{\prime}...
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Problems with a derivation of the Born-Oppenheimer approximation
I am trying to follow the derivation of the Born-Oppenheimer approximation proposed on Wikipedia.
I know this is not the sexiest source, but i thought it could give a nice first glimpse. But actually,...
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Born-Oppenheimer approximation
In B-O approximation, one of the basic assumptions is that the total many-body wavefunction can be expanded as the following:
$$\Psi(\bf{r},\bf{R})=\sum_n\phi_n(\bf{R})\psi_n(\bf{r;\bf{R}})$$
where $\...
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Born-Oppenheimer approximation and perturbation theory
In the book Molecular Physics by Demtroder there is an explanation of the Born-Oppenheimer approximation and the adiabatic approximation in terms of a perturbative series. The Hamiltonian is $H_0 + T_\...
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Why do only lattice vibrations contribute to the specific heat?
I would like to justify, that I can write the free energy of a real material (many-body system) at ambient conditions as
$$
F(T) = E_0^{\text{e}} + F^\text{v}(T)
$$
where $E_0^{\text{e}}$ is the ...
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Hamiltonian in denominator
The second-order Born approximation for the scattering amplitude is given by
$$ f^{(2)}(k_f, k_i) = -\frac{m}{2\pi\hbar^2} \left<k_f\left| V \frac{1}{E-H_0+i\eta}V \right|k_i\right>, $$
where ...
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Born-Oppenheimer approximation, electronic/nuclear wavefunction and product form
As I understand it, the main (or at least an important) statement of the Born-Oppenheimer Approximation is that the electronic and nuclear motion are separated and that the total wavefunction $\Psi$ ...
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Mathematical statement of the Born-Oppenheimer approximation
I have been looking up a formal mathematical definition of the Born-Oppenheimer approximation. I have thus far come across two (my wording):
Definition 1
The Born-Oppenheimer approximation is ...
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Resource for Born-Oppenheimer approximation
I'm studying molecular physics but I'm failing in finding a good book with an accurate description of the Born-Oppenheimer approximation. I've tried some resource on the Internet and the McQuarrie's ...
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Why does electron respond almost instantaneously on nucleus' displacement due to the difference in mass of it and the nucleus?
In Born Oppenheimer Approximation, we take note of the great difference between the mass of the electrons and nuclei. But, I have not been able to understand this statement quoted from Molecular ...
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Solving Schrödinger's Equation for the electronic energies of the Molecular Ion Hydrogen H2+ in the Elliptic coordinate system
Electronic Energies of Molecular Ion Hydrogen $H_2^{+}$
$r_1$ is the distance between the proton $1$ and the electron.
$r_2$ is the distance between the proton $2$ and the electron.
$R$ is the ...
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Form Factor in Rutherford Scattering
My question relates to Rutherford Scattering of particles. When we calculate the "differential cross-section" expression for a nucleus with finite size, it is said that the expression is almost the ...
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Why can we not apply perturbation theory in Born-Oppenheimer approximation
In Weinberg's Lectures on Quantum Mechanics, he mentions
Unfortunately, we cannot simply use first-order perturbation theory, with $T_{nuc}$ taken as the perturbation and the state vectors $\Phi_{...
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Franck Condon Principle and Born Oppenheimer approximation
My question here is purely fundamental. I am confused with the concept in Franck Condon (FC) principle and Born Oppenheimer (BO) approximation. The FC principle is in accordance with the BO ...
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Derivative with respect to the nuclear coordinates in the Born–Oppenheimer approximation
Reading few sources on the Born–Oppenheimer approximation I don't understand one particular thing.
If you look for example here (PDF, 70 KB) and focus attention on equations 14 and 15 than it is ...
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Born-Oppenheimer separation in Dirac bra-ket notation
Most derivations I have seen of the Born-Oppenheimer approximation are made using wave-functions. To understand it better, I was trying to write a derivation using Dirac notation, but I am stuck. I am ...
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Born Oppenheimer Approximation: Why can any molecular state be represented as a linear combination of electronic states?
in the Born Oppenheimer Approximation, one expands the molecular wavefunction $\Psi(x,X)$ in terms of the electronic wavefunctions $\phi(x;X)$:
$$\Psi(x,X)= \sum_k(c(X)_k\phi(x;X)_k)$$
($x$ are the ...
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Born-Oppenheimer Approximation equivalent to Tensor-product ?
If you have a wave function $\Psi$ of a system consisting of an electron and the vibrational modes of the crystal, THEN we represent the wavefunction $\Psi%$ to be in the Hilbert Space formed by the ...
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The Born-Oppenheimer approximation and muonic molecules
Does the Born-Oppenheimer approximation fail for muonic molecules (i.e. molecules where one or more electrons are replaced with muons)?
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