Dario Bressanini

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…

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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

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)

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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 and fragments:Tails and fragments: Usually neglectedUsually neglected

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Asymptotic behavior of Asymptotic behavior of

1221

22

21

1)11()(21

rrrZH

21

22

21

1)(212

rZ

rZH

r

Example with 2-e atomsExample with 2-e atoms

IE222 1/)1(

210 )( rZr

err

)( 10 r is the solution of the 1 electron problemis the solution of the 1 electron problem

9


)()( 21 rr The usual formThe usual form

)()()()()()(

2011

2102

rrrrrr

does does notnot satisfy the asymptotic conditions satisfy the asymptotic conditions

)()()()( 1221 rrrr

A closed shell determinant has the A closed shell determinant has the wrongwrong structure structure

)( 21 rrae

10


),...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

Each electron has its own orbital, Each electron has its own orbital, Multideterminant (GVB) Multideterminant (GVB) Structure!Structure!

factor ncorrelatio,function spin UN e

11

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

12

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

rbrar

es

1

2

10 r

are

rbre

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2-electron atoms2-electron atoms

12

12

2

22222

1

21111

12 1exp

1exp

1exp)ˆ1(

erdr

rrbra

rrbraP

Tails OKTails OK

12

12

2

2222

1

2111

12 12exp

1exp

1exp)ˆ1(

err

rrbZr

rrbZrP

Cusps OK Cusps OK – 3 parameters– 3 parameters

12

12

2

2222

112 12exp

1expexp)ˆ1(

err

rrbZrZrP

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 12exp

1exp

1exp)ˆ1()|2,1(

rkZ/r

r)rkZ(kZr

rrkZZr

PZΨ

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

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For larger atoms ?For larger atoms ?

He Li Be B C N O F Ne

-0.4

-0.3

-0.2

-0.1

0

Cor

rela

tion

Ener

gy (h

artr

ee)

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

Cor

rela

tion

Ener

gy (h

artr

ee)

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)(

21

1221

22

21

Critical Z for binding Critical Z for binding ZZcc=0.91103=0.91103 cc is square integrable is square integrable

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

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

Ener

gy

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

r3-r4r3-r4

r1-r2r1-r2

r1+r2r1+r2

0Exact

r1-r2r1-r2

r1+r2r1+r2

r3-r4r3-r4

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

Ener

gyLithiumBeryllium

cc

N=3N=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

Ener

gy

Close upBerylliumLithiumN=3N=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

preliminary resultspreliminary results

Critical Node Critical Node very close tovery close to

-

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Take a look at Take a look at youryour nodes nodes

The EndThe End

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Dario Bressanini