Fermion Quantum Monte Carlo based on the idea of sampling “graphs” Ali Alavi University of Cambridge Alex Thom James Spencer EPSRC.

Fermion Quantum Monte Carlobased on the idea of sampling “graphs”

Ali AlaviUniversity of Cambridge

Alex ThomJames SpencerEPSRC

Overview

Introduction and motivation

Paths integrals and the Fermion sign problem

FSP as a problem in “path counting”

A useful combinatorial formula

From path-sums to graph-sums

Applications to molecular systems

Towards application to periodic systems

Essence of idea

Express the many-electron path integral in a finite Slater Determinant basis

Resum the path integral over exponentially large numbers of paths to convert

path-sums => graph-sums

The graphs are much more stable entities which can then be sampled.

i1

i2 iP-1

iPi j

k

l

+ + + + + ….G

Q

n G

n GwQ ][)(

2-vertex 3-vertex

Each vertex is a Slater determinant

Each graph represents the sum over all paths of length P which visitall verticies of the graph

Non-pertubative expansion

A graphical, or diagrammatic, expansion of the partition function

Path Integrals

Consider the (thermal) density matrix:

Heˆ

)(ˆ

In terms of the eigenstates of the Hamiltonian:

e)temperatur zero (i.e. for

00

ˆi

iE

iie

Q

e

HeE

eQ

H

H

H

ln

][

][

][

ˆ

ˆ

ˆ

Tr

Tr

Tr

The energy can be calculated from:

xexdxQ

xexxx

H

H

ˆ

ˆ''

The density matrix can be represented in real space

For a single electron located at x:

"timeslices" or :factors P

HPHPHPH eeeeˆ)/(ˆ)/(ˆ)/(ˆ

.....

))'()()(2/()')(2/()ˆˆ)(/( .'2 xVxVPxxmPVTP eexex

KE terms:harmonic spring

'......'ˆ)/(

3

ˆ)/(22

ˆ)/(2 xexxexxexdxdxxx HP

PHPHP

P

PE terms

dxVxmxS

exDdxQ

xxx

xxxx

P

xS

P

)]([)(2

1)]([

)(

)0()(,0),(

...

0

2

)]([

2

intergral path

:bc with

function continuous a to tends

by denoted path the limit the In

PE along the pathKE along path

x

x2

xP

x3

One can simulate an electronas a ring-polymer, movingin the external field (which itselfcan be dynamic).Polarons [Parrinello Rahman]

For N electrons

P

SP

iii

iii

N

eDXdXN

Q

XUXVxmdXS

xxxXPX

X

xxxX

N

ˆ

/ˆ

2

0

21

)1)((!

1

)]([)]([2

1)]([

)...,,()0(ˆ)(

)(

),...,,(

21

Coulomb interaction

Odd permutations subtract from the Partition function: Fermion sign problem

Describes closed paths which can exchange identical particle coordinates

11 xx 22 xx

12 xx 21 xx

As N or increases, there is an exponential cancellation of contributionsarising from even and odd paths.

Slater Determinant space

]...det[!

12121 .. NN nnnnnn uuu

NDD i

},...,,{ 221 Muuu

Let Di be a Slater determinant composed out of N orthonormal spin-orbitals [e.g. Hartree-Fock orbitals, Kohn-Sham, etc]chosen out of a set of 2M:

The Di form a orthonormal set of antisymmetric functions.They are solutions to a non-interacting, or uncorrelated (mean-field)Hamiltonian H0: iii DEDH 0

0

17det

det

10

10,100 ..

2

N

NMge

N

MN

iii

rrr

DcEH

vVUTVUTH

a

N

ii

ji ji

N

ii

[

,

)](,1

,2

1, 2

Full problem:

Exact solutions are linear superpositionsof uncorrelated determinants

Paths in Slater determinant space

A closed path in S.D. space

1P21 iiii DDDD ...

space SD in steps of pathA P

PHPHPHP DeDDeDDeDDDDDwPP 1321

///)( ...],,....,,[ iiiiiiiiii 2121

i1

i2 iP-1

iP

1 2

],,...,,[...

][

121)(

i ii

ii

i

iiiiP

H

PP

H

w

DeDeTrQ

A given S.D. can occur multiple times along a path

][

])/[()(

])/[(

)1()0(

2)1(2/)(

2

)0()0(

HHH

POHP

e

POHP

extremely

PEE

ijijij

ijijij

ij

ji

:is ionapproximat better a and

ion)approximat (Primitive

:matrix sparse

and dominant-diagonally ,computable a is

jiij DeD PH /

Hamiltonian matrix elements (Slater-Condon rules)

Since H contains at most 2-body interactions:

orbitals-spin 2 than moreby differ and if jiji DDDHD 0

iD

jD

][S ][S if zz jiji DDDHD 0

)( 2NODHD ii

Hamiltonian connects only single and double excitations:224/)12)(2)(1( MNNMNMNN Maximum connectivity

orbitals-spin 1by differ and if jiji DDNODHD )(orbitals-spin 2by differ and if jiji DDODHD )1(

Spin selection rule:

Cost of calculation

Other symmetries may also existHubbard model: translational invariance;Molecules:point group symmetry

i j

a bbarijabrijDUD 1

121

12 ji

Search for a power series in ii

...3

0

3

0

3

0

2

2

0

2

0

2

1

1

2

21

3

321321

1

1

2

2121

ijkiikikkjkjjijii

ijiijijjijiiiii

P

n

nP

n

nnP

n

nnnPnnn

P

n

nP

n

nnPnnPw

i

j

i

j

k

n1P-2-n1-n2

n2

Two-hop

3-hop

ii

jiij

wQ

DeD PH /

Rearranging

ijk ii

kk

ii

jjkijkijii

ijkiikkjkjjijii

ij ii

jjjiijii

ijiijijjijii

3

0

3

0

3

0

3

3

0

3

0

3

0

3

2

0

2

0

22

0

2

0

2

1

1

2

21

3

32

1

1

2

21

3

21321

1

1

2

2

1

1

2

2121

P

n

nP

n

nnP

n

nn

P

P

n

nP

n

nnP

n

nnPnnn

P

n

nP

n

n

PP

n

nP

n

nnPnn

:terms Hop-3

:terms hop-Twosum Nested

elementsmatrixtransition

Define the nested sum:

hP

n

nhP

n

nhP

n

nh

nnh

Ph

h

ii

h

hxxxxxxZ0 0 0

2121)(

1

1

2

1

121 .....)...,,(

which appears in the h-hop term

The “hop” series

....),,,1(

),,1(

),1(1

)(44

)(33

)(22

ikljk

ij ii

ll

ii

kk

ii

jj

ii

liklkjij

ijk ii

kk

ii

jj

ii

kjij

ij ii

jj

ii

jiijiii

P

P

PP

Z

Z

Zw

ki

i

j

k

i

jk

l

)(

1

1

1

2

1)...,,( 21

)(

xzz

z

ixxxZ

C

P

hP

h

Using induction, one can show:

x1x2 x3

AJW Thom and A Alavi, J Chem Phys, 123, 204106, (2005)

Residue Theorem gives

)(

1

1

1

)(

1

1

1),(

1

1)(

..

)(

1

1

1)...,,(

..

122

2

211

121

)(2

1

11

)(1

21)(

321

xxx

x

xxx

xxxZ

x

xxZ

ge

xxx

xxxxZ

xxx

PPP

PP

h

iij

jii

Pi

hP

h

For

hhP

h xxxPhxxxZ ...,,0)...,,( 2121)( all for if

Some useful properties of Z-sums

)...,,()...,,(21

)(21

)(

hiiiP

hhP

h xxxZxxxZ

Symmetry:

Replace upper limit of sums over h to

01

1

2

1)(0

dzz

z

iZ

PP

From “hop”-expansion to “vertex” expansion

i

jk

l

i

j=lk

i=k

j

l

i=k

j=l

4-vertex

3-vertex

2-vertex

Consider the 4-hop terms:

“chain” “Star”

Analytic summation over alternating series

)(3PZ

i

j

k

i

j

k ...)(6 PZ

th. residue the into solutions feed

0)(1:

)(1

1

1

1

2

1

1

1

2

1

))()(()(:

),,1(

0

30

3

zASolve

zAz

z

iA

z

z

iS

zzzzADefine

ZS

C

P

n C

nP

n

n

n

ijk

ijkijk

kkiijjiiiiii

kijkijijk

ii

kk

ii

jj

ii

kijkij

“Cycle function”

Eg. A 2-vertex graph

i

j

b

a

ij

jjii

22

22

)1())(()(

za

b

azaaza

bzA

(For simplicity)

Next compute S2:

1)(

1)(

2

1

)/(2

)/(1)/1(

)/(2

)/(1)/1(

)/1)(/1(

)1(1

2

1

)/()1(

)1(1

2

1

)/()1(

)1(

1

1

2

1

)1(1

1

1

1

2

1

)(1

1

1

1

2

1

2222

2

22

22

P

P

P

P

PP

P

PP

PP

a

ba

a

ba

ab

abab

ab

abab

abzabz

zz

i

abz

zz

iabz

z

z

z

i

zabz

z

izAz

z

iS

)(5

PZi

j

k...)(

8 PZ

l

1

2

i

j

k

l

1

1

2

3

Star graphs

g

g

g

g

GGGC

P

nG

nG

nG

C

P

nnn g

gstar

AAAz

z

i

AAAz

z

innn

nnnS

....1

1

1

1

2

1

....1

1

2

1,...,,

...

21

2

2

1

1

21 ..,, 21

21

For a star-graph with g-spokes, G1,G2,…Gg attached to i

G2

G1

G3

Gg

)(7

PZ ...)(10 PZ

1

1

1

11

Chains graphs

g

g

g

g

G

G

GC

P

nG

nG

nG

nnnchain

A

AAz

z

i

AAAn

nn

n

nnS

1...

11

1

1

1

1

2

1

........11

2

..,, 2

32

1

21

1

2

2

1

1

21

G1G2

G12

G2

G3G3

General 3-vertex graph

i

j

k i

j

k

k

j

kk

j

Folded representation Unfolded representation:Each spoke represents anIndependent circuit on the graph

jkikijijk

jk

kj

ik

jk

ij

jk

ijk

ijk

AAAA

A

z

z

i

AA

A

A

A

Az

z

iS

c

P

c

P

21

1

1

1

2

1

11121

1

1

1

2

1)(3

Denominator is cubic polynomial in z i.e. there are 3 residues

Unfolded 4-vertex graph

Denominator is a quartic polynomial in z

G

Q

n G

n GwQ ][)(

2-vertex 3-vertex

Each vertex is a Slater determinant

Each graph represents the sum over all paths of length P which visitall verticies of the graph

A graphical, or diagrammatic, expansion of the partition function

+ + + + + ….

abcdijklDab

ijD

''''bajiD

abijD ab

ijD ''baijD

2,3, and 4-vertex graphs

+ + + +

][.][

][

11

ln

)()(

)(Gw

Gw

GwQ

QE

nn

n Gn

Monte Carlo sampling of graphsThe energy can be obtained from:

][

)(

)()(

)(][

~..

][ln][

~

Gw

n

nn

nGEE

ei

GwGE

If graphs can be sampled with an un-normalised probability given byw(n) [G], then the energy estimator is:

||

)(

||

)()(

])[(

][~

])[(

w

n

w

nn

Gwsign

GEGwsignE

For this to be useful, the denominator has to be well-behaved as

i.e. the number of positive sampled graphs should exceed the number ofnegative sampled graphs in such a way that this difference is finite anddoes not vanish.

||||||][][,][][,][][

wwwCGCfCGCfTGTf

][][][])[(||

)( CfCfTfGwsignw

n

Monitor fraction of sampled graphs which are trees, positive cyclic and negative cyclic graphs.

0

0

)(

~

~

ln~

][

lim

lim

EE

EE

wE

Gww

HF

HFHF

HFHF

n HFG

nHF

Approximation:Truncate series at 2-vertex, 3-vertex or higher-vertex graphs.

2 vertex: Double-excitations Number of graphs= [N2M2] 3 vertex: Quadruple excitations [N4M4]4 vertex: Hexatuple excitations [N6M6]

For graphs that contain the HF determinant:

[Hartree-Fock energy]

[Ground state energy]

v

n HFG

nvHF Gww ][)()(

N2 molecule HFE~

Types of sampled graphs (4-vertex level)

N2 molecule in VDZ basis

||

)( ])[(w

n Gwsign

N2 sampled energies (4-vertex level)

N2 binding curve [sampling graphs which contain the HF determinant]

Applications to periodic systems

Taking a plane-wave PP code (CPMD) which can solve for (i) KS orbitals and potential (ii) KS virtuals

-> Use these as the basis for the vertex series

KS Hamiltonian becomes the reference (single-excitations now contribute)

Need 2-index and 4-index integrals, which are computed on-the-flyusing FFTs (time consuming part)

Advantage: (i) Treatment of periodic systems (ii) No BSSE (iii) Can be used as a post-DFT method

Graphite (4 atom) primitive cell. 16el, BHS PP (Ec=90Ry)

2-vertex

Conclusions and outlook

Development of QMC methods based on graphs gives a methodto combat the Fermion sign problem

Proof of concept for small molecular systems

Major effort is now being expended on developing a periodic code…..

…..perhaps to return to surface problems in due course!

Advantage of graph-sampling algorithm

O(N2) scaling!

The observed stability at the 4-vertex level is extremely encouraging.

Current work:

(1) Extension to higher order graphs(2) Improved Monte Carlo sampling (3) Applications to large systems

Graphs

.....,, kji DDDG n distinct determinants

Connectivity of graph is determined by ij

A graph a set of n distinct elements (in no particular order) with a given connectivity

i j

k

l

i j

km

Ga Gb

n G

n GwQ ][)(

Each graph represents a sum over exponentially large numbers of pathsits weight can be expected to be much better behaved than that of individual paths.

Compactly expressed:

.....,, kji DDDG

][)( Gw nSum over all paths which visit all the determinants in G

A set of n connected determinants

A graph, G, is an object on which we can represent the paths which visit all the vertices in G

G G

PP

G

n

P

wGw1 2

],,...,,['...][ 121)()(

i ii

iiii

The weight of a given graph is the sum over all paths of lengthP which visit all the vertices of the graph:

The prime ‘ indicates that the summation indicies must be chosen in Such a way that each vertex in G is visited at least once.

i j

k

l

i j

km

Ga Gb

This condition ensures that the weights of two different graphsGa and Gb (I.e. two graphs that differ in at least one vertex)

do not double-count paths which visit only Ga Gb

][][ ba GwGw will not double-count paths which visit ][ ba GGw

Quantum Chemical applications

Dissociation of diatomic molecules:

Multiple-bond dissociation, e.g. the N2 molecule, is a major challenge to any ab initio method.

Use HF orbitals generated from MOLPRO

Gaussian basis set [cc-pVDZ or VTZ]

Two-electron primitive integrals read in from MOLPRO outputand matrix constructed on the fly.

Cost of the calculations

2-vertex <1 s3-vertex 150 secs4-vertex 1 week [over 109 4-vertex graphs to sum]

On a pentium 4 processor [2003 vintage]

How to make 4-vertex (and eventually higher vertex) calculationspractical?

][~1

lim )(t

n

tK

GEK

E

So if on step t of an MC simulation consisting of K steps we are at graph Gt

In order to perform a Metropolis MC simulation, one needs to ensure that microscopic reversibility is satisfied. In the present implementation, we generate fresh graphs at each step according to an algorithm to be shortly described.

In addition one needs to compute the generation probability of a graph usingthis algorithm, in order to unbias the Metropolis MC acceptance ratio.

)][

][

]'[

]'[,1min(]|'[

)(

)(

Gw

GP

GP

GwGGP

n

gen

gen

n

acc

Tree graphs are graphs that do not contain cycles

i

j

k

l

The weight of trees is positive definite at all

N2 VDZ

-109.4

-109.2

-109

-108.8

-108.6

-108.4

-108.2

-108

0 1 2 3 4 5

r/a.u.

E/a

.u.

FCI

RHF

v=2 b=5

v=3 b=5 all

v=4 b=5

Exactly diagonalised by Krogh, Olsen CPL 344, 578, (2001),and by Chan, Kallay and Gauss, JCP, 121, 610 (2004)

15820024220 determinants

...)()()(1 432ijklijkij

iii ijklijkij SSSw P

Approximation:Truncate series at 2, 3 or higher vertex terms.

2 vertex: Double-excitations [N2M2] 3 vertex: Quadruple excitations [N4M4]

4 vertex: Hexatuple excitations [N6M6]

By Comparison: CCSD: N2(M-N)4 Nit

CCSDT: N3(M-N)5 Nit

CCSDTQ: N4(M-N)6 Nit

CCSDTQ56: N6(M-N)8 Nit

To summarize:

8 site system at or near half-filling is strongly open-shell

-4

+4

]2/)cos[(]2/)cos[(2),( yxyxyx kkkktkk

4954

12

An iIlustration of the Monte Carlo: 8x8 Hubbard lattice with 6 e-

Momentum-space basis

(0,0) (1,0)(2,0)

(1,1)

80086

16det

N

+ + + + + ….HFD

abcdijklDab

ijD

''''bajiD

abijD ab

ijD ''baijD

Finite T

We wish to compute the energy at a finite -1=kT as

iii

iii

DeD

DHeD

e

HeE

H

H

H

H

][Tr

][Tr

Where the trace is taken over all Ndet determinants.Problem is that these sums are not “Monte Carlo-able”.

Sampling Slater determinants

iii DeDw H Letting

ii

iiiii

w

wDHeDwE

H /

)normalised and negative-non (i.e.

wordthe of sense usual the iny probabilit a is

positive are elementsmatrix

diagonal its operator, definite positive a is Since

i

i 0

i

i w

wp

e H

ii

ii

ii

i

ii

i

ii

iii

i

ii

w

HH

Ew

EwEpE

wE

wDeD

w

DHeD

~~

~

ln~

lnln

:writing

:experiment Carlo-Monte

a for suitable form a inenergy the writecan one

:Noting

Where the expectation value is taken over an ensemble of determinantssampled with probability pi.

Perform Metropolis sampling of Di chosen according to wi

simulation

MC the of step on tdeterminan the is where

t

EK

E

t

tt

K

i

i

~1lim

)/min(1,P acc ijji wwDD ,

lkjlijkiji

iiiii

...,,

...

)(

factors P

PH

w

DeDw

Discrete path integral: wildly oscillatory integrand.Can’t use Monte Carlo!

Hopeless to calculate by brute-force!

i

j

l

k

jiij DeD PH / [High-temperature DM]

Problem: the weight itself is a path-integral!

Define:

54

43

1010

1010/

P

P

Generation of graphs with a computable generation probability

ij

k

j2

j3

We adopt a Markov chain algorithm is which successive determinantsare added to a list until the desired size of graph is reached. However,since the connectivity of each determinant is not uniform, such an algorithm can produce a non-uniform generation probability.

Start at i, and selected a connected determinant, j, with probability pij. Thisresults in a 2-vertex graph, G={i,j}.

Next, select k, connected to j, with probability pjk. If k is distinct, then add kto the list: G={i,j,k}. Otherwise, select a new determinant from the currentPosition (i.e. the last visited determinant). Continue this process until n distinct verticies have been visited.

The generation probability can be calculated by examining all possibleways of generating G according to this algorithm.

For example, for a 3-vertex graph, G={i,j,k}:

kiik

ijkikjik

jiij

ikjijkij

ijkiikkjikkiik

ikjiijjkijjiij

pp

pppp

pp

pppp

ppppppp

pppppppGP

n

n

n

n

n

n

n

ngen

1

)(

1

)(

)()(

)()(][

10

10

This procedure can be generalised for a n-vertex graph (n>3).

The general case is most compactly expressed in matrix notation.Let us call our n verticies G={i1,i2,…,in}, all distinct, with i1=i.

Consider the generation probability of G in the given order (i1,i2,…,in).

According to this algorithm, we visit i2 for the first time from i1, i3 forthe first time from either i1 or i2, etc. In general we visit ik for the firsttime from any of the previous visited k-1 verticies. The algorithm terminates when we first visit the n-th vertex.

This is a first-passage problem in Markov chain theory.

1000

...

0

...0

],...,,[ 23212

121

21)( k

k

iiiiii

iiii

kk ppp

pp

iiiP

We will construct a series of transition probability matrices in which vertex ik is an absorbing state:

1000

...

0

...0

],...,,[ 23212

121

21)( k

k

iiiiii

iiii

kk ppp

pp

iiiP

Note that:kk ii

nkP ,)(

1][

represents the probability of arriving at ik in exactly n steps given weStarted from ik-1, passing through some or all of (i1,i2,…., ik-1).

So therefore the total probability of arriving at ik, is simply the geometricseries:

kkkii

nk

n PIP kk

,1)(,

)( 1][ 1

kkkkgen PIPIPI

iiiP,1

)(3,2

)3(2,1

)2(21

1...

11],...,,[

Therefore the probability to generate the graph G in the sequence:

kiii ...21

The probability to generate G in any order is given by the sum over all n!permutations:

)],...,,(ˆ[][ 21ˆ

kP

gengen iiiPPGP

iij N

p1

In current implementation we choose

Where Ni is the number of determinants connected to i. In other wordswe do not introduce an energetic bias in the selection of determinants.

ij1

j2

j3

Conclusions

A new approach for Fermion Monte Carlo is being developed, based on

sampling Slater determinant space with weights computed according to

a novel path counting scheme.

The mathematics of path-counting needs further investigation.

The scheme has been applied to the Hubbard model and the N2 problem with encouraging results.

Topology of graphs

i

j

k

i

j

k

i

j

k

Cyclic Star (tree) Chain (tree)

i,j, and k all must be single or double-excitations ofeach other.

j and k all must be s- ord-excitations of i,but not necessarily of eachother

j must be a s- or d-excitation of i, and kis a triple or quadrupleof i.

Future work

Technical

Sampling graphs to counter the scaling problem

Calculation of electron density

Parallelisation of code

Systems:

Hubbard models [e.g. stability of striped phases]

Dispersion interactions (e.g. graphitic systems)

...)()()(1 432ijklijkij

iii ijklijkij SSSw P

Contribution to the weights

10-site, N=10, U=4 weights

0

1

2

3

4

5

6

7

8

9

0 1 2 3 4 5 6

Vertex

Co

ntr

ibu

tio

n t

o t

he

We

igh

t

momentum b=1

UHF b=1

UHF b=2

UHF b=5

UHF b=10

Tentative conclusion is: for the Hubbard model with U=4, the 3-vertexapproximation is not perfect, although it is nevertheless an improvement over UHF: Captures about 20-50% of the correlation energy.

Can we estimate the contribution of the higher order graphs through a MCsampling?

=> work in this direction is in progress

[UHF basis]

Distribution of terms among the 2,3 and 4 vertex graphs

N=10, U=4, beta=1

-9

-8.5

-8

-7.5

-7

-6.5

-6

-5.5

-5

0 1 2 3 4 5 6

Vertex

En

erg

y

vertex [mom]

Exact GS

UHF

vertex UHF b=1

RHF

Convergence of Êi with the vertex approximation

1010 Hubbard Model

-1

+1

+4

-4

Half-filled system is closed-shell

184756 determinants at N=10. Exactly diagonalisable with effort on P4

Two important questions

How good is the 3-vertex approximation?

What is the best one-particle basis to use?

A 3-determinant star

2

)2(

2

)2(

2/1

2/1

0

,

2/1

2/1

2/1

,

2/1

2/1

2/1

),2(

0

0

3

3

PP babaw

aba

ab

ab

bba

ccc

i

j

k

Contour integral solution:

22

2

)1()()(

za

bzAzA ikij

PPP

P

P

P

P

PP

P

PP

PP

babaSaw

Therefore

a

ba

a

ba

ab

abab

ab

abab

abzabz

zz

i

abz

zz

iabz

z

z

z

i

zabz

z

izAzAz

z

iS

)2()2(2

1)1(

1)2(

1)2(

2

1

)/(22

)/2(1)/21(

)/(22

)/2(1)/21(

)/21)(/21(

)1(1

2

1

)/(2)1(

)1(1

2

1

)/(2)1(

)1(

1

1

2

1

)1(2

1

1

1

1

2

1

)()(1

1

1

1

2

1

3

2222

2

22

23

ikij

Again in exact agreement with the diagonalisation result

i

j

kFully connected 3-vertex graph

PP

baba

w

baba

abb

bab

bba

)(3

2

3

)2(

61

61

6/2

,

21

21

0

,

31

31

31

)(),2(

321

3,21

i

Via diagonalisation:

3

1)2(

3

1

3

2)(

3

2

)/(3

)/(1)/21(

)/3(

)/2(1)/1(

)/21)(/1(

)/1(1

2

1

)/21()/1(

)/1)(/1(1

2

1

)1()/(3)/(2)1(

)/()1(1

2

1

)1(3

)1(2

1

)1(1

1

1

2

1

21

1

1

1

2

1

2

233

22

22

2

33

3

22

2

3

P

P

P

P

PP

P

P

P

P

P

a

ba

a

ba

ab

abab

ab

abab

abzabz

abzz

i

abzabz

abzabzz

i

zababz

abzz

i

zab

zab

zab

z

z

i

AAAA

A

z

z

iS

jkikijijk

jk

Via the contour integral:

[multiply top and bottom by (z-1)3]

[Factorise]

[Cancel factors]

[Evaluate two residues atz=1-b/a, and z=1+2b/a]

PPP babaSaw

Therefore

)2(3

1)(

3

2)1( 3

N2 VDZ

-109.4

-109.2

-109

-108.8

-108.6

-108.4

-108.2

-108

-107.8

-107.6

0 2 4 6 8 10

r/a.u.

E/a

.u.

FCI

RHF

CCSD

CCSD(T)

CCSDT

v=2 b=5

v=3 b=5 all

v=3 b=5 spec

v=3 b=5 spec3

v=4 b=5 fullsum spec3

8-site Hubbard model with N=6 electrons

3-vertex weights against exact weights

Some simple examples.

(a) A two-determinant system

2

)(

2

)(

2/1

2/1,

2/1

2/1),(

2PP

Pk

kk

babaDw

baab

ba

ii

Exact solution via diagonalisation:

Solution via Contour integral formula:

First define A(z):

22

22

)1())(()(

za

b

azaaza

bzA

i

j

b

a

ij

jjii

Next compute S2:

In exact agreement with diagonalisation result

PPP

P

P

P

P

PP

P

PP

PP

babaSaw

Therefore

a

ba

a

ba

ab

abab

ab

abab

abzabz

zz

i

abz

zz

iabz

z

z

z

i

zabz

z

izAz

z

iS

)()(2

1)1(

1)(

1)(

2

1

)/(2

)/(1)/1(

)/(2

)/(1)/1(

)/1)(/1(

)1(1

2

1

)/()1(

)1(1

2

1

)/()1(

)1(

1

1

2

1

)1(1

1

1

1

2

1

)(1

1

1

1

2

1

2

2222

2

22

22

Residue theorem

)()()!1(

1

)()(2

1

0)1(

)1(

1

0

zfzzdz

d

ma

zm

dzzfi

mm

m

c

at order of pole at Residue

residues enclosed of sum

10-site, U=4

-11

-10.5

-10

-9.5

-9

-8.5

-8

-7.5

-7

-6.5

-6

0 2 4 6 8 10 12

Nel

En

erg

y/t

Ground E

EUHF

10-site, U=4

-11

-10.5

-10

-9.5

-9

-8.5

-8

-7.5

-7

-6.5

-6

0 2 4 6 8 10 12

Nel

En

erg

y/t Ground E

EUHF

v=3 [momentum]

10-site, U=4

-11

-10.5

-10

-9.5

-9

-8.5

-8

-7.5

-7

-6.5

-6

0 2 4 6 8 10 12

Nel

Ene

rgy/

t Ground E

EUHF

v=3 [momentum]

10-site, U=4

-11

-10.5

-10

-9.5

-9

-8.5

-8

-7.5

-7

-6.5

-6

0 2 4 6 8 10 12

Nel

En

erg

y/t Ground E

EUHF

v=3 [momentum]

v=3 UHF

Exact ground-state energy, UHF and lowest Êi vs particle number

The electron correlation problem

How to account for the fact that electrons move around in a correlated fashion?

Quantum chemistry approach is:Start from Hartree-Fock and try to improve systematically

Hartree-Fock [mean-field theory, N3 ~ N4] HF=D0

Coupled Cluster [CCSD(T), N7] =eT HF

+ perturbation theory

Full-configuration interaction [eN] = HF+ j cjDjExpansions in Antisymmetricfunctions

Essential feature of HF theory: maintains an orbital (one-particle)picture of electronic structure

)]()...()(det[ 2211 NNHF uuu xxx

Quantum Monte Carlo

QMC refers to stochastic methods to solve the Schrödinger Equation(or sample path integrals) based on interpreting the S.E. as a “diffusion equationin imaginary time”:

is interpreted as a probability distribution. Long-time stochastic propagation [diffusion+life/death processes] leads to sampling the nodeless eigenstateof H.

Application to Fermion systems is severely hampered by “sign problems”.Unconstrained sampling of the configuration space of Fermions leads to BosonCatastrophe.

QMC can be stabilised by the introduction of constraints:-Fixed node approximation in diffusion MC [J Anderson]-Restricted path integral MC (fixing nodes of density matrix) [Ceperley]-Positive projection and constrained path MC for auxillary field QMC [Fahy and Hamman, Zhang]

VH 2

2

1

Why not use antisymmetric spaces?

We would like to explore the possibility of using an antisymmetrized space as the basis for quantum monte carlo.

Can we sample a set of Slater determinants in such way that we can extract meaningful physical quantities (eg energy) at the end ofthe simulation?

This strategy avoids the Boson catastrophe from the outset without imposition of fixed-node type approximations.

We will show that(1) Such a method is indeed numerically stable(2) The MC weights are obtained by summing over many paths of fluctuating sign.(3) Method depends on combinatorial ideas for path counting -> So far it is not exact(4) Applications to (a) Hubbard model and (b) Dissociating molecules

A major conceptual advantage is that it allows to build directly on the one-particle picture of mean-field theory.

What’s the problem?

17det

det

10

10,100 ..

2

N

NMge

N

MN

-109.4

-109.2

-109

-108.8

-108.6

-108.4

-108.2

-108

-107.8

-107.6

0 2 4 6 8 10 12

r/a.u.

En

erg

y/H

artr

ees

FCI

RHF

v=2 b=5

HF v=3

SPEC2

SPEC3

v=4 b=5

u

g

g

SPEC

SPEC

HF

7

5

1

3

2

15820024220 determinants (exactly diagonalised by Krogh, Olsen CPL 344, 578, (2001),and by Chan, Kallay and Gauss, JCP, 121, 610 (2004) )

N2 VDZ

g

u

g (pz)

u

g

u(pz)

g

u

g (pz)

u

g

u(pz)

g

u

g (pz)

u

g

u(pz)

g

1

g

5

u

7

)()()( 441

2 SNSNNg

0~

lim

~

00

2

2

2

ii

i

i

i

ii

DEE

De

DeEE

Dew

aa

E

aa

Ea

aa

E

a

a

a

when

Formally speaking, in the eigenvalue basis of H:

Hubbard Model

ijiiiji

nnUh.c.cctH,,,,

,,

][

U

Model of itinerant magnetism for narrow-band systems. Intensively studied since the mid-80’s in the context of High Tc.

Partition function

1 2

],,...,,[...

][

121)(

i ii

ii

i

iiiiP

H

PP

H

w

DeDeTrQ

Note that the sign of w(P) is a very poorly behaved quantity:

Depends on the product of P matrix elements. Therefore small variations in the path can lead to wild fluctuations inthe sign of the path.

Exact Diagonalisation

problem eigenvalue linear a :

,

)(,1

,2

1 2

ijjj

i

iii

rrr

aa

a

N

ii

ji ji

N

ii

cEcDHD

Dc

EH

vVUT

VUTH

Exact solutions are expressed as linear superpositions of uncorrelatedDeterminants.

Conjecture: the n-vertex graph gives rise to a polynomial of degree n in the denominator of the contour integral

Contour integrals reduce to a sum over n residues

1

ˆ)/(3

ˆ)/(22

ˆ)/(121 ...... xexxexxexdxdxdxQ HP

PHPHP

P

x1

x2

xP

x3

Harmonic springs hold together a “ring polymer”

One can simulate an electronas a ring-polymer, movingin the external field (which itselfcan be dynamic).

Polarons [Parrinello Rahman]

1

ˆ)/(

3

ˆ)/(22

ˆ)/(12121 ...],..,[...

1

xex

xexxexxxxEdxdxdxQ

E

HPP

HPHPPP

Motivation

The development of stable fermion QMC algorithms which do not require fixed-node approximations, but which maintain a ~ N2 or N3 scaling.

Does working in antisymmetric spaces (eg Slater determinants) help?

Intuitively, Slater determinant spaces are the “right” spaces to be dealing with fermions: i.e. one should build in the fermion antisymmetry in the outsetin any N-particle representation.

Computational cost of electronic structure methods

Hartree Fock N3

MP2 - MP4 N4-N7

Coupled Cluster CCSD-(T) N6-N7

FCI eN

DFT N3

QMC N2-N3

Electron correlation is ubiquitous in chemistry e.g. in molecular dissociation

)2()2(

)1()1(

2

1

)(2

1)2()1(),( 2121210

gg

gg

ggxxD

)(2

1BAu ss

)(2

1BAg ss

AsBs

))2()1()1()2()2()1()2()1((2

1

))2()2())(1()1((2

1~

....

0

HH

BB

HH

BA

HH

BA

HH

AA

BABA

ssssssss

ssssD

Slater determinant: an uncorrelated wavefunction

Incorrect dissociation

Configuration Interaction

)2()2(

)1()1(

2

12

uu

uuD AsBs

Consider the doubly excited determinant:

)(2

120 DDcorr

))1()2()2()1((2

1~

.... HH

BA

HH

BA ssss

Correct dissociation

Correlated wavefunction

What is the problem with configuration interaction?

-Slowly convergent with respect to short and intermediate range correlation.

Must include many determinants: the problem grows exponentiallywith number of electrons and the number of virtuals

-(linear) Truncated CI lacks size consistency: Coupled cluster methods are nowadays preferred.