Much ado about

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Universita dellInsubria, Como, Italy
Much ado about zeroes (of wave functions)
Dario Bressanini
http//scienze-como.uninsubria.it/bressanini
Electronic Structure beyond DFT, Leiden 2004
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A little advertisement

• Besides nodes, I am interested in
• VMC improvement
• Robust optimization
• Delayed rejection VMC
• Mixed 3He/4He clusters, ground and excited states
• Sign problem
• Other QMC topics

http//scienze-como.uninsubria.it/bressanini
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Nodes and the Sign Problem

• Fixed-node QMC is efficient. If only we could
  have the exact nodes
• or at least a systematic way to improve the
  nodes ...
• we could bypass the sign problem
• How do we build a Y with good nodes?
• We know very little about nodes (very few
  analytic examples)

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Fixed Node Approximation

• Restrict random walk to a positive region bounded
  by (approximate) nodes.
• The energy is an upper bound

• Fixed Node IS efficient, but approximation is
  uncontrolled

• There is not (yet) a way to sistematically
  improve the nodes
• How do we build a Y with good nodes?

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Fixed Node Approximation
circa 1950 Rediscovered by Anderson and
Ceperly in the 70s
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Common misconception on nodes

• Nodes are not fixed by antisymmetry alone, only a
  3N-3 sub-dimensional subset

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Common misconception on nodes

• They have (almost) nothing to do with Orbital
  Nodes.
• It is (sometimes) possible to use nodeless
  orbitals

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Common misconceptions on nodes

• A common misconception is that on a node, two
  like-electrons are always close. This is not true

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Common misconceptions on nodes

• Nodal theorem is NOT VALID in N-Dimensions
• Higher energy states does not mean more nodes
  (Courant and Hilbert )
• It is only an upper bound

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Common misconceptions on nodes

• Not even for the same symmetry species

Courant counterexample
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Tiling Theorem (Ceperley)
Impossible for ground state
Nodal regions must have the same shape
The Tiling Theorem does not say how many nodal
regions we should expect
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Nodes are relevant

• Levinson Theorem
• the number of nodes of the zero-energy scattering
  wave function gives the number of bound states
• Fractional quantum Hall effect
• Quantum Chaos

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Generalized Variational Principle
Upper bound to ground state
Higher states can be above or below
Bressanini and Reynolds, to be published
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Nodes and Configurations
It is necessary to get a better understanding how
CSF influence the nodes. Flad, Caffarel and
Savin
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The (long term) Plan of Attack

• Study the nodes of exact and good approximate
  trial wave functions
• Understand their properties
• Find a way to sistematically improve the nodes of
  trial functions
• ...building them from scratch
• improving existing nodes

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The Helium triplet

• First 3S state of He is one of very few systems
  where we know the exact node
• For S states we can write

• For the Pauli Principle

• Which means that the node is

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The Helium triplet node

• Independent of r12
• The node is more symmetric than the wave function
  itself
• It is a polynomial in r1 and r2
• Present in all 3S states of two-electron atoms

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He Other states

• Other states have similar properties
• Breit (1930) showed that Y(P e) (x1 y2 y1
  x2) f(r1,r2,r12)
• 2p2 3P e f( ) symmetric node (x1 y2
  y1 x2)
• 2p3p 1P e f( ) antisymmetric node (x1 y2
  y1 x2) (r1-r2)
• 1s2p 1P o node independent from r12
  (J.B.Anderson)

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He 3S a look at non-physical regions

• Consider Y(r1,r2,q12) defined in all space
• A node in a non-physical regions appears. Using a
  simple trial function...

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He 3S a look at non-physical regions

• Consider Y(r1,r2,q12) defined in all space

• Expanding Y at second order in (0,0)
• Y (10-6 0.001 (r1r2))(r1-r2)...

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He 3S a look at non-physical regions

• If we turn off the e-e interaction we observe the
  same feature (r1r2)(r1-r2)/2...

• There is no apparent reason why even the exact
  wave function should be
• Y c (r1r2)(r1-r2)...

• It seems the nodal structure of the exact wave
  function resembles the independent electron case

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Other He states 1s2s 2 1S and 2 3S
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Helium Nodes

• Independent from r12
• Higher symmetry than the wave function
• Some are described by polynomials in distances
  and/or coordinates
• The HF Y, sometimes, has the correct node, or a
  node with the correct (higher) symmetry
• Are these general properties of nodal surfaces ?

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Lithium Atom Ground State

• The RHF node is r1 r3
• if two like-spin electrons are at the same
  distance from the nucleus then Y 0
• Node has higher symmetry than Y
• How good is the RHF node?
• YRHF is not very good, however its node is
  surprisingly good
• DMC(YRHF ) -7.47803(5) a.u. Lüchow Anderson
  JCP 1996
• Exact -7.47806032 a.u. Drake, Hylleraas
  expansion

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Li atom Study of Exact Node

• We take an almost exact Hylleraas expansion 250
  term

• The node seems to ber1 r3, taking different
  cuts, independent from r2 or rij

• a DMC simulation with r1 r3 node and good Y to
  reduce the variance gives
• DMC -7.478061(3) a.u. Exact -7.4780603
  a.u.

Is r1 r3 the exact node of Lithium ?
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Li atom Study of Exact Node

• Li exact node is more symmetric than Y

• At convergence, there is a delicate cancellation
  in order to build the node
• Crude Y has a good node (r1-r3)Exp(...)
• Increasing the expansion spoils the node, by
  including rij terms

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Nodal Symmetry Conjecture

• This observation is generalIf the symmetry of
  the nodes is higher than the symmetry of Y,
  adding terms in Y might decrease the quality of
  the nodes (which is what we often see).

WARNING Conjecture Ahead...
Symmetry of nodes of Y is higher than symmetry of
Y
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Beryllium Atom

• HF predicts 4 nodal regions Bressanini et al.
  JCP 97, 9200 (1992)
• Node (r1-r2)(r3-r4) 0

• Y factors into two determinants each one
  describing a triplet Be2. The node is the
  union of the two independent nodes.

• The HF node is wrong
• DMC energy -14.6576(4)
• Exact energy -14.6673

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Be beyond Restricted Hartree-Fock

• Hartree-Fock Y is not the most general single
  particle approximation
• Try a GVB wave function (4 determinants)

VMC energy improves, s2(H) improves... but still
the same node (r1-r2)(r3-r4) 0
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Be CI expansion

• What happens to the HF node in a good CI
  expansion?

• In 9-D space, the direct product structure opens
  up

Node is (r1-r2)(r3-r4) ...
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Be Nodal Topology
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Be nodal topology

• Now there are only two nodal regions
• It can be proved that the exact Be wave function
  has exactly two regions

Node is (r1-r2)(r3-r4) ...
See Bressanini, Ceperley and Reynolds http//scie
nze-como.uninsubria.it/bressanini/ http//archive
.ncsa.uiuc.edu/Apps/CMP/
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Hartree-Fock Nodes

• YHF has always, at least, 4 nodal regions for 4
  or more electrons
• It might have Na! Nb! Regions
• Ne atom 5! 5! 14400 possible regions
• Li2 molecule 3! 3! 36 regions

How Many ?
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Nodal Regions
Nodal Regions
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Nodal Regions
Nodal Regions
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Nodal Topology Conjecture
WARNING Conjecture Ahead...
The HF ground state of Atomic and Molecular
systems has 4 Nodal Regions, while the Exact
ground state has only 2
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Avoided crossings
Be
e- gas
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Be model node

• Second order approx.
• Gives the right topology and the right shape
• What's next?

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Be numbers

• HF node -14.6565(2) 1s2 2s2
• GVB node same 1s1s' 2s2s'
• Luechow Anderson -14.6672(2) 1s2 2p2
• Umrigar et al. -14.66718(3) 1s2 2p2
• Huang et al. -14.66726(1) 1s2 2p2 opt
• Casula Sorella -14.66728(2) 1s2 2p2 opt
• Exact -14.6673555

• Including 1s2 ns ms or 1s2 np mp configurations
  does not improve the Fixed Node energy...
• ...Why?

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Be Node considerations

• ... (I believe) they give the same contribution
  to the node expansion
• ex 1s22s2 and 1s23s2 have the same node
• ex 2px2, 2px3px and 3px2 have the same structure

• The nodes of "useful" CSFs belong to higher and
  different symmetry groups than the exact Y

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The effect of d orbitals
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Be numbers

• HF -14.6565(2) 1s2 2s2
• GVB node same 1s1s' 2s2s'
• Luechow Anderson -14.6672(2) 1s2 2p2
• Umrigar et al. -14.66718(3) 1s2 2p2
• Huang et al. -14.66726(1) 1s2 2p2 opt
• Casula Sorella -14.66728(2) 1s2 2p2 opt
• Bressanini et al. -14.66733(7) 1s2 3d2
• Exact -14.6673555

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CSF nodal conjecture
WARNING Conjecture Ahead...
If the basis is sufficiently large, only
configurations built with orbitals of different
angular momentum and symmetry contribute to the
shape of the nodes
This explains why single excitations are not
useful
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Carbon Atom Topology
Adding determinants might not be sufficient to
change the topology
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Carbon Atom Energy

• CSFs Det. Energy
• 1 1s22s2 2p2 1 -37.8303(4)
• 2 1s2 2p4 2 -37.8342(4)
• 5 1s2 2s 2p23d 18 -37.8399(1)
• 83 1s2 4 electrons in 2s 2p 3s 3p 3d
  shell 422 -37.8387(4)
• adding f orbitals
• 7 (4f2 2p34f) 34 -37.8407(1)
• Exact -37.8450
• Where is the missing energy? (g, core, optim..)

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He2 molecule
3 electrons 9-1 8 degrees of freedom
Basis 2(1s) E-4.9927(1) 5(1s) E-4.9943(2)
(almost exact) nodal surface of Y0 depends
on r1a, r1b, r2a and r2b higher symmetry than Y0
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He2 molecule
2 Determinants
EExact -4.994598
E -4.9932(2)
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He2 molecule
3 Determinants
EExact -4.994598
E -4.9778(3)
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Li2 molecule

• Adding more configuration with a small basis
  (double zeta STO)...

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Li2 molecule, large basis

• Adding CFS with a larger basis ... (1sg2 1su2
  omitted)

• GVB 8 dets -14.9907(6) 96.2(6)

Estimated n.r. limit -14.9954
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O2

• Small basis
• 1 Det. -150.268(1) Filippi Umrigar
• 7 Det. -150.277(1) .....................
• Large basis
• 1 Det. -150.2850(6) Tarasco, work in progress
• 2 Det. -150.2873(7) ..............................
  ....
• Exact -150.3268

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C2

• CSF
• 1 -75.860(1) 20 -75.900(1) Barnett et. al.
• 36 -75.9025(7) Barnett et. al.
• 1 -75.8613(8) 4 -75.8901(7) Filippi - Umrigar
• 1 -75.866(2) 32 -75.900(1) Lüchow - Fink
• Exact -75.9255
• Work in progress 5(s)4(p)2(d)
• 1 -75.8692(5) 12 -75.9032(8)
• 12 -75.9038(6) Linear opt.

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A tentative recipe

• Use a large Slater basis
• But not too large
• Try to reach HF nodes convergence
• Use the right determinants...
• ...different Angular Momentum CSFs
• And not the bad ones
• ...types already included

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Use a good basis
The nodes of HartreeFock wavefunctions and their
orbitals, Chem. Phys.Lett. 392, 55
(2004) Hachmann, Galek, Yanai, Chan and, Handy
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How to directly improve nodes?

• Fit to a functional form and optimize the
  parameters (small systems)
• IF the topology is correct, use a coordinate
  transformation (Linear? Feynmans backflow ?)

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Conclusions

• Nodes are worth studying!
• Conjectures on nodes
• have higher symmetry than Y itself
• resemble simple functions
• the ground state has only 2 nodal volumes
• HF nodes are quite good they naturally have
  these properties
• Recipe
• Use large basis, until HF nodes are converged
• Include "different kind" of CSFs with higher
  angular momentum

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Acknowledgments.. and a suggestion

• Silvia Tarasco Peter Reynolds
• Gabriele Morosi Carlos Bunge

Take a look at your nodes
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A (Nodal) song...
He deals the cards to find the answers the secret
geometry of chance the hidden law of a probable
outcome the numbers lead a dance
Sting Shape of my heart