Abstract
This chapter presents a new methodological development that represents an improvement over ECEID. Through the use of a canonical transformation on the ECEID Hamiltonian, it is possible to include the dynamics of an extra classical degree of freedom in the model, a semiclassical oscillator position. Physically, we can think of it as a representation of the oscillator centroid, but it enters the derivation just as a time-dependent reference position.
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Notes
- 1.
Where now \(\hat{a}^{(\dagger )}_\nu \) is the creation (annihilation) operator associated with the canonical displacement \(\hat{\xi }_\nu \) and momentum \(\hat{\pi }_\nu \) of oscillator \(\nu \).
- 2.
This alternative approach is also applicable to ECEID in Chap. 5. There, the formalism with the auxiliary operators was chosen for consistency with the method as it was originally developed and presented in [3, 4]. Here, the derivation follows a sleeker approach that does not involve the definition of any auxiliary operators. The dynamics resulting from the two approaches is equivalent.
- 3.
For example, the contribution of the electron-phonon coupling term \(-\hat{F}_\nu \hat{\xi }_\nu \) in Eq. (9.19) is
- 4.
In Appendix D, a recent development is shown where the double (de)excitations approximation is not invoked.
- 5.
\(\hat{\mu }_\nu (t)\) and \(\hat{\lambda }_\nu (t)\) in Eqs. (9.28) and (9.29) contain terms summed over \(\nu '\). Therefore, in ECEID xp, the correlation operators depend on all the oscillators. The parallelism of the method hinges on the fact that every operator associated with oscillator \(\nu \) depends only on purely electronic quantities and on oscillator-\(\nu \)-specific quantities. The implementation of the new formalism would reduce the effectiveness of the parallelism in the presence of many oscillators. A possible solution, consistent with the spirit of ECEID’s approximations is to insert a \(\delta _{\nu \nu '}\) in Eqs. (9.28) and (9.29) and consider only operators depending on \(\nu \).
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Rizzi, V. (2018). A New Development: ECEID xp. In: Real-Time Quantum Dynamics of Electron–Phonon Systems. Springer Theses. Springer, Cham. https://doi.org/10.1007/978-3-319-96280-1_9
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