High Accuracy Local Correlation Methods: Computer Aided Implementation

Transcription

1 High Accuracy Local Correlation Methods: Computer Aided Implementation Marcel Nooijen Alexander Auer Princeton University University of Waterloo USA Canada So Hirata, PNNL Supported by: NSF ITR (Information Technology Research) NSERC

2 Local Correlation Alleviates the n 6 /n 7 scaling laws of CCSD / CCSD(T) Strong emphasis on efficiency: Linear scaling What about accuracy and reliability we expect from CCSD(T)? 99.5% of correlation energy is not a sufficient measure. We want systematic energy differences of < kcal/mol accuracy. Can local correlation methods provide a reliable bound on the chemically relevant errors?

3 Summary & Critique of of present local correlation methods Pulay-Szaebo-Werner-Hampel-Schütz: Variable localization (Pipek-Mezey) of occupied orbitals Imposed rigid domain structure for excitations [ij] [ab]. Scuseria-Ayala: Purely AO-based algorithms: ultimate localization Dynamical screening criteria? Large basis set calculations (occ MO AO)?? Convergence issues due to redundant orbitals? Head-Gordon, Maslen (TRIM, MP2): Consistent localization (redundant occupied orbitals) Extended, but rigid domain excitations.

4 Desiderata for for local methods Consistent localization of occupied / virtual spaces, independent of nature chemical bonds. Controllable (absolute?) accuracy within a single calculation No rigid excitation domains, but dynamical selection. Reasonable efficiency for medium sized systems (10-20 atoms). Linear scaling onset within a reasonable range. Reintroduce symmetry: Qualitatively correct results, Vital for spectroscopy, Increased efficiency.

5 Proposed approach to to local correlation Localized, non-orthogonal, symmetric, set of orbitals that span atomic minimal basis set which envelops the occupied space (à la Head-Gordon): Enveloping Localized Orbitals: ELO s. Full AO basis set to represent virtual space (enveloping) Hierarchy of accuracies: Multipole expansion MBPT[2] CCSD / CCSD(T) Use lower-level results to screen on next-level amplitudes Automatic synthesis of efficient computer codes.

6 Comparison Pipek Mezey vs. vs. ELO orbitals ELO s P-M orbitals

7 Enveloping Localized Orbitals (ELO s): - Construct linear-dependent AO-centered orbitals within occupied space, of size minimal basis. - Discard AO-tails centered on other atoms and partition result into occupied and complementary localizing virtual part. - Obtain optimal subset of complementary virtual orbitals, that together with occupied MO s define a minimal basis. - Redo localization Linearly independent, non-orthogonal, localized, minimal basis that envelops occupied space. - The ELO s will span complete irreducible representations.

8 Coupled Cluster equations in in localized basis i, j, k : occupied MO : ELO s µ, ν i µ =Σµ C i i j i j i ν µ ν µ µ j µ a, b, c : Virtual MO : AO α, β a =Σβ C a β α a b α β β C S C = δ C C C S C = δ C C tµν Ca Cb tij C µ Cν (Definition) a b αβ µ ν C C t C C = t E = v t = v t β α µν i j ij ab a b β b a β 1 ij ab ij ab 1 µν αβ 4 4 αβ µν Localized CC equations: Replace all MO indices by AO/ELO indices. Multiply external indices on H by α β 1β = α ν or D µ (projectors on virtual / occupied space). D S D α β

9 Analysis of of combining CC and MBPT[2] ECC = 1 4 v t αβ µν CC µν,, α, β block ( )[block] ( ), Likewise for E(MP2) = + E( ε ) E( MP2) E( CC) E( MP2) < ε E( MP2) ε Absolute error E( ε ) ε

10 Analysis local correlation: (preliminary) conclusions If we replace CCD amplitudes by MBPT[2] amplitudes for many blocks with E (block) < 10-6, we loose only one order of accuracy in total energy (10-5 )! This holds for both P-M and ELO orbitals. We can alternatively screen on magnitude MBPT[2] amplitudes. We rely on cancellation of errors: E( MP2) E( CCD) is positive or negative for small contributions (systematic?) Use of pure AO s instead of projected AO s leads to negative and positive contributions to per block! E

11 Combining CC and PT PT equations in in practice. P αβ H 0, T + V 0 = 0: ( PT) αβ T T P e He αβ 0 = 0: ( CC) αβ µν µν µν µν Select CC vs. PT Equation. : µν αβ E(block) = v t ( MP2) µν,, α, β block if 1 4 αβ µν αβ E > ε use µν ( CC) αβ E < ε use µν ( PT) Coupled set of equations. Converges rapidly.

12 Some further issues: t αβ Not necessarily a pure excitation operator µν if coefficients are varied independently! (e.g. α might be 1s orbital) ab Test: t αβ t t αβ t αβ =? = t αβ µν ij µν µν µν αβ Update t-amplitudes: solve P H 0, T 0 = R αβ ab ab αβ At present: R R t t DIIS T = T ( PT) + T ( CC) µν ij ij µν αβ µν µν T( CC) T ( CC) αβ P e He 0 = 0: ( t CC) truncated CC αβ µν µν

13 Results for for Hexatriene in in DZ basis set set ε ε % CC amplitudes Error in milli-hartree Analysis CC/PT t-cc/pt 10(-7) 96% (-6) 89% (-5) 69% (-4) 41% (-3) 16%

14 Hexatriene in in DZ basis set setcontinued

15 Comparison # of of CC equations at at ε ε =10(-6) for for LiH, Alkane, Alkene chains.

16 Computer Aided Implementation of of Many-Body Methods: The Tensor Contraction Engine Oak Ridge National Laboratory David E. Bernholdt, Venkatesh Choppella, David Dean, Robert Harrison, Thomas Papenbrock, Michael Strayer, Trey White Pacific Northwest National Laboratory So Hirata Ohio State University Gerald Baumgartner, Daniel Cociorva, Russ Pitzer, P Sadayappan, a small army of graduate students Louisiana State University J Ramanujam Princeton University / University of Waterloo Marcel Nooijen, Alexander Auer

17 The (re-)coding Bottleneck of of Quantum Chemistry New ideas are emerging continuously. Developing and testing new ideas is time consuming. Good ideas should be incorporated (all the way) in efficient production-level codes. Much of actual coding is fairly routine. Can we teach the computer to do the job? Can we train the computer to do the job better than we could ever do it ourselves? Automation will support evolving technology.

18 Advantages of of computer aided implementations A) Develop new methodologies: Codes expected to be robust and free of errors. Develop and test new ideas quickly. Generation of many similar pieces of code. B) Develop highly efficient implementations: Explore wide variety of algorithms and select optimal strategy for specific problem. Computer codes can evolve and improve over time. Relatively easy to build in new strategies & methodologies

19 For the skeptics among us us It is more than a little embarrassing to quote my own words (MN) from about 5 years ago [88], commenting on the work by Paldus and Li and Janssen and Schaefer: We feel this complexity will be reflected in the computational Efficiency of the approach and doubt therefore, that this scheme will lead to a widely applicable computational scheme. Needless to say, I (MN) have changed my mind on these matters [88] M. Nooijen and R.J.Bartlett, J. Chem. Phys. 104, (1996), Marcel Nooijen and Victor Lotrich in Journal of Molecular Structure (Theochem), 547 (2001), , a tribute to Josef Paldus.

20 A) A) Operator Contraction Engine (OCE): Generating Many-Body equations General set of tools to derive many-body equations, based on second quantization and Wick s theorem. Additional manipulations of equations, e.g. Derive energy gradients, second derivatives. Obtain AO-based expressions. Multiply by density matrices (for multireference treatments). Preliminary (heuristic) factorization of tensor expressions. Prepare input for TCE.

21 B) B) Tensor Contraction Engine (TCE): Generate efficient computer codes Synthesize code to evaluate a sequence of tensor contractions. Optimize data flow and performance at all stages of calculation, e.g. Optimize memory vs. disk usage. Minimize cache misses. Optimize disk usage vs. recomputation of integrals. Optimize local vs. remote disk/memory on parallel machines. Optimize codes for particular application of interest (precise computational perimeters are available).

22 PNNL version of of OCE/TCE (So Hirata) Programming language Python Interfaced to NWChem (PNNL) and UTChem (University of Tokyo) Spin-orbital based, Abelian spatial symmetry Full treatment of permutational symmetry and antisymmetry. Parallellization using Global Arrays or Replicated Data Structures or Global File System. High-order canonical MO-based CC / CI / MBPT Production level codes up to quadruple excitations (CCSDTQ) in NWchem Equation-of-Motion CC and excited state properties (in progress). Relativistic Douglas-Kroll and 4-component Fock-Dirac in UTchem

23 Comparison of of timings TCE-NWChem and ACES II II MO-CCSD on on AMD Athlon processor (in (in seconds) Ethylene, CC-PVTZ RHF D2h UHF D2h RHF C1 UHF C1 ACES II TCE-NWChem Benzene, Dunning DZ RHF D2h UHF D2h RHF C1 UHF C1 ACES II TCE-NWChem

24 Parallel performance on on distributed memory machine (Intel Itanium-2 1-GHz Linux cluster) Molecule Sym. Method Basis set Algorithm CH 2 C 1 CCSDT cc-pvtz GA CH 2 C 2v CCSDT cc-pvtz GA CH 2 C 2v CCSDT cc-pvtz Repl C 10 H 8 + C 2v CCSD cc-pvdz Repl NC 4 H 5 + C 2v CCSDT cc-pvdz Repl

25 Princeton/Waterloo directions for for OCE // TCE (Alexander Auer) Local correlation & integral direct approaches: Enveloping (atomic-like) non-orthogonal occupied orbitals. Pure AO s for the virtual space. Hierarchy of methods: (Multipole Expansion) Local PT Local Coupled Cluster Use results at low level to screen amplitudes at higher level. Current status: Implementation using Hirata TCE Parallel, Integral direct, CCSD/PT using Pipek-Mezey orbitals

26 Comparison of of timings MO and integral direct AO code Full CCSD method in in TCE-NWChem Ethylene, cc-pvtz basis set on AMD Athlon Processor MO, Symmetry MO, No Symmetry AO/PM, No Symmetry 9.7 s s s AO/PM, No Symmetry Integral Direct s

27 Parallel Performance on on SGI Origin 3800 Shared Memory Machine, Global Arrays Algorithm 1,2,4,8, 12, and Processors Benzene in in Dunning DZ basis set set Integral direct AO/PM CCSD

28 Summary New Local Coupled Cluster / Perturbation Theory method: Dynamical ε selection of CC vs. PT domains. Use of AO-like symmetry-adapted set of Enveloping Localized Orbitals (ELO s), independent of chemical environment. Smooth convergence of equations. High accuracy will depend on cancellation of errors. Computer Aided Implementation of new Methodology. Efficient, Integral Direct, Parallel AO-based CCSD code. Workshop on the OCE / TCE 2004 Sanibel Symposium