Transcript Quantum Monte Carlo Simulations of Mixed 3He/4He Clusters

Universita’ degli Studi dell’Insubria
Quantum Monte Carlo
Simulations of Mixed
3He/4He Clusters
Dario Bressanini
[email protected]
http://www.unico.it/~dario
Göttingen 24/05/2002
Overview

Introduction to quantum monte carlo methods


VMC, QMC, advantages and drawbacks
Stability and structure of small 3He/4He mixed
clusters

Trimers
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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
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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
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VMC: Variational Monte Carlo

To solve H  = E  start from the Variational Principle
H

 (R ) H (R)dR


E
  (R)dR
0
2
Translate it into Monte Carlo language
H   P(R ) EL (R )dR
H (R )
EL (R ) 
 (R )
Dario Bressanini – Göttingen 24/05/2002
P(R ) 
 (R )
2
2

 (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 ) 
 (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

 (R)dR
Anyone who consider
arithmetical methods of
producing random digits
is, of course, in a state of sin.
John Von Neumann
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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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The Metropolis Algorithm
The Oracle
  (new) 

p  
  (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
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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
e
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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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First Major VMC Calculations


McMillan VMC calculation of ground state of liquid 4He (1964)
Generalized for fermions by Ceperley, Chester and Kalos
PRB 16, 3081 (1977).
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VMC drawbacks

Error bar goes down as N-1/2

It is computationally demanding

The optimization of  becomes difficult as the
number of nonlinear parameters increases

It depends critically on our skill to invent a good 

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

2 2
i

   V
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

 is a complex quantity, while C is real and positive
 (R, t )  e iEnt /   n (R)

If we let the time t be imaginary, then  can be real!

 D 2   V

Imaginary time Schrödinger equation
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 as a concentration

 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

2
 D   V

 (R, )   ai  i (R)e  ( Ei  ER )
i
(R,  )   0 (R)e ( E0  ER )
Ground State
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Diffusion Monte Carlo
SIMULATION: discretize time
•Diffusion process

 D 2 

(R,  )  e
 ( R  R 0 ) 2 / 4 D
•Kinetic process (branching)

 (V (R )  ER )

(R,  )  e
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 (V ( R )  ER ) 
(R,0)
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The DMC algorithm
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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 . Kill the walkers if
they cross a node.
Dario Bressanini – Göttingen 24/05/2002
+
-
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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


Three-body terms not important for small clusters
Standard  
N
  (r
4
i j
N
He  4 He
)  (r4 He3 He )
k
p5 p2
 (r )   (r )  exp(  5  2  p0 ln( r )  p1r )
r
r
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4He
n
Clusters Stability
4He
n
4He dimer exists
2
 4He3
All clusters
bound
Liquid: stable
bound. Efimov effect?
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Pure
4He
n
Clusters
0
DMC
VMC
Energy cm-1
-1
-2
-3
-4
-5
DMC gives exact
results.
The quality of the
VMC simulations
decreases as the
cluster increases
-6
-7
-8
2 3 4 5 6 7 8 9 10 11 12
n
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3He
n

Clusters Stability
What is the smallest 3Hem stable cluster ?
3He
3He
2
dimer unbound

m
m = ? 20 < m < 35
critically bound
Liquid: stable
Even less is known for mixed clusters.
Is 3Hem4Hen stable ?
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3He4He
n
Clusters Stability
Bonding interaction
Non-bonding interaction
3He4He
3He4He
2
dimer unbound
Trimer bound
3He4He
2
4He
3
Dario Bressanini – Göttingen 24/05/2002
3He4He
n
All clusters
up bound
E = -0.00984(5) cm-1
E = -0.08784(7) cm-1
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Mixed
0.0
(0,2)
Energy cm-1
-0.5
3He4He
n
(1,2) (2,2)
(0,3)
(m,n) = 3Hem4Hen
(1,3)
(0,4)
Bressanini et. al.
J.Chem.Phys.
112, 717 (2000)
(1,4)
(0,5)
-1.0
(1,5)
-1.5
(0,6)
4He
n
-2.0
(1,6)
(0,7)
-2.5
1
2
3
4
5
Number of atoms
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Clusters
6
7
is
destabilized by
substituting a 4He
with a 3He
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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)
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-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)
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3He/4He
Distribution Functions
Pair distribution functions
0.016
3He4He
4
He- He
3
4
He- He
0.012
g(r)
4
5
0.008
0.004
0.000
0
10
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r (u.a.)
30
40
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3He/4He
Distribution Functions
Distributions with respect to the center of mass
0.016
3He4He
0.012
5
g(r)
4
He
3
He
0.008
c.o.m
0.004
0.000
0
10
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r (u.a.)
30
40
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Distribution Functions
c.o.m. = center of mass
(4He-C.O.M.)
(3He-C.O.M.)
0.015
0.015
N=3
0.010
3He4He
N=19
 (r) (bohr-3)
 (r)
-3
(bohr )
N=19
2
0.005
0.000
0.010
3He4He
2
N=3
0.005
0.000
0
10
20
0
r (bohr)
Similar to pure clusters
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10
20
30
r (bohr)
3He
is pushed away
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What is the shape of
4He
3
?
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
We find NO sign of double
peak
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What is the shape of
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4He
3
?
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The Shape of the Trimers
Ne trimer
(Ne-center of mass)
He trimer
(4He-center of mass)
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Ne3 Angular Distributions
a
b
Ne trimer
b
b
a
a
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4He
3
a
Angular Distributions
b
b
b
a
a
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3He4He
a
2
Angular Distributions
b
b
b
a
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a
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3He 4He
2
n

Clusters Stability
Now put two 3He. Singlet state.  is positive everywhere
3He 4He
2
3He 4He
2
2
Trimer unbound
Tetramer bound
5 out of 6 unbound pairs
4He
4
3He4He
3
3He 4He
2
2
Dario Bressanini – Göttingen 24/05/2002
3He 4He
2
n
All clusters
up bound
E = -0.3886(1) cm-1
E = -0.2062(1) cm-1
E = -0.071(1) cm-1
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3He 4He
2
n

Clusters Structure
The two 3He atoms stay mainly on the surface of the
4He cluster
0.01
3He 4He
2
10
0.008
0.006
0.004
0.002
0
5
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10
15
20
25
30
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3He 4He
3
n

Clusters Stability
Adding a third fermionic helium, introduces a nodal
surface into the wave function that destabilizes the system

What is the smallest 3He34Hen stable cluster ?
 3He35
is bound, so 3He34He32 should be bound. n < 32
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The Wave Function
N
N
M
i j
k
l
   (r4 He4 He ) (r4 He3 He )Aˆ  l (r3 He3 He )

The total wave function must be antisymmetric with
respect to the fermionic helium

Consider the doublet spin eigenfunction (two a and one b)

The 4He-4He and 4He-3He functions are symmetric

The 3He-3He part is antisymmetric
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3He 4He
m
n
m = 0,1,2,3 Energies
0
4He
n
4He 3He
n
-4
4He 3He
n
2
4He 3He
3
Energy (cm-1)
n
-8
-12
-16
8
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10
12
4He
n
14
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Work in progress:
4He
n
3He
m
0
1
0
1
2
2
3
4
5
6
7
8
3He 4He
m
n
9 10 11
Bound
Unbound
3
Unknown
4
Maybe
5
Unlikely
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First 3He24Hen
bound
First 3He34Hen
bound
3He 4He
m
10
4He
m = 0,1,2,3
distribution with respect to the center of mass
(4He-C.O.M.)
0.01
0.008
0.006
c.o.m
0.004
0.002
0
5
10
15
20
25
30
0.012
0.01
The 4He distribution is
unchanged with 0,1,2 or 3 3He
0.008
0.006
0.004
0.002
0
5
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10
15
20
25
30
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3He 4He
m
10
3He
m = 0,1,2,3
distribution with respect to the center of mass
(3He-C.O.M.)
0.01
3He4He
10
0.008
0.006
c.o.m
0.004
0.002
0
5
10
15
20
25
30
0.012
0.01
3He 4He
2
10
0.008
3He 4He
3
10
0.01
0.008
0.006
0.006
0.004
0.004
0.002
0.002
0
0
5
10
15
20
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25
30
5
10
15
20
25
30
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3He 4He
m
10
3He
m = 0,1,2,3
distribution with respect to the center of mass
0.012
(3He-C.O.M.)
0.01
0.008
0.006
0.004
0.002
0
5
10
15
20
25
30
One a 3He is pushed inside the cluster, the other two (a, b) outside
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3He 4He
m
10
3He- 3He
m = 0,1,2,3
distributions (3He- 3He)
a outside b outside
on opposite sides
0.015
0.0125
0.01
a outside a inside
0.0075
0.005
b outside a inside
0.0025
0
5
10
15
20
25
30
35
40
The (tentative) picture: two 3He outside
(a, b) and one a inside, pushed away
from the other a 3He
Dario Bressanini – Göttingen 24/05/2002
a
a
b
4He
10
58
3He 4He
3
10
a
a
4He
10
b
Why ?
It is a Nodal Effect.
The wave function is zero if the two a 3He are at the
same distance from the b 3He. 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.
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More Flexible Wave Function
N
N
i j
k
   (r4 He 4 He ) (r4 HeImpurity )
p5 p2
    exp(  5  2  p0 ln( r )  p1r )
r
ri

The standard form is not very flexible

Difficult to optimize

Difficult to reproduce the shell structure
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Different wave function form
(rij )  exp( 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
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10.00
0.00
2.00
4.00
6.00
r (a.u.)
8.00
10.00
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Spline Wave Function
SF6He39
(rHe SF6 )  exp( SPLINE )
Knots of the He-SF6 spline function
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Shell Structure
Standard
(4He-SF6)
Spline
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Work in Progress and Future

Various impurities embedded in a Helium cluster
(suggestions welcome!)

Different functional forms for  (splines)

anisotropy

Analysis of 3He34Hen

What about 3He44Hen and 3He54Hen ?
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Acknowledgments
Gabriele Morosi
Mose’ Casalegno
Giordano Fabbri
Matteo Zavaglia
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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)
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