Comp. Mat. Science School 20011 Linear Scaling ‘Order-N’ Methods in Electronic Structure Theory Richard M. Martin University of Illinois Acknowledgements:

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, ...


Linear Scaling ‘Order-N’ Methodsand Car-Parrinello Simulations

• Fundamental Issues of locality in quantum mechanics

• Paradigm for view of electronic properties• Practical Algorithms • Results


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”


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


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


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


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


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


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)


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


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


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)


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


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 -


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.


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


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


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


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


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).


Wannier Function in a-SiU. Stephan


Combination of O(N) Methods


Collision of C60 Buckyballs on DiamondGalli and Mauri, PRL 73, 3471 (1994)


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).


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


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


HOMO and LUMO in DNA (SIESTA code)

• Eigenstates found by N3 method after relaxation

• Could be O(N) for each state


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


FUTURE! ---- Biological Systems• Examples of oriented pigment molecules that today are being simulated by empirical potentials


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

Comp. Mat. Science School 20011 Linear Scaling ‘Order-N’ Methods in Electronic Structure Theory Richard M. Martin University of Illinois Acknowledgements:

Documents

density matrix r

r decays

r definition

insulator r

metal r

square of r

science school

r atom positions