Recent progress with Lattice Regularized
Diffusion Monte Carlo
Sandro SorellaSISSA Democritos Trieste
In collaboration with TurboRVB developers (www.sissa.it/~sorella)
Claudio AttaccaliteUniversity of San Sebastian, Spain
Michele Casula University of Illinois at UrbanaChampaign, IL, United States
Leonardo SpanuUniversity of California, Davis, United States
Workshop on ‘’Recent developments in electronic structure methods’’ June 2008 • UrbanaChaimpain
Workshop on Recent developments in electronic structure methodsJune2008 Recent progress with Lattice Regularized Diffusion Monte Carlo (S.Sorella) 2
Outline
• Diffusion Monte Carlo (DMC, D.M. Ceperley ‘82) methods:
the algorithm on a lattice (M.Casula et al. ’05)
– Non local pseudopotentials and locality approximation
– Lattice regularization
• Semplification of the algorithm Improved efficiency.
N.B. Opposite path compared to DMC
(improved algorithm Umrigar ’93, quite involved, now standard...).
• Application to some simple test cases (still in progress).
– Water monomer all electron (no pseudo).
– Benzene molecule with pseudo Carbon.
– Bulk Silicon. with pseudo and PBC.
Workshop on Recent developments in electronic structure methodsJune2008 Recent progress with Lattice Regularized Diffusion Monte Carlo (S.Sorella) 3
Motivations• Accurate and efficient methods for Quantum Monte Carlo (petascaling).• Lattice regularization: an approximation that is used in Path integrals. ‘’ Do it with path integrals, they really are discrete’’ R. Feynmann My interpretation (extrapolated): Continuous models are sometimes not well posed mathematically and only the a0 of some lattice regularized theory is well defined: where ‘’a’’ is the minimum cutoff distance in real space.
e.g. lattice gauge theories or generic field theories.
They make sense when and because the a0 limit exists. The approximation a>0 is easy to control mathematically.
Here we want to show that : ‘’ Do it on a lattice, it’s really more efficient …!!!’
Workshop on Recent developments in electronic structure methodsJune2008 Recent progress with Lattice Regularized Diffusion Monte Carlo (S.Sorella) 4
Notations
{ }NrrrrN
Nx
,,,, :scoordinate 3by determined
electrons ofion configurat Generic
321
⋅⋅⋅
LRDMCfor space lattice Minimum aDMCfor timediffusion Minimum τ
1/2). e.g. units ( electrons for thelaplacian full Coulombion ion ,ion el , elel )(
)( of state Ground
LRDMC..or DMCfor function Guiding
0
×=∆=
+∆−=
NxV
xVH
g
ψ
ψ
Workshop on Recent developments in electronic structure methodsJune2008 Recent progress with Lattice Regularized Diffusion Monte Carlo (S.Sorella) 5
The ‘’DMC’’ algorithm on a lattice
flip)(spin for 2/ e.g.
computed becan andgiven are elementsmatrix The
, xxJxHxH xx ≠′=′=′
Heisenberg 1D for the e.g. 1
↑↓↓↑↑↓↑↑↓↓=
⋅= +∑x
SSJH ii
i
0)( such that )(
computed) be(can given isfunction guidingA
>→ xx gg ψψ
Workshop on Recent developments in electronic structure methodsJune2008 Recent progress with Lattice Regularized Diffusion Monte Carlo (S.Sorella) 6
)(/)()(
:energy local The
, xHxx
xHxe gxx
xg
g
gL ψψ
ψψ
′′
′== ∑
0s problemsign no I Case x,x ≤′
)(/)(
matter)not do (diagonals 0
,,
≠′′=′
=′
′ xxxHxxx
sgxxg
xx ψψ
Workshop on Recent developments in electronic structure methodsJune2008 Recent progress with Lattice Regularized Diffusion Monte Carlo (S.Sorella) 7
)(0)(g)(g(x)
:yprobabilit theSampling 0 xexexx E
gH ψψψψ ττ −− →∝Π
endif !!! done have We
else 2 step togo
ii)
prob. with Choose i)
if
)](exp[ )3
1)( with 10 ; ),/(ln,0 )2
1 : 0 )1
x )(walker :chain Markov
,
,
,
000
t
T
txxtt
T
tLTttt
Tx
ttt
tttN
spxx
tt
xetww
zpztNzMintsN
wxxt
w,wx
txx
T
T
txx
+=
−=′=
<+
−=
=≤<−=>−=
===
≈
′
′′′
′+
+
′′∑
τ
τ
No error in τ
Workshop on Recent developments in electronic structure methodsJune2008 Recent progress with Lattice Regularized Diffusion Monte Carlo (S.Sorella) 8
The picture of the DMC on a lattice
0.0 0.5 1.0 1.5
7.0
6.5
6.0
5.5
5.0
4.5
L
oca
l En
erg
y e L
(x(t
))
τ
2atT ≈
.0 0 .0 5 .1 0 .1 5
.7 0
.6 5
.6 0
.5 5
.5 0
.4 5
L
oca
l En
erg
y e L
(x(t
))
τ
Carlo. MonteDiffusion :name that theFrom
process.diffusion afor y probabilit Transition
0limit continuous In the
→′→
∆+ xex
aτ
Workshop on Recent developments in electronic structure methodsJune2008 Recent progress with Lattice Regularized Diffusion Monte Carlo (S.Sorella) 9
for 0
0for 0
0for
)(,,
,
,,
,
=′>+
><
=
∑≠′′
′′
′
′′
′
xxsH
s
sH
H
xxxxxx
xx
xxxx
xxeff
0)s some problem,(sign II Case x,x >′
!!same! apply thecan weandn frustratio no effH
)(/)(
matter)not do (diagonals 0
,,
≠′′=′
=′
′ xxxHxxx
sgxxg
xx ψψ
Workshop on Recent developments in electronic structure methodsJune2008 Recent progress with Lattice Regularized Diffusion Monte Carlo (S.Sorella) 10
Fixed node approximation
II with compatibleenergy possiblelowest
thehas )( 0limit continuous in the III)
nodes. Fixed why sThat' 0)()( II)
E I)
: ofenergy better a has of state ground The
0
0
00
00
FN
g0
xa
xx
HH
H
eff
effg
gg
gg
effeff
effeff
effeff
ψ
ψψ
ψψψψ
ψψ
ψψ
ψψ
→
>
≤=
Workshop on Recent developments in electronic structure methodsJune2008 Recent progress with Lattice Regularized Diffusion Monte Carlo (S.Sorella) 11
)(FN)(G)(G(x)
:yprobabilit theSampling
xexexx MAeff E
GH ψψψψ ττ −− →∝Π
endif !!! done have We
else 2 step togo
iii)
)1( prob. with Choose ii)
0)(/)()(
for i)
if
)](exp[ )3
)),(/ln( )2
1 : 0 )1
x )(walker :chain Markov
,,
,
000
t
,
T
xtxxtxxtt
tGGt
txxt
T
tLTttt
tT
ttt
ttt
ppxx
xxxV
Hpxx
tt
xetwwtxVzMint
wxxt
w,wx
T
txx
T
+=
=′=
>′−
=≠′
<+
−=−−=
===
≈
∑′
′′+
−
−
′
+
−
′ ψψ
τ
τ
Workshop on Recent developments in electronic structure methodsJune2008 Recent progress with Lattice Regularized Diffusion Monte Carlo (S.Sorella) 12
0 )()()( Indeed
:small becan operator The
0,
,
=′−= ∑>±′
′+
′ xxsxxxggG HxxVxxO
O
ψψψ
: then'91, PRB al.et Haaf T. H. definite, negativenon is O
VMCgg
gg
gg
geff
g
effeff
effeffeff
effg
effeffg E
HHHH==≤==→
ψψψψ
ψψψψ
ψψ
ψψ
ψψ
ψψ
00
00
0
0
MAE
MAeffeff
effeffeff
effeff
effeffeff
effeff
effeff
EHOHH
=≤−
==→00
00
00
00
00
00
FNE ψψ
ψψ
ψψ
ψψ
ψψ
ψψ
)()( xexEx
LMA ∑Π= Computable
Upper bound property, OHH eff −= remind
Workshop on Recent developments in electronic structure methodsJune2008 Recent progress with Lattice Regularized Diffusion Monte Carlo (S.Sorella) 13
Why O is positive definite? Just very simple algebra:
[ ]xxxHxs
xxxxsO
gxxgxx
ggsxx
xxxx
≠′′=
′′−=
′′
>′ ≥∑
′
)()( :Remind
)(/)()(/)(
,,
2
0,, 0
,'
ψψ
ψψψψψψ
Workshop on Recent developments in electronic structure methodsJune2008 Recent progress with Lattice Regularized Diffusion Monte Carlo (S.Sorella) 14
All this approach can be generalized and extended to Hamiltonian defined in the continuous space
el.~#finite is i.e.evaluated, becan
:elementsmatrix ofnumber the for that is used have weAll
xHx′
xgiven a
Workshop on Recent developments in electronic structure methodsJune2008 Recent progress with Lattice Regularized Diffusion Monte Carlo (S.Sorella) 15
Can we put the physical Hamiltonian on a lattice?
Vm
H +∆−=2
2�
Namely to find a minimum length cutoff ‘’a’’ such that:
HuaaH
aHHa
a
to'close''' is ..~ reasonablefor
and 0for →→
Motivations:• The exact Green function can be sampled for lattice hamiltonians: no approximations, no time discretization.• No restriction to non local operators appearing in H.
Workshop on Recent developments in electronic structure methodsJune2008 Recent progress with Lattice Regularized Diffusion Monte Carlo (S.Sorella) 16
Non local 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 integrationS. Fahy, X. W. Wang and Steven G. Louie, PRB 42, 3503 (1990)
Discretization of the projection Lattice Hamiltonian term
∑∑=−m
jil
ljiP lmlmrvRrV )()( ,
Workshop on Recent developments in electronic structure methodsJune2008 Recent progress with Lattice Regularized Diffusion Monte Carlo (S.Sorella) 17
The pseudo potential acts on single particle wavefunctionswith a reference centered on the pseudoatom we have:
∑∫
∫ ∑
∑
′
−−=
′′⋅+
≅′′⋅′+
=
′′′=
+=
knkkll
ml
l
lmlm
ll
l
llocP
nrnnPl
nrnnPndl
nrnYnYndrvnrV
VrvV
),()(1
4),()(
14
),()()()(),(
)(
,
ψπψπ
ψψ
nr
⋅kn′
Directions are randomized and discretized : 6 ‘’lattice’’ points are usually enough great idea !!! Borrowed
Workshop on Recent developments in electronic structure methodsJune2008 Recent progress with Lattice Regularized Diffusion Monte Carlo (S.Sorella) 18
Locality approximation
)(
)()(
,
x
xVxV
g
xg
Pxx
LA
ψ
ψ ′=
∑′
′
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
• (locality is exact only if is exact)
•
ττ
ττ
τ
τ
ψψψψ
ψψψψ HH
g
g ≠
0for of GSLALA
LA
VVH
H
++∆−=
→= τψ τ
gψLA
g Vx in sDivergence0)( →=ψ
Workshop on Recent developments in electronic structure methodsJune2008 Recent progress with Lattice Regularized Diffusion Monte Carlo (S.Sorella) 19
Locality approximation drawbacks
•non variational results:The energy may be good but we do not really know if it corresponds to a good wavefunction close to the ground state.
•simulations less stable when pseudo are included divergencies appear in the localized potential close to the nodal surface: 0)( =xgψ
Workshop on Recent developments in electronic structure methodsJune2008 Recent progress with Lattice Regularized Diffusion Monte Carlo (S.Sorella) 20
But what about the Laplacian?
Within Lattice Regularized Hamiltonians we can avoid the locality approximation!
Workshop on Recent developments in electronic structure methodsJune2008 Recent progress with Lattice Regularized Diffusion Monte Carlo (S.Sorella) 21
Lattice regularizationKinetic term: discretization of the laplacian
hopping term t→1/a2 where a is the discretization mesh
22
2 2
( ) ( ) 2 ( )( ) ( )
d f x a f x a f xf x O a
dx a
Laplacian with finite differences in the 1D case:
3 dimensional case:
)](2)()([1
,,2 rfarfarf
a zyx
a
−−++=∆→∆ ∑
=
ννυ
Workshop on Recent developments in electronic structure methodsJune2008 Recent progress with Lattice Regularized Diffusion Monte Carlo (S.Sorella) 22
In order to sample the continuous space we use randomized reference frames for the three orthogonal directions.
This is exactly the same trick to compute angular integrals with pseudopotentials and allows to sample the continuous space
in a very simple and efficient way.
The simplification: the simple LRDMC
x�y�
z�
Use of randomized directions (no real lattice) necessary for Coulomb potential: for a real lattice there is a finite
probability that two electrons occupy the same site.We simply do not know what to do in this case (yet).
Further simplification?
Workshop on Recent developments in electronic structure methodsJune2008 Recent progress with Lattice Regularized Diffusion Monte Carlo (S.Sorella) 23
Previous approach:Lattice discretization with two meshes
Separation of core and valence dynamics for heavy nuclei by means of two hopping terms in the kinetic part
),()( )1()( )( 22 aaOxpxpx aa ′+Ψ∆−+Ψ∆≈∆ Ψ ′
a finest mesh, a’ largestp is a function which sets the relative weight of the two meshes.
It can depend on the distance from the nucleus: 0)( and 1)0( , if =∞=′< ppaa
Moreover, if a’ is not a multiple of a, the random walk can sample the space more densely!
Our choice: 2 11
)(r
rpγ+
=
Double mesh for the discretized laplacian
2 / 4Z
14// 2 +=′ Zaa
Workshop on Recent developments in electronic structure methodsJune2008 Recent progress with Lattice Regularized Diffusion Monte Carlo (S.Sorella) 24
Lattice regularization (main idea):how to work with a~1 (a.u.)?
For a better continuous limit: for a0, HaH choose:
Local energy of Ha = local energy of H
Definition of the lattice regularized Hamiltonian
Much faster convergence in the lattice space a!
)( 2aOVVH aaa ++∆−=+∆−=
M. Casula, C. Filippi, S. Sorella, PRL 95, 100201 (2005)
( )
)()(
)(
)()(
2aOxV
xx
xVVg
ga
a
+=
∆−∆+=
ψψ
Workshop on Recent developments in electronic structure methodsJune2008 Recent progress with Lattice Regularized Diffusion Monte Carlo (S.Sorella) 25
Why this is a much better choice?
...
:on theoryperturbati simpleby Then
allfor xx
:condition LRDMC thefrom coming
(I) 0with
GS ingcorrespond with of
:energy state ground thecompute we0any For
002
0
a2
+∆+=
=
=∆
∆+=>
ψψ
ψψ
ψ
ψ
HaEE
xHH
H
HaHHE
a
a
gga
g
aa
)|(| :(I) usingBut 2g0g0g000 ψψψψψψψψ −=−∆−=∆ OHH
)(error LRDMC Thus2
02
0 ga aOEE ψψ −=−
Workshop on Recent developments in electronic structure methodsJune2008 Recent progress with Lattice Regularized Diffusion Monte Carlo (S.Sorella) 26
DMC vs LRDMC
extrapolation properties
Same Error with same efficiencybut different prefactor…
Error Error (my guess)
Efficiency ~1/a2Efficiency ~ 1/τ
For each awell defined Hamiltonian
Trotter approximation
LRDMCDMC
2
02 ψψ −ga
0ψψτ −g
2a≈τ
Workshop on Recent developments in electronic structure methodsJune2008 Recent progress with Lattice Regularized Diffusion Monte Carlo (S.Sorella) 27
But there is a subtle point….
( )
∞
∞
=+∆−=→<∆
∆−=
∆−∆+=
=
ofenergy GS #12
||But
surface nodal on the becan Namely
)(
)()(
)(
)()(
:potential theback tolook sLet'
0)( surface nodal theexists therefermionsFor
2
g
aaaa
a
g
ga
Lg
ga
a
VHa
el
V
xx
xex
xxVV
x
ψψ
ψψ
ψ
Workshop on Recent developments in electronic structure methodsJune2008 Recent progress with Lattice Regularized Diffusion Monte Carlo (S.Sorella) 28
( ))(
)()(
)(
)()(
OK. wasnode fixed theonly
LRDMC original in the correctly definednot was
xx
xex
xxVV
H
H
g
ga
Lg
ga
a
effa
a
ψψ
ψψ ∆
−=∆−∆
+=
0 distanceion el and elelfor finite becan
)( conditions cusp theall satisfyingBy
→xeL
∆−
′>
∆−
=′
otherwise )(
)()(
somefor 0 if )(
)()(),( ,
xx
xe
xsxx
xexVMax
V
g
ga
L
xxg
ga
L
a
ψψ
ψψ
N.B. No ad hoc parameter has been used in this simple LRDMC.
Workshop on Recent developments in electronic structure methodsJune2008 Recent progress with Lattice Regularized Diffusion Monte Carlo (S.Sorella) 29
32 )( Thus
employed. ision approximat node Fixed ifa~ esion vanish wavefunctregion the In this S a~region nodal thearound volumeThe
S. area has surface nodal The
axaSENODALa ≈≈∆ ψ
Estimation of the error due to the nodal surface
Workshop on Recent developments in electronic structure methodsJune2008 Recent progress with Lattice Regularized Diffusion Monte Carlo (S.Sorella) 30
21 Kaa +=η
S. Sorella, M. Casula, D. Rocca, J. Chem. Phys. 127, 014105 (2007)
0.0 0.1 0.2 0.3
5.44
5.43
5.42
5.41
En
erg
y (H
)
a2
LRDMC K=0,(a '/a)2=10 LRDMC PRL
LRDMC K=7.37 (a '/a)2=10
LRDMC K=3.2 (a '/a)2=5
Convergence for C
Workshop on Recent developments in electronic structure methodsJune2008 Recent progress with Lattice Regularized Diffusion Monte Carlo (S.Sorella) 31
Nodal error contr. in LRDMC ~a^3
0.00 0.05 0.100.00
0.01
0.02
<H
a ><
H>
a30.00 0.05 0.10 0.15 0.20 0.25 0.30
0.00
0.01
0.02
<H
a ><
H>
a2
aa
aa
ag
ag
ag
aa
ga
HH
HH
HH
ψψ
ψψψψ
ψψψψ
=
==
Workshop on Recent developments in electronic structure methodsJune2008 Recent progress with Lattice Regularized Diffusion Monte Carlo (S.Sorella) 32
The new approachCarbon pseudoatom
0.0 0.1 0.2 0.35.46
5.45
5.44
5.43
5.42
5.41
En
erg
y (H
)
a2
The simple LRDMC The PRL'05 choice
. because ionalantivariat It was surface. nodal thefrom coming term the wasproblem The 3
−∞→V(x)a
Workshop on Recent developments in electronic structure methodsJune2008 Recent progress with Lattice Regularized Diffusion Monte Carlo (S.Sorella) 33
What about locality approximation?
)(
)()(
,
x
xVxV
g
xg
Pxx
LA
ψ
ψ ′=
∑′
′
0)( when becan )(Again 0g0 →∞ xxV LA ψ
The locality approximation is simply not defined in LRDMC as:
ED UNBOUND)(
)()(
0 −∞==
++∆−=
xVE
xVxVHLAa
LAaaa
Workshop on Recent developments in electronic structure methodsJune2008 Recent progress with Lattice Regularized Diffusion Monte Carlo (S.Sorella) 34
The problem of UNBOUNDED ground state energy was overlooked
0 2000 4000 6000 8000 10000
5.5
Lo
cal e
ner
gy
(H)
Iterations
With vanishing guiding
The exact LRDMC energy with locality approx. should be –INFTY!!!
No big sign of instability seen
0 200 400 6001000
800
600
400
200
En
erg
y (H
)
Iterations
With non zero guiding
Workshop on Recent developments in electronic structure methodsJune2008 Recent progress with Lattice Regularized Diffusion Monte Carlo (S.Sorella) 35
Instead without Locality everything is well defined within LRDMC
0 2000 4000 6000 8000 10000
6.5
6.0
5.5
5.0
4.5
En
erg
y (H
)
Iterations
With non zero guiding With guiding vanishing at the nodes
Workshop on Recent developments in electronic structure methodsJune2008 Recent progress with Lattice Regularized Diffusion Monte Carlo (S.Sorella) 36
Summary
below. from bounded if )( if
)(
: withdefined wellis 0 LRDMC The
xV
xVH
a
a
aaa +∆−=
→
This is possible if:
3) The guiding function satisfies the cusp conditions.4) No type of locality approximation is employed or
use a ‘’bosonized’’ guiding function that never vanishes for fermions with large a0 error.
Workshop on Recent developments in electronic structure methodsJune2008 Recent progress with Lattice Regularized Diffusion Monte Carlo (S.Sorella) 37
Standard DMC is based on the discretization in time e.g. Trotter approximation.
Even in this formalism the problem of infinitely negative potentials exists.
.)(even when sense makes if seigenvalue bounded has
)( ,
:matrixfunction Green The
0
,
∞=∞+<+∆−=′= −
′
xVH
xVHxexG Hxx
τ
+∞→=→
′=∆+−
−∆+′−′
02/
0 )(
,
2/ )(2/ 2/ )(,
0
00
:(Trotter) in timetion discretiza naiveA
xexeG
xeeexGxV
xx
xVxVxx
ττ
τττ
Workshop on Recent developments in electronic structure methodsJune2008 Recent progress with Lattice Regularized Diffusion Monte Carlo (S.Sorella) 38
I just note here that: 1) There is no proof in the literature that the Green function
used with the locality approximation has a bounded spectrumonce the discretization in time is employed.
By simple inspection it should be like that in the Umrigar ’93 DMCbecause there is a cutoff in the local energy
2) Even if the simulation is stable, this does not mean that the
numbers make sense (see LRDMC example). So some care should be used before trusting a DMC energy if you do not know
what the algorithm does to avoid divergences close to the nodal surface.
1τ
≈
Workshop on Recent developments in electronic structure methodsJune2008 Recent progress with Lattice Regularized Diffusion Monte Carlo (S.Sorella) 39
Neverheless even Umrigar ’93 may failCarbon pseudoatom (He core, SBK pseudo)
LRDMC with no locality
Workshop on Recent developments in electronic structure methodsJune2008 Recent progress with Lattice Regularized Diffusion Monte Carlo (S.Sorella) 40
This is easily understood
)(
)()(
,
x
xVxV
g
xg
Pxx
LA
ψ
ψ ′=
∑′
′
)()()(
)(
, nYnYndrvV
VrvV
ml
l
lmlm
ll
l
llocP
′′=
+=
−−=
∫ ∑
∑
21 /1~)(SBK But the rrv +
3g ||/1~)( |~|)( nodes the toclose Thus ssVsx LA
±ψ32 / sVVVH s
LALA −−∂≈++∆−=
NO calculation needed the GS energy of HLA is simply See e.g. Landau, ‘’collapse of a particle in an attractive centre’’
∞−
Workshop on Recent developments in electronic structure methodsJune2008 Recent progress with Lattice Regularized Diffusion Monte Carlo (S.Sorella) 41
Stability
locality approximation large and negative attractive potential close to the nodal surface. It works for finite non local pseudo but depends strongly on the way the algorithm allow (or not) the crossing of the nodal surface. LRDMC works always as long as the non local pseudo has a bounded spectrum from below (no matter if it is infinitely positive).
++
Dangerous attraction!
nodal surface
€
−
Workshop on Recent developments in electronic structure methodsJune2008 Recent progress with Lattice Regularized Diffusion Monte Carlo (S.Sorella) 42
Error in the discretization
Discretized Laplacian
Discretized non local pseudopotential
€
e
€
e
€
Z
discretization error reduced by the randomization of the quadrature mesh
discretization error reduced by the introduction of a double mesh
Workshop on Recent developments in electronic structure methodsJune2008 Recent progress with Lattice Regularized Diffusion Monte Carlo (S.Sorella) 43
The metal insulator transition within LRDMC
15 20 25
107.0
106.5
LRDMC
P transition~19.2Gpa
64 Si Γpoint
En
erg
y (
eV
)
V/A3
βtin Diamond
Workshop on Recent developments in electronic structure methodsJune2008 Recent progress with Lattice Regularized Diffusion Monte Carlo (S.Sorella) 44
Relative efficiency for carbon atomwith SBK pseudopotentials
0.162.03.40.0256
0.111.31.50.0120
0.0830.81.00.0068
Lattice spaceLRDMC
Non local DMC
Time step
T 12σ
η ∝ Variance σ 2
CPU time T
Z effective = 4
LRDMC is slightly less efficient than the non local DMC
Workshop on Recent developments in electronic structure methodsJune2008 Recent progress with Lattice Regularized Diffusion Monte Carlo (S.Sorella) 45
Relative efficiency for iron dimer with Dolg pseudopotentials
0.062515.05.20.0039
0.03907.01.80.0015
0.02845.21.00.0008
Lattice spaceLRDMC
Non local DMC
Time step
LRDMC is 3-5 times faster than the non local DMC
When stable, the standard DMC is 1.25 faster than the non local DMC
Z effective = 16
Workshop on Recent developments in electronic structure methodsJune2008 Recent progress with Lattice Regularized Diffusion Monte Carlo (S.Sorella) 46
Water monomer (no pseudo)
0.00 0.01 0.02 0.03 0.0476.426
76.425
76.424
76.4234xDt (H 1)
En
erg
y (H
)
<Ha><H>
<H>
<Ha> DMC
2)( )( tbtaEtE DMCDMC ∆+∆+=∆
0.0 0.1 0.2 0.3 0.476.5
76.4
76.3
76.2
76.1a> <H0
a>=<H>= . ( ) 76 4245 1
En
erg
y (H
)<Ha><H>
<Ha> <H>
Workshop on Recent developments in electronic structure methodsJune2008 Recent progress with Lattice Regularized Diffusion Monte Carlo (S.Sorella) 47
(*) I. G. Gurtubay and R. Needs JCP, 127, 124306 (2007).
Similar non linear DMC extrapolation in a recent paper
Workshop on Recent developments in electronic structure methodsJune2008 Recent progress with Lattice Regularized Diffusion Monte Carlo (S.Sorella) 48
Water monomer
3/2)( )( tbtaEtE DMCDMC ∆+∆+=∆
.0 00 .0 01 .0 02 .0 03 .0 04 .76 426
.76 425
.76 424
.76 4234xDt (H 1)
E
ner
gy
(H)
<Ha><H>
<H>
<Ha> DMC
Workshop on Recent developments in electronic structure methodsJune2008 Recent progress with Lattice Regularized Diffusion Monte Carlo (S.Sorella) 49
Water monomer
76.42830(5)DMC with few par.(*)
HF+ Backflow
76.4219(1) GAUSSIAN No QMC
HartreeFock
DMC energy (Hartree)
Optimization method
Wavefunction nodes
76.438(1) CCSDT…. EXACT
76.4257(1) AGP+J2+J3+J4 AGP (+)
76.4245(1)HF+J2+J3+J4(234 bodies)
HartreeFock
76.42376(5) DMC with few parameters(*)
HartreeFock
76.4230(2) HF+J2+J3 (2and 3bodies)
HartreeFock
(*) I. G. Gurtubay and R. Needs JCP, 127, 124306 (2007). (+) M. Casula et al. JCP, 121, …(2004), with small basis -76.4175(4).
Workshop on Recent developments in electronic structure methodsJune2008 Recent progress with Lattice Regularized Diffusion Monte Carlo (S.Sorella) 50
The benzene molecule
.0 00 .0 05 .0 10.59 2
.59 3
.59 4
.59 5
.59 6
Exp.
Dis
sosi
atio
n E
ner
gy
(eV
)
a2
<H>
<Ha>
.0 00 .0 02 .0 04
.37 72
.37 70
.37 68
.37 66
.37 64
En
erg
y (H
)
<Ha><H>
<H>
<Ha>
Workshop on Recent developments in electronic structure methodsJune2008 Recent progress with Lattice Regularized Diffusion Monte Carlo (S.Sorella) 51
Non local DMC (Casula ’06)
( , )DMCG x y �( ) exp ( ) ( , )DMC loc
y
w x K V x V y x � �� �� � � �� �� �� ��
¥
( ) exp ( , )Ty
w x V y x � � � �
� �¥
( ( ) )( ) LE xw x e
diffusion + drift (with rejection)
non local move (heat bath)
weight with local energy (it includes the contribution from both diffusion and non local move)
Three steps in the evolution of the walkers: the non local move is the new one introduced in the non local DMC scheme
€
p(x →y)=T FN (y, x) /wT (x)
Workshop on Recent developments in electronic structure methodsJune2008 Recent progress with Lattice Regularized Diffusion Monte Carlo (S.Sorella) 52
8 Silicon diamond
0.000 0.001 0.002 0.003 0.004 0.00531.24
31.22
31.20
31.18
E
ner
gy
(H)
/Tcpu~a12~∆ t
DMC NL
DMC NL+
LRDMC
The CPU time to have a given error bar is proportional to Tcpu:the number of accepted single electron moves per unit time.
Workshop on Recent developments in electronic structure methodsJune2008 Recent progress with Lattice Regularized Diffusion Monte Carlo (S.Sorella) 53
LRDMC: summary
•Simple and robust method to make reliable calculations.•Simplification of the approach makes possible: to have a very good extrapolation of the energy for a0. In all cases studied very accurate numbers even compared with the standard state of the art DMC methods.• double mesh in the laplacian can help to decorrelate faster the electrons (core-valence separation). Not treated here. • Locality approximation can be avoided and variational and more stable results can be obtained with QMC.
•It is simple to control the a0 limit and compute energy Derivatives (e.g. forces) or energy differences.
Workshop on Recent developments in electronic structure methodsJune2008 Recent progress with Lattice Regularized Diffusion Monte Carlo (S.Sorella) 54
Iron dimer
PHOTOELECTRON SPECTROSCOPY
GS anion:
GS neutral:
Leopold and Lineberger, J. Chem. Phys. 85, 51(1986)
13 2 * 2 8(3 ) (4 ) (4 ) ud s s � 13 2 * 1 9(3 ) (4 ) (4 ) gd s s �
previous NUMERICAL STUDIES on the neutral iron dimer
DFT methods:
more correlated methods (CC, MRCI, DFT+U):
electron affinity very hard to compute
7u
9g
Workshop on Recent developments in electronic structure methodsJune2008 Recent progress with Lattice Regularized Diffusion Monte Carlo (S.Sorella) 55
Calculation details
Dolg pseudopotentialsDolg pseudopotentials neon core
spd non local componentsscalar relativistic corrections included
Gaussian basis set for JAGP wave function Gaussian basis set for JAGP wave function (8s5p6d)/[2s1p1d] contracted for AGPTotal independent parameters: 227
Workshop on Recent developments in electronic structure methodsJune2008 Recent progress with Lattice Regularized Diffusion Monte Carlo (S.Sorella) 56
Dispersion curves
Workshop on Recent developments in electronic structure methodsJune2008 Recent progress with Lattice Regularized Diffusion Monte Carlo (S.Sorella) 57
Neutral ground state
LRDMC gives for neutral dimer9
g
7 9( ) ( ) 0.52 (10) eVu gE E
Workshop on Recent developments in electronic structure methodsJune2008 Recent progress with Lattice Regularized Diffusion Monte Carlo (S.Sorella) 58
Neutral ground state
LRDMC DFTPP86Physical Review B, 66 (2002) 155425
The lack of correlation leads to underestimate the “on-site” repulsion in the d orbitals, and overestimate the 4σ splitting.
Workshop on Recent developments in electronic structure methodsJune2008 Recent progress with Lattice Regularized Diffusion Monte Carlo (S.Sorella) 59
Iron dimer: structural properties9
g
LRDMC equilibrium distance: 3.818(11)Experimental value: ~ 4.093(19)Harmonic frequency: 301 (15) cm1
Experimental value: ~ 300 (15) cm1
Workshop on Recent developments in electronic structure methodsJune2008 Recent progress with Lattice Regularized Diffusion Monte Carlo (S.Sorella) 60
Iron dimer: photoelectron spectrumExperiment
LRDMCIncoming photon: 2.540 eVTemperature 300K
Workshop on Recent developments in electronic structure methodsJune2008 Recent progress with Lattice Regularized Diffusion Monte Carlo (S.Sorella) 61
Benzene dimer
Binding energy (kcal/mol)
0.5(3) parallel
2.2(3) slipped parallel
0.37 ZPE
1.6(2) experiment
S. Sorella, M. Casula, D. Rocca, J. Chem. Phys. 127, 014105 (2007)
Van der Waals + interactions
Important for DNA
and protein structures
Workshop on Recent developments in electronic structure methodsJune2008 Recent progress with Lattice Regularized Diffusion Monte Carlo (S.Sorella) 62
Conclusions
• The pseudopotentials can be “safely” included in the DMC,
with the possibility to perform accurate simulations for large or
extended systems, in solid state physics or quantum chemistry.
• The fixed node approximation is still the major problem for
this zero temperature technique.
Workshop on Recent developments in electronic structure methodsJune2008 Recent progress with Lattice Regularized Diffusion Monte Carlo (S.Sorella) 63
15 20 25
107.0
106.5
LRDMC
P transition~ . Gpa19 2
Si 64 Γpoint
En
erg
y (e
V)
V/A3
βtin Diamond