Is QMC delivering its early promises?Dario Bressanini QMCI Sardagna (Trento) 2006 http://scienze-como.uninsubria.it/bressaniniUniversita dellInsubria, Como, ItalySome reflections on nodes and trial wave functions

30 years of QMC in chemistry

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

Good for Helium studiesThousands of theoretical and experimental papershave been published on Helium, in its various forms:AtomSmall ClustersDropletsBulk

3Hem4Hen Stability Chart32 0 1 2 3 4 5 6 7 8 9 10 110123453He34He8 L=0 S=1/23He24He4 L=1 S=13He24He2 L=0 S=03He34He4 L=1 S=1/2Terra Incognita

Good for vibrational problems

For electronic structure?Sign ProblemFixed Nodal error problem

The influence on the nodes of YTQMC currently relies on YT(R) and its nodes (indirectly)How are the nodes YT(R) of influenced by:The single particle basis setThe generation of the orbitals (HF, CAS, MCSCF, NO, )The number and type of configurations in the multidet. expansion?

He2+: the basis setThe ROHF wave function:1sE = -4.9905(2) hartree1s1s2s3sE = -4.9943(2) hartreeEN.R.L = -4.9945 hartree

He2+: MOsE(RHF) = -4.9943(2) hartreeE(CAS) = -4.9925(2) hartreeE(CAS-NO) = -4.9916(2) hartreeE(CI-NO) = -4.9917(2) hartree

EN.R.L = -4.9945 hartree

Bressanini et al. J. Chem. Phys. 123, 204109 (2005)

He2+: CSFs

1s1s2s3s2p2p

E(1 csf) = -4.9932(2) hartree

1s1s2s3s

E(1 csf) = -4.9943(2) hartree

Li2-14.9954 E (N.R.L.)-14.9952(1)-14.9939(2)-14.9933(1)-14.9933(2)-14.9914(2)-14.9923(2)E (hartree) CSFNot all CSF are usefulOnly 4 csf are needed to build a statistically exact nodal surface(1sg2 1su2 omitted)

A tentative recipeUse a large Slater basisBut not too largeTry to reach HF nodes convergenceOrbitals from CAS seem better than HF, or NONot worth optimizing MOs, if the basis is large enoughOnly few configurations seem to improve the FN energyUse the right determinants......different Angular Momentum CSFsAnd not the bad ones...types already included

DimersBressanini et al. J. Chem. Phys. 123, 204109 (2005)

Is QMC competitive ?

Carbon Atom: EnergyCSFsDet.Energy1 1s22s2 2p21-37.8303(4)2 + 1s2 2p42-37.8342(4)5 + 1s2 2s 2p23d 18 -37.8399(1)83 1s2 + 4 electrons in 2s 2p 3s 3p 3d shell422-37.8387(4) adding f orbitals7(4f2 + 2p34f) 34-37.8407(1)

R12-MR-CI-37.845179Exact (estimated)-37.8450

Ne AtomDrummond et al. -128.9237(2) DMCDrummond et al. -128.9290(2) DMC backflowGdanitz et al. -128.93701 R12-MR-CIExact (estimated) -128.9376

Conventional wisdom on YEVMC(YRHF) > EVMC(YUHF) > EVMC(YGVB)Single particle approximationsYRHF = |1sR 2sR 2px 2py 2pz| |1sR 2sR|YUHF = |1sU 2sU 2px 2py 2pz| |1sU 2sU|Consider the N atomEDMC(YRHF) > ? < EDMC(YUHF)

Conventional wisdom on YWe can build a YRHF with the same nodes of YUHF YUHF = |1sU 2sU 2px 2py 2pz| |1sU 2sU|YRHF = |1sU 2sU 2px 2py 2pz| |1sU 2sU|EDMC(YRHF) = EDMC(YUHF)EVMC(YRHF) > EVMC(YRHF) > EVMC(YUHF)

Conventional wisdom on YNode equivalent to a YUHF |f(r) g(r) 2p3| |1s 2s|EDMC(YGVB) = EDMC(YRHF)YGVB = |1s 2s 2p3| |1s 2s| - |1s 2s 2p3| |1s 2s| + |1s 2s 2p3| |1s 2s|- |1s 2s 2p3| |1s 2s|

Nitrogen AtomY Param. E corr. VMC E corr. DMC

Simple RHF (1 det)426.0%91.9%Simple RHF (1 det)842.7%92.6%Simple UHF (1 det)11 41.2%92.3%Simple GVB (4 det) 1142.3%92.3%

Clementi-Roetti + J2724.5%93.1% Is it worth to continue to add parameters to the wave function?

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 advertiseIf not, and pursue orthodox QMC (no pseudopotentials, no cancellation of errors, ) , can we avoid the curse of YT ?

The curse of the YTQMC currently relies on YT(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 largep = parameters per variableM = total parameters needed For M=109 and p=3 N=6The Exponential Wall

Convergence to the exact YWe must include the correct analytical structureCusps:3-body coalescence and logarithmic terms:QMC OKQMC OKTails:Often neglected

Asymptotic behavior of YExample with 2-e atomsis the solution of the 1 electron problem

Asymptotic behavior of YThe usual formdoes not satisfy the asymptotic conditionsA closed shell determinant has the wrong structure

Asymptotic behavior of YIn generalRecursively, fixing the cusps, and setting the right symmetryEach electron has its own orbital, Multideterminant (GVB) Structure!2N determinants. Again an exponential wall

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

We need new, and different, ideasDifferent representationsDifferent dimensionsDifferent equationsDifferent potentialRadically different algorithmsDifferent somethingResearch is the process of going up alleys to see if they are blind. Marston Bates

Just an exampleTry a different representationIs some QMC in the momentum representationPossible ? 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 ?

The other half of Quantum mechanicsThe Schrodinger equation in the momentum representationSome QMC (GFMC) should be possible, given the iterative formOr write the imaginary time propagator in momentum space

Better?For coulomb systems:There are NO cusps in momentum space. Y convergence should be fasterHydrogenic orbitals are simple rational functions

Another (failed so far) exampleDifferent dimensionality: HypernodesGiven HY (R) = EY (R) build The hope was that it could be better than Fixed Node

HypernodesThe energy is still an upper boundUnfortunately, it seems to recover exactly the FN energy

Why is QMC not used by chemists?A little intermezzo

DMC Top 10 reasons12. 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.edu/Apps/CMP/topten/topten.html

Chemistry and MathematicsThe 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"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

Nature and Mathematicsil Grande libro della Natura e scritto nel linguaggio della matematica, e non possiamo capirla se prima non ne capiamo i simboli Galileo GalileiEvery 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

A Quantum Chemistry ChartJ.PopleThe more accurate the calculations became, the more the concepts tended to vanish into thin air (Robert Mulliken)

Chemical conceptsMolecular structure and geometryChemical bondIonic-CovalentSinge, Double, TripleElectronegativityOxidation numberAtomic chargeLone pairsAromaticity

NodesConjectures on nodeshave higher symmetry than Y itselfresemble simple functionsthe ground state has only 2 nodal volumesHF nodes are quite good: they naturally have these propertiesShould we concentrate on nodes?Checked on small systems: L, Be, He2+. See also Mitas

Be Nodal Topology

Avoided crossingsBee- gasStadium

Nodal topologyThe conjecture (which I believe is true) claims that there are only two nodal volumes in the fermion ground stateSee, 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)

Nodal RegionsNodal Regions

Avoided nodal crossingAt a nodal crossing, Y and Y are zeroAvoided nodal crossing is the rule, not the exceptionNot (yet) a proof...

He atom with noninteracting electrons

Casual similarity ?First unstable antisymmetric stretch orbit of semiclassical linear helium along with the symmetric Wannier orbit r1 = r2 and various equipotential lines

Casual similarity ?Superimposed Hylleraas node

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

Coordinate transformationTake a wave function with the correct nodal topologyChange the nodes with a coordinate transformation (Linear? Feynmans backflow ?) preserving the topologyMiller-Good transformations

Feynman on simulating natureNature isnt classical, dammit, and if you want to make a simulation of Nature, youd better make it quantum mechanical, and by golly its a wonderful problem, because it doesnt look so easy Richard Feynman 1981

A QMC song...He deals the cards to find the answersthe sacred geometry of chancethe hidden law of a probable outcomethe numbers lead a danceSting: Shape of my heart

Think Different!