Linear scaling electronic structure methods in chemistry and physics Computing in Science

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Linear scaling electronic structure methods in
chemistry and physicsComputing in Science
Engineering, vol. 5, issue 4, 2003 (1)Stefan
Goedecker, Gustavo E. ScuseriaLinear Scaling
Density Functional Calculations with Gaussian
OrbitalsJournal of Physical Chemistry A, vol.
103, no. 25, 1999 (2) Gustavo E. Scuseria
Linear Scaling Electronic Structure
MethodsReviews of Modern Physics, Vol 71, No. 4,
July 1999 (3) Stefan Goedecker
Linear scaling electronic structure methods
DFT Journal Club METU Physics Department
Nazim
Dugan dugannaz_at_GMail.com
2
Stefan Goedecker is a professor of computational
physics at the University of Basel. His research
interests include linear scaling algorithms for
electronic structure calculations and other
atomistic simulation methods. He received a PhD
from the Swiss Federal Institute of Technology in
Lausanne. Contact him at Stefan.Goedecker_at_unibas.c
h Gustavo E. Scuseria is the Robert A. Welch
Professor of Chemistry at Rice University. His
research interests include the development of
low-order scaling electronic structure methods
and their application to molecules and solids.
His undergraduate and PhD degrees are in physics
from the University of Buenos Aires. He is a
member of the American Chemical Society and a
Fellow of the American Physical Society, the
American Association for the Advancement of
Science, and the Guggenheim Foundation. Contact
him at guscus_at_rice.edu
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Outline
Exposition of the problem Strategies for linear
scaling Benchmark Calculations
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Exposition of the problem
Strategies for linear scaling
Benchmark Calculations

Exposition of the problem
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Exposition of the problem
Strategies for linear scaling
Benchmark Calculations

Complexity of algorithms TCPU const N
k complexity O( N k ) Linear scaling means
O(N)
Most physical quantities are extensive - that is,
they grow linearly with system size. We might
therefore expect that the computational effort
will grow linearly with system size as well. An
even slower increase in computing time is
certainly not possible unless we ignore the basic
physics of the electronic system. (1)
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Exposition of the problem
Strategies for linear scaling
Benchmark Calculations

Two body interactions in many particle systems
(electrons, atoms, planets)
combinations of
particles Computation time
(quadratic) In DFT even though the
complexity of finding ground state energy has
linear scaling
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Exposition of the problem
Strategies for linear scaling
Benchmark Calculations

Why do we make approximations One electron
system ?
Matrix (2D) Evaluation of elements N 2 Matrix
diagonalization N 3 2 electron system
? 4 th
rank Tensor In General Dimensionality of
Hamiltonian 2Ne Evaluation of elements Ne
2Ne Diagonalization method not known
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Exposition of the problem
Strategies for linear scaling
Benchmark Calculations

Scaling of other electronic structure
methods Full configuration interaction
exponential Coupled-cluster
O( N 6 ) Quantum Monte Carlo O(
N 3 )
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Exposition of the problem
Strategies for linear scaling
Benchmark Calculations

Origin of cubic scaling in DFT Coulomb and XC
potential evaluation N 2 Matrix
diagonalization to solve Kohn-Sham equation
N 3 Orthonormalization of Kohn-Sham
orbitals N(N-1)/2 orbital pairs cost of each
integral is proportional to N NN(N-1)/2
N 3 TCPU cc N 2 cx N 2 cm N 3 co N 3
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Exposition of the problem
Strategies for linear scaling
Benchmark Calculations

Applicability of cubic scaling methods 10
electrons 1 seconds 100 electrons
16 minutes 1000 electeons 11.5 days
10000 electrons 32 years With linear
scaling methods up to 25000 atoms on 24 nodes of
Earth Simulator
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Exposition of the problem
Strategies for linear scaling
Benchmark Calculations

Crossover point Cubic scaling TCPU c3 N
3 Linear scaling TCPU c1 N
José M. Soler Universidad Autónoma de Madrid
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Exposition of the problem
Strategies for linear scaling
Benchmark Calculations

Strategies for linear scaling
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Exposition of the problem
Strategies for linear scaling
Benchmark Calculations

Well known low complexity algorithms Fast
Fourier Transform (FFT) FT O(N 2) ? FFT O(N
log(N) ) Cooley and Tukey algorithm published in
1965 Quick Sort Bubble Sort O(N 2) ? QS
O(N log(N) ) Divide and Conquer !!!
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Exposition of the problem
Strategies for linear scaling
Benchmark Calculations

Locality (Nearsightedness) in QM makes it
possible to apply CUTOFF
Because the extended eigenorbitals diagonalizing
the independent particle Hamiltonian, usually
referred to as canonical orbitals, do not reflect
this locality principle, they are not suitable as
the basic quantities in O(N) calculations. Linear
scaling also rules out the use of basis functions
extending over the whole computational volume,
such as plane waves. (1)
Blip Functions E Hernández, MJ Gillan, CM
Goringe, Phys Rev B 55, 13485(1997)
W. Kohn, Phys. Rev. Lett. 76, 3168(1996)
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Exposition of the problem
Strategies for linear scaling
Benchmark Calculations

Two-body problem
Neighborhood Keep neighbor information for each
particle Calculate interactions with neigbors
only ( O(N) ) Update neighbors ( O(N 2) ) Mesh
technique Divide space into subspaces Calculate
interactions only with particles of owner and
neighbor subspaces ( O(N) ) Check subspace of
particles ( O(N) )
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Exposition of the problem
Strategies for linear scaling
Benchmark Calculations

Tree-code and the Fast Multipole Method (FMM)
- Search starting from the root of the tree - If
the particle is far enough to a goup of
particles, treat the group as a single
particle at the center of mass (monopole
approximation). - If it is not, go one step
further in the tree and check again. - Use FMM
for Far Field only
Tancred Lindholm, N-body Algorithms, 1999
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Exposition of the problem
Strategies for linear scaling
Benchmark Calculations

Density Matrix approach in DFT
Traditional sequence
DM sequence
Eigenvectors of the effective Hamiltonian which
are obtained through the diagonalization step are
in practice only needed to construct the density
matrix. However, this is not the only way of
obtaining the density matrix, and one can instead
adopt direct search methods like CGDMS. (2)
W. Kohn, Phys. Rev. Lett. 76, 3168(1996)
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Exposition of the problem
Strategies for linear scaling
Benchmark Calculations

Density Matrix Minimization
- Express Total Energy in terms of density
matrix - Minimize wrt density matrix
E Hernandez, MJ Gillan, CM Goringe, Phys. Rev. B
55, 7147(1996)
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Exposition of the problem
Strategies for linear scaling
Benchmark Calculations

Sparse Density Matrix
In this regard, the size of the HOMO-LUMO gap is
connected to localization and, consequently,
sparsity in the system. It is well known that
systems showing metallic character (i.e., small
HOMO-LUMO gap) yield denser Hamiltonians and
density matrices than insulators. (2) Insulaters
exponential decay Metals at finite temperature
exponential decay Metals at zero temperature
algebraic decay
Progressive convergence Dynamical adjustment of
treshold
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Exposition of the problem
Strategies for linear scaling
Benchmark Calculations

Divide and conquer method
The idea is to calculate certain regions of the
density matrix by considering subvolumes and then
to generate the full density matrix by adding up
these parts with the appropriate weights. (3)
W. Yang, Phys. Rev. Lett. 66, 1438(1991)
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Exposition of the problem
Strategies for linear scaling
Benchmark Calculations

The Chebyshev Fermi operator expansion
The rational Fermi operator expansion
The desired linear scaling can be obtained by
introducing a localization region for each
column, outside of which the elements are
negligibly small. For the kth column, this
localization region will be centered on the kth
basis function. (3)
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Exposition of the problem
Strategies for linear scaling
Benchmark Calculations

Benchmark Calculations
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Exposition of the problem
Strategies for linear scaling
Benchmark Calculations

System Water clusters in two and three
dimensions Size Up to 1152 molecules in 2D,
1000 moelcules in 3D Functionals LSDA Becke
88 exchange, Lee-Yang-Parr correlation (BLYP)
GGA Perdew-Burke-Erznehof (PBE) Software
Gaussian 99 (Development version) Basis sets
3-21G and 6-31G (up to 15000 basis
functions) Hardware SGI Origin-2000 195 MHz 4
MB cache Up to 10 GB disk space, 180
megawords RAM
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Exposition of the problem
Strategies for linear scaling
Benchmark Calculations

Energies (Hartrees) and CPU Times (min) per SCF
Cycle Obtained by Conjugate Gradient Density
Matrix Search (CGDMS) and Diagonalization in a
Series of Two-Dimensional Water Cluster
Calculations at the LSDA/3-21G Level of Theory
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Exposition of the problem
Strategies for linear scaling
Benchmark Calculations

2D 3-21-G
2D 6-31-G
CPU Time (min)
3D 3-21-G
3D 6-31-G
number of basis functions
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Exposition of the problem
Strategies for linear scaling
Benchmark Calculations

RNA fragment 1026 atoms, 6767 basis
function LSDA/3-21G
. . . all three DFT steps for the RNA piece are
computationally more expensive than those for the
water clusters, especially CGDMS which is about 5
more costly than the 3D cluster case. These
results simply indicate that typical biomolecules
may have density matrices and Hamiltonians which
are denser than 3D water clusters, but they are
still amenable to efficient treatment by the
methods and algorithms discussed in this work.
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Exposition of the problem
Strategies for linear scaling
Benchmark Calculations

Concluding Remarks Significant factors
dimensionality, HOMO-LUMO gap, desired
accuracy, basis set The more compact a molecular
system is, the less sparse all matrices are, and
the more demanding the O(N) DFT calculation will
turn out to be. (larger prefactor) . . . if DFT
fails to deliver a next generation of
significantly more accurate functionals, it would
then be reasonable to assume that much work will
be devoted to developing fast (i.e., small
prefactor) O(N) wave function methods . .
. Linear scaling DFT codes SIESTA, CONQUEST,
ONETEP, GAUSSIAN
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References
Practical Methods for Ab Initio Calculations on
Thousands of Atoms D.R. Bowler, I.J. Bush, M.J.
Gillan International Journal of Quantum
Chemistry, Vol. 77, 831842 (2000) Large-Scale
Electronic Structure Calculations Using Linear
Scaling Methods G. Galli Physica Status. Solidi
(b) 217, 231 (2000) Recent progress in linear
scaling ab initio electronic structure
techniques D.R. Bowler, T. Miyazaki, M.J.
Gillan Journal Of Physics Condensed Matter, 14
(2002) 27812798 ONETEP linear-scaling
density-functional theory with plane-waves P.D.
Haynes, A.A. Mostofi, C.K. Skylaris, M.C.
Payne Journal of Physics Conference Series 26
(2006) 143148
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We may regard the present state of the universe
as the effect of its past and the cause of its
future. An intellect which at any given moment
knew all of the forces that animate nature and
the mutual positions of the beings that compose
it, if this intellect were vast enough to submit
the data to analysis, could condense into a
single formula the movement of the greatest
bodies of the universe and that of the lightest
atom for such an intellect nothing could be
uncertain and the future just like the past would
be present before its eyes. Marquis Pierre
Simon de Laplace, 1814
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