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Lawrence Livermore National Laboratory
Norm Tubman Jonathan DuBois, Randolph Hood, Berni Alder
Lawrence Livermore National Laboratory, P. O. Box 808, Livermore, CA 94551This work performed under the auspices of the U.S. Department of Energy by Lawrence Livermore National Laboratory under Contract DE-AC52-07NA27344
Transient Methods in Quantum Monte Carlo Calculations
7/24/2010
Northwestern University
LLNL-PRES-444177
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Solving the Schrödinger Equation
Transient methods which go beyond fixed-node DMC solve the Schrödinger equation without systematic bias, but are computationally expensive.
How many electrons can be done efficiently?
This talk has 2 main parts1) Release the nodes for as long as we can2) Project the ground state from the release data
What are the tradeoffs between the two?
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Transient/Release Node Method
1) Start with VMC/DMC population
2) Use a guiding wavefunction that is positive everywhere
3) Let the walkers relax to the Boson ground state for as long as possible
4) Project out the antisymmetric component of the energy
The eigenfunction expansion
is given by
The release energy is given by keeping
positive and negative walkersW=walker weight
R=ΨT/ΨG
EL =Local Energy
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Ideal Data (LiH)
A converged Release Node calculation flattens out
after the excited anti-symmetric states decay
Error bars grow exponentially in imaginary time.
1)The time for convergence
is determined by fermion
excited state
2)The growth of the error
bars are determined by
fermion-boson energy
difference.
Two Time Scales
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(Easy) Release Problems: Free Electron Gas
The free electron gas is one of the great successes for the release node method.
In the above paper, the authors were able to converge up to 216 electrons, which
they used for determine the stability of different phases of the free electron gas.
1984 (Ceperley, Alder)
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(Hard) Release Problems: Molecules
Potential Energy Surface
H-H-H
Dierich 1992
Li2 Dimer
Ceperley 1984
Systems
LiH, Li2, H2O: (Ceperley 1984)
LiH: (Caffarel 1991)
H2+H : (Diedrich 1992)
HF,Fluorine: (Luchow 1996)
HF
Luchow 1996
The attractive ionic potential (Z/r) is the main factor
in computational difficulty
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Release Methods: Projection Techniques
1991-1992 Lanczos,Maxent
Comparison of Release Energy, Lanczos Energy and Maxent.
The projected energies are fit to data up until the indicated time.
Release Energy
Lanczos Fit
MaxEnt Fit
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Release Methods: Cancellation
Systems
Step Potentials
(Arnow 1982)
H-H-H
(Anderson 1991)
Paralleogram
(Liu 1994)3He
(Kalos 2000)
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Current Calculations
Trial Wavefunction: First row dimers from Umrigar 1996
Guiding Wavefunction: Our own optimized node-less wavefunction
Propagator: DMC (Short-time approximation)
Other Details: No population control during release (fixed ET)
No cancellation
distance Li2 Be2 B2 C2 N2 O2 F2
R0(a.u.) 5.051 4.63 3.005 2.348 2.068 2.282 2.68
Test set : 6 electrons - 18 electrons (all electron)
Exact Estimates are available for all of the second row dimers. They are generated by precise
experimental dissociation energies and highly accurate atomic calculations.
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Release Node Results Li2
Time step test (Li2)
Ener
gy (a
.u.)
Imaginary Time (a.u.)
Li2
Exact Estimatetime step 0.002 a.u
-14.9938
-14.9940
-14.9942
-14.9944
0 0.2 0.4 0.6 0.8 1.0
Imaginary Time (a.u.)
-14.9938
-14.9943
-14.9948
-14.9953
0 1.5 3.0
time step 0.004 a.u.
time step 0.002 a.u.
time step 0.001 a.u.
time step .0005 a.u.
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Release Node Results F2
time step 0.0008 a.u.
Ener
gy (a
.u.)
Imaginary Time (a.u.)
-199.484
-199.488
-199.492
-199.496
Imaginary Time (a.u.)
Time step test (F2) F2
Experiment
0 0.010 0.020 0.030 0.040
-199.48
-199.50
-199.52
-199.54
0 0.010 0.020 0.030 0.040
time step 0.0008 a.u.
time step 0.0006 a.u.
time step 0.0004 a.u.
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Error growth
• The error grows exponentially with imaginary time as the difference of (EF-EB)
• The growth of (EF - EB) is faster than linear with electron number
e t(E0F E0
B )
O2
5 7 9 11 13 15 17 190
50
100
150
200
250
F2
N2
C2B2
Li2Be2
# of Electrons
EF-EB (a.u.)
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Summary of Results
System Li2 Be2 B2 C2 N2 O2 F2
Largest Imaginary Time 3.0 0.8 0.375 0.09 0.09 0.05 0.035
All calculated energies reported at the largest imaginary time allowed during the release
5 7 9 11 13 15 17 190
10203040506070
# of Electrons
Percentage of Expected Decay
(DMC Energy/Experimental Energy)Li2
Be2B2
C2N2 O2 F2
Percentage
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MaximumEntropy
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Projecting the ground state
€
h(0)(t) = I w(0)w(t)exp(− ELG (s)ds
0
t
∫ )
€
h(0)(t) = cn exp(−t(En − ET ))n
∞
∑
Fitting the release energy data is
hard due to its complicated form
Using a noisy estimate of h(0)(t) we want to determine eigenstates En and coefficients cn
Example (for a discrete spectrum) the following
correlator h(0)
has the functional form
Release energy
(Li2)
• We can make use of the entire decay process for a better estimate of the ground-state energy.
• It is known (Caffarel 1992) that we can calculate various correlations functions that have simple analytical forms.
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Techniques for Inverse Laplace
The problem we would like to solve (a least squares fit):
Where the covariance matrix is given by
€
hF(0)(t) = c0e
−t(E0 −ET )
Model Fits
Single Exponential
€
hF(0)(t) = c0e
−t(E l −ET )
l=0
n
∑Multiple Exponentials
€
hF(0)(t) = c0e
−t(E l −ET )
l=0
P
∑
€
χ2 = [hF (ti) − hc (ti)]Cij−1[hF (t j ) − hc (t j )]
ij
M
∑
€
Cij = hc (ti)hc (t j ) − hc (ti) hc (t j )
1)Starting data includes decay of all states
2)Ending data is noisy
3)How many exponentials to fit
4) Many possible solutions
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Example of regularization techniques: Tinkhnov, Maximum Entropy,SVD
In MaxEnt we regularize the problem with an entropic term
€
Max |eαS− χ
2
2 |The lagrange multiplier, α, determines how much of the entropy to include.
Mathematically, the problem described on the last slide this can be considered an inverse laplace transformation. The inverse laplace transform of the data we are
looking at is ill posed.
Techniques for Inverse Laplace
Ideal Decay Correlator:
1)Smooth, low noise data
2)Large dominant peak
3)Small excited state peaks
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Maximum Entropy (Analysis)Our Maximum Entropy Estimate
is derived by integrating over
all Lagrange multipliers by their
probability.
Ground State
Excited State
Energy Axis
Lagrange
Multiplier
Axis
Integrate
Least Squares Limit
Entropic LimitEnergy Axis
Spectral
Weight
Axis
Maximum Entropy Output
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Maximum Entropy (Time Step)
Time step 0.004
Time step 0.002
Time step 0.0005
and smaller
Time step 0.001
Time steps smaller than
0.0005 agree within
error bars.
h(0)
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Maximum Entropy (Efficiency)
Converged Time Step (This work)
LiH 0.001 Be2 0.0005
Li2 0.0005 B2
0.0005
GFMC Time Steps
LiH(coul) 0.087 Li2(coul) 0.043
LiH(Bes) 0.100 Be(Bes)0.005
-The time step extrapolation is non-linear and demands very small values for convergence.
Time step comparisons between GFMC and the short time approximation DMC vs GFMC
LiH: 100x
Li2: 100x
Be: 10x
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Short Time Fits– Large uncertainty Fits– Problematic Excited States
Small peak forms below
dominant ground state
peak.
No excited state peak,
single exponential fit.
Excited state forms, dozens
of a.u. above the ground
state peak.
Ground state energy can be over estimated.
A MaxEnt fit can produce
problematic results in
certain situations.
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Maximum Entropy (Large Z Computational Time)
Time Step 0.0006
Cor
rela
tor
Imaginary Time (a.u.) Imaginary Time (a.u.)
h(0) (Be2)1.0
0.997
0.994
0.991
0.988
h(0) (O2)
Cor
rela
tor
Time Step 0.0011.0
0.999
0.998
0.997
0.996
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.0 0.01 0.02 0.03 0.04 0.05
Time Step Li2 Be2 B2 C2 N2 O2 F2
Growth 100x (a.u.) 2.5 0.7 0.3 0.08 0.07 0.03 0.015
# of Sidewalks 20000 12000 12000 3000 3000 3000 3000
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Li2 Predictions
Max Ent
Release Node Energy
Exact Estimate
Exact Estimate
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Preliminary Results
Be2
MaxEnt fits can be problematic for larger Z dimers. No excited
state peaks are seen for B2 and Be2. The fits can serve as an upper bound.
B2
Release Node EnergyRelease Node Energy
Exact EstimateExact Estimate
MaxEnt MaxEnt
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Single Exponential Fits
N2 O2
DMC2009
Estimated Exact
Single Exponential
DMC 2009
Estimated Exact
Single Exponential
We can get an estimate of the ground state energy, with a single exponential fit of the correlators.
This in general will be an upper bound of the energy, when the time step is converged.
Time steps are not converged in the above plot, they are projected to zero time step.
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Current Results/Conclusions
Decaying to the ground state is virtually impossible with standard release for systems larger than 8 electrons. Time-step errors tend to increase for both higher Z and large imaginary times.
Projection techniques require data with low noise in order to work, but have the potential to give highly accurate results.
Investigations into additional methods may provide significant improvements.
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The End