Wave function approaches to non-adiabatic systems Norm Tubman.
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Transcript of Wave function approaches to non-adiabatic systems Norm Tubman.
Wave function approaches to non-adiabatic
systems
Norm 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 H2
e (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 C2
molecule. 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 H2
Ground 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