thomas halverson and bill poirier [email protected] texas tech university department of physics...

EXACT QUANTUM DYNAMICSEXACT QUANTUM DYNAMICS

USING PHASE SPACE WAVELETSUSING PHASE SPACE WAVELETS

Thomas Halverson and Bill [email protected] Tech UniversityDepartment of Physics6-9-13

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I. What we mean by exactII.Exponential scaling and

basis truncationIII.Theoretical OverviewIV.Code specificV. ResultsVI.Future work

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Exact Quantum DynamicsExact Quantum Dynamics

Nuclear dynamicsNuclear dynamicsBorn-Oppenheimer ApproximationBound statesRovibrational Spectroscopy

TISE:TISE:Exact to numerical convergenceIn general, all quantities of interest can be calculated from the eigenfunctions and eigenenergies

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Choice of potentialProblem specificAb initio methods/ DFT

Basis ChoiceDirect product vs. not-direct product

Symmetry• Point group• Permutation/ Inversion

Exponential ScalingDiagonalization Method

Exact Quantum DynamicsExact Quantum Dynamics

Choice of potentialChoice of potential

Basis ChoiceBasis Choice

Diagonalization MethodDiagonalization Method

Symmetrized Gaussians

Direct Diagonalization

(in 1-D)(in 1-D)

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Exponential Scaling and Basis TruncationExponential Scaling and Basis Truncation

For a direct product basisdirect product basis, the size of the Hamiltonian matrix grows exponentially with the degrees of freedom (D)

At large dimensionality problems become intractable

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Exponential Scaling and Basis TruncationExponential Scaling and Basis Truncation

A method for decreasing basis size, without decreasing accuracy

Increases basisbasis efficiencyefficiency::Types

Polyad truncationEnergy truncation

Phase space truncationPhase space truncation

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Theoretical OverviewTheoretical Overview

Density operator for lowest K Density operator for lowest K states of a given Hstates of a given H

Weyl symbol: Wigner PDFPhase space region RQuasi-classical approximation*

*B. Poirier and J. C. Light, J. Chem. Phys. 111, 4869 (1999); B. Poirier and J. C. Light, J. Chem. Phys. 113, 211 (2000).

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Density operator in some Density operator in some basisbasis

Phase space Region R’

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R

R’


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Each square is denoted by a two indices (m, n)

The are of each square has area 2 = (double densitydouble density)

The red square is (m=5/2, n=3/2) and is at coordinate (x = 5/2, p = 3/2

*B. Poirier and A. Salam, J. Chem. Phys. 121, 1690 (2004); B. Poirier and A. Salam, J. Chem. Phys. 121, 1704 (2004).

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Collective well-localizedVery small overlap

Can apply phase space truncated

Linearly IndependentReal valued

*B. Poirier and A. Salam, J. Chem. Phys. 121, 1690 (2004); B. Poirier and A. Salam, J. Chem. Phys. 121, 1704 (2004).

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Phase Space Truncation

Weyl-Heisenberg Lattice

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Start with DPB: Start with DPB: Fmn(x) = fm1n1(x1)×fm2n2(x2)×...

Retain:Retain: H(m1Δ, m2Δ,..., n1Δ, n2Δ) < Emax

Defeats Exponential Defeats Exponential ScalingScaling

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Code SpecificsCode Specifics

Stand alone, single codeStand alone, single codeDimension independentMassively Parallel

Utilizes standard dense matrix routines (Scalapack Library)

effective scaling tested up to 4104 cores

Designed for usability

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ResultsResults

Coupled anharmonic oscillatorA=C=0, B=1/2, D=αMultidimensional Hamiltonian is a sum of one-dimensional Hamiltonians + coupling term

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ResultsResults

Basis size

Converged states

Harmonic Oscillator Basis


3% 2% 0.2% 0.06% 3% 2% 0.2% 0.06%

~1000 820 693 336 241 1015(1018) 954(997) 178(312) 31(115)

~3000 1625 1388 703 547 2893(2931) 2755(2812) 456(786) 75(162)

~10000 3885 3361 1857 1526 9231(9378) 8268(8697) 1683(2184) 428(872)

~30000 8769 7984 4543 3876 26466(26918) 23778(24695) 6546(8625) 1775(4351)

~100000 20614 18355 11970 10698 85501(89227) 75691(81625) 20525(24364) 4105(10761)

~175000 -- -- -- -- 150276(150276) 136595(136595) 48416(48416) 25176(25176)

T. Halverson and B. Poirier, J. Chem. Phys. 137, 224101 (2012).

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ResultsResults

Basis size

Converged states

Harmonic Oscillator Basis


3% 2% 0.2% 0.06% 3% 2% 0.2% 0.06%

~1000 820 693 336 241 1015(1018) 954(997) 178(312) 31(115)

~3000 1625 1388 703 547 2893(2931) 2755(2812) 456(786) 75(162)

~10000 3885 3361 1857 1526 9231(9378) 8268(8697) 1683(2184) 428(872)

~30000 8769 7984 4543 3876 26466(26918) 23778(24695) 6546(8625) 1775(4351)

~100000 20614 18355 11970 10698 85501(89227) 75691(81625) 20525(24364) 4105(10761)

~175000 -- -- -- -- 150276(150276) 136595(136595) 48416(48416) 25176(25176)


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ResultsResults

Lowest 100000 Lowest 100000 computed energies computed energies for 3D coupled for 3D coupled harmonic oscillator harmonic oscillator

Solid line- “master list” energies (N = 302 352)

Dotted line- energy truncated HO basis energies (N = 102 340)

Dashed line- PST SG energies (N = 100 016)


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ResultsResults

DimensionalityBasis Size

Cores 2.0% 0.2% 0.06%

8 23925 24 2408 0 0

8 154323 576 88320 128 0

8 211251 1152 160713 652 1

8 242627 1728 203946 2359 11

12 79759 144 66890 12 0

12 175087 1728 168421 21932 3128

16 13321 36 8265 0 0

16 42473 60 32023 0 0

16 117993 576 110264 239 0


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ResultsResults

DimensionalityBasis Size

Cores 2.0% 0.2% 0.06%

8 23925 24 2408 0 0

8 154323 576 88320 128 0

8 211251 1152 160713 652 1

8 242627 1728 203946 2359 11

12 79759 144 66890 12 0

12 175087 1728 168421 21932 3128

16 13321 36 8265 0 0

16 42473 60 32023 0 0

16 117993 576 110264 239 0


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ResultsResults

Accuracy,(cm-1)

Number ofStates, K

Efficiency,

K/N1000 35186 60.5%

100 10877 18.7%

10 2272 3.9%

1 140 0.24%

0.1 31 0.05%

Quartic force field Quartic force field by Pouchan group*by Pouchan group*

Accurately Computed States for Accurately Computed States for N=58163 N=58163

SG calculations converged using seven different basis sizes ranging from N=310 to N=102858

*C. Pouchan, M. Aouni, and D. Bégué, Chem. Phys. Lett. 334, 352 (2001).

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ResultsResults

Accuracy,(cm-1)

Number ofStates, K

Efficiency,

K/N1000 35186 60.5%

100 10877 18.7%

10 2272 3.9%

1 140 0.24%

0.1 31 0.05%



SG calculations converged using seven different basis sizes ranging from N=310 to N=102858


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ResultsResults

Accuracy,(cm-1)

Number ofStates, K

Efficiency,

K/N1000 85450 40.3%

100 3404 1.6%

10 302 0.14%

1 11 0.005%

0.1 3 0.001%



SG calculations converged using six different basis sizes ranging from N=2042 to N=409582


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ResultsResults

Accuracy,(cm-1)

Number ofStates, K

Efficiency,

K/N1000 85450 40.3%

100 3404 1.6%

10 302 0.14%

1 11 0.005%

0.1 3 0.001%





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ResultsResults

Accuracy,(cm-1)

Number ofStates, K

Efficiency,

K/N1000 60293 39.99%

100 8192 5.4%

10 1100 0.73%

1 73 0.048%

0.1 3 0.002%





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ResultsResults

Accuracy,(cm-1)

Number ofStates, K

Efficiency,

K/N1000 60293 39.99%

100 8192 5.4%

10 1100 0.73%

1 73 0.048%

0.1 3 0.002%





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Future WorkFuture Work

1.1. Newer, Larger ClustersNewer, Larger ClustersKraken, Blue Waters

2.2. Newer, Larger PotentialsNewer, Larger PotentialsWork with experimental and

electronic structure groups

3.3. Improve accuracyImprove accuracyPhase space region operators*

*R. Lombardini and B. Poirier, Phys. Rev. E 74, 036705 (2006).

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PersonnelPersonnelPrinciple Investigator

Dr. Bill PoirierFellow Graduate Students

Corey PettyDrew Brandon

SupportSupportRobert A. Welch FoundationNational Science Foundation

OrganizerOrganizerssTerry A. MillerFrank C. DeluciaAnne B. McCoy

Computer Computer ResourcesResourcesTexas Tech University

High Performance Computing Center

University Of Texas-Austin

Texas Advanced Computing Center

thomas halverson and bill poirier [email protected] texas tech university department of physics...

Documents

basis truncationfor

direct product basis

theoretical overviewstart

decreasing basis size

basis truncationa method

exactexponential scaling

basisphase space region

hamiltonian matrix