1. Quantum Monte Carlo Simulations of Mixed 3 He/ 4 He Clusters dario.bressanini@uninsubria.it http://www.unico.it/~dario Dario Bressanini Universita’ degli Studi dell’Insubria Göttingen 24/05/2002

2. Dario Bressanini – Göttingen 24/05/2002 2 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

3. Dario Bressanini – Göttingen 24/05/2002 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

4. Dario Bressanini – Göttingen 24/05/2002 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)

5. Dario Bressanini – Göttingen 24/05/2002 5 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

6. Dario Bressanini – Göttingen 24/05/2002 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: 3-30000 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 ~ 0.001 hartree ~ 0.027 eV

7. Dario Bressanini – Göttingen 24/05/2002 7 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

8. Dario Bressanini – Göttingen 24/05/2002 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

9. Dario Bressanini – Göttingen 24/05/2002 9 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) 2 1953)...... and a powerful computer Use the Metropolis algorithm (M(RT) 2 1953)...... 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

11. Dario Bressanini – Göttingen 24/05/2002 11 The Metropolis Algorithm move rejectaccept RiRiRiRi R try R i+1 =R i R i+1 =R try Call the Oracle Compute averages

12. Dario Bressanini – Göttingen 24/05/2002 12 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

13. Dario Bressanini – Göttingen 24/05/2002 13 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

14. Dario Bressanini – Göttingen 24/05/2002 14 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

15. Dario Bressanini – Göttingen 24/05/2002 15 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 = -2.9037245 a.u. Exact = -2.90372437 a.u.

16. Dario Bressanini – Göttingen 24/05/2002 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 = -0.596235(9)a.u. E 10 terms = -0.597136(3)a.u. Exact = -0.597139 a.u.

17. Dario Bressanini – Göttingen 24/05/2002 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

18. Dario Bressanini – Göttingen 24/05/2002 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).

19. Dario Bressanini – Göttingen 24/05/2002 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...

20. Dario Bressanini – Göttingen 24/05/2002 20 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

21. Dario Bressanini – Göttingen 24/05/2002 21 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

22. Dario Bressanini – Göttingen 24/05/2002 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 Schrödinger equation

23. Dario Bressanini – Göttingen 24/05/2002 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 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

24. Dario Bressanini – Göttingen 24/05/2002 24 Diffusion Monte Carlo SIMULATION: discretize time Kinetic process (branching)Kinetic process (branching) Diffusion processDiffusion process

25. Dario Bressanini – Göttingen 24/05/2002 25 The DMC algorithm

26. Dario Bressanini – Göttingen 24/05/2002 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. + -

27. Dario Bressanini – Göttingen 24/05/2002 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

28. Dario Bressanini – Göttingen 24/05/2002 28 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

29. Dario Bressanini – Göttingen 24/05/2002 29 Helium Clusters 1. Small mass of helium atom 2. Very weak He-He interaction 0.02 Kcal/mol 0.9 * 10 -3 cm -1 0.4 * 10 -8 hartree 10 -7 eV Highly non-classical systems. No equilibrium structure. ab-initio methods and normal mode analysis useless Superfluidity High resolution spectroscopy Low temperature chemistry

30. Dario Bressanini – Göttingen 24/05/2002 30 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

31. Dario Bressanini – Göttingen 24/05/2002 31 4 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

32. Dario Bressanini – Göttingen 24/05/2002 32 DMC gives exact results. The quality of the VMC simulations decreases as the cluster increases Pure 4 He n Clusters

33. Dario Bressanini – Göttingen 24/05/2002 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

34. Dario Bressanini – Göttingen 24/05/2002 34 3 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

35. Dario Bressanini – Göttingen 24/05/2002 35 3 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 = -0.08784(7) cm -1 3 He 4 He 2 E = -0.00984(5) cm -1 Bonding interaction Non-bonding interaction

36. Dario Bressanini – Göttingen 24/05/2002 36 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

37. Dario Bressanini – Göttingen 24/05/2002 37 Helium Clusters: energy (cm -1 )

38. Dario Bressanini – Göttingen 24/05/2002 38 3 He/ 4 He Distribution Functions 3 He 4 He 5 Pair distribution functions

39. Dario Bressanini – Göttingen 24/05/2002 39 3 He/ 4 He Distribution Functions 3 He 4 He 5 Distributions with respect to the center of mass c.o.m

40. Dario Bressanini – Göttingen 24/05/2002 40 Distribution Functions in 3 He 4 He n  ( 4 He- 4 He)  ( 3 He- 4 He)

41. Dario Bressanini – Göttingen 24/05/2002 41  ( 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

42. What is the shape of 4 He 3 ?

43. Dario Bressanini – Göttingen 24/05/2002 43 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

44. Dario Bressanini – Göttingen 24/05/2002 44 What is the shape of 4 He 3 ?

45. Dario Bressanini – Göttingen 24/05/2002 45 The Shape of the Trimers Ne trimer He trimer  ( 4 He-center of mass)  (Ne-center of mass)

46. Dario Bressanini – Göttingen 24/05/2002 46 Ne 3 Angular Distributions       Ne trimer

47. Dario Bressanini – Göttingen 24/05/2002 47      4 He 3 Angular Distributions

48. Dario Bressanini – Göttingen 24/05/2002 48 3 He 4 He 2 Angular Distributions     

49. Dario Bressanini – Göttingen 24/05/2002 49 3 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 = -0.3886(1) cm -1 3 He 4 He 3 E = -0.2062(1) cm -1 3 He 2 4 He 2 E = -0.071(1) cm -1

50. Dario Bressanini – Göttingen 24/05/2002 50 3 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

51. Dario Bressanini – Göttingen 24/05/2002 51 3 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

52. Dario Bressanini – Göttingen 24/05/2002 52 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

53. Dario Bressanini – Göttingen 24/05/2002 53 3 He m 4 He n m = 0,1,2,3 Energies

54. Work in progress: 3 He m 4 He n 4 He n 4 He n 3 He m 3 He m 0 1 2 3 4 5 6 7 8 9 10 11 0 1 2 3 4 5 6 7 8 9 10 11 012345 35 First 3 He 2 4 He n bound First 3 He 3 4 He n bound BoundUnbound Unknown Maybe Unlikely

55. Dario Bressanini – Göttingen 24/05/2002 55 3 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

56. Dario Bressanini – Göttingen 24/05/2002 56 3 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

57. Dario Bressanini – Göttingen 24/05/2002 57 3 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. )

58. Dario Bressanini – Göttingen 24/05/2002 58 3 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

59. Dario Bressanini – Göttingen 24/05/2002 59 3 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.

60. Dario Bressanini – Göttingen 24/05/2002 60

61. Dario Bressanini – Göttingen 24/05/2002 61 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

62. Dario Bressanini – Göttingen 24/05/2002 62 Different wave function form 0.002.004.006.008.0010.00 r (a.u.) 0.00 0.20 0.40 0.60 f ( r )

63. Dario Bressanini – Göttingen 24/05/2002 63 Spline Wave Function SF 6 He 39 Knots of the He-SF 6 spline function

64. Dario Bressanini – Göttingen 24/05/2002 64 Shell Structure  ( 4 He-SF 6 ) Standard Spline

65. Dario Bressanini – Göttingen 24/05/2002 65  literature (Rick & Doll)  literature (Rick & Doll) E = -0.00046 cm -1 Numerical Numerical E = -0.00091 cm -1 QMC QMC E = -0.00089(1) cm -1 Optimize  Optimize  Unbound Optimize E (numerically) Optimize E (numerically) E = -0.00075 cm -1  with Exp()  with Exp() E = -0.00084 cm -1  using splines  using splines E = -0.00081 cm -1  for 4 He 2

66. Dario Bressanini – Göttingen 24/05/2002 66  literature (Rick & Doll)  literature (Rick & Doll) E = -0.0798 cm -1 E = -0.08784(7) cm -1 QMC exact QMC exact Optimize Energy Optimize Energy E = -0.0829(4) cm -1  with Exp()  with Exp() E = -0.0851(2) cm -1  using splines  using splines E = -0.0868(2) cm -1  for 4 He 3 three-body terms are not important in  for the trimer

67. Dario Bressanini – Göttingen 24/05/2002 67 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 ?

68. Dario Bressanini – Göttingen 24/05/2002 68 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

69. Dario Bressanini – Göttingen 24/05/2002 69 Acknowledgments Gabriele Morosi Mose’ Casalegno Giordano Fabbri Matteo Zavaglia

70. Dario Bressanini – Göttingen 24/05/2002 70 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)