Dario Bressanini

1
Universita dellInsubria, Como, Italy
Boundary-condition-determined wave functions (and
their nodal structure) for few-electron atomic
systems
Dario Bressanini
http//scienze-como.uninsubria.it/bressanini
Critical stability V (Erice) 2008
2
Numbers and insight

• There is no shortage of accurate calculations for
  few-electron systems
• -2.90372437703411959831115924519440444669690537
  a.u. Helium atom (Nakashima and Nakatsuji JCP
  2007)
• However

The more accurate the calculations became, the
more the concepts tended to vanish into thin air
(Robert Mulliken)
3
The curse of YT

• Currently Quantum Monte Carlo (and quantum
  chemistry in general) uses moderatly large to
  extremely large expansions for Y
• Can we ask for both accurate and compact wave
  functions?

4
VMC Variational Monte Carlo

• Use the Variational Principle

• Use Monte Carlo to estimate the integrals
• Complete freedom in the choice of the trial wave
  function
• Can use interparticle distances into Y
• But It depends critically on our skill to invent
  a good Y

5
QMC Quantum Monte Carlo

• Analogy with diffusion equation
• Wave functions for fermions have nodes
• If we knew the exact nodes of Y, we could exactly
  simulate the system by QMC
• The exact nodes are unknown. Use approximate
  nodes from a trial Y as boundary conditions

6
Long term motivations

• In QMC we only need the zeros of the wave
  function, not what is in between!
• A stochastic process of diffusing points is set
  up using the nodes as boundary conditions
• The exact wave function (for that boundary
  conditions) is sampled
• We need ways to build good approximate nodes
• We need to study their mathematical properties
  (poorly understood)

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Convergence to the exact Y

• We must include the correct analytical structure

Cusps
QMC OK
QMC OK
3-body coalescence and logarithmic terms
Usually neglected
Tails and fragments
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Asymptotic behavior of Y

• Example with 2-e atoms

is the solution of the 1 electron problem
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Asymptotic behavior of Y

• The usual form

does not satisfy the asymptotic conditions
A closed shell determinant has the wrong structure
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Asymptotic behavior of Y

• In general

Recursively, fixing the cusps, and setting the
right symmetry
Each electron has its own orbital,
Multideterminant (GVB) Structure!
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PsH Positronium Hydride

• A wave function with the correct asymptotic
  conditions

Bressanini and Morosi JCP 119, 7037 (2003)
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Basis

• In order to build compact wave functions we used
  orbital functions where the cusp and the
  asymptotic behavior are decoupled

13
2-electron atoms
Tails OK
Cusps OK 3 parameters
Fragments OK 2 parameters (coalescence wave
function)
14
Z dependence

• Best values around for compact wave functions
• D. Bressanini and G. Morosi J. Phys. B 41, 145001
  (2008)
• We can write a general wave function, with Z as a
  parameter and fixed constants ki

• Tested for Z30
• Can we use this approach to larger systems? Nodes
  for QMC become crucial

15
For larger atoms ?
16
GVB Monte Carlo for Atoms
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Nodes does not improve

• The wave function can be improved by
  incorporating the known analytical structure
  with a small number of parameters
• but the nodes do not seem to improve
• Was able to prove it mathematically up to N7
  (Nitrogen atom), but it seems a general feature

• EVMC(YRHF) gt EVMC(YGVB)
• EDMC(YRHF) EDMC(YGVB)

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Is there anything critical about the nodes of
critical wave functions?
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Critical charge Zc

• 2 electrons

• Critical Z for binding Zc0.91103
• Yc is square integrable

• llt1 infinitely many discrete bound states
• 1l lc only one bound state
• All discrete excited state are absorbed in the
  continuum exactly at l1
• Their Y become more and more diffuse

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Critical charge Zc

• N electrons atom
• l lt 1/(N-1) infinite number of discrete
  eigenvalues
• l 1/(N-1) finite number of discrete eigenvalues
• N-2 Zc N-1
• N3 Lithium atom Zc ? 2. As Z? Zc
• N4 Beryllium atom Zc? 2.85 As Z? Zc

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Lithium atom
Is r1 r2 the exact node of Lithium ?

• Even the exaxt node seems to be r1 r2, taking
  different cuts (using a very accurate Hylleraas
  expansion)

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Varying Z QMC versus Hylleraas
preliminary results
The node r1r2 seems to be valid over a wide
range of l Up to lc 1/2 ?
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Be Nodal Topology
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N4 critical charge
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N4 critical charge
lc ? 0.3502 Zc ? 2.855
Zc (Hogreve) ? 2.85
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N4 critical charge node
preliminary results
very close to lc0.3502
Critical Node very close to
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The End
Take a look at your nodes