Performance and Productivity: NWChem

1
Performance and Productivity NWChem

• Robert J. Harrison
• Oak Ridge National Laboratory,
• University of Tennessee

NWChem is funded by the U.S. Department of
Energy, Office of Science, Office of Biological
and Environmental Research, under contract
DE-AC06-76RLO 1830 with Battelle Memorial
Institute (Pacific Northwest National Laboratory,
PNNL) as part of the Environmental Molecular
Sciences Laboratory, PNNL.
2
NWChem Citation
T.H. Dunning, Jr., D.A. Dixon, M.F. Guest

• R. J. Harrison, J. A. Nichols, T. P. Straatsma,
  M. Dupuis,E. J. Bylaska, G. I. Fann, T. L.
  Windus, E. Apra, W. de Jong,S. Hirata, M. T.
  Hackler, J. Anchell, D. Bernholdt, P.
  Borowski,T. Clark, D. Clerc, H. Dachsel, M.
  Deegan, K. Dyall, D. Elwood,H. Fruchtl, E.
  Glendening, M. Gutowski, K. Hirao, A. Hess,J.
  Jaffe, B. Johnson, J. Ju, R. Kendall, R.
  Kobayashi, R. Kutteh,Z. Lin, R. Littlefield, X.
  Long, B. Meng, T. Nakajima,J. Nieplocha, S. Niu,
  M. Rosing, G. Sandrone, M. Stave, H. Taylor,G.
  Thomas, J. van Lenthe, K. Wolinski, A. Wong, and
  Z. Zhang,
• "NWChem, A Computational Chemistry Package for
  Parallel
• Computers, Version 4.1" (2002),
• Pacific Northwest National Laboratory,Richland,
  Washington 99352-0999, USA.

3
NWChem Overview

• Provides major new modeling and simulation
  capability for molecular science
• Broad range of molecules, including biomolecules
• Electronic structure of molecules
  (non-relativistic, relativistic,
  one-/two-component, ECPs, second deriv.)
• Increasingly extensive solid state capability
• Molecular dynamics, molecular mechanics
• Extensible and long-lived
• Freely distributed installed at about 1000
  sites worldwide
• Performance characteristics designed for MPP
• Single node performance comparable to best serial
  codes
• Scalability to 1000s of processors
• Portable runs on a wide range of computers

4
Molecular Science Software Suite (MS3)
http//www.emsl.pnl.gov/pub/docs/ecce/
http//www.emsl.pnl.gov/pub/docs/parsoft/
http//www.emsl.pnl.gov/pub/docs/nwchem/
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NWChem to go ...

• Compaq iPAQ
• Linux (Intimate)
• 64 Mbyte RAM,
• 16 Mbyte flash,
• 2 Gbyte
• PCMCIA disk
• Strongarm CPU
• Sadly, no FPU

8
Higher-level composition

• Modular, hierarchical design
• Easy to access high level features
• Easy to extend with new high level features
• Standardized interfaces
• Reuse of low-level functionality without side
  effects
• Distributed-shared memory parallel programming
  model
• Non-uniform memory access (NUMA) aware algorithms
  
• Python interface
• Users developers can write NWChem programs in
  Python
• Cool stuff now becoming available
• Automatic code generation compose with
  many-body theory and/or tensor expressions
  (already in NWChem 4.5)
• Multiresolution quantum chemistry compose with
  operators and functions

9
NWChem Architecture

• Object-oriented design
• abstraction, data hiding, APIs
• Parallel programming model
• non-uniform memory access, global arrays, MPI
• Infrastructure
• GA, Parallel I/O, RTDB, MA, ...
• Program modules
• communication only through the database
• persistence for easy restart

10
Issues in Parallel Computing

• Expressing and managing concurrency
• The memory hierarchy
• Efficient sequential execution

11
The Memory Hierarchy

• Non-uniform memory access - NUMA
• Your workstation is NUMA - registers, cache, main
  memory, virtual memory
• Parallel computers just add non-local memory(s)
• Unites sequential and parallel computation
• Differ only in expression and management of
  concurrency
• Distributed data
• Do not limit calculation by resources of one node
• Exploit aggregate resources of the whole machine
• SCF and DFT can distribute all data gt O(N)
• MP2 gradients distribute all data gt O(N2)

12
Parallel Programming ModelGlobal Arrays MPI

• J. Nieplocha
• Supported by DOE/ASCR/MICS
• Shared-memory-like model
• Fast local access
• NUMA aware and easy to use
• MIMD and data-parallel modes
• Inter-operates with MPI,
• BLAS and linear algebra interface
• Used by most major chemistry codes, also in
  financial futures forecasting, astrophysics,
  computer graphics,
• Ported to major parallel machines
• IBM, Cray, SGI, clusters, ...

http//www.emsl.pnl.gov/pub/docs/global
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Non-uniform memory access model of computation
Shared Object
Shared Object
1-sided communication
1-sided communication
copy to shared object
copy to local memory
compute/update
local memory
local memory
local memory
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O(1) programmers O(1000) nodes O(100,000)
processors O(10,000,000) threads

• Expressing/managing concurrency at the petascale
• It is too trite to say that the parallelism is in
  the physics
• Must express and discover parallelism at more
  levels
• Low level tools (MPI, Co-Array Fortran, UPC, )
  dont discover parallelism or hide complexity or
  facilitate abstraction
• Management of the memory hierarchy
• Sending data from one multiprocessor chip to
  another will be like us taking a trip to Europe
• Memory will be deeper less uniformity between
  vendors
• Need tools to automate and manage this, even at
  runtime

15
Synthesis of High Performance Algorithms for
Electronic Structure Calculations

• Sadayappan, Baumgartner, Cociorva, Pitzer (OSU)
  Ramanujam (LSU)Bernholdt, Dean, White
  III, Harrison (ORNL)Hirata (PNNL)Nooijen
  (Waterloo)
• Objective
• Automate the implementation of optimized parallel
  computer programs for many-electron methods
  expressed as tensor contractions
• Multi-disciplinary, multi-institution project
• Collaboration between NSF ITR, DOE SciDAC, and
  ORNL LDRD projects

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CCSD Doubles Equation

• hbara,b,i,j sumfb,cti,j,a,c,c
  -sumfk,ctk,bti,j,a,c,k,c
  sumfa,cti,j,c,b,c -sumfk,ctk,ati
  ,j,c,b,k,c -sumfk,jti,k,a,b,k
  -sumfk,ctj,cti,k,a,b,k,c
  -sumfk,itj,k,b,a,k -sumfk,cti,ctj
  ,k,b,a,k,c sumti,ctj,dva,b,c,d,c,d
  sumti,j,c,dva,b,c,d,c,d
  sumtj,cva,b,i,c,c -sumtk,bva,k,i,j
  ,k sumti,cvb,a,j,c,c
  -sumtk,avb,k,j,i,k -sumtk,dti,j,c,b
  vk,a,c,d,k,c,d -sumti,ctj,k,b,dvk,a,
  c,d,k,c,d -sumtj,ctk,bvk,a,c,i,k,c
  2sumtj,k,b,cvk,a,c,i,k,c
  -sumtj,k,c,bvk,a,c,i,k,c
  -sumti,ctj,dtk,bvk,a,d,c,k,c,d
  2sumtk,dti,j,c,bvk,a,d,c,k,c,d
  -sumtk,bti,j,c,dvk,a,d,c,k,c,d
  -sumtj,dti,k,c,bvk,a,d,c,k,c,d
  2sumti,ctj,k,b,dvk,a,d,c,k,c,d
  -sumti,ctj,k,d,bvk,a,d,c,k,c,d
  -sumtj,k,b,cvk,a,i,c,k,c
  -sumti,ctk,bvk,a,j,c,k,c
  -sumti,k,c,bvk,a,j,c,k,c
  -sumti,ctj,dtk,avk,b,c,d,k,c,d
  -sumtk,dti,j,a,cvk,b,c,d,k,c,d
  -sumtk,ati,j,c,dvk,b,c,d,k,c,d
  2sumtj,dti,k,a,cvk,b,c,d,k,c,d
  -sumtj,dti,k,c,avk,b,c,d,k,c,d
  -sumti,ctj,k,d,avk,b,c,d,k,c,d
  -sumti,ctk,avk,b,c,j,k,c
  2sumti,k,a,cvk,b,c,j,k,c
  -sumti,k,c,avk,b,c,j,k,c
  2sumtk,dti,j,a,cvk,b,d,c,k,c,d
  -sumtj,dti,k,a,cvk,b,d,c,k,c,d
  -sumtj,ctk,avk,b,i,c,k,c
  -sumtj,k,c,avk,b,i,c,k,c
  -sumti,k,a,cvk,b,j,c,k,c
  sumti,ctj,dtk,atl,bvk,l,c,d,k,l,c
  ,d -2sumtk,btl,dti,j,a,cvk,l,c,d,k
  ,l,c,d -2sumtk,atl,dti,j,c,bvk,l,c,d
  ,k,l,c,d sumtk,atl,bti,j,c,dvk,l,c
  ,d,k,l,c,d -2sumtj,ctl,dti,k,a,bvk
  ,l,c,d,k,l,c,d -2sumtj,dtl,bti,k,a,c
  vk,l,c,d,k,l,c,d sumtj,dtl,bti,k,c,
  avk,l,c,d,k,l,c,d -2sumti,ctl,dtj,
  k,b,avk,l,c,d,k,l,c,d sumti,ctl,at
  j,k,b,dvk,l,c,d,k,l,c,d sumti,ctl,b
  tj,k,d,avk,l,c,d,k,l,c,d
  sumti,k,c,dtj,l,b,avk,l,c,d,k,l,c,d
  4sumti,k,a,ctj,l,b,dvk,l,c,d,k,l,c,d
  -2sumti,k,c,atj,l,b,dvk,l,c,d,k,l,c,d
  -2sumti,k,a,btj,l,c,dvk,l,c,d,k,l,c,d
  -2sumti,k,a,ctj,l,d,bvk,l,c,d,k,l,c,
  d sumti,k,c,atj,l,d,bvk,l,c,d,k,l,c,d
  sumti,ctj,dtk,l,a,bvk,l,c,d,k,l,c
  ,d sumti,j,c,dtk,l,a,bvk,l,c,d,k,l,c,
  d -2sumti,j,c,btk,l,a,dvk,l,c,d,k,l,c
  ,d -2sumti,j,a,ctk,l,b,dvk,l,c,d,k,l,
  c,d sumtj,ctk,btl,avk,l,c,i,k,l,c
  sumtl,ctj,k,b,avk,l,c,i,k,l,c
  -2sumtl,atj,k,b,cvk,l,c,i,k,l,c
  sumtl,atj,k,c,bvk,l,c,i,k,l,c
  -2sumtk,ctj,l,b,avk,l,c,i,k,l,c
  sumtk,atj,l,b,cvk,l,c,i,k,l,c
  sumtk,btj,l,c,avk,l,c,i,k,l,c
  sumtj,ctl,k,a,bvk,l,c,i,k,l,c
  sumti,ctk,atl,bvk,l,c,j,k,l,c
  sumtl,cti,k,a,bvk,l,c,j,k,l,c
  -2sumtl,bti,k,a,cvk,l,c,j,k,l,c
  sumtl,bti,k,c,avk,l,c,j,k,l,c
  sumti,ctk,l,a,bvk,l,c,j,k,l,c
  sumtj,ctl,dti,k,a,bvk,l,d,c,k,l,c,d
  sumtj,dtl,bti,k,a,cvk,l,d,c,k,l,c,
  d sumtj,dtl,ati,k,c,bvk,l,d,c,k,l,
  c,d -2sumti,k,c,dtj,l,b,avk,l,d,c,k,l
  ,c,d -2sumti,k,a,ctj,l,b,dvk,l,d,c,k,
  l,c,d sumti,k,c,atj,l,b,dvk,l,d,c,k,l
  ,c,d sumti,k,a,btj,l,c,dvk,l,d,c,k,l,
  c,d sumti,k,c,btj,l,d,avk,l,d,c,k,l,c
  ,d sumti,k,a,ctj,l,d,bvk,l,d,c,k,l,c,
  d sumtk,atl,bvk,l,i,j,k,l
  sumtk,l,a,bvk,l,i,j,k,l
  sumtk,btl,dti,j,a,cvl,k,c,d,k,l,c,d
  sumtk,atl,dti,j,c,bvl,k,c,d,k,l,c,
  d sumti,ctl,dtj,k,b,avl,k,c,d,k,l,
  c,d -2sumti,ctl,atj,k,b,dvl,k,c,d,
  k,l,c,d sumti,ctl,atj,k,d,bvl,k,c,d
  ,k,l,c,d sumti,j,c,btk,l,a,dvl,k,c,d,
  k,l,c,d sumti,j,a,ctk,l,b,dvl,k,c,d,
  k,l,c,d -2sumtl,cti,k,a,bvl,k,c,j,k,l
  ,c sumtl,bti,k,a,cvl,k,c,j,k,l,c
  sumtl,ati,k,c,bvl,k,c,j,k,l,c
  va,b,i,j

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TCE Components
Sequence of Matrix Products Element-wise Matrix
Operations Element-wise Function Eval.
Tensor Expressions
Algebraic Transformations

• Algebraic Transformations
• Minimize operation count
• Memory Minimization
• Reduce intermediate storage
• Space-Time Transformation
• Trade storage for recomputation
• Storage Management and Data Locality Optimization
• Optimize use of storage hierarchy
• Data Distribution and Partitioning
• Optimize parallel layout

System Memory Specification
No soln fits disk
Memory Minimization
No soln fits disk
Soln fits disk, not mem.
Soln fits mem.
Space-Time Trade-Offs
Storage and Data Locality Management
Soln fits mem.
Data Distribution and Partitioning
Performance Model
Parallel Code Fortran/C/ OpenMP/MPI/Global Arrays
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Multiresolution Quantum Chemistry Robert J.
Harrison, George I. Fann, Takeshi Yanai,
Zhengting GanOak Ridge National Laboratory
andUniversity of Tennessee, KnoxvilleandGregory
BeylkinUniversity of Coloradoharrisonrj_at_ornl.
gov
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The funding

• This work is funded by the U.S. Department of
  Energy, the division of Basic Energy Science,
  Office of Science, under contract
  DE-AC05-00OR22725 with Oak Ridge National
  Laboratory. This research was performed in part
  using
• the Molecular Science Computing Facility in the
  Environmental Molecular Sciences Laboratory at
  the Pacific Northwest National Laboratory under
  contract DE-AC06-76RLO 1830 with Battelle
  Memorial Institute,
• resources of the National Energy Scientific
  Computing Center which is supported by the Office
  of Energy Research of the U.S. Department of
  Energy under contract DE-AC03-76SF0098,
• and the Center for Computational Sciences at Oak
  Ridge National Laboratory under contract
  DE-AC05-00OR22725 .
• ORNL LDRD

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Outline

• Brief introduction to methodology
• Practical computation in higher dimensions
• Separated form for operators
• Analytic derivatives
• Initial results
• Accuracy, timing and scaling
• MP2
• Path to basis set limit results?

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Objectives

• Complete elimination of the basis error
• One-electron models (e.g., HF, DFT)
• Pair models (e.g., MP2, CCSD, )
• Correct scaling of cost with system size
• General approach
• Readily accessible by students and researchers
• Much smaller computer code than Gaussians
• No two-electron integrals replaced by fast
  application of integral operators
• Fast algorithms with guaranteed precision

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References

• The (multi)wavelet methods in this work are
  primarily based upon
• Alpert, Beylkin, Grimes, Vozovoi (J. Comp. Phys.,
  in press)
• B. Alpert (SIAM Journal on Mathematical Analysis
  24, 246-262, 1993).
• Beylkin, Coifman, Rokhlin (Communications on Pure
  and Applied Mathematics, 44, 141-183, 1991.)
• The following are useful further reading
• Daubechies, Ten lectures on wavelets
• Walnut, An introduction to wavelets
• Meyer, Wavelets, algorithms and applications
• Burrus et al, Wavelets and Wavelet transforms

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Linear Combination of Atomic Orbitals (LCAO)

• Molecules are composed of (weakly) perturbed
  atoms
• Use finite set of atomic wave functions as the
  basis
• Hydrogen-like wave functions are exponentials
• E.g., hydrogen molecule (H2)
• Smooth function ofmolecular geometry
• MOs cusp at nucleuswith exponential decay

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LCAO

• A fantastic success, but
• Basis functions have extended support
• causes great inefficiency in high accuracy
  calculations
• origin of non-physical density matrix
• Basis set superposition error (BSSE)
• incomplete basis on each center leads to
  over-binding as atoms are brought together
• Linear dependence problems
• accurate calculations require balanced approach
  to a complete basis on every atom
• Must extrapolate to complete basis limit
• unsatisfactory and not feasible for large systems

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Why think multiresolution?

• It is everywhere in nature/chemistry/physics
• Core/valence high/low frequency short/long
  range smooth/non-smooth atomic/nano/micro/macro
  scale
• Common to separate just two scales
• E.g., core orbital heavily contracted, valence
  flexible
• More efficient, compact, and numerically stable
• Multiresolution
• Recursively separate all length/time scales
• Computationally efficient and numerically stable
• Coarse-scale models that capture fine-scale detail

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How to think multiresolution

• Consider a ladder of function spaces
• E.g., increasing quality atomic basis sets, or
  finer resolution grids,
• Telescoping series
• Instead of using the most accurate
  representation, use the difference between
  successive approximations
• Representation on V0 small/dense differences
  sparse
• Computationally efficient possible insights

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Scaling Function Basis

• Divide domain into 2n pieces (level n)
• Adaptive sub-division (local refinement)
• lth sub-interval l2-n,(l1)2-n l0,,n-1
• In each sub-interval define a polynomial basis
• First k Legendre polynomials
• Orthonormal, disjoint support

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Scaling Function Basis - II
i1
i0
i3
i2
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Multiwavelet Basis

• Space of polynomials on level n is Vn
• Wavelets - an orthonormal basis to span
• Currently use Alperts basis
• Vanishing moments
• Critically important property
• Since Wn is orthogonal to Vn the first k moments
  of functions in Wn vanish, i.e.,
• Sparse representations of many physically
  important kernels

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Some Consequences of Vanishing Moments

• Compact representation of smooth functions
• Consider Taylor series the first k terms vanish
  and smooth implies higher order terms are small
• Compact representation of integral operators
• E.g., 1/r-s
• Consider double Taylor series or multipole
  expansion
• Interaction between wavelets decays as r-2k-1
• Derivatives at origin vanish in Fourier space
• Diminishes effect of singularities at that point

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• Slice thru grid used to represent the nuclear
  potential for H2 using k7 to a precision of
  10-5.
• Automatically adapts it does not know a priori
  where the nuclei are.
• Nuclei at dyadic points on level 5 refinement
  stops at level 8
• If were at non-dyadic points refinement
  continues (to level ??) but the precision is
  still guaranteed.
• In future will unevenly subdivide boxes to force
  nuclei to dyadic points.

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Integral Formulation

• E.g., used by Kalos, 1962

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Integral operators in 3D

• Non-standard matrix elements easy to evaluate
  from compressed form of kernel K(x)
• Application in 1-d is fairly efficient
• O(Nboxk2) operations
• In 3-d seems to need O(Nboxk6) operations
• Prohibitively expensive
• Separated form
• Beylkin, Cramer, Mohlenkamp, Monzon
• O(Nboxk4) or better in 3D

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Low Separation Rank Representation

• Many functions/operators have short expansions
• Different from low operator rank
• E.g., identity has full operator rank, but unit
  separation rank.

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Separated form for integral operators

• Approach in current prototype code
• Represent the kernel over a finite range as a sum
  of Gaussians
• Only need compute 1D transition matrices (X,Y,Z)
• SVD the 1-D operators (low rank away from
  singularity)
• Apply most efficient choice of low/full rank 1-D
  operator
• Even better algorithms not yet implemented

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Accurate Quadratures

• Trapezoidal quadrature
• Geometric precision for periodic functions with
  sufficient smoothness.

The kernel for x1e-4,1e-3,1e-2,1e-,1e0. The
curve for x1e-4 is the rightmost
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Automatically generated representations
of exp(-30r)/r accurate to 1e-10, 1e-8, 1e-6,
1e-4, and 1e-2 (measured by the weighted error
r(exp(-30r)/r - fit(r))) for r in 1e-8,1 were
formed with 92, 74, 57, 39 and 21 terms,
respectively. Note logarithmic dependence
upon precision.
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Smoothed Nuclear Potential

• u(r/c)/c shifts error to rltc
• e0.00435Z5c3
• ltVgt accurate
• ltTgt main source of error

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Translational Invariance

• Dyadic
• 10-3 -75.9139
• 10-5 -75.913564
• 10-7 -75.91355634

• Non-dyadic
• -75.9139
• -75.913564
• -75.91355635

• Uncontracted aug-cc-pVQZ 75.913002
• Solving with e1e-3, 1e-5, 1e-7 (k7,9,11)
• Demonstrates translation invariance and that
  forcing to dyadic points is only an optimization
  and does not change the obtained precision.
• Average orbital sizes 1.6Mb, 8Mb, 56Mb

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Analytic Derivatives

• Hellman-Feynman theorem applies

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N2 Hartree-Fock R2.0 a.u.

• Basis Grad.Err. EnergyErr.
• cc-pVDZ 5e-2 4e-2
• aug-cc-pVDZ 5e-2 4e-2
• cc-pVTZ 7e-3 1e-2
• aug-cc-pVTZ 6e-3 9e-3
• cc-pVQZ 8e-4 2e-3
• aug-cc-pVQZ 9e-4 2e-3
• cc-pV5Z 1e-4 4e-4
• aug-cc-pV5Z 2e-5 2e-4
• k5 6e-3 1e-2
• k7 4e-5 2e-5
• k9 3e-7 -2e-7
• k11 0.0 0.0
• 0.026839623 -108.9964232

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Sources of error in the gradient

• Partially converged orbitals
• Same as for conventional methods
• Smoothed potential
• Numerical errors in the density/potential
• Higher-order convergence except where the
  functions are not sufficiently smooth
• Inadequate refinement (clearly adequate for the
  energy, but not necessarily for other properties)
• Exacerbated by nuclei at non-dyadic points
• Gradient measures loss of spherical symmetry
  around the nucleus the large value of the
  derivative potential amplifies small errors

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Dependence on potential smoothing parameter
(c) Absolute errors ofderivatives for
diatomics with the nuclei at dyadic points. For
energy accuracyof 1e-6 H 0.039 Li 0.0062 B 0.0026
N 0.0015 O 0.0012 F 0.00099
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Dependence on potential smoothing parameter
(c) Absolute errors ofderivatives for
diatomics with the nuclei at non-dyadic
points. For energy accuracyof
1e-6 H 0.039 Li 0.0062 B 0.0026 N 0.0015 O 0.0012
F 0.00099
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Comparison with NUMOL and aug-cc-pVTZ

• H2, Li2, LiH, CO, N2, Be2, HF, BH, F2, P2, BH3,
  CH2, CH4, C2H2, C2H4, C2H6, NH3, H2O, CO2, H2CO,
  SiH4, SiO, PH3, HCP
• NUMOL, Dickson Becke JCP 99 (1993) 3898
• Dyadic points (0.001a.u.) Newton correction
• Agrees with NUMOL to available precision
• LDA (k7,0.002 k9, 0.0006)
• k9 vs. aug-cc-pVTZ rms error
• Hartree-Fock 0.004 a.u. (0.019 SiO)
• LDA 0.003 a.u. (0.018 SiO)

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High-precision Hartree-Fock geometry for water

• Pahl and Handy Mol. Phys. 100 (2002) 3199
• Plane waves polynomials for the core
• Finite box (L18) requires extrapolation
• Estimated error 3mH, 1e-5 Angstrom
• k11, conv.tol1e-8,e1e-9, L40
• Max. gradient 3e-8, RMS step5e-8
• Difference to Pahl 10mH, 4e-6 Angstrom, 0.0012
• Basis OH HOH Energy
• k11 0.939594 106.3375 -76.06818006
• Pahl 0.939598 106.3387 -76.068170
• cc-pVQZ 0.93980 106.329 -76.066676

48
Energy Timing

• Water LDA with energy error of 1e-5
• Initial prototype code with lots of Python
  overhead
• 450s on 2.4 GHz Pentium IV processor
• Current version (revised tensor class, integral
  operators)
• 96s on 2.4 GHz Pentium IV processor
• Predicted future performance
• lt 30s with known algorithmic improvements
• faster still with better representations of the
  separated operators, alternative basis sets,
  improved iterative solution

49
Asymptotic Scaling

• Current implementation
• Based upon canonical orbitals O(N) to O(N2)
  currently dominant ( O(N3) linear algebra)
• Density matrix/spectral projector
• Well established O(Natomlogm(e)) to any finite
  precision (Goedecker, Beylkin, )
• This is not possible with conventional AO
  Gaussians
• Need separated representation for efficiency
• Gradient
• each dV/dx requires O(-log(e)log(vol.)) terms
• All gradients evaluated in O(-Natomlog(e)log(vol.)
  )

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Water dimer LDAaug-cc-pVTZ geometry, kcal/mol.
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Benzene dimer LDAaug-cc-pVDZ geometry, kcal/mol.
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Benzene dimer timings(Sequential Pentium IV 2.4
GHz)
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Benzene monomer, dimer and trimer

• (aug-cc-pVDZ LDA geometry)
• Dimer binding energy -0.96 kcal/mol.
• Trimer -1.67 kcal/mol.
• Single processor times for k9 energy(energy
  accurate to about 1e-6).
• Monomer 56 minutes
• Dimer 200 (3.6x 21.84)
• Trimer 457 (2.3x (3/2)2.05)

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Also working

• Takeshi Yanai
• Analytic derivatives
• Fast (O(N)) Hartree-Fock exchange
• TDDFT within Tamm-Damcoff approximation
• GGA
• Abelian point group symmetry (D2h subgroups)
• Thanks also to
• So Hirata for guidance with TDDFT
• Edo Apra for insights into DFT

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Putting it all together A path to O(N) exact
MP2

• HF provably O(N) to arbitrary finite precision
• Based upon the density matrix
• Localized orbitals also possible (Bernholc)
• Need an MP2 scheme based upon density matrices

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The Resolvent has low separation rank

• Already known Almlöf Laplace factorization

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Density matrix form of MP2
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Summary

• Multiresolution provides a general framework for
  computational chemistry
• Accurate and efficient with a very small code
• Multiwavelets provide high-order convergence and
  accommodate singularities
• Familiar orthonormal basis (Legendre polynomials)
• Compression and reconstruction (c.f., FFT)
• Fast integral operators (c.f., FMM)
• Separated form for operators and functions
• Critical for efficient computation in higher
  dimension
• Expect speed competitive to Gaussians in near
  future
• Optimal separated forms for kernels, multi-scale
  non-linear solver, better implementation
• Real impact will be application to many-body
  models