Quantum Monte Carlo Simulations of Mixed 3 He/ 4 He ClustersDario Bressanini Universita degli Studi dellInsubria Dario Bressanini 2 Overview Introduction to quantum monte carlo methods Introduction to quantum monte carlo methods VMC, QMC, advantages and drawbacks Helium clusters simulations Helium clusters simulations Problems, solutions Mixed 3 He/ 4 He clusters Mixed 3 He/ 4 He clusters Trimers 3 He 4 He N Future directions Future directions Dario Bressanini 3 Monte Carlo Methods How to solve a deterministic problem using a Monte Carlo method? How to solve a deterministic problem using a Monte Carlo method? Rephrase the problem using a probability distribution Rephrase the problem using a probability distribution Measure A by sampling the probability distribution Measure A by sampling the probability distribution Dario Bressanini 4 Monte Carlo Methods The points R i are generated using random numbers The points R i are generated using random numbers We introduce noise into the problem!! We introduce noise into the problem!! Our results have error bars... ... Nevertheless it might be a good way to proceed This is why the methods are called Monte Carlo methods Metropolis, Ulam, Fermi, Von Neumann (-1945) Metropolis, Ulam, Fermi, Von Neumann (-1945) Dario Bressanini 5 Monte Carlo Methods Not necessarily... Not necessarily... ... It might be the only way to proceed ... It might reduce considerably the problems complexity ... It might scale better than other methods Dario Bressanini 6 Quantum Mechanics We wish to solve H = E to high accuracy We wish to solve H = E to high accuracy The solution usually involves computing integrals in high dimensions: The classic approach (from 1929): The classic approach (from 1929): Find approximate (... but good...) ... whose integrals are analitically computable (gaussians) Compute the approximate energy chemical accuracy ~ hartree ~ eV Dario Bressanini 7 VMC: Variational Monte Carlo Start from the Variational Principle Start from the Variational Principle Translate it into Monte Carlo language Translate it into Monte Carlo language Dario Bressanini 8 VMC: Variational Monte Carlo E is a statistical average of the local energy E L over P(R) E is a statistical average of the local energy E L over P(R) Recipe: Recipe: take an appropriate trial wave function distribute N points according to P(R) compute the average of the local energy Dario Bressanini 9 VMC: Variational Monte Carlo There is no need to analytically compute integrals, so there is complete freedom in the choice of the trial wave function There is no need to analytically compute integrals, so there is complete freedom in the choice of the trial wave function r1r1 r2r2 r 12 He atom Quantum chemistry uses a product of single particle functions. With VMC we can use any function: explicitly correlated wave functions can be used Quantum chemistry uses a product of single particle functions. With VMC we can use any function: explicitly correlated wave functions can be used Dario Bressanini 10 The Metropolis Algorithm How do we sample How do we sample Anyone who consider arithmetical methods of producing random digits is, of course, in a state of sin. John Von Neumann John Von Neumann Use the Metropolis algorithm (M(RT) ) and a powerful computer Use the Metropolis algorithm (M(RT) ) and a powerful computer ? The algorithm is a random walk (markov chain) in configuration space The algorithm is a random walk (markov chain) in configuration space Dario Bressanini 12 The Metropolis Algorithm move rejectaccept RiRiRiRi R try R i+1 =R i R i+1 =R try Call the Oracle Compute averages Dario Bressanini 13 if p 1 /* accept always */ accept move If 0 p 1 /* accept with probability p */ if p > rnd() accept move else reject move The Metropolis Algorithm The Oracle Dario Bressanini 14 No need to make the single-particle approximation No need to make the single-particle approximation Can use for which no analytical integrals exist Can use for which no analytical integrals exist Use explicitly correlated wave functions Can satisfy the cusp conditions VMC advantages He atom ground state E 19 terms = a.u. Exact = a.u. Dario Bressanini 15 VMC advantages Can compute difficult quantities, e.g. Can compute difficult quantities, e.g. Can compute lower bounds Can compute lower bounds Dario Bressanini 16 VMC advantages Can easily go beyond the Born-Oppenheimer approximation. Can easily go beyond the Born-Oppenheimer approximation. H 2 + molecule ground state E 1 term = (9)a.u. E 10 terms = (3)a.u. Exact = a.u. Dario Bressanini 17 VMC advantages Can work with ANY potential, in ANY number of dimensions. Can work with ANY potential, in ANY number of dimensions. Ps 2 molecule (e + e + e - e - ) in 2D and 3D Optimization of nonlinear parameters Optimization of nonlinear parameters Numerically stable Minimum known in advance (0) Can be used for excited states with same symmetry too Dario Bressanini 18 First Major VMC Calculations McMillan VMC calculation of ground state of liquid 4 He (1964) McMillan VMC calculation of ground state of liquid 4 He (1964) Generalized for fermions by Ceperley, Chester and Kalos PRB 16, 3081 (1977). Generalized for fermions by Ceperley, Chester and Kalos PRB 16, 3081 (1977). Dario Bressanini 19 VMC drawbacks Error bar goes down as N -1/2 Error bar goes down as N -1/2 It is computationally demanding It is computationally demanding The optimization of becomes difficult as the number of nonlinear parameters increases The optimization of becomes difficult as the number of nonlinear parameters increases It depends critically on our skill to invent a good It depends critically on our skill to invent a good There exist exact, automatic ways to get better wave functions. There exist exact, automatic ways to get better wave functions. Let the computer do the work... Dario Bressanini 20 Diffusion Monte Carlo Suggested by Fermi in 1945, but implemented only in the 70s Suggested by Fermi in 1945, but implemented only in the 70s Nature is not 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 P. Feynman VMC is a classical simulation method VMC is a classical simulation method Dario Bressanini 21 The time dependent Schrdinger equation is similar to a diffusion equation The time dependent Schrdinger equation is similar to a diffusion equation Time evolution DiffusionBranch The diffusion equation can be solved by directly simulating the system The diffusion equation can be solved by directly simulating the system Can we do the same with the Schrdinger equation ? Diffusion equation analogy Dario Bressanini 22 The analogy is only formal The analogy is only formal is a complex quantity, while C is real and positive Imaginary Time Sch. Equation If we let the time t be imaginary, then can be real! If we let the time t be imaginary, then can be real! Imaginary time Schrdinger equation Dario Bressanini 23 as a concentration is interpreted as a concentration of fictitious particles, called walkers is interpreted as a concentration of fictitious particles, called walkers The schrdinger equation is simulated by a process of diffusion, growth and disappearance of walkers The schrdinger equation is simulated by a process of diffusion, growth and disappearance of walkers Ground State Dario Bressanini 24 Diffusion Monte Carlo SIMULATION: discretize time Kinetic process (branching)Kinetic process (branching) Diffusion processDiffusion process Dario Bressanini 25 The DMC algorithm Dario Bressanini 26 The Fermion Problem Wave functions for fermions have nodes. Wave functions for fermions have nodes. Diffusion equation analogy is lost. Need to introduce positive and negative walkers. The (In)famous Sign Problem Restrict random walk to a positive region bounded by nodes. Unfortunately, the exact nodes are unknown. Restrict random walk to a positive region bounded by nodes. Unfortunately, the exact nodes are unknown. Use approximate nodes from a trial . Kill the walkers if they cross a node. Use approximate nodes from a trial . Kill the walkers if they cross a node. + - Dario Bressanini 27 Helium A helium atom is an elementary particle. A weakly interacting hard sphere. A helium atom is an elementary particle. A weakly interacting hard sphere. Interatomic potential is known more accurately than any other atom. Interatomic potential is known more accurately than any other atom. Two isotopes: 3 He (fermion: antisymmetric trial function, spin 1/2) 3 He (fermion: antisymmetric trial function, spin 1/2) 4 He (boson: symmetric trial function, spin zero) 4 He (boson: symmetric trial function, spin zero) The interaction potential is the same The interaction potential is the same Dario Bressanini 28 Helium Clusters They are not rigid, and show large-amplitude motion They are not rigid, and show large-amplitude motion Normal mode analysis useless Normal mode analysis useless Equilibrium structure is ill-defined Equilibrium structure is ill-defined Stochastic methods well suited to study helium clusters, both pure or with impurities Stochastic methods well suited to study helium clusters, both pure or with impurities VMC, DMC, GFMC, PIMC etc... Dario Bressanini 29 Adiabatic expansion cools helium to below the critical point, forming droplets. Adiabatic expansion cools helium to below the critical point, forming droplets. The droplets are sent through a scattering chamber to pick up impurities, and are detected either with a mass spectrometer The droplets are sent through a scattering chamber to pick up impurities, and are detected either with a mass spectrometer Toennies and Vilesov, Ann. Rev. Phys. Chem. 49, 1 (1998) Experiment on He droplets Dario Bressanini 30 Helium Clusters 1. Small mass of helium atom 2. Very weak He-He interaction 0.02 Kcal/mol 0.9 * cm * hartree eV Highly non-classical systems Superfluidity High resolution spectroscopy Low temperature chemistry Dario Bressanini 31 The Simulations Both VMC and DMC simulations Both VMC and DMC simulations Standard Standard Potential = sum of two-body TTY pair-potential Potential = sum of two-body TTY pair-potential Three-body terms not important for small clusters Three-body terms not important for small clusters Dario Bressanini 32 The quality of the VMC simulations decreases as the cluster increases Pure 4 He n Clusters Dario Bressanini 33 Wave function quality decreases as N increases Wave function quality decreases as N increases It was optimized to get minimum (H), not minimum It was optimized to get minimum (H), not minimum Are three- and many-body terms in important ? Very difficult to optimize. Unstable process especially for the trimers. Can we improve ? for 4 He n Clusters Dario Bressanini 34 Mixed 3 He/ 4 He Clusters (m,n) = 3 He m 4 He n Bressanini et. al. J.Chem.Phys. 112, 717 (2000) Dario Bressanini 35 Helium Clusters: energy (cm -1 ) Dario Bressanini 36 Helium Clusters: stability 4 He N is destabilized by substituting a 4 He with a 3 He 4 He N is destabilized by substituting a 4 He with a 3 He The structure is only weakly perturbed. The structure is only weakly perturbed. 4 He 4 He Dimers 4 He 3 He 3 He 3 He BoundUnboundUnbound 4 He 3 Trimers 4 He 2 3 He 4 He 3 He 2 BoundBoundUnbound 4 He 4 Tetramers 4 He 3 3 He 4 He 2 3 He 2 BoundBoundBound Dario Bressanini 37 Trimers and Tetramers Stability 4 He 3 E = (7) cm -1 4 He 2 3 He E = (5) cm -1 Five out of six unbound pairs! 4 He 4 E = (1) cm -1 4 He 3 3 He E = (1) cm -1 4 He 2 3 He 2 E = (1) cm -1 Bonding interaction Non-bonding interaction Dario Bressanini 38 3 He/ 4 He Distribution Functions 3 He( 4 He) 5 Pair distribution functions Dario Bressanini 39 3 He/ 4 He Distribution Functions 3 He( 4 He) 5 Distributions with respect to the center of mass c.o.m Dario Bressanini 40 Distribution Functions in 4 He N 3 He ( 4 He- 4 He) ( 3 He- 4 He) Dario Bressanini 41 Distribution Functions in 4 He N 3 He ( 4 He- C.O.M. ) ( 3 He- C.O.M. ) c.o.m. = center of mass Similar to pure clusters Fermion is pushed away Dario Bressanini 42 Some people say is an equilateral triangle... Some people say is an equilateral triangle some say it is linear (almost) some say it is linear (almost) some say it is both.... some say it is both. What is the shape of 4 He 3 ? Pair distribution function Dario Bressanini 43 What is the shape of 4 He 3 ? Dario Bressanini 44 The Shape of the Trimers Ne trimer He trimer ( 4 He-center of mass) (Ne-center of mass) Dario Bressanini 45 Ne 3 Angular Distributions Ne trimer Dario Bressanini 46 4 He 3 Angular Distributions Dario Bressanini 47 3 He 4 He 2 Angular Distributions Dario Bressanini 48 Different wave function form Spline r (a.u.) f ( r ) Dario Bressanini 49 literature (Rick & Doll) literature (Rick & Doll) E = cm -1 Numerical Numerical E = cm -1 QMC QMC E = (1) cm -1 Optimize Optimize Unbound Optimize E (numerically) Optimize E (numerically) E = cm -1 with Exp() with Exp() E = cm -1 using splines using splines E = cm -1 for 4 He 2 Dario Bressanini 50 literature (Rick & Doll) literature (Rick & Doll) E = cm -1 E = (7) cm -1 QMC exact QMC exact Optimize Energy Optimize Energy E = (4) cm -1 with Exp() with Exp() E = (2) cm -1 using splines using splines E = (2) cm -1 for 4 He 3 three-body terms are not important in for the trimer Dario Bressanini 51 Work in Progress Various impurities embedded in a Helium cluster Various impurities embedded in a Helium cluster for bigger clusters using splines for bigger clusters using splines Optimize the energy instead of the variance of the local energy. Optimize the energy instead of the variance of the local energy. Geometric structure of trimers and tetramers Geometric structure of trimers and tetramers Dario Bressanini 52 Conclusions The substitution of a 4 He with a 3 He leads to an energetic destabilization. The substitution of a 4 He with a 3 He leads to an energetic destabilization. 3 He weakly perturbes the 4 He atoms distribution. 3 He weakly perturbes the 4 He atoms distribution. 3 He moves on the surface of the cluster. 3 He moves on the surface of the cluster. 4 He 2 3 He bound, 4 He 3 He 2 unbound. 4 He 2 3 He bound, 4 He 3 He 2 unbound. 4 He 3 3 He and 4 He 2 3 He 2 bound. 4 He 3 3 He and 4 He 2 3 He 2 bound. QMC gives accurate energies and structural information QMC gives accurate energies and structural information Dario Bressanini 53 Acknowledgments Gabriele Morosi Massimo Mella Mose Casalegno Giordano Fabbri Matteo Zavaglia Dario Bressanini 54 A reflection... A new method is initially not as well formulated or understood as existing methods It can seldom offer results of a comparable quality before a considerable amount of development has taken place Only rarely do new methods differ in major ways from previous approaches A new method for calculating properties in nuclei, atoms, molecules, or solids automatically provokes three sorts of negative reactions: Nonetheless, new methods need to be developed to handle problems that are vexing to or beyond the scope of the current approaches ( Slightly modified from Steven R. White, John W. Wilkins and Kenneth G. Wilson)