Fermion Quantum Monte Carlo based on the idea of sampling “graphs” Ali Alavi University of...
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Transcript of Fermion Quantum Monte Carlo based on the idea of sampling “graphs” Ali Alavi University of...
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”
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
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
][.][
][
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 ][)()(
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
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
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
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]
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
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.
-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
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.