Practical quantum Monte Carlo
calculations: QWalk
Lucas Wagner
• Used by >100 people
• Open source: http://www.qwalk.org
• Solids, liquid, gas phase
• Scales to >20,000 processor cores
Simple separable architecture
Distributed development
Lattice constant
Experiment (eV)
Experiment (Angstroms)
Cohesive energy
FN-DMC (eV)
FN-DMC (Angstroms)
Kolorenc and Mitas
Rep. Prog. Phys. 74 (2011) 026502
Petruzielo, Toulouse,
and Umrigar
J. Chem. Phys. 136,
124116 (2012)
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
The GAMESS-QWalk pipeline
Most developed interface for molecules.
5 steps to accurate calculations
1. Choose pseudopotentials and basis sets
2. Run GAMESS
3. Run gamess2qmc
4. Add a Jastrow factor and optimize
5. Run diffusion Monte Carlo
Step 4a: Jastrow factor
• General form of wave function
YT (R) = Det[j i (rj )]exp(U)
– Slater determinant (Hartree-Fock)
– Two-body Jastrow
U =0
U = å åc kei ak (riI ) +å åc kee bk (rij )
iI
k
ij
k
– Three-body Jastrow
eei
U = two - body +ååc klm
[ak (riI )al (rjI ) + ak (rjI )al (riI )]bk (rij )
ijI klm
• We optimize only the c coefficients
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
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 SlaterJastrow wave function.
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.
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?
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.
Using QWalk
Evaluate Slater determinant properties
Optimize a Jastrow factor
->filename.wfout
Run diffusion Monte Carlo
with optimized SlaterJastrow trial function
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.
Units
• Energy: 1 Hartree=27.216 eV
• Distance: 1 Bohr=0.529177 Angstrom
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.
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
1s
Hartree-Fock ground
state
2p
Almost the same
energy
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