Much ado about… zeroes (of wave functions) Universita’ dell’Insubria, Como, Italy Much ado about… zeroes (of wave functions) Dario Bressanini http://scienze-como.uninsubria.it/bressanini Electronic Structure beyond DFT, Leiden 2004

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

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?

Fixed Node Approximation circa 1950 Rediscovered by Anderson and Ceperly in the ’70s

Common misconception on nodes Nodes are not fixed by antisymmetry alone, only a 3N-3 sub-dimensional subset

Common misconception on nodes They have (almost) nothing to do with Orbital Nodes. It is (sometimes) possible to use nodeless orbitals

Common misconceptions on nodes A common misconception is that on a node, two like-electrons are always close. This is not true 2 1 1 2

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

Common misconceptions on nodes Not even for the same symmetry species Courant counterexample

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

Nodes are relevant Levinson Theorem: Fractional quantum Hall effect the number of nodes of the zero-energy scattering wave function gives the number of bound states Fractional quantum Hall effect Quantum Chaos Integrable system Chaotic system

Generalized Variational Principle Upper bound to ground state Higher states can be above or below Bressanini and Reynolds, to be published

Nodes and Configurations A better Y does not mean better nodes Why? What can we do about it? 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 Understand their properties Find a way to sistematically improve the nodes of trial functions ...building them from scratch …improving existing nodes

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

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 r1 r2

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)

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

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 ?

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

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 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 Node is (r1-r2)(r3-r4) + ... See Bressanini, Ceperley and Reynolds http://scienze-como.uninsubria.it/bressanini/ http://archive.ncsa.uiuc.edu/Apps/CMP/

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 Nodal Regions 2 4 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

Avoided crossings Be e- gas

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

The effect of d orbitals

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

Carbon Atom: Topology 4 Nodal Regions HF GVB 4 Nodal Regions Adding determinants might not be sufficient to change the topology 2 Nodal Regions CI

Carbon Atom: Energy CSFs Det. Energy 1 1s22s2 2p2 1 -37.8303(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..)

He2+ molecule Basis: 2(1s) E=-4.9927(1) 5(1s) E=-4.9943(2) 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

He2+ molecule 2 Determinants EExact = -4.994598 E = -4.9932(2)

He2+ molecule 3 Determinants EExact = -4.994598 E = -4.9778(3)

Li2 molecule Adding more configuration with a small basis (double zeta STO)... Filippi & Umrigar JCP 1996

Li2 molecule, large basis Adding CFS with a larger 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

C2 CSF 1 -75.860(1) 20 -75.900(1) 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.

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

Use a good basis The nodes of Hartree–Fock wavefunctions and their orbitals, Chem. Phys.Lett. 392, 55 (2004) Hachmann, Galek, Yanai, Chan and, Handy

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? Feynman’s backflow ?)

Conclusions Nodes are worth studying! Conjectures on nodes Recipe: 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

Acknowledgments.. and a suggestion Silvia Tarasco Peter Reynolds Gabriele Morosi Carlos Bunge Take a look at your nodes

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