Title: ETH_Sem4 / Advanced_Kinetics / Quick_Concepts

Q: [...]: [...]
A: Adiabatic approximation; The electronic system is much faster than the lattice system and can almost instantaneously adapt to new ion positions.

Q: [...] Hamilton operator:
A: Born-Oppenheimer

Q: How do you justify the Born-Oppenheimer approximation (multiple Ways)?
- [...]
- [...]
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A: Mass of the nuclei much larger than the mass of the electron. Therefore their dynamics are on a very short time scale and the Nuclei only feel the "mean elctron field". tHe later is rather a consequence of BOA as the electronic WAve function is effectifly assuemd to be independant of the nucelar coordiantes. This is the reiher Argument "infinit masses" effectfly means very slow (relation of velocity and momentum), therefore it is related in to the LZ-Argument; LZ-Transition Propabiliy Argument: Strong Coupling between Diabats if v small therfore we stay on one adiabatic surface i.e. BOA is valid; Argument using Non-Adiabtic Couplings \(\hat{\Lambda}_{mn} \propto \hat{F}_{mn} \propto \frac{1}{V_n-V_m}\): If the two surfaces are "close to each other" BOA is violated as we need to consider more then one surface.

Q: If the potential-energy surface exhibits a minimum at \(\vec R_0\), and the dynamics is known [...], it is a commonly used approximation to replace the PES by a [...]. This is typically done by [...].
A: to oscillate with small amplitudes around it; harmonic oscillator; truncating the Taylor series at second order

The description of molecular dynamics in electronically excited states almost systematically requires a description beyond the Born-Oppenheimer approximation. Such descriptions can be made either in the (unique) adiabatic representation or in one (of infinitely many) diabatic representations. The ensemble of electronic states of any of these representations forms a complete basis.

Q: Adiabatic representation:
It is possible to go beyond the Born-Oppenheimer approximation by writing a formally exact solution \[\psi(\vec r, \vec R) = [...]\]where \(\varphi_n\) are the set of all electronic states solving\[\hat H_e \varphi_n(\vec r, \vec R) = [...]\]
A: \sum_n \varphi_n(\vec r, \vec R) \chi_n(\vec R); V_n(\vec R) \varphi_n(\vec r, \vec R)

Q: [...]
The alternative to the adiabatic basis is a diabatic basis which is chosen such that [...]. There are infinitely many ways of choosing a diabatic basis. The simplest version consists in choosing the electronic basis functions to be [...]
A: Diabatic representation; all electronic matrix elements of the derivatives \(\dede{}{R}\) and \(\dede{^2}{R^2}\) are as small as possible; independent of the nuclear geometry \(\varphi(\vec r, \vec R) \to \varphi(\vec r, \vec R_0)\).

Q: Converting adiabatic representation to diabatic
Consider simplest case with only two relevant adiabatic states and suggest trfs:\[\bigg(\matrix{\tilde \varphi_1 \\ \tilde \varphi_2}\bigg) = [...]\bigg(\matrix{\varphi_1 \\ \varphi_2}\bigg)\]where we choose \(\gamma(\vec R)\) such that \[\gamma(\vec R) = [...]\]using \(\hat F_{mn} = \braket{\varphi_m | \nabla \varphi_n}\). Then the derivative terms vanish and in the diabatic basis our SE becomes\[[...] \bigg(\matrix{\tilde \chi_1 \\ \tilde \chi_2} \bigg) = E \bigg( \matrix{\tilde \chi_1 \\ \tilde \chi_2} \bigg)\]
A: \bigg(\matrix{\cos(\gamma(\vec R)) & \sin(\gamma(\vec R)) \\ -\sin(\gamma(\vec R)) & \cos(\gamma(\vec R))}\bigg); \int_0^R F_{12}(R') dR'; \bigg(\matrix{\hat T_N + U_{11} (\vec R) & U_{12}(\vec R) \\ U_{21}(\vec R) & \hat T_N + U_{22} (\vec R)} \bigg)