The random Green's function (rGF) method of Tang, Liu, Zhang, Zhang, Zhai, Yuan and Ke (Chin. Phys. Lett. 41, 053102 (2024)) computes the one-particle density matrix of large systems at linear-scaling cost by evaluating the resolvent Green's function on stochastic random states projected onto a small Krylov subspace, and was demonstrated with density-functional tight-binding (DFTB) on water clusters up to 9984 H2O at about 1 meV per molecule accuracy. The authors explicitly deferred two coupled developments: (i) preconditioned and multiple-Krylov-space constructions that remove the inner conjugate-gradient loop for O−1HO^{-1}H-vector products on non-orthogonal bases; and (ii) variance reduction beyond simple deflation via better deterministic approximations D0\mathcal{D}_0 and probing; and they claimed, without demonstration, general applicability to Kohn-Sham DFT, tight-binding, nonequilibrium Green's function and many-body formalisms. The same group's 2026 chromatic-superposition-state preprint advances the broader program but replaces, rather than extends, the stochastic representation and is again demonstrated only with DFTB. The problem — establishing rGF as a practical general-purpose linear-scaling stochastic electronic-structure method — therefore remains open.

Background

Kohn-Sham density functional theory (DFT) is normally solved by diagonalizing the Hamiltonian, a step whose cost grows as the cube of the number of electrons, which blocks first-principles simulation of systems beyond a few thousand atoms. Linear-scaling (O(N)O(N)) methods exploit Kohn's nearsightedness principle — the decay of the density matrix with distance — to compute the density matrix directly. One recent route is the stochastic (random-state) approach: the trace and density matrix are estimated by averaging over NsN_s random vectors with random ±1\pm 1 entries, giving an unbiased statistical error that scales as 1/Ns1/\sqrt{N_s} and, for spatially averaged observables such as the energy per electron, as 1/NeNs1/\sqrt{N_e N_s} (self-averaging). Stochastic DFT in this sense was introduced by Baer, Neuhauser and Rabani (Phys. Rev. Lett. 111, 106402 (2013)) and developed into mixed stochastic-deterministic warm-dense-matter algorithms by White and Collins (Phys. Rev. Lett. 125, 055002 (2020)). These implementations expand the Fermi-Dirac occupation operator f(H^)f(\hat{H}) (the finite-temperature occupation function of the Hamiltonian) in Chebyshev polynomials, which is efficient at high temperature but requires thousands of polynomial terms for narrow-gap systems at low temperature. The rGF method of Tang et al. (2024) addresses this by working with the retarded Green's function G(z)=(z−O−1H)−1G(z)=(z-O^{-1}H)^{-1} (the resolvent; HH is the Hamiltonian and OO the overlap matrix of a non-orthogonal localized atomic basis) evaluated on random states: the random states are projected onto a small Krylov subspace (the span of successive applications of the operator to a starting vector), inside which the Fermi-Dirac operator is obtained by exact diagonalization and a contour integration. The construction needs products of O−1HO^{-1}H with vectors, which the authors implement through an inner conjugate-gradient (CG) solve for each O−1O^{-1}-vector product, since O−1O^{-1} is dense. A deflation technique subtracts a sparse deterministic approximation D0\mathcal{D}_0 of the density matrix (built from fragment density matrices) before the stochastic sampling, which reduces the statistical error by orders of magnitude; an alternative variance-reduction technique in the stochastic literature is probing, which colors matrix entries so that selected elements can be recovered exactly from few structured vectors. Demonstrated with density-functional tight-binding (DFTB, an approximation to DFT with a minimal non-orthogonal basis and tabulated matrix elements) on gapped clusters (H2O with a 12.63 eV gap, Si with 2.0 eV and 0.89 eV) at low temperature, the method reaches about 1 meV per H2O in total energy for 9984-molecule water clusters; a metallic tungsten cluster appears only in a CPU-time crossover comparison, and the abstract scopes the small-Krylov claim to gapped materials at low temperature. The paper closes by claiming that rGF is generally applicable to density functional theory, tight-binding, the nonequilibrium Green's function (NEGF) formalism used for nanoelectronics transport, and quantum many-body methods, while providing no implementation, stability analysis, or performance study in any of those settings. The Krylov acceleration it defers has a precedent in the order-NN generalized-eigenvalue method of Hoshi, Yamamoto, Fujiwara, Sogabe and Zhang (J. Phys.: Condens. Matter 24, 165502 (2012)), which avoids inner solves through multiple Krylov spaces. These deferred developments form one coherent objective: turning rGF from a DFTB demonstration into a practical, general-purpose linear-scaling stochastic electronic-structure method.

Problem Statement

Develop the random Green's function (rGF) method of Tang et al. (2024) into a practical, general-purpose linear-scaling stochastic electronic-structure method. The single overarching objective — rGF's practical generality — comprises three source-grounded strands: (1) algorithmic acceleration on non-orthogonal localized bases that substantially reduces the computational prefactor while preserving linear scaling; (2) systematic stochastic variance reduction beyond the present fragment deflation, with quantified error reduction; and (3) concrete implementation of rGF in the formalisms for which the paper claims general applicability but demonstrates it only with density-functional tight-binding on gapped clusters at low temperature — Kohn-Sham DFT, tight-binding, nonequilibrium Green's function transport for nanoelectronics, and Green's-function-based quantum many-body methods. All three strands are jointly required: strands (1) and (2) must be demonstrated on a shared non-orthogonal-basis rGF implementation, and strand (3) requires a concrete implementation and numerical demonstration in at least one of the listed formalisms beyond density-functional tight-binding; completing a single strand on its own does not resolve the problem.

References

  1. Mingfa Tang, Chang Liu, Aixia Zhang, Qingyun Zhang, Shengjun Yuan, Youqi Ke, 'Random Green's function method for large-scale electronic structure calculation', arXiv:2311.18161 (v2, 2024), https://arxiv.org/abs/2311.18161
  2. Mingfa Tang, Chang Liu, Aixia Zhang, Qingyun Zhang, Jiayu Zhai, Shengjun Yuan, Youqi Ke, 'Random Green's Function Method for Large-Scale Electronic Structure Calculation', Chinese Physics Letters 41, 053102 (2024), DOI: 10.1088/0256-307X/41/5/053102, https://doi.org/10.1088/0256-307X/41/5/053102
  3. Zhikang Jiang, Zhizhi Xiao, Mingfa Tang, Weiyu Li, Zhaoru Sun, Ke Xia, Youqi Ke, 'High-performance linear-scaling electronic structure method via chromatic superposition states', arXiv:2605.20918 (2026), https://arxiv.org/abs/2605.20918
  4. Methawee Nukunudompanich, Ye min Thant, Htet Arkar Kyaw, Manabu Ihara, Sergei Manzhos, 'Ab initio-based modeling of plasticity and related phenomena', Chemical Physics Reviews 7, 031304 (2026), DOI: 10.1063/5.0341141, https://doi.org/10.1063/5.0341141
  5. Zhichang Fu, Yunhai Li, Weiqing Zhou, Shengjun Yuan, 'Large-scale stochastic propagation method beyond the sequential approach', arXiv:2510.17432 (2025), https://arxiv.org/abs/2510.17432
  6. Roi Baer, Daniel Neuhauser, Eran Rabani, 'Self-Averaging Stochastic Kohn-Sham Density-Functional Theory', Physical Review Letters 111, 106402 (2013), DOI: 10.1103/PhysRevLett.111.106402, https://doi.org/10.1103/PhysRevLett.111.106402
  7. A. J. White, L. A. Collins, 'Fast and Universal Kohn-Sham Density Functional Theory Algorithm for Warm Dense Matter to Hot Dense Plasma', Physical Review Letters 125, 055002 (2020), DOI: 10.1103/PhysRevLett.125.055002, https://doi.org/10.1103/PhysRevLett.125.055002
  8. T. Hoshi, S. Yamamoto, T. Fujiwara, T. Sogabe, S.-L. Zhang, 'An order-N electronic structure theory with generalized eigenvalue equations and its application to a ten-million-atom system', Journal of Physics: Condensed Matter 24, 165502 (2012), DOI: 10.1088/0953-8984/24/16/165502, https://doi.org/10.1088/0953-8984/24/16/165502