is qmc delivering its early promises? dario bressanini qmci sardagna (trento) 2006 universita’...
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Is QMC delivering its early promises?Is QMC delivering its early promises?Dario Dario BressaniniBressanini
QMCI Sardagna QMCI Sardagna ((TrentoTrento) ) 2006 2006
http://scienze-como.uninsubria.it/http://scienze-como.uninsubria.it/bressaninibressanini
Universita’ dell’Insubria, Como, ItalyUniversita’ dell’Insubria, Como, Italy
Some reflections on nodes and trial wave functionsSome reflections on nodes and 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)
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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
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33HeHemm44HeHenn Stability Chart Stability Chart
3232
44HeHenn 33HeHemm 0 1 2 3 4 5 6 70 1 2 3 4 5 6 7 8 9 10 11 8 9 10 11
00
11
22
33
44
55
33HeHe3344HeHe88 L=0 S=1/2 L=0 S=1/2
33HeHe2244HeHe44 L=1 S=1 L=1 S=1
33HeHe2244HeHe22 L=0 S=0 L=0 S=0
33HeHe3344HeHe44 L=1 S=1/2 L=1 S=1/2
Terra IncognitaTerra IncognitaTerra IncognitaTerra Incognita
Bound L=0Bound L=0
UnboundUnbound
UnknownUnknown
L=1 S=1/2L=1 S=1/2
L=1 S=1L=1 S=1
BoundBound
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Good for vibrational Good for vibrational problemsproblems
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For electronic structure?For electronic structure?
Sign ProblemSign Problem
Fixed Nodal error problemFixed Nodal error problem
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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 ??
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HeHe22++: the basis set: the basis set
)2())1()3()3()1(()( gugugRHF R )2())1()3()3()1(()( gugugRHF R
The ROHF wave function:The ROHF wave function:
1s1s
E = -4.99E = -4.990505(2) hartree(2) hartree1s1s’2s3s1s1s’2s3s
E = -4.99E = -4.994343(2) hartree(2) hartree
EEN.R.LN.R.L = -4.99 = -4.994545 hartree hartree
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HeHe22++: MO’s: MO’s
E(RHF) = -4.99E(RHF) = -4.994343(2) (2) hartreehartree
E(CAS) = -4.99E(CAS) = -4.992525(2) (2) hartreehartree
E(CAS-NO) = -4.99E(CAS-NO) = -4.991616(2) (2) hartreehartree
E(CI-NO) = -4.99E(CI-NO) = -4.991717(2) (2) hartreehartree
EEN.R.LN.R.L = -4.99 = -4.994545 hartree hartree
Bressanini et al. J. Chem. Phys. 123, 204109 (2005) Bressanini et al. J. Chem. Phys. 123, 204109 (2005)
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HeHe22++: CSF’s: CSF’s
1s1s’2s3s2p2p’1s1s’2s3s2p2p’
E(1 csf) = -4.99E(1 csf) = -4.993232(2) hartree(2) hartree
1s1s’2s3s1s1s’2s3s
E(1 csf) = -4.99E(1 csf) = -4.994343(2) hartree(2) hartree
+ +
E(2 csf) = -4.99E(2 csf) = -4.994646(2) hartree(2) hartree
2121 1111 uyuuxu 2121 1111 uyuuxu
+ +
E(2 csf) = -4.99E(2 csf) = -4.992525(2) hartree(2) hartree
111 211 gug 111 211 gug
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LiLi22
-14.9954 E (N.R.L.)E (N.R.L.)
-14.9952(1)
-14.9939(2)
-14.9933(1)
-14.9933(2)
-14.9914(2)
-14.9923(2)
E (hartree) CSF22 g
222 9...43 ggg 22 11 uyux
224 uyux nn 222 211 uuyux
2222 3211 guuyux
Not all CSF are usefulNot all CSF are useful Only 4 csf are needed to build a statistically Only 4 csf are needed to build a statistically
exact nodal surfaceexact nodal surface
(1(1gg22 1 1uu
22 omitted) omitted)
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A tentative recipeA tentative recipe Use a large Slater basisUse a large Slater basis
But not too largeBut not too large Try to reach HF Try to reach HF nodesnodes convergence convergence
Orbitals from CAS Orbitals from CAS seemseem better than HF, or NO better than HF, or NO 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...
...different Angular Momentum CSFs...different Angular Momentum CSFs
And not the bad onesAnd not the bad ones ...types already included...types already included
222234
222234
222234
31)31(ˆ
21)21(ˆ
21)21(ˆ
ssssi
pspsi
ssssi
222234
222234
222234
31)31(ˆ
21)21(ˆ
21)21(ˆ
ssssi
pspsi
ssssi
14
DimersDimers
Bressanini et al. J. Chem. Phys. 123, 204109 (2005) Bressanini et al. J. Chem. Phys. 123, 204109 (2005)
15
Is QMC competitive ?Is QMC competitive ?
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Carbon Atom: EnergyCarbon Atom: Energy CSFsCSFs Det.Det. EnergyEnergy 1 1 1s1s222s2s2 2 2p2p2211 -37.8303(4)-37.8303(4) 2 2 + 1s+ 1s22 2p 2p44 22 -37.8342(4)-37.8342(4) 5 5 + 1s+ 1s22 2s 2s 2p2p223d3d 1818 -37.8399(1)-37.8399(1) 83 83 1s1s22 + 4 electrons in + 4 electrons in 2s 2p 3s 3p 3d2s 2p 3s 3p 3d shell shell
422422 -37.8387(4)-37.8387(4)
adding f orbitalsadding f orbitals 77 (4f(4f22 + 2p + 2p334f)4f) 3434 -37.8407(1)-37.8407(1)
R12-MR-CIR12-MR-CI -37.845179-37.845179ExactExact (estimated) (estimated) -37.8450-37.8450
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Ne AtomNe Atom
Drummond Drummond et al. et al. -128.9237(2) -128.9237(2) DMCDMCDrummond Drummond et al. et al. -128.9290(2) -128.9290(2) DMC backflowDMC backflowGdanitz Gdanitz et al. et al. -128.93701 -128.93701 R12-MR-CIR12-MR-CIExact (estimated)Exact (estimated) -128.9376-128.9376
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Conventional wisdom on Conventional wisdom on
EEVMCVMC((RHFRHF) > E) > EVMCVMC((UHFUHF) > E) > EVMCVMC((GVBGVB))
Single particle approximationsSingle particle approximations
RHFRHF = = |1s|1sRR 2s 2sRR 2p 2pxx 2p 2pyy 2p 2pzz| |1s| |1sRR 2s 2sRR||
UHFUHF = = |1s|1sUU 2s 2sUU 2p 2pxx 2p 2pyy 2p 2pzz| |1s’| |1s’UU 2s’ 2s’UU||
Consider the Consider the NN atom atom
EEDMCDMC((RHFRHF) ) > ? <> ? < E EDMCDMC((UHFUHF))
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Conventional wisdom on Conventional wisdom on
We can build a We can build a RHFRHF with the same nodes of with the same nodes of UHFUHF
UHFUHF = = |1s|1sUU 2s 2sUU 2p 2pxx 2p 2pyy 2p 2pzz| |1s’| |1s’UU 2s’ 2s’UU||
’’RHFRHF = = |1s|1sUU 2s 2sUU 2p 2pxx 2p 2pyy 2p 2pzz| |1s| |1sUU 2s 2sUU||
EEDMCDMC((’’RHFRHF) ) == EEDMCDMC((UHFUHF))
EEVMCVMC((’’RHFRHF) ) >> E EVMCVMC((RHFRHF) ) >> E EVMCVMC((UHFUHF))
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Conventional wisdom on Conventional wisdom on
Node equivalent to a Node equivalent to a UHF UHF |f(r) g(r) 2p|f(r) g(r) 2p33| |1s | |1s 2s|2s|
EEDMCDMC((GVBGVB) ) == E EDMCDMC((’’’’RHFRHF))
GVBGVB = = |1s 2s 2p|1s 2s 2p33| |1s’ 2s’| - |1s’ 2s 2p| |1s’ 2s’| - |1s’ 2s 2p33| |1s 2s’| + | |1s 2s’| +
|1s’ 2s’ 2p |1s’ 2s’ 2p33| |1s 2s|- |1s 2s’ 2p| |1s 2s|- |1s 2s’ 2p33| |1s’ 2s|| |1s’ 2s|
Same NodeSame Node
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Nitrogen AtomNitrogen Atom Param. Param. E corr. VMCE corr. VMC E E
corr. DMCcorr. DMC
Simple RHF (1 det)Simple RHF (1 det) 44 26.0%26.0%91.9%91.9%
Simple RHF (1 det)Simple RHF (1 det) 88 42.7%42.7%92.6%92.6%
Simple UHF (1 det)Simple UHF (1 det) 1111 41.2%41.2%92.3%92.3%
Simple GVB (4 det) Simple GVB (4 det) 1111 42.3%42.3%92.3%92.3%
Clementi-Roetti + JClementi-Roetti + J 2727 24.5%24.5%93.1% 93.1%
Is it worth to continue to add parametersIs it worth to continue to add parametersto the wave function?to the wave function?
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What to do?What to do?
Should we be happy with the “cancellation Should we be happy with the “cancellation of error”, and pursue it?of error”, and pursue it?
If so:If so: Is there the risk, in this case, that QMC Is there the risk, in this case, that QMC
becomes becomes Yet Another Computational ToolYet Another Computational Tool, and , and not particularly efficient nor reliable?not particularly efficient nor reliable?
VMC seems to be much more VMC seems to be much more robustrobust, , easy to easy to “advertise”“advertise”
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 ? ?
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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
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Convergence to the exact Convergence to the exact We must include the correct analytical structureWe must include the correct analytical structure
Cusps:Cusps:2
1)0( 1212
rr Zrr 1)0(
3-body coalescence and logarithmic terms:3-body coalescence and logarithmic terms:
QMC OKQMC OK
QMC OKQMC OK
Tails:Tails:Often neglectedOften neglected
25
Asymptotic behavior of Asymptotic behavior of
1221
22
21
1)
11()(
2
1
rrrZH
1221
22
21
1)
11()(
2
1
rrrZH
21
22
21
1)(
2
12
r
Z
r
ZH
r
21
22
21
1)(
2
12
r
Z
r
ZH
r
Example with 2-e atomsExample with 2-e atoms
IE2 IE222 1/)1(
210 )( rZr
err
22 1/)1(
210 )( rZr
err
)( 10 r )( 10 r is the solution of the 1 electron problemis the solution of the 1 electron problem
26
Asymptotic behavior of Asymptotic behavior of
)()()( 1221 rJrr )()()( 1221 rJrr The usual formThe usual form
)()()(
)()()(
2011
2102
rrr
rrr
)()()(
)()()(
2011
2102
rrr
rrr
does does notnot satisfy the asymptotic conditions satisfy the asymptotic conditions
)())()()()(( 121221 rJrrrr )())()()()(( 121221 rJrrrr
A closed shell determinant has the A closed shell determinant has the wrongwrong structure structure
27
Asymptotic behavior of Asymptotic behavior of
),...2()())(1( 101
11
/21
11110
1111
NYerOrcr Nml
brar
N
r ),...2()())(1( 101
11
/21
11110
1111
NYerOrcr Nml
brar
N
r In generalIn general
Recursively, fixing the cusps, and setting the right symmetry…Recursively, fixing the cusps, and setting the right symmetry…
UNN eNfffA ))()...2()1((ˆ
21 UNN eNfffA ))()...2()1((ˆ
21
Each electron has its own orbital, Each electron has its own orbital, Multideterminant (GVB) Structure!Multideterminant (GVB) Structure!
Take 2N coupled electronsTake 2N coupled electrons )...)(( 434321212 N )...)(( 434321212 N
22NN determinants. Again an determinants. Again an exponential wallexponential wall
28
PsH – Positronium PsH – Positronium HydrideHydride
A wave function with the correct asymptotic A wave function with the correct asymptotic conditions:conditions:
Bressanini and Morosi: JCP Bressanini and Morosi: JCP 119119, 7037 (2003), 7037 (2003)
)()()()()ˆ1(),2,1( 112 pp rgPsrfHPp )()()()()ˆ1(),2,1( 112 pp rgPsrfHPp
29
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
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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 ?
31
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
32
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
33
Another Another (failed so far) (failed so far) exampleexample
Different dimensionality: HypernodesDifferent dimensionality: Hypernodes Given Given HH((RR) = ) = EE((RR)) build build
dimensionsNHHH 6)()( 21 RR dimensionsNHHH 6)()( 21 RR
)()()()( 2121 RRRR BFFBT )()()()( 2121 RRRR BFFBT
•Use the Use the HypernodeHypernode of of TT
• The hope was that it could be better than Fixed Node The hope was that it could be better than Fixed Node
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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
Why is QMC not Why is QMC not used by used by
chemists?chemists?
A little A little intermezzointermezzo
36
DMC Top 10 reasonsDMC Top 10 reasons 12. We need forces, dummy! 12. We need forces, dummy!
11. Try getting O11. Try getting O22 to bind at the variational level. to bind at the variational level.
10. How many graduate students lives have been lost optimizing 10. How many graduate students lives have been lost optimizing wavefunctions? wavefunctions?
9. It is hard to get 0.01 eV accuracy by throwing dice. 9. It is hard to get 0.01 eV accuracy by throwing dice. 8. Most chemical problems have more than 50 electrons. 8. Most chemical problems have more than 50 electrons. 7. Who thought LDA or HF pseudopotentials would be any good? 7. Who thought LDA or HF pseudopotentials would be any good? 6. How many spectra have you seen computed by QMC? 6. How many spectra have you seen computed by QMC? 5. QMC is only exact for energies. 5. QMC is only exact for energies. 4. Multiple determinants. We can't live with them, we can't live 4. Multiple determinants. We can't live with them, we can't live
without them. without them. 3. After all, electrons are fermions. 3. After all, electrons are fermions. 2. Electrons move. 2. Electrons move. 1. QMC isn't included in Gaussian 90. Who programs anyway? 1. QMC isn't included in Gaussian 90. Who programs anyway?
http://web.archive.org/web/20021019141714/http://web.archive.org/web/20021019141714/archive.ncsa.uiuc.edu/Apps/CMP/topten/topten.htmlarchive.ncsa.uiuc.edu/Apps/CMP/topten/topten.html
37
Chemistry and Chemistry and MathematicsMathematics
“The underlying physical laws necessary for the mathematical theory of a large part of physics and the whole of chemistry are thus completely known, and the difficulty is only that the exact application of these equations leads to equations much too complicated to be soluble”
P.A.M. Dirac - 1929
“The underlying physical laws necessary for the mathematical theory of a large part of physics and the whole of chemistry are thus completely known, and the difficulty is only that the exact application of these equations leads to equations much too complicated to be soluble”
P.A.M. Dirac - 1929
"We are perhaps not far removed from the time, when we shall be able to submit the bulk of chemical phenomena to calculation”
Joseph Louis Gay-Lussac - 1808
"We are perhaps not far removed from the time, when we shall be able to submit the bulk of chemical phenomena to calculation”
Joseph Louis Gay-Lussac - 1808
38
Nature and MathematicsNature and Mathematics
“il Grande libro della Natura e’ scritto nel linguaggio della matematica, e non possiamo capirla se prima non ne capiamo i simboli“
Galileo Galilei
“il Grande libro della Natura e’ scritto nel linguaggio della matematica, e non possiamo capirla se prima non ne capiamo i simboli“
Galileo Galilei
Every attempt to employ mathematical methods in the study of chemical questions must be considered profoundly irrational and contrary to the spirit of chemistry… If mathematical analysis should ever hold a prominent place in chemistry – an aberration which is happily almost impossible – it would occasion a rapid and widespread degeneration of that science.
Auguste Compte
Every attempt to employ mathematical methods in the study of chemical questions must be considered profoundly irrational and contrary to the spirit of chemistry… If mathematical analysis should ever hold a prominent place in chemistry – an aberration which is happily almost impossible – it would occasion a rapid and widespread degeneration of that science.
Auguste Compte
39
A Quantum Chemistry A Quantum Chemistry ChartChart
J.PopleJ.Pople
The more accurate the calculations became, the The more accurate the calculations became, the more the concepts tended to vanish into thin air more the concepts tended to vanish into thin air (Robert Mulliken)(Robert Mulliken)
Orthodox QMCOrthodox QMC
40
Chemical conceptsChemical concepts
Molecular structure and geometryMolecular structure and geometry Chemical bondChemical bond
Ionic-CovalentIonic-Covalent Singe, Double, TripleSinge, Double, Triple
ElectronegativityElectronegativity Oxidation numberOxidation number Atomic chargeAtomic charge Lone pairsLone pairs AromaticityAromaticity
C OO C OO
NOT DIRECTLY OBSERVABLESNOT DIRECTLY OBSERVABLESILL-DEFINED CONCEPTSILL-DEFINED CONCEPTS
41
NodesNodes
Should we concentrate on Should we concentrate on nodes?nodes?
Checked on small systems: L, Be, HeChecked on small systems: L, Be, He22++. See also . See also
MitasMitas
Conjectures on nodesConjectures on nodes have higher symmetry than have higher symmetry than itself itself resemble simple functionsresemble simple functions the ground state has only 2 nodal volumesthe ground state has only 2 nodal volumes HF nodes are quite good: they “naturally” have HF nodes are quite good: they “naturally” have
these propertiesthese properties
43
Avoided crossingsAvoided crossings
BeBe
ee-- gas gas
StadiumStadium
44
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)
46
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...) a proof...
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
47
He atom with noninteracting He atom with noninteracting electronselectrons
Sss 153 Sss 153
48
49
Casual similarity ?Casual similarity ?
First unstable antisymmetric stretch orbit of semiclassical linear helium along with the symmetric Wannier orbit r1 = r2 and various equipotential lines
HeliumSss 121
50
Superimposed Hylleraas node
Casual similarity ?Casual similarity ?
51
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
52
Coordinate transformationCoordinate transformation
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
Miller-Good transformationsMiller-Good transformations
53
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
54
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 probable outcomeof a probable outcome
the the numbersnumbers lead a dance lead a dance
Sting: Shape of my heartSting: Shape of my heart
55
Think Different!Think Different!