Trento 2003 – Few Body Critical Stability What do we know about nodes of wave functions ? Dario Bressanini Universita’ dell’Insubria, Como, Italy http://www.unico.it/~dario Trento 2003 – Few Body Critical Stability

Quantum Monte Carlo Time evolution Diffusion Birth-Death The time dependent Schrödinger equation is similar to a diffusion equation We simulate exactly the imaginary-time Schrödinger equation Y is interpreted as a concentration of fictitious particles, called walkers. Computes the exact energy

The Fermion Problem + - Wave functions for fermions have nodes. Diffusion equation analogy is lost. Need to introduce positive and negative walkers. The (In)famous Sign Problem Restrict random walk to a positive region bounded by nodes. Unfortunately, the exact nodes are unknown. + - Use approximate nodes from a trial Y. Kill the walkers if they cross a node. The energy depends ONLY from the nodes of Y.

Nodes What do we know about wave function nodes? Very little .... NOT fixed by (anti)symmetry alone. Only a 3N-3 subset Very very few analytic examples 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.

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

Nodes and Configurations A better Y does not mean better nodes Why? What can we do about it?

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 Find a way to parametrize the nodes using simple functions, and optimize the nodes directly minimizing the Fixed-Node energy

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

The Helium Triplet r1 r12 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 r1 r2

Other He states: 1s2s 2 1S and 2 3S Although , the node does not depend on q12 (or does very weakly) r1 q12 r2 Surface contour plot of the node A very good approximation of the node is The second triplet has similar properties

He: Other states 1s2s 3S : (r1-r2) f(r1,r2,r12) 1s2p 1P o : node independent from r12 (J.B.Anderson) 2p2 3P e : Y = (x1 y2 – y1 x2) f(r1,r2,r12) 2p3p 1P e : Y = (x1 y2 – y1 x2) (r1-r2) f(r1,r2,r12) 1s2s 1S : node independent from r12 1s3s 3S : node independent from r12

Helium Nodes Independent from r12 More “symmetric” than the wave function Some are described by polynomials in distances and/or coordinates The same node is present in different states The HF Y, sometimes, has the correct node, or a node with the correct (higher) symmetry Are these general properties of nodal surfaces ?

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 This is the same node present in the He 3S Again, node has higher symmetry than Y How good is the RHF node? YRHF is not very good, however its node is surprisingly good (might it be the exact one?) DMC(YRHF ) = -7.47803(5) a.u. Lüchow & Anderson JCP 1996 Exact = -7.47806032 a.u. Drake, Hylleraas expansion

Li atom: Study of Exact Node We take an “almost exact” Hylleraas expansion 250 term r3 r1 r2 The node seems to be r1 = 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 ?

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

Nodal Symmetry Conjecture This observation is general: If 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

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 Be+2. The node is the union of the two independent nodes. Plot cuts of (r1-r2) vs (r3-r4) The HF node is wrong DMC energy -14.6576(4) Exact energy -14.6673

Be: CI expansion Plot cuts of (r1-r2) vs (r3-r4) What happens to the HF node in a good CI expansion? Plot cuts of (r1-r2) vs (r3-r4) In 9-D space, the direct product structure “opens up” Node is (r1-r2)(r3-r4) + ...

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

Be nodal topology Now there are only two nodal regions It can be proved that the exact Be wave function has exactly two regions See Bressanini, Ceperley and Reynolds http://www.unico.it/~dario/ http://archive.ncsa.uiuc.edu/Apps/CMP/ Node is (r1-r2) (r3-r4) + ???

Hartree-Fock Nodes How Many ? 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 ?

Nodal Regions 2 4 Nodal Regions 2 Ne Li Be B C Li2

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

Be model node Second order approx. r1-r2 r1+r2 r3-r4 Second order approx. Gives the right topology and the right shape What's next?

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?

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

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

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

Li2 molecule, large basis Adding CFS with a large basis ... (1sg2 1su2 omitted) HF -14.9919(1) 97.2(1) %CE +8 -14.9914(1) 96.7(1) GVB 8 dets -14.9907(6) 96.2(6) + -14.9933(1) 98.3(1) +4 -14.9933(1) 98.3(1) + -14.9952(1) 99.8(1) Estimated n.r. limit -14.9954

Conclusions Exact or good nodes (at least for simple systems) seem to depend on few variables have higher symmetry than Y itself resemble simple functions Possible explanation on why HF nodes are quite good: they “naturally” have these properties Use large basis, until HF nodes are converged Include "different" CSFs Has the ground state only 2 nodal volumes?

Acknowledgments Silvia Tarasco Peter Reynolds Gabriele Morosi Carlos Bunge

Take a look at your nodes ... a suggestion Take a look at your nodes