Quantum Monte Carlo Simulations of Mixed 3 He/ 4 He Clusters Dario Bressanini Universita’ degli Studi dell’Insubria

© Dario Bressanini 2 Overview Introduction to quantum monte carlo methods Introduction to quantum monte carlo methods Mixed 3 He/ 4 He clusters simulations Mixed 3 He/ 4 He clusters simulations

© 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 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 6 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 7 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 8 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 10 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 11 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 12 VMC: Variational Monte Carlo No need to analytically compute integrals: complete freedom in the choice of the trial wave function. No need to analytically compute integrals: complete freedom in the choice of the trial wave function. r1r1 r2r2 r 12 He atom Can use explicitly correlated wave functions Can use explicitly correlated wave functions Can satisfy the cusp conditions Can satisfy the cusp conditions

© Dario Bressanini 13 VMC advantages Can go beyond the Born-Oppenheimer approximation, with ANY potential, in ANY number of dimensions. Can go beyond the Born-Oppenheimer approximation, with ANY potential, in ANY number of dimensions. Ps 2 molecule (e + e + e - e - ) in 2D and 3D M + m + M - m - as a function of M/m Can compute lower bounds Can compute lower bounds

© Dario Bressanini 14 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 15 Diffusion Monte Carlo Suggested by Fermi in 1945, but implemented only in the 70’s Suggested by Fermi in 1945, but implemented only in the 70’s 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 16 The time dependent Schrödinger equation is similar to a diffusion equation The time dependent Schrödinger 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 simulate the Schrödinger equation? Diffusion equation analogy

© Dario Bressanini 17 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 Schrödinger equation

© Dario Bressanini 18  as a concentration  is interpreted as a concentration of fictitious particles, called walkers  is interpreted as a concentration of fictitious particles, called walkers The schrödinger equation is simulated by a process of diffusion, growth and disappearance of walkers The schrödinger equation is simulated by a process of diffusion, growth and disappearance of walkers Ground State

© Dario Bressanini 19 Diffusion Monte Carlo SIMULATION: discretize time Kinetic process (branching)Kinetic process (branching) Diffusion processDiffusion process

© Dario Bressanini 20 The DMC algorithm

© Dario Bressanini 21 QMC: a simple and useful tool Yukawa potential Yukawa potential  Plasma physics, solid-state physics,... Stability of screened H, H 2 + and H 2 as a function of, without Born-Oppenheimer approximation Stability of screened H, H 2 + and H 2 as a function of, without Born-Oppenheimer approximation (preliminary results) (preliminary results)  =1.19  1.2 Borromean H bound H unbound H 2 + bound H unbound H 2 + unbound

© Dario Bressanini 22 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 23 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 24 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 25 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. No equilibrium structure. ab-initio methods and normal mode analysis useless Superfluidity High resolution spectroscopy Low temperature chemistry

© Dario Bressanini 26 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

© Dario Bressanini 27 Pure 4 He n Clusters

© Dario Bressanini 28 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 29 Helium Clusters: energy (cm -1 )

© Dario Bressanini 30 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 31 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 32 3 He/ 4 He Distribution Functions 3 He( 4 He) 5 Pair distribution functions

© Dario Bressanini 33 3 He/ 4 He Distribution Functions 3 He( 4 He) 5 Distributions with respect to the center of mass c.o.m

© Dario Bressanini 34 Distribution Functions in 4 He N 3 He  ( 4 He- 4 He)  ( 3 He- 4 He)

© Dario Bressanini 35 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 36 The Shape of the Trimers Ne trimer He trimer  ( 4 He-center of mass)  (Ne-center of mass)

© Dario Bressanini 37 4 He 3 Angular Distributions    

© Dario Bressanini 38 Ne 3 Angular Distributions       Ne trimer

© Dario Bressanini 39 Helium Cluster Stability Is 3 He m 4 He n stable ? Is 3 He m 4 He n stable ? What is the smallest 3 He m stable cluster ? What is the smallest 3 He m stable cluster ? Liquid: stable Dimer: unbound 3 He m m = ? 20 < m < 35 critically bound

© Dario Bressanini 40 4 He n 4 He n 3 He m 3 He m Bound Unbound Unknown 35 Probably unbound Work in progress: 3 He m 4 He n

© Dario Bressanini 41 Work in Progress Various impurities embedded in a Helium cluster Various impurities embedded in a Helium cluster Different functional forms for  splines) Different functional forms for  splines) Stability of 3 He m 4 He n Stability of 3 He m 4 He n

© Dario Bressanini 42 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 43 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)