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Numerical Methods for Kohn-Sham DensityFunctional Theory
Jianfeng Lu (้ฒๅ้)
Duke [email protected]
January 2019, Banff Workshop on โOptimal Transport Methods in DensityFunctional Theoryโ
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 Schrรถdinger equation
(โ12
๐
โ๐=1
ฮ๐ฅ๐ +๐
โ๐=1
๐ext(๐ฅ๐) + โ1โค๐<๐โค๐
1|๐ฅ๐ โ ๐ฅ๐|)ฮจ(๐ฅ1, โฏ , ๐ฅ๐ ) = ๐ธฮจ
under non-relativistic and Born-Oppenheimer approximations.Here the atom types and positions enter ๐ext 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 Schrรถdinger equation
(โ12
๐
โ๐=1
ฮ๐ฅ๐ +๐
โ๐=1
๐ext(๐ฅ๐) + โ1โค๐<๐โค๐
1|๐ฅ๐ โ ๐ฅ๐|)ฮจ(๐ฅ1, โฏ , ๐ฅ๐ ) = ๐ธฮจ
under non-relativistic and Born-Oppenheimer approximations.Here the atom types and positions enter ๐ext as parameters.
Still too complex to be solved, even after almost 90 years:โข Curse of dimensionality;โข Anti-symmetry of ฮจ due to Pauliโs exclusion principle
ฮจ(๐ฅ1, โฏ , ๐ฅ๐, โฏ , ๐ฅ๐ , โฏ , ๐ฅ๐ ) = โฮจ(๐ฅ1, โฏ , ๐ฅ๐ , โฏ , ๐ฅ๐, โฏ , ๐ฅ๐ );
โข ฮจ has complicated singularity structure.
Solving electronic Schrรถdinger equations:โข Tight-binding approximations (LCAO)โข Density functional theory
Orbital-free DFTKohn-Sham DFT
โข Wavefunction methodsHartree-FockMรธller-Plesset perturbation theoryConfiguration interactionCoupled clusterMulti-configuration self-consistent fieldNeural network ansatz
โข GW approximation; Bethe-Salpeter equationโข Quantum Monte Carlo (VMC, DMC, etc.)โข Density matrix renormalization group (DMRG) / tensor networksโข ...
Solving electronic Schrรถdinger equations:โข Tight-binding approximations (LCAO)โข 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 ๐ โถ โ3 โ โโฅ0:
๐(๐ฅ) = ๐ โซ|ฮจ|2(๐ฅ, ๐ฅ2, โฏ , ๐ฅ๐ ) d๐ฅ2 โฏ d๐ฅ๐ .
Levy-Lieb variational principle [Levy 1979, Lieb 1983]:
๐ธ0 = infฮจ
โจฮจ, ๐ปฮจโฉ = inf๐
infฮจ,ฮจโฆ๐
โจฮจ, ๐ปฮจโฉ = inf๐
๐ธ(๐).
The energy functional has the form
๐ธ(๐) = ๐๐ (๐) + โซ ๐๐ext + 12 โฌ
๐(๐ฅ)๐(๐ฆ)|๐ฅ โ ๐ฆ| + ๐ธxc(๐)
๐๐ (๐): Kinetic energy of non-interacting electrons;๐ธxc(๐): 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 ๐ โถ โ3 โ โโฅ0:
๐(๐ฅ) = ๐ โซ|ฮจ|2(๐ฅ, ๐ฅ2, โฏ , ๐ฅ๐ ) d๐ฅ2 โฏ d๐ฅ๐ .
Levy-Lieb variational principle [Levy 1979, Lieb 1983]:
๐ธ0 = infฮจ
โจฮจ, ๐ปฮจโฉ = inf๐
infฮจ,ฮจโฆ๐
โจฮจ, ๐ปฮจโฉ
= inf๐
๐ธ(๐).
The energy functional has the form
๐ธ(๐) = ๐๐ (๐) + โซ ๐๐ext + 12 โฌ
๐(๐ฅ)๐(๐ฆ)|๐ฅ โ ๐ฆ| + ๐ธxc(๐)
๐๐ (๐): Kinetic energy of non-interacting electrons;๐ธxc(๐): 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 ๐ โถ โ3 โ โโฅ0:
๐(๐ฅ) = ๐ โซ|ฮจ|2(๐ฅ, ๐ฅ2, โฏ , ๐ฅ๐ ) d๐ฅ2 โฏ d๐ฅ๐ .
Levy-Lieb variational principle [Levy 1979, Lieb 1983]:
๐ธ0 = infฮจ
โจฮจ, ๐ปฮจโฉ = inf๐
infฮจ,ฮจโฆ๐
โจฮจ, ๐ปฮจโฉ = inf๐
๐ธ(๐).
The energy functional has the form
๐ธ(๐) = ๐๐ (๐) + โซ ๐๐ext + 12 โฌ
๐(๐ฅ)๐(๐ฆ)|๐ฅ โ ๐ฆ| + ๐ธxc(๐)
๐๐ (๐): Kinetic energy of non-interacting electrons;๐ธxc(๐): 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 ๐ โถ โ3 โ โโฅ0:
๐(๐ฅ) = ๐ โซ|ฮจ|2(๐ฅ, ๐ฅ2, โฏ , ๐ฅ๐ ) d๐ฅ2 โฏ d๐ฅ๐ .
Levy-Lieb variational principle [Levy 1979, Lieb 1983]:
๐ธ0 = infฮจ
โจฮจ, ๐ปฮจโฉ = inf๐
infฮจ,ฮจโฆ๐
โจฮจ, ๐ปฮจโฉ = inf๐
๐ธ(๐).
The energy functional has the form
๐ธ(๐) = ๐๐ (๐) + โซ ๐๐ext + 12 โฌ
๐(๐ฅ)๐(๐ฆ)|๐ฅ โ ๐ฆ| + ๐ธxc(๐)
๐๐ (๐): Kinetic energy of non-interacting electrons;๐ธxc(๐): 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 ๐ orbitals {๐๐} โ ๐ป1(โ3).
๐ธKS({๐๐}) =๐
โ๐=1
12 โซ|โ๐๐|2 + โซ ๐๐ext + 1
2 โฌ๐(๐ฅ)๐(๐ฆ)|๐ฅ โ ๐ฆ| + ๐ธxc(๐)
where (we consider โspin-lessโ electrons through the talk)
๐(๐ฅ) =๐
โ๐=1
|๐๐|2(๐ฅ).
Note that ๐ธKS can be still viewed as a functional of ๐, implicitly.
Kohn-Sham density functional theory
๐ธKS({๐๐}) =๐
โ๐=1
12 โซ|โ๐๐|2 + โซ ๐๐ext + 1
2 โฌ๐(๐ฅ)๐(๐ฆ)|๐ฅ โ ๐ฆ| + ๐ธxc
The exchange-correlation functionals counts for the corrections from themany-body interactions between electrons:
โข Semi-local exchange-correlation functionalsLocal density approximation: ๐ธxc = โซ ๐xc(๐(๐ฅ))Generalized gradient approximation:
๐ธxc = โซ ๐xc(๐(๐ฅ), 12 |โโ๐(๐ฅ)|2)
โข Nonlocal exchange-correlation functionals
Kohn-Sham density functional theory
๐ธKS({๐๐}) =๐
โ๐=1
12 โซ|โ๐๐|2 + โซ ๐๐ext + 1
2 โฌ๐(๐ฅ)๐(๐ฆ)|๐ฅ โ ๐ฆ| + ๐ธxc
โข Nonlocal exchange-correlation functionalse.g., exact exchange + RPA correlation, ๐ธxc = ๐ธ๐ฅ + ๐ธ๐:
๐ธ๐ฅ = โ12
๐
โ๐,๐=1 โฌ
๐โ๐ (๐ฅ)๐๐(๐ฅ)๐๐(๐ฆ)๐โ
๐ (๐ฆ)|๐ฅ โ ๐ฆ| d๐ฅ d๐ฆ;
๐ธ๐ = 12๐ โซ
โ
0tr[ln(1 โ ๐0(๐๐)๐) + ๐0(๐๐)๐] d๐,
with ๐ the Coulomb operator and ๐0 Kohn-Sham polarizabilityoperator:
๐0(๐ฅ, ๐ฆ, ๐๐) = 2occ
โ๐
unocc
โ๐
๐โ๐ (๐ฅ)๐๐(๐ฅ)๐โ
๐ (๐ฆ)๐๐(๐ฆ)๐๐ โ ๐๐ โ ๐๐ .
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:
๐ปeff(๐)๐๐ = ๐๐๐๐, ๐(๐ฅ) =occ
โ๐
|๐๐|2(๐ฅ);
where the first ๐ orbitals are occupied (aufbau principle). The effectiveHamiltonian captures the interactions of electrons:
๐ปeff(๐) = โ12ฮ + ๐eff(๐);
๐eff(๐) = ๐๐(๐) + ๐xc(๐).
Note that this is a nonlinear eigenvalue problem. We can view it as afixed-point equation for the density ๐:
๐ = ๐นKS(๐).
Kohn-Sham mapKohn-Sham fixed-point equation
๐ = ๐นKS(๐),
where ๐นKS is known as the Kohn-Sham map, defined through theeigenvalue problem associated with ๐ปeff(๐).Given an effective Hamiltonian ๐ปeff, we ask for its low-lying eigenspace:the range of the spectral projection
๐ = ๐(โโ,๐๐น ](๐ปeff),
thus ๐ is given as the diagonal of the kernel of the operator ๐ .
It is often more advantageous to represent ๐ in terms of Greenโs functions(๐ โ ๐ปeff)โ1, so to turn the eigenvalue problem into solving equations.Let ๐ be a contour around the occupied spectrum, we have
๐ = ๐(โโ,๐๐น ](๐ปeff) = 12๐๐ค โฎ๐
(๐ โ ๐ปeff)โ1 d๐.
Kohn-Sham mapKohn-Sham fixed-point equation
๐ = ๐นKS(๐),
where ๐นKS is known as the Kohn-Sham map, defined through theeigenvalue problem associated with ๐ปeff(๐).Given an effective Hamiltonian ๐ปeff, we ask for its low-lying eigenspace:the range of the spectral projection
๐ = ๐(โโ,๐๐น ](๐ปeff),
thus ๐ is given as the diagonal of the kernel of the operator ๐ .It is often more advantageous to represent ๐ in terms of Greenโs functions(๐ โ ๐ปeff)โ1, so to turn the eigenvalue problem into solving equations.Let ๐ be a contour around the occupied spectrum, we have
๐ = ๐(โโ,๐๐น ](๐ปeff) = 12๐๐ค โฎ๐
(๐ โ ๐ปeff)โ1 d๐.
Numerical methods
Kohn-Sham fixed-point equation
๐ = ๐นKS(๐).
Usually solved numerically based on self-consistent field (SCF) iteration(alternative methods, such as direct minimization, are not as widely used)
โ see the talk of Ziad Musslimani
Numerical methods
Kohn-Sham fixed-point equation
๐ = ๐นKS(๐).
Usually solved numerically based on self-consistent field (SCF) iteration(alternative methods, such as direct minimization, are not as widely used)
โข Self-consistent iterationLinearize the nonlinear problem to a linear one at each step
โข DiscretizationMaking the problem finite dimensional
โข Evaluation of the Kohn-Sham mapFormation of effective Hamiltonian: ๐ โฆ ๐ปeff(๐)Evaluation of density: ๐ปeff โฆ ๐Most of the actual computational work.
Self-consistent iterationSolving for the fixed point equation ๐ = ๐นKS(๐).(Note: often more convenient to view the iteration in terms of ๐eff)
โข First idea: Fixed point iteration๐๐+1 = ๐นKS(๐๐)
Usually does not converge as ๐นKS is not necessarily a contraction.
โข Simple mixing๐๐+1 = (1 โ ๐ผ)๐๐ + ๐ผ๐นKS(๐๐)
= ๐๐ โ ๐ผ(๐๐ โ ๐นKS(๐๐))where ๐๐ โ ๐นKS(๐๐) is the residual.๐ผ = 1 corresponds to fixed-point iteration.
โข Preconditioning๐๐+1 = ๐๐ โ ๐ซ (๐๐ โ ๐นKS(๐๐))
e.g., ๐ซ = ๐ผ๐ผ corresponds to simple mixing.Kerker mixing for simple metals (jellium like)โ see the talk of Antoine Levitt
Self-consistent iterationSolving for the fixed point equation ๐ = ๐นKS(๐).(Note: often more convenient to view the iteration in terms of ๐eff)
โข First idea: Fixed point iteration๐๐+1 = ๐นKS(๐๐)
Usually does not converge as ๐นKS is not necessarily a contraction.โข Simple mixing
๐๐+1 = (1 โ ๐ผ)๐๐ + ๐ผ๐นKS(๐๐)= ๐๐ โ ๐ผ(๐๐ โ ๐นKS(๐๐))
where ๐๐ โ ๐นKS(๐๐) is the residual.๐ผ = 1 corresponds to fixed-point iteration.
โข Preconditioning๐๐+1 = ๐๐ โ ๐ซ (๐๐ โ ๐นKS(๐๐))
e.g., ๐ซ = ๐ผ๐ผ corresponds to simple mixing.Kerker mixing for simple metals (jellium like)โ see the talk of Antoine Levitt
Self-consistent iterationSolving for the fixed point equation ๐ = ๐นKS(๐).(Note: often more convenient to view the iteration in terms of ๐eff)
โข First idea: Fixed point iteration๐๐+1 = ๐นKS(๐๐)
Usually does not converge as ๐นKS is not necessarily a contraction.โข Simple mixing
๐๐+1 = (1 โ ๐ผ)๐๐ + ๐ผ๐นKS(๐๐)= ๐๐ โ ๐ผ(๐๐ โ ๐นKS(๐๐))
where ๐๐ โ ๐นKS(๐๐) is the residual.๐ผ = 1 corresponds to fixed-point iteration.
โข Preconditioning๐๐+1 = ๐๐ โ ๐ซ (๐๐ โ ๐นKS(๐๐))
e.g., ๐ซ = ๐ผ๐ผ corresponds to simple mixing.Kerker mixing for simple metals (jellium like)โ see the talk of Antoine Levitt
โข Newton method
๐๐+1 = ๐๐ โ ๐ฅ โ1๐ (๐๐ โ ๐นKS(๐๐)
where ๐ฅ๐ 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)
โข Quasi-Newton type algorithms (the most widely used approach)Broydenโs second method [Fang, Saad 2009; Marks, Luke 2008]Low-rank update to ฬ๐ฅ โ1
๐ (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 ๐ปeff[๐ ] and ๐
โข Newton method
๐๐+1 = ๐๐ โ ๐ฅ โ1๐ (๐๐ โ ๐นKS(๐๐)
where ๐ฅ๐ 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)
โข Quasi-Newton type algorithms (the most widely used approach)Broydenโs second method [Fang, Saad 2009; Marks, Luke 2008]Low-rank update to ฬ๐ฅ โ1
๐ (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 ๐ปeff[๐ ] and ๐
Discretization
โข Large basis sets (๐ช(100) โผ ๐ช(10, 000) basis functions per atom)Planewave method (Fourier basis set / pseudospectral method)[Payne et al 1992, Kresse and Furthmรผller 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]
โข Small basis sets (๐ช(10) โผ ๐ช(100) basis functions per atom)Gaussian-type orbitals (GTO) (see review [Jensen 2013])Numerical atomic-orbitals (NAO) [Blum et al 2009]
โข Adaptive / hybrid basis sets (๐ช(10) โผ ๐ช(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 ฮฉ = [0, ๐ฟ1] ร [0, ๐ฟ2] ร [0, ๐ฟ3] with periodicboundary condition (i.e., ฮ-point for simplicity).Reciprocal lattice in the frequency space
๐โ = {๐ = (2๐๐ฟ1
๐1, 2๐๐ฟ2
๐2, 2๐๐ฟ3
๐3), ๐ โ โค3}
Basis functions๐๐(๐) = 1
|ฮฉ|1/2 exp(๐ค๐ โ ๐)
with ๐ โ ๐พcut = {๐ โ ๐โ โถ 12 |๐|2 โค ๐ธcut}.
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]
๐๐โฒ(๐) = 1โ๐๐|ฮฉ| โ
๐โ๐บexp(๐ค๐ โ (๐ โ ๐โฒ)).
This is a numerical ๐ฟ-function on the discrete set ๐, satisfying
๐๐โฒ(๐) = โ๐๐|ฮฉ|๐ฟ๐,๐โฒ , ๐, ๐โฒ โ ๐.
Thus a function in the finite dimensional approximation space can berepresented by its values on the grid ๐.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
{๐๐๐๐(๐ โ ๐ ๐ผ ) = ๐ข๐(|๐ โ ๐ ๐ผ |)|๐ โ ๐ ๐ผ | ๐๐๐(
๐ โ ๐ ๐ผ|๐ โ ๐ ๐ผ |) | ๐ = 1, โฏ , ๐๐ผ , ๐ผ = 1, โฆ , ๐},
where ๐๐๐ are spherical harmonics and the radial part is obtained bysolving (numerically) Schrรถdinger-like radial equation
(โ12
d2
d๐2 + ๐(๐ + 1)๐2 + ๐ฃ๐(๐) + ๐ฃcut(๐))๐ข๐(๐) = ๐๐๐ข๐(๐)
โข ๐ฃ๐(๐): a radial potential chosen to control the main behavior of ๐ข๐;โข ๐ฃcut(๐): a confining potential to ensure the rapid decay of ๐ข๐ 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 ...โข Large basis set
pro: accuracy is systematically improvable;con: higher number of DOF per atom;
โข 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:โข Augmented planewave method [Slater 1937, Andersen 1975]โข Nonorthogonal generalized Wannier function [Skylaris et al 2005]โข Adaptive local basis set [Lin et al 2012]โข 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.
โข Based on the discontinuous Galerkin framework for flexibility of basisset choices (alternatives, such as partition of unit FEM, can be alsoused);
โข Partition the computational domain into non-overlapping patches{๐ 1, ๐ 2, โฆ , ๐ ๐};
โข For each patch ๐ , solve on an extended element ๐ eigenvalue problemlocally and take the first few eigenfunctions
(โ12ฮ + ๐ ๐
eff + ๐ ๐ nl )๐๐ ,๐ = ๐๐,๐๐๐ ,๐
โข Restrict ๐๐ ,๐ to ๐ 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 ๐ธ10: (a) ๐1, (b) ๐2, (c) ๐3, and (d) the electrondensity ๐ 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 ฮ๐ธ (Ha/atom). (b) Maximum atomicforce error ฮ๐น (Ha/Bohr). The dashed green line corresponds to chemicalaccuracy. Energy cutoff ๐ธcut = 60 Ha and penalty parameter ๐ผ = 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 ๐ป , solve for the electron density ๐.
โข EigensolverDirect diagonalization: ScaLAPACK, ELPAIterative methods:
โ Traditional iterative diagonalization [Davidson 1975, Liu 1978]โ Modern Krylov subspace method (with reduced orthogonalization)
[Knyazev 2001, Vecharynski et al 2015]โ Chebyshev filtering method [Zhou et al 2006]โ Orbital minimization method [Mauri et al 1993, Ordejรณn et al 1993]
โข Linear scaling methodsDivide-and-conquer [Yang 1991]Density matrix minimization [Li, Nunes, Vanderbilt 1993; Lai et al2015]Density matrix purification [McWeeny 1960]
โข 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 ๐ป , but only a representation for theoccupied space.
Accelerated subspace iteration
๐๐+1 = ๐๐(๐ป)(๐๐).
๐๐ is a ๐-th degree Chebyshev polynomial to filter out the higher spectrum.
โข Orthogonalization of ๐๐+1 is needed to avoid collapsing(Rayleigh-Ritz rotation is used);
โข Might be bypassed using 2-level Chebyshev filtering (inner Chebyshevfor spectrum near Fermi level) [Banerjee et al 2018]
โข 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 ๐ป๐๐ = ๐๐๐๐.Input: Hamiltonian matrix ๐ป and an orthonormal matrix ๐ โ โ๐๐ร๐๐
Output: Eigenvalues {๐๐}๐๐=1 and wave functions {๐๐}๐
๐=11: Estimate ๐๐+1 and ๐๐๐ using a few steps of the Lanczos algorithm.2: while convergence not reached do3: Apply the Chebyshev polynomial ๐๐(๐ป) to ๐: ๐ = ๐๐(๐ป)๐.4: Orthonormalize columns of ๐ .5: Compute the projected Hamiltonian matrix ๏ฟฝฬ๏ฟฝ = ๐ โ๐ป๐ and solve
the eigenproblem ๏ฟฝฬ๏ฟฝฮจฬ = ฮจฬ๏ฟฝฬ๏ฟฝ.6: Subspace rotation ๐ = ๐ ฮจฬ.7: end while8: Update {๐๐}๐
๐=1 from the first ๐ columns of ๐.
Linear scaling methods
The conventional diagonalization algorithm scales as ๐ช(๐3), known as theโcubic scaling wallโ, which limits the applicability of Kohn-Sham DFT.For insulating systems (or metallic system at high temperature),๐ช(๐)-scaling algorithms are available, based on the locality of theelectronic structure problem:
โข Exponential decay of the density matrix;โข Exponentially localized basis of the occupied space (i.e., localized
molecular orbitals, Wannier functions)
Side: Localization methods for Wannier functionsโข Optimization based approaches [Foster, Boys 1960; Marzari,
Vanderbilt 1998; E, Li, Lu 2010; Mustafa et al 2015];โข Gauge smoothing approach [Cances et al 2017];โข SCDM (selected column of density matrix) [Damle et al 2015]
Linear scaling methods
The conventional diagonalization algorithm scales as ๐ช(๐3), known as theโcubic scaling wallโ, which limits the applicability of Kohn-Sham DFT.For insulating systems (or metallic system at high temperature),๐ช(๐)-scaling algorithms are available, based on the locality of theelectronic structure problem:
โข Exponential decay of the density matrix;โข Exponentially localized basis of the occupied space (i.e., localized
molecular orbitals, Wannier functions)
Side: Localization methods for Wannier functionsโข Optimization based approaches [Foster, Boys 1960; Marzari,
Vanderbilt 1998; E, Li, Lu 2010; Mustafa et al 2015];โข Gauge smoothing approach [Cances et al 2017];โข SCDM (selected column of density matrix) [Damle et al 2015]
Density matrix purificationMcWeenyโs purification
๐๐+1 = ๐McW(๐๐) = 3๐ 2๐ โ 2๐ 3
๐
with initial iterate (๐ผ = min{(๐max โ ๐)โ1, (๐ โ ๐min)โ1})
๐0 = ๐ผ2 (๐ โ ๐ป) + 1
2๐ผ.
Fixed point of ๐McW = 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 ๐ป andalso the chemical potential.
๐0 = ๐ผ2 (๐ โ ๐ป) + 1
2๐ผ.
where ๐ผ = min{(๐max โ ๐)โ1, (๐ โ ๐min)โ1}.Several ways have been proposed for auto-tuning the chemical potential(reviewed in [Niklasson 2011]):
โข Canonical purification [Paler, Manolopoulos 1998]โข Trace correcting purification [Niklasson 2002]โข Trace resetting purification [Niklasson et al 2003]โข 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]โข Reduced scaling algorithm based on Fermi operator expansion and
sparse direct linear algebra๐ช(๐) for quasi-1D system;๐ช(๐3/2) for quasi-2D system;๐ช(๐2) for bulk 3D system.
โข Applicable to general systems (insulating or metallic) with highaccuracy;
โข Integrated into packages include BigDFT, CP2K, DFTB+, DGDFT,FHI-aims, SIESTA;
โข Part of the Electronic Structure Infrastructure (ELSI);โข 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]
๐๐ฝ(๐ป โ ๐) = 12๐๐ค โฎ๐
๐๐ฝ(๐ง)((๐ง + ๐)๐ผ โ ๐ป)โ1 d๐ง
Discretization with ๐ช(log(๐ฝฮ๐ธ)) terms [Lin et al 2009; Moussa 2016]
๐ โ๐
โ๐=1
๐๐(๐ป โ ๐ง๐)โ1.
Selected inversionKey observation: We donโt need every entry of (๐ป โ ๐ง๐)โ1; only those nearthe diagonal.Assume ๐ด partitioned into a 2 ร 2 block form
๐ด = (๐ผ ๐โค
๐ ๐ด )
Pivoting by ๐ผ (with ๐ = ๐ด โ ๐๐ผโ1๐โค known as the Schur complement)
๐ด = (1โ ๐ผ) (
๐ผ๐ด โ ๐๐ผโ1๐โค) (
1 โโค
๐ผ )
๐ดโ1 can be expressed by
๐ดโ1 = (๐ผโ1 + โโค๐โ1โ โโโค๐โ1
โ๐โ1โ ๐โ1 )
Thus the calculation can be organized in a recursive fashion based on thehierarchical Schur complement.
Algorithm 2: Selected inversion based on ๐ฟ๐ท๐ฟโค factorization.
Input: ๐ฟ๐ท๐ฟโค factorization of a symmetric matrix ๐ด โ โ๐๐ร๐๐ .Output: Selected elements of ๐ดโ1, i.e., {๐ดโ1
๐,๐ โฃ (๐ฟ + ๐ฟโค)๐,๐ โ 0}.1: Calculate ๐ดโ1
๐๐,๐๐โ (๐ท๐๐,๐๐)โ1.
2: for ๐ = ๐๐ โ 1, ..., 1 do3: Find the collection of indices ๐ถ = {๐ | ๐ > ๐, ๐ฟ๐,๐ โ 0}.4: Calculate ๐ดโ1
๐ถ,๐ โ โ๐ดโ1๐ถ,๐ถ๐ฟ๐ถ,๐.
5: Calculate ๐ดโ1๐,๐ถ โ (๐ดโ1
๐ถ,๐)โค.6: Calculate ๐ดโ1
๐,๐ โ (๐ท๐,๐)โ1 โ ๐ดโ1๐,๐ถ๐ฟ๐ถ,๐.
7: end for
In practice, as in LDLT packages, columns of ๐ด 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?โข Massively scalable software development
e.g., the ELSI project [Yu et al 2018]; the ESL project;many actively maintained packages available
โข 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]....
โข Excited states; time-dependent DFT; highly correlated systems;โข Numerical analysis; robust and black-box algorithms for
high-throughput calculations;โข Machine learning techniques for electronic structure;โข Quantum computing for quantum chemistry
What is happening now and perhaps in the near future?โข Massively scalable software development
e.g., the ELSI project [Yu et al 2018]; the ESL project;many actively maintained packages available
โข 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]....
โข Excited states; time-dependent DFT; highly correlated systems;โข Numerical analysis; robust and black-box algorithms for
high-throughput calculations;โข Machine learning techniques for electronic structure;โข Quantum computing for quantum chemistry
What is happening now and perhaps in the near future?โข Massively scalable software development
e.g., the ELSI project [Yu et al 2018]; the ESL project;many actively maintained packages available
โข 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]....
โข Excited states; time-dependent DFT; highly correlated systems;
โข Numerical analysis; robust and black-box algorithms forhigh-throughput calculations;
โข Machine learning techniques for electronic structure;โข Quantum computing for quantum chemistry
What is happening now and perhaps in the near future?โข Massively scalable software development
e.g., the ELSI project [Yu et al 2018]; the ESL project;many actively maintained packages available
โข 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]....
โข Excited states; time-dependent DFT; highly correlated systems;โข Numerical analysis; robust and black-box algorithms for
high-throughput calculations;
โข Machine learning techniques for electronic structure;โข Quantum computing for quantum chemistry
What is happening now and perhaps in the near future?โข Massively scalable software development
e.g., the ELSI project [Yu et al 2018]; the ESL project;many actively maintained packages available
โข 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]....
โข Excited states; time-dependent DFT; highly correlated systems;โข Numerical analysis; robust and black-box algorithms for
high-throughput calculations;โข Machine learning techniques for electronic structure;
โข Quantum computing for quantum chemistry
What is happening now and perhaps in the near future?โข Massively scalable software development
e.g., the ELSI project [Yu et al 2018]; the ESL project;many actively maintained packages available
โข 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]....
โข Excited states; time-dependent DFT; highly correlated systems;โข Numerical analysis; robust and black-box algorithms for
high-throughput calculations;โข Machine learning techniques for electronic structure;โข Quantum computing for quantum chemistry
Thank you for your attention
Email: [email protected]
url: http://www.math.duke.edu/~jianfeng/
Reference:โข Lin Lin, L., and Lexing Ying, Numerical Methods for Kohn-Sham
Density Functional Theory, Acta Numerica, 2019