Post on 04-Jul-2020
Numerical Methods for Kohn-Sham DensityFunctional Theory
Jianfeng Lu (é²åé)
Duke Universityjianfeng@math.duke.edu
January 2019, Banff Workshop on âOptimal Transport Methods in DensityFunctional Theoryâ
ReferenceLin Lin, L., and Lexing Ying, Numerical Methods for Kohn-Sham DensityFunctional Theory, Acta Numerica, 2019 (under review)
Electronic structure theory
We consider the electronic structure, which is given by the many-bodyelectronic Schrödinger equation
(â12
ð
âð=1
Îð¥ð +ð
âð=1
ðext(ð¥ð) + â1â€ð<ðâ€ð
1|ð¥ð â ð¥ð|)Κ(ð¥1, ⯠, ð¥ð ) = ðžÎš
under non-relativistic and Born-Oppenheimer approximations.Here the atom types and positions enter ðext as parameters.
P.A.M. Dirac, 1929The fundamental laws necessary to the mathematical treatment of largeparts of physics and the whole of chemistry are thus fully known, and thedifficult lies only in the fact that application of these laws leads toequations that are too complex to be solved.
Electronic structure theory
We consider the electronic structure, which is given by the many-bodyelectronic Schrödinger equation
(â12
ð
âð=1
Îð¥ð +ð
âð=1
ðext(ð¥ð) + â1â€ð<ðâ€ð
1|ð¥ð â ð¥ð|)Κ(ð¥1, ⯠, ð¥ð ) = ðžÎš
under non-relativistic and Born-Oppenheimer approximations.Here the atom types and positions enter ðext as parameters.
Still too complex to be solved, even after almost 90 years:⢠Curse of dimensionality;⢠Anti-symmetry of Κ due to Pauliâs exclusion principle
Κ(ð¥1, ⯠, ð¥ð, ⯠, ð¥ð , ⯠, ð¥ð ) = âΚ(ð¥1, ⯠, ð¥ð , ⯠, ð¥ð, ⯠, ð¥ð );
⢠Κ has complicated singularity structure.
Solving electronic Schrödinger equations:⢠Tight-binding approximations (LCAO)⢠Density functional theory
Orbital-free DFTKohn-Sham DFT
⢠Wavefunction methodsHartree-FockMÞller-Plesset perturbation theoryConfiguration interactionCoupled clusterMulti-configuration self-consistent fieldNeural network ansatz
⢠GW approximation; Bethe-Salpeter equation⢠Quantum Monte Carlo (VMC, DMC, etc.)⢠Density matrix renormalization group (DMRG) / tensor networks⢠...
Solving electronic Schrödinger equations:⢠Tight-binding approximations (LCAO)⢠Density functional theory
Orbital-free DFTKohn-Sham DFT
10 of 18 most cited papers in physics [Perdew 2010]More than 50, 000 citations on Google scholar
Figure: DFT papers count [Burke 2012].
Density functional theory [Hohenberg-Kohn 1964, Kohn-Sham 1965]: Viewenergy as a functional of the one-body electron density ð ⶠâ3 â ââ¥0:
ð(ð¥) = ð â«|Κ|2(ð¥, ð¥2, ⯠, ð¥ð ) dð¥2 ⯠dð¥ð .
Levy-Lieb variational principle [Levy 1979, Lieb 1983]:
ðž0 = infΚ
âšÎš, ð»Îšâ© = infð
infΚ,ΚâŠð
âšÎš, ð»Îšâ© = infð
ðž(ð).
The energy functional has the form
ðž(ð) = ðð (ð) + â« ððext + 12 â¬
ð(ð¥)ð(ðŠ)|ð¥ â ðŠ| + ðžxc(ð)
ðð (ð): Kinetic energy of non-interacting electrons;ðžxc(ð): Exchange-correlation energy, which encodes the many-bodyinteraction between electrons (chemistry).
Density functional theory [Hohenberg-Kohn 1964, Kohn-Sham 1965]: Viewenergy as a functional of the one-body electron density ð ⶠâ3 â ââ¥0:
ð(ð¥) = ð â«|Κ|2(ð¥, ð¥2, ⯠, ð¥ð ) dð¥2 ⯠dð¥ð .
Levy-Lieb variational principle [Levy 1979, Lieb 1983]:
ðž0 = infΚ
âšÎš, ð»Îšâ© = infð
infΚ,ΚâŠð
âšÎš, ð»Îšâ©
= infð
ðž(ð).
The energy functional has the form
ðž(ð) = ðð (ð) + â« ððext + 12 â¬
ð(ð¥)ð(ðŠ)|ð¥ â ðŠ| + ðžxc(ð)
ðð (ð): Kinetic energy of non-interacting electrons;ðžxc(ð): Exchange-correlation energy, which encodes the many-bodyinteraction between electrons (chemistry).
Density functional theory [Hohenberg-Kohn 1964, Kohn-Sham 1965]: Viewenergy as a functional of the one-body electron density ð ⶠâ3 â ââ¥0:
ð(ð¥) = ð â«|Κ|2(ð¥, ð¥2, ⯠, ð¥ð ) dð¥2 ⯠dð¥ð .
Levy-Lieb variational principle [Levy 1979, Lieb 1983]:
ðž0 = infΚ
âšÎš, ð»Îšâ© = infð
infΚ,ΚâŠð
âšÎš, ð»Îšâ© = infð
ðž(ð).
The energy functional has the form
ðž(ð) = ðð (ð) + â« ððext + 12 â¬
ð(ð¥)ð(ðŠ)|ð¥ â ðŠ| + ðžxc(ð)
ðð (ð): Kinetic energy of non-interacting electrons;ðžxc(ð): Exchange-correlation energy, which encodes the many-bodyinteraction between electrons (chemistry).
Density functional theory [Hohenberg-Kohn 1964, Kohn-Sham 1965]: Viewenergy as a functional of the one-body electron density ð ⶠâ3 â ââ¥0:
ð(ð¥) = ð â«|Κ|2(ð¥, ð¥2, ⯠, ð¥ð ) dð¥2 ⯠dð¥ð .
Levy-Lieb variational principle [Levy 1979, Lieb 1983]:
ðž0 = infΚ
âšÎš, ð»Îšâ© = infð
infΚ,ΚâŠð
âšÎš, ð»Îšâ© = infð
ðž(ð).
The energy functional has the form
ðž(ð) = ðð (ð) + â« ððext + 12 â¬
ð(ð¥)ð(ðŠ)|ð¥ â ðŠ| + ðžxc(ð)
ðð (ð): Kinetic energy of non-interacting electrons;ðžxc(ð): Exchange-correlation energy, which encodes the many-bodyinteraction between electrons (chemistry).
Kohn-Sham density functional theoryKohn-Sham density functional theory introduces one-particle orbitals tobetter approximate the kinetic and exchange-correlation energies.
It is today the most widely used electronic structure theory, which achievesthe best compromise between accuracy and cost.
The energy functional is minimized for ð orbitals {ðð} â ð»1(â3).
ðžKS({ðð}) =ð
âð=1
12 â«|âðð|2 + â« ððext + 1
2 â¬ð(ð¥)ð(ðŠ)|ð¥ â ðŠ| + ðžxc(ð)
where (we consider âspin-lessâ electrons through the talk)
ð(ð¥) =ð
âð=1
|ðð|2(ð¥).
Note that ðžKS can be still viewed as a functional of ð, implicitly.
Kohn-Sham density functional theory
ðžKS({ðð}) =ð
âð=1
12 â«|âðð|2 + â« ððext + 1
2 â¬ð(ð¥)ð(ðŠ)|ð¥ â ðŠ| + ðžxc
The exchange-correlation functionals counts for the corrections from themany-body interactions between electrons:
⢠Semi-local exchange-correlation functionalsLocal density approximation: ðžxc = â« ðxc(ð(ð¥))Generalized gradient approximation:
ðžxc = â« ðxc(ð(ð¥), 12 |ââð(ð¥)|2)
⢠Nonlocal exchange-correlation functionals
Kohn-Sham density functional theory
ðžKS({ðð}) =ð
âð=1
12 â«|âðð|2 + â« ððext + 1
2 â¬ð(ð¥)ð(ðŠ)|ð¥ â ðŠ| + ðžxc
⢠Nonlocal exchange-correlation functionalse.g., exact exchange + RPA correlation, ðžxc = ðžð¥ + ðžð:
ðžð¥ = â12
ð
âð,ð=1 â¬
ðâð (ð¥)ðð(ð¥)ðð(ðŠ)ðâ
ð (ðŠ)|ð¥ â ðŠ| dð¥ dðŠ;
ðžð = 12ð â«
â
0tr[ln(1 â ð0(ðð)ð) + ð0(ðð)ð] dð,
with ð the Coulomb operator and ð0 Kohn-Sham polarizabilityoperator:
ð0(ð¥, ðŠ, ðð) = 2occ
âð
unocc
âð
ðâð (ð¥)ðð(ð¥)ðâ
ð (ðŠ)ðð(ðŠ)ðð â ðð â ðð .
Kohn-Sham density functional theory has a similar structure as a meanfield type theory (at least for semilocal xc): The electrons interact throughan effective potential.The electron density is given by an effective Hamiltonian:
ð»eff(ð)ðð = ðððð, ð(ð¥) =occ
âð
|ðð|2(ð¥);
where the first ð orbitals are occupied (aufbau principle). The effectiveHamiltonian captures the interactions of electrons:
ð»eff(ð) = â12Î + ðeff(ð);
ðeff(ð) = ðð(ð) + ðxc(ð).
Note that this is a nonlinear eigenvalue problem. We can view it as afixed-point equation for the density ð:
ð = ð¹KS(ð).
Kohn-Sham mapKohn-Sham fixed-point equation
ð = ð¹KS(ð),
where ð¹KS is known as the Kohn-Sham map, defined through theeigenvalue problem associated with ð»eff(ð).Given an effective Hamiltonian ð»eff, we ask for its low-lying eigenspace:the range of the spectral projection
ð = ð(ââ,ðð¹ ](ð»eff),
thus ð is given as the diagonal of the kernel of the operator ð .
It is often more advantageous to represent ð in terms of Greenâs functions(ð â ð»eff)â1, so to turn the eigenvalue problem into solving equations.Let ð be a contour around the occupied spectrum, we have
ð = ð(ââ,ðð¹ ](ð»eff) = 12ðð€ â®ð
(ð â ð»eff)â1 dð.
Kohn-Sham mapKohn-Sham fixed-point equation
ð = ð¹KS(ð),
where ð¹KS is known as the Kohn-Sham map, defined through theeigenvalue problem associated with ð»eff(ð).Given an effective Hamiltonian ð»eff, we ask for its low-lying eigenspace:the range of the spectral projection
ð = ð(ââ,ðð¹ ](ð»eff),
thus ð is given as the diagonal of the kernel of the operator ð .It is often more advantageous to represent ð in terms of Greenâs functions(ð â ð»eff)â1, so to turn the eigenvalue problem into solving equations.Let ð be a contour around the occupied spectrum, we have
ð = ð(ââ,ðð¹ ](ð»eff) = 12ðð€ â®ð
(ð â ð»eff)â1 dð.
Numerical methods
Kohn-Sham fixed-point equation
ð = ð¹KS(ð).
Usually solved numerically based on self-consistent field (SCF) iteration(alternative methods, such as direct minimization, are not as widely used)
â see the talk of Ziad Musslimani
Numerical methods
Kohn-Sham fixed-point equation
ð = ð¹KS(ð).
Usually solved numerically based on self-consistent field (SCF) iteration(alternative methods, such as direct minimization, are not as widely used)
⢠Self-consistent iterationLinearize the nonlinear problem to a linear one at each step
⢠DiscretizationMaking the problem finite dimensional
⢠Evaluation of the Kohn-Sham mapFormation of effective Hamiltonian: ð ⊠ð»eff(ð)Evaluation of density: ð»eff ⊠ðMost of the actual computational work.
Self-consistent iterationSolving for the fixed point equation ð = ð¹KS(ð).(Note: often more convenient to view the iteration in terms of ðeff)
⢠First idea: Fixed point iterationðð+1 = ð¹KS(ðð)
Usually does not converge as ð¹KS is not necessarily a contraction.
⢠Simple mixingðð+1 = (1 â ðŒ)ðð + ðŒð¹KS(ðð)
= ðð â ðŒ(ðð â ð¹KS(ðð))where ðð â ð¹KS(ðð) is the residual.ðŒ = 1 corresponds to fixed-point iteration.
⢠Preconditioningðð+1 = ðð â ð« (ðð â ð¹KS(ðð))
e.g., ð« = ðŒðŒ corresponds to simple mixing.Kerker mixing for simple metals (jellium like)â see the talk of Antoine Levitt
Self-consistent iterationSolving for the fixed point equation ð = ð¹KS(ð).(Note: often more convenient to view the iteration in terms of ðeff)
⢠First idea: Fixed point iterationðð+1 = ð¹KS(ðð)
Usually does not converge as ð¹KS is not necessarily a contraction.⢠Simple mixing
ðð+1 = (1 â ðŒ)ðð + ðŒð¹KS(ðð)= ðð â ðŒ(ðð â ð¹KS(ðð))
where ðð â ð¹KS(ðð) is the residual.ðŒ = 1 corresponds to fixed-point iteration.
⢠Preconditioningðð+1 = ðð â ð« (ðð â ð¹KS(ðð))
e.g., ð« = ðŒðŒ corresponds to simple mixing.Kerker mixing for simple metals (jellium like)â see the talk of Antoine Levitt
Self-consistent iterationSolving for the fixed point equation ð = ð¹KS(ð).(Note: often more convenient to view the iteration in terms of ðeff)
⢠First idea: Fixed point iterationðð+1 = ð¹KS(ðð)
Usually does not converge as ð¹KS is not necessarily a contraction.⢠Simple mixing
ðð+1 = (1 â ðŒ)ðð + ðŒð¹KS(ðð)= ðð â ðŒ(ðð â ð¹KS(ðð))
where ðð â ð¹KS(ðð) is the residual.ðŒ = 1 corresponds to fixed-point iteration.
⢠Preconditioningðð+1 = ðð â ð« (ðð â ð¹KS(ðð))
e.g., ð« = ðŒðŒ corresponds to simple mixing.Kerker mixing for simple metals (jellium like)â see the talk of Antoine Levitt
⢠Newton method
ðð+1 = ðð â ð¥ â1ð (ðð â ð¹KS(ðð)
where ð¥ð is the Jacobian matrix.Locally quadratic convergence, but each iteration is quite expensive(involves iterative methods e.g., GMRES and finite differenceapproximation to Jacobian matrix)
⢠Quasi-Newton type algorithms (the most widely used approach)Broydenâs second method [Fang, Saad 2009; Marks, Luke 2008]Low-rank update to Ìð¥ â1
ð (proxy for inverse Jacobian)Anderson mixing [Anderson 1965] (a variant of Broyden)Direct Inversion of Iterative Subspace (DIIS) [Pulay 1980]Minimize a linear combination of residual to get linear combinationcoefficients; other variants include E-DIIS [Cances, Le Bris 2000].commutator-DIIS [Pulay 1983] (for nonlocal functionals)Choice of residual as the commutator of ð»eff[ð ] and ð
⢠Newton method
ðð+1 = ðð â ð¥ â1ð (ðð â ð¹KS(ðð)
where ð¥ð is the Jacobian matrix.Locally quadratic convergence, but each iteration is quite expensive(involves iterative methods e.g., GMRES and finite differenceapproximation to Jacobian matrix)
⢠Quasi-Newton type algorithms (the most widely used approach)Broydenâs second method [Fang, Saad 2009; Marks, Luke 2008]Low-rank update to Ìð¥ â1
ð (proxy for inverse Jacobian)Anderson mixing [Anderson 1965] (a variant of Broyden)Direct Inversion of Iterative Subspace (DIIS) [Pulay 1980]Minimize a linear combination of residual to get linear combinationcoefficients; other variants include E-DIIS [Cances, Le Bris 2000].commutator-DIIS [Pulay 1983] (for nonlocal functionals)Choice of residual as the commutator of ð»eff[ð ] and ð
Discretization
⢠Large basis sets (ðª(100) ⌠ðª(10, 000) basis functions per atom)Planewave method (Fourier basis set / pseudospectral method)[Payne et al 1992, Kresse and FurthmÃŒller 1996]Finite element method[Tsuchida and Tsukada 1995, Suryanarayana et al 2010, Bao et al 2012]Wavelet method [Genovese et al 2008]Finite different method [Chelikowski et al 1994]
⢠Small basis sets (ðª(10) ⌠ðª(100) basis functions per atom)Gaussian-type orbitals (GTO) (see review [Jensen 2013])Numerical atomic-orbitals (NAO) [Blum et al 2009]
⢠Adaptive / hybrid basis sets (ðª(10) ⌠ðª(100) basis fctns per atom)Augmented planewave method [Slater 1937, Andersen 1975]Nonorthogonal generalized Wannier function [Skylaris et al 2005]Adaptive local basis set [Lin et al 2012]Adaptive minimal basis in BigDFT [Mohr et al 2014]
Fourier basis setComputational domain Ω = [0, ð¿1] à [0, ð¿2] à [0, ð¿3] with periodicboundary condition (i.e., Î-point for simplicity).Reciprocal lattice in the frequency space
ðâ = {ð = (2ðð¿1
ð1, 2ðð¿2
ð2, 2ðð¿3
ð3), ð â â€3}
Basis functionsðð(ð) = 1
|Ω|1/2 exp(ð€ð â ð)
with ð â ðŸcut = {ð â ðâ ⶠ12 |ð|2 †ðžcut}.
Real space representation
Fourier basis set (without energy cutoff) can be equivalently viewed in thereal space using psinc function as basis set [Skylaris et al 2005]
ððâ²(ð) = 1âðð|Ω| â
ðâðºexp(ð€ð â (ð â ðâ²)).
This is a numerical ð¿-function on the discrete set ð, satisfying
ððâ²(ð) = âðð|Ω|ð¿ð,ðâ² , ð, ðâ² â ð.
Thus a function in the finite dimensional approximation space can berepresented by its values on the grid ð.We can keep in mind discretization based on real space representation forthe discussion of algorithms below (pseudopotential assumed).
Numerical atomic-orbitals
Basis functions are given by
{ðððð(ð â ð ðŒ ) = ð¢ð(|ð â ð ðŒ |)|ð â ð ðŒ | ððð(
ð â ð ðŒ|ð â ð ðŒ |) | ð = 1, ⯠, ððŒ , ðŒ = 1, ⊠, ð},
where ððð are spherical harmonics and the radial part is obtained bysolving (numerically) Schrödinger-like radial equation
(â12
d2
dð2 + ð(ð + 1)ð2 + ð£ð(ð) + ð£cut(ð))ð¢ð(ð) = ððð¢ð(ð)
⢠ð£ð(ð): a radial potential chosen to control the main behavior of ð¢ð;⢠ð£cut(ð): a confining potential to ensure the rapid decay of ð¢ð beyond a
certain radius and can be treated as compactly supported.NAO is used in FHI-aims [Blum et al 2009], which is one of the mostaccurate all-electron DFT package.
Adaptive / hybrid basis set
Recall ...⢠Large basis set
pro: accuracy is systematically improvable;con: higher number of DOF per atom;
⢠Small basis setpro: much smaller number of DOF per atom;con: more difficult to improve its quality in a systematic fashion (replyheavily on tuning and experience).
Adaptive basis aims at combining the best of two worlds. Examples:⢠Augmented planewave method [Slater 1937, Andersen 1975]⢠Nonorthogonal generalized Wannier function [Skylaris et al 2005]⢠Adaptive local basis set [Lin et al 2012]⢠Adaptive minimal basis in BigDFT [Mohr et al 2014]
Adaptive local basis set [Lin, Lu, Ying, E, 2012]
Key idea: Use local basis sets numerically obtained on local patches of thedomain so that it captures the local information of the Hamiltonian.
⢠Based on the discontinuous Galerkin framework for flexibility of basisset choices (alternatives, such as partition of unit FEM, can be alsoused);
⢠Partition the computational domain into non-overlapping patches{ð 1, ð 2, ⊠, ð ð};
⢠For each patch ð , solve on an extended element ð eigenvalue problemlocally and take the first few eigenfunctions
(â12Î + ð ð
eff + ð ð nl )ðð ,ð = ðð,ððð ,ð
⢠Restrict ðð ,ð to ð and use SVD to obtain an orthogonal set of localbasis functions.
Remark: Similar in spirit to GFEM, GMsFEM, reduced basis functions, etc.
Figure: The isosurfaces (0.04 Hartree/Bohr3) of the first three ALB functionsbelonging to the tenth element ðž10: (a) ð1, (b) ð2, (c) ð3, and (d) the electrondensity ð across in top and side views in the global domain in the example ofP140. There are 64 elements and 80 ALB functions in each element, whichcorresponds to 37 basis functions per atom [Hu et al, 2015].
DGDFT package
Figure: Convergence of DGDFT total energy and forces for quasi-1D and 3D Sisystems to reference planewave results with increasing number of ALB functionsper atom. (a) Total energy error per atom Îðž (Ha/atom). (b) Maximum atomicforce error Îð¹ (Ha/Bohr). The dashed green line corresponds to chemicalaccuracy. Energy cutoff ðžcut = 60 Ha and penalty parameter ðŒ = 40. When thenumber of basis functions per atom is sufficiently large, the error is well below thetarget accuracy (green dashed line) [Zhang et al 2017].
Evaluation of the Kohn-Sham mapProblem: Given ð» , solve for the electron density ð.
⢠EigensolverDirect diagonalization: ScaLAPACK, ELPAIterative methods:
â Traditional iterative diagonalization [Davidson 1975, Liu 1978]â Modern Krylov subspace method (with reduced orthogonalization)
[Knyazev 2001, Vecharynski et al 2015]â Chebyshev filtering method [Zhou et al 2006]â Orbital minimization method [Mauri et al 1993, Ordejón et al 1993]
⢠Linear scaling methodsDivide-and-conquer [Yang 1991]Density matrix minimization [Li, Nunes, Vanderbilt 1993; Lai et al2015]Density matrix purification [McWeeny 1960]
⢠Pole expansion and selected inversion (PEXSI) method [Lin et al2009]
Common library approach for Kohn-Sham map:ELSI (ELectronic Structure Infrastructure) [Yu et al 2018]
https://wordpress.elsi-interchange.org/
Chebyshev filtering [Zhou et al 2006]
Key observation: For Kohn-Sham DFT, we do not need individualeigenvalues and eigenvectors of ð» , but only a representation for theoccupied space.
Accelerated subspace iteration
ðð+1 = ðð(ð»)(ðð).
ðð is a ð-th degree Chebyshev polynomial to filter out the higher spectrum.
⢠Orthogonalization of ðð+1 is needed to avoid collapsing(Rayleigh-Ritz rotation is used);
⢠Might be bypassed using 2-level Chebyshev filtering (inner Chebyshevfor spectrum near Fermi level) [Banerjee et al 2018]
⢠For insulating systems, can be replace by localization procedure toachieve linear scaling [E, Li, Lu 2010]
Algorithm 1: Chebyshev filtering method for solving the Kohn-ShamDFT eigenvalue problems ð»ðð = ðððð.Input: Hamiltonian matrix ð» and an orthonormal matrix ð â âððÃðð
Output: Eigenvalues {ðð}ðð=1 and wave functions {ðð}ð
ð=11: Estimate ðð+1 and ððð using a few steps of the Lanczos algorithm.2: while convergence not reached do3: Apply the Chebyshev polynomial ðð(ð») to ð: ð = ðð(ð»)ð.4: Orthonormalize columns of ð .5: Compute the projected Hamiltonian matrix ï¿œÌï¿œ = ð âð»ð and solve
the eigenproblem ï¿œÌï¿œÎšÌ = ΚÌï¿œÌï¿œ.6: Subspace rotation ð = ð ΚÌ.7: end while8: Update {ðð}ð
ð=1 from the first ð columns of ð.
Linear scaling methods
The conventional diagonalization algorithm scales as ðª(ð3), known as theâcubic scaling wallâ, which limits the applicability of Kohn-Sham DFT.For insulating systems (or metallic system at high temperature),ðª(ð)-scaling algorithms are available, based on the locality of theelectronic structure problem:
⢠Exponential decay of the density matrix;⢠Exponentially localized basis of the occupied space (i.e., localized
molecular orbitals, Wannier functions)
Side: Localization methods for Wannier functions⢠Optimization based approaches [Foster, Boys 1960; Marzari,
Vanderbilt 1998; E, Li, Lu 2010; Mustafa et al 2015];⢠Gauge smoothing approach [Cances et al 2017];⢠SCDM (selected column of density matrix) [Damle et al 2015]
Linear scaling methods
The conventional diagonalization algorithm scales as ðª(ð3), known as theâcubic scaling wallâ, which limits the applicability of Kohn-Sham DFT.For insulating systems (or metallic system at high temperature),ðª(ð)-scaling algorithms are available, based on the locality of theelectronic structure problem:
⢠Exponential decay of the density matrix;⢠Exponentially localized basis of the occupied space (i.e., localized
molecular orbitals, Wannier functions)
Side: Localization methods for Wannier functions⢠Optimization based approaches [Foster, Boys 1960; Marzari,
Vanderbilt 1998; E, Li, Lu 2010; Mustafa et al 2015];⢠Gauge smoothing approach [Cances et al 2017];⢠SCDM (selected column of density matrix) [Damle et al 2015]
Density matrix purificationMcWeenyâs purification
ðð+1 = ðMcW(ðð) = 3ð 2ð â 2ð 3
ð
with initial iterate (ðŒ = min{(ðmax â ð)â1, (ð â ðmin)â1})
ð0 = ðŒ2 (ð â ð») + 1
2ðŒ.
Fixed point of ðMcW = 0, 12 , 1:
Density matrix purification
Linear scaling can be achieved by truncating the density matrix at eachiteration; see e.g., NTPoly [Dawson, Nakajima 2018]The initialization requires estimate of the extreme eigenvalues of ð» andalso the chemical potential.
ð0 = ðŒ2 (ð â ð») + 1
2ðŒ.
where ðŒ = min{(ðmax â ð)â1, (ð â ðmin)â1}.Several ways have been proposed for auto-tuning the chemical potential(reviewed in [Niklasson 2011]):
⢠Canonical purification [Paler, Manolopoulos 1998]⢠Trace correcting purification [Niklasson 2002]⢠Trace resetting purification [Niklasson et al 2003]⢠Generalized canonical purification [Truflandier et al 2016]
PEXSI (Pole EXpansion and Selected Inversion)
PEXSI [Lin et al 2009, Lin et al 2011, Jacquelin et al 2016]⢠Reduced scaling algorithm based on Fermi operator expansion and
sparse direct linear algebraðª(ð) for quasi-1D system;ðª(ð3/2) for quasi-2D system;ðª(ð2) for bulk 3D system.
⢠Applicable to general systems (insulating or metallic) with highaccuracy;
⢠Integrated into packages include BigDFT, CP2K, DFTB+, DGDFT,FHI-aims, SIESTA;
⢠Part of the Electronic Structure Infrastructure (ELSI);⢠Available at http://www.pexsi.org/ under the BSD-3 license.
Pole expansionContour integral representation for density matrix (at finite temperature);chemical potential can be estimated on-the-fly during SCF [Jia, Lin 2017]
ððœ(ð» â ð) = 12ðð€ â®ð
ððœ(ð§)((ð§ + ð)ðŒ â ð»)â1 dð§
Discretization with ðª(log(ðœÎðž)) terms [Lin et al 2009; Moussa 2016]
ð âð
âð=1
ðð(ð» â ð§ð)â1.
Selected inversionKey observation: We donât need every entry of (ð» â ð§ð)â1; only those nearthe diagonal.Assume ðŽ partitioned into a 2 à 2 block form
ðŽ = (ðŒ ðâ€
ð ðŽ )
Pivoting by ðŒ (with ð = ðŽ â ððŒâ1ð†known as the Schur complement)
ðŽ = (1â ðŒ) (
ðŒðŽ â ððŒâ1ðâ€) (
1 ââ€
ðŒ )
ðŽâ1 can be expressed by
ðŽâ1 = (ðŒâ1 + ââ€ðâ1â âââ€ðâ1
âðâ1â ðâ1 )
Thus the calculation can be organized in a recursive fashion based on thehierarchical Schur complement.
Algorithm 2: Selected inversion based on ð¿ð·ð¿â€ factorization.
Input: ð¿ð·ð¿â€ factorization of a symmetric matrix ðŽ â âððÃðð .Output: Selected elements of ðŽâ1, i.e., {ðŽâ1
ð,ð ⣠(ð¿ + ð¿â€)ð,ð â 0}.1: Calculate ðŽâ1
ðð,ððâ (ð·ðð,ðð)â1.
2: for ð = ðð â 1, ..., 1 do3: Find the collection of indices ð¶ = {ð | ð > ð, ð¿ð,ð â 0}.4: Calculate ðŽâ1
ð¶,ð â âðŽâ1ð¶,ð¶ð¿ð¶,ð.
5: Calculate ðŽâ1ð,ð¶ â (ðŽâ1
ð¶,ð)â€.6: Calculate ðŽâ1
ð,ð â (ð·ð,ð)â1 â ðŽâ1ð,ð¶ð¿ð¶,ð.
7: end for
In practice, as in LDLT packages, columns of ðŽ are partitioned intosupernodes (set of contiguous columns) to enable level-3 BLAS forefficiency [Jacquelin et al 2016].
Performance and scalability of PEXSI
320 1280 5120 20480 81920 32768010
0
101
102
103
104
Proc #
Tim
e[s]
D isordered graphene 2048
Diagonalize
PEXSIPSe lInv
5120 20480 81920 327680 131072010
1
102
103
104
105
Proc #
Tim
e[s]
D isordered graphene 8192
Diagonalize
PEXSIPSe lInv
1024 20480 81920 327680 1310720
102
103
104
105
106
Proc #
Tim
e[s]
D isordered graphene 32768
Diagonalize ( ideal)
PEXSI
PSelInv
Figure: Wall clock time versus the number of cores for a graphene systems with2048, 8192, and 32768 atoms [Jacquelin et al 2016]
241 second wall clock time for graphene with 32, 768 atoms (DG ALBdiscretization); infeasible for traditional solvers.
What is happening now and perhaps in the near future?⢠Massively scalable software development
e.g., the ELSI project [Yu et al 2018]; the ESL project;many actively maintained packages available
⢠Efficient algorithms for nonlocal functionals (and beyond)Some recent developments of algorithmic tools(low-rankness seems to be useful):
Interpolative separable density fitting (ISDF) for pair products oforbital functions [Lu, Ying 2015];Projective eigendecomposition of the dielectric screening (PDEP) forGW calculations [Govoni, Galli 2015]Adaptive compressed exchange operator (ACE) method [Lin 2016]....
⢠Excited states; time-dependent DFT; highly correlated systems;⢠Numerical analysis; robust and black-box algorithms for
high-throughput calculations;⢠Machine learning techniques for electronic structure;⢠Quantum computing for quantum chemistry
What is happening now and perhaps in the near future?⢠Massively scalable software development
e.g., the ELSI project [Yu et al 2018]; the ESL project;many actively maintained packages available
⢠Efficient algorithms for nonlocal functionals (and beyond)Some recent developments of algorithmic tools(low-rankness seems to be useful):
Interpolative separable density fitting (ISDF) for pair products oforbital functions [Lu, Ying 2015];Projective eigendecomposition of the dielectric screening (PDEP) forGW calculations [Govoni, Galli 2015]Adaptive compressed exchange operator (ACE) method [Lin 2016]....
⢠Excited states; time-dependent DFT; highly correlated systems;⢠Numerical analysis; robust and black-box algorithms for
high-throughput calculations;⢠Machine learning techniques for electronic structure;⢠Quantum computing for quantum chemistry
What is happening now and perhaps in the near future?⢠Massively scalable software development
e.g., the ELSI project [Yu et al 2018]; the ESL project;many actively maintained packages available
⢠Efficient algorithms for nonlocal functionals (and beyond)Some recent developments of algorithmic tools(low-rankness seems to be useful):
Interpolative separable density fitting (ISDF) for pair products oforbital functions [Lu, Ying 2015];Projective eigendecomposition of the dielectric screening (PDEP) forGW calculations [Govoni, Galli 2015]Adaptive compressed exchange operator (ACE) method [Lin 2016]....
⢠Excited states; time-dependent DFT; highly correlated systems;
⢠Numerical analysis; robust and black-box algorithms forhigh-throughput calculations;
⢠Machine learning techniques for electronic structure;⢠Quantum computing for quantum chemistry
What is happening now and perhaps in the near future?⢠Massively scalable software development
e.g., the ELSI project [Yu et al 2018]; the ESL project;many actively maintained packages available
⢠Efficient algorithms for nonlocal functionals (and beyond)Some recent developments of algorithmic tools(low-rankness seems to be useful):
Interpolative separable density fitting (ISDF) for pair products oforbital functions [Lu, Ying 2015];Projective eigendecomposition of the dielectric screening (PDEP) forGW calculations [Govoni, Galli 2015]Adaptive compressed exchange operator (ACE) method [Lin 2016]....
⢠Excited states; time-dependent DFT; highly correlated systems;⢠Numerical analysis; robust and black-box algorithms for
high-throughput calculations;
⢠Machine learning techniques for electronic structure;⢠Quantum computing for quantum chemistry
What is happening now and perhaps in the near future?⢠Massively scalable software development
e.g., the ELSI project [Yu et al 2018]; the ESL project;many actively maintained packages available
⢠Efficient algorithms for nonlocal functionals (and beyond)Some recent developments of algorithmic tools(low-rankness seems to be useful):
Interpolative separable density fitting (ISDF) for pair products oforbital functions [Lu, Ying 2015];Projective eigendecomposition of the dielectric screening (PDEP) forGW calculations [Govoni, Galli 2015]Adaptive compressed exchange operator (ACE) method [Lin 2016]....
⢠Excited states; time-dependent DFT; highly correlated systems;⢠Numerical analysis; robust and black-box algorithms for
high-throughput calculations;⢠Machine learning techniques for electronic structure;
⢠Quantum computing for quantum chemistry
What is happening now and perhaps in the near future?⢠Massively scalable software development
e.g., the ELSI project [Yu et al 2018]; the ESL project;many actively maintained packages available
⢠Efficient algorithms for nonlocal functionals (and beyond)Some recent developments of algorithmic tools(low-rankness seems to be useful):
Interpolative separable density fitting (ISDF) for pair products oforbital functions [Lu, Ying 2015];Projective eigendecomposition of the dielectric screening (PDEP) forGW calculations [Govoni, Galli 2015]Adaptive compressed exchange operator (ACE) method [Lin 2016]....
⢠Excited states; time-dependent DFT; highly correlated systems;⢠Numerical analysis; robust and black-box algorithms for
high-throughput calculations;⢠Machine learning techniques for electronic structure;⢠Quantum computing for quantum chemistry
Thank you for your attention
Email: jianfeng@math.duke.edu
url: http://www.math.duke.edu/~jianfeng/
Reference:⢠Lin Lin, L., and Lexing Ying, Numerical Methods for Kohn-Sham
Density Functional Theory, Acta Numerica, 2019