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Adaptively Compressed Exchange Operator
Lin Lin
TL;DR
The computational cost of Fock exchange hinders Hartree-Fock and hybrid-functional calculations. The paper develops ACE, an adaptive low-rank operator matching exchange on occupied orbitals, and reports near-GGA SCF costs with more than 50 times speedup for a 216-atom silicon system.
Problem
The large computational cost of Fock exchange hinders Hartree-Fock and hybrid-functional calculations, including systems with hundreds of atoms.
Method
ACE adaptively constructs a low-rank exchange operator that reproduces exchange-operator results on occupied orbitals during self-consistent iterations.
Results
The ACE SCF iteration cost is only marginally larger than GGA, with more than 50 times speedup for a 216-atom silicon system.
Takeaways & Limitations
ACE can reduce exchange cost without loss of accuracy, independently of band-gap size, for insulators, semiconductors, and metals.
Takeaways & Limitations
Standard low-rank decompositions fail because the exchange operator is not itself low rank and can produce unphysical results when forcefully compressed.
Abstract
from arXiv · showhide
The Fock exchange operator plays a central role in modern quantum chemistry. The large computational cost associated with the Fock exchange operator hinders Hartree-Fock calculations and Kohn-Sham density functional theory calculations with hybrid exchange-correlation functionals, even for systems consisting of hundreds of atoms. We develop the adaptively compressed exchange operator (ACE) formulation, which greatly reduces the computational cost associated with the Fock exchange operator without loss of accuracy. The ACE formulation does not depend on the size of the band gap, and thus can be applied to insulating, semiconducting as well as metallic systems. In an iterative framework for solving Hartree-Fock-like systems, the ACE formulation only requires moderate modification of the code, and can be potentially beneficial for all electronic structure software packages involving exchange calculations. Numerical results indicate that the ACE formulation can become advantageous even for small systems with tens of atoms. In particular, the cost of each self-consistent field iteration for the electron density in the ACE formulation is only marginally larger than that of the generalized gradient approximation (GGA) calculation, and thus offers orders of magnitude speedup for Hartree-Fock-like calculations.
1 Introduction
The Fock exchange operator makes Hartree-Fock-like calculations computationally difficult because it depends on the full density matrix. This work introduces ACE, an adaptive low-rank exchange representation that preserves relevant occupied-orbital results while substantially reducing cost.
- Motivation: Hybrid-functional calculations are computationally demanding because Fock exchange depends on the entire density matrix rather than only the electron density.This limits calculations for systems containing hundreds of atoms or fewer.
- Existing approaches: Existing linear-scaling exchange methods exploit localized occupied orbitals but rely on a finite HOMO-LUMO gap and are most successful for systems with substantial band gaps.Their compressed representations arise from localization in insulating systems.
- ACE formulation: ACE constructs a low-rank exchange operator that matches the exchange operator on occupied orbitals, adapting as those orbitals change during self-consistent iterations.The occupied-orbital subspace has known rank, enabling the targeted compression.
- ACE advantages: ACE is strictly low rank without loss of accuracy for physical quantities such as total energies and band gaps, and its effectiveness does not depend on band-gap size.The formulation is therefore applicable to insulators, semiconductors, and metals.
- ACE advantages: After construction, ACE application has low cost, and integrating it into iterative Hartree-Fock-like solvers requires only moderate code changes.The authors state that the approach could benefit electronic-structure software involving exchange calculations.
- Numerical results: The cost of each ACE hybrid-functional SCF iteration is only marginally larger than for GGA, with more than 50 times speedup for a 216-atom silicon system.The formulation is also reported to be advantageous for systems with tens of atoms.
2 Iterative methods for solving Hartree-Fock-like equa-
Iterative methods avoid explicitly constructing the exchange operator by repeatedly applying it to orbitals and solving linear eigenvalue problems within nested self-consistent iterations. Their cubic scaling can be preferable to quartic integral-based methods, especially for large-basis calculations.
- Exchange-operator structure: The exchange operator is full-rank and nonlocal, depending on the occupied orbitals rather than only the electron density.This dependence makes direct treatment substantially more difficult than local potential terms.
- Iterative treatment: Iterative algorithms apply VX to orbitals instead of explicitly constructing VX, using Poisson solves accelerated by FFTs.In planewave calculations, each Poisson solve costs O(Ng log Ng).
- Iterative treatment: The total cost of applying VX to all occupied orbitals scales cubically, making iterative algorithms asymptotically less expensive than quartic methods.This can make iterative methods attractive even with small basis sets for large systems.
- Self-consistent solution: Nested self-consistency separates electron-density updates from exchange-operator updates, using inner density iterations and outer exchange-energy iterations.With VX fixed, standard charge-mixing schemes can efficiently converge the electron density.
- Hybrid functionals: Hybrid-functionals calculations replace the full exchange contribution with a scaled or screened exchange operator, depending on the functional.For PBE0, the exchange operator is one quarter of the Hartree-Fock exchange operator; HSE replaces the Coulomb kernel with a screened kernel.
X and EP BE,LR
Screened-exchange functionals separate short- and long-range exchange contributions and control the short-range interaction with a screening parameter. Their exchange operator uses a screened Coulomb kernel instead of the ordinary Coulomb kernel.
- Exchange-range decomposition: In the PBE functional, X denotes the short-range and long-range parts of the exchange contribution.The short-range component is identified as the short-range part of the Fock exchange.
- Screening: The parameter µ controls the screening length of the short-range Fock exchange interaction.The screened interaction is defined through the complementary error function.
- Screening: HSE changes the exchange construction by replacing the Coulomb kernel with a screened Coulomb kernel.The screened kernel is then used to define the HSE exchange operator.
3 Adaptively compressed exchange operator
ACE replaces the full Fock exchange operator with an adaptive low-rank operator designed to agree with exchange on the occupied-orbital subspace. At self-consistency, it preserves the relevant physical quantities while substantially reducing repeated application costs.
- Motivation: The Fock exchange operator is expensive because iterative methods repeatedly apply it to orbitals.These applications include matrix-vector products and Poisson-equation solves.
- Construction: ACE constructs a strict-rank-N_e effective operator whose accuracy is enforced on the occupied-orbital subspace rather than on arbitrary orbitals.This adaptive condition is sufficient for solving Hartree-Fock-like equations.
- Accuracy: At orbital self-consistency, ACE yields the same physical quantities and correct eigenvalues as standard Hartree-Fock-like methods.The result holds even though the operator discrepancy is not controlled for arbitrary orbitals.
- Cost reduction: After construction, applying ACE to any orbital costs about as much as applying a nonlocal pseudopotential because of its low-rank structure.ACE can be reused across inner SCF iterations and linear-eigenvalue iterations.
- Cost reduction: 10 ∼100 or more Hamiltonian applications per outer iteration make ACE's reuse especially effective for reducing Poisson-problem cost.The implementation requires only moderate code modification.
- Practical implications: ACE can reduce exchange-energy cost without extra Poisson equations and shows computational advantages even for small systems.Its construction cost is the same as applying the Fock exchange interaction.
4 Numerical results
The numerical experiments validate ACE accuracy against Quantum ESPRESSO and show that it preserves convergence while substantially reducing hybrid-functional computational cost. For silicon systems, ACE brings per-iteration HSE costs close to GGA costs and yields substantial speedups over standard exchange calculations.
- 9 meV and 11 meV are the Fock exchange-energy differences between Quantum ESPRESSO and PWDFT for the water and silicon systems, respectively.
- 8 meV and 5 meV are the corresponding HOMO-LUMO-gap differences for the water and silicon systems, respectively.
- The PWDFT results differ negligibly between the standard hybrid-functional implementation and ACE for both systems.
- ACE does not hinder hybrid-functional convergence: with oversampling ratio r = 1, its convergence rate is very close to that without ACE.
- 42 and 58 times are the standard-HSE-to-PBE per-SCF-iteration cost ratios for the 64- and 216-atom silicon systems, respectively.
- 1.18 and 1.05 are the corresponding ACE-HSE-to-PBE ratios for the 64- and 216-atom systems, making ACE-HSE costs only marginally larger than GGA costs.
- For the 216-atom system, ACE achieves an 8.8 times speedup over standard implementation even when the inner iteration contains one SCF iteration.
5 Conclusion
The conclusion presents ACE as an accurate low-rank compression of the Fock exchange operator that substantially reduces hybrid-functional costs after construction. Its effectiveness is independent of band-gap size, and the formulation is applicable across insulating, semiconducting, and metallic systems with moderate code modification.
- ACE compresses the Fock exchange operator without loss of accuracy and without dependence on band-gap size.
- ACE can be used for insulators, semiconductors, and metals in iterative Hartree-Fock and hybrid-functional Kohn-Sham calculations.
- The formulation requires only moderate code modification and could potentially be applied across electronic-structure software packages that treat exchange.
- ACE construction costs the same as applying the Fock exchange operator, while subsequent reuse reduces the cost of repeated SCF iterations.
- After one construction of ACE, each hybrid-functional electron-density SCF iteration costs only marginally more than a GGA iteration.
- Combining ACE with linear-scaling methods or localization for screened interactions could further reduce construction and application costs.