Dario Bressanini

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Universita dellInsubria, Como, Italy
Some considerations on nodes and trial wave
functions
Dario Bressanini
http//scienze-como.uninsubria.it/bressanini
QMCI Sardagna (Trento) 2008
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30 years of QMC in chemistry
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The Early promises?

• Solve the Schrödinger equation exactly without
  approximation (very strong)
• Solve the Schrödinger equation with controlled
  approximations, and converge to the exact
  solution (strong)
• Solve the Schrödinger equation with some
  approximation, and do better than other methods
  (weak)

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Good for Helium studies

• Thousands of theoretical and experimental papers

have been published on Helium, in its various
forms
Small Clusters
Droplets
Bulk
Atom
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Good for vibrational problems
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For electronic structure?
Sign Problem Fixed Nodal error problem
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The influence on the nodes of YT

• QMC currently relies on YT(R) and its nodes
  (indirectly)
• How are the nodes YT(R) of influenced by
• The single particle basis set
• The generation of the orbitals (HF, CAS, MCSCF,
  NO, )
• The number and type of configurations in the
  multidet. Expansion
• The functional form of YT(R)

?
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Improving YT

• Current Quantum Monte Carlo research focuses on
• Optimizing the energy
• Adding more determinants (large number of
  parameters)
• Exploring new trial wave function forms
  (moderately large number of parameters)
• Pfaffians, Geminals, Backflow ...
• Node are improved (but not always) only indirectly

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Adding more determinants

• Use a large Slater basis
• Try to reach HF nodes convergence
• Orbitals from MCSCF are good
• Not worth optimizing MOs, if the basis is large
  enough
• Only few configurations seem to improve the FN
  energy
• Use the right determinants...
• ...different Angular Momentum CSFs
• And not the bad ones
• ...types already included

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Li2
E (hartree)
CSF
(1sg2 1su2 omitted)
-14.9923(2)
-14.9914(2)
-14.9933(2)
-14.9933(1)
-14.9939(2)
-14.9952(1)
-14.9954
E (N.R.L.)

• Not all CSF are useful
• Only 4 csf are needed to build a statistically
  exact nodal surface Bressanini et al. J. Chem.
  Phys. 123, 204109 (2005)

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Dimers
Bressanini et al. J. Chem. Phys. 123, 204109
(2005)
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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
Often neglected
Tails
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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!
2N determinants. An exponential wall
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Basis

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

• Use one function per electron plus a simple
  Jastrow

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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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GVB for atoms
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GVB for atoms
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GVB for atoms
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GVB for atoms
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GVB for atoms
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GVB for molecules

• Correct asymptotic structure
• Is there a nodal error component in HF wave
  function coming from incorrect dissociation?

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GVB for molecules
Localized orbitals
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GVB Li2
VMC
Wave functions
DMC
HF 1 det compact
-14.9523(2)
-14.9916(1)
GVB 8 det compact
-14.9688(1)
-14.9915(1)
CI 3 det compact
-14.9632(1)
-14.9931(1)
GVB CI 24 det compact
-14.9782(1)
-14.9936(1)
CI 3 det large basis
-14.9933(2)
CI 5 det large basis
-14.9952(1)
E (N.R.L.)
-14.9954
Improvement in the wave function but irrelevant
on the nodes,
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GVB in QMC

• Conclusions
• The quality of the wave function improves, giving
  better VMC energies
• but the nodes are not changed, giving the same
  QMC energies
• Bressanini and Morosi J. Chem. Phys. 129,
  054103 (2008)

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Conventional wisdom on Y
Single particle approximations

• EVMC(YRHF) gt EVMC(YUHF) gt EVMC(YGVB)

Consider the N atom

• YRHF 1sR 2sR 2px 2py 2pz 1sR 2sR
• YUHF 1sU 2sU 2px 2py 2pz 1sU 2sU

EDMC(YRHF) gt ? lt EDMC(YUHF)
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Conventional wisdom on Y
We can build a YRHF with the same nodes of YUHF

• YUHF 1sU 2sU 2px 2py 2pz 1sU 2sU
• YRHF 1sU 2sU 2px 2py 2pz 1sU 2sU

EDMC(YRHF) EDMC(YUHF)
EVMC(YRHF) gt EVMC(YRHF) gt EVMC(YUHF)
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Conventional wisdom on Y
YGVB 1s 2s 2p3 1s 2s - 1s 2s 2p3 1s
2s 1s 2s 2p3 1s 2s- 1s
2s 2p3 1s 2s
Node equivalent to a YUHF f(r) g(r) 2p3 1s 2s
EDMC(YGVB) EDMC(YRHF)
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What to do?

• Should we be happy with the cancellation of
  error, and pursue it?
• After all, the whole quantum chemistry is built
  on it!
• If not, and pursue orthodox QMC (no
  pseudopotentials, no cancellation of errors, ) ,
  can we avoid the curse of YT ?

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The curse of the YT

• QMC currently relies on YT(R)
• Walter Kohn in its Nobel lecture (R.M.P. 71, 1253
  (1999)) discredited the wave function as a non
  legitimate concept when N (number of electrons)
  is large

For M109 and p3 ? N6
p parameters per variable M total parameters
needed
The Exponential Wall
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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)
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Ne Atom
Drummond et al. -128.9237(2) DMC
Drummond et al. -128.9290(2)
DMC backflow
Gdanitz et al. -128.93701
R12-MR-CI
Exact (estimated) -128.9376
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We need new, and different, ideas

• A little intermezzo (for the students)

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We need new, and different, ideas

• Different representations
• Different dimensions
• Different equations
• Different potential
• Radically different algorithms
• Different something

Research is the process of going up alleys to see
if they are blind.  Marston Bates
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Just an example

• Try a different representation
• Is some QMC in the momentum representation
• Possible ? And if so, is it
• Practical ?
• Useful/Advantageus ?
• Eventually better than plain vanilla QMC ?
• Better suited for some problems/systems ?
• Less plagued by the usual problems ?

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The other half of Quantum mechanics
The Schrodinger equation in the momentum
representation
Some QMC (GFMC) should be possible, given the
iterative form
Or write the imaginary time propagator in
momentum space
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Better?

• For coulomb systems

• There are NO cusps in momentum space. Y
  convergence should be faster
• Hydrogenic orbitals are simple rational functions

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Another (failed so far) example

• Different dimensionality Hypernodes
• Given HY (R) EY (R) build

• The hope was that it could be better than Fixed
  Node

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Hypernodes

• The energy is still an upper bound
• Unfortunately, it seems to recover exactly the
  FN energy

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Feynman on simulating nature

• Nature isnt classical, dammit, and if you want
  to make a simulation of Nature, youd better make
  it quantum mechanical, and by golly its a
  wonderful problem, because it doesnt look so
  easy
• Richard Feynman 1981

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Nodes
Should we concentrate on nodes?

• Conjectures on nodes
• have higher symmetry than Y itself
• resemble simple functions
• the ground state has only 2 nodal volumes
• HF nodes are often a god starting point

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How to directly improve nodes?

• Fit to a functional form and optimize the
  parameters (maybe for small systems)
• IF the topology is correct, use a coordinate
  transformation

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He2 expanding the node

• It is a one parameter Y !!

Exact
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expanding nodes

• This was only a kind of proof of concept
• It remains to be seen if it can be applied to
  larger systems
• Writing simple (algebraic?) trial nodes is not
  difficult .
• The goal is to have only few linear parameters to
  optimize
• Will it work???????

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Coordinate transformation

• Take a wave function with the correct nodal
  topology

• Change the nodes with a coordinate transformation
  (Linear? Feynmans backflow ?) preserving the
  topology

Miller-Good transformations
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The need for the correct topology

• Using Backflow alone, on a single determinant Y
  is not sufficient, since the topology is still
  wrong
• More determinants are necessary (only two?)

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Be Nodal Topology
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Avoided crossings
Be
e- gas
Stadium
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Nodal topology

• The conjecture (which I believe is true) claims
  that there are only two nodal volumes in the
  fermion ground state
• See, among others
• Ceperley J.Stat.Phys 63, 1237 (1991)
• Bressanini and coworkers. JCP 97, 9200 (1992)
• Bressanini, Ceperley, Reynolds, What do we know
  about wave function nodes?, in Recent Advances
  in Quantum Monte Carlo Methods II, ed. S.
  Rothstein, World Scientfic (2001)
• Mitas and coworkers PRB 72, 075131 (2005)
• Mitas PRL 96, 240402 (2006)

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Nodal Regions
Nodal Regions
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Avoided nodal crossing

• At a nodal crossing, Y and ?Y are zero
• Avoided nodal crossing is the rule, not the
  exception
• Not (yet) a proof... (any help is appreciated)

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He atom with noninteracting electrons
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(No Transcript)
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Casual similarity ?
First unstable antisymmetric stretch orbit of
semiclassical linear helium along with the
symmetric Wannier orbit r1 r2 and various
equipotential lines
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Casual similarity ?
Superimposed Hylleraas node
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A QMC song...
He deals the cards to find the answers the sacred
geometry of chance the hidden law of a probable
outcome the numbers lead a dance
Sting Shape of my heart
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Think Different
Take a look at your nodes!