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 DFT Journal Club METU Physics Department Nazım Dugan M ail.com 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 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 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 T CPU = 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 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 General Dimensionality of Hamiltonian = 2Ne Evaluation of elements ~ Ne 2Ne Diagonalization method not known 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 T CPU = c c N 2 + c x N 2 + c m N 3 + c o 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 seconds 100 electrons ~ 16 minutes 1000 electeons ~ 11.5 days electrons ~ 32 years With linear scaling methods up to 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 T CPU = c 3 N 3 Linear scaling T CPU = c 1 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 DFT Journal Club METU Physics Department 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) ) 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 MatrixIn 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

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 k th column, this localization region will be centered on the k th 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 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 basis functions) Hardware : SGI Origin 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 2D 3-21-G 2D 6-31-G** 3D 3-21-G 3D 6-31-G** CPU Time (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 Atoms D.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 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) 2781–2798 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) 143–148

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