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

Distributed development

Simple separable architecture

Cohesive energy

FN-DMC (eV)

Expe

rimen

t (eV

)Lattice constant

Expe

rimen

t (An

gstr

oms)

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 sets2. Run GAMESS3. Run gamess2qmc4. Add a Jastrow factor and optimize5. Run diffusion Monte Carlo

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

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

2-body or 3-body?

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.

How to know if if a wave function is good

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.

Step 5: Diffusion Monte Carlo

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?

How to immediately recognize that your run is messed up:

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.

A discussion on error bars

Using QWalkEvaluate Slater determinant properties

Optimize a Jastrow factor->filename.wfout

Run diffusion Monte Carlo with optimized Slater-Jastrow 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.

Lucas K. Wagner, NSE C242 & Phys C203, Spring 2009, U.C. Berkeley

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.

HF trial nodes: ~85% of the correlation energyIncluding the 2p orbitals: ~99% -- almost exact!

1s

2s

2p

Hartree-Fock ground state

Almost the same energy