Universita’ dell’Insubria, Como, Italy. Some considerations on nodes and trial wave functions. Dario Bressanini. http://scienze-como.uninsubria.it/ bressanini. QMCI Sardagna ( Trento ) 2008. 30+ years of QMC in chemistry. The Early promises?. - PowerPoint PPT Presentation
This document is posted to help you gain knowledge. Please leave a comment to let me know what you think about it! Share it to your friends and learn new things together.
Some considerations on nodes Some considerations on nodes and trial wave functionsand trial wave functions
2
30+ years of QMC in 30+ years of QMC in chemistrychemistry
3
The Early promises?The Early promises?
Solve the Schrödinger equation Solve the Schrödinger equation exactly exactly withoutwithout approximationapproximation (very strong)(very strong)
Solve the Schrödinger equation with Solve the Schrödinger equation with controlled approximationscontrolled approximations, and converge , and converge to the exact solution to the exact solution (strong)(strong)
Solve the Schrödinger equation with Solve the Schrödinger equation with some some approximationapproximation, and do better than other , and do better than other methods methods (weak)(weak)
4
Good for Helium studiesGood for Helium studies
ThousandsThousands of theoretical and experimental of theoretical and experimental paperspapers
)()(ˆ RR nnn EH )()(ˆ RR nnn EH
have been published on Helium, in its various forms:have been published on Helium, in its various forms:
AtomAtom Small ClustersSmall Clusters DropletsDroplets BulkBulk
5
Good for vibrational Good for vibrational problemsproblems
6
For electronic structure?For electronic structure?
Sign ProblemSign Problem
Fixed Nodal error problemFixed Nodal error problem
7
The influence on the nodes The influence on the nodes of of
QMC currently relies on QMC currently relies on TT(R)(R) and its nodes and its nodes
((indirectlyindirectly))
How are the nodes How are the nodes TT(R)(R) of influenced by: of influenced by: The single particle basis setThe single particle basis set The generation of the orbitals (HF, CAS, MCSCF, NO, The generation of the orbitals (HF, CAS, MCSCF, NO,
…)…) The number and type of configurations in the multidet. The number and type of configurations in the multidet.
ExpansionExpansion
The functional form of The functional form of TT(R)(R) ??
8
Improving Improving
Current Quantum Monte Carlo research Current Quantum Monte Carlo research focusesfocuses on on Optimizing the energy Optimizing the energy Adding more determinants (large number of Adding more determinants (large number of
parameters)parameters) Exploring new trial wave function forms Exploring new trial wave function forms
(moderately large number of parameters)(moderately large number of parameters)» Pfaffians, Geminals, Backflow ...Pfaffians, Geminals, Backflow ...
Node are improved (but not always) only Node are improved (but not always) only indirectlyindirectly
9
Adding more Adding more determinantsdeterminants
Use a large Slater basisUse a large Slater basis Try to reach HF Try to reach HF nodesnodes convergence convergence
Orbitals from MCSCF are goodOrbitals from MCSCF are good Not worth optimizing MOs, if the basis is large Not worth optimizing MOs, if the basis is large
enoughenough Only few configurationsOnly few configurations seem to improve the FN seem to improve the FN
energyenergy Use the right determinants...Use the right determinants...
Is there a Is there a nodal nodal error error componentcomponent in in HF wave HF wave function coming function coming from incorrect from incorrect dissociation?dissociation?
24
GVB for moleculesGVB for molecules
Localized orbitalsLocalized orbitals
25
GVB LiGVB Li22
E (N.R.L.)E (N.R.L.)
-14.9936(1)-14.9936(1)GVB CI 24 det compactGVB CI 24 det compact
-14.9632(1)-14.9632(1)CI 3 det compactCI 3 det compactCI 3 det compactCI 3 det compact
-14.9688(1)-14.9688(1)GVB 8 det compactGVB 8 det compact
-14.9523(2)-14.9523(2)HF 1 det compactHF 1 det compact
Should we be happy with the “cancellation Should we be happy with the “cancellation of error”, and pursue it?of error”, and pursue it?
After all, the whole quantum chemistry is After all, the whole quantum chemistry is built on it!built on it!
If not, and pursue If not, and pursue orthodox QMCorthodox QMC (no (no
pseudopotentials, no cancellation of errors, …)pseudopotentials, no cancellation of errors, …) , can we , can we avoid theavoid the curse of curse of TT ? ?
31
The curse of the The curse of the
QMC currently relies on QMC currently relies on TT(R)(R)
Walter Kohn in its Nobel lecture Walter Kohn in its Nobel lecture (R.M.P. (R.M.P. 7171, 1253 , 1253
(1999))(1999)) “discredited” the wave function as a “discredited” the wave function as a non non legitimate conceptlegitimate concept when when NN (number of (number of electrons) is largeelectrons) is large
1033 ppM N 1033 ppM N
pp = parameters per variable = parameters per variable
MM = total parameters needed = total parameters needed
For For MM=10=1099 and and pp=3=3 NN=6=6
The Exponential WallThe Exponential WallThe Exponential WallThe Exponential Wall
32
Numbers and insightNumbers and insight
““The more accurate the The more accurate the calculations became, the more the calculations became, the more the concepts tended to vanish into concepts tended to vanish into thin air “thin air “(Robert Mulliken)(Robert Mulliken)
There is no shortage of accurate calculations There is no shortage of accurate calculations for few-electron systemsfor few-electron systems
−−2.90372437703411959831115924519440444662.903724377034119598311159245194404446696905379690537 a.u.a.u. Helium atom (Nakashima Helium atom (Nakashima and Nakatsuji JCP 2007)and Nakatsuji JCP 2007)
However…However…
We need new,We need new, and and differentdifferent, ,
ideasideas
A little A little intermezzo intermezzo (for the (for the students)students)
35
We need new, and We need new, and differentdifferent, , ideasideas
Research is the process of going up alleys to see if Research is the process of going up alleys to see if they are blind. they are blind.
Marston BatesMarston Bates
Research is the process of going up alleys to see if Research is the process of going up alleys to see if they are blind. they are blind.
Marston BatesMarston Bates
Different representationsDifferent representations Different dimensionsDifferent dimensions Different equationsDifferent equations Different potentialDifferent potential Radically different algorithmsRadically different algorithms Different Different somethingsomething
36
JustJust an example an example
Try a different representationTry a different representation Is Is somesome QMC in the momentum QMC in the momentum
representationrepresentation Possible ? And if so, is it:Possible ? And if so, is it: Practical ?Practical ? Useful/Advantageus ?Useful/Advantageus ? Eventually better than Eventually better than plain vanillaplain vanilla QMC ? QMC ? Better suited for some problems/systems ?Better suited for some problems/systems ? Less plagued by the usual problems ?Less plagued by the usual problems ?
37
The other half of Quantum The other half of Quantum mechanicsmechanics
))((ˆ)( rFp ))((ˆ)( rFp The Schrodinger equation in the momentum representationThe Schrodinger equation in the momentum representation
pdpppVpm
pE )()(ˆ)2()()
2( 2/1
2
pdpppVpm
pE )()(ˆ)2()()
2( 2/1
2
SomeSome QMC (GFMC) should be possible, given the iterative form QMC (GFMC) should be possible, given the iterative form
OrOr write the imaginary time propagator in momentum space write the imaginary time propagator in momentum space
38
Better?Better? For coulomb systems:For coulomb systems:
2
2)1(ˆ)(ˆ
jiij pprFpV
2
2)1(ˆ)(ˆ
jiij pprFpV
There are There are NO cuspsNO cusps in momentum space. in momentum space. convergence should be fasterconvergence should be faster
Hydrogenic orbitals are Hydrogenic orbitals are simple rational functionssimple rational functions
222
2/15
1 )(
)8()(
Zp
Zps
222
2/15
1 )(
)8()(
Zp
Zps
39
Another Another (failed so far) (failed so far) exampleexample
Different dimensionality: HypernodesDifferent dimensionality: Hypernodes Given Given HH((RR) = ) = EE((RR)) build build
• The hope was that it could be better than Fixed Node The hope was that it could be better than Fixed Node
40
HypernodesHypernodes
Exact nodeExact node
Trial nodeTrial node
Fixed NodeFixed Node
Exact nodeExact node
Trial nodeTrial node
Fixed HyperNodeFixed HyperNode
The energy is still an upper boundThe energy is still an upper bound Unfortunately, it seems to recover exactly the FN Unfortunately, it seems to recover exactly the FN
energyenergy
0)( RT 0)( RTThe intuitive idea was that the system The intuitive idea was that the system couldcould correct correct the wrong fixed nodes, by exploring regions wherethe wrong fixed nodes, by exploring regions where
41
Feynman on simulating Feynman on simulating naturenature
Nature isn’t classical, dammit, and if you Nature isn’t classical, dammit, and if you want to make a simulation of Nature, you’d want to make a simulation of Nature, you’d better make it quantum mechanical, and better make it quantum mechanical, and by golly it’s a wonderful problem, because by golly it’s a wonderful problem, because it doesn’t look so easyit doesn’t look so easy””
Richard Feynman Richard Feynman 19811981
42
NodesNodes
Should we concentrate on Should we concentrate on nodes?nodes?
Conjectures on nodesConjectures on nodes have higher symmetry than have higher symmetry than itself itself resemble simple functionsresemble simple functions the ground state has the ground state has only 2 nodal only 2 nodal
volumesvolumes HF nodes are often a god starting pointHF nodes are often a god starting point
43
How to How to directlydirectly improve improve nodes?nodes?
Fit to a functional form and optimize the Fit to a functional form and optimize the parameters parameters ((maybe for small systemsmaybe for small systems))
IFIF the topology is correct, use a coordinate the topology is correct, use a coordinate transformationtransformation
)(RR T )(RR T
44
HeHe22++: “expanding” the node: “expanding” the node
0:)( 1 cNode 0:)( 1 cNode
BABA rrrr 3311 BABA rrrr 3311
1:)( 2 cNode 1:)( 2 cNode
031 zz 031 zz
Exact
It is a one It is a one parameter parameter
45
““expanding” nodesexpanding” nodes
This was only a kind of “This was only a kind of “proof of conceptproof of concept”” It remains to be seen if it can be applied to It remains to be seen if it can be applied to
larger systemslarger systems Writing “simple” (algebraic?) trial nodes is not Writing “simple” (algebraic?) trial nodes is not
difficult ….difficult …. The goal is to have only few linear parameters to The goal is to have only few linear parameters to
optimizeoptimize Will it work???????Will it work???????
Take a wave function with the correct nodal Take a wave function with the correct nodal topologytopology
)(RR T )(RR T
HF HF
Change the nodes with a coordinate Change the nodes with a coordinate transformation transformation (Linear? Feynman’s backflow ?) (Linear? Feynman’s backflow ?) preserving the topologypreserving the topology
The need for the correct The need for the correct topologytopology
Using Backflow alone, on a single Using Backflow alone, on a single determinant determinant is not sufficientis not sufficient, since the , since the topology is still wrongtopology is still wrong
More determinants are necessary (only More determinants are necessary (only two?)two?)
)()( RR TTBF )()( RR TTBF
nodestwoleastatTT 0)()(
RR nodestwoleastatTT 0)()(
RR
48
Be Nodal TopologyBe Nodal Topology
0HF 0HF
r3-r4r3-r4
r1-r2r1-r2
r1+r2r1+r2
0CI 0CI
r1-r2r1-r2
r1+r2r1+r2
r3-r4r3-r4
2222 2121 pscss 2222 2121 pscss
49
Avoided crossingsAvoided crossings
BeBe
ee-- gas gas
StadiumStadium
50
Nodal topologyNodal topology
The conjecture The conjecture (which I believe is true)(which I believe is true) claims claims that there are only two nodal volumes in that there are only two nodal volumes in the fermion ground statethe fermion ground state
See, among others:See, among others: Ceperley Ceperley J.Stat.PhysJ.Stat.Phys 6363, 1237 (1991), 1237 (1991) Bressanini and coworkers. Bressanini and coworkers. JCPJCP 9797, 9200 (1992), 9200 (1992) Bressanini, Ceperley, Reynolds, “Bressanini, Ceperley, Reynolds, “What do we know about What do we know about
wave function nodes?wave function nodes?”, in ”, in Recent Advances in Quantum Recent Advances in Quantum Monte Carlo Methods IIMonte Carlo Methods II, ed. S. Rothstein, World , ed. S. Rothstein, World Scientfic (2001)Scientfic (2001)
Mitas and coworkers Mitas and coworkers PRBPRB 7272, 075131 (2005), 075131 (2005) Mitas Mitas PRL PRL 9696, 240402 (2006), 240402 (2006)
52
Avoided nodal crossingAvoided nodal crossing
At a nodal crossing, At a nodal crossing, and and are zero are zero Avoided nodal crossing is the rule, not the Avoided nodal crossing is the rule, not the
exceptionexception Not (Not (yetyet) a proof... (any help is appreciated)) a proof... (any help is appreciated)
0
0
0
0variablesNwithN
eqsN
eq313
.3
.1
variablesNwithNeqsN
eq313
.3
.1
IfIf HFHF has 4 nodeshas 4 nodes HF HF has 2 nodes, with a properhas 2 nodes, with a proper
53
He atom with noninteracting He atom with noninteracting electronselectrons
Sss 153 Sss 153
54
57
A QMC song...A QMC song...
He deals the cards to find the answersHe deals the cards to find the answers
the sacredthe sacred geometry of chancegeometry of chance
thethe hidden lawhidden law of a of a probableprobable outcome outcome
the the numbersnumbers lead a dance lead a dance
Sting: Shape of my heartSting: Shape of my heart
58
Take a look at your nodes!Take a look at your nodes!