Wave function approaches to non-adiabatic systems

1. Wave function approaches to non-adiabatic systems Norm Tubman

2. Full electron ion dynamics with H2 Quantum mechanics only for the electrons Electrons Ions Clamped nuclei Energy:-1.7447 Equilibrium distance: 1.4011 We can write down full Hamiltonian for the electrons and the ions

3. Solving the electron-ion Hamiltonian for H2 • We have 4 particles, 2 species, and 2 spins per species. • This problem is sign free, none of the particles need to • be anti-symmetrized in space • Any starting wave function (almost), will give the exact • ground state energy • - QMCPACK can get the exact energy for this problem

4. Setup the particles for H2 e (spin down) e (spin down) H (Spin up) H (Spin down)

5. Setup the Hamiltonian and wavefunction H2 QMCPACK knows how to compute the kinetic energy andd potential energy from the previously defined parameters for the charge and the mass. Now try to make a wave function…. Any wave function will give you the exact answer with FN-DMC. But, the question still remains, how good of a wave function can be made in QMCPACK.

6. Breakdown of Born-Oppenheimer • There are many physical systems • that require theory beyond the • Born Oppenheimer approximation • in order to be treated accurately. One quantum hydrogen transferring between two carbon systems Phenoxyl-phenol Touluene From Sirjoosingh et. al. JPCA

7. Breakdown of Born-Oppenheimer Phenoxyl-phenol Touluene From Sirjoosingh et. al. JPCA

8. Born Oppenheimer Approximation • The full Hamiltonian should • have kinetic energy for both the • electrons and the ions • The clamped nuclei Hamiltonian • is obtained by setting the nuclear • kinetic energy equal to zero. • The full wave function can be • expanded in terms of the solution • of the clamped nuclei Hamiltonian • and nuclear functions that are • can be considered expansion • coefficients This expansion is expected to be exact, although it has never been proven

9. Born Oppenheimer Approximation • The full Hamiltonian can be • expanded in this basis set. The • Lambda terms are the • non-adiabatic coupling operators • The Born Oppenheimer • approximation is obtained by • reducing the wave function ansatz • from a sum over states to just one • state. This definition is not unique! • The adiabatic approximation is • obtained by setting the non-adiabatic • coupling operators equal to 0

10. The adiabatic approximation • Binding curves for the C2 • molecule. This is calculated • by solving the electronic • Hamiltonianat different • ionic coordinates • Different potential energy • surfaces arise form the • excited states

11. Born Oppenheimer Approximation • We can try to solve the full Hamiltonian with no • approximations, but it is very difficult • We can rewrite Lambda in terms of energy differences between the separate potential energy surfaces • When the difference in energy between states becomes small, then • Lambda diverges, and it does not • make sense to use the Born • Oppenheimer approximation

12. Born Oppenheimer Approximation From Sirjoosingh et. al. JPCA The coupling is large for phenoxl/phenol and Therefore the Born-Oppenheimer approximation is not valid

13. Approaches to going beyond Born Oppenheimer • Nuclear Electron Orbital Methods (HF, CASSCF, XCHF, CI) • Basis set techniques that make explicit use of the Born Oppenheimer • approximation to generate efficient basis sets for wave function generation • Correlated Basis (Hylleraas, Hyperspherical, ECG) • Generic basis set technique that uses explicitly correlated • basis sets to solve the electron-ion Hamiltonian to high accuracy. • Path Integral Monte Carlo • Finite temperature Monte Carlo technique based on thermal • density matrices • Fixed-Node diffusion Monte Carlo • Ground state method that is based on generating high quality • wave functions and projecting to the ground state wave function • Multi-component density functional theory • Density functional theory for electrons and ions simultaneously

14. Explicitly correlated basis The techniques to work with explicitly correlated basis sets provide a different way of constructing wave functions from basis sets based on single particle constructions A single (spherically symmetric) ECG is given as A linear combination of ECGs can be used to construct a trial wave function

15. Outside perspective on QMC It is important to use the right methods for the right problem. From Mitroy et al. RMP 2013

16. An Example H2 Ground state energy of H2 (QMC) • Quantum Monte Carlo can • treat para-hydrogen exactly • in its ground state. Chen and • Anderson calculated one of • the most highly accurate • QMC solutions with a simple • wave function. • QMC is exact, but…… Chen-Anderson JCP 1995

17. An Example H2 Ground state energy of H2 (QMC) • Quantum Monte Carlo can • treat para-hydrogen exactly • in its ground state. Chen and • Anderson calculated one of • the most highly accurate • QMC solutions with a simple • wave function. • QMC is exact, but…… Chen-Anderson JCP 1995 The best current ECG result

18. How is convergence determine? The ECG method employs a basis set that is complete, and therefore can be extrapolated to the the complete basis set limit First H2 ECG Paper: Kinghorn and Adamowicz 1999 Latest H2 ECG Paper: Bubin, S., et al. 2009

19. What about finite temperatures Kylanpaa Thesis 2011

20. Finite temperatures? It is possible to simulate many excited states also with the ECG method. Bubin, S., et al. 2009

21. High accuracy simulations From Mitroy et al. RMP 2013

22. High accuracy simulations From Mitroy et al. RMP 2013

23. High accuracy simulations From Mitroy et al. RMP 2013

24. Fixed-ion SystemsECG/HYL He atom From Mitroy et al. RMP 2013

25. Fixed-ion SystemsECG/HYL He atom From Mitroy et al. RMP 2013

26. Fixed-Ion SystemsECG/HYL/CI From Mitroy et al. RMP 2013

27. Fixed-Ion SystemsECG/HYL/CI From Mitroy et al. RMP 2013

28. Fixed-Ion SystemsECG/HYL/CI From Mitroy et al. RMP 2013

29. Fixed-Ion SystemsECG/CI/DMC From Seth et al. 2011

30. ECG Non-adiabatic GS energies Accuracy drops orders of magnitudes as systems get larger, for specialized basis set calculations From Mitroy et al. RMP 2013

31. What has been done with full electron-ion QMC

32. What has been done with full electron-ion QMC

33. QMC electron/ion wave functions • We consider three forms of • electron-ion wave functions • Ion independent • determinants • Ion dependence • introduced through the • basis set • Full ion dependence

34. Benefits of using local orbitals -A simple way to perform non adiabatic calculations is to make use Of the localized basis set and drag the orbitals when the ions move Move the Ion and Drag the Orbital

35. Problems of using non-symmetric orbitals

36. Problems of using non-symmetric orbitals

37. Problems of using non-symmetric orbitals

38. FN-DMC H2 • Three different forms • of the wave function • considered • The “nr” wave functions • are currently in the release • version of QMCPACK. • FN-DMC fixes a lot of • deficiencies in this form • of the wave function • What are the limits of • accuracy for FN-DMC?

39. FN-DMC LiH • FN-DMC and ECG are • well above experimental • energy. But ECG is • converged to very high • accuracy. • Symmetrizing the wave • function is incredibly • important for VMC. • Not as important for DMC. • Larger molecules also • calculatedsuch as H2O • and FHF-.

40. Improving wave functions It is important to capture large changes in the electronic wave functions as the ions move Other Wave function to explore: -Grid Based Wave functions -Wannier functions and FLAPW -Multi-determinant electron-ion wfs From Sirjoosingh et. al. JPCA

41. Conclusions • FN-QMC might be one of the only methods right now that can tackle non-adiabatic systems of more than 6 quantum particles with high accuracy • For small systems it is possible to make use of quantum chemistry techniques to calculate highly accurate non-adiabatic wave functions • There are many possibilities for improving wave function quality and running large systems with FN-QMC

42. The End