Is QMC delivering its early promises? Dario Bressanini QMCI Sardagna (Trento) Universita’ dell’Insubria, Como, Italy Some reflections on nodes and trial wave functions

2 30 years of QMC in chemistry

3 The Early promises? Solve the Schrödinger equation exactly without approximation (very strong) 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 controlled approximations, and converge to the exact solution (strong) Solve the Schrödinger equation with some approximation, and do better than other methods (weak) Solve the Schrödinger equation with some approximation, and do better than other methods (weak)

4 Good for Helium studies Thousands of theoretical and experimental papers Thousands of theoretical and experimental papers have been published on Helium, in its various forms: Atom Small Clusters DropletsBulk

5 3 He m 4 He n Stability Chart 32 4 He n 4 He n 3 He m 3 He m He 3 4 He 8 L=0 S=1/2 3 He 2 4 He 4 L=1 S=1 3 He 2 4 He 2 L=0 S=0 3 He 3 4 He 4 L=1 S=1/2 Terra Incognita Bound L=0 Unbound Unknown L=1 S=1/2 L=1 S=1 Bound

6 Good for vibrational problems

7 For electronic structure? Sign Problem Fixed Nodal error problem

8 The influence on the nodes of   QMC currently relies on  T (R) and its nodes (indirectly) QMC currently relies on  T (R) and its nodes (indirectly) How are the nodes  T (R) of influenced by: 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 ?

9 He 2 + : the basis set The ROHF wave function: 1s E = (2) hartree 1s1s’2s3s E = (2) hartree E N.R.L = hartree

10 He 2 + : MO’s E(RHF) = (2) hartree E(RHF) = (2) hartree E(CAS) = (2) hartree E(CAS) = (2) hartree E(CAS-NO) = (2) hartree E(CAS-NO) = (2) hartree E(CI-NO) = (2) hartree E(CI-NO) = (2) hartree E N.R.L = hartree E N.R.L = hartree Bressanini et al. J. Chem. Phys. 123, (2005)

11 He 2 + : CSF’s 1s1s’2s3s2p2p’ E(1 csf) = (2) hartree E(1 csf) = (2) hartree 1s1s’2s3s E(1 csf) = (2) hartree E(1 csf) = (2) hartree+ E(2 csf) = (2) hartree + E(2 csf) = (2) hartree

12 Li E (N.R.L.) E (N.R.L.) (1) (2) (1) (2) (2) (2) E (hartree) CSF Not all CSF are useful Not all CSF are useful Only 4 csf are needed to build a statistically exact nodal surface Only 4 csf are needed to build a statistically exact nodal surface (1  g 2 1  u 2 omitted)

13 A tentative recipe Use a large Slater basis Use a large Slater basis  But not too large  Try to reach HF nodes convergence Orbitals from CAS seem better than HF, or NO Orbitals from CAS seem better than HF, or NO Not worth optimizing MOs, if the basis is large enough Not worth optimizing MOs, if the basis is large enough Only few configurations seem to improve the FN energy Only few configurations seem to improve the FN energy Use the right determinants... Use the right determinants... ...different Angular Momentum CSFs And not the bad ones And not the bad ones ...types already included

14 Dimers Bressanini et al. J. Chem. Phys. 123, (2005)

15 Is QMC competitive ?

16 Carbon Atom: Energy CSFsDet.Energy CSFsDet.Energy 1 1s 2 2s 2 2p (4) 1 1s 2 2s 2 2p (4) 2 + 1s 2 2p (4) 2 + 1s 2 2p (4) 5 + 1s 2 2s 2p 2 3d (1) 5 + 1s 2 2s 2p 2 3d (1) 83 1s electrons in 2s 2p 3s 3p 3d shell (4) 83 1s electrons in 2s 2p 3s 3p 3d shell (4) adding f orbitals 7(4f 2 + 2p 3 4f) (1) 7(4f 2 + 2p 3 4f) (1) R12-MR-CI Exact (estimated)

17 Ne Atom Drummond et al (2) DMC Drummond et al (2) DMC backflow Gdanitz et al R12-MR-CI Exact (estimated)

18 Conventional wisdom on  E VMC (  RHF ) > E VMC (  UHF ) > E VMC (  GVB ) E VMC (  RHF ) > E VMC (  UHF ) > E VMC (  GVB ) Single particle approximations  RHF = |1s R 2s R 2p x 2p y 2p z | |1s R 2s R |  RHF = |1s R 2s R 2p x 2p y 2p z | |1s R 2s R |  UHF = |1s U 2s U 2p x 2p y 2p z | |1s’ U 2s’ U |  UHF = |1s U 2s U 2p x 2p y 2p z | |1s’ U 2s’ U | Consider the N atom E DMC (  RHF ) > ? ? < E DMC (  UHF )

19 Conventional wisdom on  We can build a  RHF with the same nodes of  UHF  UHF = |1s U 2s U 2p x 2p y 2p z | |1s’ U 2s’ U |  UHF = |1s U 2s U 2p x 2p y 2p z | |1s’ U 2s’ U |  ’ RHF = |1s U 2s U 2p x 2p y 2p z | |1s U 2s U |  ’ RHF = |1s U 2s U 2p x 2p y 2p z | |1s U 2s U | E DMC (  ’ RHF ) = E DMC (  UHF ) E VMC (  ’ RHF ) > E VMC (  RHF ) > E VMC (  UHF )

20 Conventional wisdom on  Node equivalent to a  UHF |f(r) g(r) 2p 3 | |1s 2s| E DMC (  GVB ) = E DMC (  ’’ RHF )  GVB = |1s 2s 2p 3 | |1s’ 2s’| - |1s’ 2s 2p 3 | |1s 2s’| + |1s’ 2s’ 2p 3 | |1s 2s|- |1s 2s’ 2p 3 | |1s’ 2s| Same Node

21 Nitrogen Atom  Param. E corr. VMC E corr. DMC  Param. E corr. VMC E corr. DMC Simple RHF (1 det)426.0%91.9% Simple RHF (1 det)426.0%91.9% Simple RHF (1 det)842.7%92.6% Simple RHF (1 det)842.7%92.6% Simple UHF (1 det) %92.3% Simple UHF (1 det) %92.3% Simple GVB (4 det) %92.3% Simple GVB (4 det) %92.3% Clementi-Roetti + J2724.5%93.1% Clementi-Roetti + J2724.5%93.1% Is it worth to continue to add parameters to the wave function?

22 What to do? Should we be happy with the “cancellation of error”, and pursue it? Should we be happy with the “cancellation of error”, and pursue it? If so: 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 ? If not, and pursue orthodox QMC (no pseudopotentials, no cancellation of errors, …), can we avoid the curse of  T ?

23 The curse of the   QMC currently relies on  T (R) 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 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 p = parameters per variable M = total parameters needed For M=10 9 and p=3  N=6 The Exponential Wall

24 Convergence to the exact  We must include the correct analytical structure We must include the correct analytical structure Cusps: 3-body coalescence and logarithmic terms: QMC OK Tails: Often neglected

25 Asymptotic behavior of  Example with 2-e atoms Example with 2-e atoms is the solution of the 1 electron problem

26 Asymptotic behavior of  The usual form The usual form does not satisfy the asymptotic conditions A closed shell determinant has the wrong structure

27 Asymptotic behavior of  In general In general Recursively, fixing the cusps, and setting the right symmetry… Each electron has its own orbital, Multideterminant (GVB) Structure! Take 2N coupled electrons 2 N determinants. Again an exponential wall

28 PsH – Positronium Hydride A wave function with the correct asymptotic conditions: A wave function with the correct asymptotic conditions: Bressanini and Morosi: JCP 119, 7037 (2003)

29 We need new, and different, ideas Research is the process of going up alleys to see if they are blind. Marston Bates Different representations Different representations Different dimensions Different dimensions Different equations Different equations Different potential Different potential Radically different algorithms Radically different algorithms Different something Different something

30 Just an example Try a different representation Try a different representation Is some QMC in the momentum 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 ?

31 The other half of Quantum mechanics The Schrodinger equation in the momentum representation Some QMC (GFMC) should be possible, given the iterative form Or write the imaginary time propagator in momentum space

32 Better? For coulomb systems: For coulomb systems: There are NO cusps in momentum space.  convergence should be faster There are NO cusps in momentum space.  convergence should be faster Hydrogenic orbitals are simple rational functions Hydrogenic orbitals are simple rational functions

33 Another (failed so far) example Different dimensionality: Hypernodes Different dimensionality: Hypernodes Given H  (R) = E  (R) build Given H  (R) = E  (R) build Use the Hypernode of Use the Hypernode of The hope was that it could be better than Fixed Node The hope was that it could be better than Fixed Node

34 Hypernodes Exact node Trial node Fixed Node Exact node Trial node Fixed HyperNode The energy is still an upper bound The energy is still an upper bound Unfortunately, it seems to recover exactly the FN energy Unfortunately, it seems to recover exactly the FN energy The intuitive idea was that the system could correct the wrong fixed nodes, by exploring regions where

Why is QMC not used by chemists? A little intermezzo

36 DMC Top 10 reasons 12. We need forces, dummy! 12. We need forces, dummy! 11. Try getting O 2 to bind at the variational level. 11. Try getting O 2 to bind at the variational level. 10. How many graduate students lives have been lost optimizing wavefunctions? 10. How many graduate students lives have been lost optimizing wavefunctions? 9. It is hard to get 0.01 eV accuracy by throwing dice. 9. It is hard to get 0.01 eV accuracy by throwing dice. 8. Most chemical problems have more than 50 electrons. 8. Most chemical problems have more than 50 electrons. 7. Who thought LDA or HF pseudopotentials would be any good? 7. Who thought LDA or HF pseudopotentials would be any good? 6. How many spectra have you seen computed by QMC? 6. How many spectra have you seen computed by QMC? 5. QMC is only exact for energies. 5. QMC is only exact for energies. 4. Multiple determinants. We can't live with them, we can't live without them. 4. Multiple determinants. We can't live with them, we can't live without them. 3. After all, electrons are fermions. 3. After all, electrons are fermions. 2. Electrons move. 2. Electrons move. 1. QMC isn't included in Gaussian 90. Who programs anyway? 1. QMC isn't included in Gaussian 90. Who programs anyway? du/Apps/CMP/topten/topten.html

37 Chemistry and Mathematics “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 “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 "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 "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

38 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 “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 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

39 A Quantum Chemistry Chart J.Pople The more accurate the calculations became, the more the concepts tended to vanish into thin air (Robert Mulliken) Orthodox QMC

40 Chemical concepts Molecular structure and geometry Molecular structure and geometry Chemical bond Chemical bond  Ionic-Covalent  Singe, Double, Triple Electronegativity Electronegativity Oxidation number Oxidation number Atomic charge Atomic charge Lone pairs Lone pairs Aromaticity Aromaticity      NOT DIRECTLY OBSERVABLES ILL-DEFINED CONCEPTS

41 Nodes Should we concentrate on nodes? Checked on small systems: L, Be, He 2 +. See also Mitas Conjectures 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

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

43 Avoided crossings Be e - gas Stadium

44 Nodal topology The conjecture (which I believe is true) claims that there are only two nodal volumes in the fermion ground state The conjecture (which I believe is true) claims that there are only two nodal volumes in the fermion ground state See, among others: 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, (2005)  Mitas PRL 96, (2006)

45 Nodal Regions NeLiBe B C Li

46 Avoided nodal crossing At a nodal crossing,  and  are zero At a nodal crossing,  and  are zero Avoided nodal crossing is the rule, not the exception Avoided nodal crossing is the rule, not the exception Not (yet) a proof... Not (yet) a proof...If has 4 nodes has 2 nodes, with a proper

47 He atom with noninteracting electrons

48

49 Casual similarity ? First unstable antisymmetric stretch orbit of semiclassical linear helium along with the symmetric Wannier orbit r 1 = r 2 and various equipotential lines

50 Superimposed Hylleraas node Casual similarity ?

51 How to directly improve nodes? Fit to a functional form and optimize the parameters ( maybe for small systems ) Fit to a functional form and optimize the parameters ( maybe for small systems ) IF the topology is correct, use a coordinate transformation IF the topology is correct, use a coordinate transformation

52 Coordinate transformation Take a wave function with the correct nodal topology Take a wave function with the correct nodal topology Change the nodes with a coordinate transformation (Linear? Feynman’s backflow ?) preserving the topology Change the nodes with a coordinate transformation (Linear? Feynman’s backflow ?) preserving the topology Miller-Good transformations

53 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” 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 Richard Feynman 1981

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

55 Think Different!