1. 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

2. Outline Review of Diffusion Monte Carlo and Motivations
Review of Lattice Green function Monte Carlo
Lattice regularized Hamiltonian
Applications
Conlcusions

3. 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
MIXED AVERAGE ESTIMATE
computed by DMC
“PURE” EXPECTATION VALUE if

4. 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

5. 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 )

6. 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?

7. Lattice GFMC Lattice hamiltonian: Propagator: Hopping:
 importance sampling
transition probability
Hopping:
weight
For fermions, lattice fixed node approx
to have a well defined transition probability

8. 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, (1995)

9. Lattice regularization I
Kinetic term: discretization of the laplacian
One dimension:
General case:
where
hopping term t1/a2

10. 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!

11. Lattice regularized H Faster convergence in a!
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!

12. Given x and Ha finite number of x’
LRDMC: Algorithm
START
Configuration x, weight w, time T
Given x and Ha finite number of x’
Transition probability px,x’ = Gx,x’/Nx
Generations loop
Walkers and time loops
Configuration x’, weight w’, time T’
Branching
END

13. DMC vs LRDMC extrapolation properties with two different meshes
same diffusion constant
with two different meshes
gain in decorrelation
CPU time

14. Examples
Carbon atom

15. 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)

16. 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)

17. 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)

18. Stability (I)
Carbon pseudoatom: 4 electrons (SBK pseudo)

19. 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

20. Efficiency Iron pseudoatom: 16 electrons (Dolg pseudo) DMC unstable
LRDMC / DMC

21. LRDMC and locality More general effective Hamiltonian
off diagonal pseudo (with FN approximation)
locality approximation + FN approximation
interpolates between two regimes: we can check the quality of the FN state given by the locality approx.

22. Si pseudoatom
LRDMC accesses the pure expectation values!

23. Scandium 4s23dn  4s13dn+1 excitation energies eV VMC DMC LRDMC 2 body
1.099(30)
1.381(15)
1.441(25)
3 body
1.303(29)
1.436(22)
1.478(22)
Experimental value: 1.43 eV
LRDMC: two simulations with for and

24. Chemical Physics Letter, 358 (2002) 442
Iron dimer
Ground state
LRDMC (Dolg pseudo) gives:
MRCI
Chemical Physics Letter, 358 (2002) 442
DFT-PP86
Physical Review B, 66 (2002)

25. Iron dimer (II) Harmonic frequency: 284 (24) cm-1
Experimental value: ~ 300 (15) cm-1
LRDMC equilibrium distance: 4.22(5)
Experimental value: ~ 3.8

26. 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/

27. 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