Wave function approaches to non-adiabatic

systemsNorm Tubman

Full electron ion dynamics with H2

We can write down full Hamiltonian for the electrons and the ions

Clamped nuclei Energy:-1.7447Equilibrium distance: 1.4011

Quantum mechanics only for theelectronsElectrons

Ions

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 tobe anti-symmetrized in space

- Any starting wave function (almost), will give the exactground state energy

- QMCPACK can get the exact energy for this problem

Setup the particles for H2e (spin down)

e (spin down)

H (Spin up)

H (Spin down)

Setup the Hamiltonian and wavefunction H2

Now try to make a wave function….

Any wave function will give you the exact answerwith FN-DMC.

But, the question still remains, how good ofa wave function can be made in QMCPACK.

QMCPACK knows how to compute the kinetic energyandd potential energy from the previously defined parameters for the charge and the mass.

Breakdown of Born-Oppenheimer

• There are many physical systems that require theory beyond the Born Oppenheimer approximationin order to be treated accurately.

Phenoxyl-phenol ToulueneOne quantum hydrogen transferringbetween two carbon systems

From Sirjoosingh et. al. JPCA

Breakdown of Born-Oppenheimer

Phenoxyl-phenol Touluene

From Sirjoosingh et. al. JPCA

Born Oppenheimer Approximation

• The full Hamiltonian should have kinetic energy for both the electrons and the ions

• The clamped nuclei Hamiltonianis obtained by setting the nuclearkinetic energy equal to zero.

• The full wave function can beexpanded in terms of the solutionof the clamped nuclei Hamiltonianand nuclear functions that are can be considered expansion coefficients

This expansion is expected to be exact, although it has neverbeen proven

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 ansatzfrom a sum over states to just onestate. This definition is not unique!

• The adiabatic approximation isobtained by setting the non-adiabatic coupling operators equal to 0

The adiabatic approximation

• Binding curves for the C2molecule. This is calculatedby solving the electronic Hamiltonian at different ionic coordinates

• Different potential energy surfaces arise form the excited states

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

Born Oppenheimer Approximation

The coupling is large for phenoxl/phenol andTherefore the Born-Oppenheimer approximationis not valid

From Sirjoosingh et. al. JPCA

Approaches to going beyond Born Oppenheimer

• Nuclear Electron Orbital Methods (HF, CASSCF, XCHF, CI)Basis set techniques that make explicit use of the Born Oppenheimerapproximation 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 CarloFinite temperature Monte Carlo technique based on thermaldensity matrices• Fixed-Node diffusion Monte CarloGround state method that is based on generating high qualitywave functions and projecting to the ground state wave function• Multi-component density functional theoryDensity functional theory for electrons and ions simultaneously

Explicitly correlated basisThe techniques to work with explicitly correlated basis setsprovide a different way of constructing wave functions frombasis sets based on single particle constructions

A single (spherically symmetric) ECG is given as

A linear combination of ECGs can be used toconstruct a trial wave function

Outside perspective on QMC

It is important to use the right methods for the right problem.

From Mitroy et al. RMP 2013

An Example H2Ground state energy of H2 (QMC)

• Quantum Monte Carlo cantreat para-hydrogen exactlyin its ground state. Chen andAnderson calculated one of the most highly accurate QMC solutions with a simplewave function.

• QMC is exact, but……Chen-Anderson JCP 1995

An Example H2

• Quantum Monte Carlo cantreat para-hydrogen exactlyin its ground state. Chen andAnderson calculated one of the most highly accurate QMC solutions with a simplewave function.

• QMC is exact, but……

Ground state energy of H2 (QMC)

The best current ECG result

Chen-Anderson JCP 1995

How is convergence determine?

The ECG method employs a basis set that is complete,and therefore can be extrapolated to the the complete basisset limit

First H2 ECG Paper: Kinghorn and Adamowicz 1999

Latest H2 ECG Paper: Bubin, S., et al. 2009

What about finite temperatures

Kylanpaa Thesis 2011

Finite temperatures?It is possible to simulate many excited states also with the ECG method.

Bubin, S., et al. 2009

High accuracy simulations

From Mitroy et al. RMP 2013

High accuracy simulations

From Mitroy et al. RMP 2013

High accuracy simulations

From Mitroy et al. RMP 2013

Fixed-ion SystemsECG/HYL

He atom

From Mitroy et al. RMP 2013

Fixed-ion SystemsECG/HYL

From Mitroy et al. RMP 2013

He atom

Fixed-Ion SystemsECG/HYL/CI

From Mitroy et al. RMP 2013

Fixed-Ion SystemsECG/HYL/CI

From Mitroy et al. RMP 2013

Fixed-Ion SystemsECG/HYL/CI

From Mitroy et al. RMP 2013

Fixed-Ion SystemsECG/CI/DMC

From Seth et al. 2011

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

What has been done with full electron-ion QMC

What has been done with full electron-ion QMC

QMC electron/ion wave functions

We consider three forms ofelectron-ion wave functions

• Ion independent determinants

• Ion dependence introduced through the basis set

• Full ion dependence

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

Problems of using non-symmetric orbitals

Problems of using non-symmetric orbitals

Problems of using non-symmetric orbitals

FN-DMC H2

• Three different formsof the wave function considered

• The “nr” wave functionsare currently in the releaseversion 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?

FN-DMC LiH• FN-DMC and ECG arewell above experimentalenergy. But ECG is converged to very high accuracy.

• Symmetrizing the wavefunction is incredibly important for VMC. Not as important for DMC.

• Larger molecules alsocalculated such as H2O and FHF-.

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

ConclusionsFN-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

The End