Linear scaling electronic structure methods in chemistry and physics Computing in Science & Engineering, vol. 5, issue 4, 2003 (1) Stefan Goedecker, Gustavo E. Scuseria Linear Scaling Density Functional Calculations with Gaussian Orbitals Journal of Physical Chemistry A, vol. 103, no. 25, 1999 (2) Gustavo E. Scuseria Linear Scaling Electronic Structure Methods Reviews of Modern Physics, Vol 71, No. 4, July 1999 (3) Stefan Goedecker METU Physics Department Nazım Dugan Linear scaling electronic structure methods
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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. Scuseria
Linear 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
DFT Journal Club METU Physics Department Nazım Dugan [email protected]
Linear scaling electronic structure methods
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 [email protected]
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 [email protected]
DFT Journal Club METU Physics Department
Exposition of the problem
Strategies for linear scaling
Benchmark Calculations
DFT Journal Club METU Physics Department
Outline
Exposition of the problem
DFT Journal Club METU Physics Department
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)
DFT Journal Club METU Physics Department
Exposition of the problem Strategies for linear scaling Benchmark Calculations
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)
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
2( 1)
2
N NN
2T N
3T N
DFT Journal Club METU Physics Department
Exposition of the problem Strategies for linear scaling Benchmark Calculations
DFT Journal Club METU Physics Department
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 GeneralDimensionality of Hamiltonian = 2NeEvaluation of elements ~ Ne 2Ne
Diagonalization method not known
| |i jH
| |i j k lH
Exposition of the problem Strategies for linear scaling Benchmark Calculations
DFT Journal Club METU Physics Department
Scaling of other electronic structure methods
Full configuration interaction exponential
Coupled-cluster O( N 6 )
Quantum Monte Carlo O( N 3 )
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
N*N*(N-1)/2 ~ N 3
TCPU = cc N 2 + cx N 2 + cm N 3 + co N 3
DFT Journal Club METU Physics Department
Exposition of the problem Strategies for linear scaling Benchmark Calculations
Applicability of cubic scaling methods
10 electrons ~ 1 seconds100 electrons ~ 16 minutes1000 electeons ~ 11.5 days 10000 electrons ~ 32 years
With linear scaling methodsup to 25000 atoms on 24 nodes of Earth Simulator
DFT Journal Club METU Physics Department
Exposition of the problem Strategies for linear scaling Benchmark Calculations
DFT Journal Club METU Physics Department
Crossover point
Cubic scaling TCPU = c3 N 3
Linear scaling TCPU = c1 N
José M. Soler
Universidad Autónoma de Madrid
Exposition of the problem Strategies for linear scaling Benchmark Calculations
DFT Journal Club METU Physics Department
Strategies for linear scaling
Exposition of the problem Strategies for linear scaling Benchmark Calculations
DFT Journal Club METU Physics Department
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 !!!
DFT Journal Club METU Physics Department
Locality (Nearsightedness) in QM
makes it possible to apply CUTOFF
W. Kohn, Phys. Rev. Lett. 76, 3168(1996)
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)
Exposition of the problem Strategies for linear scaling Benchmark Calculations
Two-body problem
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NeighborhoodKeep neighbor information for each particleCalculate interactions with neigbors only ( O(N) )Update neighbors ( O(N 2) )
Mesh techniqueDivide space into subspacesCalculate interactions only with particles
of owner and neighbor subspaces ( O(N) )Check subspace of particles ( O(N) )
Exposition of the problem Strategies for linear scaling Benchmark Calculations
DFT Journal Club METU Physics Department
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
Exposition of the problem Strategies for linear scaling Benchmark Calculations
Tancred Lindholm, N-body Algorithms, 1999
DFT Journal Club METU Physics Department
Traditional sequence
DM sequence
Density Matrix approach in DFT
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)
Exposition of the problem Strategies for linear scaling Benchmark Calculations
DFT Journal Club METU Physics Department
Exposition of the problem Strategies for linear scaling Benchmark Calculations
Density Matrix Minimization
E Hernandez, MJ Gillan, CM Goringe, Phys. Rev. B 55, 7147(1996)
- Express Total Energy in terms of density matrix - Minimize wrt density matrix
DFT Journal Club METU Physics Department
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 decayMetals at finite temperature – exponential decayMetals at zero temperature – algebraic decay
Progressive convergenceDynamical adjustment of treshold
DFT Journal Club METU Physics Department
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)
DFT Journal Club METU Physics Department
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)
DFT Journal Club METU Physics Department
Exposition of the problem Strategies for linear scaling Benchmark Calculations
Benchmark Calculations
DFT Journal Club METU Physics Department
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
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
DFT Journal Club METU Physics Department
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
DFT Journal Club METU Physics Department
Exposition of the problem Strategies for linear scaling Benchmark Calculations
2D3-21-G
2D6-31-G**
3D3-21-G
3D6-31-G**
CP
U T
ime
(min
)
number of basis functions
DFT Journal Club METU Physics Department
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.
DFT Journal Club METU Physics Department
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
DFT Journal Club METU Physics Department
References
Practical Methods for Ab Initio Calculations on Thousands of AtomsD.R. Bowler, I.J. Bush, M.J. Gillan International Journal of Quantum Chemistry, Vol. 77, 831–842 (2000)
Large-Scale Electronic Structure Calculations Using Linear Scaling MethodsG. GalliPhysica Status. Solidi (b) 217, 231 (2000)
Recent progress in linear scaling ab initio electronic structure techniquesD.R. Bowler, T. Miyazaki, M.J. GillanJournal Of Physics: Condensed Matter, 14 (2002) 2781–2798
ONETEP: linear-scaling density-functional theory with plane-wavesP.D. Haynes, A.A. Mostofi, C.K. Skylaris, M.C. PayneJournal of Physics: Conference Series 26 (2006) 143–148
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