Lattice regularized diffusion Monte CarloMichele Casula, Claudia Filippi, Sandro Sorella

International School for Advanced Studies, Trieste, Italy

National Center for Research in Atomistic Simulation

Outline

Review of Diffusion Monte Carlo and Motivations

Review of Lattice Green function Monte Carlo

Lattice regularized Hamiltonian

Applications

Conlcusions

21( ln ) ( )

2G

FNG

Hff f f E f

t

Standard DMCstochastic method to solve H with boundary conditions given by the nodes of (fixed node approximation)G

( , ) ( , ) ( )FN Gf R t R t R

DIFFUSION WITH DRIFT BRANCHING

Imaginary time Schroedinger equation with importance sampling

FN GDMC

FN G

HE

FN FN

FN FN

H

MIXED AVERAGE ESTIMATE

computed by DMC“PURE” EXPECTATION VALUE

if GS of FN H

• bad scaling of DMC with the atomic number

• locality approximation needed in the presence of non local potentials (pseudopotentials)

Motivations

5.5cpu time Z

Major drawbacks of thestandard Diffusion Monte Carlo

non variational resultssimulations less stable when pseudo are includedgreat dependence on the guidance wave function usedhowever approximation is exact if guidance is exact

D. M. Ceperley, J. Stat. Phys. 43, 815(1986)A. Ma et al., to appear in PRA

Non local potentialsLocality approximation in DMC

Mitas et al. J. Chem. Phys. 95, 3467 (1991)

, '' ( ')( )

( )

x x GLA

G

dx V xV x

x

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 )

G

LA LA LAG FN FN FN

LA LA LAG FN FN FN

H H

GS of LA LA

FN FNH

G

Pseudopotentials For heavy atoms pseudopotentials are necessary to

reduce the computational time

Usually they are non local ( ) ( )P i l il m

V x v x lm lm

In QMC angular momentum projection is calculated by using a quadrature rule for the integrationS. Fahy, X. W. Wang and Steven G. Louie, PRB 42, 3503 (1990)

Natural discretization of the projection

Can a lattice scheme be applied?

Lattice GFMC

Propagator:

Lattice hamiltonian: ji

jiijai

iai nnVchcctH,,

†

2

1.).(

, , ,

( )( )

( )G

x x x x x xG

xG H

x

importance sampling

Hopping: x’

x

transition probability

)(,

,

,, xe

G

G

Gp

L

xx

xxx

xxxx

weight ))((1 xeww Lii

For fermions, lattice fixed node approx to have a well defined transition probability

Effective Hamiltonian

Hop with sign change replaced by a positive diagonal potential

( ) / ( ) Green function

ˆ ˆ if G 0 OFF DIAGONAL TERMS

ˆ 0 otherwise

ˆ ( ) ( ) DIAGONAL TERM

( )

x y x y x y G G

effx y x y x y

effx y

effx x sf

sf x y

G H x y

H H

H

H V x v x

v x G

with 0 SIGN FLIP TERMx yy

G

LATTICE UPPER BOUND THEOREM !

0 0 0 0eff eff eff eff effH H

D.F.B. ten Haaf et al. PRB 51, 13039 (1995)

0 GS of eff effH

Lattice regularization IKinetic term: discretization of the laplacian

22

1

2 ( ) i i

da a

ai i

T T IO a

a

hopping term t1/a2

ˆ ˆ( ) ( )a T TT x x a

22

2 2

( ) ( ) 2 ( )( ) ( )

d f x a f x a f xf x O a

dx a

One dimension:

General case:

where

Separation of core and valence dynamics for heavy nuclei two hopping terms in the kinetic part

)()()1()()( 2aOxpxpx ba p can depend on the distance from the nucleus

0)( and 1)0( , if ppba

Moreover, if b is not a multiple of a, the random walk can sample all the space!

Our choice: 2 1

1)(

rrp

Lattice regularization IIDouble mesh for the discretized laplacian

2Z

2 1b a Z

Continuous limit: for a0, HaH

Local energy of Ha = local energy of H

Discretized kinetic energy = continuous kinetic energy

Lattice regularized H

,

( )( ) ( )

( )G

L x x Lx G

H xe x G E x

x

( ) ( )

( ) ( ) ( )( ) ( )

a a G G

G G

x xV x V x V x

x x

Definition of lattice regularized Hamiltonian

a a aH V

Faster convergence in a!

Given x and Ha finite number of x’Transition probability px,x’ = Gx,x’/Nx

LRDMC: Algorithm

Configuration x, weight w, time T

','( )x x xx xN G

Configuration x’, weight w’, time T’

exp( ( ))x L

x

w w e x

T T

'

log( )

]0,1] randomx x

x x

r N

r

Wal

kers

and

tim

e lo

ops

Branching

START

Gen

erat

ions

loop

END

DMC vs LRDMCextrapolation properties

DMC LRDMC

Trotter approximation For each a well defined effective H

extrapolation a extrapolation

behaviour a4 behaviour

same diffusion constanta

CPU time 1 2( )a with two different meshesgain in decorrelation

core

NN

Examples

Carbon atom

LRDMC with pseudo IOff diagonal matrix elements

( ) / ( ) propagator x y x y x y G GG H x y

From the discretized Laplacian

From the non local pseudopotential

2

( )

( )G

x yG

xpG x y a

a y

2

( )1

( )G

x yG

xpG x y b

b y

( )2 1v ( ) cos

4 ( )G

x y l l x yl G

xlG y P x y c

y

c quadrature mesh (rotation around a nucleus)

a and b: translation vectors

LRDMC with pseudo II

ˆ ˆ if G 0 OFF DIAGONAL TERMS

ˆ 0 otherwise

ˆ ( ) ( ) DIAGONAL TERM

( ) with 0 SIGN FLIP TERM

effx y x y x y

effx y

effx x sf

sf x y x yy

H H

H

H V x v x

v x G G

Effective lattice regularized Hamiltonian

• Mixed average is variational

Now kinetic & pseudo! ˆ

x yH

positive constanteffH H

(1 ) eff effdH H H

d

FN MA GE E E

x yH

(1 ) ( )sfv x

• Pure expectation value of H can be estimated

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

Pure energy estimate

(0) ( ( ) (0)) (0)FN MA MA MA MAE E E E E

Hellmann-Feynman theorem

0(1 ) (0) ( )eff eff

FN MA MA

d dH H H E E E

d d

Different ways to estimate the derivative: Finite differences Correlated sampling

( )E Variational due to the convexity of

Exact for reachable only with correlated sampling (without losing efficiency)

0

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

Stability (II)

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

Nodal surface

Efficiency Iron pseudoatom: 16 electrons (Dolg pseudo)

DMC unstable

Efficiency LRDMC / DMC 2 - 4

interpolates between two regimes: we can check the quality of the FN state given by the locality approx.

'

'

ˆ ( ) (1 ) ( ) (1 ) ( ) DIAGONAL

ˆ ˆ ˆ if ( ) ( ) 0

ˆ ˆ(1 (1 )) if ( ) ( ) 0

ˆ ˆ

eff Px x sf sf

effx y x y G x y G

eff Px y x y G x y G

effx y x y

H V x v x v x

H H x H x

H H x V x

H H

'

'( )

otherwise

( ) ( ) ( ) 0P Psf G x y Gx x

v x x V x

LRDMC and locality More general effective Hamiltonian

off diagonal pseudo (with FN approximation)locality approximation + FN approximation

0 1

Si pseudoatomLRDMC accesses the pure expectation values!

LRDMC: two simulations with for and

Scandium

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

4s23dn 4s13dn+1 excitation energies

0.5 0 1

Iron dimer

MRCIChemical Physics Letter, 358 (2002) 442

DFT-PP86Physical Review B, 66 (2002) 155425

LRDMC (Dolg pseudo) gives: 9g

7 9( ) ( ) 0.61 (14) eVu gE E

Ground state

Iron dimer (II)

9g

LRDMC equilibrium distance: 4.22(5)Experimental value: ~ 3.8 Harmonic frequency: 284 (24) cm-1

Experimental value: ~ 300 (15) cm-1

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

G

Reference:cond-mat/0502388

Limit On the continuous, usually H not bounded from above!

, , 0 x x x xG H

','( )','( )

( )

x xx xx xx x

L

Gq G

E x

1exp G H

1( ) 1

kf k q q

x k

','( )

log( ) ]0,1] randomx

x xx x

rr

G

Probability of leaving x

k distributed accordingly to f

Green function expansion

Probabilty of leaving x after k time slices