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Universita’ dell’Insubria, Como, Italy Some considerations on nodes and trial wave functions Dario Bressanini http://scienze-como.uninsubria.it/bressanini QMCI Sardagna (Trento) 2008 30+ years of QMC in chemistry 2 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) 3 Good for Helium studies Thousands of theoretical and experimental papers Hˆ n (R) En n (R) have been published on Helium, in its various forms: Atom Small Clusters Droplets Bulk 4 Good for vibrational problems 5 For electronic structure? Sign Problem Fixed Nodal error problem 6 The influence on the nodes of T QMC currently relies on T(R) and its nodes (indirectly) How are the nodes T(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 T(R) ? 7 Improving T 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 8 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 iˆ34 (1s 2 2s 2 ) 1s 2 2s 2 iˆ34 (1s 2 2 p 2 ) 1s 2 2 p 2 iˆ34 (1s 2 3s 2 ) 1s 2 3s 2 9 Li2 CSF (1g2 1u2 omitted) E (hartree) 2 g2 3 g2 4 g2 ... 9 g2 -14.9923(2) 1 ux2 1 uy2 4 n ux2 n uy2 -14.9933(2) 1 ux2 1 uy2 2 u2 -14.9939(2) 1 ux2 1 uy2 2 u2 3 g2 -14.9952(1) E (N.R.L.) -14.9954 -14.9914(2) -14.9933(1) 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) 10 Dimers Bressanini et al. J. Chem. Phys. 123, 204109 (2005) 11 Convergence to the exact We must include the correct analytical structure Cusps: r12 (r12 0) 1 2 (r 0) 1 Zr QMC OK 3-body coalescence and logarithmic terms: Tails: QMC OK Often neglected 12 Asymptotic behavior of Example with 2-e atoms 1 2 1 1 1 2 H (1 2 ) Z ( ) 2 r1 r2 r12 1 2 Z Z 1 2 H (1 2 ) 2 r1 r2 r2 r2 0 (r1 )r2 0 (r1 ) ( Z 1) / 1 r2 e 2 EI is the solution of the 1 electron problem 13 Asymptotic behavior of The usual form (r1 ) (r2 ) J (r12 ) does not satisfy the asymptotic conditions (r2 ) 0 (r1 ) (r2 ) (r1 ) (r1 ) 0 (r2 ) A closed shell determinant has the wrong structure ( (r1 ) (r2 ) (r2 ) (r1 )) J (r12 ) 14 Asymptotic behavior of r1 In general N r a1 (1 c r 1 O(r 2 ))e r1 / b1Y m1 (r ) N 1 (2,...N ) 0 1 1 1 1 l1 1 0 Recursively, fixing the cusps, and setting the right symmetry… U ˆ A( f1 (1) f 2 (2)... f N ( N ) N )e Each electron has its own orbital, Multideterminant (GVB) Structure! Take 2N coupled electrons 2 N (1 2 1 2 )( 3 4 3 4 )... 2N determinants. An exponential wall 15 GVB for atoms 18 GVB for atoms 19 GVB for atoms 20 GVB for atoms 21 GVB for molecules Correct asymptotic structure Is there a nodal error component in HF wave function coming from incorrect dissociation? 23 GVB for molecules Localized orbitals 24 GVB Li2 Wave functions VMC 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) -14.9933(2) CI 3 det large basis CI 5 det large basis E (N.R.L.) 1 ux2 1 uy2 2 u2 3 g2 -14.9952(1) -14.9954 Improvement in the wave function but irrelevant on the nodes, 25 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) 26 Conventional wisdom on Single particle approximations EVMC(RHF) > EVMC(UHF) > EVMC(GVB) Consider the N atom RHF = |1sR 2sR 2px 2py 2pz| |1sR 2sR| UHF = |1sU 2sU 2px 2py 2pz| |1s’U 2s’U| EDMC(RHF) > ? < EDMC(UHF) 27 Conventional wisdom on We can build a RHF with the same nodes of UHF UHF = |1sU 2sU 2px 2py 2pz| |1s’U 2s’U| ’RHF = |1sU 2sU 2px 2py 2pz| |1sU 2sU| EDMC(’RHF) = EDMC(UHF) EVMC(’RHF) > EVMC(RHF) > EVMC(UHF) 28 Conventional wisdom on GVB = |1s 2s 2p3| |1s’ 2s’| - |1s’ 2s 2p3| |1s 2s’| + |1s’ 2s’ 2p3| |1s 2s|- |1s 2s’ 2p3| |1s’ 2s| Same Node Node equivalent to a UHF |f(r) g(r) 2p3| |1s 2s| EDMC(GVB) = EDMC(’’RHF) 29 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 T ? 30 The curse of the T QMC currently relies on T(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 M p3N 3 p 10 p = parameters per variable For M=109 and p=3 N=6 M = total parameters needed The Exponential Wall 31 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) 32 A little intermezzo (for the students) We need new, and different, ideas 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 35 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 ? 36 The other half of Quantum mechanics ( p) Fˆ ( (r )) The Schrodinger equation in the momentum representation 2 p (E ) ( p) (2 ) 1/ 2 Vˆ ( p p) ( p)dp 2m Some QMC (GFMC) should be possible, given the iterative form Or write the imaginary time propagator in momentum space 37 Better? For coulomb systems: 1 2 ˆ ˆ V ( p) F ( ) rij pi p j 2 There are NO cusps in momentum space. convergence should be faster Hydrogenic orbitals are simple rational functions (8Z 5 )1/ 2 1s ( p) 2 2 2 (p Z ) 38 Another (failed so far) example Different dimensionality: Hypernodes Given H (R) = E (R) build H H (R1 ) H (R 2 ) 6 N dimensions T B (R1 )F (R 2 ) F (R1 )B (R 2 ) •Use the Hypernode of T • The hope was that it could be better than Fixed Node 39 Hypernodes The intuitive idea was that the system could correct the wrong fixed nodes, by exploring regions where T ( R) 0 Fixed Node Trial node Fixed HyperNode Trial node Exact node Exact node The energy is still an upper bound Unfortunately, it seems to recover exactly the FN energy 40 Feynman on simulating nature Nature isn’t classical, dammit, and if you want to make a simulation of Nature, you’d better make it quantum mechanical, and by golly it’s a wonderful problem, because it doesn’t look so easy” Richard Feynman 1981 41 Nodes Should we concentrate on nodes? Conjectures on nodes have higher symmetry than itself resemble simple functions the ground state has only 2 nodal volumes HF nodes are often a god starting point 42 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 R T (R) 43 He2+: “expanding” the node Node (1 ) : c 0 r1A r1B r3 A r3 B Node (2 ) : c 1 z1 z3 0 It is a one parameter !! Exact 44 “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??????? 45 Coordinate transformation Take a wave function with the correct nodal topology HF Change the nodes with a coordinate transformation (Linear? Feynman’s backflow ?) preserving the topology R T (R) Miller-Good transformations 46 The need for the correct topology Using Backflow alone, on a single determinant is not sufficient, since the topology is still wrong More determinants are necessary (only two?) BF T (R ) T (R ) T (R ) T (R ) 0 at least two nodes 47 Be Nodal Topology r1+r2 r1+r2 r3-r4 r3-r4 r1-r2 HF 0 r1-r2 CI 0 1s 2 2s 2 c 1s 2 2 p 2 48 Avoided crossings Be e- gas Stadium 49 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) 50 Avoided nodal crossing At a nodal crossing, and are zero Avoided nodal crossing is the rule, not the exception Not (yet) a proof... (any help is appreciated) 0 0 1 eq. 3N 1 with 3N variables 3N eqs. If HF has 4 nodes HF has 2 nodes, with a proper 52 He atom with noninteracting electrons 1 3s5s S 53 54 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 57 Think Different Take a look at your nodes! 58