Transient Methods in Quantum Monte Carlo Calculations 7/24/2010

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

2Lawrence Livermore National Laboratory

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?


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


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


(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)


(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


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


Release Methods: Cancellation

Systems

Step Potentials

(Arnow 1982)

H-H-H

(Anderson 1991)

Paralleogram

(Liu 1994)3He

(Kalos 2000)


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.


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


-14.9938

-14.9943

-14.9948

-14.9953

0 1.5 3.0

time step 0.004 a.u.



time step .0005 a.u.


Release Node Results F2


Ener

gy (a

.u.)


-199.484

-199.488

-199.492

-199.496


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





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.)


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


MaximumEntropy


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.


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


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


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


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)


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


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.


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


Li2 Predictions

Max Ent

Release Node Energy

Exact Estimate

Exact Estimate


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


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.


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.


The End

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Transient Methods in Quantum Monte Carlo Calculations 7/24/2010

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