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![Page 1: Comp. Mat. Science School 20011 Linear Scaling ‘Order-N’ Methods in Electronic Structure Theory Richard M. Martin University of Illinois Acknowledgements:](https://reader035.fdocuments.in/reader035/viewer/2022081516/56649f485503460f94c6a11d/html5/thumbnails/1.jpg)
Comp. Mat. Science School 2001 1
Linear Scaling ‘Order-N’ Methodsin Electronic Structure Theory
Richard M. Martin
University of Illinois
Acknowledgements:Pablo Ordejon David Drabold
Matthew Grumbach Uwe StephanDaniel Sanchez-PortalSatoshi Itoh
Thanks to: Jose Soler, Emilio Artacho, Giulia Galli, ...
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Comp. Mat. Science School 2001 2
Linear Scaling ‘Order-N’ Methodsand Car-Parrinello Simulations
• Fundamental Issues of locality in quantum mechanics
• Paradigm for view of electronic properties• Practical Algorithms • Results
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Comp. Mat. Science School 2001 3
Locality in Quantum Mechanics
• V. Heine (Sol. St. Phys. Vol. 35, 1980)“Throwing out k-space”Based on ideas of Friedel (1954) , . . .
• Many properties of electrons in any region are independent of distant regions
• Walter Kohn “Nearsightness”
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Comp. Mat. Science School 2001 4
Locality in Quantum Mechanics
• Which properties of electrons are independent of distant regions?
• Total integrated quantitiesDensity, Forceson atoms, . . .
• Coulomb Forces are long range but they can be handled in O(N) fashion just as in classical systems
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Comp. Mat. Science School 2001 5
Non-Locality in Quantum Mechanics
• Which properties of electrons are non-local?• Individual Eigenstates in crystals• Sharp features of the Fermi surface at low T • Electrical Conductivity at T=0
Metals vs insulators: distinguished by delocalization of eigenstates at the Fermi energy (metals) vs localization of the entire many-electron system (insulators)
• Approach in the Order-N methods: Identify localized and delocalized aspects
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Comp. Mat. Science School 2001 6
Density Matrix I
• Key property that describes the range of the non-locality is the density matrix (r,r’)
• In an insulator (r,r’) is exponentially localized• In a metal (r,r’) decays as a power law at T = 0, exponentially
for T > 0. (Goedecker, Ismail-Beigi)
• For non-interacting Bosons or Fermions, Landau and Lifshitz show that the correlation function g(r,r’) is uniquely related to the square of (r,r’)
• Thus correlation lengths and the density matrix generally become shorter range at high T
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Comp. Mat. Science School 2001 7
Density Matrix II
• Key property that describes the range of the non-locality is the density matrix (r,r’)
• Definition: (r,r’) = i i *(r) i (r’)
• Can be localized even if each i *(r) is not!
r fixed at r =0
r’
Atom positions
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Comp. Mat. Science School 2001 8
Toward Working Algorithms I
(My own personal view)
Heine and Haydock laid the groundwork - but it was applied only to limited Hamiltonians, ….
1985 - Car-Parrinello Methods changed the picture
Key quantity is the total energy E[{i}] which does not require eigenstates - only traces over the occupied states - the {i} can be linear combinations of eigenstates
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Comp. Mat. Science School 2001 9
Toward Working Algorithms II
How can we use the advantages of the Car-Parrinello and the local approaches?
1992 - Galli and Parrinello pointed out the key idea -
to make a Car-Parrinello algorithm that takes advantage of the locality
Require that the states in localized.
Note this does not require a localized basis - it may be very convenient, but a localized basis is not essential to construct localized states (example: sum of plane waves can be localized)
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Comp. Mat. Science School 2001 10
Toward Working Algorithms III
What are localized combinations of the eigenfunctions?
Wannier Functions (generalized)!
Wannier Functions span the same space as the eigenstates - all traces are the same
Wannier FunctionsOne localized Wannier Ftn centered on each site
Extended Bloch Eigenfunctions
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Comp. Mat. Science School 2001 11
Toward Working Algorithms IV
Can work with either localized Wannier functions wi (r)
or
localized density matrix (r,r’) = i i
*(r) i (r’) = i wi *(r) wi (r’)
Functions of one variableBut not unique
Functions of two variables - more complexBut unique
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Comp. Mat. Science School 2001 12
Linear Scaling ‘Order-N’ Methods
• Computational complexity ~ N = number of atoms (Current methods scale as N2 or N3)
• Intrinsically Parallel• “Divide and Conquer” • Green’s Functions• Fermi Operator Expansion• Density matrix “purification”• Generalized Wannier Functions• Spectral “Telescoping”
(Review by S. Goedecker in Rev Mod Phys)
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Comp. Mat. Science School 2001 13
Divide and Conquer (Yang, 1991)
• Divide System into (Overlapping) Spatial Regions. Solve each region in terms only of its neighbors.(Terminate regions suitably)
• Use standard methods for each region
• Sum charge densities to get total density, Coulomb terms
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Comp. Mat. Science School 2001 14
Expansion of the Fermi function
• Sankey, et al (1994); Goedecker, Colombo (1994); Wang et al (1995)
• Explicit T nonzero• Projection into the occupied Subspace• Multiply trial function by “Fermi operator”:
F = [(H - EF)/KBT +1]-1 • Localized leads to localized projection since the Fermi
operator (density matrix) is localized• Accomplish by expanding F in power series in H operator -
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Comp. Mat. Science School 2001 15
Density Matrix “Purification”
• Li, Nunes, Vanderbilt (1993); Daw (1993)Hernandez, Gillan (1995)
• Idea: A density matrix at T=0 has eigenvalues = occupation = 0 or 1
• Suppose we have an approximate that does not have this property
• The relation n+1 = 3 (n )2 - 2 (n )3 always produces a new matrix with eigenvalues closer to 0 or 1.
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Comp. Mat. Science School 2001 16
Density Matrix “Purification”
• The relation n+1 = 3 (n )2 - 2 (n )3 always produces a new matrix with eigenvalues closer to 0 or 1.
x
3 x2 - 2 x3
1
1
instability
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Comp. Mat. Science School 2001 17
Generalized Wannier Functions
• Divide System into (Overlapping) Spatial Regions.
• Require each Wannier function to be non-zero only in a given region
• Solve for the functions in each region requiring each to be orthogonal to the neighboring functions
• New functional invented to allow direct minimization without explicitly requiring orthogonalization
• Mauri, et al.; Ordejon, et al; 1993; Stechel, et al 1994; Kim et al 1995
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Comp. Mat. Science School 2001 18
Generalized Wannier Functions
• Factorization of the density matrix (r,r’) = i wi* (r ) wi(r’)
• Can chose localized Wannier functions (really linear combinations of Wannier functions)
• Minimize functional:E = Tr [ (2 - S) H]
• Since this is a variational functional, the Car-Parrinello method can be used to use one calculation as the input to the next
• Mauri, et al.; Ordejon, et al; 1993; Stechel, et al 1994; Kim et al 1995
Overlap matrix
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Comp. Mat. Science School 2001 19
Functional (2-S)(H - EF)
• Minimization leads to orthonormal filled orbitals focres empty orbitals to have zero amplitude
• Each matrix element (S and H) contains two factors of the wavefunction - amplitude ~ x.
• For occupied states (eigenvalues below EF)
x
- ( 2 x2 - x4 )
1
1
Minimum for normalizedwavefunction (x = 1)
Minimum at zero for empty states above EF
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Comp. Mat. Science School 2001 20
Example of Our workPrediction of Shapes of Giant Fullerenes
S. Itoh, P. Ordejon, D. A. Drabold and R. M. Martin, Phys Rev B 53, 2132 (1996).See also C. Xu and G. Scuceria, Chem. Phys. Lett. 262, 219 (1996).
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Comp. Mat. Science School 2001 21
Wannier Function in a-SiU. Stephan
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Comp. Mat. Science School 2001 22
Combination of O(N) Methods
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Comp. Mat. Science School 2001 23
Collision of C60 Buckyballs on DiamondGalli and Mauri, PRL 73, 3471 (1994)
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Comp. Mat. Science School 2001 24
Deposition of C28 Buckyballs on Diamond
• Simulations with ~ 5000 atoms, TB Hamiltonian from Xu, et al. ( A. Canning, G.~Galli and J .Kim, Phys.Rev.Lett. 78, 4442 (1997).
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Comp. Mat. Science School 2001 25
Example of DFT Simulation (not order N)
• Daniel Sanchez-Portal(Phys. Rev. Lett. 1999)
• Simulation of a gold nanowire pulled between two gold tips
• Full DFT simulation
• Explanation for very puzzling experiment! Thermal motion of the atoms makes some appear sharp, others weak in electron microscope
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Comp. Mat. Science School 2001 26
Simulations of DNA with the SIESTA code
• Machado, Ordejon, Artacho, Sanchez-Portal, Soler (preprint)
• Self-Consistent Local Orbital O(N) Code
• Relaxation - ~15-60 min/step (~ 1 day with diagonalization)
Iso-density surfaces
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Comp. Mat. Science School 2001 27
HOMO and LUMO in DNA (SIESTA code)
• Eigenstates found by N3 method after relaxation
• Could be O(N) for each state
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Comp. Mat. Science School 2001 28
O(N) Simulation of Magnets at T > 0
• Collaboration at ORNL, Ames, Brookhaven• Snapshot of magnetic order in a finite temperature simulation of paramagnetic Fe. These
calculations represent significant progress towards the goal of full implementation of a first principles theory of the finite temperature and non-equilibrium properties of magnetic materials.
• Record setting performance for large unit cell models (up to 1024-atoms) led to the award of the 1998 Gordon Bell prize.
• The calculations that were the basis for the award were performed using the locally self-consistent multiple scattering method, which is an O(N) Density Functional method
• Web Site: http://oldpc.ms.ornl.gov/~gms/MShome.html
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Comp. Mat. Science School 2001 29
FUTURE! ---- Biological Systems• Examples of oriented pigment molecules that today are being simulated by empirical potentials
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Comp. Mat. Science School 2001 30
Conclusions• It is possible to treat many thousands of atoms in a
full simulation - on a workstation with approximate methods - intrinsically parallel for a supercomputer
• Why treat many thousands of atoms?
• Large scale structures in materials - defects, boundaries, ….
• Biological molecules
• The ideas are also relevant to understanding even small systems