1. Lawrence Livermore National Laboratory Norm Tubman Jonathan DuBois, Randolph Hood, Berni Alder Lawrence Livermore National Laboratory, P. O. Box 808, Livermore, CA 94551 This 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

2. 2 Lawrence 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 parts 1) Release the nodes for as long as we can 2) Project the ground state from the release data What are the tradeoffs between the two?

3. 3 Lawrence Livermore National Laboratory 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 walkers W=walker weight R=Ψ T /Ψ G E L =Local Energy

4. 4 Lawrence Livermore National Laboratory 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

5. 5 Lawrence Livermore National Laboratory (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)

6. 6 Lawrence Livermore National Laboratory (Hard) Release Problems: Molecules Potential Energy Surface H-H-H Dierich 1992 Potential Energy Surface H-H-H Dierich 1992 Li 2 Dimer Ceperley 1984 Li 2 Dimer Ceperley 1984 Systems LiH, Li 2, H 2 O: (Ceperley 1984) LiH: (Caffarel 1991) H 2 +H : (Diedrich 1992) HF,Fluorine: (Luchow 1996) HF Luchow 1996 HF Luchow 1996 The attractive ionic potential (Z/r) is the main factor in computational difficulty

7. 7 Lawrence Livermore National Laboratory 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

8. 8 Lawrence Livermore National Laboratory Release Methods: Cancellation Systems Step Potentials (Arnow 1982) H-H-H (Anderson 1991) Paralleogram (Liu 1994) 3 He (Kalos 2000)

9. 9 Lawrence Livermore National Laboratory 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 E T ) No cancellation distanceLi 2 Be 2 B2B2 C2C2 N2N2 O2O2 F2F2 R 0 (a.u.) 5.0514.633.0052.3482.0682.2822.68 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.

10. 10 Lawrence Livermore National Laboratory Release Node Results Li 2 Time step test (Li 2 ) Energy (a.u.) Imaginary Time (a.u.) Li 2 Exact Estimate time 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. time step 0.004 a.u. time step 0.002 a.u. time step 0.001 a.u. time step.0005 a.u.

11. 11 Lawrence Livermore National Laboratory Release Node Results F 2 time step 0.0008 a.u. Energy (a.u.) Imaginary Time (a.u.) -199.484 -199.488 -199.492 -199.496 Imaginary Time (a.u.) Time step test (F 2 )F2F2 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. time step 0.0008 a.u. time step 0.0006 a.u. time step 0.0004 a.u.

12. 12 Lawrence Livermore National Laboratory Error growth The error grows exponentially with imaginary time as the difference of (E F -E B ) The growth of (E F - E B ) is faster than linear with electron number O2O2 F2F2 N2N2 C2C2 B2B2 Li 2 Be 2 # of Electrons E F -E B (a.u.)

13. 13 Lawrence Livermore National Laboratory Summary of Results SystemLi 2 Be 2 B2B2 C2C2 N2N2 O2O2 F2F2 Largest Imaginary Time 3.00.80.3750.09 0.050.035 All calculated energies reported at the largest imaginary time allowed during the release # of Electrons Percentage of Expected Decay (DMC Energy/Experimental Energy) Percentage of Expected Decay (DMC Energy/Experimental Energy) Li 2 Be 2 B2B2 C2C2 N2N2 O2O2 F2F2 Percentage

14. 14 Lawrence Livermore National Laboratory Maximum Entropy

15. 15 Lawrence Livermore National Laboratory Projecting the ground state 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 E n and coefficients c n Example (for a discrete spectrum) the following correlator h (0) has the functional form Release energy (Li 2 ) Release energy (Li 2 ) 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.

16. 16 Lawrence Livermore National Laboratory Techniques for Inverse Laplace The problem we would like to solve (a least squares fit) : Where the covariance matrix is given by Model Fits Single Exponential Multiple Exponentials 1)Starting data includes decay of all states 2)Ending data is noisy 3)How many exponentials to fit 4) Many possible solutions

17. 17 Lawrence Livermore National Laboratory Example of regularization techniques: Tinkhnov, Maximum Entropy,SVD In MaxEnt we regularize the problem with an entropic term 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

18. 18 Lawrence Livermore National Laboratory Maximum Entropy (Analysis) Our Maximum Entropy Estimate is derived by integrating over all Lagrange multipliers by their probability. 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 Limit Energy Axis Spectral Weight Axis Maximum Entropy Output

19. 19 Lawrence Livermore National Laboratory 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)

20. 20 Lawrence Livermore National Laboratory Maximum Entropy (Efficiency) Converged Time Step (This work) LiH0.001Be 2 0.0005 Li 2 0.0005B 2 0.0005 GFMC Time Steps LiH(coul)0.087Li 2 (coul)0.043 LiH(Bes)0.100Be(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 Li 2 : 100x Be: 10x

21. 21 Lawrence Livermore National Laboratory 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.

22. 22 Lawrence Livermore National Laboratory Maximum Entropy (Large Z Computational Time) Time Step 0.0006 Correlator Imaginary Time (a.u.) h (0) (Be 2 ) 1.0 0.997 0.994 0.991 0.988 h (0) (O 2 ) Correlator Time Step 0.001 1.0 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 StepLi 2 Be 2 B2B2 C2C2 N2N2 O2O2 F2F2 Growth 100x (a.u.) 2.50.70.30.080.070.030.015 # of Sidewalks 2000012000 3000

23. 23 Lawrence Livermore National Laboratory Li 2 Predictions Exact Estimate

24. 24 Lawrence Livermore National Laboratory Preliminary Results Be 2 MaxEnt fits can be problematic for larger Z dimers. No excited state peaks are seen for B 2 and Be 2. The fits can serve as an upper bound. B2B2

25. 25 Lawrence Livermore National Laboratory Single Exponential Fits N2N2 O2O2 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.

26. 26 Lawrence Livermore National Laboratory 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.

27. 27 Lawrence Livermore National Laboratory The End