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:
Equilibrium distance:
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
Hamiltonian at 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 Systems ECG/HYL
He atom
From Mitroy et al. RMP 2013

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

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

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

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

29. Fixed-Ion Systems ECG/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
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
calculated such 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