Dario Dario BressaniniBressanini
Critical stability V Critical stability V ((EriceErice) ) 2008 2008
http://scienze-como.uninsubria.it/http://scienze-como.uninsubria.it/bressaninibressanini
Universita’ dell’Insubria, Como, ItalyUniversita’ dell’Insubria, Como, Italy
Boundary-condition-determined wave functions (and their nodal structure) for few-electron atomic systems
2
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…
3
The curse of The curse of
Currently Quantum Monte Carlo (and Currently Quantum Monte Carlo (and quantum chemistry in general) uses quantum chemistry in general) uses moderatly large to extremely large moderatly large to extremely large expansions for expansions for
Can we ask for both Can we ask for both accurateaccurate and and compactcompact wave functions?wave functions?
4
VMC: Variational Monte VMC: Variational Monte CarloCarlo
02 )(
)()(E
d
dHH
RR
RRR02 )(
)()(E
d
dHH
RR
RRR
Use the Variational PrincipleUse the Variational Principle
Use Monte Carlo to estimate the integralsUse Monte Carlo to estimate the integrals CompleteComplete freedom in the choice of the trial wave function freedom in the choice of the trial wave function Can use interparticle distances into Can use interparticle distances into But It depends But It depends criticallycritically on our skill to invent a good on our skill to invent a good
5
QMC: Quantum Monte QMC: Quantum Monte CarloCarlo
Analogy with diffusion equationAnalogy with diffusion equation Wave functions for fermions have nodesWave functions for fermions have nodes If we knew the If we knew the exact nodesexact nodes of of , we could , we could
exactly simulateexactly simulate the system by QMC the system by QMC The The exactexact nodes are unknown. Use nodes are unknown. Use
approximate nodes from a approximate nodes from a trial trial as as boundary conditionsboundary conditions++ --
6
Long term motivationsLong term motivations
In In QMC we only need the zerosQMC we only need the zeros of the wave of the wave function, not what is in between!function, not what is in between!
A stochastic process of diffusing points is A stochastic process of diffusing points is set up using the nodes as boundary set up using the nodes as boundary conditionsconditions
The The exactexact wave function (for that wave function (for that boundary conditions) is boundary conditions) is sampledsampled
We need ways toWe need ways to build good build good approximate nodesapproximate nodes
We need to studyWe need to study their mathematical their mathematical properties properties (poorly understood)(poorly understood)
7
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
21)0( 12
12
rr Zrr 1)0( Zrr 1)0(
3-body coalescence and logarithmic terms:3-body coalescence and logarithmic terms:
QMC OKQMC OK
QMC OKQMC OK
Tails and fragments:Tails and fragments: Usually neglectedUsually neglected
8
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
9
Asymptotic behavior of Asymptotic behavior of
)()( 21 rr )()( 21 rr The usual formThe usual form
)()()(
)()()(
2011
2102
rrr
rrr
)()()(
)()()(
2011
2102
rrr
rrr
does does notnot satisfy the asymptotic conditions satisfy the asymptotic conditions
)()()()( 1221 rrrr )()()()( 1221 rrrr
A closed shell determinant has the A closed shell determinant has the wrongwrong structure structure
)( 21 rrae )( 21 rrae
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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) Multideterminant (GVB)
Structure!Structure!
factor ncorrelatio,function spin UN e factor ncorrelatio,function spin UN e
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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 ee
rgPsrfHPe )()()()()ˆ1(),2,1(112 ee
rgPsrfHPe
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BasisBasis
In order to build In order to build compactcompact wave functions we wave functions we used orbital functions where used orbital functions where the cuspthe cusp and the and the asymptotic behaviorasymptotic behavior are decoupled are decoupled
r
brar
es
1
2
1 r
brar
es
1
2
10 r
are 0 rare
rbre rbre
13
2-electron atoms2-electron atoms
12
12
2
22222
1
21111
12 1exp
1exp
1exp)ˆ1(
er
dr
r
rbra
r
rbraP
12
12
2
22222
1
21111
12 1exp
1exp
1exp)ˆ1(
er
dr
r
rbra
r
rbraP
Tails OKTails OK
12
12
2
2222
1
2111
12 1
2exp
1exp
1exp)ˆ1(
er
r
r
rbZr
r
rbZrP
12
12
2
2222
1
2111
12 1
2exp
1exp
1exp)ˆ1(
er
r
r
rbZr
r
rbZrP
Cusps OK Cusps OK – 3 parameters– 3 parameters
12
12
2
2222
112 1
2exp
1expexp)ˆ1(
er
r
r
rbZrZrP
12
12
2
2222
112 1
2exp
1expexp)ˆ1(
er
r
r
rbZrZrP
Fragments OK Fragments OK – 2 parameters (coalescence wave function)– 2 parameters (coalescence wave function)
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Z dependenceZ dependence
Best values around for compact wave functionsBest values around for compact wave functions D. Bressanini and G. MorosiD. Bressanini and G. Morosi J. Phys. B J. Phys. B 4141, 145001 (2008), 145001 (2008)
We can write a general wave function, with Z as a We can write a general wave function, with Z as a parameter and fixed constants parameter and fixed constants kkii
121
12
2
22432
1
2121
12 1
2exp
1exp
1exp)ˆ1()|2,1(
rkZ
/r
r
)rkZ(kZr
r
rkZZrPZΨ
121
12
2
22432
1
2121
12 1
2exp
1exp
1exp)ˆ1()|2,1(
rkZ
/r
r
)rkZ(kZr
r
rkZZrPZΨ
Tested for Z=30Tested for Z=30
Can we use this approach to larger Can we use this approach to larger systems? Nodes for QMC become crucialsystems? Nodes for QMC become crucial
15
For larger atoms ?For larger atoms ?
He Li Be B C N O F Ne
-0.4
-0.3
-0.2
-0.1
0
Co
rrel
ati
on
En
erg
y (
ha
rtre
e)
ExactHF VMCGVB VMCHF QMCGVB QMC
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GVB Monte Carlo for AtomsGVB Monte Carlo for Atoms
He Li Be B C N O F Ne
-0.4
-0.3
-0.2
-0.1
0
Co
rrel
ati
on
En
erg
y (
ha
rtre
e)
ExactHF VMCGVB VMCHF QMCGVB QMC
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Nodes does not improveNodes does not improve
The wave function can be improved by The wave function can be improved by incorporating the known analytical incorporating the known analytical structure… with a small number of structure… with a small number of parametersparameters
… … but the nodes do not seem to improvebut the nodes do not seem to improve Was able to prove it mathematically Was able to prove it mathematically up to up to
N=7N=7 (Nitrogen atom), but it seems a (Nitrogen atom), but it seems a general featuregeneral feature EEVMCVMC((RHFRHF) > E) > EVMCVMC((GVBGVB))
EEDMCDMC((RHFRHF) ) == E EDMCDMC((GVBGVB))
Is there anything Is there anything “critical” about the “critical” about the
nodes of critical wave nodes of critical wave functions?functions?
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Critical charge ZCritical charge Zcc
2 electrons:2 electrons: ZrrrH
111)(
2
1
1221
22
21
ZrrrH
111)(
2
1
1221
22
21
Critical Z for binding Critical Z for binding ZZcc=0.91103=0.91103
cc is square integrable is square integrable 2c 2c
<1 : infinitely many discrete bound states<1 : infinitely many discrete bound states
1≤1≤≤≤ cc: only one bound state: only one bound state
All discrete excited state are absorbed in All discrete excited state are absorbed in the continuum the continuum exactlyexactly at at =1=1
Their Their become more and more diffusebecome more and more diffuse
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Critical charge ZCritical charge Zcc
N electrons atom N electrons atom < 1/(N-1) infinite number of discrete < 1/(N-1) infinite number of discrete
eigenvalueseigenvalues ≥ ≥ 1/(N-1) finite number of discrete 1/(N-1) finite number of discrete
eigenvalueseigenvalues
N-2 N-2 ≤ Z≤ Zcc ≤ N-1 ≤ N-1
N=3 “Lithium” atom ZN=3 “Lithium” atom Zc c 2. As Z→ Z 2. As Z→ Zc c
N=4 “Beryllium” atom ZN=4 “Beryllium” atom Zcc 2.85 As Z→ Z 2.85 As Z→ Zc c
2Z 2Z
2c 2c
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Lithium atomLithium atom
021 FockHartreerrrr11
rr22
rr1212rr33
Spin Spin
Spin Spin
Spin Spin rr1313
r2r1
r3• Even the exaxt node Even the exaxt node
seemsseems to be r to be r11 = r = r22, taking , taking different cuts different cuts (using a very (using a very accurate Hylleraas expansion)accurate Hylleraas expansion)
Is r1 = r2 the exact node of Lithium ?
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Varying Z: QMC versus Varying Z: QMC versus HylleraasHylleraas
0 0.1 0.2 0.3 0.4 0.5
-1 .2
-1 .1
-1
-0 .9
-0 .8
-0 .7
En
erg
y
Quantum Monte CarloHylleraas expansion
The node The node r1=r2 seems r1=r2 seems to be valid to be valid over a wide over a wide range of range of
Up to Up to cc =1/2 =1/2 ??
preliminary resultspreliminary results
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Be Nodal TopologyBe Nodal Topology
0HF 0HF
r3-r4r3-r4
r1-r2r1-r2
r1+r2r1+r2
0Exact 0Exact
r1-r2r1-r2
r1+r2r1+r2
r3-r4r3-r4
2222 2121 pscss 2222 2121 pscss
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N=4 critical chargeN=4 critical charge
0 0.1 0.2 0.3 0.4 0.5
-1 .4
-1 .2
-1
-0 .8
-0 .6
En
erg
yLithiumBeryllium
cc
N=3
N=4
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N=4 critical chargeN=4 critical charge
cc 0.35020.3502
ZZcc 2.855 2.855
Zc Zc (Hogreve)(Hogreve) 2.852.850.348 0.349 0.35 0.351
-0 .821
-0 .82
-0 .819
-0 .818
-0 .817
En
erg
y
Close upBerylliumLithiumN=3
N=4
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N=4 critical charge nodeN=4 critical charge node
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4
-0 .0004
0
0.0004
0.0008
0.0012
E H
F-C
I
very close to very close to cc=0.3502=0.3502
0))(( 4321 rrrr 0))(( 4321 rrrr
preliminary resultspreliminary results
Critical Node Critical Node very close tovery close to
--
27
Take a look at Take a look at youryour nodes nodes
The EndThe End
