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Large-Scale Density Functional Calculations
James E. Raynolds, College of Nanoscale Science and Engineering
Lenore R. Mullin, College of Computing and Information
Overview
• Using computers to carry out “numerical experiments”
in Materials Science, Chemistry and Physics• Quantum Mechanical equations solved for a
system of atoms in a representative unit cell• Measurable properties obtained from
“first-principles”– mechanical, thermodynamic, electronic– optical, magnetic, transport
Example: Transport in molecular wire
Benzene
Phenolate/Benzenediazonium+ V
Peierls DistortionPi stacked pair dimerized pair
metal insulator
mechanical relaxation
Frontier Problems
• Non-equilibrium spin-transport in metals and semiconductors (Spintronics)
• Transport and coupled mechanical / electronic interactions in molecules (metal - insulator transition due to mechanical relaxation)
• Industrial applications: Modeling Chemical Vapor Deposition (CVD) processes atom by atom
• Challenges: correlated motion of electrons• Coupled electron-phonon interactions
(electron - vibration coupling)
Density Functional Theory
• Density Functional Theory (DFT) is a “mean-field” solution to the many-electron problem.
• Each electron interacts with an effective average field produced by all of the other electrons
• Non-linear set of coupled differential equations
Density Functional Equations
€
−∇2 + V (r r )( )Ψ j (
r r ) = E jΨ j (
r r )
Looks linear but...
€
V (r r ) depends on the charge density
€
ρ(r r )
through:
€
ρ(r r ) = Ψ j (
r r )
j
∑2
Example: Local density approximation
€
V (r r ) = d3s
ρ (r s )
r r −
r s
∫ +δ
δρ (r r )
d3∫ sρ (r r )εxc (ρ(
r r ))
DFT solution approach
• Expand the wave-functions in a basis set:
• Matrix eigenvalue-eigenvector problem:
• Orthogonality:
• Iterative solution to “self-consistency” (i.e. output V(r) coincides with input)
€
Ψj (r r ) = Cl
j
l
∑ ϕ l (r r )
€
H jl
l
∑ Clj = EC j
j
€
Clk
( )l
∑*Cl
j = δkj
Popular implementations
• Plane wave basis functions (Fourier Series):
– Drawback: – Benefit: easy to code, sophisticated non-linear
response calculations possible
• Localized “atomic-like” basis functions
€
ϕ j (r r ) =
1
Vexp(ik j •
r r )
€
O(N 3) scaling
€
ϕ j (r r ) = a j (
r r ) - exponential distance decay for
insulators- power law distance decay for - metals
Contrasting Implementations
• Abinit: www.abinit.org– Very sophisticated array of calculated properties– Calculations become prohibitive for more than a few dozen
atoms • VASP (Vienna Ab-Initio Simulation Package)
– Less sophisticated by much faster– few hundred atoms possible
• Siesta: (Spanish Initiative for Electronic Simulations with Thousands of Atoms)– O(N) scaling: fast but less sophisticated– few thousand atoms possible
€
O(N 3)
€
O(N 3)
Public Access
• Many codes are freely available: go to http://psi-k.dl.ac.uk/data/codes.html for a list of more than 20
• Most codes still not user-friendly and take months to years to master
The Brick Wall!!
• All of these methods run out of steam very quickly in terms of run time and memory
• Calculations with scaling take days or weeks to run!!
• Even calculations with scaling run into memory bottlenecks
• Materials Science simulations require thousands of atoms for thousands of time steps
€
O(N 3)
€
O(N)
Key Algorithms
• For plane wave based codes:
the Fast Fourier Transform– We have gained factor’s of 4 improvement in
speed and storage using Conformal Computing– A number of new developments are being
implemented for further increases
• Matrix diagonalization routines for very large matrices
Conformal Computing
• Density Functional Calculations are an ideal setting for Conformal Computing!
• In fact: any array (matrix) based computational setting is ripe for Conformal Computing
• Why? Conformal Computing eliminates temporary arrays and un-necessary loops!
Opportunities
• Current electronic band structures fairly fast (on the order of one hour):
Contrasting: electron-phonon
• Electron-phonon calculations: on the order of 1 day for small systems
• Superconductivity in “conventional” materialsdetermined by the electron -phonon interaction• Aluminum (1 atom) takes roughly 1 day of computing• Imagine several dozen atomswith scaling
€
O(N 3)
Electron-Phonon improvements
• Many quantities currently written to files then later combined
• The size and number of these files is becoming prohibitively expensive
• Opportunities for parallelization of integrals
• Opportunities to eliminate temporaries through the use of direct indexing
Grid Computing
• Even with highly optimized code (which is still a way off) there is always a need for more and more resources
• For example: electron-phonon calculations involve dozens of separate calculations that could be run on independent machines
• Grid computing allows many independent calculations to be run in parallel
Grid Computing: First Steps
• QMolDyn GAT: a template for submitting Density Functional Calculations over the grid
• Vision: QMolDyn will eventually have a variety of codes (modules)
• Presently: Siesta ( ) running on the grid, 8, 16, 32, 64, 128, 256, 512- atom systems
€
O(N)
Summary / Conclusions
• There is a great demand for large-scale array (matrix) based calculations in materials science
• Quantum calculations are increasingly important for Materials Science, Chemistry and Physics
• Grid computing combined with Conformal Computing techniques is very promising