linear scaling solvers based on wannier-like functions
DESCRIPTION
Linear scaling solvers based on Wannier-like functions. P. Ordejón Institut de Ciència de Materials de Barcelona (CSIC). Linear scaling = Order(N). CPU load. 3. ~ N. ~ N. Early 90’s. ~ 100. N (# atoms). Order-N DFT. Find density and hamiltonian (80% of code) - PowerPoint PPT PresentationTRANSCRIPT
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Linear scaling solvers based on Wannier-
like functions
P. Ordejón
Institut de Ciència de Materials de Barcelona (CSIC)
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Linear scaling = Order(N)
N (# atoms)
CPU load
~ 100
Early90’s
~ N
~ N3
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Order-N DFT
1. Find density and hamiltonian (80% of code)2. Find “eigenvectors” and energy (20% of
code)3. Iterate SCF loop
Steps 1 and 3 spared in tight-binding schemes
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Key to O(N): locality
``Divide and conquer’’ W. Yang, Phys. Rev. Lett. 66, 1438 (1992)
``Nearsightedness’’ W. Kohn, Phys. Rev. Lett. 76, 3168 (1996)
Large system
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Locality of Wave Functions
Ψ1
Ψ21 = 1/2 (Ψ1+Ψ2)
2 = 1/2 (Ψ1-Ψ2)
Wannier functions (crystals)Localized Molecular Orbitals (molecules)
occocc ψUχ
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Locality of Wave Functions
Energy:)(2211 HTrHHE occ
Unitary Transformation:2211)( HHHTrE occ
We do NOT need eigenstates! We can compute energy with Loc. Wavefuncs.
ii χψ
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Locality of Wave Functions
Exponential localization (insulators):
610-21
7.6
Wannier function in Carbon (diamond)Drabold et al.
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Locality of Wave Functions
Insulators vs Metals:
Carbon (diamond)
Aluminium
Goedecker & Teter, PRB 51, 9455 (1995)
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Linear Scaling
Localization + Truncation
1 2 34
5jiij HH ˆˆ
• Sparse Matrices
• Truncation errors
cRError expIn systems with a gap.Decay rate depends on gap Eg
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Linear Scaling Approaches
(Localized) object which is computed:- wave functions- density matrix
Approach to obtain the solution:- minimization- projection- spectral
Reviews on O(N) Methods: Goedecker, RMP ’98Ordejón, Comp. Mat. Sci.’98
rr exp)( )'(exp)'( rrrr
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Basis sets for linear-scaling DFT
• LCAO: - Gaussian based + QC machinery M. Challacombe, G. Scuseria, M. Head-
Gordon ... - Numerical atomic orbitals (NAO) SIESTA S. Kenny &. A Horsfield (PLATO) OpenMX• Hybrid PW – Localized orbitals - Gaussians J. Hutter, M. Parrinello
- “Localized PWs” C. Skylaris, P, Haynes & M. Payne• B-splines in 3D grid D. Bowler & M. Gillan• Finite-differences (nearly O(N)) J. Bernholc
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Divide and conquer
buffercentral
buffer
b c bx
x’
central buffer
Weitao Yang (1992)
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Fermi operator/projector
Goedecker & Colombo (1994)
f(E) = 1/(1+eE/kT) n cn En
F cn Hn
Etot = Tr[ F H ]
Ntot = Tr[ F ]
^^
^ ^Emin EF Emax
1
0
^
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Density matrix functional
-0.5 0 1 1.5
1
0
Li, Nunes & Vanderbilt (1993)
= 3 2 - 2 3
Etot() = H = min
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Wannier O(N) functional• Mauri, Galli & Car, PRB 47, 9973 (1993)
• Ordejón et al, PRB 48, 14646 (1993)
Sij = < i | j > | ’k > = j | j > Sjk-1/2
EKS = k < ’k | H | ’k >
= ijk Ski-1/2 < i | H | j > Sjk
-1/2
= Trocc[ S-1 H ] Kohn-Sham
EOM = Trocc[ (2I-S) H ] Order-N
= Trocc[ H] + Trocc[(I-S) H ]
^
^
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Order-N vs KS functionals
O(N)
KS
Non-orthogonality
penalty
Sij = ij EOM = EKS
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Chemical potentialKim, Mauri & Galli, PRB 52, 1640 (1995)
(r) = 2ij i(r) (2ij-Sij) j(r)
EOM = Trocc[ (2I-S) H ] # states = # electron pairs
Local minima
EKMG = Trocc+[ (2I-S) (H-S) ] # states > # electron pairs
= chemical potential (Fermi energy)
Ei > |i| 0
Ei < |i| 1
Difficulties Solutions
Stability of N() Initial diagonalization / Estimate of
First minimization of EKMG Reuse previous solutions
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Orbital localization
i
Rcrc
i(r) = ci (r)
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Convergence with localisation radius
Rc (Ang)
Relative
Error
(%)
Si supercell, 512 atoms
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Sparse vectors and matrices
2.1271.8535.372xi
02.12
000
1.855.37
0
1.1583.1448.293yi
8.29 1.85 = 15.343.14 0 = 0
1.15 0 = 0 ------- Sum 15.34
x
Restore to zero xi 0 only
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Actual linear scaling
Single Pentium III 800 MHz. 1 Gb RAM
c-Si supercells, single-
132.000 atoms in 64 nodes
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Linear scaling solver: practicalities in
SIESTA
P. Ordejón
Institut de Ciència de Materials de Barcelona (CSIC)
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Order-N in SIESTA (1)
Calculate Hamiltonian
Minimize EKS with respect to WFs (GC minimization)
Build new charge density from WFs
SCF
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Energy Functional Minimization• Start from initial LWFs (from scratch or from previous step)
• Minimize Energy Functional w.r.t. ci
EOM = Trocc[ (2I-S) H ] or
EKMG = Trocc+[ (2I-S) (H-S) ]
• Obtain new density
(r) = 2ij i(r) (2ij-Sij) j(r)
i(r) = ci (r)
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Orbital localization
i
Rcrc
i(r) = ci (r)
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Order-N in SIESTA (2)• Practical problems:
– Minimization of E versus WFs:
• First minimization is hard!!! (~1000 CG iterations)• Next minimizations are much faster (next SCF and MD steps)• ALWAYS save SystemName.LWF and SystemName.DM files!!!!
– The Chemical Potential (in Kim’s functional):
• Data on input (ON.Eta). Problem: can change during SCF and dynamics.
• Possibility to estimate the chemical potential in O(N) operations• If chosen ON.Eta is inside a band (conduction or valence), the
minimization often becomes unstable and diverges• Solution I: use chemical potential estimated on the run• Solution II: do a previous diagonalization
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Example of instability related to a wrong chemical potential
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Order-N in SIESTA (3)
• SolutionMethod OrderN• ON.Functional Ordejon-Mauri or Kim
(def)• ON.MaxNumIter Max. iterations in
CG minim. (WFs)• ON.Etol Tolerance in the energy
minimization2(En-En-1)/(En+En-1) < ON.Etol
• ON.RcLWF Localisation radius of WFs
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Order-N in SIESTA (4)
• ON.Eta (energy units) Chemical Potential (Kim) Shift of Hamiltonian (Ordejon-Mauri)
• ON.ChemicalPotential• ON.ChemicalPotentialUse• ON.ChemicalPotentialRc• ON.ChemicalPotentialTemperature• ON.ChemicalPotentialOrder
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Fermi operator/projectorGoedecker & Colombo (1994)
f(E) = 1/(1+eE/kT) n cn En
F cn Hn
Etot = Tr[ F H ]
Ntot = Tr[ F ]
^^
^ ^Emin EF Emax
1
0
^