Download - Lattice regularized diffusion Monte Carlo
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