Nazim Dugan dugannazgmail'com Introduction to the Diffusion Monte Carlo Method Ioan Kosztin, Byron F

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Nazim Dugan dugannaz_at_gmail.comIntroduction
to the Diffusion Monte Carlo MethodIoan Kosztin,
Byron Faber and Klaus Schulten American Journal
of Physics, 64, 1996
Quantum Monte Carlo methods for fermionic
systems
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Variational Monte Carlo
Diffusion Monte Carlo
Fixed-node approximation
Electronic structure methods Configuration
Interaction Coupled Cluster Hartree-Fock
method Density Functional Theory Quantum Monte
Carlo methods (QMC) Variational Monte Carlo
(VMC) Diffusion Monte Carlo (DMC) Greens
function Monte Carlo Path integral Monte Carlo
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Variational Monte Carlo
Diffusion Monte Carlo
Fixed-node approximation
Monte Carlo Random processes to solve
problems Quantum Monte Carlo (QMC) Random walk
methods to solve electronic structure
problem. Variational Monte Carlo
(VMC) Expectation values are calculated via Monte
Carlo integration over the 3N-dimensional space
of electron coordinates. Use variational
principle to find the ground state.
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Variational Monte Carlo
Diffusion Monte Carlo
Fixed-node approximation
Monte Carlo Integration
Probability density
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Variational Monte Carlo
Diffusion Monte Carlo
Fixed-node approximation
Monte Carlo integration is similar to Riemman sum
However, sample points are not taken uniformly
but chosen randomly according to a probability
distribution.
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Variational Monte Carlo
Diffusion Monte Carlo
Fixed-node approximation
Test of MC integration in 1D
Gaussian
Analytic calculation 10000 sample points
Riemman sum MC uniform MC Gaussian
MC Derivative
2.50659 2.50659 2.51269 2.50997
2.50779
whole function
MC integration is advantageous for dimensions
larger than 4 (Rubin Landau, Oregon State
University )
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Variational Monte Carlo
Diffusion Monte Carlo
Fixed-node approximation
Monte Carlo integration in VMC
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Variational Monte Carlo
Diffusion Monte Carlo
Fixed-node approximation
Metropolis rejection algorithm allows an
arbitrarily complex distribution to be sampled in
a straightforward way without knowledge of its
normalization
Animation for benzene molecule Atomic nuclei are
fixed. Electron positions are generated by using
Metropolis algorithm.
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Variational Monte Carlo
Diffusion Monte Carlo
Fixed-node approximation
Diffusion Monte Carlo (DMC) Solve time-dependent
Schrödinger Equation in the long time limit to
find the ground state.
diffusion equation
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Variational Monte Carlo
Diffusion Monte Carlo
Fixed-node approximation
DMC formalizm Kinetic energy term Treat
as a density distribution of artificial particles
diffusing in 3N dimensional configuration space
of the many body system.
In this way we can simulate the effect of kinetic
energy term in the Schrödinger equation.
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Variational Monte Carlo
Diffusion Monte Carlo
Fixed-node approximation
DMC formalizm Potential energy term Effect of
the potential energy term can be inserted into
the simulation by a branching term in the
diffusion process. Number of artifical particles
may change !
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Variational Monte Carlo
Diffusion Monte Carlo
Fixed-node approximation
DMC formalizm Solve time-dependent Schrödinger
equation for Set ERE0 for

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Variational Monte Carlo
Diffusion Monte Carlo
Fixed-node approximation
DMC formalizm Branching occurs according to
the quantity Adjust ER to keep the number of
artificial particles constant
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Variational Monte Carlo
Diffusion Monte Carlo
Fixed-node approximation
DMC formalizm Repeat until ER becomes
constant. Then, and distribution of artificial
particles gives ground-state wavefunction.
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Variational Monte Carlo
Diffusion Monte Carlo
Fixed-node approximation
Importance Sampling Direct application of DMC
results in large population fluctuations. Use a
guide function to improve the efficiency of the
algorithm. A drift velocity according to the
guide function is added to the diffusion
process. Guide function is usually taken from a
VMC run.
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Variational Monte Carlo
Diffusion Monte Carlo
Fixed-node approximation
Parallelization Istatistical error reduces
as Thus, usually large number of time steps are
necessary to achive the desired
accuracy. However DMC calculations are suitable
for parallel computing. Artificial particles
diffusing in the phase space may be easily
distributed to the available computing nodes
since they move independently. Small amount of
nodel communication is necessary for population
control.
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Variational Monte Carlo
Diffusion Monte Carlo
Fixed-node approximation
Linear scaling QMC DFT and QMC has cubic scaling
with number of electrons. However, scaling can
be reduced to linear with some reasonable
approximations for most of the systems. Linear
scaling methods are used for large systems.
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Variational Monte Carlo
Diffusion Monte Carlo
Fixed-node approximation
Fermion sign problem In DMC, wavefunction is
treated as a probability distribution which is
positive definite. However fermionic ground
states may have negative regions. Ground state
wavefunctions
bosonic
fermionic
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Variational Monte Carlo
Diffusion Monte Carlo
Fixed-node approximation
DMCview java application Simple harmonic
oscillator
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Variational Monte Carlo
Diffusion Monte Carlo
Fixed-node approximation
DMCview java application H2 molecule Total wave
function should be antisymmetric Spin part is
antisymmetric So, Spatial part is symmetric. (no
sign problem)
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Variational Monte Carlo
Diffusion Monte Carlo
Fixed-node approximation
There is no exact solution to the fermion sign
problem up to now. Fixed-node
approxiamtion Take 3N-1 dimensional node
information from a trial wavefunction. Usually,
the guide function used for importance sampling
is used also for the node information. Fix the
nodes of the wavefunction and make DMC run for
each nodal packet separately
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Variational Monte Carlo
Diffusion Monte Carlo
Fixed-node approximation
Within each nodal packet wavefunction is always
positive or always negative. So DMC can be
applied without the minus sign problem.
Error arising from nodes being not exact is
second order. However, nodes may be released and
accuracy may be improved by variational
principle Energy is minimum when nodes are
exact. (Energy is exact when nodes are exact.)
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Variational Monte Carlo
Diffusion Monte Carlo
Fixed-node approximation
Tiling Theorem All nodal pockets are infact
equivalent and therefore one only need solve the
Schrödinger equation in one of them. D.
Ceperley, J. Stat. Phys. 63, 1237(1991)
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Variational Monte Carlo
Diffusion Monte Carlo
Fixed-node approximation
Fixed-node approximation in practice Take a
trial wavefunction from a DFT run. Improve the
wavefunction using a short VMC run DMC run Use
the result of VMC for the guide function of
importance sampling and also determine nodes of
the ground state wavefunction according to this
guide function. Make DMC run for one nodal
packet until the desired accuracy is reached.
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Variational Monte Carlo
Diffusion Monte Carlo
Fixed-node approximation
Trial wavefunctions
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Variational Monte Carlo
Diffusion Monte Carlo
Fixed-node approximation
QMC packages CASINO - Cavendish Laboratory at
Cambridge UK (FREE) qmcPACK - University of
Illinois at Urbana-Champaign (GPL) ZORI -
Lester Group, University of California at
Berkeley (GPL) QMCBeaver - Chip Kent and Mike
Feldmann at Caltech (GPL) QUMAX - Claudio
Attaccalite at SISSA Italy (GPL)
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Variational Monte Carlo
Diffusion Monte Carlo
Fixed-node approximation
CASINO (Cavendish Laboratory, Cambridge
UK) Parallel code for QMC electronic structure
calculations. Functionality Wavefunction
optimization using VMC Fixed-node DMC Can take
wavefunction information from CRYSTAL 9X/03,
GAUSSIAN 9X/03, TURBOMOLE (gaussian
orbitals) PWSCF, ABINIT, GP, CASTEP, K207
(plane-wave orbitals)
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Variational Monte Carlo
Diffusion Monte Carlo
Fixed-node approximation
Functionality Computation of various expectation
values density, spin-density, non-collinear
spin-density matrix, reciprocal space and
spherical real space pair correlation functions,
localization tensor, structure factor, and
spherically-averaged structure factor. 2D/3D
electron phases and electron-hole with fluid or
crystal wave functions, with arbitrary cell
shape/spin polarization/density (including
excited-state capability).
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Variational Monte Carlo
Diffusion Monte Carlo
Fixed-node approximation
Benzene molecule with CASINO GAUSSIAN DFT
STO-3G -229.406 a.u CASINO DMC
-232.038 a.u GAUSSIAN DFT 6.31G
-232.258 a.u
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Variational Monte Carlo
Diffusion Monte Carlo
Fixed-node approximation
C24H12 with CASINO GAUSSIAN DFT STO-3G
-910.613 a.u CASINO DMC
-920.958 a.u GAUSSIAN DFT 6.31G
-921.920 a.u
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Variational Monte Carlo
Diffusion Monte Carlo
Fixed-node approximation
David Ceperleys reasons for not using QMC
(1996) 1. We need forces, dummy! 2. Try getting
O2 to bind at the variational level. 3. How many
graduate students lives have been lost optimizing
wavefunctions? 4. It is hard to get 0.01 eV
accuracy by throwing dice. 5. Most chemical
problems have more than 50 electrons. 6. Who
thought LDA or HF pseudopotentials would be any
good? 7. How many spectra have you seen computed
by QMC? 8. QMC is only exact for energies. 9.
Multiple determinants. We can't live with them,
we can't live without them. 10. After all,
electrons are fermions. 11. Electrons move. 12.
QMC isn't included in Gaussian 90. Who programs
anyway? M.D. Towler (developer of CASINO) has
counter arguments (See Quantum Monte Carlo and
the CASINO program)
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Suggested reading

• Introduction to the Diffusion Monte Carlo Method
• Ioan Kosztin, Byron Faber and Klaus Schulten,
  American Journal of Physics, 64, 1996
• Quantum Monte Carlo simulations of solids
• WMC Foulkes, L Mitas, RJ Needs, G Rajagopal,
  Reviews of Modern Physics, 73, 2001
• QMC Methods for the Solution of Schrödinger
  Equation for Molecular Systems
• A Aspuru, WA Lester, arXiv cond-mat / 0204486 v2
  12 Jun 2002
• Quantum Monte Carlo Methods in the Study of
  Nanostructures
• J Shumway and DM Ceperley, in Handbook of
  Theoretical and Computational Nanotechnology, 2005

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