Wave function approaches to non-adiabatic systems Norm Tubman

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

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

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

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.

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

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 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

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

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

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

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

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

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

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

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

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

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 Systems ECG/HYL He atom From Mitroy et al. RMP 2013

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

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

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

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

Fixed-Ion Systems ECG/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 of electron-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 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?

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-.

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

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

The End