1. Practical quantum Monte Carlo calculations: QWalk
Lucas Wagner

2. Open source: http://www.qwalk.org Solids, liquid, gas phase
Simple separable architecture
Used by >100 people
Open source:
Solids, liquid, gas phase
Scales to >20,000 processor cores
Distributed development

3. Cohesive energy
FN-DMC (eV)
Experiment (eV)
Lattice constant
Experiment (Angstroms)
FN-DMC (Angstroms)
Kolorenc and Mitas
Rep. Prog. Phys. 74 (2011)
Petruzielo, Toulouse,
and Umrigar
J. Chem. Phys. 136,
(2012)

4. Information passed from DFT/Hartree-Fock program to QMC code:
Positions of atoms
Pseudopotentials
The one-particle orbitals and their occupations
QWalk supports reading this information from several DFT/quantum chemistry codes:
GAMESS
Gaussian
NWChem
SIESTA
ABINIT
CRYSTAL

5. 5 steps to accurate calculations
The GAMESS-QWalk pipeline
Most developed interface for molecules.
5 steps to accurate calculations
Choose pseudopotentials and basis sets
Run GAMESS
Run gamess2qmc
Add a Jastrow factor and optimize
Run diffusion Monte Carlo

6. Step 4a: Jastrow factor General form of wave function
Slater determinant (Hartree-Fock)
Two-body Jastrow
Three-body Jastrow
We optimize only the c coefficients

7. 2-body or 3-body? 2-body: Homogeneous systems (silicon, hydrogen, etc)
Very cheap
3-body:
Strongly inhomogeneous systems
More expensive
Can always check how much it improves the wave function

8. How to know if if a wave function is good
Properties of an exact ground state wave function:
Energy is minimized
Variance of the local energy is zero
Usually the variance decreases by a factor of ~2 between the Slater determinant and the Slater-Jastrow wave function.

9. Step 5: Diffusion Monte Carlo
Timestep: you must extrapolate this to zero
The ultimate accuracy of DMC calculations is determined by the nodes, the zeros of your trial wave function.

10. How to immediately recognize that your run is messed up:
The variance (sigma in QWalk) is high(>10). Causes:
Poor basis
Unconverged DFT/HF run
Bad geometry
The kinetic energy should match the DFT/HF kinetic energy.
Conceptual questions:
How does the total energy of QMC relate to:
DFT?
Hartree-Fock?
Coupled-cluster?

11. A discussion on error bars
All numbers in QWalk are reported with one-sigma stochastic errors.
There is a 33% chance that the true average is outside this range.
Errors are reduced as
1/sqrt(T), where T is the
computer time.

12. Using QWalk Evaluate Slater determinant properties
Optimize a Jastrow factor
->filename.wfout
Run diffusion Monte Carlo with optimized Slater-Jastrow trial function

13. Dealing with stochastic simulation (VMC/DMC)
Calculation is divided into blocks of moves
Averaged information for each block is appended to filename.log
Checkpoint is written every block to filename.config
To decrease error bars, just rerun the input file, the calculation will continue where it left off.

14. Units
Energy: 1 Hartree= eV
Distance: 1 Bohr= Angstrom

15. Discussion points When might one want to use QMC?
What questions can it answer?
When is it easy?
When is it hard?
When might fixed node error be large?
Much of the challenge in QMC calculations is setting up the pseudopotentials, getting DFT converged, etc. Not so much the actual run.

16. Where does the fixed-node approximation fail?
Most of the time, the approximation is good.
Let’s look at a classic case where it fails: Be atom. 2s
Almost the same energy 2p 1s
Hartree-Fock ground state
HF trial nodes: ~85% of the correlation energy Including the 2p orbitals: ~99% -- almost exact!
Lucas K. Wagner, NSE C242 & Phys C203, Spring 2009, U.C. Berkeley

17. Setting up the Hamiltonian: choosing pseudopotentials.
What is a pseudopotential?
Guidelines:
Want the potential to be soft: variance costs efficiency
Hartree-Fock/Dirac-Fock potentials are usually better than DFT potentials  why?
For transition metals:
Traditional DFT potentials are not usually good. Two reasons: DFT isn’t the best approximation for the core and more importantly, current pseudopotential generation codes do not handle for example, 3s and 4s electrons in existence together.
Kleinman-Bylandar projection can be a problem for transition metals. Abinit, Q-Espresso, etc, need special consideration to make accurate potentials.

18. Basis set and pseudopotential libraries
Several good sets generated by chemists:
Burkatski-Filippi-Dolg (specifically for QMC)
John Trail/Cambridge group pseudopotentials (quite hard!)
Stuttgart RSC (need some modification for soft core)
Usually written in a Gaussian representation

19. Step 1A: Pseudopotentials
For the first two rows, GGA pseudopotentials are ok, download them from SIESTA’s site (check them!)

20. Basis Sets
Do not want kinks in basis
Splitgauss type basis in SIESTA avoids this.
Basis functions for oxygen
Bad!
Good!