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Universita’ degli Studi dell’Insubria Quantum Monte Carlo Simulations of Mixed 3He/4He Clusters Dario Bressanini [email protected] http://www.unico.it/~dario Overview Introduction to quantum monte carlo methods Helium clusters simulations VMC, QMC, advantages and drawbacks Problems, solutions Mixed 3He/4He clusters Trimers 3 He4HeN Future directions © Dario Bressanini 2 Monte Carlo Methods How to solve a deterministic problem using a Monte Carlo method? Rephrase the problem using a probability distribution A P(R) f (R)dR R N “Measure” A by sampling the probability distribution 1 A N © Dario Bressanini N f (R ) i 1 i R i ~ P(R ) 3 Monte Carlo Methods The points Ri are generated using random numbers This is why the methods are called Monte Carlo methods Metropolis, Ulam, Fermi, Von Neumann (-1945) We introduce noise into the problem!! Our results have error bars... ... Nevertheless it might be a good way to proceed © Dario Bressanini 4 Quantum Mechanics We wish to solve H Y = E Y to high accuracy The solution usually involves computing integrals in high dimensions: 3-30000 The “classic” approach (from 1929): Find approximate Y ( ... but good ...) ... whose integrals are analitically computable (gaussians) Compute the approximate energy chemical accuracy ~ 0.001 hartree ~ 0.027 eV © Dario Bressanini 6 VMC: Variational Monte Carlo Start from the Variational Principle H Y (R ) HY (R)dR E Y (R)dR 0 2 Translate it into Monte Carlo language H P(R ) EL (R )dR HY (R ) EL (R ) Y (R ) © Dario Bressanini P(R ) Y (R ) 2 2 Y (R)dR 7 VMC: Variational Monte Carlo E H P(R) EL (R )dR E is a statistical average of the local energy EL over P(R) 1 E H N N E i 1 L (R i ) R i ~ P(R ) Recipe: take an appropriate trial wave function distribute N points according to P(R) compute the average of the local energy © Dario Bressanini 8 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 Quantum chemistry uses a product of single particle functions. With VMC we can use any function: explicitly correlated wave functions can be used r12 r1 r2 He atom Ye © Dario Bressanini a r1 b r2 g r12 9 The Metropolis Algorithm P(R ) Y (R ) 2 ? How do we sample Use the Metropolis algorithm (M(RT)2 1953) ... ... and a powerful computer The algorithm is a random walk (markov chain) in configuration space 2 Y (R)dR Anyone who consider arithmetical methods of producing random digits is, of course, in a state of sin. John Von Neumann © Dario Bressanini 10 The Metropolis Algorithm move Ri Call the Oracle Rtry reject accept Ri+1=Ri Ri+1=Rtry Compute averages © Dario Bressanini 12 The Metropolis Algorithm The Oracle Y (new) p Y (old ) 2 if p 1 /* accept always */ accept move If 0 p 1 /* accept with probability p */ if p > rnd() accept move else reject move © Dario Bressanini 13 VMC advantages No need to make the single-particle approximation Can use for which no analytical integrals exist Use explicitly correlated wave functions Can satisfy the cusp conditions He atom ground state E19 terms = -2.9037245 a.u. Exact = -2.90372437 a.u. © Dario Bressanini Y ci e ai r1 bi r2 g i r12 i 14 VMC advantages Can compute difficult quantities, e.g. 1 H N H 2 N E L (R i ) i 2 H 2 H2 N H 2 1 N N E 2 L (R i ) i Can compute lower bounds H E0 H © Dario Bressanini 15 VMC advantages Can easily go beyond the Born-Oppenheimer approximation. H2+ molecule ground state Y ci e 2 ai rA bi rB g i rAB f i rAB i E1 term = -0.596235(9)a.u. E10 terms = -0.597136(3)a.u. Exact = -0.597139 a.u. © Dario Bressanini 16 VMC advantages Can work with ANY potential, in ANY number of dimensions. Ps2 molecule (e+e+e-e-) in 2D and 3D Optimization of nonlinear parameters min ( H ) min H 2 2 H 2 Numerically stable Minimum known in advance (0) Can be used for excited states with same symmetry too © Dario Bressanini 17 VMC drawbacks Error bar goes down as N-1/2 It is computationally demanding The optimization of Y becomes difficult as the number of nonlinear parameters increases It depends critically on our skill to invent a good Y There exist exact, automatic ways to get better wave functions. Let the computer do the work ... © Dario Bressanini 19 Diffusion Monte Carlo VMC is a “classical” simulation method 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 Suggested by Fermi in 1945, but implemented only in the 70’s © Dario Bressanini 20 Diffusion equation analogy The time dependent Schrödinger equation is similar to a diffusion equation The diffusion equation can be “solved” by directly simulating the system Y 2 2 i Y VY t 2m C 2 D C kC t Time evolution Diffusion Branch Can we do the same with the Schrödinger equation ? © Dario Bressanini 21 Imaginary Time Sch. Equation The analogy is only formal Y is a complex quantity, while C is real and positive Y (R, t ) e iEnt / n (R) If we let the time t be imaginary, then Y can be real! Y D 2 Y VY Imaginary time Schrödinger equation © Dario Bressanini 22 Y as a concentration Y 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 Y 2 D Y VY Y (R, ) ai i (R)e ( Ei ER ) i Y(R, ) 0 (R)e ( E0 ER ) Ground State © Dario Bressanini 23 Diffusion Monte Carlo SIMULATION: discretize time •Diffusion process Y D 2 Y Y(R, ) e ( R R 0 ) 2 / 4 D •Kinetic process (branching) Y (V (R ) ER )Y Y(R, ) e © Dario Bressanini (V ( R ) ER ) Y(R,0) 24 The DMC algorithm © Dario Bressanini 25 The Fermion Problem 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. Use approximate nodes from a trial Y. Kill the walkers if they cross a node. © Dario Bressanini + - 26 Helium A helium atom is an elementary particle. A weakly interacting hard sphere. Interatomic potential is known more accurately than any other atom. Two isotopes: • 3He (fermion: antisymmetric trial function, spin 1/2) • 4He (boson: symmetric trial function, spin zero) • The interaction potential is the same © Dario Bressanini 27 Helium Clusters They are not rigid, and show large-amplitude motion Normal mode analysis useless “Equilibrium structure” is ill-defined Stochastic methods well suited to study helium clusters, both pure or with impurities VMC, DMC, GFMC, PIMC etc... © Dario Bressanini 28 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 Superfluidity High resolution spectroscopy Low temperature chemistry © Dario Bressanini 30 The Simulations Both VMC and DMC simulations Potential = sum of two-body TTY pair-potential V (R ) VHe He (rij ) i j Three-body terms not important for small clusters Standard Y N (r 4 i j N He 4 He ) (r4 He3 He ) k p5 p2 (r ) (r ) exp( 5 2 p0 ln( r ) p1r ) r r © Dario Bressanini 31 Pure 4He n Clusters 0 DMC VMC Energy cm-1 -1 -2 -3 -4 The quality of the VMC simulations decreases as the cluster increases -5 -6 -7 -8 2 3 4 5 6 7 8 9 10 11 12 n © Dario Bressanini 32 Y for 4He n Clusters Wave function quality decreases as N increases It was optimized to get minimum (H), not minimum <H> Are three- and many-body terms in Y important ? Very difficult to optimize. Unstable process especially for the trimers. Can we improve Y ? © Dario Bressanini 33 Mixed 0.0 (0,2) Energy cm-1 -0.5 3He/4He Clusters (1,2) (2,2) (0,3) (m,n) = 3Hem4Hen (1,3) (0,4) (1,4) Bressanini et. al. J.Chem.Phys. 112, 717 (2000) (0,5) -1.0 (1,5) -1.5 (0,6) -2.0 (1,6) (0,7) -2.5 1 © Dario Bressanini 2 3 4 5 Number of atoms 6 7 34 Helium Clusters: energy (cm-1) N DMC 4HeN VMC 4HeN-13He DMC 4HeN-13He 2 3 4 5 6 7 11 20 -0.00089(1) -0.08784(7) -0.3886(1) -0.9015(3) -1.6077(4) -2.4805(7) -7.286(1) -23.04(1) © Dario Bressanini -0.00666(2) -0.19199(2) -0.57484(6) -1.1505(2) -1.8595(2) -5.975(3) -19.98(1) -0.00984(5) -0.2062(1) -0.6326(2) -1.2626(4) -2.0718(5) -6.679(4) -22.234(9) 35 Helium Clusters: stability 4HeN is destabilized by substituting a 4He with a 3He The structure is only weakly perturbed. Dimers 4He4He Bound Trimers 4He 3 Bound Tetramers 4He 4 Bound © Dario Bressanini 4He3He 3He3He Unbound Unbound 4He 3He 2 4He3He 2 Bound Unbound 4He 3He 3 Bound 4He 3He 2 2 Bound 36 Trimers and Tetramers Stability 4He 3 E = -0.08784(7) cm-1 4He 3He 2 E = -0.00984(5) cm-1 Bonding interaction Non-bonding interaction 4He 4 E = -0.3886(1) cm-1 4He 3He 3 E = -0.2062(1) cm-1 4He 3He 2 2 E = -0.071(1) cm-1 Five out of six unbound pairs! © Dario Bressanini 37 3He/4He Distribution Functions Pair distribution functions 0.016 4 4 He- He 3 4 He- He 0.012 g(r) 3He(4He) 5 0.008 0.004 0.000 0 © Dario Bressanini 10 20 r (u.a.) 30 40 38 3He/4He Distribution Functions Distributions with respect to the center of mass 0.016 3He(4He) 5 0.012 g(r) 4 He 3 He 0.008 c.o.m 0.004 0.000 0 © Dario Bressanini 10 20 r (u.a.) 30 40 39 4He 3He N Distribution Functions in r(4He-4He) r(3He-4He) 0.015 0.3 N=4 N=19 N=5 N=6 N=10 N=3 0.010 N=19 P(r) P(r) 0.2 N=2 0.005 0.1 N=2 0.0 0 0.000 10 20 r (bohr) © Dario Bressanini 30 0 10 20 30 r (bohr) 40 Distribution Functions in 4HeN3He c.o.m. = center of mass r(4He-C.O.M.) r(3He-C.O.M.) 0.015 0.015 N=3 0.010 3He4He N=19 r (r) (bohr-3) r (r) (bohr-3) N=19 2 0.005 0.000 0.010 3He4He 2 N=3 0.005 0.000 0 10 r (bohr) Similar to pure clusters © Dario Bressanini 20 0 10 20 30 r (bohr) Fermion is pushed away 41 What is the shape of 4He Some people say is an equilateral triangle ... ... some say it is linear (almost) ... ... some say it is both. 3 ? Pair distribution function © Dario Bressanini 42 What is the shape of © Dario Bressanini 4He 3 ? 43 The Shape of the Trimers Ne trimer r(Ne-center of mass) He trimer r(4He-center of mass) © Dario Bressanini 44 Ne3 Angular Distributions a b Ne trimer b b a © Dario Bressanini a 45 4He 3 a b Angular Distributions b b a a © Dario Bressanini b a 46 3He4He a 2 Angular Distributions b b b a © Dario Bressanini a 47 Different wave function form p5 p2 ( b rij ) (r ) exp( 5 2 p0 ln( rij ) p1rij a e ) rij rij Spline 0.60 0.60 0.40 0.40 f(r) f(r) 0.20 0.20 0.00 0.00 0.00 2.00 4.00 r 6.00 8.00 Optimize heights © Dario Bressanini 10.00 0.00 2.00 4.00 6.00 r (a.u.) 8.00 10.00 48 Y for 4He 2 Y literature (Rick & Doll) E = -0.00046 cm-1 Optimize Unbound Optimize E (numerically) E = -0.00075 cm-1 Y with Exp() E = -0.00084 cm-1 Y using splines E = -0.00081 cm-1 QMC E = -0.00089(1) cm-1 Numerical E = -0.00091 cm-1 © Dario Bressanini 49 Y for 4He 3 Y literature (Rick & Doll) E = -0.0798 cm-1 Optimize Energy E = -0.0829(4) cm-1 Y with Exp() E = -0.0851(2) cm-1 Y using splines E = -0.0868(2) cm-1 QMC exact E = -0.08784(7) cm-1 three-body terms are not important in Y for the trimer © Dario Bressanini 50 Work in Progress Various impurities embedded in a Helium cluster Y for bigger clusters using splines Optimize the energy instead of the variance of the local energy. Geometric structure of trimers and tetramers © Dario Bressanini 51 Conclusions The substitution of a 4He with a 3He leads to an energetic destabilization. 3He weakly perturbes the 4He atoms distribution. 3He moves on the surface of the cluster. 4He23He bound, 4He3He2 unbound. 4He33He and 4He23He2 bound. QMC gives accurate energies and structural information © Dario Bressanini 52 Acknowledgments Gabriele Morosi Massimo Mella Mose’ Casalegno Giordano Fabbri Matteo Zavaglia © Dario Bressanini 53 A reflection... A new method for calculating properties in nuclei, atoms, molecules, or solids automatically provokes three sorts of negative reactions: 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 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) © Dario Bressanini 54