The nodes of trial and exact wave functions in Quantum Monte Carlo Dario Bressanini Universita’ dell’Insubria, Como, Italy Peter J. Reynolds Office of Naval Research CECAM Lyon

Nodes and the Sign Problem So far, solutions to the sign problem have not proven to be efficient So far, solutions to the sign problem have not proven to be efficient Fixed-node approach is efficient. If only we could have the exact nodes … Fixed-node approach is efficient. If only we could have the exact nodes … … or at least a systematic way to improve the nodes... … or at least a systematic way to improve the nodes... … we could bypass the sign problem … we could bypass the sign problem How do we build a  with good nodes? How do we build a  with good nodes?  CI? MCSCF? Natural Orbitals (Lüchow) ?

Nodes What do we know about wave function nodes? What do we know about wave function nodes?  Very little.... NOT fixed by (anti)symmetry alone. Only a 3N-3 subset NOT fixed by (anti)symmetry alone. Only a 3N-3 subset Very very few analytic examples Very very few analytic examples Nodal theorem is NOT VALID Nodal theorem is NOT VALID  Higher energy states does not mean more nodes ( Courant and Hilbert ) They have (almost) nothing to do with Orbital Nodes. It is possible to use nodeless orbitals. They have (almost) nothing to do with Orbital Nodes. It is possible to use nodeless orbitals. Tiling theorem Tiling theorem

Tiling Theorem (Ceperley) Impossible for ground state The Tiling Theorem does not say how many nodal regions we should expect

Nodes and Configurations It is necessary to get a better understanding how CSF influence the nodes. Flad, Caffarel and Savin

The (long term) Plan of Attack Study the nodes of exact and good approximate trial wave functions Study the nodes of exact and good approximate trial wave functions Understand their properties Understand their properties Find a way to sistematically improve the nodes of trial functions, or... Find a way to sistematically improve the nodes of trial functions, or... …find a way to parametrize the nodes using simple functions, and optimize the nodes directly minimizing the Fixed-Node energy …find a way to parametrize the nodes using simple functions, and optimize the nodes directly minimizing the Fixed-Node energy

The Helium Triplet First 3 S state of He is one of very few systems where we know exact node First 3 S state of He is one of very few systems where we know exact node For S states we can write For S states we can write Which means that the node is Which means that the node is For the Pauli Principle For the Pauli Principle

The Helium Triplet Independent of r 12 Independent of r 12 The node is more symmetric than the wave function itself The node is more symmetric than the wave function itself It is a polynomial in r 1 and r 2 It is a polynomial in r 1 and r 2 Present in all 3 S states of two-electron atoms Present in all 3 S states of two-electron atoms r1r1 r2r2 r 12 r1r1 r2r2 The wave function is not factorizable but The wave function is not factorizable but

The Helium Triplet This is NOT trivial This is NOT trivial N = r 1 -r 2, Antisymmetric Nodal Function N = r 1 -r 2, Antisymmetric Nodal Function f = unknown, totally symmetric f = unknown, totally symmetric The HF function has the exact node The HF function has the exact node Which of these properties are present in other systems? Which of these properties are present in other systems? Are these be general properties of the nodal surfaces ? Are these be general properties of the nodal surfaces ? For a generic system, what can we say about N ? For a generic system, what can we say about N ?

Helium Singlet 1s2s 2 1 S A very good approximation of the node is A very good approximation of the node is The second triplet has similar properties The second triplet has similar properties r1r1   r2r2 Surface contour plot of the node Although, the node does not depend on   (or does very weakly) Although, the node does not depend on   (or does very weakly)

He: Other states 1s2s 3 S : ( r 1 - r 2 ) f( r 1, r 2, r 12 ) 1s2s 3 S : ( r 1 - r 2 ) f( r 1, r 2, r 12 ) 1s2p 1 P o : node independent from r 12 (J.B.Anderson) 1s2p 1 P o : node independent from r 12 (J.B.Anderson) 2 p 2 3 P e :  = ( x 1 y 2 – y 1 x 2 ) f( r 1, r 2, r 12 ) 2 p 2 3 P e :  = ( x 1 y 2 – y 1 x 2 ) f( r 1, r 2, r 12 ) 2 p 3 p 1 P e :  = ( x 1 y 2 – y 1 x 2 ) ( r 1 - r 2 ) f( r 1, r 2, r 12 ) 2 p 3 p 1 P e :  = ( x 1 y 2 – y 1 x 2 ) ( r 1 - r 2 ) f( r 1, r 2, r 12 ) 1s2s 1 S : node independent from r 12 1s2s 1 S : node independent from r 12  Similar to 1s3s 3 S : node independent from r 12 1s3s 3 S : node independent from r 12  Similar to

Helium Nodes Independent from r 12 Independent from r 12 More “symmetric” than the wave function More “symmetric” than the wave function Some are described by polynomials in distances and/or coordinates Some are described by polynomials in distances and/or coordinates The same node is present in different states (as if Helium were separable) The same node is present in different states (as if Helium were separable) The HF , sometimes, has the correct node, or a node with the correct (higher) symmetry The HF , sometimes, has the correct node, or a node with the correct (higher) symmetry

Lithium Atom Ground State The RHF node is r 1 = r 3 The RHF node is r 1 = r 3 if two like-spin electrons are at the same distance from the nucleus then  =0 This is the same node we found in the He 3 S This is the same node we found in the He 3 S Again, node has higher symmetry Again, node has higher symmetry How good is the RHF node? How good is the RHF node?  RHF is not very good, however its node is surprisingly good ( might it be the exact one? ) DMC(  RHF ) = (5) a.u. Lüchow & Anderson JCP 1996 Exact = a.u. Drake, Hylleraas expansion

We take an “almost exact” Hylleraas expansion 250 terms Energy  Hy = a.u. Exact = a.u. How different is its node from r 1 = r 3 ?? Li atom: Study of Exact Node r3r3 r1r1 r2r2 The node seems to be r 1 = r 3, taking different cuts, independent from r 2 or r ij The node seems to be r 1 = r 3, taking different cuts, independent from r 2 or r ij

Li atom: Study of Exact Node Numerically, we found only very small deviations from the HF node, or artifacts of the linear expansion a DMC simulation with r 1 = r 3 node and good  to reduce the variance gives a DMC simulation with r 1 = r 3 node and good  to reduce the variance gives DMC a.u. Exact a.u. DMC (3) a.u. Exact a.u. Is r 1 = r 3 the exact node of Lithium ?

Beryllium Atom Ground State 12 D After spin assignment 9 D Node  (R) = 0 8-dimensional hypersurface Factor external angles

Beryllium Atom  HF predicts 4 nodal regions Bressanini et al. JCP 97, 9200 (1992)  Node: (r 1 -r 2 )(r 3 -r 4 ) = 0   factors into two determinants each one “describing” a triplet Be +2. The node is the union of the two independent nodes.  The HF node is wrong DMC energy (4)DMC energy (4) Exact energy Exact energy

Hartree-Fock Nodes  HF has always, at least, 4 nodal regions for 4 or more electrons  HF has always, at least, 4 nodal regions for 4 or more electrons It might have N  ! N  ! Regions It might have N  ! N  ! Regions Ne atom: 5! 5! = possible regions Ne atom: 5! 5! = possible regions Li 2 molecule: 3! 3! = 36 regions Li 2 molecule: 3! 3! = 36 regions How Many ?

Clustering Algorithm 2 4 Nodal Regions Ne Li Be B C Li 2

Be: beyond Restricted Hartree-Fock Hartree-Fock  is not the most general single particle approximation Hartree-Fock  is not the most general single particle approximation Try a GVB wave function (each electron in its own orbital) Try a GVB wave function (each electron in its own orbital)  GVB (Be) = sum of 4 Determinants

Be: beyond Hartree-Fock  GVB (Be) = sum of 4 Determinants  GVB (Be) = sum of 4 Determinants VMC energy improves VMC energy improves  2 (H) improves  2 (H) improves …but still the same node …but still the same node (r 1 -r 2 )(r 3 -r 4 ) = 0 for any f 1, f 2, f 3, f 4 (r 1 -r 2 )(r 3 -r 4 ) = 0 for any f 1, f 2, f 3, f 4

Be: CI expansion What happens to the HF node in a CI expansion? What happens to the HF node in a CI expansion? Plot cuts of (r 1 -r 2 ) vs (r 3 -r 4 ) Plot cuts of (r 1 -r 2 ) vs (r 3 -r 4 ) Still the same topology and the same node (same in lithium) Still the same topology and the same node (same in lithium) In DMC, CI is not necessarily better than HF Of course, one would first use Of course, one would first use

Be: CI expansion What happens to the HF node in a good CI expansion? What happens to the HF node in a good CI expansion? Node is (r 1 -r 2 )(r 3 -r 4 ) +... Plot cuts of (r 1 -r 2 ) vs (r 3 -r 4 ) Plot cuts of (r 1 -r 2 ) vs (r 3 -r 4 ) In 9-D space, the direct product structure “opens up” In 9-D space, the direct product structure “opens up”

Be Nodal Topology r3-r4 r1-r2 r1+r2 r1-r2 r1+r2 r3-r4

Be nodal topology The clustering algorithm confirms that now there are only two nodal regions The clustering algorithm confirms that now there are only two nodal regions It can be proved that the exact Be wave function has exactly two regions It can be proved that the exact Be wave function has exactly two regions Node is (r 1 -r 2 ) (r 3 -r 4 ) + ??? See Bressanini, Ceperley and Reynolds

Fitting Nodes We would like a simple analytical approximation of the node We would like a simple analytical approximation of the node Ultimate goal: parametrize the node with few parameters, and directly optimize them Ultimate goal: parametrize the node with few parameters, and directly optimize them Useful also for diagnostic. To see if the node changes, and how, by changing basis, or expansion, or functional form. Useful also for diagnostic. To see if the node changes, and how, by changing basis, or expansion, or functional form. How can we model the implicit surface  =0 ? How can we model the implicit surface  =0 ?  The studied (simple) systems suggest polynomials of distances (plus x, y and z for L  0 states)

Fitting Nodes Model Node: Polynomials of r i and r ij (or other simple functions) with the correct spin-space symmetry. To find c : To find c : Node f (Node,c) line integral

Fitting Nodes Collect points on the nodal surface during a DMC or VMC walk Collect points on the nodal surface during a DMC or VMC walk Linear Fit: Minimize least square deviation  2 Minimize least square deviation  2 Discard null solution c i = 0 Discard null solution c i = 0

Be model node Second order approx. Second order approx. Gives the right topology and the right shape Gives the right topology and the right shape r1-r2 r1+r2 r3-r4

Be Node: considerations HF and GVB give the wrong topology HF and GVB give the wrong topology In CI, it seems useless to include 1s 2 ns ms CSFs In CI, it seems useless to include 1s 2 ns ms CSFs 1s 2 2s 2 + 1s 2 2p 2 give already the right topology 1s 2 2s 2 + 1s 2 2p 2 give already the right topology Exact  has only 2 nodal volumes Exact  has only 2 nodal volumes The nodes of the individual CSFs belong to higher symmetry groups than the exact  The nodes of the individual CSFs belong to higher symmetry groups than the exact  The nodes of 1s 2 2s 2 + 1s 2 2p 2 belong to different symmetry groups The nodes of 1s 2 2s 2 + 1s 2 2p 2 belong to different symmetry groups

CSFs Nodal Conjecture Include configurations built with orbitals of different angular momentum and symmetry If we consider the nodes of the individual CSFs as a basis for the node of the exact , IF we can generalize from Be, it seems necessary (maybe not sufficient) to If we consider the nodes of the individual CSFs as a basis for the node of the exact , IF we can generalize from Be, it seems necessary (maybe not sufficient) to ?

Boron Atom 4 Nodal Regions HF 2 Nodal Regions CI GVBGVB 4 Nodal Regions Is it possible to change the topology of the nodal hypersurfaces by adding particular CSFs or do they merely generate deformations? Flad, Caffarel and Savin Both!

Li 2 molecule 4 Nodal Regions HF GVBGVB 2 Nodal Regions 8 determinants But energy not different than HF 2 Nodal Regions CI But energy not different than HF

Nodal Topology Conjecture The HF ground state of Atomic and Molecular systems has 4 Nodal Regions, while the Exact ground state has only 2 ?

Node Optimization ? LiH LiH Exact LiH Exact Simple node –8.0673(6) optimized with derivatives Simple node –8.0673(6) optimized with derivatives Higher terms (6) optimized with derivatives Higher terms (6) optimized with derivatives However However  HF nodes gives practically the exact result  Big fluctuations, due to poor S ( R )

Conclusions Algorithms to study topology and shape of nodes Algorithms to study topology and shape of nodes “Nodes are weird” M. Foulkes. Seattle meeting 1999 “...maybe not” Bressanini, CECAM workshop 2002 “Nodes are weird” M. Foulkes. Seattle meeting 1999 “...maybe not” Bressanini, CECAM workshop 2002 Exact or good nodes (at least for simple systems) seem to Exact or good nodes (at least for simple systems) seem to  depend on few variables  have higher symmetry than  itself  resemble polynomial functions Possible explanation on why HF nodes are quite good: they “naturally” have these properties Possible explanation on why HF nodes are quite good: they “naturally” have these properties Hints on how to build compact MultiDet. expansions Hints on how to build compact MultiDet. expansions It seems possible to optimize nodes directly It seems possible to optimize nodes directly Has the ground state only 2 nodal volumes? Has the ground state only 2 nodal volumes?

Acknowledgments Peter Reynolds Gabriele Morosi Mose’ Casalegno Silvia Tarasco