# Questions tagged [born-oppenheimer-approximation]

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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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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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### Existence of higher derivatives of molecular potential energy surfaces

I would like to know if anything is known about the mathematical regularity properties (continuity and existence/continuity of higher derivatives) of the (exact) Born-Oppenheimer potential energy ...
126 views

### 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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First-order Born Approximation in Quantum Mechanics states that scattering amplitude is a Fourier transform of potential: $$f(\theta) = \int d^3 r^{\prime} e^{-i (\bf k - k_i)r^{\prime}} V(r^{\prime}... 1answer 203 views ### Problems with a derivation of the Born-Oppenheimer approximation I am trying to follow the derivation of the Born-Oppenheimer approximation proposed on Wikipedia. I know this is not the sexiest source, but i thought it could give a nice first glimpse. But actually,... 2answers 215 views ### Born-Oppenheimer approximation In B-O approximation, one of the basic assumptions is that the total many-body wavefunction can be expanded as the following:$$\Psi(\bf{r},\bf{R})=\sum_n\phi_n(\bf{R})\psi_n(\bf{r;\bf{R}})$$where \... 1answer 143 views ### Born Oppenheimer approximation and perturbation theory In the book Molecular Physics by Demtroder there is an explanation of the Born Oppenheimer approximation and the adiabatic approximation in terms of a perturbative series. The Hamiltonian is H_0 + T_\... 0answers 75 views ### Why do only lattice vibrations contribute to the specific heat? I would like to justify, that I can write the free energy of a real material (many-body system) at ambient conditions as$$ F(T) = E_0^{\text{e}} + F^\text{v}(T) $$where E_0^{\text{e}} is the ... 1answer 181 views ### Hamiltonian in denominator The second-order Born approximation for the scattering amplitude is given by$$ 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 ... 2answers 236 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 ... 2answers 643 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 ... 0answers 108 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 ... 2answers 94 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 ... 1answer 3k 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 ... 1answer 765 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 ... 1answer 410 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_{... 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 ... 1answer 327 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 ... 1answer 822 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 ... 2answers 555 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 ...
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 ...