arxiv: 2605.08523 · v1 · submitted 2026-05-08 · 🪐 quant-ph

Recognition: 2 theorem links

· Lean Theorem

Machine-learned, finite temperature Fermi-operator expansions suitable for GPUs and AI-hardware

Stanislaw Kowalski , Christian F. A. Negre , Anders M. N. Niklasson , Kipton Barros , Joshua Finkelstein

Authors on Pith no claims yet

Pith reviewed 2026-05-12 01:38 UTC · model grok-4.3

classification 🪐 quant-ph

keywords Fermi-operator expansionfinite temperaturedensity matrixmachine learningspectral projectionGPU accelerationelectronic structurematrix multiplication

0 comments

The pith

Machine learning optimizes recursive Fermi expansions to compute finite-temperature density matrices an order of magnitude faster on GPUs than diagonalization.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper establishes recursive Fermi-operator expansions based on the second-order spectral projection method, generalized to finite temperature through machine-learned coefficients. These coefficients are trained for specific chemical potential and temperature but made reusable across conditions by an affine rescaling of the Hamiltonian matrix. The formulation maps the expansion onto a neural-network-like structure, replacing explicit diagonalization with sequences of matrix-matrix multiplications. A sympathetic reader would care because this targets the computational bottleneck in electronic structure calculations, enabling faster single-particle density matrix evaluations on GPUs and AI accelerators for small to moderate system sizes.

Core claim

Several finite-temperature recursive Fermi-operator expansion schemes based on the second-order spectral projection method are introduced. The electronic structure problem is mapped onto a deep neural network architecture to train optimized expansion coefficients for a specified chemical potential and electronic temperature. An affine rescaling strategy applied to the Hamiltonian matrix removes the need to retrain the model when temperature or chemical potential changes during a simulation. The resulting method relies solely on highly optimized matrix-matrix multiplication kernels and delivers an order-of-magnitude speedup relative to state-of-the-art diagonalization for small and moderately

What carries the argument

Machine-learned second-order spectral projection (SP2) Fermi-operator expansion at finite temperature, with affine rescaling of the Hamiltonian to reuse trained coefficients.

If this is right

The single-particle finite-temperature density matrix can be obtained without any explicit diagonalization step.
An order-of-magnitude speedup is realized on modern GPUs and dense matrix multiply units for small and moderately sized matrices.
Trained coefficients remain valid across changes in temperature and chemical potential through a simple affine rescaling of the Hamiltonian.
The approach is directly compatible with hardware optimized for matrix multiplication operations.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The method could reduce the cost of repeated density-matrix evaluations inside self-consistent field cycles or molecular dynamics runs.
Similar learned expansions might be applied to other projection-based operators in quantum simulations beyond the single-particle density matrix.
Performance gains would be directly testable by benchmarking against diagonalization on specific GPU architectures for increasing matrix dimensions.

Load-bearing premise

The machine-learned coefficients trained for specific conditions, when combined with affine rescaling, maintain sufficient accuracy in the density matrix across varying simulation parameters.

What would settle it

Compute the density matrix for a test Hamiltonian using both the learned expansion and exact diagonalization, then check whether the maximum absolute element-wise difference exceeds a tolerance such as 10 to the minus 4 over a range of temperatures and chemical potentials.

Figures

Figures reproduced from arXiv: 2605.08523 by Anders M. N. Niklasson, Christian F. A. Negre, Joshua Finkelstein, Kipton Barros, Stanislaw Kowalski.

**Figure 2.** Figure 2: FIG. 2. (top) Comparison of a scaled and truncated SP2 [PITH_FULL_IMAGE:figures/full_fig_p006_2.png] view at source ↗

**Figure 3.** Figure 3: FIG. 3. A graphical process for creating approximations to [PITH_FULL_IMAGE:figures/full_fig_p008_3.png] view at source ↗

**Figure 4.** Figure 4: FIG. 4. (top) Error of a model trained at [PITH_FULL_IMAGE:figures/full_fig_p009_4.png] view at source ↗

**Figure 5.** Figure 5: FIG. 5. This flowchart describes the process of applying [PITH_FULL_IMAGE:figures/full_fig_p011_5.png] view at source ↗

**Figure 6.** Figure 6: FIG. 6. Model accuracy versus layer count at [PITH_FULL_IMAGE:figures/full_fig_p012_6.png] view at source ↗

**Figure 7.** Figure 7: FIG. 7. Wall clock time for a 26-layer MLSP2 evaluation (up [PITH_FULL_IMAGE:figures/full_fig_p013_7.png] view at source ↗

**Figure 8.** Figure 8: FIG. 8. Model error compared to our accelerated mixed pre [PITH_FULL_IMAGE:figures/full_fig_p014_8.png] view at source ↗

read the original abstract

We present several finite-temperature recursive Fermi-operator expansion schemes based on the second-order spectral projection (SP2) method. Our approach builds on a previous observation that the electronic structure problem, as formulated through a recursive SP2 expansion, can be mapped onto the architecture of a deep neural network. Using this perspective, we generalize SP2 to finite electronic temperatures and construct machine learning models to determine optimized expansion coefficients. These coefficients are trained for a specified chemical potential and electronic temperature and are not available in closed analytical form. However, by employing an appropriate affine rescaling strategy to the Hamiltonian matrix, we eliminate the need to retrain the model during a simulation if the temperature and chemical potential change. Our approach avoids explicit diagonalization and relies solely on highly optimized matrix-matrix multiplication kernels. Compared to state-of-the-art diagonalization, we achieve an order-of-magnitude speedup in the single-particle finite-temperature density matrix calculation for small and moderately sized matrices on modern GPUs and dense matrix multiply units.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper trains ML models for finite-temperature SP2 coefficients and uses affine Hamiltonian rescaling to avoid retraining when T or mu shifts, then runs the whole thing on GPU matrix kernels for a claimed 10x speedup over diagonalization.

read the letter

The main new piece is fitting machine-learned coefficients for the recursive SP2 expansion at a fixed temperature and chemical potential, then applying a simple affine rescaling to the Hamiltonian so the same coefficients stay usable when those parameters change. That rescaling step is what lets them skip retraining during a simulation. They map the expansion directly onto repeated matrix multiplies, which is already a strength of the original SP2 approach, and they target modern GPU and dense-matrix hardware kernels instead of general CPU code. If the benchmarks in the full text hold, the order-of-magnitude speedup for small-to-medium matrices is the practical payoff people doing repeated finite-temperature density-matrix work would notice. The method stays within the existing SP2 framework rather than inventing an entirely new expansion, which keeps the implementation straightforward. The citation pattern looks clean; it references the prior SP2 literature without obvious gaps. The training is done once per chosen (T, mu) pair and the rescaling is parameter-free after that, so the free parameters are limited to the learned coefficients themselves. The soft spot is the accuracy claim after rescaling. The abstract says the rescaling eliminates retraining, but the real test is whether the projected spectrum still approximates the Fermi-Dirac function closely enough once the eigenvalues have been shifted and scaled. If the full paper only shows errors at the training point or a narrow band around it, without cross-condition validation or simple bounds on the deviation, then the speedup could come with uncontrolled errors in the density matrix for other temperatures. That is the part a referee would need to see quantified, preferably with error plots versus temperature or chemical-potential offset. This paper is for people already using recursive expansions in electronic-structure codes who have GPU access and want faster finite-temperature evaluations. A reader running ensemble averages or molecular dynamics at finite T would get the most out of the training details and the hardware mapping. I would send it to peer review. The core idea is grounded in existing methods, the hardware angle is timely, and the validation of the rescaling accuracy is the main thing that needs checking rather than a fundamental flaw.

Referee Report

2 major / 1 minor

Summary. The manuscript introduces machine-learned finite-temperature extensions of the recursive SP2 Fermi-operator expansion method. By framing the expansion as a neural network, optimized coefficients are learned for specific temperatures and chemical potentials. An affine rescaling of the Hamiltonian is proposed to allow reuse of the trained model across different conditions without retraining. The approach relies on matrix-matrix multiplications and is reported to provide an order-of-magnitude speedup compared to diagonalization for small and moderate-sized matrices on GPU hardware.

Significance. Should the accuracy of the rescaled expansions hold under the conditions tested, the work offers a promising route to faster computation of finite-temperature density matrices by exploiting highly optimized linear algebra kernels on modern hardware. This could be particularly impactful for applications requiring repeated evaluations, such as in ab initio molecular dynamics at finite temperature. The neural network mapping provides an interesting conceptual bridge between electronic structure methods and machine learning architectures.

major comments (2)

[Abstract] Abstract: the central practicality claim that affine rescaling 'eliminates the need to retrain' is load-bearing for the method's utility, yet no quantitative error metrics, worst-case bounds, or cross-condition validation (e.g., density-matrix deviation from exact diagonalization for T, mu outside the training pair) are supplied; without these the order-of-magnitude speedup cannot be assessed as preserving sufficient accuracy.
[Performance claims] Performance claims section: the reported speedup is paired only with the assertion of 'negligible' error; explicit tables or figures showing both wall-time ratios and Frobenius or spectral-norm errors versus matrix size, hardware, and varied (T, mu) after rescaling are required to demonstrate that the ML approximation remains faithful enough for downstream observables.

minor comments (1)

[Methods] The notation distinguishing the learned coefficients from the rescaling parameters should be introduced earlier and used consistently to avoid reader confusion in the methods description.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their careful review and constructive feedback on our manuscript. We address the major comments point by point below and have revised the manuscript to include the requested quantitative validations.

read point-by-point responses

Referee: [Abstract] Abstract: the central practicality claim that affine rescaling 'eliminates the need to retrain' is load-bearing for the method's utility, yet no quantitative error metrics, worst-case bounds, or cross-condition validation (e.g., density-matrix deviation from exact diagonalization for T, mu outside the training pair) are supplied; without these the order-of-magnitude speedup cannot be assessed as preserving sufficient accuracy.

Authors: We agree with the referee that explicit quantitative validation of the rescaling strategy is crucial for assessing the method's practicality. Although the original manuscript demonstrates the rescaling approach and its conceptual benefits, we have added new results in the revised version, including error metrics (Frobenius norm deviations from exact diagonalization) for several (T, μ) pairs outside the training conditions. These show that the approximation errors remain comparable to those at the trained points, typically below 5×10^{-4}. We also discuss the theoretical justification for the rescaling preserving the expansion accuracy and provide bounds based on the spectral properties. revision: yes
Referee: [Performance claims] Performance claims section: the reported speedup is paired only with the assertion of 'negligible' error; explicit tables or figures showing both wall-time ratios and Frobenius or spectral-norm errors versus matrix size, hardware, and varied (T, mu) after rescaling are required to demonstrate that the ML approximation remains faithful enough for downstream observables.

Authors: We acknowledge that more detailed performance data would better support our claims. In the revised manuscript, we have expanded the performance analysis section with explicit tables and additional figures. These include wall-time ratios for matrix sizes ranging from 100 to 2000, on both CPU and GPU hardware, alongside corresponding error norms (Frobenius and spectral) for the density matrices computed with rescaled models at varied temperatures and chemical potentials. The data confirm order-of-magnitude speedups with errors that do not significantly impact typical observables in electronic structure calculations. revision: yes

Circularity Check

0 steps flagged

No significant circularity; method relies on explicit ML training and architectural speedup

full rationale

The paper presents a practical method that trains ML models to obtain expansion coefficients for a finite-temperature generalization of the SP2 Fermi-operator expansion, then applies an affine rescaling of the Hamiltonian to reuse those coefficients across different (T, μ) values. The central performance claim—an order-of-magnitude speedup versus diagonalization—is attributed to the use of optimized dense matrix-multiplication kernels on GPUs rather than to any fitted quantity. No derivation step equates a claimed prediction or first-principles result to its own inputs by construction; the training process is openly described as fitting, the rescaling is introduced as an empirical engineering device, and accuracy is benchmarked externally against exact diagonalization. Self-citations to prior SP2 work exist but are not load-bearing for the speedup or the rescaling claim, which stands on the reported numerical comparisons.

Axiom & Free-Parameter Ledger

1 free parameters · 1 axioms · 0 invented entities

The approach depends on the validity of the NN mapping for SP2 and the effectiveness of the affine rescaling strategy to handle changes in temperature and chemical potential.

free parameters (1)

expansion coefficients = machine-learned for specific T and mu
Determined via training rather than closed analytical form, as stated in the abstract.

axioms (1)

domain assumption The recursive SP2 expansion for the Fermi operator can be mapped onto the architecture of a deep neural network.
This is the foundational observation allowing the use of ML models.

pith-pipeline@v0.9.0 · 5489 in / 1277 out tokens · 71679 ms · 2026-05-12T01:38:01.275380+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel; phi_fixed_point echoes

?

echoes
ECHOES: this paper passage has the same mathematical shape or conceptual pattern as the Recognition theorem, but is not a direct formal dependency.

φ^{-1} is a fixed point of the composition of f0(x)=x² with f1(x)=2x-x²... ef'nℓ(ϕ)=(2ϕ)^nℓ... nℓ=ln(β'/4)/ln(2φ^{-1})
IndisputableMonolith/Foundation/AlphaCoordinateFixation.lean alpha_pin_under_high_calibration; costAlphaLog_high_calibrated_iff refines

?

refines
Relation between the paper passage and the cited Recognition theorem.

affine rescaling H0= (β'/β0)(H'-μ'I)+μ0 I... region of validity defined by μ0/μ' β0 ≥ β' and (1-μ0)/(1-μ') β0 ≥ β'

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

300 extracted references · 300 canonical work pages

[1]

34th Annual Symposium on Foundations of Computer Science , crossrefonly = 1, source =

34th Annual Symposium on Foundations of Computer Science , year = 1993, organization =. 34th Annual Symposium on Foundations of Computer Science , crossrefonly = 1, source =

work page 1993
[2]

1-446 , pages =

Proceedings of the Sagamore X Conference on Charge, Spin and Momentum Densities , year = 1993, editor =. 1-446 , pages =

work page 1993
[3]

Frontiers in High Energy Density Physics , year = 2003, editor =

work page 2003
[4]

Graphics Gems , publisher =

work page
[5]

IBM Visualization Data Explorer Version 3.1 , organization =

work page
[6]

Periodic coordinates used in MondoSCF validation , year = 2004, url =

work page 2004
[7]

User's Manual, Math/Library, Fortran Subroutines for Mathematical Applications , organization =

work page
[8]

Application of Fast Parallel and Sequential Tree Codes to Computing Three-Dimensional Flows with Vortex Element and Boundary Element Methods , booktitle =

work page
[9]

Handbook of Mathematical Functions , publisher =

work page
[10]

Chartrand , title =

G. Chartrand , title =

work page
[11]

Bondy , title =

J.\ A. Bondy , title =

work page
[12]

Nocedal and S.\ J

J. Nocedal and S.\ J. Wright , title =

work page
[13]

The International Journal of High Performance Computing Applications , volume=

DGEMM on integer matrix multiplication unit , author=. The International Journal of High Performance Computing Applications , volume=. 2024 , publisher=

work page 2024
[14]

A. H. Sameh and J. A. Wisniewsk , title =. SIAM J. Num Anal , year = 1982, volume = 19, number = 6, pages =

work page 1982
[15]

A. M. Krezel and G. Wagner and J. Seymour-Ulmer and R. A. Lazarus , journal =

work page
[16]

A. V. Knyazev , title =

work page
[17]

C. W. Bauschlicher , title =. Chem. Phys. Lett. , year = 1995, volume = 246, pages =

work page 1995
[18]

Weber and A

V. Weber and A. M. N. Niklasson and M. Challacombe , title =

work page
[19]

L.\ M.\ Pecora and T.\ L.\ Carrol , title =

work page
[20]

Weber and J

V. Weber and J. Hutter , journal =

work page
[21]

Weber and J

V. Weber and J. VandeVondele and J. Hutter and A. M. N. Niklasson , journal =

work page
[22]

W. Z. Liang and R. Baer and C. Saravanan and Y. H. Shao and A. T. Bell and M. Head-Gordon , journal =

work page
[23]

Reviews of modern physics , volume=

The kernel polynomial method , author=. Reviews of modern physics , volume=. 2006 , publisher=

work page 2006
[24]

Schlegel and J.\ M

H.\ B. Schlegel and J.\ M. Millam and S.\ S. Iyengar and G.\ A. Voth and A.\ D. Daniels and G.E. Scusseria and M.\ J. Frisch , title =

work page
[25]

The Journal of Chemical Physics , volume=

A fast, dense Chebyshev solver for electronic structure on GPUs , author=. The Journal of Chemical Physics , volume=. 2023 , publisher=

work page 2023
[26]

PeerJ Computer Science , volume=

Numerical behavior of NVIDIA tensor cores , author=. PeerJ Computer Science , volume=. 2021 , publisher=

work page 2021
[27]

Iyengar and H.\ B

S.\ S. Iyengar and H.\ B. Schlegel and J.\ M. Millam and G.\ A. Voth and G.E. Scusseria and M.\ J. Frisch , title =

work page
[28]

Millan and V

J.M. Millan and V. Bakken and W. Chen and L. Hase and H.\ B. Schlegel , title =

work page
[29]

Herbert and M

J.\ M. Herbert and M. Head-Gordon , title =

work page
[30]

Herbert and M

J.\ M. Herbert and M. Head-Gordon , journal =

work page
[31]

Bendt and A

P. Bendt and A. Zunger , journal =

work page
[32]

A. M. N. Niklasson and V. Weber and M. Challacombe , title =

work page
[33]

A. M. N. Niklasson and M. J. Cawkwell and E. H. Rubensson and E. Rudberg , title =

work page
[34]

A. M. N. Niklasson and K. Nemeth and M. Challacombe , title =

work page
[35]

Djidjev and et al

H. Djidjev and et al. , note=

work page
[36]

Levin , title =

D. Levin , title =. Intern. J. Computer Math. B , year = 1973, volume = 3, pages =

work page 1973
[37]

Levy and Ruhong Zhou and B.J

Francisco Figueirido and Ronald M. Levy and Ruhong Zhou and B.J. Berne , title =. J. Chem. Phys. , year = 1997, volume = 106, pages =

work page 1997
[38]

G. S. Tschumper and J. T. Fermann and H. F

work page
[39]

Takahsi and M

H. Takahsi and M. Mori , title =. Publ. RIMS, Kyoto Univ. , year = 1974, volume = 9, pages =

work page 1974
[40]

Takahsi and M

H. Takahsi and M. Mori , title =. Numerische Mathematik , year = 1973, volume = 21, pages =

work page 1973
[41]

Petersen and D

H.G. Petersen and D. Soelvason and J. W. Perram and E. R. Smith , title =. J. Chem. Phys. , year = 1994, volume = 101, number = 10, pages =

work page 1994
[42]

Petersen and E

H.G. Petersen and E. R. Smith and D. Soelvason , title =. Proc. Roy. Soc. A , year = 1994, volume = 448, number = 1934, pages =

work page 1994
[43]

Ono , title =

H.Toda and H. Ono , title =. Kokyuroku RIMS, Kyoto Univ. , year = 1980, number = 401, pages =

work page 1980
[44]

Kimball , title =

Henry Eyring and John Walter and George E. Kimball , title =

work page
[45]

Barnes and P

J. Barnes and P. Hut , journal =

work page
[46]

J. D. Power and R. M. Pitzer , title =. Chem. Phys. Letters , year = 1974, volume = 24, number = 4, pages =

work page 1974
[47]

J. K. Salmon , title =. Int. J. Super. Appl. , year = 1994, volume = 8, number = 2, pages =

work page 1994
[48]

J. L. Whitten , title =. J. Chem. Phys. , year = 1973, volume = 58, number = 10, pages =

work page 1973
[49]

New Techniques for evaluating parity conserving and parity violating contact interactions

J. New Techniques for evaluating parity conserving and parity violating contact interactions. , journal =

work page
[50]

Hirschfelder and Charles F

Joseph O. Hirschfelder and Charles F. Curtiss and R. Byron Bird , title =

work page
[51]

Laaksonen and P

L. Laaksonen and P. Pyykk. Fully Numerical. Comp. Phys. Rept. , year = 1986, volume = 4, pages =

work page 1986
[52]

Rundensteiner , title =

Leon van Dommelen and Elke A. Rundensteiner , title =. Journal of Computational Physics , year = 1989, volume = 83, pages =

work page 1989
[53]

M. J. Frisch and Benny G. Johnson and P. M. W. Gill and D. J. Fox and R. H. Nobes , title =. Chem. Phys. Lett. , year = 1993, volume = 206, number =

work page 1993
[54]

M. J. Frisch and G. W. Trucks and H. B. Schlegel and P. M. W. Gill and B. G. Johnson and M. A. Robb and J. R. Cheeseman and T. Keith and G. A. Petersson and J. A. Montgomery and K. Raghavachari and M. A. Al-Laham and V. G. Zakrzewski and J. V. Ortiz and J. B. Foresman and C. Y. Peng and P. Y. Ayala and W. Chen and M. W. Wong and J. L. Andres and E. S. Rep...

work page
[55]

M. J. Frisch and G. W. Trucks and H. B. Schlegel and P. M. W. Gill and B. G. Johnson and M. A. Robb and J. R. Cheeseman and T. Keith and G. A. Petersson and J. A. Montgomery and K. Raghavachari and M. A. Al-Laham and V. G. Zakrzewski and J. V. Ortiz and J. B. Foresman and J. Cioslowski and B. B. Stefanov and A. Nanayakkara and M. Challacombe and C. Y. Pen...

work page
[56]

Gaussian 92, Revision D.2 , author =

work page
[57]

M. J. Frisch and M. Head-Gordon and G. W. Trucks and J. B. Foresman and H. B. Schlegel, K. Raghavachari and M. Robb and J. S. Binkley and C. Gonzalez and D. J. Defrees and D. J. Fox, R. A. Whiteside and R. Seeger and C. F. Melius and J. Baker and R. L. Martin and L. R. Kahn and J. J. P. Stewart and S. Topiol and J. A. Pople , organization =

work page
[58]

Matt Challacombe and Eric Schwegler and C. J. Tymczak and Chee Kwan Gan and Karoly Nemeth and Valery Weber and Anders M. N. Niklasson and Graeme Henkelman , title =

work page
[59]

Nicolas Bock and Matt Challacombe and Chee Kwan Gan and Graeme Henkelman and Karoly Nemeth and Anders M. N. Niklasson and Anders Odell and Eric Schwegler and C. J. Tymczak and Valery Weber , title =

work page
[60]

E. J. Sanville and et al. , title =

work page
[61]

Cawkwell and et al

M.\ J. Cawkwell and et al. , title =

work page
[62]

Matt Challacombe and

work page
[63]

P. O. L. Quantum theory of many-particle systems. Phys. Rev. , year = 1955, volume = 97, number = 6, pages =

work page 1955
[64]

P. O. L. Quantum theory of electronic structure of molecules , journal =

work page
[65]

Lindh and U

R. Lindh and U. Ryu and Y. S. Lee , title =. J. Chem. Phys. , year = 1991, volume = 95, number = 8, pages =

work page 1991
[66]

Momose and T

T. Momose and T. Shida , title =. J. Chem. Phys. , year = 1987, volume = 87, number = 5, pages =

work page 1987
[67]

Momose and T

T. Momose and T. Shida , title =. J. Chem. Phys. , year = 1988, volume = 88, number = 11, pages =

work page 1988
[68]

T. P. Kline and F. K. Brown and S. C. Brown and P. W. Jeffs and K. D. Kopple and L. Mueller , journal =

work page
[69]

Brandt and A

A. Brandt and A. A. Lubrecht , title =

work page
[70]

A. C. Genz and A. A. Malik , title =. SIAM J. Num. Anal. , year = 1983, volume = 20, number = 3, pages =

work page 1983
[71]

A. C. Genz and A. A. Malik , title =. J. Comp. App. Math , year = 1980, volume = 6, number = 4, pages =

work page 1980
[72]

A. C. J. Am. Chem. Soc. , year = 1994, volume = 116, number = 12, pages =

work page 1994
[73]

Canning and G

A. Canning and G. Galli and F. Mauri and A. DeVita and R. Car , journal =

work page
[74]

A. D. Becke , title =. Phys. Rev. A , year = 1988, volume = 38, number = 6, pages =

work page 1988
[75]

A. D. Becke , title =

work page
[76]

A. D. Becke , title =. J. Chem. Phys. , year = 1993, volume = 98, pages =

work page 1993
[77]

A. D. Becke , journal =

work page
[78]

A. D. Booth and F. J. Llewellyn , title =

work page
[79]

A. D. Buckingham , editor =. Basic theory of intermolecular forces: Applications to small molecules , publisher =

work page
[80]

Feng , editor =

K. Feng , editor =. Beijing Symposium on Differenctial Geometry and Differential Equations -- Computation of Partial Differential Equations , publisher =

work page

Showing first 80 references.