Introduction
Numerical method
Implementation
Validation and Results
Conclusions and further work
Efficient object orientation for many-body
physics problems
Islen Vallejo Henao
Department of physics
University of Oslo
November 2009
Islen Vallejo Henao
Efficient object orientation for many-body physics problems
Introduction
Numerical method
Implementation
Validation and Results
Conclusions and further work
Physical problem
Mathematical model
Why computational physics?
Advantages
Better understanding of the processes.
Cheaper and less dangerous than
experiments.
Allow experiments impossible practically.
Computation
Need to routinely predict processes.
Theory
Experiment
What are the difficulties?
Numerical method: which one?
efficiency?,...
.
Programming: style? ...
Hardware: Speed, memory (storage)?
Islen Vallejo Henao
Efficient object orientation for many-body physics problems
Introduction
Numerical method
Implementation
Validation and Results
Conclusions and further work
Physical problem
Mathematical model
Methodology in computational physics
Physical problem
Mathematical model
Numerical model
Computer code
Simulation
(hardware)
Figure: Simulation chart flow
Islen Vallejo Henao
Efficient object orientation for many-body physics problems
Introduction
Numerical method
Implementation
Validation and Results
Conclusions and further work
Physical problem
Mathematical model
The many-body problem in quantum mechanics
Goal
Solve the many-electron Schrödinger equation to obtain the ground state energy in
quantum mechanical systems with the so-called closed shell model.
Time-independent Schrödinger eq.
For a system of N-particles:
Physical systems
b = EΨ
HΨ
(1)
Quantum dots.
Hamiltonian: Total energy
b = K(x)
b
b
H
+ V(x)
Atoms: helium and beryllium.
(2)
Islen Vallejo Henao
Efficient object orientation for many-body physics problems
Introduction
Numerical method
Implementation
Validation and Results
Conclusions and further work
Physical problem
Mathematical model
How to define our Hamiltonian...for atoms?
Assumptions
1
Assume non-relativistic energies.
2
Use Born-Oppenheimer approximation:
Nuclei are much more massive than electrons (∼ 2000 : 1 or more).
Electron motions much faster than nuclear motions.
Assume that electrons move instantly compared to nuclei.
3
Go from a molecular to an electronic Schrödinger equation.
Hamiltonian for hydrogen-like atoms.
z
Potential energy
}|
{
N
N
N X
N
2 X
2
2
X
X
h̄
1
Ze
1
e
b =−
H
∇2 −
+
2m i=1 ri 4π0 i=1 |ri − R| 4π0 i=1 j=i+1 |ri − rj |
|
{z
} |
{z
} |
{z
}
Electronic
kinetic energy
Attraction of electron i
by nucleus
Islen Vallejo Henao
Repulsion between
electrons i and j
Efficient object orientation for many-body physics problems
Introduction
Numerical method
Implementation
Validation and Results
Conclusions and further work
Physical problem
Mathematical model
How to define our Hamiltonian...for quantum dots?
Assumptions
1
Assume non-relativistic energies.
2
Confining potential modelled by an harmonic oscillator potential.
3
Use the electronic Schröndinger equation.
Hamiltonian for quantum dots.
b =
H
N
X
i=1
|
N X
N
X
e2
h̄2
1 ∗ 2 2 2
1
2
∇
m
ω
(x
+y
)]
+
+
i
i
i
2m∗
2
4π
|r
−
rj |
r
0
i
|
{z
}
i=1 j=i+1
{z
}
|
{z
}
Confining
[−
Electronic
kinetic energy
potential
Islen Vallejo Henao
Repulsion
Efficient object orientation for many-body physics problems
Introduction
Numerical method
Implementation
Validation and Results
Conclusions and further work
Why numerical methods?
Trial wave function
QVMC algorithm
Optimization of the trial wave function
Table of contents
Islen Vallejo Henao
Efficient object orientation for many-body physics problems
Introduction
Numerical method
Implementation
Validation and Results
Conclusions and further work
Why numerical methods?
Trial wave function
QVMC algorithm
Optimization of the trial wave function
Why is it a hard problem to solve?
P.A.M. Dirac:
The fundamental laws necessary for the mathematical treatment of a large part of physics and
the whole of chemistry are thus completely known, and the difficulty lies only in the fact that
application of these laws leads to equations that are too complex to be solved.
Hamiltonian term
Quantum nature
Kinetic energy
Nuclei-electron
Electron-electron
One-body
One-body
Two-body (hard)
Table: Nature of the terms in the Hamiltonian.
The wave function contains explicit correlations that leads to
non-factoring multi-dimensional integrals.
Islen Vallejo Henao
Efficient object orientation for many-body physics problems
Introduction
Numerical method
Implementation
Validation and Results
Conclusions and further work
Why numerical methods?
Trial wave function
QVMC algorithm
Optimization of the trial wave function
Quantum Variational Monte Carlo
Variational principle
b given a (parametrized) trial
The expectation value of the energy computed for a Hamiltonian H
wave function ΨT is an upper bound to the ground state energy E0 .
R
b =
EVMC = hHi
b T dR
b Ti
Ψ∗T HΨ
hΨT |H|Ψ
R 2
=
≥ E0
hΨT |ΨT i
ΨT dR
How to compute high dimensional integrals efficiently?
Solution: Translate the problem into a Monte Carlo language.
R
Evmc =
T
Ψ∗T ΨT HΨ
dR
Ψ
R 2 T
=
ΨT dR
b
R
hb i
T
Z
|ΨT |2 HΨ
dR
Ψ
R 2 T
= P(R)EL dR
ΨT dR
Sample configurations, R, stochastically.
Islen Vallejo Henao
Efficient object orientation for many-body physics problems
Introduction
Numerical method
Implementation
Validation and Results
Conclusions and further work
Why numerical methods?
Trial wave function
QVMC algorithm
Optimization of the trial wave function
...How to compute high dimensional integrals efficiently? [continued]
Average of the local energy EL over the probability distribution function P(R).
Z
hEi =
P(R)EL dR ≈
M
1 X
EL (Ri ),
M i=1
where M is the number of Monte Carlo samples.
Statistical variance (assuming uncorrelated data set)
var(E) = hE2 i − hEi2
Zero variance principle.
The optimization of the trial wave function has as goal to find the optimal set of
parameters that gives the minimum energy/variance.
BUT: We just need a (parametric) trial wave function...!
Islen Vallejo Henao
Efficient object orientation for many-body physics problems
Introduction
Numerical method
Implementation
Validation and Results
Conclusions and further work
Why numerical methods?
Trial wave function
QVMC algorithm
Optimization of the trial wave function
Trial wave function: ΨT = ΨD ΨJ (Slater-Jastrow)
Slater determinant
˛
˛ φ1 (r1 )
˛
˛ φ1 (r2 )
˛
ΨD ∝ ˛ .
˛ ..
˛
˛
φ1 (rN )
φ2 (r1 )
φ2 (r2 )
..
.
φ2 (rN )
···
···
..
.
···
˛
φN (r1 ) ˛
˛
φN (r2 ) ˛
˛
˛
..
˛
.
˛
˛
φN (rN )
Pauli exclusion principle.
Spin-independent Hamiltonian?
ΨD = |D|↑ |D|↓ .
Electron-nucleus cusp
conditions.
Linear Padé-Jastrow correlation function
0
ΨJ = exp @
X
j<i
aij rij
1 + βij rij
1
Linear correlation.
A
Two-body correlation.
Fullfils electron-electron cusp
conditions.
rij = |rj − ri |
Islen Vallejo Henao
Efficient object orientation for many-body physics problems
Introduction
Numerical method
Implementation
Validation and Results
Conclusions and further work
Why numerical methods?
Trial wave function
QVMC algorithm
Optimization of the trial wave function
Initialize:
Set R, α and
ΨT−α (R)
Suggest a move
Require: nel, nmc, nes, δt, R and Ψα (R).
Ensure: hEα i.
for c = 1 to nmc do
for p = 1 to nel do
cur
xnew
= xcur
p
p + χ + DF(xp )δt
Generate an
uniformly
distributed
variable r
Compute
acceptance ratio
Is
R ≥ r?
no
Reject move:
xnew
= xold
i
i
yes
T
Compute F(xnew ) = ∇Ψ
ΨT
Accept trial move with probability
h
i
ω(xcur ,xnew ) |Ψ(xnew )|2
min 1, ω(xnew ,xcur ) |Ψ(xcur )|2
end for
2
b T
Compute EL = HΨ
= − 12 ∇ΨΨT + V.
ΨT
T
end for
1 Pnmc
Compute hEi = nmc c=1 EL and
σ2 = hEi2 − hE2 i.
Accept move:
xold
= xnew
i
i
Last
move?
yes
Get local
energy EL
no
Last
MC
step?
yes
Collect samples
End
Islen Vallejo Henao
Efficient object orientation for many-body physics problems
Introduction
Numerical method
Implementation
Validation and Results
Conclusions and further work
Why numerical methods?
Trial wave function
QVMC algorithm
Optimization of the trial wave function
Wave function optimization, why and how?
Goal
Find an optimal set of parameters α in ΨT to minimized the estimated energy.
Approaches
1
Minimization of the variance of the local energy.
2
Minimization of the energy.
3
A combination of both.
Examples of optimization algorithms
1
Stochastic gradient approximation (SGA).
2
Quasi-Newton method.
Islen Vallejo Henao
Efficient object orientation for many-body physics problems
Introduction
Numerical method
Implementation
Validation and Results
Conclusions and further work
Why numerical methods?
Trial wave function
QVMC algorithm
Optimization of the trial wave function
Minimization of energy with quasi-Newton method
Objective function and its derivative
Expectation value of the
energy:
»
–
R
Evmc =
|ΨT |2
R
b
HΨ
T
ΨT
dR
Ψ2T dR
Derivative of the expectation value of the
energy: 2
* ∂ΨT +3
* ∂ΨT +
∂E
∂cm
= 2 4 EL
cm
∂cm
ΨTc
−E
m
cm
∂cm
ΨTc
m
5
Strategies for computing the derivative
Direct analytical differentiation:
i
h
∂ΨTc
Q
QN
m
= ∂c∂ |D|↑ |D|↓ N
i=1
i=j+1 J(rij ) = ...?
∂c
m
m
Enjoy it!
Numerical derivative: central differences: Just for ΨSD we have to do
dΨSD
dαm
=
ΨSD (αm +∆αm )−ΨSD (αm −∆αm )
2∆αm
+ O(∆α2m )
(...to be done per parameter).
Islen Vallejo Henao
Efficient object orientation for many-body physics problems
Introduction
Numerical method
Implementation
Validation and Results
Conclusions and further work
Why numerical methods?
Trial wave function
QVMC algorithm
Optimization of the trial wave function
Minimization of energy with quasi-Newton method
Trick to compute the derivative of the energy
Goal:
∂E
∂cm
2*
= 2 4 EL
∂ΨT
cm
∂cm
ΨTc
m
+
*
−E
∂ΨT
cm
∂cm
ΨTc
m
+3
5=2
»fi
fl
fi
fl–
∂ ln ΨT
∂ ln ΨTc
m
EL ∂c cm − E
∂c
m
m
1
Split the trial wave function: ΨTcm = ΨSDcm ΨJcm = ΨSDc ↑ ΨSDc ↓ ΨJcm .
m
m
2
Rewrite the derivatives as:
3
4
∂ ln ΨTc
m
∂cm
=
∂ ln(ΨSD
cm ↑
∂cm
)
+
∂ ln(ΨSD
cm ↓
)
∂cm
+
∂ ln(ΨJc )
m
∂cm
«
„
d
Divide the standard expression dt
(det A) = (det A) tr A−1 dA
by det A to get:
dt
“
” P
P
N
N
d
dA
−1
ln det A(t) = tr A−1 dt = i=1 j=1 Aij Ȧji
dt
This expression inserted in step (??) gives:
∂ ln ΨTcm
∂cm
=(
N X
N
X
D−1
ij Ḋji )↑ + (
i=1 j=1
Islen Vallejo Henao
N X
N
X
i=1 j=1
D−1
ij Ḋji )↓ +
∂ ln(ΨJcm )
∂cm
Efficient object orientation for many-body physics problems
Introduction
Numerical method
Implementation
Validation and Results
Conclusions and further work
Implementation in Python
Implementation in mixed Python/C++
Implementation in C++
Table of contents
Islen Vallejo Henao
Efficient object orientation for many-body physics problems
Introduction
Numerical method
Implementation
Validation and Results
Conclusions and further work
Implementation in Python
Implementation in mixed Python/C++
Implementation in C++
Quick design of a QVMC simulator in Python
Main parts of a QVMC simulator: VMC, Energy, PsiTrial and
Particle.
#Import some packages
...
class VMC():
def init ( self , _dim, _np, _charge, ...,_parameters):
...
particle = Particle(_dim, _np, _step)
psi
= Psi(_np, _dim, _parameters)
energy = Energy(..., particle, psi, _charge)
self . mc = MonteCarlo(psi, _ncycles, particle, energy,...)
def doVariationalLoop(self):
...
for var in xrange(nVar):
self . mc.doMonteCarloImportanceSampling()
self . mc.psi.updateVariationalParameters()
Islen Vallejo Henao
Efficient object orientation for many-body physics problems
Introduction
Numerical method
Implementation
Validation and Results
Conclusions and further work
Implementation in Python
Implementation in mixed Python/C++
Implementation in C++
Easy creation/manipulation of matrices in Python
...
class Particle () :
def init ( self , _dim, _np, _step):
# Initialize matrices for configuration space
r_old = zeros((_np, _dim))
...
def acceptMove(self, i):
self . r_old[i,0:dim] = self.r_new[i,0:dim]
...
def setTrialPositionsBF ( self ) :
dt = self . step
r_old = dt*random.uniform(−0.5,0.5,size=np*dim)
r_old.reshape(np,dim)
...
Islen Vallejo Henao
Efficient object orientation for many-body physics problems
Introduction
Numerical method
Implementation
Validation and Results
Conclusions and further work
Implementation in Python
Implementation in mixed Python/C++
Implementation in C++
Calling the code
from VMC import *
# Set parameters of simulation
nsd = 3
# Number of spatial dimensions
nVar= 10
# Number of variations (optimization method)
nmc = 10000 # Number of monte Carlo cycles
nel = 2
# Number of electrons
Z = 2.0
# Nuclear charge
...
vmc = VMC(nsd, nel, Z,... , nmc, dt, nVar, varPar)
vmc.doVariationalLoop()
vmc.mc.energy.printResults()
Islen Vallejo Henao
Efficient object orientation for many-body physics problems
Introduction
Numerical method
Implementation
Validation and Results
Conclusions and further work
Implementation in Python
Implementation in mixed Python/C++
Implementation in C++
How well does Python perform?
High level language?
Clear and compact syntax.
Support all the major program styles.
Runs on all major platforms.
Free, open source (e.g. vs Matlab).
Comprehensive standard library.
Huge collection of free modules on the web.
Good support for scientific computing.
However...
How well performs Python with respect to C++?
Islen Vallejo Henao
Efficient object orientation for many-body physics problems
Introduction
Numerical method
Implementation
Validation and Results
Conclusions and further work
Implementation in Python
Implementation in mixed Python/C++
Implementation in C++
Comparing Python to C++ computing VMC
2.5
180
160
2
Execution time, s
Execution time, s
140
120
100
80
60
40
1.5
1
0.5
20
0
20000
40000
60000
80000
Monte Carlo cycles
100000
0
20000
40000
60000
80000
Monte Carlo cycles
100000
Figure: Execution time as a function of the number of Monte Carlo cycles for a
Python (blue) and C++ (green) simulators implementing the QVMC method with
importance sampling for the He atom.
CONCLUSION: Python is SLOW, except when it is not running!
Islen Vallejo Henao
Efficient object orientation for many-body physics problems
Introduction
Numerical method
Implementation
Validation and Results
Conclusions and further work
Implementation in Python
Implementation in mixed Python/C++
Implementation in C++
Detecting bottlenecks in Python
# calls
Total time
Cum. time
1
1
1910014
100001
1910014
50000
1910014
50000
6180049
900002
300010
50000
9.153
0.000
23.794
12.473
58.956
0.864
57.476
8.153
21.489
4.968
2.072
2.272
207.061
207.061
159.910
117.223
71.704
66.208
64.412
62.548
21.489
4.968
2.072
2.796
Class:function
MonteCarlo.py:(doMCISampling)
VMC.py:(doVariationalLoop)
Psi.py:(getPsiTrial)
Psi.py:(getQuantumForce)
Psi.py:(getModelWaveFunctionHe)
Energy.py:(getLocalEnergy)
Psi.py:(getCorrelationFactor)
Energy.py:(getKineticEnergy)
:0(sqrt)
:0(copy)
:0(zeros)
Energy.py:(getPotentialEnergy)
Table: Profile of a QVMC simulator with importance sampling for the He atom
implemented in Python. The run was done with 50000 Monte Carlo cycles.
Islen Vallejo Henao
Efficient object orientation for many-body physics problems
Introduction
Numerical method
Implementation
Validation and Results
Conclusions and further work
Implementation in Python
Implementation in mixed Python/C++
Implementation in C++
Can ”Python” do better?: Extending Python with C++
sys.path.insert(0, './extensions') # Set the path to the extensions
import ext_QVMC
# Extension module
class Vmc():
def init ( self , _Parameters):
# Create an object of the ' conversion class '
self . convert = ext_QVMC.Convert()
# Get the paramters of the currrent simulation
simParameters = _Parameters.getParameters()
alpha = simParameters[6]
self . varpar = array([alpha, beta])
# Convert a Python array to a MyArray object
self . v_p = self . convert.py2my_copy(self.varpar)
# Create objects to be extended in C++
self . psi = ext_QVMC.Psi(self.v_p, self.nel, self.nsd) ...
Islen Vallejo Henao
Efficient object orientation for many-body physics problems
Introduction
Numerical method
Implementation
Validation and Results
Conclusions and further work
Implementation in Python
Implementation in mixed Python/C++
Implementation in C++
Calling code for the mixed Python/C++ simulator
from SimParameters import * # Class encapsulating the \
# parameters of simulation
from Vmc import *
# Import the simulator box.
# Create an object containing the
# parameters of the current simulation
simpar = SimParameters('Be.data')
# Create a Variational Monte Carlo simulation
vmc = Vmc(simpar)
vmc.doVariationalLoop()
vmc.energy.doStatistics(”resultsBe.data”, 1.0)
Islen Vallejo Henao
Efficient object orientation for many-body physics problems
Introduction
Numerical method
Implementation
Validation and Results
Conclusions and further work
Implementation in Python
Implementation in mixed Python/C++
Implementation in C++
Comparing Python to C++
30
Execution time, s
25
20
Mixed -O1
C++ -O1
Mixed -O2
C++ -O2
Mixed -O3
C++ -O3
15
10
5
0
500000
1e+06
1.5e+06
Monte Carlo cycles
2e+06
Figure: Execution time as a function of the number of Monte Carlo cycles for mixed
Python/C++ and pure C++ simulators implementing the QVMC method with
importance sampling for He atom.
Islen Vallejo Henao
Efficient object orientation for many-body physics problems
Introduction
Numerical method
Implementation
Validation and Results
Conclusions and further work
Implementation in Python
Implementation in mixed Python/C++
Implementation in C++
5 steps for structuring a simulator in OOP
Defining classes
1
From the mathematical model and the algorithms, identify
the main components of the problem (classes).
Vmc simulation: VmcSimulation.h
Parameters in simulation: Parameters.h
Monte Carlo method: MonteCarlo.h
Energy: Energy.h
Potential:Potential.h
Trial wave function: PsiTrial ...
2
Identify the mathematical and algorithmic parts
changing from problem to problem (superclasses).
Potential:Potential.h
Trial wave function: PsiTrial
Monte Carlo method: MonteCarlo.h ...
Islen Vallejo Henao
Efficient object orientation for many-body physics problems
Introduction
Numerical method
Implementation
Validation and Results
Conclusions and further work
Implementation in Python
Implementation in mixed Python/C++
Implementation in C++
Enhance flexibility
3
List what each of these classes should do, IN GENERAL
(member functions), e.g., for PsiTrial.h:
Compute the acceptance ratio: getPsiPsiRatio().
Compute the quantum force: getQuantumForce(). ...
#include "SomeClass.h"
class PsiTrial{
public:
virtual double getAcceptanceRatio()=0;
virtual MyArray<double> getQuantumForce()=0;
virtual getLapPsiRatio()=0;
}
Islen Vallejo Henao
Efficient object orientation for many-body physics problems
Introduction
Numerical method
Implementation
Validation and Results
Conclusions and further work
Implementation in Python
Implementation in mixed Python/C++
Implementation in C++
Inheritance (specializing behaviours): “is a”-relationship
4
Be specific with the behaviour , i.e., create subclases by
finding ”is a” -relationships.(subclasses) For example:
Slater-Jastrow (SlaterJastrow.h) is a (kind of) trial wave function
(PsiTrial.h).
Slater alone (SlaterAlone.h) is a (kind of) trial wave function
(PsiTrial.h).
...
Coulomb one-body (OneBodyCoulomb.h) is a (kind of) potential
(Potential.h).
PsiTrial *s = new SlaterJastrow(sd, pj);
or
PsiTrial *sj = new SlaterAlone(sd);
Islen Vallejo Henao
Efficient object orientation for many-body physics problems
Introduction
Numerical method
Implementation
Validation and Results
Conclusions and further work
Implementation in Python
Implementation in mixed Python/C++
Implementation in C++
Composition: has a-relationship
5
Connect the whole structure with relatonships of type has
a (composition).
ΨT = |D|↑ |D|↓ |J(rij)
For example: Slater-Jastrow wave function (SlaterJastrow.h)
has a Slater determinant (SlaterDeterminant.h) and a
correlation function (CorrelationFnc.h).
#include "SlaterDeterminant.h"
#include "CorrelationFnc.h"
class SlaterJastrow: public PsiTrial{
private:
SlaterDeterminant *slater;
CorrelationFnc *correlation;
public:
...
};
Islen Vallejo Henao
Efficient object orientation for many-body physics problems
Introduction
Numerical method
Implementation
Validation and Results
Conclusions and further work
Validation
Optimization of the trial wave function
Extrapolation to zero dt
Influence of the # MC cycles
Table of contents
Islen Vallejo Henao
Efficient object orientation for many-body physics problems
Introduction
Numerical method
Implementation
Validation and Results
Conclusions and further work
Validation
Optimization of the trial wave function
Extrapolation to zero dt
Influence of the # MC cycles
Analytical GS for atoms (without correlation)
−3
−15.2
−16
Energy, ⟨ E ⟩ (au)
Energy, ⟨ E ⟩ (au)
−3.2
−3.4
−3.6
−3.8
−17.6
−18.4
−19.2
−4
−4.2
1
−16.8
1.5
2
α
2.5
3
−20
2
3
4
α
5
6
Figure: Dependence of the energy on α for He(left) and Be(right)
atoms.
Islen Vallejo Henao
Efficient object orientation for many-body physics problems
Introduction
Numerical method
Implementation
Validation and Results
Conclusions and further work
Validation
Optimization of the trial wave function
Extrapolation to zero dt
Influence of the # MC cycles
Analytical GS for quantum dots (without correlation)
3
14
Energy, ⟨ E ⟩ (au)
Energy, ⟨ E ⟩ (au)
2.75
2.5
2.25
12.4
11.6
10.8
10
2
0.4
13.2
0.6
0.8
1
1.2
α
1.4
1.6
1.8
2
0.4
0.6
0.8
1
1.2
α
1.4
1.6
1.8
2
Figure: Dependence of the energy on α for a 2D quantum dot with
2-and 6-electron without interactions.
Islen Vallejo Henao
Efficient object orientation for many-body physics problems
Introduction
Numerical method
Implementation
Validation and Results
Conclusions and further work
Validation
Optimization of the trial wave function
Extrapolation to zero dt
Influence of the # MC cycles
Graphical estimation of the GS energy for atoms
−14
Mean energy, E (au)
Mean energy, E (au)
−1.5
−2
−2.5
−3
1
−14.1
−14.2
−14.3
−14.4
−14.5
−14.6
0.8
3
2.5
0.5
2
4.4
0.6
0
1
α
4
0.2
1.5
β
4.2
0.4
β
3.8
0
3.6
α
Figure: Energy as a function of α and β for He (left) and Be (right) atoms.
Experimental set up: 107 Monte Carlo cycles, 10% equilibration steps and dt = 0.01.
Islen Vallejo Henao
Efficient object orientation for many-body physics problems
Introduction
Numerical method
Implementation
Validation and Results
Conclusions and further work
Validation
Optimization of the trial wave function
Extrapolation to zero dt
Influence of the # MC cycles
Graphical estimation of the GS energy for QD
Figure: Energy as a function of α and β for 2D quantum dots with two (left) and six
electrons (right). Experimental set up: 107 Monte Carlo cycles, 10% equilibration steps
and dt = 0.01.
Islen Vallejo Henao
Efficient object orientation for many-body physics problems
Introduction
Numerical method
Implementation
Validation and Results
Conclusions and further work
Validation
Optimization of the trial wave function
Extrapolation to zero dt
Influence of the # MC cycles
Graphical estimation of the GS energy for helium
−1.4
−1.6
−2.886
Mean energy, E (au)
Mean energy, E (au)
−1.8
−2
−2.2
−2.4
−2.888
−2.6
−2.89
−2.8
−3
1
1.5
2
α
2.5
3
0.18
0.26
0.34
β
0.42
0.5
0.58
Figure: Dependence of the energy on α (left) along the value of β
(right) that gives the minimum variational energy for a He atom.
Islen Vallejo Henao
Efficient object orientation for many-body physics problems
Introduction
Numerical method
Implementation
Validation and Results
Conclusions and further work
Validation
Optimization of the trial wave function
Extrapolation to zero dt
Influence of the # MC cycles
Graphical estimation of the GS energy for beryllium
−14.485
−14
−14.1
Mean energy, E (au)
Mean energy, E (au)
−14.49
−14.2
−14.3
−14.4
−14.495
−14.5
−14.5
−14.6
3.6
3.7
3.8
3.9
α
4
4.1
4.2
0.06
0.07
0.08
0.09
β
0.1
0.11
0.12
Figure: Dependence of the energy on α (left) along the value of β
(right) that gives the minimum variational energy for a Be atom.
Islen Vallejo Henao
Efficient object orientation for many-body physics problems
Introduction
Numerical method
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Validation and Results
Conclusions and further work
Validation
Optimization of the trial wave function
Extrapolation to zero dt
Influence of the # MC cycles
Graphical estimation of the GS energy for 2-e− QD
3.6
3.25
3.2
Mean energy, E (au)
Mean energy, E (au)
3.5
3.4
3.3
3.2
3.15
3.1
3.05
3.1
3
3
0.8
0.9
1
1.1
α
1.2
1.3
0
0.1
0.2
0.3
β
0.4
0.5
Figure: Dependence of the energy on α (left) along the value of β
(right) that gives the minimum variational energy for a 2D quantum
dot with two electrons.
Islen Vallejo Henao
Efficient object orientation for many-body physics problems
Introduction
Numerical method
Implementation
Validation and Results
Conclusions and further work
Validation
Optimization of the trial wave function
Extrapolation to zero dt
Influence of the # MC cycles
Graphical estimation of the GS energy for 6-e− QD
23
20.45
22
20.4
Mean energy, E (au)
Mean energy, E (au)
22.5
21.5
21
20.35
20.3
20.25
20.5
20.2
20
0.5
1
α
1.5
20.15
0.35
0.4
0.45
0.5
0.55
β
0.6
0.65
0.7
0.75
Figure: Dependence of the energy on α (left) along the value of β
(right) that gives the minimum variational energy for a 2D quantum
dot with six electrons.
Islen Vallejo Henao
Efficient object orientation for many-body physics problems
Introduction
Numerical method
Implementation
Validation and Results
Conclusions and further work
Validation
Optimization of the trial wave function
Extrapolation to zero dt
Influence of the # MC cycles
Summary of the results with graphical estimation
System
He
Be
2DQDot2e (ω = 1.0)
2DQDot6e (ω = 1.0)
αoptimal
βoptimal
Energy, hEi (au)
1.85
3.96
0.98
0.9
0.35
0.09
0.41
0.6
−2.8901 ± 1.0 × 10−4
−14.5043 ± 4.0 × 10−4
3.0003 ± 1.2 × 10−5
20.19 ± 1.2 × 10−4
Table: Ground state energy and corresponding variational
parameters α and β estimated graphically. (2DQDot2e = 2D quantum
dot with two electrons).
Islen Vallejo Henao
Efficient object orientation for many-body physics problems
Introduction
Numerical method
Implementation
Validation and Results
Conclusions and further work
Validation
Optimization of the trial wave function
Extrapolation to zero dt
Influence of the # MC cycles
Optimization with Quasi-Newton method
System
He
Be
α0
β0
αopt
βopt
1.564
3.85
0.134
0.08
1.838
3.983
0.370
0.104
Energy, (au)
-2.891
-14.503
Table: Optimized variational parameters and corresponding energy
minimization using a quasi-Newton method. Simulation parameters:
107 Monte Carlo cycles with 10 % equilibration steps, dt = 0.01.
Islen Vallejo Henao
Efficient object orientation for many-body physics problems
Introduction
Numerical method
Implementation
Validation and Results
Conclusions and further work
Validation
Optimization of the trial wave function
Extrapolation to zero dt
Influence of the # MC cycles
Convergence to the optimal parameters for helium
2.3
0.8
α0 = 1.564,
αopt = 1.8378900818459
2.2
Variational parameter, β
Variational parameter, α
2.1
2
1.9
1.8
1.7
0.6
0.5
0.4
0.3
0.2
1.6
1.5
0
β0 = 0.1342,
βopt = 0.3703844012544
0.7
2
4
6
8
Number of iterations
10
12
0.1
0
2
4
6
8
Number of iterations
10
12
Figure: Evolution of α (left) and β (right) with the number of
iterations for He atom. Experimental set up: 107 Monte Carlo cycles,
10% equilibration steps in four nodes.
Islen Vallejo Henao
Efficient object orientation for many-body physics problems
Introduction
Numerical method
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Validation and Results
Conclusions and further work
Validation
Optimization of the trial wave function
Extrapolation to zero dt
Influence of the # MC cycles
Convergence to the minimal energy for helium
−2.65
α0 = 1.564,
β0 = 0.1342,
α = 1.8378900818459,
opt
βopt = 0.3703844012544,
Emin = −2.8908185137222
Mean energy, E (au)
−2.7
−2.75
−2.8
−2.85
−2.9
0
2
4
6
8
Number of iterations
10
12
Figure: Evolution of the energy with the number of iterations for He
atom. Experimental set up: 107 Monte Carlo cycles and 10%
equilibration steps in four nodes.
Islen Vallejo Henao
Efficient object orientation for many-body physics problems
Introduction
Numerical method
Implementation
Validation and Results
Conclusions and further work
Validation
Optimization of the trial wave function
Extrapolation to zero dt
Influence of the # MC cycles
Convergence to the optimized parameters for Be
4.25
0.8
α0 = 3.85,
α = 3.9826281304910
opt
4.2
0.6
Variational parameter, β
Variational parameter, α
4.15
4.1
4.05
4
0.5
0.4
0.3
3.95
0.2
3.9
0.1
3.85
0
β0 = 0.08,
βopt = 0.1039459074823
0.7
5
10
15
Number of iterations
20
25
0
0
5
10
15
Number of iterations
20
25
Figure: Evolution of α (left) and β (right) with the number of
iterations for Be atom. Experimental set up: 107 Monte Carlo cycles
and 10% equilibration steps in four nodes.
Islen Vallejo Henao
Efficient object orientation for many-body physics problems
Introduction
Numerical method
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Validation and Results
Conclusions and further work
Validation
Optimization of the trial wave function
Extrapolation to zero dt
Influence of the # MC cycles
Convergence to the minimal energy for Be
−13.9
α0 = 3.85,
β0 = 0.08,
α = 3.9826281304910,
opt
βopt = 0.1039459074823,
Emin = −14.5034420665939
Mean energy, E (au)
−14
−14.1
−14.2
−14.3
−14.4
−14.5
−14.6
0
5
10
15
Number of iterations
20
25
Figure: Evolution of the energy with the number of iterations for Be
atom. Experimental set up: 107 Monte Carlo cycles and 10%
equilibration steps in four nodes.
Islen Vallejo Henao
Efficient object orientation for many-body physics problems
Introduction
Numerical method
Implementation
Validation and Results
Conclusions and further work
Validation
Optimization of the trial wave function
Extrapolation to zero dt
Influence of the # MC cycles
Extrapolation of energy to zero dt for He
Time step
Energy, (au)
Error
Accepted moves, (%)
0.002
0.003
0.004
0.005
0.006
0.007
0.008
-2.891412637854
-2.890797649433
-2.890386198895
-2.890078440930
-2.890463490951
-2.890100432462
-2.889659923905
5.5e-4
4.5e-4
4.0e-4
3.5e-4
3.2e-4
2.8e-4
2.7e-4
99.97
99.94
99.91
99.88
99.84
99.81
99.77
Table: Energy computed for the He atom and the error associated as a
function of the time step. Experimental set up: 107 Monte Carlo
cycles with 10 % equilibration steps, α = 1.8379 and β = 0.3704.
Islen Vallejo Henao
Efficient object orientation for many-body physics problems
Introduction
Numerical method
Implementation
Validation and Results
Conclusions and further work
Validation
Optimization of the trial wave function
Extrapolation to zero dt
Influence of the # MC cycles
Extrapolation of energy to zero dt for He
−4
−2.8885
3
x 10
−2.889
2
−2.89
Error
Mean energy, E (au)
2.5
−2.8895
−2.8905
1.5
−2.891
1
−2.8915
−2.892
1
2
3
4
5
6
Time step, dt
7
8
9
−3
x 10
0.5
0
0.5
1
Number of blocks
1.5
2
4
x 10
Figure: Extrapolation (left) of the energy to zero-dt and blocking
(right) at dt = 0.01 where the energy −2.89039 ± 2.5 × 10−4 au.
Parameters: α = 1.8379, β = 0.3704.
Islen Vallejo Henao
Efficient object orientation for many-body physics problems
Introduction
Numerical method
Implementation
Validation and Results
Conclusions and further work
Validation
Optimization of the trial wave function
Extrapolation to zero dt
Influence of the # MC cycles
Extrapolation of energy to zero dt for Be
Time step
Energy, (au)
Error
Accepted moves, (%)
0.004
0.005
0.006
0.007
0.008
0.009
0.01
-14.50321303316
-14.50266236227
-14.50136820967
-14.50314292468
-14.50206184582
-14.50164368104
-14.50145748870
1.3e-3
1.2e-3
1.1e-3
1.0e-3
9.5e-4
8.5e-4
8.0e-4
99.61
99.47
99.32
99.17
99.01
98.85
98.68
Table: Energy computed for the Be atom and the error associated as a
function of the time step. Parameters: α = 3.983 and β = 0.103.
Islen Vallejo Henao
Efficient object orientation for many-body physics problems
Introduction
Numerical method
Implementation
Validation and Results
Conclusions and further work
Validation
Optimization of the trial wave function
Extrapolation to zero dt
Influence of the # MC cycles
Extrapolation of energy to zero dt for Be
−4
−14.499
10
x 10
9
−14.5
7
Error
Mean energy, E(au)
8
−14.501
−14.502
6
5
−14.503
4
−14.504
3
−14.505
0
0.002
0.004
0.006
Time step, dt
0.008
0.01
2
0
0.5
1
Number of blocks
1.5
2
4
x 10
Figure: Extrapolation (left) of the energy to dt−zero and blocking
(right) at dt = 0.01 for the Be atom where the energy
E = −14.50146 ± 8.5 × 10−4 au. Parameters: α = 3.983 and β = 0.103.
Islen Vallejo Henao
Efficient object orientation for many-body physics problems
Introduction
Numerical method
Implementation
Validation and Results
Conclusions and further work
Validation
Optimization of the trial wave function
Extrapolation to zero dt
Influence of the # MC cycles
Extrapolation of energy to zero dt for 2-e− electron QD
Time step
Energy, (au)
Error
Accepted moves, (%)
0.01
0.02
0.03
0.04
0.05
0.06
0.07
3.000340072477
3.000357900850
3.000364180564
3.000384908560
3.000370330692
3.000380980039
3.000402836533
4.5e-5
3.2e-5
2.6e-5
2.2e-5
2.0e-5
1.8e-5
1.7e-5
99.95
99.87
99.77
99.65
99.52
99.37
99.21
Table: Dependence of the energy on dt for a 2D quantum dot with
two electrons. Parameters: α = 0.99044, β = 0.39994 taken from
Albrigtsen(2009).
Islen Vallejo Henao
Efficient object orientation for many-body physics problems
Introduction
Numerical method
Implementation
Validation and Results
Conclusions and further work
Validation
Optimization of the trial wave function
Extrapolation to zero dt
Influence of the # MC cycles
Extrapolation of energy to zero dt for 2-e− electron QD
−5
2
3,00044
x 10
1.8
1.6
3,0004
1.4
3,00038
Error
Mean energy, E (au)
3,00042
3,00036
1.2
3,00034
1
3,00032
0.8
3,0003
0
0.01
0.02
0.03
0.04
0.05
Time step, dt
0.06
0.07
0.08
0.6
0
0.5
1
Number of blocks
1.5
2
4
x 10
Figure: Extrapolation (left) of the energy to zero-dt for a 2D quantum dot with 2e−
(left) and blocking (right)(Emin = 3.000381 ± 1.8 × 10−5 au) at dt = 0.06. Parameters:
α = 0.99044 and β = 0.39994.
Islen Vallejo Henao
Efficient object orientation for many-body physics problems
Introduction
Numerical method
Implementation
Validation and Results
Conclusions and further work
Validation
Optimization of the trial wave function
Extrapolation to zero dt
Influence of the # MC cycles
Extrapolation of energy to zero dt for 6-e− electron QD
Time step
Energy, (au)
Error
Accepted moves, (%)
0.01
0.02
0.03
0.04
0.05
0.06
0.07
20.19048030567
20.19059799459
20.19045049792
20.19069748408
20.19066469178
20.19064491561
20.19078449010
4.0e-4
2.8e-4
2.4e-4
2.0e-4
1.8e-4
1.7e-4
1.6e-4
99.90
99.75
99.55
99.34
99.10
98.85
98.58
Table: Energy as a function of the time step for a 2D-quantum dot
with six electrons. Parameters: ω = 1.0, α = 0.926273 and
β = 0.561221 taken from Albrigtsen (2009).
Islen Vallejo Henao
Efficient object orientation for many-body physics problems
Introduction
Numerical method
Implementation
Validation and Results
Conclusions and further work
Validation
Optimization of the trial wave function
Extrapolation to zero dt
Influence of the # MC cycles
Extrapolation of energy to zero dt for 6-e− electron QD
−4
20.1912
1.8
1.6
20.191
1.4
20.1908
1.2
Error
Mean energy, E (au)
x 10
20.1906
1
20.1904
0.8
20.1902
20.19
0
0.6
0.01
0.02
0.03
0.04
0.05
Time step, dt
0.06
0.07
0.08
0.4
0
0.5
1
Number of blocks
1.5
2
4
x 10
Figure: Extrapolation (left) of the energy to zero−dt 2D quantum dot with 6e− (left)
and blocking (right)(Emin = 20.190645 ± 1.7 × 10−4 au) at dt = 0.06 . Parameters of
simulation: 107 Monte Carlo steps with 10 % equilibration steps, α = 0.926273 and
β = 0.561221 running with four nodes.
Islen Vallejo Henao
Efficient object orientation for many-body physics problems
Introduction
Numerical method
Implementation
Validation and Results
Conclusions and further work
Validation
Optimization of the trial wave function
Extrapolation to zero dt
Influence of the # MC cycles
Extrapolated energy to zero dt for atoms and QD
System
He
Be
2DQDot2e
2DQDot6e
Evmc , (au),
EHF
Eexact
-2.8913
-14.5039
3.0003
20.1904
-2.862
-14.573
-
-2.904
-14.667
3.000
-
Table: Energies estimated using zero-dt extrapolation.
Islen Vallejo Henao
Efficient object orientation for many-body physics problems
Introduction
Numerical method
Implementation
Validation and Results
Conclusions and further work
Validation
Optimization of the trial wave function
Extrapolation to zero dt
Influence of the # MC cycles
Influence of the # MC cycles on the statistical error
−3
8
x 10
7
6
Error
5
4
3
2
1
0
0
1
2
3
4
5
Monte Carlo cycles
6
7
8
7
x 10
Figure: Error in the energy as a function of the number of Monte
Carlo cycles for a Be atom. Parameters:dt = 0.01, α = 3.981,
β = 0.09271, 10 % equilibration steps and four processors per run.
Islen Vallejo Henao
Efficient object orientation for many-body physics problems
Introduction
Numerical method
Implementation
Validation and Results
Conclusions and further work
Main results
Discussion
Future work and perspectives
Table of contents
Islen Vallejo Henao
Efficient object orientation for many-body physics problems
Introduction
Numerical method
Implementation
Validation and Results
Conclusions and further work
Main results
Discussion
Future work and perspectives
Conclusion: main results
What has be done?
Prototyping of a QVMC simulator with Python.
Extension of the Python simulator with C++.
Development of a applied to atoms and quantum dots
within the so-called closed shell model.
Proposal of an efficient algorithm to compute the
analytical derivative of the energy in QVMC.
Islen Vallejo Henao
Efficient object orientation for many-body physics problems
Introduction
Numerical method
Implementation
Validation and Results
Conclusions and further work
Main results
Discussion
Future work and perspectives
Conclusion: discussion
Some reflections
Potential use of Python in scientific computing
(prototyping, testing and limitations) vs C++.
Parametric optimization of ΨT (α, β).
Graphical method (manual optimization of parameters).
Quasi-Newton method (automatic optimization of
parameters).
Choice of dt in QVMC.
Expensive extrapolation of the energy to zero − dt.
Importance of locating the region with quasilinear energy − dt plot.
Islen Vallejo Henao
Efficient object orientation for many-body physics problems
Introduction
Numerical method
Implementation
Validation and Results
Conclusions and further work
Main results
Discussion
Future work and perspectives
Conclusion: Future work and perspectives
Improve SWIG (Simplified Wrapper Interface Generator) to enhance
compatibility Python/C++.
Make Python faster (Google-Python at SIMULA).
Try alternative algorithms.
Updating the inverse of the Slater matrix.
Evaluation of quantum force and kinetic energy.
Sampling of core and valence electrons with different δt.
Methods adapted for handling stochastic noise:
Stochastic Newton-based methods.
Stochastic gradient approximation (SGA).
Improve flexibility using functors and templates.
Islen Vallejo Henao
Efficient object orientation for many-body physics problems
Introduction
Numerical method
Implementation
Validation and Results
Conclusions and further work
Main results
Discussion
Future work and perspectives
Acknowledgements
Morten Hjorth-Jensen.
Rune Algbritsen.
Johannes Rekkedal.
Patrick Merlot.
Magnus Pedersen.
Audience.
Islen Vallejo Henao
Efficient object orientation for many-body physics problems
Introduction
Numerical method
Implementation
Validation and Results
Conclusions and further work
Main results
Discussion
Future work and perspectives
Figure: Patrick and Johannes ”drinking coffee”.
Islen Vallejo Henao
Efficient object orientation for many-body physics problems
Introduction
Numerical method
Implementation
Validation and Results
Conclusions and further work
Main results
Discussion
Future work and perspectives
Cusp conditions
For the exact solution of the Schrödinger equation the local energy EL = HΨ
Ψ
must be constant (finite).
The electronic Hamiltonian has singularities at points where two particles
collidate, i.e., ri → 0 and/or rij → 0. For helium atom:
b
b = − 1 ∇2 − 1 ∇2 − 2 − 2 + 1
H
2 1 2 2 r1
r2
r12
The infinite potential terms must be cancelled by infinite kinetic terms.
The first derivatives must be discontinuous at the singularities.
Nuclear cusp conditions
“
”
1
∂Ψ
= −Z, l = 0.
ΨSD
∂ri r =0
“
”i
1
∂Ψ
Z
= − l+1
, l > 0.
Ψ
∂r
SD
i
ri =0
Islen Vallejo Henao
Electronic cusp conditions (2D and 3D)
“
”
1 ulike-spin
∂ΨC
1
=
1
ΨC
∂rij
like-spin.
rij =0
3
1
“
”
ulike-spin
∂Ψ
1
C
= 21
ΨC
∂rij
like-spin.
rij =0
4
Efficient object orientation for many-body physics problems
Introduction
Numerical method
Implementation
Validation and Results
Conclusions and further work
Main results
Discussion
Future work and perspectives
Implementation of a QVMC simulator in C++
QVMC algorithm with drift diffusion
Requirements
1
Numerical efficiency.
2
Flexibility.
Extensibility.
Independent of the nsd.
3
Readability.
How?
1
Alternative algorithms.
2
Object oriented programming.
Classes, inheritance,
templates, etc.
3
Operator overloading, functors,
etc.
Islen Vallejo Henao
Require: nel, nmc, nes, δt, R and Ψα (R).
Ensure: hEα i.
for c = 1 to nmc do
for p = 1 to nel do
cur
xnew
= xcur
p
p + χ + DF(xp )δt
T
Compute F(xnew ) = 2.0 ∇Ψ
ΨT
Accept trial move with probability
h
i
ω(xcur ,xnew ) |Ψ(xnew )|2
min 1, ω(xnew ,xcur ) |Ψ(xcur )|2
end for
Compute
2
T
= − 12 ∇ΨΨT + V.
EL = HΨ
ΨT
T
end for
P
nmc
1
Compute hEi = nmc
c=1 EL and
σ2 = hEi2 − hE2 i.
b
Efficient object orientation for many-body physics problems
Introduction
Numerical method
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Validation and Results
Conclusions and further work
Main results
Discussion
Future work and perspectives
Algorithm analysis and performance improvement
No optimization
Ψnew
T
Ψcur
T
Optimization
new
=
|D|↑
new
|D|↓
cur
cur
Ψnew
C
Ψcur
C
|D|↑ |D|↓
|
{z
} | {z }
RC
RSD
∇Ψnew
T
Ψnew
T
∇2 Ψnew
T
Ψnew
T
=
.
∇(|D|↑ |D|↓ ΨC )
|D|↑ |D|↓ ΨC
2
=
∇ (|D|↑ |D|↓ ΨC )
|D|↑ |D|↓ ΨC
R ≡ RSD
PRC
new )D−1 (xcur ).
RSD = N
j=1 φj (xi
ji
RC = · · · ?
∇Ψ
Ψ
=
∇(|D|↑ )
∇i |D(x)|
|D(x)|
∇i |D(y)|
|D(y)|
...
|D|↑
=
=
PN
1
R
+
∇(|D|↓ )
|D|↓
+
∇ΨC
.
ΨC
−1
j=1 ∇i φj (xi )Dji (x)
PN
−1
j=1 ∇i φj (yi )Dji (x).
Inverse updating

−1 cur

D−1 (xcur ) − Dki (x ) PN D (xnew )D−1 (xcur ) if j 6= i
l=1 il
kj
lj
R
new ) =
D−1
(x
−1 cur P
kj

 Dki (x ) N Dil (xcur )D−1 (xcur )
if j = i
l=1
lj
R
Islen Vallejo Henao
Efficient object orientation for many-body physics problems