recent progress with lattice regularized diffusion monte … · workshop on recent developments in...

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.


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 …!!!’


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


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


)(/)()(

: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


)(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


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


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


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

→

>

≤=


)(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


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


Why O is positive definite? Just very simple algebra:

[ ]xxxHxs

xxxxsO

gxxgxx

ggsxx

xxxx

≠′′=

′′−=

′′

>′ ≥∑

′

)()( :Remind

)(/)()(/)(

,,

2

0,, 0

,'

ψψ

ψψψψψψ


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


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.


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 )()( ,


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


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


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


But what about the Laplacian?

Within Lattice Regularized Hamiltonians we can avoid the locality approximation!


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


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?


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


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


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 −=−


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


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


( ))(

)()(

)(

)()(

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.


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


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


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

=

==


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


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


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


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


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.


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


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

≈


Neverheless even Umrigar ’93 may failCarbon pseudoatom (He core, SBK pseudo)

LRDMC with no locality


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’’

∞−


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

€

−


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


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


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


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


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>


(*) I. G. Gurtubay and R. Needs JCP, 127, 124306 (2007).

Similar non linear DMC extrapolation in a recent paper


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


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).


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>


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)


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.


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.


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


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


Dispersion curves


Neutral ground state

LRDMC gives for neutral dimer9

g

7 9( ) ( ) 0.52 (10) eVu gE E


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.


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


Iron dimer: photoelectron spectrumExperiment

LRDMCIncoming photon: 2.540 eVTemperature 300K


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


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.


15 20 25

107.0

106.5

LRDMC

P transition~ . Gpa19 2

Si 64 Γpoint

En

erg

y (e

V)

V/A3

βtin Diamond


The LRDMC upper bound theorem

recent progress with lattice regularized diffusion monte … · workshop on recent developments in...

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