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Universita’ dell’Insubria, Como, Italy Some reflections on nodes and trial wave functions Is QMC delivering its early promises? Dario Bressanini http://scienze-como.uninsubria.it/bressanini QMCI Sardagna (Trento) 2006 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 3He 4He m n 3He 4He n 0 1 2 3 4 m Stability Chart 5 6 7 8 9 10 11 0 Bound L=0 1 Unbound 2 3 Unknown 4 L=1 S=1/2 5 L=1 S=1 Terra Incognita Bound 32 3He 4He 2 2 L=0 S=0 3He 4He 2 4 L=1 S=1 3He 4He 3 8 3He 4He 3 4 L=0 S=1/2 L=1 S=1/2 5 Good for vibrational problems 6 For electronic structure? Sign Problem Fixed Nodal error problem 7 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 ? 8 He2+: the basis set The ROHF wave function: RHF (R) ( g (1) u (3) g (3) u (1)) g (2) 1s E = -4.9905(2) hartree 1s1s’2s3s E = -4.9943(2) hartree EN.R.L = -4.9945 hartree 9 He2+: MO’s Bressanini et al. J. Chem. Phys. 123, 204109 (2005) E(RHF) = -4.9943(2) hartree E(CAS) = -4.9925(2) hartree E(CAS-NO) = -4.9916(2) hartree E(CI-NO) = -4.9917(2) hartree EN.R.L = -4.9945 hartree 10 He2+: CSF’s 1s1s’2s3s2p2p’ E(1 csf) = -4.9932(2) hartree + 2 2 1 u11 ux 1 u11 uy E(2 csf) = -4.9946(2) hartree 1s1s’2s3s E(1 csf) = -4.9943(2) hartree + 1 1g 1 u1 2 1g E(2 csf) = -4.9925(2) hartree 11 Li2 CSF (1g2 1u2 omitted) 2 g2 3 g2 4 g2 ... 9 g2 E (hartree) -14.9923(2) -14.9914(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 -14.9933(1) Not all CSF are useful Only 4 csf are needed to build a statistically exact nodal surface 12 A tentative recipe Use a large Slater basis But not too large Try to reach HF nodes convergence Orbitals from CAS seem better than HF, or NO 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 13 Dimers Bressanini et al. J. Chem. Phys. 123, 204109 (2005) 14 Is QMC competitive ? 15 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) R12-MR-CI Exact (estimated) 34 -37.8407(1) -37.845179 -37.8450 16 Ne Atom Drummond et al. -128.9237(2) DMC Drummond et al. -128.9290(2) DMC backflow Gdanitz et al. -128.93701 Exact (estimated) -128.9376 R12-MR-CI 17 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) 18 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) 19 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) 20 Nitrogen Atom Simple RHF (1 det) Simple RHF (1 det) Simple UHF (1 det) Simple GVB (4 det) 4 8 11 11 26.0% 42.7% 41.2% 42.3% 91.9% 92.6% 92.3% 92.3% Clementi-Roetti + J 27 24.5% 93.1% Param. E corr. VMC E corr. DMC Is it worth to continue to add parameters to the wave function? 21 What to do? Should we be happy with the “cancellation of error”, and pursue it? If so: Is there the risk, in this case, that QMC becomes Yet Another Computational Tool, and not particularly efficient nor reliable? VMC seems to be much more robust, easy to “advertise” If not, and pursue orthodox QMC (no pseudopotentials, no cancellation of errors, …) , can we avoid the curse of T ? 22 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 23 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 24 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 25 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 ) 26 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. Again an exponential wall 27 PsH – Positronium Hydride A wave function with the correct asymptotic conditions: (1,2, p) (1 Pˆ12 )( H ) f (rp )( Ps) g (r1 p ) Bressanini and Morosi: JCP 119, 7037 (2003) 28 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 29 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 ? 30 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 31 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 ) 32 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 33 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 34 A little intermezzo Why is QMC not used by chemists? DMC Top 10 reasons 12. We need forces, dummy! 11. Try getting O2 to bind at the variational level. 10. How many graduate students lives have been lost optimizing wavefunctions? 9. It is hard to get 0.01 eV accuracy by throwing dice. 8. Most chemical problems have more than 50 electrons. 7. Who thought LDA or HF pseudopotentials would be any good? 6. How many spectra have you seen computed by QMC? 5. QMC is only exact for energies. 4. Multiple determinants. We can't live with them, we can't live without them. 3. After all, electrons are fermions. 2. Electrons move. 1. QMC isn't included in Gaussian 90. Who programs anyway? http://web.archive.org/web/20021019141714/archive.ncsa.uiuc.e du/Apps/CMP/topten/topten.html 36 Chemistry and Mathematics "We are perhaps not far removed from the time, when we shall be able to submit the bulk of chemical phenomena to calculation” Joseph Louis Gay-Lussac - 1808 “The underlying physical laws necessary for the mathematical theory of a large part of physics and the whole of chemistry are thus completely known, and the difficulty is only that the exact application of these equations leads to equations much too complicated to be soluble” P.A.M. Dirac - 1929 37 Nature and Mathematics “il Grande libro della Natura e’ scritto nel linguaggio della matematica, e non possiamo capirla se prima non ne capiamo i simboli“ Galileo Galilei Every attempt to employ mathematical methods in the study of chemical questions must be considered profoundly irrational and contrary to the spirit of chemistry… If mathematical analysis should ever hold a prominent place in chemistry – an aberration which is happily almost impossible – it would occasion a rapid and widespread degeneration of that science. Auguste Compte 38 A Quantum Chemistry Chart Orthodox QMC J.Pople The more accurate the calculations became, the more the concepts tended to vanish into thin air (Robert Mulliken) 39 Chemical concepts Molecular structure and geometry Chemical bond Ionic-Covalent Singe, Double, Triple d NOT DIRECTLY OBSERVABLES Electronegativity ILL-DEFINED CONCEPTS Oxidation number Atomic charge Lone pairs Aromaticity d d d O C d O 40 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 quite good: they “naturally” have these properties Checked on small systems: L, Be, He2+. See also Mitas 41 Avoided crossings Be e- gas Stadium 43 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) 44 Avoided nodal crossing At a nodal crossing, and are zero Avoided nodal crossing is the rule, not the exception Not (yet) a proof... 0 0 3N 1 with 3N variables 3N eqs. 1 eq. If HF has 4 nodes HF has 2 nodes, with a proper 46 He atom with noninteracting electrons 1 3s5s S 47 48 Casual similarity ? 1s2s 1S Helium First unstable antisymmetric stretch orbit of semiclassical linear helium along with the symmetric Wannier orbit r1 = r2 and various equipotential lines 49 Casual similarity ? Superimposed Hylleraas node 50 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) 51 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 52 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 53 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 54 Think Different! 55