Transcript Document

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

Mixed 3He/4He clusters simulations
© Dario Bressanini
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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 )
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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
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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
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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
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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
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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
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The Metropolis Algorithm
move
Ri
Call the Oracle
Rtry
reject
accept
Ri+1=Ri
Ri+1=Rtry
Compute
averages
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VMC: Variational Monte Carlo

No need to analytically compute integrals: complete freedom in
the choice of the trial wave function.

Can use explicitly correlated
wave functions

r12
r1
Can satisfy the cusp conditions
r2
He atom
Ye
© Dario Bressanini
a r1 b r2  c r12
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VMC advantages

Can compute lower bounds
H  E0  H  
  H
2

2
 H
2
Can go beyond the Born-Oppenheimer approximation,
with ANY potential, in ANY number of dimensions.
Ps2 molecule (e+e+e-e-) in 2D and 3D
M+m+M-m- as a function of M/m
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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 ...
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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
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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 simulate the Schrödinger equation?
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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
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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
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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)
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The DMC algorithm
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QMC: a simple and useful tool

Yukawa potential


e
Plasma physics, solid-state physics, ...
 r
/r
Stability of screened H, H2+ and H2 as a function
of , without Born-Oppenheimer approximation
(preliminary results)
H unbound
H2+ bound
H bound
 =1.19
© Dario Bressanini
Borromean
H unbound
H2+ unbound
  1.2
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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
+
-
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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
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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
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The Simulations

Both VMC and DMC simulations

Potential = sum of two-body TTY pair-potential
V (R )  VHe He (rij )
i j

Standard Y 
N
  (r
4
i j
© Dario Bressanini
N
He  4 He
)  (r4 He3 He )
k
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Pure
4He
n
Clusters
0
Energy cm-1
-1
DMC
VMC
-2
-3
-4
-5
-6
-7
-8
2 3 4 5 6 7 8 9 10 11 12
n
© Dario Bressanini
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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
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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
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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
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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
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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
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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)
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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)
-3
(bohr )
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
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0
10
20
30
r (bohr)
Fermion is pushed away
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4He
3


Angular Distributions




© Dario Bressanini


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Ne3 Angular Distributions


Ne trimer



© Dario Bressanini

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Helium Cluster Stability

What is the smallest 3Hem stable cluster ?
3He
Dimer: unbound

m
m = ? 20 < m < 35
critically bound
Liquid: stable
Is 3Hem4Hen stable ?
© Dario Bressanini
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Work in progress:
3He 4He
m
n
4He
n
3He
m
0
0
1
2
3
4
5
6
7
Bound
Unbound
1
2
Unknown
3
4
5
Probably
unbound
35
© Dario Bressanini
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Work in Progress

Various impurities embedded in a Helium cluster

Different functional forms for Y (splines)

Stability of
© Dario Bressanini
3He 4He
m
n
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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
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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
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