Quantum Monte Carlo Simulations of Mixed 3 He/ 4 He Clusters Dario Bressanini Universita’ degli Studi dell’Insubria Göttingen 24/05/2002

Dario Bressanini – Göttingen 24/05/ Overview Introduction to quantum monte carlo methods Introduction to quantum monte carlo methods  VMC, QMC, advantages and drawbacks Stability and structure of small 3 He/ 4 He mixed clusters Stability and structure of small 3 He/ 4 He mixed clusters  Trimers

Dario Bressanini – Göttingen 24/05/ 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 – Göttingen 24/05/ 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 – Göttingen 24/05/ Monte Carlo Methods Not necessarily... Not necessarily... ... It might be the only way to proceed ... It might reduce considerably the problem’s complexity ... It might scale better than other methods

Dario Bressanini – Göttingen 24/05/ 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 – Göttingen 24/05/ VMC: Variational Monte Carlo To solve H  = E  start from the Variational Principle To solve H  = E  start from the Variational Principle Translate it into Monte Carlo language Translate it into Monte Carlo language

Dario Bressanini – Göttingen 24/05/ 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 – Göttingen 24/05/ 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 – Göttingen 24/05/ 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 – Göttingen 24/05/ 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 – Göttingen 24/05/ 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 – Göttingen 24/05/ 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 – Göttingen 24/05/ 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 – Göttingen 24/05/ 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 – Göttingen 24/05/ 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 – Göttingen 24/05/ 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 – Göttingen 24/05/ 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 – Göttingen 24/05/ 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 – Göttingen 24/05/ The time dependent Schrödinger equation is similar to a diffusion equation The time dependent Schrödinger equation is similar to a diffusion equation Time evolution Diffusion Branch 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 – Göttingen 24/05/ 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 – Göttingen 24/05/  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 – Göttingen 24/05/ Diffusion Monte Carlo SIMULATION: discretize time Kinetic process (branching)Kinetic process (branching) Diffusion processDiffusion process

Dario Bressanini – Göttingen 24/05/ The DMC algorithm

Dario Bressanini – Göttingen 24/05/ 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 – Göttingen 24/05/ 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 – Göttingen 24/05/ 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 – Göttingen 24/05/ 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 – Göttingen 24/05/ 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 – Göttingen 24/05/ He n Clusters Stability 4 He 3 bound. Efimov effect? 4 He 3 bound. Efimov effect? Liquid: stable 4 He 2 dimer exists 4 He n All clusters bound

Dario Bressanini – Göttingen 24/05/ DMC gives exact results. The quality of the VMC simulations decreases as the cluster increases Pure 4 He n Clusters

Dario Bressanini – Göttingen 24/05/ 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 – Göttingen 24/05/ He n Clusters Stability Even less is known for mixed clusters. Is 3 He m 4 He n stable ? Even less is known for mixed clusters. 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 3 He 2 dimer unbound 3 He m m = ? 20 < m < 35 critically bound

Dario Bressanini – Göttingen 24/05/ He 4 He n Clusters Stability 3 He 4 He dimer unbound 3 He 4 He 2 Trimer bound 3 He 4 He n All clusters up bound 4 He 3 E = (7) cm -1 3 He 4 He 2 E = (5) cm -1 Bonding interaction Non-bonding interaction

Dario Bressanini – Göttingen 24/05/ Mixed 3 He 4 He n Clusters (m,n) = 3 He m 4 He n Bressanini et. al. J.Chem.Phys. 112, 717 (2000) 4 He n is destabilized by substituting a 4 He with a 3 He

Dario Bressanini – Göttingen 24/05/ Helium Clusters: energy (cm -1 )

Dario Bressanini – Göttingen 24/05/ He/ 4 He Distribution Functions 3 He 4 He 5 Pair distribution functions

Dario Bressanini – Göttingen 24/05/ He/ 4 He Distribution Functions 3 He 4 He 5 Distributions with respect to the center of mass c.o.m

Dario Bressanini – Göttingen 24/05/ Distribution Functions in 3 He 4 He n  ( 4 He- 4 He)  ( 3 He- 4 He)

Dario Bressanini – Göttingen 24/05/  ( 4 He- C.O.M. )  ( 3 He- C.O.M. ) c.o.m. = center of mass Similar to pure clusters 3 He is pushed away Distribution Functions

What is the shape of 4 He 3 ?

Dario Bressanini – Göttingen 24/05/ 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 We find NO sign of double peak

Dario Bressanini – Göttingen 24/05/ What is the shape of 4 He 3 ?

Dario Bressanini – Göttingen 24/05/ The Shape of the Trimers Ne trimer He trimer  ( 4 He-center of mass)  (Ne-center of mass)

Dario Bressanini – Göttingen 24/05/ Ne 3 Angular Distributions       Ne trimer

Dario Bressanini – Göttingen 24/05/      4 He 3 Angular Distributions

Dario Bressanini – Göttingen 24/05/ He 4 He 2 Angular Distributions     

Dario Bressanini – Göttingen 24/05/ He 2 4 He n Clusters Stability Now put two 3 He. Singlet state.  is positive everywhere Now put two 3 He. Singlet state.  is positive everywhere 3 He 2 4 He n All clusters up bound 3 He 2 4 He Trimer unbound 3 He 2 4 He 2 Tetramer bound 5 out of 6 unbound pairs 4 He 4 E = (1) cm -1 3 He 4 He 3 E = (1) cm -1 3 He 2 4 He 2 E = (1) cm -1

Dario Bressanini – Göttingen 24/05/ He 2 4 He n Clusters Structure The two 3 He atoms stay mainly on the surface of the 4 He cluster The two 3 He atoms stay mainly on the surface of the 4 He cluster 3 He 2 4 He 10

Dario Bressanini – Göttingen 24/05/ He 3 4 He n Clusters Stability Adding a third fermionic helium, introduces a nodal surface into the wave function that destabilizes the system Adding a third fermionic helium, introduces a nodal surface into the wave function that destabilizes the system What is the smallest 3 He 3 4 He n stable cluster ? What is the smallest 3 He 3 4 He n stable cluster ? 3 He 35 is bound, so 3 He 3 4 He 32 should be bound. n < 32 3 He 35 is bound, so 3 He 3 4 He 32 should be bound. n < 32

Dario Bressanini – Göttingen 24/05/ The Wave Function The total wave function must be antisymmetric with respect to the fermionic helium The total wave function must be antisymmetric with respect to the fermionic helium Consider the doublet spin eigenfunction (two  and one  ) Consider the doublet spin eigenfunction (two  and one  )  The 4 He- 4 He and 4 He- 3 He functions are symmetric  The 3 He- 3 He part is antisymmetric

Dario Bressanini – Göttingen 24/05/ He m 4 He n m = 0,1,2,3 Energies

Work in progress: 3 He m 4 He n 4 He n 4 He n 3 He m 3 He m First 3 He 2 4 He n bound First 3 He 3 4 He n bound BoundUnbound Unknown Maybe Unlikely

Dario Bressanini – Göttingen 24/05/ He m 4 He 10 m = 0,1,2,3 4 He distribution with respect to the center of mass c.o.m  ( 4 He- C.O.M. ) The 4 He distribution is unchanged with 0,1,2 or 3 3 He

Dario Bressanini – Göttingen 24/05/ He m 4 He 10 m = 0,1,2,3 3 He distribution with respect to the center of mass  ( 3 He- C.O.M. ) c.o.m 3 He 4 He 10 3 He 2 4 He 10 3 He 3 4 He 10

Dario Bressanini – Göttingen 24/05/ He m 4 He 10 m = 0,1,2,3 3 He distribution with respect to the center of mass One  3 He is pushed inside the cluster, the other two ( ,  ) outside  ( 3 He- C.O.M. )

Dario Bressanini – Göttingen 24/05/ He m 4 He 10 m = 0,1,2,3 3 He- 3 He distributions  ( 3 He- 3 He) The (tentative) picture: two 3 He outside ( ,  ) and one  inside, pushed away from the other  3 He    4 He 10  outside  inside  outside  inside  outside  outside on opposite sides

Dario Bressanini – Göttingen 24/05/ He 3 4 He 10 Why ?    4 He 10 It is a Nodal Effect. The wave function is zero if the two  3 He are at the same distance from the  3 He. For this reason the three atoms are not free to move on the surface of the cluster. One is pushed inside to avoid the wave function node.

Dario Bressanini – Göttingen 24/05/

Dario Bressanini – Göttingen 24/05/ More Flexible Wave Function The standard form is not very flexible The standard form is not very flexible Difficult to optimize Difficult to optimize Difficult to reproduce the shell structure Difficult to reproduce the shell structure

Dario Bressanini – Göttingen 24/05/ Different wave function form r (a.u.) f ( r )

Dario Bressanini – Göttingen 24/05/ Spline Wave Function SF 6 He 39 Knots of the He-SF 6 spline function

Dario Bressanini – Göttingen 24/05/ Shell Structure  ( 4 He-SF 6 ) Standard Spline

Dario Bressanini – Göttingen 24/05/  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 – Göttingen 24/05/  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 – Göttingen 24/05/ Work in Progress and Future Various impurities embedded in a Helium cluster (suggestions welcome!) Various impurities embedded in a Helium cluster (suggestions welcome!) Different functional forms for  splines) Different functional forms for  splines)  anisotropy Analysis of 3 He 3 4 He n Analysis of 3 He 3 4 He n What about 3 He 4 4 He n and 3 He 5 4 He n ? What about 3 He 4 4 He n and 3 He 5 4 He n ?

Dario Bressanini – Göttingen 24/05/ 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. 4 He 3 3 He. 4 He n 3 He 2 bound for n > 1 4 He n 3 He 2 bound for n > 1

Dario Bressanini – Göttingen 24/05/ Acknowledgments Gabriele Morosi Mose’ Casalegno Giordano Fabbri Matteo Zavaglia

Dario Bressanini – Göttingen 24/05/ 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)