1. How do you build a good Hamiltonian for CEID? Andrew Horsfield, Lorenzo Stella, Andrew Fisher

2. Simplification of Chinese Sounds “A prominent and peculiar phenomenon in Chinese is that during last 3,000 years or so, the language has experienced repeated and massive losses of important phonological distinctions, and become increasingly homophonous. http://www.pinyinology.com/diaoHao2a/group/simple2m.html Something similar is happening to Correlated Electron-Ion Dynamics, except for the Hamiltonian.

3. Dynamics: adiabatic and non- adiabatic

4. Adiabatic dynamics For molecular dynamics we assume atoms move classically on a fixed energy surface. Electrons take a known configuration defined by instantaneous nuclear coordinates Do not respond to the nuclear velocity Can be computed accurately with ground state DFT Job done.

5. Non-adiabatic dynamics Heat can flow between electrons and nuclei From hot electrons to nuclei Heating of light bulbs Energy transfer in photoexcited conjugated polymers From hot nuclei to electrons Drag on atoms in radiation damage cascades Finnis et al, PRB 44, 567 (1991)

6. Ehrenfest Dynamics

7. Ehrenfest dynamics: the idea Simplest non-adiabatic method Relatively easy to implement efficiently Fast ions can excite electrons But hot electrons cannot heat ions Newton Schrödinger

8. Ehrenfest dynamics: the problem Ions see electrons as cold fluid (no fluctuations) Electrons see ionic fluctuations (could be hot) Number of electrons Force per electron

9. Ehrenfest dynamics: the problem

10. Correlated Electron- Ion Dynamics

11. Correlated Electron-Ion Dynamics (CEID) Current flows past one dynamic atom Classical kinetic energy Quantum kinetic energy J. Phys.: Condens. Matter 16 (2004) 8251–8266

12. Basis set formulation of CEID Original moment formulation does not converge systematically Now use harmonic oscillator basis for ions L. Stella, M. Meister, A. J. Fisher, and A. P. Horsfield, J. Chem. Phys. 127, 214104 (2007). L. Stella. R. P. Miranda, A. P. Horsfield, and A. J. Fisher, J. Chem. Phys. 134, 194105 (2011)

13. © Imperial College London Correlated Electron-Ion Dynamics (CEID) Retain form of molecular dynamics Note: these are exact, but insoluble without approximation.

14. To develop approximate scheme, use narrowness of nuclear wavefunction Expand Hamiltonian in Taylor series about average position of nucleus Correlated Electron-Ion Dynamics (CEID)

15. Basis set formulation of CEID For the electrons we use the Ehrenfest states For the nuclei we use moving harmonic oscillator states Note: the formalism of Stella (2011) is more general than this.

16. Basis set formulation of CEID The density matrix then has the following equation of motion where the fluctuation Hamiltonian is given by Makes nuclei quantum Couples electron and nuclear fluctuations Constrains spread of nuclear packets

17. Correlated electron ion dynamics (CEID)

18. Basis set formulation of CEID Matrix elements of the Hamiltonian become easy to evaluate if use ladder operators Can select just few modes to undergo quantum fluctuations: great reduction in computational cost

19. Basis set formulation of CEID L. Stella, M. Meister, A. J. Fisher, and A. P. Horsfield, J. Chem. Phys. 127, 214104 (2007).

20. Basis set formulation of CEID L. Stella. R. P. Miranda, A. P. Horsfield, and A. J. Fisher, J. Chem. Phys. 134, 194105 (2011)

21. Basis set formulation of CEID L. Stella. R. P. Miranda, A. P. Horsfield, and A. J. Fisher, J. Chem. Phys. 134, 194105 (2011)

22. The Hamiltonian

23. What is the problem with the Hamiltonian? How do we compute these terms?

24. What is the problem with the Hamiltonian? What basis set do we use for the electrons? Planewaves Removes many problems associated with atom centered orbitals Often very large number of functions Inefficient for molecules Atom centered (e.g. gaussians) Efficient for many static problems But being atom centered creates problems in dynamic simulations Even for Ehrenfest, overlap matrix is problematic For CEID also need to decide where to put the orbitals

25. What is the problem with the Hamiltonian? Need to take matrix elements with basis before differentiating

26. What is the problem with the Hamiltonian? Interacting electrons In past tried to reduce interacting electron problem to effective independent electron problem Based on Hartree-Fock Coupling between electrons and nuclear fluctuations results in proliferation of matrices Not well controlled

27. What is the problem with the Hamiltonian? Solution? Could construct energy surfaces But opposes philosophy of CEID How do we extend to metals? Very labour intensive