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Questions tagged [born-oppenheimer-approximation]

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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 ...
2
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1answer
363 views

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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1answer
127 views

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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2answers
171 views

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 $\...
0
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1answer
96 views

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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0answers
70 views

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 ...
1
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1answer
138 views

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 ...
1
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1answer
192 views

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 $\Psi$ ...
5
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2answers
599 views

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 ...
1
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0answers
102 views

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 ...
0
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2answers
91 views

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 ...
10
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1answer
2k views

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 ...
2
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1answer
595 views

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 ...
5
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1answer
383 views

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_{...
2
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3answers
5k views

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 ...
3
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1answer
300 views

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 ...
7
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1answer
753 views

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 ...
4
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2answers
492 views

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 ...
6
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1answer
505 views

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 ...
6
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1answer
330 views

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)?