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:\[H = [...]\]
A: Born-Oppenheimer; H_e+H_N+H_{e-N}

Q: How do you justify the Born-Oppenheimer approximation?
[...]
A: Mass of the nuclei much larger than the mass of the electron.

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

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

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)