Lattice regularized diffusion Monte Carlo

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Lattice regularized diffusion Monte Carlo

• Michele Casula, Claudia Filippi, Sandro Sorella

International School for Advanced Studies,
Trieste, Italy
National Center for Research in Atomistic
Simulation
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Outline

• Review of Diffusion Monte Carlo and Motivations
• Review of Lattice Green function Monte Carlo
• Lattice regularized Hamiltonian
• Applications
• Conlcusions

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Standard DMC
stochastic method to solve H with boundary
conditions given by the nodes of (fixed
node approximation)
DIFFUSION WITH DRIFT
BRANCHING
Imaginary time Schroedinger equation with
importance sampling
PURE EXPECTATION VALUE if
MIXED AVERAGE ESTIMATE computed by DMC
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Motivations
Major drawbacks of the standard Diffusion Monte
Carlo

• bad scaling of DMC with the atomic number
• locality approximation needed in the presence of
  
• non local potentials (pseudopotentials)

D. M. Ceperley, J. Stat. Phys. 43, 815(1986) A.
Ma et al., to appear in PRA
non variational results simulations less stable
when pseudo are included great dependence on the
guidance wave function used however approximation
is exact if guidance is exact
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Non local potentials
Locality approximation in DMC Mitas et al. J.
Chem. Phys. 95, 3467 (1991)
Effective Hamiltonian HLA containing the
localized potential

• the mixed estimate is not variational since
• if is exact, the approximation is exact
• (in general it will depend on the shape of
  )

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Pseudopotentials

• For heavy atoms pseudopotentials are necessary
  to reduce the computational time
• Usually they are non local

In QMC angular momentum projection is calculated
by using a quadrature rule for the integration S.
Fahy, X. W. Wang and Steven G. Louie, PRB 42,
3503 (1990)
Natural discretization of the projection
Can a lattice scheme be applied?
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Lattice GFMC
Lattice hamiltonian
Propagator
? importance sampling
transition probability
Hopping
weight
For fermions, lattice fixed node approx to
have a well defined transition probability
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Effective Hamiltonian
Hop with sign change replaced by a positive
diagonal potential
LATTICE UPPER BOUND THEOREM !
D.F.B. ten Haaf et al. PRB 51, 13039 (1995)
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Lattice regularization I
Kinetic term discretization of the laplacian
One dimension
General case
where
hopping term t?1/a2
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Lattice regularization II
Double mesh for the discretized laplacian
Separation of core and valence dynamics for heavy
nuclei ? two hopping terms in the kinetic part
p can depend on the distance from the nucleus
Our choice
Moreover, if b is not a multiple of a, the random
walk can sample all the space!
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Lattice regularized H
Definition of lattice regularized Hamiltonian

• Continuous limit for a?0, Ha?H
• Local energy of Ha local energy of H
• Discretized kinetic energy continuous kinetic
  energy

Faster convergence in a!
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LRDMC Algorithm
START
Configuration x, weight w, time T
Given x and Ha finite number of x Transition
probability px,x Gx,x/Nx
Walkers and time loops
Generations loop
Configuration x, weight w, time T
Branching
END
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DMC vs LRDMC
extrapolation properties
same diffusion constant
with two different meshes gain in decorrelation
CPU time
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Examples
Carbon atom
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LRDMC with pseudo I
Off diagonal matrix elements
From the discretized Laplacian
a and b translation vectors
From the non local pseudopotential
c quadrature mesh (rotation around a nucleus)
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LRDMC with pseudo II
Effective lattice regularized Hamiltonian
Now kinetic pseudo!

• Mixed average is variational

• Pure expectation value of H can be estimated

• Much more stable than the locality approximation
  (less statistical fluctuations)

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Pure energy estimate
Hellmann-Feynman theorem

• Different ways to estimate the derivative
• Finite differences
• Correlated sampling

Variational due to the convexity of
Exact for reachable only with correlated
sampling (without losing efficiency)
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Stability (I)
Carbon pseudoatom 4 electrons (SBK pseudo)
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Stability (II)
Nodal surface
non local move

• locality approximation ? infinitely negative
  attractive potential close to the nodal surface
  (It works for good trial functions / small time
  steps)
• non local move ? escape from nodes

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Efficiency
Iron pseudoatom 16 electrons (Dolg pseudo)
DMC unstable
Efficiency LRDMC / DMC
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LRDMC and locality
More general effective Hamiltonian
off diagonal pseudo (with FN approximation) locali
ty approximation FN approximation
interpolates between two regimes we can
check the quality of the FN state given by the
locality approx.
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Si pseudoatom
LRDMC accesses the pure expectation values!
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Scandium
4s23dn ? 4s13dn1 excitation energies
Experimental value 1.43 eV
LRDMC two simulations with for
and
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Iron dimer
Ground state
LRDMC (Dolg pseudo) gives
DFT-PP86 Physical Review B, 66 (2002) 155425
MRCI Chemical Physics Letter, 358 (2002) 442
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Iron dimer (II)
LRDMC equilibrium distance 4.22(5) Experimental
value 3.8
Harmonic frequency 284 (24) cm-1 Experimental
value 300 (15) cm-1
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Conclusions

• LRDMC as an alternative variational approach for
  dealing with non local potentials
• Pure energy expectation values accessible
• The FN energy depends only on the nodes and very
  weakly on the amplitudes of
• Very stable simulation also for poor wave
  functions
• Double mesh more efficient for heavy nuclei

Reference cond-mat/0502388
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Limit L ??
On the continuous, usually H not bounded from
above!
Green function expansion
Probability of leaving x
Probabilty of leaving x after k time slices
k distributed accordingly to f