Marcel NooijenAlexander Auer

Princeton University University of WaterlooUSA Canada

Supported by: NSF ITR (Information Technology Research)NSERC

So Hirata, PNNL

High Accuracy Local Correlation Methods:Computer Aided Implementation

High Accuracy Local Correlation Methods:Computer Aided Implementation

Local CorrelationLocal Correlation

Alleviates the n6/n7 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?

Summary & Critique of present local correlation methods Summary & Critique of present local correlation methods

Pulay-Szaebo-Werner-Hampel-Schütz:Variable localization (Pipek-Mezey) of occupied orbitalsImposed rigid domain structure for excitations [ij] [ab].

Scuseria-Ayala:Purely AO-based algorithms: ultimate localizationDynamical 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.

Desiderata for local methodsDesiderata for local methods

Consistent localization of occupied / virtual spaces,independent of nature chemical bonds.

Controllable (absolute ?) accuracy within a single calculationNo 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.

Proposed approach to local correlationProposed approach 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.

Comparison Pipek Mezey vs. ELO orbitalsComparison Pipek Mezey vs. ELO orbitals

P-M orbitals

ELO’s

Enveloping Localized Orbitals (ELO’s):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 resultinto 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.

Coupled Cluster equations in localized basisCoupled Cluster equations in localized basis

a, b, c : Virtual MO: AOα β,

a Ca= Σ β β

i, j, k : occupied MO: ELO’sµ ν,

i Ci= Σ µ µ

C S C C Ci

j ji i

j†ν µ

ν µµ

µδ= ≡ C S C C Cba

ba

ba†α

αβ

ββ

βδ= ≡

∆E v t v tabij

ijab= =1

414 αβ

µνµναβ

t C C t C Ca b ijab i j

µναβ β α

µ ν≡

C C t C C ta bi j ij

abβ α µν

αβ µ ν =

(Definition)

Localized CC equations: Replace all MO indices by AO/ELO indices. Multiply external indices on H byor (projectors on virtual / occupied space).

D S Dαβ

αβ

αβ= −−1

Dµν

Analysis of combining CC and MBPT[2]Analysis of combining CC and MBPT[2]

14

, , , block

( )[block] ( ), Likewise for E(MP2)E CC v t CCµν αβαβ µν

µ ν α β∈

∆ = ∆∑

( 2) ( 2)

( ) ( 2) ( )E MP E MP

E E MP E CCε ε

ε∆ < ∆ ≥

= ∆ + ∆∑ ∑

( )E εAbsolute error

ε→

Analysis local correlation: (preliminary) conclusionsAnalysis local correlation: (preliminary) conclusions

If we replace CCD amplitudes by MBPT[2] amplitudesfor many blocks with (block) < 10-6, we loose only one order of accuracy in total energy (10-5)!

E∆

This holds for both P-M and ELO orbitals. We can alternatively screen on magnitude MBPT[2] amplitudes.

We rely on cancellation of errors: is positive or negative for small contributions (systematic?)

( 2) ( )E MP E CCD∆ −∆

Use of pure AO’s instead of projected AO’s leads to negative and positive contributions to per block!E∆

Combining CC and PT equations in practice.Combining CC and PT equations in practice.

0 , 0 0: ( )

0 0 : ( )T T

P H T V PT

P e He CC

αβ αβµν µν

αβ αβµν µν

−

+ =

=

Select CC vs. PT Equation. :

14

, , , block(block) ( 2)

( )if

( )

E v t MP

E use CCE use PT

µν αβαβ µν

µ ν α β

αβµναβµν

εε

∈

∆ =

∆ > →∆ < →

∑

Coupled set of equations. Converges rapidly.

Some further issues:Some further issues:

tαβµν Not necessarily a pure excitation operator if coefficients are varied independently!

(e.g. α might be 1s orbital)

Update t-amplitudes: solve 0 , 0P H T Rαβ αβ

µν µν − ∆ =

At present: ab abij ijR R t t DIISαβ αβ

µν µν→ → ∆ →∆ →

Test: ?abijt t t t tαβ αβ αβ αβ

µν µν µν µν→ → = =

( ) ( )

( ) ( )

0 0 : ( )T CC T CC

T T PT T CC

P e He t CCαβ αβµν µν

−

= +

= − truncated CC

Results for Hexatriene in DZ basis setResults for Hexatriene in DZ basis set

3.73.341.7216%10(-3)

.612.02.9241%10(-4)

.45.571.2369%10(-5)

.21.03.0689%10(-6)

.02.01.0196%10(-7)

ε Error in milli-HartreeAnalysis CC/PT t-CC/PT

% CCamplitudes

ε

Hexatriene in DZ basis set continuedHexatriene in DZ basis set continued

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

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

Oak Ridge National LaboratoryDavid 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 UniversityJ Ramanujam

Princeton University / University of Waterloo

Marcel Nooijen, Alexander Auer

Computer Aided Implementation of Many-Body Methods:

The Tensor Contraction Engine

Computer Aided Implementation of Many-Body Methods:

The Tensor Contraction Engine

Can we teach the computer to do the job?

Can we train the computer to do the jobbetter than we could ever do it ourselves?

Automation will support evolving technology.

The (re-)Coding Bottleneck of Quantum ChemistryThe (re-)Coding Bottleneck 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.

• 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

Advantages of computer aided implementationsAdvantages of computer aided implementations

For the skeptics among usFor the skeptics among 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 schemewill lead to a widely applicable computational scheme.”…Needless to say, I (MN) have changed my mind on these matters “

Marcel Nooijen and Victor Lotrich in Journal of Molecular Structure (Theochem), 547 (2001), 253-267, a tribute to Josef Paldus.

[88] M. Nooijen and R.J.Bartlett, J. Chem. Phys. 104, (1996), 2652-2668

• 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.

A) Operator Contraction Engine (OCE):Generating Many-Body equations

A) Operator Contraction Engine (OCE):Generating Many-Body equations

• 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).

B) Tensor Contraction Engine (TCE):Generate efficient computer codes

B) Tensor Contraction Engine (TCE):Generate efficient computer codes

PNNL version of OCE/TCE (So Hirata)PNNL version of OCE/TCE (So Hirata)

Programming language Python Interfaced to NWChem (PNNL) and UTChem (University of Tokyo)

Spin-orbital based, Abelian spatial symmetryFull treatment of permutational symmetry and antisymmetry.Parallellization using Global Arrays or Replicated Data Structures orGlobal File System.

High-order canonical MO-based CC / CI / MBPTProduction level codes up to quadruple excitations (CCSDTQ) in NWchemEquation-of-Motion CC and excited state properties (in progress).

Relativistic Douglas-Kroll and 4-component Fock-Dirac in UTchem

Comparison of timings TCE-NWChem and ACES IIMO-CCSD on AMD Athlon 2100+ processor (in seconds)

Comparison of timings TCE-NWChem and ACES IIMO-CCSD on AMD Athlon 2100+ processor (in seconds)

1951421210TCE-NWChem

30913874ACES II

UHFC1

“RHF”C1

UHFD2h

“RHF”D2h

Ethylene,CC-PVTZ

1671211310TCE-NWChem

437137104ACES II

UHFC1

“RHF”C1

UHFD2h

“RHF”D2h

Benzene,Dunning DZ

110140190320550Repl.CCSDTcc-PVDZ

C2vNC4H5+

1.72.12.64.27.2Repl.CCSDcc-PVDZ

C2vC10H8+

0.10.20.40.71.6Repl.CCSDTcc-PVTZ

C2vCH2

0.20.20.40.61.6GACCSDTcc-PVTZ

C2vCH2

1.11.52.03.76.9GACCSDTcc-PVTZ

C1CH2

168421Algo-rithm

MethodBasis set

Sym.Molecule

Parallel performance on distributed memory machine(Intel Itanium-2 1-GHz Linux cluster)

Parallel performance on distributed memory machine(Intel Itanium-2 1-GHz Linux cluster)

Princeton/Waterloo directions for OCE / TCE(Alexander Auer)

Princeton/Waterloo directions 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 PTLocal 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

Comparison of timings MO and integral direct AO codeFull CCSD method in TCE-NWChem

Comparison of timings MO and integral direct AO codeFull CCSD method in TCE-NWChem

Ethylene, cc-PVTZ basis set on AMD Athlon 2100+ Processor

687.7 sAO/PM, No SymmetryIntegral Direct

259.4 sAO/PM, No Symmetry

141.7 sMO, No Symmetry

9.7 sMO, Symmetry

Parallel Performance on SGI Origin 3800Shared Memory Machine, Global Arrays Algorithm

1,2,4,8, 12, and 16 Processors Benzene in Dunning DZ basis set

Integral direct AO/PM CCSD

Parallel Performance on SGI Origin 3800Shared Memory Machine, Global Arrays Algorithm

1,2,4,8, 12, and 16 Processors Benzene in Dunning DZ basis set

Integral direct AO/PM CCSD

SummarySummary

• 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 / TCE2004 Sanibel Symposium