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Compressed representation of Kohn-Sham orbitals via selected columns of the density matrix

Anil Damle, Lin Lin, Lexing Ying

arXiv:1408.4926v2physics.comp-phphysics.chem-ph

TL;DR

Large Kohn–Sham orbital sets are costly to store and compute, and hybrid-functional exchange adds substantial cost. The paper uses selected density-matrix columns to construct localized basis functions for insulating systems; with Hockney’s algorithm, exchange calculations are reported to be substantially faster than using Kohn–Sham orbitals.

  • Problem

    Large, spatially delocalized Kohn–Sham orbitals are costly to store and compute, and exact exchange in hybrid functionals has cubic scaling and a large prefactor.

    Source: INTRODUCTION · p. 1

  • Method

    For insulating systems, the method selects density-matrix columns to form localized basis functions spanning the Kohn–Sham subspace, with optional orthogonalization by a linear transformation.

    Source: INTRODUCTION · p. 1, THEORY · p. 2

  • Results

    At the largest problem size, localized basis functions with Hockney’s algorithm gave about 300-fold speedup at 4%–5% relative error and 90-fold speedup at 0.3%–0.4% relative error versus Kohn–Sham orbitals.

    Source: Computation of Hartree-Fock exchange… · p. 5

  • Takeaways & Limitations

    The orthogonalized SCDM can be built immediately after computing the orbitals as a localized orthogonal basis for their subspace.

    Source: CONCLUSION · p. 6

    The work requires orbitals represented on a real-space grid; directly using localized-basis density matrices instead is proposed, but its numerical consequences need careful testing.

    Source: CONCLUSION · p. 6

Abstract

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Given a set of Kohn-Sham orbitals from an insulating system, we present a simple, robust, efficient and highly parallelizable method to construct a set of, optionally orthogonal, localized basis functions for the associated subspace. Our method explicitly uses the fact that density matrices associated with insulating systems decay exponentially along the off-diagonal direction in the real space representation. Our method avoids the usage of an optimization procedure, and the localized basis functions are constructed directly from a set of selected columns of the density matrix (SCDM). Consequently, the only adjustable parameter in our method is the truncation threshold of the localized basis functions. Our method can be used in any electronic structure software package with an arbitrary basis set. We demonstrate the numerical accuracy and parallel scalability of the SCDM procedure using orbitals generated by the Quantum ESPRESSO software package. We also demonstrate a procedure for combining SCDM with Hockney's algorithm to efficiently perform Hartree-Fock exchange energy calculations with near linear scaling.

INTRODUCTION

Large Kohn–Sham calculations face substantial orbital storage and computation costs, while hybrid-functional exchange is especially costly. The section introduces SCDM as a localized representation of the Kohn–Sham subspace and describes its scope and implementation.

INTRODUCTION

  • Kohn–Sham orbitals are expensive to store and compute for large systems, while hybrid-functional exact exchange has cubic scaling and a large prefactor.The latter cost limits hybrid-functional calculations to hundreds of atoms, as described in the passage.Source: INTRODUCTION · p. 1
  • Maximally localized Wannier functions use nonlinear optimization, whereas localized orbitals can support linear-scaling methods for local, semilocal, and hybrid functionals.The cited locality principle applies to insulating systems with a finite HOMO-LUMO gap.Source: INTRODUCTION · p. 1
  • SCDM constructs localized basis functions directly from selected density-matrix columns, with a simple linear transformation available to obtain an orthonormal localized basis.The approach uses the exponential off-diagonal decay of the density matrix in insulating systems.Source: INTRODUCTION · p. 1
  • The method has one adjustable parameter, the truncation threshold, and supports arbitrary basis sets when orbitals can be represented on a real-space grid.The stated scope includes planewave and Gaussian basis sets.Source: INTRODUCTION · p. 1
  • A prototype uses rank-revealing QR and matrix multiplication, and the tested parallel implementation scaled to more than 1,000 processors.The method is intended to support localized basis functions for self-consistent Hartree–Fock terms and exchange-energy calculations combined with Hockney’s algorithm.Source: INTRODUCTION · p. 2

THEORY

For insulating systems represented on a real-space grid, localized density-matrix columns can form a basis for the Kohn–Sham subspace. The method selects a well-conditioned set of columns using rank-revealing QR, with optional Cholesky orthogonalization.

THEORY

  • Rank-revealing QR selects density-matrix columns that form a well-conditioned localized basis; arbitrary independent columns may be poorly conditioned.For insulating systems, each density-matrix column is localized.Source: THEORY · pp. 2–3
  • Factoring Ψ* instead of explicitly forming P yields the column-selection permutation and scales as O(N^2 Ne).Directly storing P costs N^2, and its partial rank-revealing QR costs O(Ne N^2).Source: THEORY · p. 3
  • Cholesky orthogonalization of PC,C produces an orthogonal basis without significantly reducing localization.The selected columns make PC,C sparse, with significant entries of its Cholesky factor near the diagonal.Source: THEORY · p. 3
  • The selected density-matrix columns P:,C provide the localized SCDM basis for the span of the Kohn–Sham orbitals.The paper gives P:,C = Ψ(ΨC,:)∗ as its construction.Source: THEORY · pp. 2–3
  • The deterministic procedure applies to local or nonlocal basis sets when orbitals are represented on a real-space grid.The authors describe this grid representation as an explicit requirement for the method in this paper.Source: THEORY · pp. 2–3
  • Its construction uses standard linear-algebra routines that can be parallelized without specialized code.The authors identify parallel versions of several algorithm steps and note that the small Cholesky factorization often need not be parallelized.Source: THEORY · p. 3

Localized basis functions

The SCDM basis functions are highly localized in silicon and water, and orthogonalization preserves that locality. Their centers track silicon bonds and oxygen atoms in the tested systems.

Localized basis functions

  • The localized functions resemble MLWFs, with centers between Si–Si bonds in silicon and closer to oxygen than hydrogen in water.Source: Localized basis functions · p. 4
  • At 10^-2 relative truncation, each SCDM support occupies 1–2% of the domain; orthogonalization preserves locality and can further localize functions.The locality ratio is the truncated support volume divided by the whole domain volume.Source: Localized basis functions · p. 4
  • The PC,C condition numbers are 3.18 for silicon and 2.83 for water, consistent with the reported well-conditioned matrices.Source: Localized basis functions · p. 4
  • Figure 1 depicts orthogonalized SCDM isosurfaces for a 512-atom silicon crystal and a 64-molecule water snapshot.Source: Localized basis functions · p. 4, Figure 1 · p. 4

Computation of Hartree-Fock exchange energy

The section uses orthogonalized, truncated SCDM functions and Hockney’s algorithm to reduce Hartree–Fock exchange costs. In quasi-one-dimensional silicon tests, this approach produced large speedups at both tested error levels.

Computation of Hartree-Fock exchange energy

  • Because exchange is invariant under unitary transformations, orthogonalized, truncated SCDM can replace delocalized Kohn–Sham orbitals while introducing controllable truncation error.The relative truncation threshold can be chosen to target a desired accuracy.Source: Computation of Hartree-Fock exchange… · pp. 4–5
  • Discarding disjoint or negligible orbital pairs leaves O(Ne) significant terms, while Hockney’s algorithm confines Poisson solves to pair supports without added approximation error.The figure compares localized-basis timings using full FFTs and small FFTs with Kohn–Sham timings.Source: Computation of Hartree-Fock exchange… · p. 5
  • Overall cost is O(Ne) when basis-function support remains constant as system size grows; the scaling constant depends mainly on Hockney FFT sizes.The authors describe this as holding under mild assumptions.Source: Computation of Hartree-Fock exchange… · p. 5
  • At the largest tested size, Hockney-based exchange was about 300× faster at 4%–5% error and 90× faster at 0.3%–0.4% error.The tests used quasi-one-dimensional silicon systems with 32–512 atoms.Source: Computation of Hartree-Fock exchange… · p. 5, Figure 3 · p. 5

Parallel computation

The parallel implementation computes orthogonalized SCDM close to ideally through 1,024 processors for the tested problem. Its reported runtime excludes matrix distribution and disk reading.

Parallel computation

  • The method scaled close to ideally through 1,024 processors for the tested problem.The largest case used a 777600 × 1024 orbital matrix.Source: Parallel computation · p. 6
  • Computing orthogonalized SCDM took under 1.5 seconds on 1,024 processors, excluding 1.3 seconds to distribute the matrix and the time to read it from disk.Source: Parallel computation · p. 6
  • The FORTRAN implementation used ScaLAPACK for parallel pivoted QR, matrix multiplication, and triangular solves.The reported parallel timings omit matrix-distribution time.Source: Parallel computation · p. 6

CONCLUSION

SCDM provides a fast route to a well-localized orthogonal basis for the Kohn–Sham subspace of insulating systems. Its scalability supports hybrid-functional calculations, while real-space-grid representation and randomized variants mark important scope and quality considerations.

CONCLUSION

  • For insulating systems, the nearsightedness principle guarantees SCDM locality, and orthogonalized SCDM can quickly build a well-localized basis immediately after computing Ψ.The basis represents the Kohn–Sham orbital subspace.Source: CONCLUSION · p. 6
  • SCDM’s high parallel scalability can support localized basis functions for self-consistent Hartree–Fock exchange in hybrid-functional calculations.The conclusion presents this as an extension beyond computing the Hartree–Fock exchange energy.Source: CONCLUSION · p. 6
  • This work requires Kohn–Sham orbitals on a real-space grid; directly finding a localized representation in localized basis sets is a possible alternative whose numerical consequences need testing.The grid requirement may be less natural for localized atomic orbitals than for planewaves or finite differences.Source: CONCLUSION · p. 6
  • A randomized SCDM algorithm may reduce computational cost, but it slightly reduced localized-basis quality in the examples and may still benefit very large problems.The randomized method was not used for the examples presented.Source: CONCLUSION · p. 6
  • SCDM may also be useful for linear-scaling calculations in insulating systems, post-DFT quantum-chemical treatment, and compression of orbital pair products for excited-state calculations.Source: CONCLUSION · p. 6
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