quantum-many-body-leaderboard
mcp
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QMBL: the Quantum Many-Body Leaderboard. The best published ground-state energies for lattice models, hydrogen chains and the electron gas, with a JSON API and an MCP server for agents.
README.md
A live record of SOTA published ground-state energies for lattice models, hydrogen chains and the electron gas.
Organised literature for fellow researchers and AI systems. So that we can see the works of the wider research community at one glance, build on each other's work more efficiently and advance frontier science together.
AI agents and crawlers: read qmbl.org/llms.txt for what the data means and where it is, and the JSON API under qmbl.org/api/.
Use QMBL with your agent
Paste this prompt into your favorite agent:
Use QMBL: the Quantum Many-Body Leaderboard (qmbl.org) to answer my questions about ground-state energies. First read https://qmbl.org/llms.txt. Then use the JSON API it describes: https://qmbl.org/api/models.json lists the models, https://qmbl.org/api/m/<model>.json gives every instance of one model with its exact and best energies, and https://qmbl.org/api/i/<instance_id>.json gives every published energy of one instance.Or add the MCP server. Paste this prompt into your favorite agent:
Add the QMBL MCP server: download https://raw.githubusercontent.com/quantum-many-body-leaderboard/quantum-many-body-leaderboard/main/mcp/server.mjs and register it as a local stdio MCP server named "qmbl" that runs `node <path to server.mjs>`.
Leaderboard
All 1121 Hamiltonian instances, 31 of them infinite lattices (click a model to expand)
Heisenberg: 121 instances, energies asE/N (S.S)
| instance | value | energy | method | code |
|---|---|---|---|---|
| chain, 20 sites, open | exact | -0.4341237 | ED Wu et al. (2024) | run script |
| best variational | -0.4341237 (+<1e-7) | DMRG Wu et al. (2024) | run script | |
| chain, 20 sites | exact | -0.4452193 | ED Wu et al. (2024) | run script |
| best variational | -0.4452193 (+<1e-7) | DMRG Wu et al. (2024) | run script | |
| chain, 200 sites, open | exact | -0.4422078 | Exact solution McCulloch (code) | Uni20-dev/bethe, QMBL check |
| best variational | -0.4422078 (+<1e-7) | DMRG Wu et al. (2024) | run script | |
| chain, 200 sites | exact | -0.4431678 | Exact solution McCulloch (code) | Uni20-dev/bethe, QMBL check |
| best variational | -0.4431678 (+<1e-7) | DMRG Wu et al. (2024) | run script | |
| checkerboard, infinite lattice | best variational | -0.513442(1) | variational wave function Schäfer et al. (2023) | |
| best extrapolated | -0.5132(2) | DMRG Schäfer et al. (2023) | ||
| best approximate | -0.5138(2) | NLCE Schäfer et al. (2023) | ||
| checkerboard, infinite lattice, S = 1 | best variational | -1.5339(5) | variational wave function Schäfer et al. (2023) | |
| best extrapolated | -1.532(1) | DMRG Schäfer et al. (2023) | ||
| best approximate | -1.533(2) | NLCE Schäfer et al. (2023) | ||
| crossedruby, infinite lattice | best variational | -0.48946(5) | variational wave function Schäfer et al. (2023) | |
| best extrapolated | -0.4865(7) | DMRG Schäfer et al. (2023) | ||
| best approximate | -0.492(1) | NLCE Schäfer et al. (2023) | ||
| crossedruby, infinite lattice, S = 1 | best variational | -1.490(1) | variational wave function Schäfer et al. (2023) | |
| best extrapolated | -1.4764(5) | DMRG Schäfer et al. (2023) | ||
| best approximate | -1.489(5) | NLCE Schäfer et al. (2023) | ||
| honeycomb 7x7 (98 sites) | best variational | -0.5421(4) | VMC (LCN) Fu et al. (2022) | |
| kagome, infinite lattice | best extrapolated | -0.4386(5) | DMRG Depenbrock et al. (2012) | |
| kagome-24, 24 sites | exact | -0.4412486 | ED Läuchli et al. (2011) | |
| kagome 2x2 (12 sites) | exact | -0.4537396 | ED Läuchli et al. (2011) | |
| kagome 2x3 (18 sites) | exact | -0.4471262 | ED Wu et al. (2024) | run script |
| best variational | -0.4471262 (+<1e-7) | DMRG Wu et al. (2024) | run script | |
| kagome-30, 30 sites | exact | -0.4384773 | ED Läuchli et al. (2011) | |
| kagome-36a, 36 sites | exact | -0.4385521 | ED Läuchli et al. (2011) | |
| kagome-36b, 36 sites | exact | -0.4390813 | ED Läuchli et al. (2011) | |
| kagome-36c, 36 sites | exact | -0.4392871 | ED Läuchli et al. (2011) | |
| kagome-36d, 36 sites | exact | -0.4383765 | ED Läuchli et al. (2011) | |
| kagome-42a, 42 sites | exact | -0.437999 | ED Läuchli et al. (2011) | |
| kagome-42b, 42 sites | exact | -0.4381426 | ED Läuchli et al. (2011) | |
| kagome 4x4 (48 sites) | exact | -0.4387039 | ED Wietek & Läuchli (2018) | |
| best variational | -0.4375(2) (+1.2e-3) | VMC (GCNN) Đurić et al. (2024) | uses NetKet | |
| best extrapolated | -0.4383(2) (+4.0e-4) | DMRG Depenbrock et al. (2012) | ||
| kagome 6x6 (108 sites) | best extrapolated | -0.4383(3) | DMRG Depenbrock et al. (2012) | |
| kagome 8x8 (192 sites) | best variational | -0.42987(1) | VMC (Gutzwiller) Wu et al. (2024) | run script |
| pyrochlore, infinite lattice | best variational | -0.489472(8) | variational wave function Schäfer et al. (2023) | |
| best extrapolated | -0.490(6) | DMRG Hagymási et al. (2020) | ||
| best approximate | -0.49(1) | NLCE Hagymási et al. (2020) | ||
| pyrochlore, infinite lattice, S = 1 | best variational | -1.490(1) | variational wave function Schäfer et al. (2023) | |
| best approximate | -1.489(5) | NLCE Schäfer et al. (2023) | ||
| pyrochlore 2x2x2 (32 sites) | exact | -0.5168588 | ED QMBL check, checks/pyrochlore-32-ed/ | QMBL code |
| best variational | -0.5162(1) (+6.6e-4) | VMC (PP) Astrakhantsev et al. (2021) | uses mVMC | |
| best extrapolated | -0.5168 ○ (+5.9e-5) | DMRG Hagymási et al. (2020) | ||
| pyrochlore 2x2x2 (128 sites) | best variational | -0.49229(7) | VMC (PP + RBM) Pohle et al. (2023) | uses mVMC |
| best extrapolated | -0.49434(5) | VMC (PP) Pohle et al. (2023) | uses mVMC | |
| pyrochlore 2x2x3 (48 sites) | best variational | -0.5028(2) | VMC (PP) Astrakhantsev et al. (2021) | uses mVMC |
| pyrochlore 2x2x4 (64 sites) | best variational | -0.4924(2) | VMC (PP) Astrakhantsev et al. (2021) | uses mVMC |
| pyrochlore 2x2x6 (96 sites) | best variational | -0.4917(4) | VMC (PP) Astrakhantsev et al. (2021) | uses mVMC |
| pyrochlore 2x4x6 (192 sites) | best variational | -0.4915(2) | VMC (PP) Astrakhantsev et al. (2021) | uses mVMC |
| pyrochlore 3x3x3 (108 sites) | best variational | -0.4871(1) | VMC (PP) Astrakhantsev et al. (2021) | uses mVMC |
| pyrochlore 3x3x3 (432 sites) | best variational | -0.48851(3) | VMC (PP) Pohle et al. (2023) | uses mVMC |
| best extrapolated | -0.4924(2) | VMC (PP) Pohle et al. (2023) | uses mVMC | |
| pyrochlore 4x4x4 (256 sites) | best variational | -0.485584(3) | VMC (RVB) Cheng & Li (2026) | |
| pyrochlore 4x4x4 (1024 sites) | best variational | -0.48800(1) | VMC (PP) Pohle et al. (2023) | run script |
| best extrapolated | -0.4921(2) | VMC (PP) Pohle et al. (2023) | uses mVMC | |
| rectangular 10x20, periodic/open | exact (stochastic) | -0.6599522(4) | SSE QMC Sandvik (2026) | |
| rectangular 12x24, periodic/open | exact (stochastic) | -0.6612549(4) | SSE QMC Sandvik (2026) | |
| rectangular 14x28, periodic/open | exact (stochastic) | -0.6622809(4) | SSE QMC Sandvik (2026) | |
| rectangular 16x32, periodic/open | exact (stochastic) | -0.6630939(4) | SSE QMC Sandvik (2026) | |
| rectangular 18x36, periodic/open | exact (stochastic) | -0.6637496(4) | SSE QMC Sandvik (2026) | |
| rectangular 20x40, periodic/open | exact (stochastic) | -0.6642865(4) | SSE QMC Sandvik (2026) | |
| rectangular 24x48, periodic/open | exact (stochastic) | -0.6651098(6) | SSE QMC Sandvik (2026) | |
| rectangular 32x64, periodic/open | exact (stochastic) | -0.6661668(6) | SSE QMC Sandvik (2026) | |
| rectangular 48x96, periodic/open | exact (stochastic) | -0.6672462(6) | SSE QMC Sandvik (2026) | |
| rectangular 4x8, periodic/open | exact (stochastic) | -0.6623181(4) | SSE QMC Sandvik (2026) | |
| rectangular 64x128, periodic/open | exact (stochastic) | -0.6677918(4) | SSE QMC Sandvik (2026) | |
| rectangular 6x12, periodic/open | exact (stochastic) | -0.6570101(4) | SSE QMC Sandvik (2026) | |
| rectangular 6x8 | exact | -0.6759867 | ED Wietek & Läuchli (2018) | |
| rectangular 8x16, periodic/open | exact (stochastic) | -0.6583592(4) | SSE QMC Sandvik (2026) | |
| shuriken, 24 sites | exact | -0.4483291 | ED Astrakhantsev et al. (2021) | |
| shuriken, 96 sites | best variational | -0.43826(1) | VMC (PP) Wu et al. (2024) | uses mVMC |
| shuriken, 216 sites | best variational | -0.4376(1) | VMC (PP) Wu et al. (2024) | uses mVMC |
| shuriken, 384 sites | best variational | -0.4371(1) | VMC (PP) Wu et al. (2024) | uses mVMC |
| square 4x4 | exact | -0.7017802 | ED Wu et al. (2024) | run script |
| best variational | -0.7017449 (+3.5e-5) | VQE Wu et al. (2024) | run script | |
| square 6x6, open | exact | -0.6035218 | ED Wu et al. (2024) | run script |
| best variational | -0.6035218 ○ (+<1e-7) | DMRG Huang et al. (2016) | ||
| best extrapolated | -0.60351496(1) (+6.9e-6) | VMC (2D RNN) Moss et al. (2025) | mschuylermoss/HeisenbergRNN | |
| square 6x6 | exact | -0.6788721 | ED Richter et al. (2004) | |
| best variational | -0.678871(3) (+1.1e-6) | VMC (CNN BF + RBM) Hendry et al. (2025) | uses NetKet | |
| best extrapolated | -0.67887177(7) (+3.8e-7) | VMC (2D RNN) Moss et al. (2025) | mschuylermoss/HeisenbergRNN | |
| square, 38 sites | exact | -0.6779156 | ED Richter et al. (2004) | |
| square, 40 sites | exact | -0.6773713 | ED Richter et al. (2004) | |
| square, 50 sites | exact | -0.675102 | ED Wietek & Läuchli (2018) | |
| square 8x8, open | exact (stochastic) | -0.6190371(2) | SSE QMC Sandvik (2026) | |
| best variational | -0.619034(6) (+2.7e-6) | VMC (2D RNN) Moss et al. (2025) | mschuylermoss/HeisenbergRNN | |
| square 8x8 | exact (stochastic) | -0.67349005(2) | SSE QMC Sandvik (2026) | |
| best variational | -0.6735 ○ (−1.0e-5) | VMC (HQT) Guo et al. (2026) | uses NetKet | |
| square 10x10, open | exact (stochastic) | -0.6286561(2) | SSE QMC Sandvik (2026) | |
| best variational | -0.628656(9) (+1.2e-7) | VMC (2D RNN) Moss et al. (2025) | mschuylermoss/HeisenbergRNN | |
| best extrapolated | -0.628657(5) (−8.8e-7) | DMRG Wu et al. (2024) | run script | |
| square 10x10 | exact (stochastic) | -0.67155266(2) | SSE QMC Sandvik (2026) | |
| best variational | -0.67155260(3) (+6.0e-8) | VMC (ResNet) Chen & Heyl (2023) | ChenAo-Phys/MinSR | |
| square 12x12, open | exact (stochastic) | -0.6352007(2) | SSE QMC Sandvik (2026) | |
| best variational | -0.635201(7) (+1.3e-7) | VMC (2D RNN) Moss et al. (2025) | mschuylermoss/HeisenbergRNN | |
| square 12x12 | exact (stochastic) | -0.67068192(2) | SSE QMC Sandvik (2026) | |
| best variational | -0.67062(2) (+6.3e-5) | VMC (2D RNN) Moss et al. (2025) | mschuylermoss/HeisenbergRNN | |
| square 14x14, open | exact (stochastic) | -0.6399410(2) | SSE QMC Sandvik (2026) | |
| best variational | -0.639929(6) (+1.2e-5) | VMC (2D RNN) Moss et al. (2025) | mschuylermoss/HeisenbergRNN | |
| square 14x14 | exact (stochastic) | -0.67023225(2) | SSE QMC Sandvik (2026) | |
| best variational | -0.67014(2) (+9.4e-5) | VMC (2D RNN) Moss et al. (2025) | mschuylermoss/HeisenbergRNN | |
| square 16x16, open | exact (stochastic) | -0.6435317(2) | SSE QMC Sandvik (2026) | |
| best variational | -0.643522(7) (+9.8e-6) | VMC (2D RNN) Moss et al. (2025) | mschuylermoss/HeisenbergRNN | |
| square 16x16 | exact (stochastic) | -0.66997660(2) | SSE QMC Sandvik (2026) | |
| best variational | -0.66987(2) (+1.1e-4) | VMC (2D RNN) Moss et al. (2025) | mschuylermoss/HeisenbergRNN | |
| best extrapolated | -0.6699687(9) (+7.9e-6) | VMC (2D RNN) Moss et al. (2025) | mschuylermoss/HeisenbergRNN | |
| square 18x18, open | exact (stochastic) | -0.6463451(4) | SSE QMC Sandvik (2026) | |
| best variational | -0.646337(5) (+8.6e-6) | VMC (2D RNN) Moss et al. (2025) | mschuylermoss/HeisenbergRNN | |
| square 18x18 | exact (stochastic) | -0.66982043(2) | SSE QMC Sandvik (2026) | |
| best variational | -0.66968(2) (+1.4e-4) | VMC (2D RNN) Moss et al. (2025) | mschuylermoss/HeisenbergRNN | |
| best extrapolated | -0.6697819(5) (+3.9e-5) | VMC (2D RNN) Moss et al. (2025) | mschuylermoss/HeisenbergRNN | |
| square 20x20, open | exact (stochastic) | -0.6486091(4) | SSE QMC Sandvik (2026) | |
| best variational | -0.648590(5) (+1.9e-5) | VMC (2D RNN) Moss et al. (2025) | mschuylermoss/HeisenbergRNN | |
| square 20x20 | exact (stochastic) | -0.66971967(2) | SSE QMC Sandvik (2026) | |
| best variational | -0.66959(1) (+1.3e-4) | VMC (2D RNN) Moss et al. (2025) | mschuylermoss/HeisenbergRNN | |
| square 22x22, open | exact (stochastic) | -0.6504689(4) | SSE QMC Sandvik (2026) | |
| best variational | -0.650253(5) (+2.2e-4) | VMC (minGRU) Merali et al. (2026) | ParallelScan-RNNs/PSR-NQS | |
| square 22x22 | exact (stochastic) | -0.66965176(2) | SSE QMC Sandvik (2026) | |
| square 24x24, open | exact (stochastic) | -0.6520251(4) | SSE QMC Sandvik (2026) | |
| best variational | -0.651982(7) (+4.3e-5) | VMC (2D RNN) Moss et al. (2025) | mschuylermoss/HeisenbergRNN | |
| square 24x24 | exact (stochastic) | -0.66960434(2) | SSE QMC Sandvik (2026) | |
| best variational | -0.66944(2) (+1.7e-4) | VMC (2D RNN) Moss et al. (2025) | mschuylermoss/HeisenbergRNN | |
| square 26x26, open | exact (stochastic) | -0.6533456(4) | SSE QMC Sandvik (2026) | |
| best variational | -0.653129(5) (+2.2e-4) | VMC (minGRU) Merali et al. (2026) | ParallelScan-RNNs/PSR-NQS | |
| square 26x26 | exact (stochastic) | -0.66957017(3) | SSE QMC Sandvik (2026) | |
| square 28x28, open | exact (stochastic) | -0.6544806(4) | SSE QMC Sandvik (2026) | |
| best variational | -0.654261(4) (+2.2e-4) | VMC (minGRU) Merali et al. (2026) | ParallelScan-RNNs/PSR-NQS | |
| square 28x28 | exact (stochastic) | -0.66954496(3) | SSE QMC Sandvik (2026) | |
| best variational | -0.66937(1) (+1.7e-4) | VMC (2D RNN) Moss et al. (2025) | mschuylermoss/HeisenbergRNN | |
| square 30x30, open | best variational | -0.655241(4) | VMC (minGRU) Merali et al. (2026) | ParallelScan-RNNs/PSR-NQS |
| square 30x30 | exact (stochastic) | -0.66952595(3) | SSE QMC Sandvik (2026) | |
| square 32x32, open | exact (stochastic) | -0.6563289(4) | SSE QMC Sandvik (2026) | |
| best variational | -0.656089(4) (+2.4e-4) | VMC (minGRU) Merali et al. (2026) | ParallelScan-RNNs/PSR-NQS | |
| square 32x32 | exact (stochastic) | -0.66951133(3) | SSE QMC Sandvik (2026) | |
| best variational | -0.66930(1) (+2.2e-4) | VMC (2D RNN) Moss et al. (2025) | mschuylermoss/HeisenbergRNN | |
| best extrapolated | -0.6694877(4) (+2.4e-5) | VMC (2D RNN) Moss et al. (2025) | mschuylermoss/HeisenbergRNN | |
| square 36x36, open | best variational | -0.657523(4) | VMC (minGRU) Merali et al. (2026) | ParallelScan-RNNs/PSR-NQS |
| square 36x36 | exact (stochastic) | -0.66949091(3) | SSE QMC Sandvik (2026) | |
| square 40x40, open | best variational | -0.658657(4) | VMC (minGRU) Merali et al. (2026) | ParallelScan-RNNs/PSR-NQS |
| square 40x40 | exact (stochastic) | -0.66947776(3) | SSE QMC Sandvik (2026) | |
| square 44x44, open | best variational | -0.659565(4) | VMC (minGRU) Merali et al. (2026) | ParallelScan-RNNs/PSR-NQS |
| square 44x44 | exact (stochastic) | -0.66946889(3) | SSE QMC Sandvik (2026) | |
| square 46x46, open | best variational | -0.659941(4) | VMC (minGRU) Merali et al. (2026) | ParallelScan-RNNs/PSR-NQS |
| square 48x48, open | exact (stochastic) | -0.6606690(4) | SSE QMC Sandvik (2026) | |
| best variational | -0.660323(4) (+3.5e-4) | VMC (minGRU) Merali et al. (2026) | ParallelScan-RNNs/PSR-NQS | |
| square 48x48 | exact (stochastic) | -0.66946274(3) | SSE QMC Sandvik (2026) | |
| square 50x50, open | best variational | -0.660649(3) | VMC (minGRU) Merali et al. (2026) | ParallelScan-RNNs/PSR-NQS |
| square 52x52, open | best variational | -0.660967(4) | VMC (minGRU) Merali et al. (2026) | ParallelScan-RNNs/PSR-NQS |
| square 52x52 | exact (stochastic) | -0.66945833(3) | SSE QMC Sandvik (2026) | |
| square 56x56 | exact (stochastic) | -0.66945502(3) | SSE QMC Sandvik (2026) | |
| square 60x60 | exact (stochastic) | -0.66945260(3) | SSE QMC Sandvik (2026) | |
| square 64x64, open | exact (stochastic) | -0.6628515(4) | SSE QMC Sandvik (2026) | |
| square 64x64 | exact (stochastic) | -0.66945072(3) | SSE QMC Sandvik (2026) | |
| square 72x72 | exact (stochastic) | -0.66944815(3) | SSE QMC Sandvik (2026) | |
| square 80x80 | exact (stochastic) | -0.66944645(3) | SSE QMC Sandvik (2026) | |
| square 88x88 | exact (stochastic) | -0.66944530(3) | SSE QMC Sandvik (2026) | |
| square 96x96, open | exact (stochastic) | -0.6650415(2) | SSE QMC Sandvik (2026) | |
| square 96x96 | exact (stochastic) | -0.66944454(3) | SSE QMC Sandvik (2026) | |
| square 128x128, open | exact (stochastic) | -0.6661389(2) | SSE QMC Sandvik (2026) | |
| square, infinite lattice | best variational | -0.6693781 ○ | iPEPS Rader & Läuchli (2018) | |
| best extrapolated | -0.669441878(9) | SSE QMC Sandvik (2026) | ||
| triangular 4x4 | exact | -0.5347197 | ED Wu et al. (2024) | run script |
| best variational | -0.5347197 (−<1e-7) | DMRG Wu et al. (2024) | run script | |
| triangular 6x6 | exact | -0.5603734 | ED Wu et al. (2024) | run script |
| best variational | -0.560313(3) (+6.0e-5) | VMC (GCNN) Roth et al. (2022) | uses NetKet | |
| triangular, 48 sites | exact | -0.558603 | ED Wietek & Läuchli (2018) | |
| triangular 10x10, open | best variational | -0.5147668 | DMRG Wu et al. (2024) | run script |
| triangular, 108 sites | best variational | -0.55315(3) | VMC (GCNN) Roth et al. (2022) | uses NetKet |
| triangular 12x12, open | best variational | -0.5150648 | DMRG Wu et al. (2024) | run script |
| triangular 12x12 | best variational | -0.5493(1) | VMC (2D RNN) Moss et al. (2025) | mschuylermoss/HeisenbergRNN |
| triangular 14x14, open | best variational | -0.5138(1) | VMC (2D RNN) Wu et al. (2024) | run script |
| triangular 16x16, open | best variational | -0.51668(9) | VMC (2D RNN) Wu et al. (2024) | run script |
| triangular 18x18 | best variational | -0.551703(1) | VMC (ViT) Viteritti et al. (2026) | |
| best extrapolated | -0.55200(1) | VMC (ViT) Viteritti et al. (2026) | ||
| triangular 24x24 | best variational | -0.551602(1) | VMC (ViT) Viteritti et al. (2026) | |
| best extrapolated | -0.55189(1) | VMC (ViT) Viteritti et al. (2026) | ||
| triangular 30x30 | best variational | -0.551528(1) | VMC (ViT) Viteritti et al. (2026) | |
| best extrapolated | -0.55180(1) | VMC (ViT) Viteritti et al. (2026) | ||
| triangular 42x42 | best variational | -0.551429(1) | VMC (ViT) Viteritti et al. (2026) | |
| best extrapolated | -0.55171(1) | VMC (ViT) Viteritti et al. (2026) |
E/N (S.S)
| instance | value | energy | method | code |
|---|---|---|---|---|
| honeycomb 7x7 (98 sites), J2 = 0.2 | best variational | -0.4538(4) | VMC (LCN) Fu et al. (2022) | |
| rectangular 4x6, J2 = 0.5 | exact | -0.5225249 | ED Wu et al. (2024) | run script |
| best variational | -0.5225249 (+<1e-7) | DMRG Wu et al. (2024) | run script | |
| square 4x4, J2 = 0.05 | exact | -0.68059 | ED Wu et al. (2024) | run script |
| best variational | -0.6805657 (+2.4e-5) | VQE Wu et al. (2024) | run script | |
| square 4x4, J2 = 0.1 | exact | -0.6598172 | ED Wu et al. (2024) | run script |
| best variational | -0.6597832 (+3.4e-5) | VQE Wu et al. (2024) | run script | |
| square 4x4, J2 = 0.15 | exact | -0.6395429 | ED Wu et al. (2024) | run script |
| best variational | -0.6395107 (+3.2e-5) | VQE Wu et al. (2024) | run script | |
| square 4x4, J2 = 0.2 | exact | -0.6198739 | ED Wu et al. (2024) | run script |
| best variational | -0.6198639 (+1.0e-5) | VQE Wu et al. (2024) | run script | |
| square 4x4, J2 = 0.25 | exact | -0.6009545 | ED Wu et al. (2024) | run script |
| best variational | -0.6009332 (+2.1e-5) | VQE Wu et al. (2024) | run script | |
| square 4x4, J2 = 0.3 | exact | -0.5829838 | ED Wu et al. (2024) | run script |
| best variational | -0.5829522 (+3.2e-5) | VQE Wu et al. (2024) | run script | |
| square 4x4, J2 = 0.35 | exact | -0.5662448 | ED Wu et al. (2024) | run script |
| best variational | -0.5662285 (+1.6e-5) | VQE Wu et al. (2024) | run script | |
| square 4x4, J2 = 0.4 | exact | -0.5511478 | ED Wu et al. (2024) | run script |
| best variational | -0.5511135 (+3.4e-5) | VQE Wu et al. (2024) | run script | |
| square 4x4, J2 = 0.45 | exact | -0.5382998 | ED Wu et al. (2024) | run script |
| best variational | -0.5382766 (+2.3e-5) | VQE Wu et al. (2024) | run script | |
| best extrapolated | -0.5370(4) (+1.3e-3) | cluster update (TPS) Wang et al. (2011) | ||
| square 4x4, J2 = 0.5 | exact | -0.5286202 | ED Wu et al. (2024) | run script |
| best variational | -0.5286202 (−<1e-7) | DMRG Wu et al. (2024) | run script | |
| best extrapolated | -0.5268(4) (+1.8e-3) | cluster update (TPS) Wang et al. (2011) | ||
| square 4x4, J2 = 0.6 | exact | -0.5258958 | ED Wu et al. (2024) | run script |
| best variational | -0.5258158 (+8.0e-5) | VQE Wu et al. (2024) | run script | |
| best extrapolated | -0.518(2) (+8.2e-3) | cluster update (TPS) Wang et al. (2011) | ||
| square 4x4, J2 = 0.65 | exact | -0.5393825 | ED Wu et al. (2024) | run script |
| best variational | -0.5393444 (+3.8e-5) | VQE Wu et al. (2024) | run script | |
| square 4x4, J2 = 0.7 | exact | -0.5638581 | ED Wu et al. (2024) | run script |
| best variational | -0.5637716 (+8.7e-5) | VQE Wu et al. (2024) | run script | |
| square 4x4, J2 = 0.75 | exact | -0.5942731 | ED Wu et al. (2024) | run script |
| best variational | -0.5942152 (+5.8e-5) | VQE Wu et al. (2024) | run script | |
| square 4x4, J2 = 0.8 | exact | -0.6273351 | ED Wu et al. (2024) | run script |
| best variational | -0.6273137 (+2.1e-5) | VQE Wu et al. (2024) | run script | |
| square 4x4, J2 = 0.85 | exact | -0.6617197 | ED Wu et al. (2024) | run script |
| best variational | -0.6617069 (+1.3e-5) | VQE Wu et al. (2024) | run script | |
| square 4x4, J2 = 0.9 | exact | -0.6968656 | ED Wu et al. (2024) | run script |
| best variational | -0.6967967 (+6.9e-5) | VQE Wu et al. (2024) | run script | |
| square 4x4, J2 = 0.95 | exact | -0.7324989 | ED Wu et al. (2024) | run script |
| best variational | -0.7324804 (+1.8e-5) | VQE Wu et al. (2024) | run script | |
| square 4x4, J2 = 1 | exact | -0.7684681 | ED Wu et al. (2024) | run script |
| best variational | -0.7684501 (+1.8e-5) | VQE Wu et al. (2024) | run script | |
| square 6x6, J2 = 0.1 | exact | -0.638096 | ED Schulz et al. (1996) | |
| best variational | -0.638078 ○ (+1.8e-5) | VMC (ViT) Golubev et al. (2026) | uses NetKet | |
| square 6x6, J2 = 0.2 | exact | -0.5990463 | ED Schulz et al. (1996) | |
| best variational | -0.599044 ○ (+2.3e-6) | VMC (ViT) Golubev et al. (2026) | uses NetKet | |
| square 6x6, J2 = 0.3 | exact | -0.5624595 | ED Wu et al. (2024) | run script |
| best variational | -0.562453 ○ (+6.5e-6) | VMC (ViT) Golubev et al. (2026) | uses NetKet | |
| square 6x6, J2 = 0.4 | exact | -0.529745 | ED Wu et al. (2024) | run script |
| best variational | -0.529744 ○ (+1.0e-6) | DMRG Gong et al. (2013) | ||
| best extrapolated | -0.529747(1) (−2.0e-6) | DMRG Gong et al. (2013) | ||
| square 6x6, J2 = 0.45 | exact | -0.5156574 | ED QMBL check, checks/exact-recompute-2026-09/ | QMBL code |
| best variational | -0.515655 ○ (+2.4e-6) | DMRG Gong et al. (2013) | ||
| best extrapolated | -0.51566(1) (−2.6e-6) | VMC (Gutzwiller) Hu et al. (2013) | ||
| square 6x6, J2 = 0.5 | exact | -0.5038097 | ED Wu et al. (2024) | run script |
| best variational | -0.503805 ○ (+4.7e-6) | DMRG Gong et al. (2013) | ||
| best projected | -0.5033(4) (+5.1e-4) | FN-GFMC (PEPS) Lin et al. (2024) | ||
| best extrapolated | -0.50382(1) (−1.0e-5) | VMC (Gutzwiller) Hu et al. (2013) | ||
| square 6x6, J2 = 0.55 | exact | -0.4951777 | ED Schulz et al. (1996) | |
| best variational | -0.495167 ○ (+1.1e-5) | DMRG Gong et al. (2013) | ||
| best projected | -0.4939(5) (+1.3e-3) | FN-GFMC (PEPS) Lin et al. (2024) | ||
| best extrapolated | -0.49521(7) (−3.2e-5) | VMC (Gutzwiller) Hu et al. (2013) | ||
| square 6x6, J2 = 0.6 | exact | -0.4932386 | ED Wu et al. (2024) | run script |
| best variational | -0.49318(5) (+5.9e-5) | VMC (RBM) Chen et al. (2022) | ||
| best extrapolated | -0.488(1) (+5.3e-3) | cluster update (TPS) Wang et al. (2011) | ||
| square 6x6, J2 = 0.65 | exact | -0.5065879 | ED Schulz et al. (1996) | |
| best variational | -0.506299 ○ (+2.9e-4) | VMC (ViT) Golubev et al. (2026) | uses NetKet | |
| square 6x6, J2 = 0.7 | exact | -0.5300012 | ED Schulz et al. (1996) | run script |
| best variational | -0.5299243 (+7.7e-5) | DMRG Wu et al. (2024) | run script | |
| square 6x6, J2 = 0.8 | exact | -0.5864866 | ED Schulz et al. (1996) | run script |
| best variational | -0.586411(9) (+7.6e-5) | VMC (RBM) Chen et al. (2022) | ||
| square 6x6, J2 = 0.9 | exact | -0.649052 | ED Schulz et al. (1996) | run script |
| best variational | -0.6478705 (+1.2e-3) | DMRG Wu et al. (2024) | run script | |
| square 6x6, J2 = 1 | exact | -0.7143604 | ED Schulz et al. (1996) | run script |
| best variational | -0.71429(1) (+7.0e-5) | VMC (RBM) Chen et al. (2022) | ||
| square, 40 sites, J2 = 0.1 | exact | -0.6366151 | ED Richter & Schulenburg (2009) | |
| square, 40 sites, J2 = 0.2 | exact | -0.5975117 | ED Richter & Schulenburg (2009) | |
| square, 40 sites, J2 = 0.3 | exact | -0.5606822 | ED Richter & Schulenburg (2009) | |
| square, 40 sites, J2 = 0.4 | exact | -0.5272092 | ED Richter & Schulenburg (2009) | |
| square, 40 sites, J2 = 0.5 | exact | -0.4990762 | ED Richter & Schulenburg (2009) | |
| square, 40 sites, J2 = 0.55 | exact | -0.4879479 | ED Richter & Schulenburg (2009) | |
| square, 40 sites, J2 = 0.6 | exact | -0.4796232 | ED QMBL check, checks/j1j2-40-sector-ed/ | QMBL code |
| square, 40 sites, J2 = 0.65 | exact | -0.5011508 | ED Richter & Schulenburg (2009) | |
| square, 40 sites, J2 = 0.7 | exact | -0.5263826 | ED Richter & Schulenburg (2009) | |
| square, 40 sites, J2 = 0.8 | exact | -0.5835051 | ED Richter & Schulenburg (2009) | |
| square, 40 sites, J2 = 0.9 | exact | -0.6459228 | ED Richter & Schulenburg (2009) | |
| square, 40 sites, J2 = 1 | exact | -0.7109702 | ED Richter & Schulenburg (2009) | |
| square 8x8, J2 = 0.1 | best variational | -0.633093 ○ | VMC (ViT) Golubev et al. (2026) | uses NetKet |
| square 8x8, J2 = 0.2 | best variational | -0.594508 ○ | VMC (ViT) Golubev et al. (2026) | uses NetKet |
| square 8x8, J2 = 0.3 | best variational | -0.558295 ○ | VMC (ViT) Golubev et al. (2026) | uses NetKet |
| square 8x8, J2 = 0.4 | best variational | -0.525653(1) | VMC (RBM + PP) Nomura & Imada (2021) | uses mVMC |
| best extrapolated | -0.5262(1) | DMRG Gong et al. (2013) | ||
| square 8x8, J2 = 0.45 | best variational | -0.511331(1) | VMC (RBM + PP) Nomura & Imada (2021) | uses mVMC |
| best extrapolated | -0.5116(1) | DMRG Gong et al. (2013) | ||
| square 8x8, J2 = 0.48 | best extrapolated | -0.5034(4) | cluster update (TPS) Wang et al. (2011) | |
| square 8x8, J2 = 0.5 | best variational | -0.498963(2) | VMC (RBM + PP) Wu et al. (2024) | uses mVMC |
| best extrapolated | -0.4992(1) | DMRG Gong et al. (2013) | ||
| square 8x8, J2 = 0.52 | best extrapolated | -0.4941(1) | cluster update (TPS) Wang et al. (2011) | |
| square 8x8, J2 = 0.55 | best variational | -0.488820(2) | VMC (RBM + PP) Nomura & Imada (2021) | uses mVMC |
| best extrapolated | -0.4891(1) | DMRG Gong et al. (2013) | ||
| square 8x8, J2 = 0.6 | best variational | -0.481606 ○ | VMC (ViT) Golubev et al. (2026) | uses NetKet |
| best extrapolated | -0.4808(9) | cluster update (TPS) Wang et al. (2011) | ||
| square 8x8, J2 = 0.65 | best variational | -0.498312 ○ | VMC (ViT) Golubev et al. (2026) | uses NetKet |
| square 8x8, J2 = 0.7 | best variational | -0.522091 ○ | VMC (ViT) Golubev et al. (2026) | uses NetKet |
| square 8x8, J2 = 0.8 | best variational | -0.57731 ○ | VMC (ViT) Golubev et al. (2026) | uses NetKet |
| square 8x8, J2 = 0.9 | best variational | -0.637518 ○ | VMC (ViT) Golubev et al. (2026) | uses NetKet |
| square 8x8, J2 = 1 | best variational | -0.700508 ○ | VMC (ViT) Golubev et al. (2026) | uses NetKet |
| square 10x10, J2 = 0.1 | best variational | -0.631365 † | VMC (ViT) Golubev et al. (2026) | uses NetKet |
| square 10x10, J2 = 0.2 | best variational | -0.592982 † | VMC (ViT) Golubev et al. (2026) | uses NetKet |
| square 10x10, J2 = 0.3 | best variational | -0.556912 † | VMC (ViT) Golubev et al. (2026) | uses NetKet |
| square 10x10, J2 = 0.4 | best variational | -0.524361 † | VMC (ViT) Golubev et al. (2026) | uses NetKet |
| best extrapolated | -0.5253 ○ | DMRG Gong et al. (2013) | ||
| square 10x10, J2 = 0.44 | best variational | -0.51273 ○ | VMC (CNN) Li et al. (2022) | |
| square 10x10, J2 = 0.45 | best variational | -0.51001(1) | VMC (Gutzwiller) Hu et al. (2013) | |
| best extrapolated | -0.511 ○ | DMRG Gong et al. (2013) | ||
| square 10x10, J2 = 0.46 | best variational | -0.50744 ○ | VMC (CNN) Liang et al. (2023) | |
| square 10x10, J2 = 0.48 | best variational | -0.50231 ○ | VMC (CNN) Liang et al. (2023) | |
| square 10x10, J2 = 0.5 | best variational | -0.4976939(2) | VMC (CNN-MPS) Fan et al. (2026) | |
| best projected | -0.4957(2) | FN-GFMC (PEPS) Lin et al. (2024) | ||
| best extrapolated | -0.4988 ○ | DMRG Gong et al. (2013) | ||
| square 10x10, J2 = 0.55 | best variational | -0.48693(3) | VMC (Gutzwiller) Hu et al. (2013) | |
| best projected | -0.4853(4) | FN-GFMC (PEPS) Lin et al. (2024) | ||
| best extrapolated | -0.488 ○ | DMRG Gong et al. (2013) | ||
| square 10x10, J2 = 0.6 | best variational | -0.47839 ○ | VMC (CNN) Liang et al. (2023) | |
| square 10x10, J2 = 0.65 | best variational | -0.494672 † | VMC (ViT) Golubev et al. (2026) | uses NetKet |
| square 10x10, J2 = 0.7 | best variational | -0.519035 † | VMC (ViT) Golubev et al. (2026) | uses NetKet |
| square 10x10, J2 = 0.8 | best variational | -0.57413 † | VMC (ViT) Golubev et al. (2026) | uses NetKet |
| square 10x10, J2 = 0.9 | best variational | -0.63417 † | VMC (ViT) Golubev et al. (2026) | uses NetKet |
| square 10x10, J2 = 1 | best variational | -0.69670(2) | VMC (RBM) Chen et al. (2022) | |
| square 12x12, J2 = 0.44 | best variational | -0.51225 ○ | VMC (CNN) Li et al. (2022) | |
| square 12x12, J2 = 0.45 | best extrapolated | -0.5088(1) | cluster update (TPS) Wang et al. (2011) | |
| square 12x12, J2 = 0.46 | best variational | -0.50697 ○ | VMC (CNN) Liang et al. (2023) | |
| square 12x12, J2 = 0.48 | best variational | -0.50183 ○ | VMC (CNN) Liang et al. (2023) | |
| best extrapolated | -0.5008(1) | cluster update (TPS) Wang et al. (2011) | ||
| square 12x12, J2 = 0.5 | best variational | -0.496989 ○ | VMC (CNN) Liang et al. (2023) | |
| best extrapolated | -0.4963(2) | cluster update (TPS) Wang et al. (2011) | ||
| square 12x12, J2 = 0.52 | best extrapolated | -0.4920(1) | cluster update (TPS) Wang et al. (2011) | |
| square 12x12, J2 = 0.55 | best variational | -0.48607(1) | VMC (GCNN) Roth et al. (2022) | uses NetKet |
| best extrapolated | -0.4862(2) | cluster update (TPS) Wang et al. (2011) | ||
| square 12x12, J2 = 0.6 | best variational | -0.47715 ○ | VMC (CNN) Liang et al. (2023) | |
| best extrapolated | -0.4784(3) | cluster update (TPS) Wang et al. (2011) | ||
| square 14x14, J2 = 0.4 | best variational | -0.52287(1) | VMC (Gutzwiller) Hu et al. (2013) | |
| best extrapolated | -0.52351(2) | VMC (Gutzwiller) Hu et al. (2013) | ||
| square 14x14, J2 = 0.44 | best variational | -0.512 ○ | VMC (CNN) Li et al. (2022) | |
| square 14x14, J2 = 0.45 | best variational | -0.50899(1) | VMC (Gutzwiller) Hu et al. (2013) | |
| best extrapolated | -0.50953(1) | VMC (Gutzwiller) Hu et al. (2013) | ||
| square 14x14, J2 = 0.46 | best variational | -0.50675 ○ | VMC (CNN) Liang et al. (2023) | |
| square 14x14, J2 = 0.48 | best variational | -0.50159 ○ | VMC (CNN) Liang et al. (2023) | |
| square 14x14, J2 = 0.5 | best variational | -0.49674 ○ | VMC (CNN) Liang et al. (2023) | |
| best extrapolated | -0.49722(2) | VMC (Gutzwiller) Hu et al. (2013) | ||
| square 14x14, J2 = 0.55 | best variational | -0.48581 ○ | VMC (CNN) Liang et al. (2023) | |
| best extrapolated | -0.48696(5) | VMC (Gutzwiller) Hu et al. (2013) | ||
| square 14x14, J2 = 0.6 | best variational | -0.47678 ○ | VMC (CNN) Liang et al. (2023) | |
| square 16x16, J2 = 0.44 | best variational | -0.51188 ○ | VMC (CNN) Li et al. (2022) | |
| square 16x16, J2 = 0.45 | best extrapolated | -0.5087(1) | cluster update (TPS) Wang et al. (2011) | |
| square 16x16, J2 = 0.46 | best variational | -0.50662 ○ | VMC (CNN) Liang et al. (2023) | |
| square 16x16, J2 = 0.48 | best variational | -0.50147 ○ | VMC (CNN) Liang et al. (2023) | |
| best extrapolated | -0.5008(2) | cluster update (TPS) Wang et al. (2011) | ||
| square 16x16, J2 = 0.5 | best variational | -0.4969140(5) | VMC (CNN-MPS) Fan et al. (2026) | |
| best extrapolated | -0.4966(3) | cluster update (TPS) Wang et al. (2011) | ||
| square 16x16, J2 = 0.52 | best extrapolated | -0.4920(1) | cluster update (TPS) Wang et al. (2011) | |
| square 16x16, J2 = 0.55 | best variational | -0.48565 ○ | VMC (CNN) Liang et al. (2023) | |
| best extrapolated | -0.4859(4) | cluster update (TPS) Wang et al. (2011) | ||
| square 16x16, J2 = 0.6 | best variational | -0.47656 ○ | VMC (CNN) Liang et al. (2023) | |
| best extrapolated | -0.4777(3) | cluster update (TPS) Wang et al. (2011) | ||
| square 18x18, J2 = 0.4 | best variational | -0.52259(1) | VMC (Gutzwiller) Hu et al. (2013) | |
| best extrapolated | -0.52333(1) | VMC (Gutzwiller) Hu et al. (2013) | ||
| square 18x18, J2 = 0.44 | best variational | -0.51179 ○ | VMC (CNN) Li et al. (2022) | |
| square 18x18, J2 = 0.45 | best variational | -0.50874(1) | VMC (Gutzwiller) Hu et al. (2013) | |
| best extrapolated | -0.50940(1) | VMC (Gutzwiller) Hu et al. (2013) | ||
| square 18x18, J2 = 0.46 | best variational | -0.50651 ○ | VMC (CNN) Liang et al. (2023) | |
| square 18x18, J2 = 0.48 | best variational | -0.50139 ○ | VMC (CNN) Liang et al. (2023) | |
| square 18x18, J2 = 0.5 | best variational | -0.496513 ○ | VMC (CNN) Liang et al. (2023) | |
| best extrapolated | -0.49717(2) | VMC (Gutzwiller) Hu et al. (2013) | ||
| square 18x18, J2 = 0.55 | best variational | -0.48557 ○ | VMC (CNN) Liang et al. (2023) | |
| best extrapolated | -0.48698(5) | VMC (Gutzwiller) Hu et al. (2013) | ||
| square 18x18, J2 = 0.6 | best variational | -0.47652 ○ | VMC (CNN) Liang et al. (2023) | |
| square 20x20, J2 = 0.5 | best variational | -0.4967987(6) | VMC (CNN-MPS) Fan et al. (2026) | |
| best extrapolated | -0.49684(1) | VMC (ViT) Viteritti et al. (2026) | ||
| triangular, 48 sites, J2 = 0.125 | exact | -0.5173144 | ED Wu et al. (2024) | |
| triangular, 108 sites, J2 = 0.125 | best variational | -0.51268(9) | VMC (GCNN) Roth et al. (2022) | uses NetKet |
| triangular 12x12, J2 = 0.125 | best variational | -0.51218(9) | VMC (GCNN) Roth et al. (2022) | uses NetKet |
E/N (S.S)
| instance | value | energy | method | code |
|---|---|---|---|---|
| honeycomb 11x11 (242 sites), open, K = -1, J = 0, Gamma = 0 | exact | -0.1882584 | Exact solution Lin et al. (2022) | |
| best variational | -0.1820213 ○ (+6.2e-3) | DMRG Lin et al. (2022) | ||
| best projected | -0.1867058 ○ (+1.6e-3) | DMRG² (isoTNS) Lin et al. (2022) | ||
| honeycomb 3x4 (24 sites), K = -1, J = 0, Gamma = 0 | exact | -0.1988727 | ED Kurita et al. (2015) | |
| best variational | -0.19887(2) (−2.3e-6) | VMC (PP) Kurita et al. (2015) | ||
| honeycomb 4x4 (32 sites), K = -1, J = 0, Gamma = 0 | exact | -0.1963139 | ED Kurita et al. (2015) | |
| best variational | -0.19617(8) (+1.4e-4) | VMC (PP) Kurita et al. (2015) | ||
| honeycomb 5x5 (50 sites), open, K = -1, J = 0, Gamma = 0 | exact | -0.178506 | Exact solution Lin et al. (2022) | |
| best variational | -0.1785027 ○ (+3.3e-6) | DMRG Lin et al. (2022) | ||
| best projected | -0.1767248 ○ (+1.8e-3) | DMRG² (isoTNS) Lin et al. (2022) | ||
| honeycomb 6x6 (72 sites), periodic/open, K = 1, J = 0, Gamma = 0 | exact | -0.1891314 | Exact solution Jin et al. (2021) | |
| honeycomb 7x7 (98 sites), open, K = -1, J = 0, Gamma = 0 | exact | -0.1835418 | Exact solution Lin et al. (2022) | |
| best variational | -0.1826 ○ (+9.4e-4) | DMRG Lin et al. (2022) | ||
| best projected | -0.1824684 ○ (+1.1e-3) | DMRG² (isoTNS) Lin et al. (2022) | ||
| honeycomb-hex24, 24 sites, K = -1, J = 0, Gamma = 0 | exact | -0.1985125 | ED Kurita et al. (2015) | |
| best variational | -0.198510(6) (+2.1e-6) | VMC (PP) Kurita et al. (2015) |
E/N (S.S)
| instance | value | energy | method | code |
|---|---|---|---|---|
| square 4x4, J = 0.7, Jd = 1 | exact | -0.3968907 | ED Munehisa & Munehisa (2004) | |
| square 6x6, J = 0.5, Jd = 1 | exact | -0.375 | ED Sindermann et al. (2006) | |
| square 6x6, J = 0.66697792, Jd = 1 | exact | -0.375 | ED Sindermann et al. (2006) | |
| square 6x6, J = 0.67441, Jd = 1 | exact | -0.375004 | ED Sindermann et al. (2006) | |
| square 6x6, J = 0.675, Jd = 1 | exact | -0.3752862 | ED Sindermann et al. (2006) | |
| square 6x6, J = 0.68, Jd = 1 | exact | -0.3777047 | ED Sindermann et al. (2006) | |
| square 6x6, J = 0.7, Jd = 1 | exact | -0.3885784 | ED Sindermann et al. (2006) | |
| square 6x6, J = 0.8, Jd = 1 | exact | -0.4517531 | ED Sindermann et al. (2006) | |
| best variational | -0.451717(6) (+3.6e-5) | VMC (ConvNeXt) Nutakki et al. (2025) | NeuralQXLab/convnextnnqs | |
| best extrapolated | -0.45175 ○ (+3.1e-6) | VMC (Transformer) Viteritti et al. (2025) | ||
| square 6x6, J = 0.85, Jd = 1 | exact | -0.4843341 | ED Sindermann et al. (2006) | |
| square 8x8, J = 0.8, Jd = 1 | best variational | -0.449829 ○ | VMC (Transformer) Viteritti et al. (2025) | |
| best extrapolated | -0.44995 ○ | VMC (Transformer) Viteritti et al. (2025) | ||
| square 10x10, J = 0.8, Jd = 1 | best variational | -0.449342(4) | VMC (ConvNeXt) Nutakki et al. (2025) | NeuralQXLab/convnextnnqs |
| best extrapolated | -0.44947 ○ | VMC (Transformer) Viteritti et al. (2025) | ||
| square 12x12, J = 0.7, Jd = 1 | best variational | -0.37627 ○ | VMC (Gutzwiller) Chen & Liu (2026) | |
| square 12x12, J = 0.8, Jd = 1 | best variational | -0.449078 ○ | VMC (Transformer) Viteritti et al. (2025) | |
| best extrapolated | -0.44931 ○ | VMC (Transformer) Viteritti et al. (2025) | ||
| square 14x14, J = 0.8, Jd = 1 | best variational | -0.448929 ○ | VMC (Transformer) Viteritti et al. (2025) | |
| best extrapolated | -0.4492 ○ | VMC (Transformer) Viteritti et al. (2025) | ||
| square 16x16, J = 0.7, Jd = 1 | best variational | -0.37642 ○ | VMC (Gutzwiller) Chen & Liu (2026) | |
| square 16x16, J = 0.8, Jd = 1 | best variational | -0.448881 ○ | VMC (Transformer) Viteritti et al. (2025) | |
| square 18x18, J = 0.8, Jd = 1 | best variational | -0.448859 ○ | VMC (Transformer) Viteritti et al. (2025) | |
| square 20x20, open, J = 0.8, Jd = 1 | best variational | -0.442928(3) | VMC (PEPS) Liu et al. (2024) | |
| square-tilted, 32 sites, J = 0.677, Jd = 1 | exact | -0.375 | ED Sindermann et al. (2006) | |
| square-tilted, 32 sites, J = 0.6772, Jd = 1 | exact | -0.375 | ED Sindermann et al. (2006) | |
| square-tilted, 32 sites, J = 0.67724, Jd = 1 | exact | -0.375 | ED Sindermann et al. (2006) | |
| square-tilted, 32 sites, J = 0.677244, Jd = 1 | exact | -0.375 | ED Sindermann et al. (2006) | |
| square-tilted, 32 sites, J = 0.677246, Jd = 1 | exact | -0.375 | ED Sindermann et al. (2006) | |
| square-tilted, 32 sites, J = 0.677248, Jd = 1 | exact | -0.375 | ED Sindermann et al. (2006) | |
| square-tilted, 32 sites, J = 0.6773, Jd = 1 | exact | -0.3750238 | ED Sindermann et al. (2006) | |
| square-tilted, 32 sites, J = 0.6775, Jd = 1 | exact | -0.3751152 | ED Sindermann et al. (2006) | |
| square-tilted, 32 sites, J = 0.6778, Jd = 1 | exact | -0.3752526 | ED Sindermann et al. (2006) | |
| square-tilted, 32 sites, J = 0.68, Jd = 1 | exact | -0.3762717 | ED Sindermann et al. (2006) | |
| square-tilted, 32 sites, J = 0.7, Jd = 1 | exact | -0.3867209 | ED Sindermann et al. (2006) |
E/site
| instance | value | energy | method | code |
|---|---|---|---|---|
| chain, 14 sites, U = 1, n ≈ 0.5714 | exact | -0.9116302 | ED Wu et al. (2024) | run script |
| best variational | -0.9116293 (+9.1e-7) | DMRG Wu et al. (2024) | run script | |
| best extrapolated | -0.9116302(1) (−<1e-7) | DMRG Wu et al. (2024) | run script | |
| chain, 14 sites, U = 1.66810054, n ≈ 0.5714 | exact | -0.8761535 | ED Wu et al. (2024) | run script |
| best variational | -0.8761535 (+<1e-7) | DMRG Wu et al. (2024) | run script | |
| best extrapolated | -0.876153552(4) (−<1e-7) | DMRG Wu et al. (2024) | run script | |
| chain, 14 sites, U = 2.15443469, n ≈ 0.5714 | exact | -0.8542284 | ED Wu et al. (2024) | run script |
| best variational | -0.8542284 (+<1e-7) | DMRG Wu et al. (2024) | run script | |
| chain, 14 sites, U = 2.7825594, n ≈ 0.5714 | exact | -0.8300032 | ED Wu et al. (2024) | run script |
| best variational | -0.8300032 (+<1e-7) | DMRG Wu et al. (2024) | run script | |
| chain, 14 sites, U = 3.59381366, n ≈ 0.5714 | exact | -0.804323 | ED Wu et al. (2024) | run script |
| best variational | -0.804323 (+<1e-6) | DMRG Wu et al. (2024) | run script | |
| chain, 14 sites, U = 4.64158882, n ≈ 0.5714 | exact | -0.7783541 | ED QMBL check, checks/hubbard-u-labels/ | QMBL code |
| best variational | -0.7783541 (+<1e-7) | DMRG Wu et al. (2024) | run script | |
| chain, 14 sites, U = 4.64158883, n ≈ 0.5714 | exact | -0.7783541 | ED Wu et al. (2024) | run script |
| chain, 14 sites, U = 7.74263683, n ≈ 0.5714 | exact | -0.7304717 | ED Wu et al. (2024) | run script |
| best variational | -0.7304717 (+<1e-7) | DMRG Wu et al. (2024) | run script | |
| chain, 14 sites, U = 10, n ≈ 0.5714 | exact | -0.7103639 | ED Wu et al. (2024) | run script |
| best variational | -0.7103637 (+1.4e-7) | DMRG Wu et al. (2024) | run script | |
| best extrapolated | -0.71036386(2) (−<1e-7) | DMRG Wu et al. (2024) | run script | |
| rectangular 10x16, open, U = 8, n = 0.875 | best variational | -0.7163(3) | simple update (fPEPS) 10.1103/r4q9-4yvj | |
| rectangular 10x16, U = 8, n = 0.875, t12 (t2 = -0.2) | best variational | -0.7434 ○ | VMC (transformer BF) Gu et al. (2026) | bytedance/laqx |
| rectangular 12x16, open, U = 8, n = 0.875 | best variational | -0.72212 ○ | VMC (transformer BF) Gu et al. (2026) | |
| rectangular 12x16, U = 8, n = 0.875 | best variational | -0.7562 ○ | VMC (transformer BF) Gu et al. (2026) | bytedance/laqx |
| rectangular 12x16, U = 8, n = 0.875, t12 (t2 = -0.2) | best variational | -0.744 ○ | VMC (transformer BF) Gu et al. (2026) | bytedance/laqx |
| rectangular 14x16, U = 8, n = 0.875 | best variational | -0.75830(4) | VAFQMC Wu et al. (2024) | |
| rectangular 16x32, U = 8, n = 0.875 | best variational | -0.7507 ○ | VMC (SCALE) Gu et al. (2026) | uses LaQX |
| best projected | -0.7547 ○ | FN-GFMC (SCALE) Gu et al. (2026) | ||
| rectangular 16x32, U = 8, n = 0.875, t12 (t2 = -0.2) | best variational | -0.7371 ○ | VMC (SCALE) Gu et al. (2026) | uses LaQX |
| best projected | -0.7406 ○ | FN-GFMC (SCALE) Gu et al. (2026) | ||
| rectangular 4x12, U = 8, n = 0.875 | best variational | -0.76608(1) | VMC (HFPS) Chen et al. (2026) | uses Quantax |
| rectangular 4x16, open, U = 8, n = 0.875 | best variational | -0.6853698 ○ | DMRG 10.1103/r4q9-4yvj | hczhai/2d-hubbard-dmrg-2024 |
| best extrapolated | -0.68554(4) | DMRG 10.1103/r4q9-4yvj | hczhai/2d-hubbard-dmrg-2024 | |
| rectangular 4x16, U = 8, n = 0.875 | best variational | -0.76623(1) | VMC (ACE) Gu et al. (2026) | uses LaQX |
| best projected | -0.76647(2) | FN-GFMC (ACE) Gu et al. (2026) | ||
| best extrapolated | -0.76748 ○ | VMC (NNBF) Loehr & Clark (2025) | ||
| rectangular 4x16, periodic/open, U = 2, n = 0.8125 | best variational | -1.2852656 | DMRG Wu et al. (2024) | run script |
| rectangular 4x16, periodic/open, U = 8, n = 0.8125 | best variational | -0.853 | DMRG Wu et al. (2024) | run script |
| rectangular 4x16, periodic/open, U = 2, n = 0.875 | best variational | -1.2611875 | DMRG Wu et al. (2024) | run script |
| rectangular 4x16, periodic/open, U = 8, n = 0.875 | best variational | -0.7508328 | DMRG Wu et al. (2024) | run script |
| best extrapolated | -0.75114(6) | DMRG Wu et al. (2024) | run script | |
| rectangular 4x16, periodic/open, U = 2, n = 0.9375 | best variational | -1.2199062 | DMRG Wu et al. (2024) | run script |
| rectangular 4x16, periodic/open, U = 8, n = 0.9375 | best variational | -0.637875 | DMRG Wu et al. (2024) | run script |
| rectangular 4x16, periodic/open, U = 2, n = 1 | best variational | -1.1710781 | DMRG Wu et al. (2024) | run script |
| rectangular 4x16, periodic/open, U = 8, n = 1 | best variational | -0.5144219 | DMRG Wu et al. (2024) | run script |
| rectangular 4x24, U = 4, n ≈ 0.8333 | best variational | -1.07872(3) | VMC (HFPS) Roth et al. (2025) | uses Quantax |
| rectangular 4x8, U = 8, n = 0.875 | best variational | -0.768337(2) | VMC (NNBF) Loehr & Clark (2025) | |
| best extrapolated | -0.768416 ○ | VMC (NNBF) Loehr & Clark (2025) | ||
| rectangular 4x8, periodic/open, U = 8, n = 0.875 | best variational | -0.7363906 | DMRG Wu et al. (2024) | run script |
| best extrapolated | -0.736431(9) | DMRG Wu et al. (2024) | run script | |
| rectangular 6x16, open, U = 8, n = 0.875 | best variational | -0.70307 ○ | VMC (transformer BF) Gu et al. (2026) | |
| rectangular 6x16, U = 8, n = 0.875 | best variational | -0.7486 ○ | VMC (Jastrow + BF) Marino et al. (2022) | |
| rectangular 6x32, U = 8, n = 0.875 | best variational | -0.7483 ○ | VMC (Jastrow + BF) Marino et al. (2022) | |
| rectangular 6x40, U = 12, n = 0.875, t12 (t2 = -0.1) | best variational | -0.6206 ○ | VMC (Jastrow + BF) Marino et al. (2022) | |
| rectangular 6x40, U = 12, n = 0.875, t12 (t2 = -0.2) | best variational | -0.6187 ○ | VMC (Jastrow + BF) Marino et al. (2022) | |
| rectangular 6x40, U = 12, n = 0.875, t12 (t2 = -0.3) | best variational | -0.619 ○ | VMC (Jastrow + BF) Marino et al. (2022) | |
| rectangular 6x40, U = 12, n = 0.875, t12 (t2 = -0.35) | best variational | -0.6202 ○ | VMC (Jastrow + BF) Marino et al. (2022) | |
| rectangular 6x40, U = 12, n = 0.875, t12 (t2 = -0.4) | best variational | -0.622 ○ | VMC (Jastrow + BF) Marino et al. (2022) | |
| rectangular 6x40, U = 12, n = 0.875, t12 (t2 = -0.45) | best variational | -0.624 ○ | VMC (Jastrow + BF) Marino et al. (2022) | |
| rectangular 6x40, U = 12, n = 0.875, t12 (t2 = -0.5) | best variational | -0.626 ○ | VMC (Jastrow + BF) Marino et al. (2022) | |
| rectangular 6x48, U = 8, n ≈ 0.8333 | best variational | -0.82075(3) | VMC (Jastrow + BF) Wu et al. (2024) | run script |
| best projected | -0.82575(2) | FN-GFMC (Jastrow + BF) Wu et al. (2024) | run script | |
| rectangular 6x48, U = 8, n = 0.875 | best variational | -0.74834(3) | VMC (Jastrow + BF) Wu et al. (2024) | run script |
| best projected | -0.75398(2) | FN-GFMC (Jastrow + BF) Wu et al. (2024) | run script | |
| rectangular 6x48, U = 12, n = 0.875 | best variational | -0.62556(3) | VMC (Jastrow + BF) Wu et al. (2024) | run script |
| best projected | -0.63217(2) | FN-GFMC (Jastrow + BF) Wu et al. (2024) | run script | |
| rectangular 6x48, U = 12, n = 0.875, t12 (t2 = -0.1) | best variational | -0.6147 ○ | VMC (Jastrow + BF) Marino et al. (2022) | |
| rectangular 6x48, U = 12, n = 0.875, t12 (t2 = -0.2) | best variational | -0.6113 ○ | VMC (Jastrow + BF) Marino et al. (2022) | |
| rectangular 6x48, U = 12, n = 0.875, t12 (t2 = -0.3) | best variational | -0.6118 ○ | VMC (Jastrow + BF) Marino et al. (2022) | |
| rectangular 6x48, U = 12, n = 0.875, t12 (t2 = -0.35) | best variational | -0.6135 ○ | VMC (Jastrow + BF) Marino et al. (2022) | |
| rectangular 6x48, U = 12, n = 0.875, t12 (t2 = -0.4) | best variational | -0.6162 ○ | VMC (Jastrow + BF) Marino et al. (2022) | |
| rectangular 6x48, U = 12, n = 0.875, t12 (t2 = -0.45) | best variational | -0.6202 ○ | VMC (Jastrow + BF) Marino et al. (2022) | |
| rectangular 6x48, U = 12, n = 0.875, t12 (t2 = -0.5) | best variational | -0.6261 ○ | VMC (Jastrow + BF) Marino et al. (2022) | |
| rectangular 6x48, U = 12, n = 0.875, t12 (t2 = 0.1) | best variational | -0.6375 ○ | VMC (Jastrow + BF) Marino et al. (2022) | |
| rectangular 6x48, U = 12, n = 0.875, t12 (t2 = 0.15) | best variational | -0.6464 ○ | VMC (Jastrow + BF) Marino et al. (2022) | |
| rectangular 6x48, U = 12, n = 0.875, t12 (t2 = 0.2) | best variational | -0.6562 ○ | VMC (Jastrow + BF) Marino et al. (2022) | |
| rectangular 6x48, U = 12, n = 0.875, t12 (t2 = 0.25) | best variational | -0.6669 ○ | VMC (Jastrow + BF) Marino et al. (2022) | |
| rectangular 6x48, U = 8, n ≈ 0.9167 | best variational | -0.67269(3) | VMC (Jastrow + BF) Wu et al. (2024) | run script |
| best projected | -0.6862(2) | CP-AFQMC (stripe) Xu et al. (2022) | ||
| rectangular 6x64, U = 8, n = 0.875 | best variational | -0.7482 ○ | VMC (Jastrow + BF) Marino et al. (2022) | |
| rectangular 8x10, U = 8, n = 0.875, t12 (t2 = -0.2) | best variational | -0.7427 ○ | VMC (transformer BF) Gu et al. (2026) | bytedance/laqx |
| rectangular 8x12, U = 8, n = 0.875, t12 (t2 = -0.2) | best variational | -0.7456 ○ | VMC (transformer BF) Gu et al. (2026) | bytedance/laqx |
| rectangular 8x16, open, U = 8, n = 0.875 | best variational | -0.71309 ○ | VMC (transformer BF) Gu et al. (2026) | |
| best extrapolated | -0.718(1) | DMRG 10.1103/r4q9-4yvj | hczhai/2d-hubbard-dmrg-2024 | |
| rectangular 8x16, open, U = 8, n = 0.9375 | best variational | -0.6019(2) | simple update (fPEPS) 10.1103/r4q9-4yvj | |
| best extrapolated | -0.6043(6) | DMRG 10.1103/r4q9-4yvj | hczhai/2d-hubbard-dmrg-2024 | |
| rectangular 8x16, U = 8, n = 0.875 | best variational | -0.7568 ○ | VMC (tensor BF) Liang (2025) | |
| rectangular 8x16, U = 8, n = 0.875, t12 (t2 = -0.2) | best variational | -0.7419 ○ | VMC (transformer BF) Gu et al. (2026) | bytedance/laqx |
| rectangular 8x18, U = 8, n = 0.875, t12 (t2 = -0.2) | best variational | -0.7421 ○ | VMC (transformer BF) Gu et al. (2026) | bytedance/laqx |
| rectangular 8x20, U = 8, n = 0.875, t12 (t2 = -0.2) | best variational | -0.7433 ○ | VMC (transformer BF) Gu et al. (2026) | bytedance/laqx |
| rectangular 8x24, U = 8, n = 0.875 | best variational | -0.7546 ○ | VMC (tensor BF) Liang (2025) | |
| rectangular 8x24, U = 8, n = 0.875, t12 (t2 = -0.2) | best variational | -0.7423 ○ | VMC (transformer BF) Gu et al. (2026) | bytedance/laqx |
| rectangular 8x32, U = 8, n = 0.875 | best variational | -0.7547 ○ | VMC (tensor BF) Liang (2025) | |
| rectangular 8x32, U = 8, n = 0.875, t12 (t2 = -0.2) | best variational | -0.742 ○ | VMC (transformer BF) Gu et al. (2026) | bytedance/laqx |
| best extrapolated | -0.7491 ○ | VMC (transformer BF) Gu et al. (2026) | bytedance/laqx | |
| rectangular 8x48, U = 8, n ≈ 0.9167 | best projected | -0.6854(2) | CP-AFQMC (stripe) Xu et al. (2022) | |
| square 4x4, open, U = 8, n = 0.875 | best variational | -0.6326183 ○ | DMRG 10.1103/r4q9-4yvj | hczhai/2d-hubbard-dmrg-2024 |
| best extrapolated | -0.6326190(1) | DMRG 10.1103/r4q9-4yvj | hczhai/2d-hubbard-dmrg-2024 | |
| square 4x4, open, U = 8, n = 1 | best variational | -0.4255259 ○ | DMRG 10.1103/r4q9-4yvj | hczhai/2d-hubbard-dmrg-2024 |
| best extrapolated | -0.42552596(1) | DMRG 10.1103/r4q9-4yvj | hczhai/2d-hubbard-dmrg-2024 | |
| square 4x4, U = 0.1, n = 0.5 | exact | -1.2442068 | ED Moreno et al. (2022) | |
| best variational | -1.2441738(2) (+3.3e-5) | VMC (HFDS) Moreno et al. (2022) | uses NetKet | |
| square 4x4, U = 0.59948425, n = 0.5 | exact | -1.2171727 | ED Moreno et al. (2022) | |
| best variational | -1.2171594(6) (+1.3e-5) | VMC (HFDS) Moreno et al. (2022) | uses NetKet | |
| square 4x4, U = 1, n = 0.5 | exact | -1.1976241 | ED Moreno et al. (2022) | |
| best variational | -1.197511(2) (+1.1e-4) | VMC (HFDS) Moreno et al. (2022) | uses NetKet | |
| square 4x4, U = 2, n = 0.5 | exact | -1.1559648 | ED Wu et al. (2024) | run script |
| best variational | -1.1559648 (+<1e-7) | DMRG Wu et al. (2024) | run script | |
| square 4x4, U = 2.15443469, n = 0.5 | exact | -1.1503325 | ED Moreno et al. (2022) | |
| best variational | -1.148627(3) (+1.7e-3) | VMC (HFDS) Moreno et al. (2022) | uses NetKet | |
| square 4x4, U = 3.59381366, n = 0.5 | exact | -1.1061269 | ED Moreno et al. (2022) | |
| best variational | -1.105848(3) (+2.8e-4) | VMC (HFDS) Moreno et al. (2022) | uses NetKet | |
| square 4x4, U = 3.5981, n = 0.5 | exact | -1.1060147 | ED QMBL check, checks/hubbard-u-labels/ | QMBL code |
| best variational | -1.1060144 (+3.2e-7) | DMRG Wu et al. (2024) | run script | |
| square 4x4, U = 4, n = 0.5 | exact | -1.0959311 | ED Wu et al. (2024) | run script |
| best variational | -1.095931 (+<1e-7) | DMRG Wu et al. (2024) | run script | |
| square 4x4, U = 6, n = 0.5 | exact | -1.0562454 | ED Wu et al. (2024) | run script |
| best variational | -1.0562454 (+<1e-7) | DMRG Wu et al. (2024) | run script | |
| square 4x4, U = 7.74263683, n = 0.5 | exact | -1.0318222 | ED Moreno et al. (2022) | |
| best variational | -1.031562(6) (+2.6e-4) | VMC (HFDS) Moreno et al. (2022) | uses NetKet | |
| square 4x4, U = 7.74264, n = 0.5 | exact | -1.0318221 | ED QMBL check, checks/hubbard-u-labels/ | QMBL code |
| best variational | -1.0318221 (+<1e-7) | DMRG Wu et al. (2024) | run script | |
| square 4x4, U = 8, n = 0.5 | exact | -1.0287892 | ED Wu et al. (2024) | run script |
| best variational | -1.0287892 (+<1e-7) | DMRG Wu et al. (2024) | run script | |
| square 4x4, U = 10, n = 0.5 | exact | -1.0089505 | ED Wu et al. (2024) | run script |
| best variational | -1.0089505 (+<1e-7) | DMRG Wu et al. (2024) | run script | |
| square 4x4, U = 2, n = 0.625 | exact | -1.3360594 | ED Wu et al. (2024) | run script |
| best variational | -1.3360594 (+<1e-7) | DMRG Wu et al. (2024) | run script | |
| square 4x4, U = 2.1544, n = 0.625 | exact | -1.325767 | ED QMBL check, checks/hubbard-u-labels/ | QMBL code |
| best variational | -1.325767 (+<1e-6) | DMRG Wu et al. (2024) | run script | |
| square 4x4, U = 2.15443469, n = 0.625 | exact | -1.3257647 | ED Moreno et al. (2022) | |
| best variational | -1.3257435(1) (+2.1e-5) | VMC (HFDS) Moreno et al. (2022) | uses NetKet | |
| square 4x4, U = 3.59381366, n = 0.625 | exact | -1.2432273 | ED Moreno et al. (2022) | |
| best variational | -1.2431527(4) (+7.5e-5) | VMC (HFDS) Moreno et al. (2022) | uses NetKet | |
| square 4x4, U = 3.5981, n = 0.625 | exact | -1.2430143 | ED QMBL check, checks/hubbard-u-labels/ | QMBL code |
| best variational | -1.2430143 (+<1e-7) | DMRG Wu et al. (2024) | run script | |
| square 4x4, U = 4, n = 0.625 | exact | -1.2238086 | ED Wu et al. (2024) | run script |
| best variational | -1.2238086 (+<1e-7) | DMRG Wu et al. (2024) | run script | |
| square 4x4, U = 6, n = 0.625 | exact | -1.1473978 | ED Wu et al. (2024) | run script |
| best variational | -1.1473978 ○ (+<1e-7) | DMRG Wu et al. (2024) | run script | |
| square 4x4, U = 7.74263683, n = 0.625 | exact | -1.1002332 | ED Moreno et al. (2022) | |
| best variational | -1.1001250(6) (+1.1e-4) | VMC (HFDS) Moreno et al. (2022) | uses NetKet | |
| square 4x4, U = 7.74264, n = 0.625 | exact | -1.1002331 | ED QMBL check, checks/hubbard-u-labels/ | QMBL code |
| best variational | -1.1002331 (+<1e-7) | DMRG Wu et al. (2024) | run script | |
| square 4x4, U = 8, n = 0.625 | exact | -1.0943979 | ED Wu et al. (2024) | run script |
| best variational | -1.0943979 (+<1e-7) | DMRG Wu et al. (2024) | run script | |
| square 4x4, U = 10, n = 0.625 | exact | -1.0564725 | ED Wu et al. (2024) | run script |
| best variational | -1.0564725 (+<1e-7) | DMRG Wu et al. (2024) | run script | |
| square 4x4, U = 8, n = 0.875 | exact | -0.7418022 | ED QMBL check, checks/hubbard-4x4-n78-ed/ | QMBL code |
| best variational | -0.74177(5) (+3.2e-5) | VMC (HFPS) Chen et al. (2026) | uses Quantax | |
| square 4x4, U = 2, n = 1 | exact (stochastic) | -1.1265(4) | AFQMC Qin et al. (2016) | |
| best variational | -1.1248 ○ (+1.7e-3) | VMC (HB) Zhou et al. (2026) | ||
| square 4x4, U = 4, n = 1 | exact | -0.8513659 | ED QMBL check, checks/hubbard-4x4-half-filling-ed/ | QMBL code |
| best variational | -0.8511(1) (+2.3e-4) | VMC (TQS) Yamazaki et al. (2026) | ||
| square 4x4, U = 6, n = 1 | exact (stochastic) | -0.6588(3) | AFQMC Qin et al. (2016) | |
| best variational | -0.6577 ○ (+1.1e-3) | VMC (HB) Zhou et al. (2026) | ||
| square 4x4, U = 8, n = 1 | exact | -0.5293047 | ED QMBL check, checks/hubbard-4x4-half-filling-ed/ | QMBL code |
| best variational | -0.5292(3) (+1.1e-4) | VMC (PITQS) Yamazaki et al. (2026) | Zenodo 18334851 | |
| square 4x4, periodic/antiperiodic, U = 2, n = 1 | exact (stochastic) | -1.2571(1) | AFQMC Qin et al. (2016) | |
| best variational | -1.25693(1) (+1.9e-4) | VMC (HFDS) Moreno et al. (2022) | uses NetKet | |
| square 4x4, periodic/antiperiodic, U = 4, n = 1 | exact (stochastic) | -0.9121(2) | AFQMC Qin et al. (2016) | |
| best variational | -0.9120(1) (+1.2e-4) | VMC (HFDS) Moreno et al. (2022) | uses NetKet | |
| square 4x4, periodic/antiperiodic, U = 6, n = 1 | exact (stochastic) | -0.6814(4) | AFQMC Qin et al. (2016) | |
| best variational | -0.68135(1) (+<1e-4) | VMC (HFDS) Moreno et al. (2022) | uses NetKet | |
| square 4x4, periodic/antiperiodic, U = 8, n = 1 | exact (stochastic) | -0.5404(5) | AFQMC Qin et al. (2016) | |
| best variational | -0.5401(1) (+2.8e-4) | VMC (HFDS) Moreno et al. (2022) | uses NetKet | |
| square 6x6, open, U = 8, n ≈ 0.8889 | best variational | -0.6573063 ○ | DMRG 10.1103/r4q9-4yvj | hczhai/2d-hubbard-dmrg-2024 |
| best extrapolated | -0.65768(8) | DMRG 10.1103/r4q9-4yvj | hczhai/2d-hubbard-dmrg-2024 | |
| square 6x6, open, U = 8, n ≈ 0.9444 | best variational | -0.5599001 ○ | DMRG 10.1103/r4q9-4yvj | hczhai/2d-hubbard-dmrg-2024 |
| best extrapolated | -0.55997(2) | DMRG 10.1103/r4q9-4yvj | hczhai/2d-hubbard-dmrg-2024 | |
| square 6x6, U = 4, n ≈ 0.6667 | best variational | -1.1834(1) | VAFQMC Levy et al. (2023) | |
| best projected | -1.18525(4) | CP-AFQMC Shi & Zhang (2013) | ||
| square 6x6, U = 2, n = 1 | exact (stochastic) | -1.1516(1) | AFQMC Qin et al. (2016) | |
| best variational | -1.1506 ○ (+9.8e-4) | VMC (HB) Zhou et al. (2026) | ||
| square 6x6, U = 4, n = 1 | exact (stochastic) | -0.8574(3) | AFQMC Qin et al. (2016) | |
| best variational | -0.8554 ○ (+2.0e-3) | VMC (HB) Zhou et al. (2026) | ||
| square 6x6, U = 6, n = 1 | exact (stochastic) | -0.6594(3) | AFQMC Qin et al. (2016) | |
| square 6x6, U = 8, n = 1 | exact (stochastic) | -0.5278(6) | AFQMC Qin et al. (2016) | |
| best variational | -0.5266 ○ (+1.2e-3) | VMC (HB) Zhou et al. (2026) | ||
| square 6x6, U = 4, n ≈ 1.3333 | best projected | 0.14809(4) | CP-AFQMC Shi & Zhang (2013) | |
| square 6x6, periodic/antiperiodic, U = 2, n = 1 | exact (stochastic) | -1.20831(6) | AFQMC Qin et al. (2016) | |
| best variational | -1.2079(1) (+4.1e-4) | VMC (HFDS) Moreno et al. (2022) | uses NetKet | |
| square 6x6, periodic/antiperiodic, U = 4, n = 1 | exact (stochastic) | -0.8731(6) | AFQMC Qin et al. (2016) | |
| best variational | -0.87173(2) (+1.3e-3) | VMC (HFDS) Moreno et al. (2022) | uses NetKet | |
| square 6x6, periodic/antiperiodic, U = 6, n = 1 | exact (stochastic) | -0.6622(3) | AFQMC Qin et al. (2016) | |
| best variational | -0.66094(3) (+1.3e-3) | VMC (HFDS) Moreno et al. (2022) | uses NetKet | |
| square 6x6, periodic/antiperiodic, U = 8, n = 1 | exact (stochastic) | -0.5281(3) | AFQMC Qin et al. (2016) | |
| best variational | -0.52705(3) (+1.0e-3) | VMC (HFDS) Moreno et al. (2022) | uses NetKet | |
| square, 50 sites, U = 8, n = 0.84 | best variational | -0.82328(3) | VAFQMC Wu et al. (2024) | |
| square 8x8, open, U = 8, n = 0.875 | best variational | -0.69952 ○ | VMC (transformer BF) Gu et al. (2026) | |
| best extrapolated | -0.704(1) | DMRG 10.1103/r4q9-4yvj | hczhai/2d-hubbard-dmrg-2024 | |
| square 8x8, open, U = 8, n ≈ 0.9688 | best variational | -0.5322603 ○ | DMRG 10.1103/r4q9-4yvj | hczhai/2d-hubbard-dmrg-2024 |
| best extrapolated | -0.5335(3) | DMRG 10.1103/r4q9-4yvj | hczhai/2d-hubbard-dmrg-2024 | |
| square 8x8, U = 4, n = 0.4375 | best projected | -1.07180(2) | CP-AFQMC Shi & Zhang (2013) | |
| square 8x8, U = 4, n = 0.6875 | best variational | -1.1846 ○ | VMC (tensor BF) Liang (2025) | |
| best projected | -1.1858(2) | CP-AFQMC Shi & Zhang (2013) | ||
| square 8x8, U = -8, n ≈ 0.7813 | exact (stochastic) | -3.6258(1) | AFQMC Wu et al. (2024) | run scripts |
| square 8x8, U = -4, n ≈ 0.7813 | exact (stochastic) | -2.39630(9) | AFQMC Wu et al. (2024) | run scripts |
| square 8x8, U = 4, n ≈ 0.7813 | best variational | -1.13048(8) | VAFQMC Levy et al. (2023) | |
| best projected | -1.13253(3) | CP-AFQMC Levy et al. (2023) | ||
| square 8x8, U = 8, n ≈ 0.7813 | best variational | -0.91483(2) | VMC (PP + RBM) Wu et al. (2024) | run script |
| best projected | -0.9252(1) | CP-AFQMC Levy et al. (2023) | ||
| square 8x8, U = -8, n = 0.875 | exact (stochastic) | -4.01700(9) | AFQMC Wu et al. (2024) | run scripts |
| best variational | -4.0169(1) (+1.0e-4) | VMC (transformer BF) Viteritti et al. (2026) | ||
| best projected | -4.0168 ○ (+2.0e-4) | FN-GFMC (ACE) Gu et al. (2026) | ||
| square 8x8, U = -4, n = 0.875 | exact (stochastic) | -2.60123(8) | AFQMC Wu et al. (2024) | run scripts |
| square 8x8, U = -2, n = 0.875 | exact (stochastic) | -2.03430(7) | DQMC Gu et al. (2026) | |
| best variational | -2.0338 ○ (+5.0e-4) | VMC (ACE) Gu et al. (2026) | uses LaQX | |
| best projected | -2.0339 ○ (+4.0e-4) | FN-GFMC (ACE) Gu et al. (2026) | ||
| square 8x8, U = 4, n = 0.875 | best variational | -1.01391(8) | VAFQMC Levy et al. (2023) | |
| best projected | -1.01923(6) | CP-AFQMC Levy et al. (2023) | ||
| square 8x8, U = 8, n = 0.875 | best variational | -0.7532 ○ | VMC (transformer BF) Gu et al. (2026) | bytedance/laqx |
| best projected | -0.7616(1) | CP-AFQMC Levy et al. (2023) | ||
| square 8x8, U = 8, n = 0.875, t12 (t2 = -0.2) | best variational | -0.75058(1) | VMC (transformer BF) Viteritti et al. (2026) | |
| best extrapolated | -0.7540(4) | VMC (transformer BF) Viteritti et al. (2026) | ||
| square 8x8, U = -8, n = 1 | exact (stochastic) | -4.52589(9) | AFQMC Wu et al. (2024) | run scripts |
| best variational | -4.525834 † (+5.7e-5) | VMC (HFPS) Chen et al. (2026) | uses Quantax | |
| square 8x8, U = -4, n = 1 | exact (stochastic) | -2.86037(6) | AFQMC Wu et al. (2024) | run scripts |
| best variational | -2.859578 † (+8.0e-4) | VMC (HFPS) Chen et al. (2026) | uses Quantax | |
| square 8x8, U = 2, n = 1 | exact (stochastic) | -1.16359(8) | AFQMC Qin et al. (2016) | |
| best variational | -1.1626 ○ (+9.9e-4) | VMC (HB) Zhou et al. (2026) | ||
| square 8x8, U = 4, n = 1 | exact (stochastic) | -0.86036(6) | AFQMC Wu et al. (2024) | |
| best variational | -0.859156(8) (+1.2e-3) | VMC (PP + RBM) Wu et al. (2024) | run script | |
| best projected | -0.857653(9) (+2.7e-3) | FN-GFMC (Jastrow + BF) Wu et al. (2024) | run script | |
| square 8x8, U = 6, n = 1 | exact (stochastic) | -0.6587(3) | AFQMC Qin et al. (2016) | |
| best variational | -0.6554 ○ (+3.3e-3) | VMC (HB) Zhou et al. (2026) | ||
| square 8x8, U = 8, n = 1 | exact (stochastic) | -0.52566(8) | AFQMC Wu et al. (2024) | |
| best variational | -0.52459(1) (+1.1e-3) | VMC (PP + RBM) Wu et al. (2024) | run script | |
| square 8x8, periodic/antiperiodic, U = 2, n = 1 | exact (stochastic) | -1.19231(5) | AFQMC Qin et al. (2016) | |
| best variational | -1.19003(9) (+2.3e-3) | VMC (HFDS) Moreno et al. (2022) | uses NetKet | |
| square 8x8, periodic/antiperiodic, U = 4, n = 1 | exact (stochastic) | -0.8642(2) | AFQMC Qin et al. (2016) | |
| best variational | -0.8622(2) (+2.0e-3) | VMC (HFDS) Moreno et al. (2022) | uses NetKet | |
| square 8x8, periodic/antiperiodic, U = 6, n = 1 | exact (stochastic) | -0.6589(3) | AFQMC Qin et al. (2016) | |
| best variational | -0.6574(2) (+1.5e-3) | VMC (HFDS) Moreno et al. (2022) | uses NetKet | |
| square 8x8, periodic/antiperiodic, U = 8, n = 1 | exact (stochastic) | -0.5259(3) | AFQMC Qin et al. (2016) | |
| best variational | -0.5244(2) (+1.5e-3) | VMC (HFDS) Moreno et al. (2022) | uses NetKet | |
| square 8x8, periodic/open, U = 8, n = 1 | exact (stochastic) | -0.5005(2) | AFQMC Wu et al. (2024) | run scripts |
| best variational | -0.49944(4) (+1.0e-3) | VAFQMC Wu et al. (2024) | ||
| best extrapolated | -0.5000(3) (+4.1e-4) | DMRG Wu et al. (2024) | run script | |
| square 10x10, U = 4, n = 0.8 | best variational | -1.11097(6) | VAFQMC Levy et al. (2023) | |
| best projected | -1.1135(2) | CP-AFQMC Shi & Zhang (2013) | ||
| square 10x10, U = 2, n = 1 | exact (stochastic) | -1.16908(4) | AFQMC Qin et al. (2016) | |
| best variational | -1.1677 ○ (+1.4e-3) | VMC (HB) Zhou et al. (2026) | ||
| square 10x10, U = 4, n = 1 | exact (stochastic) | -0.8612(4) | AFQMC Qin et al. (2016) | |
| best variational | -0.8585 ○ (+2.7e-3) | VMC (HB) Zhou et al. (2026) | ||
| square 10x10, U = 6, n = 1 | exact (stochastic) | -0.6580(2) | AFQMC Qin et al. (2016) | |
| best variational | -0.655 ○ (+3.0e-3) | VMC (HB) Zhou et al. (2026) | ||
| square 10x10, U = 8, n = 1 | exact (stochastic) | -0.5254(3) | AFQMC Qin et al. (2016) | |
| best variational | -0.52492 ○ (+4.8e-4) | VMC (transformer BF) Gu et al. (2026) | bytedance/laqx | |
| square 10x10, periodic/antiperiodic, U = 2, n = 1 | exact (stochastic) | -1.18505(4) | AFQMC Qin et al. (2016) | |
| square 10x10, periodic/antiperiodic, U = 4, n = 1 | exact (stochastic) | -0.8620(2) | AFQMC Qin et al. (2016) | |
| square 10x10, periodic/antiperiodic, U = 6, n = 1 | exact (stochastic) | -0.6576(2) | AFQMC Qin et al. (2016) | |
| square 10x10, periodic/antiperiodic, U = 8, n = 1 | exact (stochastic) | -0.5249(2) | AFQMC Qin et al. (2016) | |
| square 12x12, U = 4, n ≈ 0.8056 | best variational | -1.1061 ○ | VMC (tensor BF) Liang (2025) | |
| best projected | -1.1089(2) | CP-AFQMC Shi & Zhang (2013) | ||
| square 12x12, U = 2, n ≈ 0.8333 | best variational | -1.29500(3) | VMC (HFPS) Roth et al. (2025) | uses Quantax |
| square 12x12, U = 4, n ≈ 0.8333 | best variational | -1.0751(1) | VMC (HFPS) Roth et al. (2025) | uses Quantax |
| square 12x12, U = 6, n ≈ 0.8333 | best variational | -0.92516(5) | VMC (HFPS) Roth et al. (2025) | uses Quantax |
| square 12x12, U = 10, n ≈ 0.8333 | best variational | -0.76179(5) | VMC (HFPS) Roth et al. (2025) | uses Quantax |
| square 12x12, U = 15, n ≈ 0.8333 | best variational | -0.66690(5) | VMC (HFPS) Roth et al. (2025) | uses Quantax |
| square 12x12, U = -8, n = 0.875 | best variational | -4.0159 ○ | VMC (ACE) Gu et al. (2026) | uses LaQX |
| best projected | -4.0161 ○ | FN-GFMC (ACE) Gu et al. (2026) | ||
| square 12x12, U = -2, n = 0.875 | exact (stochastic) | -2.04160(5) | DQMC Gu et al. (2026) | |
| best variational | -2.0405 ○ (+1.1e-3) | VMC (ACE) Gu et al. (2026) | uses LaQX | |
| best projected | -2.0407 ○ (+9.0e-4) | FN-GFMC (ACE) Gu et al. (2026) | ||
| square 12x12, U = 8, n = 0.875 | best variational | -0.7567 ○ | VMC (transformer BF) Gu et al. (2026) | bytedance/laqx |
| square 12x12, U = 8, n = 0.875, t12 (t2 = -0.2) | best variational | -0.74425(1) | VMC (SBP) Rende et al. (2026) | |
| square 12x12, U = 2, n = 1 | exact (stochastic) | -1.17187(5) | AFQMC Qin et al. (2016) | |
| square 12x12, U = 4, n = 1 | exact (stochastic) | -0.8608(1) | AFQMC Qin et al. (2016) | |
| square 12x12, U = 6, n = 1 | exact (stochastic) | -0.6574(1) | AFQMC Qin et al. (2016) | |
| square 12x12, U = 8, n = 1 | exact (stochastic) | -0.5246(1) | AFQMC Qin et al. (2016) | |
| best variational | -0.5244 ○ (+1.8e-4) | VMC (transformer BF) Gu et al. (2026) | bytedance/laqx | |
| square 12x12, periodic/antiperiodic, U = 2, n = 1 | exact (stochastic) | -1.18133(2) | AFQMC Qin et al. (2016) | |
| square 12x12, periodic/antiperiodic, U = 4, n = 1 | exact (stochastic) | -0.8610(2) | AFQMC Qin et al. (2016) | |
| square 12x12, periodic/antiperiodic, U = 6, n = 1 | exact (stochastic) | -0.6574(1) | AFQMC Qin et al. (2016) | |
| square 12x12, periodic/antiperiodic, U = 8, n = 1 | exact (stochastic) | -0.5249(2) | AFQMC Qin et al. (2016) | |
| best variational | -0.5224 ○ (+2.5e-3) | VMC (tensor BF) Liang (2025) | ||
| square 14x14, U = 2, n = 1 | exact (stochastic) | -1.17337(3) | AFQMC Qin et al. (2016) | |
| square 14x14, U = 4, n = 1 | exact (stochastic) | -0.8606(1) | AFQMC Qin et al. (2016) | |
| square 14x14, U = 6, n = 1 | exact (stochastic) | -0.6569(1) | AFQMC Qin et al. (2016) | |
| square 14x14, U = 8, n = 1 | exact (stochastic) | -0.5247(2) | AFQMC Qin et al. (2016) | |
| square 14x14, periodic/antiperiodic, U = 2, n = 1 | exact (stochastic) | -1.17926(2) | AFQMC Qin et al. (2016) | |
| square 14x14, periodic/antiperiodic, U = 4, n = 1 | exact (stochastic) | -0.8607(2) | AFQMC Qin et al. (2016) | |
| square 14x14, periodic/antiperiodic, U = 6, n = 1 | exact (stochastic) | -0.6570(2) | AFQMC Qin et al. (2016) | |
| square 14x14, periodic/antiperiodic, U = 8, n = 1 | exact (stochastic) | -0.5246(2) | AFQMC Qin et al. (2016) | |
| square 16x16, open, U = 8, n = 0.875 | best variational | -0.728 ○ | VMC (ACE) Gu et al. (2026) | uses LaQX |
| best projected | -0.7288 ○ | FN-GFMC (ACE) Gu et al. (2026) | ||
| square 16x16, open, U = 8, n = 0.875, t12 (t2 = -0.2) | best variational | -0.7097(1) | VMC (tensor BF) Liang (2026) | |
| square 16x16, U = 8, n ≈ 0.8281 | best variational | -0.83788(6) | VAFQMC Wu et al. (2024) | |
| square 16x16, U = 8, n ≈ 0.8438 | best variational | -0.81198(6) | VAFQMC Wu et al. (2024) | |
| square 16x16, U = 8, n ≈ 0.8516 | best variational | -0.79896(5) | VAFQMC Wu et al. (2024) | |
| square 16x16, U = 8, n ≈ 0.8594 | best variational | -0.78530(6) | VAFQMC Wu et al. (2024) | |
| square 16x16, U = 8, n ≈ 0.8672 | best variational | -0.77196(6) | VAFQMC Wu et al. (2024) | |
| square 16x16, U = -8, n = 0.875 | exact (stochastic) | -4.01629(5) | AFQMC Wu et al. (2024) | run scripts |
| square 16x16, U = -4, n = 0.875 | exact (stochastic) | -2.60067(6) | AFQMC Wu et al. (2024) | run scripts |
| square 16x16, U = 4, n = 0.875 | best variational | -1.02916(2) | VAFQMC Wu et al. (2024) | |
| square 16x16, U = 6, n = 0.875 | best variational | -0.86592(2) | VAFQMC Wu et al. (2024) | |
| square 16x16, U = 8, n = 0.875 | best variational | -0.75865(3) | VAFQMC Wu et al. (2024) | |
| best projected | -0.7583 ○ | FN-GFMC (ACE) Gu et al. (2026) | ||
| square 16x16, U = 8, n = 0.875, t12 (t2 = -0.1) | best variational | -0.7370(1) | VMC (HFPS) Roth et al. (2025) | uses Quantax |
| square 16x16, U = 8, n = 0.875, t12 (t2 = -0.2) | best variational | -0.74411(1) | VMC (SBP) Rende et al. (2026) | |
| best projected | -0.744 ○ | FN-GFMC (ACE) Gu et al. (2026) | ||
| square 16x16, U = 8, n = 0.875, t12 (t2 = -0.3) | best variational | -0.7375(1) | VMC (HFPS) Roth et al. (2025) | uses Quantax |
| square 16x16, U = 8, n ≈ 0.8828 | best variational | -0.74455(3) | VAFQMC Wu et al. (2024) | |
| square 16x16, U = 4, n ≈ 0.8906 | best variational | -1.00849(2) | VAFQMC Wu et al. (2024) | |
| square 16x16, U = 6, n ≈ 0.8906 | best variational | -0.84058(2) | VAFQMC Wu et al. (2024) | |
| square 16x16, U = 8, n ≈ 0.8906 | best variational | -0.73038(5) | VAFQMC Wu et al. (2024) | |
| square 16x16, U = 8, n ≈ 0.8984 | best variational | -0.71584(5) | VAFQMC Wu et al. (2024) | |
| square 16x16, U = 8, n ≈ 0.9063 | best variational | -0.70135(3) | VAFQMC Wu et al. (2024) | |
| square 16x16, U = 8, n ≈ 0.9141 | best variational | -0.68678(5) | VAFQMC Wu et al. (2024) | |
| square 16x16, U = 8, n ≈ 0.9219 | best variational | -0.67215(5) | VAFQMC Wu et al. (2024) | |
| square 16x16, U = 8, n = 0.9375 | best variational | -0.64053(5) | VAFQMC Wu et al. (2024) | |
| square 16x16, U = -8, n = 1 | exact (stochastic) | -4.52446(6) | AFQMC Wu et al. (2024) | run scripts |
| square 16x16, U = -4, n = 1 | exact (stochastic) | -2.8605(1) | AFQMC Wu et al. (2024) | run scripts |
| square 16x16, U = 2, n = 1 | exact (stochastic) | -1.17420(2) | AFQMC Qin et al. (2016) | |
| square 16x16, U = 4, n = 1 | exact (stochastic) | -0.8605(2) | AFQMC Qin et al. (2016) | |
| square 16x16, U = 6, n = 1 | exact (stochastic) | -0.6570(1) | AFQMC Qin et al. (2016) | |
| square 16x16, U = 8, n = 1 | exact (stochastic) | -0.5243(1) | AFQMC Qin et al. (2016) | |
| square 16x16, periodic/antiperiodic, U = 8, n ≈ 0.8359 | best variational | -0.82333(3) | VAFQMC Wu et al. (2024) | |
| square 16x16, periodic/antiperiodic, U = 8, n ≈ 0.8359, t12 (t2 = -0.25) | best variational | -0.79937(4) | VMC (PP + RBM) Wu et al. (2024) | run script |
| square 16x16, periodic/antiperiodic, U = 8, n ≈ 0.8359, t12 (t2 = -0.25), UV1V2 | best variational | 1.25918(3) | VMC (PP + RBM) Wu et al. (2024) | run script |
| square 16x16, periodic/antiperiodic, U = 2, n = 1 | exact (stochastic) | -1.17798(2) | AFQMC Qin et al. (2016) | |
| square 16x16, periodic/antiperiodic, U = 4, n = 1 | exact (stochastic) | -0.8605(2) | AFQMC Qin et al. (2016) | |
| square 16x16, periodic/antiperiodic, U = 6, n = 1 | exact (stochastic) | -0.6571(2) | AFQMC Qin et al. (2016) | |
| square 16x16, periodic/antiperiodic, U = 8, n = 1 | exact (stochastic) | -0.5244(1) | AFQMC Qin et al. (2016) | |
| best variational | -0.52427(4) (+1.5e-4) | VAFQMC Wu et al. (2024) | ||
| square, 288 sites, U = 2, n ≈ 0.8333 | best variational | -1.29563(3) | VMC (HFPS) Roth et al. (2025) | uses Quantax |
| square, 288 sites, U = 4, n ≈ 0.8333 | best variational | -1.07256(5) | VMC (HFPS) Roth et al. (2025) | uses Quantax |
| square, 288 sites, U = 6, n ≈ 0.8333 | best variational | -0.91859(6) | VMC (HFPS) Roth et al. (2025) | uses Quantax |
| square 20x20, U = 8, n = 0.875, t12 (t2 = -0.2) | best variational | -0.74382(1) | VMC (SBP) Rende et al. (2026) | |
| square 24x24, U = 8, n = 0.875, t12 (t2 = -0.2) | best variational | -0.74397(1) | VMC (SBP) Rende et al. (2026) | |
| square 32x32, U = 8, n = 0.875 | best variational | -0.7501 ○ | VMC (SCALE) Gu et al. (2026) | uses LaQX |
| best projected | -0.7548 ○ | FN-GFMC (SCALE) Gu et al. (2026) | ||
| square, infinite lattice, U = 2, n = 0.8 | best extrapolated | -1.306(2) | CP-AFQMC LeBlanc et al. (2015) | |
| best approximate | -1.3062(4) | DMET LeBlanc et al. (2015) | ||
| square, infinite lattice, U = 2, n = 0.875 | best extrapolated | -1.270(2) | FN-DMC LeBlanc et al. (2015) | |
| best approximate | -1.2721(6) | DMET LeBlanc et al. (2015) | ||
| square, infinite lattice, U = 2, n = 1 | best extrapolated | -1.1763(2) | AFQMC LeBlanc et al. (2015) | |
| best approximate | -1.1764(3) | DMET LeBlanc et al. (2015) | ||
| square, infinite lattice, U = 4, n = 0.3 | best extrapolated | -0.8793(2) | CP-AFQMC Qin et al. (2016) | |
| square, infinite lattice, U = 4, n = 0.5 | best extrapolated | -1.141(2) | CP-AFQMC Qin et al. (2016) | |
| square, infinite lattice, U = 4, n = 0.6 | best extrapolated | -1.1845(5) | CP-AFQMC Qin et al. (2016) | |
| square, infinite lattice, U = 4, n = 0.75 | best extrapolated | -1.1491(2) | CP-AFQMC Qin et al. (2016) | |
| square, infinite lattice, U = 4, n = 0.8 | best extrapolated | -1.110(3) | CP-AFQMC LeBlanc et al. (2015) | |
| best approximate | -1.108(2) | DMET LeBlanc et al. (2015) | ||
| square, infinite lattice, U = 4, n = 0.875 | best extrapolated | -1.032(1) | CP-AFQMC Qin et al. (2016) | |
| best approximate | -1.031(3) | DMET LeBlanc et al. (2015) | ||
| square, infinite lattice, U = 4, n = 1 | best extrapolated | -0.8605(5) | DMRG LeBlanc et al. (2015) | |
| best approximate | -0.861 ○ | UCCSDTQ* LeBlanc et al. (2015) | ||
| square, infinite lattice, U = 6, n = 0.8 | best extrapolated | -0.967(1) | FN-DMC LeBlanc et al. (2015) | |
| best approximate | -0.977(4) | DMET LeBlanc et al. (2015) | ||
| square, infinite lattice, U = 6, n = 0.875 | best extrapolated | -0.854(2) | FN-DMC LeBlanc et al. (2015) | |
| best approximate | -0.8767(7) | DMET Zheng et al. (2017) | ||
| square, infinite lattice, U = 6, n = 1 | best extrapolated | -0.6568(3) | AFQMC LeBlanc et al. (2015) | |
| best approximate | -0.6582 ○ | UCCSDTQ* LeBlanc et al. (2015) | ||
| square, infinite lattice, U = 8, n = 0.3 | best extrapolated | -0.8534(1) | CP-AFQMC Qin et al. (2016) | |
| square, infinite lattice, U = 8, n = 0.5 | best extrapolated | -1.066(2) | CP-AFQMC Qin et al. (2016) | |
| square, infinite lattice, U = 8, n = 0.6 | best extrapolated | -1.0729(1) | CP-AFQMC Qin et al. (2016) | |
| square, infinite lattice, U = 8, n = 0.75 | best extrapolated | -0.9666(4) | CP-AFQMC Qin et al. (2016) | |
| square, infinite lattice, U = 8, n = 0.8 | best extrapolated | -0.877(1) | FN-DMC LeBlanc et al. (2015) | |
| best approximate | -0.88(3) | DMET LeBlanc et al. (2015) | ||
| square, infinite lattice, U = 8, n = 0.875 | best extrapolated | -0.773(1) | CP-AFQMC Qin et al. (2016) | |
| best approximate | -0.77063(1) | DMET Zheng et al. (2017) | ||
| square, infinite lattice, U = 8, n = 1 | best extrapolated | -0.5247(2) | AFQMC LeBlanc et al. (2015) | |
| best approximate | -0.5255 ○ | UCCSDTQ* LeBlanc et al. (2015) | ||
| square, infinite lattice, U = 12, n = 0.875 | best approximate | -0.6531(1) | DMET Zheng et al. (2017) | |
| square, infinite lattice, U = 12, n = 1 | best extrapolated | -0.3693(2) | AFQMC LeBlanc et al. (2015) | |
| best approximate | -0.3696 ○ | UCCSDTQ* LeBlanc et al. (2015) |
E/site
| instance | value | energy | method | code |
|---|---|---|---|---|
| chain, 32 sites, V = 1, n = 0.5 | exact | -0.498339 | ED run script, Wu et al. (2024) | |
| best variational | -0.498339 (+<1e-6) | DMRG Wu et al. (2024) | run script | |
| chain, 32 sites, V = 2, n = 0.5 | exact | -0.3852719 | ED run script, Wu et al. (2024) | |
| best variational | -0.3852719 ○ (+<1e-7) | DMRG Wu et al. (2024) | run script | |
| chain, 32 sites, V = 4, n = 0.5 | exact | -0.2344426 | ED run script, Wu et al. (2024) | |
| best variational | -0.2344426 ○ (+<1e-7) | DMRG Wu et al. (2024) | run script | |
| square 4x4, V = 0.01, n = 0.3125 | exact | -0.7487516 | ED Wu et al. (2024) | run script |
| best variational | -0.7487521(6) (−4.2e-7) | VMC (SJ + RBM) Wu et al. (2024) | run script | |
| square 4x4, V = 0.1, n = 0.3125 | exact | -0.7376632 | ED Wu et al. (2024) | run script |
| best variational | -0.7376632(3) (−<1e-7) | VMC (SJB + RBM) Wu et al. (2024) | run script | |
| square 4x4, V = 1, n = 0.3125 | exact | -0.6400407 | ED Wu et al. (2024) | run script |
| best variational | -0.6400407 (+<1e-7) | DMRG Wu et al. (2024) | run script | |
| square 4x4, V = 10, n = 0.3125 | exact | -0.2532535 | ED Wu et al. (2024) | run script |
| best variational | -0.2532535 (−<1e-7) | DMRG Wu et al. (2024) | run script | |
| square 6x6, V = 0.01, n ≈ 0.3611 | exact | -0.7759279 | ED Humeniuk et al. (2022) | uses QuSpin |
| best variational | -0.7759280(1) (−<1e-7) | VMC (SJ + RBM) Wu et al. (2024) | run script | |
| square 6x6, V = 0.1, n ≈ 0.3611 | exact | -0.7594596 | ED Humeniuk et al. (2022) | uses QuSpin |
| best variational | -0.759460(1) (−2.4e-7) | VMC (SJ + RBM) Wu et al. (2024) | run script | |
| square 6x6, V = 1, n ≈ 0.3611 | exact | -0.6132603 | ED Humeniuk et al. (2022) | uses QuSpin |
| best variational | -0.6132263 (+3.4e-5) | DMRG Wu et al. (2024) | run script | |
| square 6x6, V = 10, n ≈ 0.3611 | exact | -0.220223 | ED Humeniuk et al. (2022) | uses QuSpin |
| best variational | -0.2201239 (+9.9e-5) | DMRG Wu et al. (2024) | run script | |
| square 8x8, V = 1, n = 0.5 | exact (stochastic) | -0.4607(4) | CT-QMC Wu et al. (2024) | run script |
| best variational | -0.4590635 (+1.6e-3) | DMRG Wu et al. (2024) | run script | |
| square 8x8, V = 2, n = 0.5 | exact (stochastic) | -0.292(1) | CT-QMC Wu et al. (2024) | run script |
| best variational | -0.2911555 (+1.2e-3) | DMRG Wu et al. (2024) | run script | |
| square 8x8, V = 4, n = 0.5 | best variational | -0.1601344 | DMRG Wu et al. (2024) | run script |
E/site
| instance | value | energy | method | code |
|---|---|---|---|---|
| honeycomb-flake24, 24 sites, open, J = 0.333333, 1 hole | exact | -0.3441916 | ED Cui et al. (2024) | |
| best variational | -0.3366806 ○ (+7.5e-3) | VMC (phase string) Cui et al. (2024) | ||
| honeycomb-flake24, 24 sites, open, J = 0.333333, 2 holes | exact | -0.4143583 | ED Cui et al. (2024) | |
| best variational | -0.398 ○ (+1.6e-2) | VMC (phase string) Cui et al. (2024) | ||
| ladder 2x11, J = 0.5, 2 holes | exact | -0.5908993 | ED Chiappa et al. (2001) | |
| ladder 2x13, J = 0.5, 2 holes | exact | -0.5733713 | ED Chiappa et al. (2001) | |
| ladder 3x6, antiperiodic legs, J = 0.5, 1 hole | exact | -0.5781986 | ED Rice et al. (1997) | |
| ladder 3x6, antiperiodic legs, J = 0.5, 2 holes | exact | -0.6559335 | ED Rice et al. (1997) | |
| rectangular 6x16, periodic/open, J = 0.35, 8 holes | best variational | -0.5470833 ○ | DMRG McCulloch et al. (2001) | |
| best extrapolated | -0.5484(5) | DMRG McCulloch et al. (2001) | ||
| square 4x4, open, J = 0.4, 2 holes | exact | -0.5643668 | ED Dong et al. (2020) | |
| best variational | -0.56428(2) (+8.7e-5) | VMC (fPEPS) Dong et al. (2020) | ||
| square, 18 sites, J = 0.4, 2 holes | exact | -0.6590431 | ED Becca et al. (2001) | |
| best variational | -0.65904 (+3.1e-6) | VMC (Gutzwiller) Becca et al. (2001) | ||
| square, 26 sites, J = 0.3, 2 holes | exact | -0.5009718 | ED Lugas et al. (2006) | |
| best variational | -0.48476(1) (+1.6e-2) | VMC (Pfaffian) Lugas et al. (2006) | ||
| best projected | -0.49445(2) (+6.5e-3) | FN-GFMC (Pfaffian) Lugas et al. (2006) | ||
| square, 26 sites, J = 0.4, 2 holes | exact | -0.5945167 | ED Lugas et al. (2006) | |
| best variational | -0.57978(1) (+1.5e-2) | VMC (Pfaffian) Lugas et al. (2006) | ||
| best projected | -0.58881(2) (+5.7e-3) | FN-GFMC (Pfaffian) Lugas et al. (2006) | ||
| square, 26 sites, J = 0.5, 2 holes | exact | -0.6894488 | ED Lugas et al. (2006) | |
| best variational | -0.67568(1) (+1.4e-2) | VMC (Pfaffian) Lugas et al. (2006) | ||
| best projected | -0.68434(3) (+5.1e-3) | FN-GFMC (Pfaffian) Lugas et al. (2006) | ||
| square, 26 sites, J = 0.6, 2 holes | exact | -0.7853745 | ED Lugas et al. (2006) | |
| best variational | -0.77228(1) (+1.3e-2) | VMC (Pfaffian) Lugas et al. (2006) | ||
| best projected | -0.78062(3) (+4.8e-3) | FN-GFMC (Pfaffian) Lugas et al. (2006) | ||
| square, 26 sites, J = 0.8, 2 holes | exact | -0.9793539 | ED Lugas et al. (2006) | |
| best variational | -0.96706(1) (+1.2e-2) | VMC (Pfaffian) Lugas et al. (2006) | ||
| best projected | -0.97505(3) (+4.3e-3) | FN-GFMC (Pfaffian) Lugas et al. (2006) | ||
| square, 26 sites, J = 1, 2 holes | exact | -1.1753789 | ED Lugas et al. (2006) | |
| best variational | -1.16352(1) (+1.2e-2) | VMC (Pfaffian) Lugas et al. (2006) | ||
| best projected | -1.17136(2) (+4.0e-3) | FN-GFMC (Pfaffian) Lugas et al. (2006) | ||
| square, 26 sites, J = 0.3, 4 holes | exact | -0.6426164 | ED D. Poilblanc (unpublished), quoted in Sorella (2001) | |
| best variational | -0.6387(1) (+3.9e-3) | VMC (Gutzwiller) Sorella (2001) | ||
| best projected | -0.6371(1) (+5.5e-3) | FN-GFMC (Gutzwiller) Sorella (2001) | ||
| square, 26 sites, J = 0.4, 4 holes | exact | -0.7143669 | ED Lugas et al. (2006) | |
| best variational | -0.68946(1) (+2.5e-2) | VMC (Pfaffian) Lugas et al. (2006) | ||
| best projected | -0.70292(2) (+1.1e-2) | FN-GFMC (Pfaffian) Lugas et al. (2006) | ||
| square, 26 sites, J = 0.5, 4 holes | exact | -0.7881242 | ED D. Poilblanc (unpublished), quoted in Sorella (2001) | |
| best variational | -0.7855(1) (+2.6e-3) | VMC (Gutzwiller) Sorella (2001) | ||
| best projected | -0.7841(1) (+4.0e-3) | FN-GFMC (Gutzwiller) Sorella (2001) | ||
| square, 26 sites, J = 0.6, 4 holes | exact | -0.8633713 | ED Lugas et al. (2006) | |
| best variational | -0.84170(1) (+2.2e-2) | VMC (Pfaffian) Lugas et al. (2006) | ||
| best projected | -0.85348(3) (+9.9e-3) | FN-GFMC (Pfaffian) Lugas et al. (2006) | ||
| square, 26 sites, J = 0.8, 4 holes | exact | -1.0173302 | ED Lugas et al. (2006) | |
| best variational | -0.99709(1) (+2.0e-2) | VMC (Pfaffian) Lugas et al. (2006) | ||
| best projected | -1.00806(2) (+9.3e-3) | FN-GFMC (Pfaffian) Lugas et al. (2006) | ||
| square, 26 sites, J = 1, 4 holes | exact | -1.1749259 | ED D. Poilblanc (unpublished), quoted in Sorella (2001) | |
| best variational | -1.1724(1) (+2.5e-3) | VMC (Gutzwiller) Sorella (2001) | ||
| best projected | -1.1706(1) (+4.3e-3) | FN-GFMC (Gutzwiller) Sorella (2001) | ||
| square, 32 sites, J = 0.3, t′ = 0.3, t″ = -0.2, 1 hole | exact | -0.4348005 | ED Leung (2006) | |
| square, 32 sites, J = 0.3, t′ = 0.3, t″ = -0.2, 2 holes | exact | -0.5188028 | ED Leung (2006) | |
| square, 32 sites, J = 0.3, t′ = 0.3, t″ = -0.2, 4 holes | exact | -0.6394265 | ED Leung (2006) | |
| square 8x8, open, J = 0.333333, 1 hole | best variational | -0.380099 ○ | DMRG Cui et al. (2026) | |
| square 8x8, J = 0.2, 14 holes | best variational | -0.685(1) | Power-Lanczos Shih et al. (2001) | |
| best extrapolated | -0.690(5) | Power-Lanczos Shih et al. (2001) | ||
| square 8x8, J = 0.4, 8 holes | best variational | -0.64266(8) | VMC (Gutzwiller) Sorella et al. (2002) | |
| best projected | -0.66672(6) | FN-GFMC (Gutzwiller) Sorella et al. (2002) | ||
| best extrapolated | -0.671(1) | VMC (Gutzwiller) Sorella et al. (2002) |
E/N (Pauli)
| instance | value | energy | method | code |
|---|---|---|---|---|
| chain, 10 sites, open, h = 1 | exact | -1.238149 | ED Wu et al. (2024) | run script |
| best variational | -1.238149 (−<1e-6) | DMRG Wu et al. (2024) | run script | |
| chain, 10 sites, h = 1 | exact | -1.2784906 | Exact solution Wu et al. (2024) | run script |
| best variational | -1.2784906 (−<1e-7) | DMRG Wu et al. (2024) | run script | |
| chain, 32 sites, h = 0.5 | exact | -1.0635444 | Exact solution Wu et al. (2024) | run script |
| best variational | -1.0635444 (+<1e-7) | DMRG Wu et al. (2024) | run script | |
| chain, 32 sites, h = 1 | exact | -1.273751 | Exact solution Wu et al. (2024) | run script |
| best variational | -1.273751 (+<1e-6) | DMRG Wu et al. (2024) | run script | |
| square 6x6, h = 3 | exact | -3.2009085 | ED Wu et al. (2024) | run script |
| best variational | -3.2009084 (+1.6e-7) | DMRG Wu et al. (2024) | run script | |
| square 8x8, open, h = 3 | best variational | -3.1641828 ○ | DMRG Dao (2026) | |
| square 10x10, open, h = 3 | best variational | -3.16977 ○ | DMRG Dao (2026) | |
| square 12x12, open, h = 2 | best variational | -2.4096026 ○ | DMRG Hibat-Allah et al. (2020) | |
| square 12x12, open, h = 3 | best variational | -3.1739018(2) | VMC (2D RNN) Hibat-Allah et al. (2020) | run script |
| square 12x12, open, h = 4 | best variational | -4.1217979 ○ | DMRG Hibat-Allah et al. (2020) | |
| square 20x20, open, h = 3 | best variational | -3.18197 ○ | DMRG Ganahl et al. (2023) | |
| square, infinite lattice, h = 3.04438 | best variational | -3.2342607 ○ | iPEPS Rader & Läuchli (2018) | |
| best extrapolated | -3.2342623 ○ | iPEPS Rader & Läuchli (2018) |
E (total)
| instance | value | energy | method | code |
|---|---|---|---|---|
| SB-DMFT-MI-HF, size 10 | exact | -34.5805205 | ED run script, Wu et al. (2024) | |
| best variational | -34.5805205 (+<1e-7) | DMRG run script, Wu et al. (2024) | ||
| SB-DMFT-MT-AHF, size 9 | exact | -13.8893249 | ED run script, Wu et al. (2024) | |
| best variational | -13.8893249 (+<1e-7) | DMRG Wu et al. (2024) | run script | |
| SB-DMFT-MT-HF, size 9 | exact | -23.3009472 | ED run script, Wu et al. (2024) | |
| best variational | -23.3009472 (+<1e-7) | DMRG Wu et al. (2024) | run script | |
| SB-IMP, size 9 | exact | -10.4871285 | ED run script, Wu et al. (2024) | |
| best variational | -10.4871285 (+<1e-7) | DMRG Wu et al. (2024) | run script | |
| TB-DMFT, size 9 | best variational | -112.9042852 | DMRG Wu et al. (2024) | run script |
| TB-DMFT, size 19 | best variational | -224.3360713 | DMRG Wu et al. (2024) | run script |
| TB-DMFT, size 119 | best variational | -1307.5645417 | DMRG Wu et al. (2024) | run script |
| TB-DMFT, size 309 | best variational | -3378.6762368 | DMRG Wu et al. (2024) | run script |
| TB-DMFT-SOC, size 9 | best variational | -112.8717703 | DMRG Wu et al. (2024) | run script |
| TB-DMFT-SOC, size 19 | best variational | -219.766911 | DMRG Wu et al. (2024) | run script |
| TB-DMFT-SOC, size 119 | best variational | -1305.3902308 | DMRG Wu et al. (2024) | run script |
| TB-DMFT-SOC, size 309 | best variational | -3351.7573592 | DMRG Wu et al. (2024) | run script |
E/N (Ha)
| instance | value | energy | method | code |
|---|---|---|---|---|
| bcc cell, N = 27, rs = 0.2, polarized | best variational | 44.18067(2) | VMC (FermiNet) Cassella et al. (2023) | uses FermiNet |
| best projected | 44.18224(1) | FN-DMC (SJB) Cassella et al. (2023) | uses CASINO | |
| bcc cell, N = 27, rs = 0.5, polarized | best variational | 6.318704(2) | VMC (FermiNet) Cassella et al. (2023) | uses FermiNet |
| best projected | 6.319754(7) | FN-DMC (SJB) Cassella et al. (2023) | uses CASINO | |
| bcc cell, N = 27, rs = 1, polarized | best variational | 1.261547(1) | VMC (FermiNet) Cassella et al. (2023) | uses FermiNet |
| best projected | 1.261975(2) | FN-DMC (SJB) Cassella et al. (2023) | uses CASINO | |
| bcc cell, N = 27, rs = 10, polarized | best variational | -0.050320(5) | VMC (FermiNet) Cassella et al. (2023) | uses FermiNet |
| best projected | -0.0503349(8) | FN-DMC (SJB) Cassella et al. (2023) | uses CASINO | |
| bcc cell, N = 27, rs = 2, polarized | best variational | 0.152086(1) | VMC (FermiNet) Cassella et al. (2023) | uses FermiNet |
| best projected | 0.152077(4) | FN-DMC (SJB) Cassella et al. (2023) | uses CASINO | |
| bcc cell, N = 27, rs = 3, polarized | best variational | -0.006961(2) | VMC (FermiNet) Cassella et al. (2023) | uses FermiNet |
| best projected | -0.006966(2) | FN-DMC (SJB) Cassella et al. (2023) | uses CASINO | |
| bcc cell, N = 27, rs = 30, polarized | best variational | -0.022541(5) | VMC (FermiNet) Cassella et al. (2023) | uses FermiNet |
| best projected | -0.0225620(2) | FN-DMC (SJB) Cassella et al. (2023) | uses CASINO | |
| bcc cell, N = 27, rs = 5, polarized | best variational | -0.057691(4) | VMC (FermiNet) Cassella et al. (2023) | uses FermiNet |
| best projected | -0.057702(1) | FN-DMC (SJB) Cassella et al. (2023) | uses CASINO | |
| bcc cell, N = 27, rs = 50, polarized | best variational | -0.014478(4) | VMC (FermiNet) Cassella et al. (2023) | uses FermiNet |
| best projected | -0.0144918(1) | FN-DMC (SJB) Cassella et al. (2023) | uses CASINO | |
| bcc cell, N = 27, rs = 70, polarized | best variational | -0.010693(3) | VMC (FermiNet) Cassella et al. (2023) | uses FermiNet |
| best projected | -0.0107118(1) | FN-DMC (SJB) Cassella et al. (2023) | uses CASINO | |
| bcc cell, N = 27, rs = 90, polarized | best variational | -0.008496(1) | VMC (FermiNet) Cassella et al. (2023) | uses FermiNet |
| best projected | -0.0085137(2) | FN-DMC (SJB) Cassella et al. (2023) | uses CASINO | |
| bcc cell, N = 64, rs = 100, unpolarized | best variational | -0.00767129(8) | VMC (SJ) Azadi & Drummond (2022) | uses CASINO |
| best projected | -0.0076974(1) | FN-DMC (SJ) Azadi & Drummond (2022) | uses CASINO | |
| bcc cell, N = 64, rs = 125, unpolarized | best variational | -0.0062378(1) | VMC (SJ) Azadi & Drummond (2022) | uses CASINO |
| best projected | -0.0062597(1) | FN-DMC (SJ) Azadi & Drummond (2022) | uses CASINO | |
| bcc cell, N = 64, rs = 80, unpolarized | best variational | -0.0093993(3) | VMC (SJ) Azadi & Drummond (2022) | uses CASINO |
| best projected | -0.0094484(2) | FN-DMC (SJ) Azadi & Drummond (2022) | uses CASINO | |
| bcc cell, N = 64, rs = 90, unpolarized | best variational | -0.00844873(6) | VMC (SJ) Azadi & Drummond (2022) | uses CASINO |
| best projected | -0.0084813(1) | FN-DMC (SJ) Azadi & Drummond (2022) | uses CASINO | |
| bcc cell, N = 64, rs = 100, polarized | best variational | -0.0076803(2) | VMC (SJB) Azadi & Drummond (2022) | uses CASINO |
| best projected | -0.00769619(9) | FN-DMC Azadi & Drummond (2022) | uses CASINO | |
| bcc cell, N = 64, rs = 125, polarized | best variational | -0.00624309(3) | VMC (SJ) Azadi & Drummond (2022) | uses CASINO |
| best projected | -0.0062592(1) | FN-DMC (SJ) Azadi & Drummond (2022) | uses CASINO | |
| bcc cell, N = 64, rs = 80, polarized | best variational | -0.0094086(1) | VMC (SJ) Azadi & Drummond (2022) | uses CASINO |
| best projected | -0.00944340(6) | FN-DMC (SJ) Azadi & Drummond (2022) | uses CASINO | |
| bcc cell, N = 64, rs = 90, polarized | best variational | -0.00845026(5) | VMC (SJ) Azadi & Drummond (2022) | uses CASINO |
| best projected | -0.00847899(9) | FN-DMC (SJ) Azadi & Drummond (2022) | uses CASINO | |
| bcc cell, N = 216, rs = 100, unpolarized | best variational | -0.00765917(9) | VMC (SJ) Azadi & Drummond (2022) | uses CASINO |
| best projected | -0.0076828(1) | FN-DMC (SJ) Azadi & Drummond (2022) | uses CASINO | |
| bcc cell, N = 216, rs = 125, unpolarized | best variational | -0.00623503(4) | VMC (SJ) Azadi & Drummond (2022) | uses CASINO |
| best projected | -0.00624956(9) | FN-DMC (SJ) Azadi & Drummond (2022) | uses CASINO | |
| bcc cell, N = 216, rs = 80, unpolarized | best variational | -0.0093763(1) | VMC (SJ) Azadi & Drummond (2022) | uses CASINO |
| best projected | -0.0094299(3) | FN-DMC (SJ) Azadi & Drummond (2022) | uses CASINO | |
| bcc cell, N = 216, rs = 90, unpolarized | best variational | -0.00843572(7) | VMC (SJ) Azadi & Drummond (2022) | uses CASINO |
| best projected | -0.0084658(1) | FN-DMC (SJ) Azadi & Drummond (2022) | uses CASINO | |
| bcc cell, N = 216, rs = 100, polarized | best variational | -0.0076584(2) | VMC (SJ) Azadi & Drummond (2022) | uses CASINO |
| best projected | -0.0076816(4) | FN-DMC (SJ) Azadi & Drummond (2022) | uses CASINO | |
| bcc cell, N = 216, rs = 125, polarized | best variational | -0.00623504(4) | VMC (SJ) Azadi & Drummond (2022) | uses CASINO |
| best projected | -0.00624968(7) | FN-DMC (SJ) Azadi & Drummond (2022) | uses CASINO | |
| bcc cell, N = 216, rs = 80, polarized | best variational | -0.00939055(8) | VMC (SJ) Azadi & Drummond (2022) | uses CASINO |
| best projected | -0.0094277(1) | FN-DMC (SJ) Azadi & Drummond (2022) | uses CASINO | |
| bcc cell, N = 216, rs = 90, polarized | best variational | -0.00843484(6) | VMC (SJ) Azadi & Drummond (2022) | uses CASINO |
| best projected | -0.0084646(1) | FN-DMC (SJ) Azadi & Drummond (2022) | uses CASINO | |
| bcc cell, N = 512, rs = 100, unpolarized | best variational | -0.00765566(9) | VMC (SJ) Azadi & Drummond (2022) | uses CASINO |
| best projected | -0.0076790(1) | FN-DMC (SJ) Azadi & Drummond (2022) | uses CASINO | |
| bcc cell, N = 512, rs = 125, unpolarized | best variational | -0.00623185(4) | VMC (SJ) Azadi & Drummond (2022) | uses CASINO |
| best projected | -0.00624652(5) | FN-DMC (SJ) Azadi & Drummond (2022) | uses CASINO | |
| bcc cell, N = 512, rs = 80, unpolarized | best variational | -0.0093843(1) | VMC (SJ) Azadi & Drummond (2022) | uses CASINO |
| best projected | -0.0094232(2) | FN-DMC (SJ) Azadi & Drummond (2022) | uses CASINO | |
| bcc cell, N = 512, rs = 90, unpolarized | best variational | -0.00843112(8) | VMC (SJ) Azadi & Drummond (2022) | uses CASINO |
| best projected | -0.0084611(1) | FN-DMC (SJ) Azadi & Drummond (2022) | uses CASINO | |
| bcc cell, N = 512, rs = 100, polarized | best variational | -0.0076578(4) | VMC (SJ) Azadi & Drummond (2022) | uses CASINO |
| best projected | -0.0076795(1) | FN-DMC (SJ) Azadi & Drummond (2022) | uses CASINO | |
| bcc cell, N = 512, rs = 125, polarized | best variational | -0.00623263(4) | VMC (SJ) Azadi & Drummond (2022) | uses CASINO |
| best projected | -0.00624666(6) | FN-DMC (SJ) Azadi & Drummond (2022) | uses CASINO | |
| bcc cell, N = 512, rs = 80, polarized | best variational | -0.00938865(8) | VMC (SJ) Azadi & Drummond (2022) | uses CASINO |
| best projected | -0.00942260(9) | FN-DMC (SJ) Azadi & Drummond (2022) | uses CASINO | |
| bcc cell, N = 512, rs = 90, polarized | best variational | -0.00843318(7) | VMC (SJ) Azadi & Drummond (2022) | uses CASINO |
| best projected | -0.0084605(1) | FN-DMC (SJ) Azadi & Drummond (2022) | uses CASINO | |
| fcc cell, N = 54, rs = 2, unpolarized | best projected | 0.00426(2) | FN-DMC Gurtubay et al. (2010) | uses CASINO 2.1.400 |
| fcc cell, N = 102, rs = 2, unpolarized | best projected | 0.00081(2) | FN-DMC Gurtubay et al. (2010) | uses CASINO 2.1.400 |
| fcc cell, N = 130, rs = 100, unpolarized | best variational | -0.0076434(2) | VMC (SJB) Azadi & Drummond (2022) | |
| best projected | -0.00767914(4) | FN-DMC (SJB) Azadi & Drummond (2022) | ||
| fcc cell, N = 130, rs = 30, unpolarized | best variational | -0.0225999(8) | VMC (SJB) Azadi & Drummond (2022) | |
| best projected | -0.0226541(1) | FN-DMC (SJB) Azadi & Drummond (2022) | ||
| fcc cell, N = 130, rs = 40, unpolarized | best variational | -0.0175766(8) | VMC (SJB) Azadi & Drummond (2022) | |
| best projected | -0.01762938(7) | FN-DMC (SJB) Azadi & Drummond (2022) | ||
| fcc cell, N = 130, rs = 50, unpolarized | best variational | -0.0143571(4) | VMC (SJ) Azadi & Drummond (2022) | |
| best projected | -0.0144253(2) | FN-DMC (SJ) Azadi & Drummond (2022) | ||
| fcc cell, N = 130, rs = 60, unpolarized | best variational | -0.0122189(4) | VMC (SJB) Azadi & Drummond (2022) | |
| best projected | -0.01226761(6) | FN-DMC (SJB) Azadi & Drummond (2022) | ||
| fcc cell, N = 130, rs = 70, unpolarized | best variational | -0.0105888(4) | VMC (SJ) Azadi & Drummond (2022) | |
| best projected | -0.0106437(1) | FN-DMC (SJ) Azadi & Drummond (2022) | ||
| fcc cell, N = 130, rs = 80, unpolarized | best variational | -0.0093925(4) | VMC (SJB) Azadi & Drummond (2022) | |
| best projected | -0.00943534(6) | FN-DMC (SJB) Azadi & Drummond (2022) | ||
| fcc cell, N = 136, rs = 30, N↑ = 85, N↓ = 51 | best variational | -0.0224390(7) | VMC (SJ) Azadi & Drummond (2022) | |
| best projected | -0.0225752(2) | FN-DMC (SJ) Azadi & Drummond (2022) | ||
| fcc cell, N = 136, rs = 40, N↑ = 85, N↓ = 51 | best variational | -0.0174776(5) | VMC (SJ) Azadi & Drummond (2022) | |
| best projected | -0.0175820(1) | FN-DMC (SJ) Azadi & Drummond (2022) | ||
| fcc cell, N = 136, rs = 50, N↑ = 85, N↓ = 51 | best variational | -0.0143356(4) | VMC (SJ) Azadi & Drummond (2022) | |
| best projected | -0.0144239(1) | FN-DMC (SJ) Azadi & Drummond (2022) | ||
| fcc cell, N = 136, rs = 60, N↑ = 85, N↓ = 51 | best variational | -0.0121693(5) | VMC (SJ) Azadi & Drummond (2022) | |
| best projected | -0.01224281(8) | FN-DMC (SJ) Azadi & Drummond (2022) | ||
| fcc cell, N = 136, rs = 70, N↑ = 85, N↓ = 51 | best variational | -0.0105784(4) | VMC (SJ) Azadi & Drummond (2022) | |
| best projected | -0.01064357(8) | FN-DMC (SJ) Azadi & Drummond (2022) | ||
| fcc cell, N = 136, rs = 80, N↑ = 85, N↓ = 51 | best variational | -0.0093583(4) | VMC (SJ) Azadi & Drummond (2022) | |
| best projected | -0.00941978(8) | FN-DMC (SJ) Azadi & Drummond (2022) | ||
| fcc cell, N = 137, rs = 100, polarized | best variational | -0.0076530(2) | VMC (SJB) Azadi & Drummond (2022) | |
| best projected | -0.00767943(5) | FN-DMC (SJB) Azadi & Drummond (2022) | ||
| fcc cell, N = 137, rs = 30, polarized | best variational | -0.0224803(7) | VMC (SJB) Azadi & Drummond (2022) | |
| best projected | -0.02251468(9) | FN-DMC (SJB) Azadi & Drummond (2022) | ||
| fcc cell, N = 137, rs = 40, polarized | best variational | -0.0175368(5) | VMC (SJB) Azadi & Drummond (2022) | |
| best projected | -0.01757424(6) | FN-DMC (SJB) Azadi & Drummond (2022) | ||
| fcc cell, N = 137, rs = 50, polarized | best variational | -0.0143754(4) | VMC (SJ) Azadi & Drummond (2022) | |
| best projected | -0.0144184(3) | FN-DMC (SJ) Azadi & Drummond (2022) | ||
| fcc cell, N = 137, rs = 60, polarized | best variational | -0.0122233(3) | VMC (SJB) Azadi & Drummond (2022) | |
| best projected | -0.01225634(5) | FN-DMC (SJB) Azadi & Drummond (2022) | ||
| fcc cell, N = 137, rs = 70, polarized | best variational | -0.0106119(3) | VMC (SJ) Azadi & Drummond (2022) | |
| best projected | -0.0106489(1) | FN-DMC (SJ) Azadi & Drummond (2022) | ||
| fcc cell, N = 137, rs = 80, polarized | best variational | -0.0094043(2) | VMC (SJB) Azadi & Drummond (2022) | |
| best projected | -0.00943313(4) | FN-DMC (SJB) Azadi & Drummond (2022) | ||
| fcc cell, N = 140, rs = 100, N↑ = 105, N↓ = 35 | best variational | -0.0076402(2) | VMC (SJB) Azadi & Drummond (2022) | |
| best projected | -0.00767946(6) | FN-DMC (SJB) Azadi & Drummond (2022) | ||
| fcc cell, N = 140, rs = 30, N↑ = 105, N↓ = 35 | best variational | -0.0225221(6) | VMC (SJB) Azadi & Drummond (2022) | |
| best projected | -0.02262347(5) | FN-DMC (SJB) Azadi & Drummond (2022) | ||
| fcc cell, N = 140, rs = 40, N↑ = 105, N↓ = 35 | best variational | -0.0175382(4) | VMC (SJB) Azadi & Drummond (2022) | |
| best projected | -0.01761925(4) | FN-DMC (SJB) Azadi & Drummond (2022) | ||
| fcc cell, N = 140, rs = 50, N↑ = 105, N↓ = 35 | best variational | -0.0143415(4) | VMC (SJ) Azadi & Drummond (2022) | |
| best projected | -0.0144259(2) | FN-DMC (SJ) Azadi & Drummond (2022) | ||
| fcc cell, N = 140, rs = 60, N↑ = 105, N↓ = 35 | best variational | -0.0122063(3) | VMC (SJB) Azadi & Drummond (2022) | |
| best projected | -0.01226566(4) | FN-DMC (SJB) Azadi & Drummond (2022) | ||
| fcc cell, N = 140, rs = 70, N↑ = 105, N↓ = 35 | best variational | -0.0105822(3) | VMC (SJ) Azadi & Drummond (2022) | |
| best projected | -0.0106464(1) | FN-DMC (SJ) Azadi & Drummond (2022) | ||
| fcc cell, N = 140, rs = 80, N↑ = 105, N↓ = 35 | best variational | -0.0093864(3) | VMC (SJB) Azadi & Drummond (2022) | |
| best projected | -0.00943526(5) | FN-DMC (SJB) Azadi & Drummond (2022) | ||
| fcc cell, N = 152, rs = 30, N↑ = 133, N↓ = 19 | best variational | -0.0224543(6) | VMC (SJ) Azadi & Drummond (2022) | |
| best projected | -0.0225363(3) | FN-DMC (SJ) Azadi & Drummond (2022) | ||
| fcc cell, N = 152, rs = 40, N↑ = 133, N↓ = 19 | best variational | -0.0175016(4) | VMC (SJ) Azadi & Drummond (2022) | |
| best projected | -0.0175731(1) | FN-DMC (SJ) Azadi & Drummond (2022) | ||
| fcc cell, N = 152, rs = 50, N↑ = 133, N↓ = 19 | best variational | -0.0143617(3) | VMC (SJ) Azadi & Drummond (2022) | |
| best projected | -0.0144243(1) | FN-DMC (SJ) Azadi & Drummond (2022) | ||
| fcc cell, N = 152, rs = 60, N↑ = 133, N↓ = 19 | best variational | -0.0121888(3) | VMC (SJ) Azadi & Drummond (2022) | |
| best projected | -0.0122460(1) | FN-DMC (SJ) Azadi & Drummond (2022) | ||
| fcc cell, N = 152, rs = 70, N↑ = 133, N↓ = 19 | best variational | -0.0105960(2) | VMC (SJ) Azadi & Drummond (2022) | |
| best projected | -0.0106479(1) | FN-DMC (SJ) Azadi & Drummond (2022) | ||
| fcc cell, N = 152, rs = 80, N↑ = 133, N↓ = 19 | best variational | -0.0093769(3) | VMC (SJ) Azadi & Drummond (2022) | |
| best projected | -0.0094245(1) | FN-DMC (SJ) Azadi & Drummond (2022) | ||
| fcc cell, N = 169, rs = 30, polarized | best variational | -0.0224527(6) | VMC (SJ) Azadi & Drummond (2022) | |
| best projected | -0.0224795(5) | FN-DMC (SJ) Azadi & Drummond (2022) | ||
| fcc cell, N = 169, rs = 40, polarized | best variational | -0.0175196(5) | VMC (SJ) Azadi & Drummond (2022) | |
| best projected | -0.0175515(3) | FN-DMC (SJ) Azadi & Drummond (2022) | ||
| fcc cell, N = 169, rs = 50, polarized | best variational | -0.0143793(4) | VMC (SJ) Azadi & Drummond (2022) | |
| best projected | -0.0144154(2) | FN-DMC (SJ) Azadi & Drummond (2022) | ||
| fcc cell, N = 169, rs = 60, polarized | best variational | -0.0122080(3) | VMC (SJ) Azadi & Drummond (2022) | |
| best projected | -0.0122425(1) | FN-DMC (SJ) Azadi & Drummond (2022) | ||
| fcc cell, N = 169, rs = 70, polarized | best variational | -0.0106158(3) | VMC (SJ) Azadi & Drummond (2022) | |
| best projected | -0.0106470(1) | FN-DMC (SJ) Azadi & Drummond (2022) | ||
| fcc cell, N = 169, rs = 80, polarized | best variational | -0.0093933(3) | VMC (SJ) Azadi & Drummond (2022) | |
| best projected | -0.0094248(1) | FN-DMC (SJ) Azadi & Drummond (2022) | ||
| fcc cell, N = 178, rs = 100, unpolarized | best variational | -0.0076356(2) | VMC (SJB) Azadi & Drummond (2022) | |
| best projected | -0.00767661(9) | FN-DMC (SJB) Azadi & Drummond (2022) | ||
| fcc cell, N = 178, rs = 2, unpolarized | best projected | 0.00892(2) | FN-DMC Gurtubay et al. (2010) | uses CASINO 2.1.400 |
| fcc cell, N = 178, rs = 30, unpolarized | best variational | -0.0225404(4) | VMC (SJB) Azadi & Drummond (2022) | |
| best projected | -0.0226461(1) | FN-DMC (SJB) Azadi & Drummond (2022) | ||
| fcc cell, N = 178, rs = 40, unpolarized | best variational | -0.0175445(6) | VMC (SJB) Azadi & Drummond (2022) | |
| best projected | -0.01762589(7) | FN-DMC (SJB) Azadi & Drummond (2022) | ||
| fcc cell, N = 178, rs = 50, unpolarized | best variational | -0.0143317(4) | VMC (SJ) Azadi & Drummond (2022) | |
| best projected | -0.0144184(1) | FN-DMC (SJ) Azadi & Drummond (2022) | ||
| fcc cell, N = 178, rs = 60, unpolarized | best variational | -0.0122063(5) | VMC (SJB) Azadi & Drummond (2022) | |
| best projected | -0.01226479(7) | FN-DMC (SJB) Azadi & Drummond (2022) | ||
| fcc cell, N = 178, rs = 70, unpolarized | best variational | -0.0105749(3) | VMC (SJ) Azadi & Drummond (2022) | |
| best projected | -0.0106391(1) | FN-DMC (SJ) Azadi & Drummond (2022) | ||
| fcc cell, N = 178, rs = 80, unpolarized | best variational | -0.0093822(3) | VMC (SJB) Azadi & Drummond (2022) | |
| best projected | -0.00943287(9) | FN-DMC (SJB) Azadi & Drummond (2022) | ||
| fcc cell, N = 181, rs = 100, polarized | best variational | -0.0076512(2) | VMC (SJB) Azadi & Drummond (2022) | |
| best projected | -0.0076780(1) | FN-DMC (SJB) Azadi & Drummond (2022) | ||
| fcc cell, N = 181, rs = 30, polarized | best variational | -0.0224736(4) | VMC (SJB) Azadi & Drummond (2022) | |
| best projected | -0.0225066(1) | FN-DMC (SJB) Azadi & Drummond (2022) | ||
| fcc cell, N = 181, rs = 40, polarized | best variational | -0.0175354(4) | VMC (SJB) Azadi & Drummond (2022) | |
| best projected | -0.01756983(5) | FN-DMC (SJB) Azadi & Drummond (2022) | ||
| fcc cell, N = 181, rs = 50, polarized | best variational | -0.0143712(4) | VMC (SJ) Azadi & Drummond (2022) | |
| best projected | -0.0144139(1) | FN-DMC (SJ) Azadi & Drummond (2022) | ||
| fcc cell, N = 181, rs = 60, polarized | best variational | -0.0122180(3) | VMC (SJB) Azadi & Drummond (2022) | |
| best projected | -0.01225212(6) | FN-DMC (SJB) Azadi & Drummond (2022) | ||
| fcc cell, N = 181, rs = 70, polarized | best variational | -0.0106097(3) | VMC (SJ) Azadi & Drummond (2022) | |
| best projected | -0.0106461(1) | FN-DMC (SJ) Azadi & Drummond (2022) | ||
| fcc cell, N = 181, rs = 80, polarized | best variational | -0.0094012(2) | VMC (SJB) Azadi & Drummond (2022) | |
| best projected | -0.00943101(6) | FN-DMC (SJB) Azadi & Drummond (2022) | ||
| fcc cell, N = 184, rs = 30, N↑ = 115, N↓ = 69 | best variational | -0.0224577(6) | VMC (SJ) Azadi & Drummond (2022) | |
| best projected | -0.0225651(1) | FN-DMC (SJ) Azadi & Drummond (2022) | ||
| fcc cell, N = 184, rs = 40, N↑ = 115, N↓ = 69 | best variational | -0.0174818(5) | VMC (SJ) Azadi & Drummond (2022) | |
| best projected | -0.0175749(1) | FN-DMC (SJ) Azadi & Drummond (2022) | ||
| fcc cell, N = 184, rs = 50, N↑ = 115, N↓ = 69 | best variational | -0.0143354(4) | VMC (SJ) Azadi & Drummond (2022) | |
| best projected | -0.0144186(1) | FN-DMC (SJ) Azadi & Drummond (2022) | ||
| fcc cell, N = 184, rs = 60, N↑ = 115, N↓ = 69 | best variational | -0.0121663(3) | VMC (SJ) Azadi & Drummond (2022) | |
| best projected | -0.0122379(1) | FN-DMC (SJ) Azadi & Drummond (2022) | ||
| fcc cell, N = 184, rs = 70, N↑ = 115, N↓ = 69 | best variational | -0.0105739(4) | VMC (SJ) Azadi & Drummond (2022) | |
| best projected | -0.01064014(1) | FN-DMC (SJ) Azadi & Drummond (2022) | ||
| fcc cell, N = 184, rs = 80, N↑ = 115, N↓ = 69 | best variational | -0.0093549(3) | VMC (SJ) Azadi & Drummond (2022) | |
| best projected | -0.00941649(1) | FN-DMC (SJ) Azadi & Drummond (2022) | ||
| fcc cell, N = 184, rs = 30, N↑ = 161, N↓ = 23 | best variational | -0.0224608(5) | VMC (SJ) Azadi & Drummond (2022) | |
| best projected | -0.0225319(3) | FN-DMC (SJ) Azadi & Drummond (2022) | ||
| fcc cell, N = 184, rs = 40, N↑ = 161, N↓ = 23 | best variational | -0.0175027(4) | VMC (SJ) Azadi & Drummond (2022) | |
| best projected | -0.0175701(2) | FN-DMC (SJ) Azadi & Drummond (2022) | ||
| fcc cell, N = 184, rs = 50, N↑ = 161, N↓ = 23 | best variational | -0.0143616(3) | VMC (SJ) Azadi & Drummond (2022) | |
| best projected | -0.0144220(1) | FN-DMC (SJ) Azadi & Drummond (2022) | ||
| fcc cell, N = 184, rs = 60, N↑ = 161, N↓ = 23 | best variational | -0.0121877(3) | VMC (SJ) Azadi & Drummond (2022) | |
| best projected | -0.0122440(1) | FN-DMC (SJ) Azadi & Drummond (2022) | ||
| fcc cell, N = 184, rs = 70, N↑ = 161, N↓ = 23 | best variational | -0.0105942(2) | VMC (SJ) Azadi & Drummond (2022) | |
| best projected | -0.0106463(1) | FN-DMC (SJ) Azadi & Drummond (2022) | ||
| fcc cell, N = 184, rs = 80, N↑ = 161, N↓ = 23 | best variational | -0.0093734(3) | VMC (SJ) Azadi & Drummond (2022) | |
| best projected | -0.0094229(1) | FN-DMC (SJ) Azadi & Drummond (2022) | ||
| fcc cell, N = 188, rs = 100, N↑ = 141, N↓ = 47 | best variational | -0.0076410(1) | VMC (SJB) Azadi & Drummond (2022) | |
| best projected | -0.00767704(8) | FN-DMC (SJB) Azadi & Drummond (2022) | ||
| fcc cell, N = 188, rs = 30, N↑ = 141, N↓ = 47 | best variational | -0.0225381(5) | VMC (SJB) Azadi & Drummond (2022) | |
| best projected | -0.0226138(1) | FN-DMC (SJB) Azadi & Drummond (2022) | ||
| fcc cell, N = 188, rs = 40, N↑ = 141, N↓ = 47 | best variational | -0.0175518(5) | VMC (SJB) Azadi & Drummond (2022) | |
| best projected | -0.01761386(6) | FN-DMC (SJB) Azadi & Drummond (2022) | ||
| fcc cell, N = 188, rs = 50, N↑ = 141, N↓ = 47 | best variational | -0.0143414(3) | VMC (SJ) Azadi & Drummond (2022) | |
| best projected | -0.0144203(1) | FN-DMC (SJ) Azadi & Drummond (2022) | ||
| fcc cell, N = 188, rs = 60, N↑ = 141, N↓ = 47 | best variational | -0.0122054(2) | VMC (SJB) Azadi & Drummond (2022) | |
| best projected | -0.01226154(7) | FN-DMC (SJB) Azadi & Drummond (2022) | ||
| fcc cell, N = 188, rs = 70, N↑ = 141, N↓ = 47 | best variational | -0.0105783(3) | VMC (SJ) Azadi & Drummond (2022) | |
| best projected | -0.0106423(1) | FN-DMC (SJ) Azadi & Drummond (2022) | ||
| fcc cell, N = 188, rs = 80, N↑ = 141, N↓ = 47 | best variational | -0.0093829(2) | VMC (SJB) Azadi & Drummond (2022) | |
| best projected | -0.00943141(9) | FN-DMC (SJB) Azadi & Drummond (2022) | ||
| fcc cell, N = 216, rs = 30, N↑ = 189, N↓ = 27 | best variational | -0.0224319(5) | VMC (SJ) Azadi & Drummond (2022) | |
| best projected | -0.0225259(1) | FN-DMC (SJ) Azadi & Drummond (2022) | ||
| fcc cell, N = 216, rs = 40, N↑ = 189, N↓ = 27 | best variational | -0.0174868(4) | VMC (SJ) Azadi & Drummond (2022) | |
| best projected | -0.0175662(1) | FN-DMC (SJ) Azadi & Drummond (2022) | ||
| fcc cell, N = 216, rs = 50, N↑ = 189, N↓ = 27 | best variational | -0.0143535(3) | VMC (SJ) Azadi & Drummond (2022) | |
| best projected | -0.0144192(1) | FN-DMC (SJ) Azadi & Drummond (2022) | ||
| fcc cell, N = 216, rs = 60, N↑ = 189, N↓ = 27 | best variational | -0.0121833(3) | VMC (SJ) Azadi & Drummond (2022) | |
| best projected | -0.0122417(1) | FN-DMC (SJ) Azadi & Drummond (2022) | ||
| fcc cell, N = 216, rs = 70, N↑ = 189, N↓ = 27 | best variational | -0.0105860(2) | VMC (SJ) Azadi & Drummond (2022) | |
| best projected | -0.0106436(1) | FN-DMC (SJ) Azadi & Drummond (2022) | ||
| fcc cell, N = 216, rs = 80, N↑ = 189, N↓ = 27 | best variational | -0.0093712(2) | VMC (SJ) Azadi & Drummond (2022) | |
| best projected | -0.0094211(1) | FN-DMC (SJ) Azadi & Drummond (2022) | ||
| fcc cell, N = 226, rs = 100, unpolarized | best variational | -0.0076375(1) | VMC (SJB) Azadi & Drummond (2022) | |
| best projected | -0.0076755(1) | FN-DMC (SJB) Azadi & Drummond (2022) | ||
| fcc cell, N = 226, rs = 2, unpolarized | best projected | 0.00202(2) | FN-DMC Gurtubay et al. (2010) | uses CASINO 2.1.400 |
| fcc cell, N = 226, rs = 30, unpolarized | best variational | -0.0225531(4) | VMC (SJB) Azadi & Drummond (2022) | |
| best projected | -0.0226358(1) | FN-DMC (SJB) Azadi & Drummond (2022) | ||
| fcc cell, N = 226, rs = 40, unpolarized | best variational | -0.0175548(6) | VMC (SJB) Azadi & Drummond (2022) | |
| best projected | -0.01762088(9) | FN-DMC (SJB) Azadi & Drummond (2022) | ||
| fcc cell, N = 226, rs = 50, unpolarized | best variational | -0.0143403(4) | VMC (SJ) Azadi & Drummond (2022) | |
| best projected | -0.0144151(2) | FN-DMC (SJ) Azadi & Drummond (2022) | ||
| fcc cell, N = 226, rs = 60, unpolarized | best variational | -0.0122056(4) | VMC (SJB) Azadi & Drummond (2022) | |
| best projected | -0.0122608(1) | FN-DMC (SJB) Azadi & Drummond (2022) | ||
| fcc cell, N = 226, rs = 70, unpolarized | best variational | -0.0105783(3) | VMC (SJ) Azadi & Drummond (2022) | |
| best projected | -0.0106371(1) | FN-DMC (SJ) Azadi & Drummond (2022) | ||
| fcc cell, N = 226, rs = 80, unpolarized | best variational | -0.0093862(2) | VMC (SJB) Azadi & Drummond (2022) | |
| best projected | -0.0094307(1) | FN-DMC (SJB) Azadi & Drummond (2022) | ||
| fcc cell, N = 229, rs = 100, polarized | best variational | -0.0076428(2) | VMC (SJB) Azadi & Drummond (2022) | |
| best projected | -0.0076762(1) | FN-DMC (SJB) Azadi & Drummond (2022) | ||
| fcc cell, N = 229, rs = 30, polarized | best variational | -0.0224174(4) | VMC (SJB) Azadi & Drummond (2022) | |
| best projected | -0.02250086(8) | FN-DMC (SJB) Azadi & Drummond (2022) | ||
| fcc cell, N = 229, rs = 40, polarized | best variational | -0.0175008(3) | VMC (SJB) Azadi & Drummond (2022) | |
| best projected | -0.01756580(1) | FN-DMC (SJB) Azadi & Drummond (2022) | ||
| fcc cell, N = 229, rs = 50, polarized | best variational | -0.0143463(4) | VMC (SJ) Azadi & Drummond (2022) | |
| best projected | -0.0144109(2) | FN-DMC (SJ) Azadi & Drummond (2022) | ||
| fcc cell, N = 229, rs = 60, polarized | best variational | -0.0122010(3) | VMC (SJB) Azadi & Drummond (2022) | |
| best projected | -0.0122499(1) | FN-DMC (SJB) Azadi & Drummond (2022) | ||
| fcc cell, N = 229, rs = 70, polarized | best variational | -0.0105951(3) | VMC (SJ) Azadi & Drummond (2022) | |
| best projected | -0.0106443(1) | FN-DMC (SJ) Azadi & Drummond (2022) | ||
| fcc cell, N = 229, rs = 80, polarized | best variational | -0.0093899(1) | VMC (SJB) Azadi & Drummond (2022) | |
| best projected | -0.0094294(1) | FN-DMC (SJB) Azadi & Drummond (2022) | ||
| fcc cell, N = 232, rs = 30, N↑ = 145, N↓ = 87 | best variational | -0.0224340(6) | VMC (SJ) Azadi & Drummond (2022) | |
| best projected | -0.0225577(2) | FN-DMC (SJ) Azadi & Drummond (2022) | ||
| fcc cell, N = 232, rs = 40, N↑ = 145, N↓ = 87 | best variational | -0.0174479(4) | VMC (SJ) Azadi & Drummond (2022) | |
| best projected | -0.0175674(2) | FN-DMC (SJ) Azadi & Drummond (2022) | ||
| fcc cell, N = 232, rs = 50, N↑ = 145, N↓ = 87 | best variational | -0.0143387(4) | VMC (SJ) Azadi & Drummond (2022) | |
| best projected | -0.0144154(1) | FN-DMC (SJ) Azadi & Drummond (2022) | ||
| fcc cell, N = 232, rs = 60, N↑ = 145, N↓ = 87 | best variational | -0.0121571(4) | VMC (SJ) Azadi & Drummond (2022) | |
| best projected | -0.0122349(1) | FN-DMC (SJ) Azadi & Drummond (2022) | ||
| fcc cell, N = 232, rs = 70, N↑ = 145, N↓ = 87 | best variational | -0.0105717(3) | VMC (SJ) Azadi & Drummond (2022) | |
| best projected | -0.01063722(1) | FN-DMC (SJ) Azadi & Drummond (2022) | ||
| fcc cell, N = 232, rs = 80, N↑ = 145, N↓ = 87 | best variational | -0.0093482(3) | VMC (SJ) Azadi & Drummond (2022) | |
| best projected | -0.00941375(1) | FN-DMC (SJ) Azadi & Drummond (2022) | ||
| fcc cell, N = 244, rs = 100, N↑ = 183, N↓ = 61 | best variational | -0.0076400(1) | VMC (SJB) Azadi & Drummond (2022) | |
| best projected | -0.0076754(1) | FN-DMC (SJB) Azadi & Drummond (2022) | ||
| fcc cell, N = 244, rs = 30, N↑ = 183, N↓ = 61 | best variational | -0.0225369(5) | VMC (SJB) Azadi & Drummond (2022) | |
| best projected | -0.0226077(1) | FN-DMC (SJB) Azadi & Drummond (2022) | ||
| fcc cell, N = 244, rs = 40, N↑ = 183, N↓ = 61 | best variational | -0.0175366(6) | VMC (SJB) Azadi & Drummond (2022) | |
| best projected | -0.0176088(1) | FN-DMC (SJB) Azadi & Drummond (2022) | ||
| fcc cell, N = 244, rs = 50, N↑ = 183, N↓ = 61 | best variational | -0.0143394(3) | VMC (SJ) Azadi & Drummond (2022) | |
| best projected | -0.0144168(1) | FN-DMC (SJ) Azadi & Drummond (2022) | ||
| fcc cell, N = 244, rs = 60, N↑ = 183, N↓ = 61 | best variational | -0.0122020(4) | VMC (SJB) Azadi & Drummond (2022) | |
| best projected | -0.0122585(1) | FN-DMC (SJB) Azadi & Drummond (2022) | ||
| fcc cell, N = 244, rs = 70, N↑ = 183, N↓ = 61 | best variational | -0.0105734(3) | VMC (SJ) Azadi & Drummond (2022) | |
| best projected | -0.0106399(1) | FN-DMC (SJ) Azadi & Drummond (2022) | ||
| fcc cell, N = 244, rs = 80, N↑ = 183, N↓ = 61 | best variational | -0.0093813(3) | VMC (SJB) Azadi & Drummond (2022) | |
| best projected | -0.0094298(1) | FN-DMC (SJB) Azadi & Drummond (2022) | ||
| fcc cell, N = 274, rs = 30, unpolarized | best variational | -0.0224531(3) | VMC (SJ) Azadi & Drummond (2022) | |
| best projected | -0.0225566(3) | FN-DMC (SJ) Azadi & Drummond (2022) | ||
| fcc cell, N = 274, rs = 40, unpolarized | best variational | -0.0174824(6) | VMC (SJ) Azadi & Drummond (2022) | |
| best projected | -0.0175674(1) | FN-DMC (SJ) Azadi & Drummond (2022) | ||
| fcc cell, N = 274, rs = 50, unpolarized | best variational | -0.0143348(3) | VMC (SJ) Azadi & Drummond (2022) | |
| best projected | -0.0144121(1) | FN-DMC (SJ) Azadi & Drummond (2022) | ||
| fcc cell, N = 274, rs = 60, unpolarized | best variational | -0.0121603(4) | VMC (SJ) Azadi & Drummond (2022) | |
| best projected | -0.0122328(1) | FN-DMC (SJ) Azadi & Drummond (2022) | ||
| fcc cell, N = 274, rs = 70, unpolarized | best variational | -0.0105725(4) | VMC (SJ) Azadi & Drummond (2022) | |
| best projected | -0.0106352(2) | FN-DMC (SJ) Azadi & Drummond (2022) | ||
| fcc cell, N = 274, rs = 80, unpolarized | best variational | -0.0093431(2) | VMC (SJ) Azadi & Drummond (2022) | |
| best projected | -0.0094113(1) | FN-DMC (SJ) Azadi & Drummond (2022) | ||
| rect-3x3x2 cell, N = 36, rs = 10, unpolarized | best variational | -0.0537396(1) | VMC (MP-NQS) Liu et al. (2026) | |
| rect-3x3x2 cell, N = 36, rs = 100, unpolarized | best variational | -0.00769287(1) | VMC (MP-NQS) Liu et al. (2026) | |
| rect-3x3x2 cell, N = 36, rs = 30, unpolarized | best variational | -0.02272554(2) | VMC (MP-NQS) Liu et al. (2026) | |
| rect-3x3x2 cell, N = 36, rs = 5, unpolarized | best variational | -0.0756267(6) | VMC (MP-NQS) Liu et al. (2026) | |
| rect-3x3x2 cell, N = 36, rs = 50, unpolarized | best variational | -0.01449795(2) | VMC (MP-NQS) Liu et al. (2026) | |
| rect-3x3x2 cell, N = 36, rs = 80, unpolarized | best variational | -0.00945620(2) | VMC (MP-NQS) Liu et al. (2026) | |
| rect-3x3x2 cell, N = 36, rs = 86.6, unpolarized | best variational | -0.00879026(2) | VMC (MP-NQS) Liu et al. (2026) | |
| rect-3x3x2 cell, N = 36, rs = 86.7, unpolarized | best variational | -0.00878092(1) | VMC (MP-NQS) Liu et al. (2026) | |
| rect-3x3x2 cell, N = 36, rs = 86.8, unpolarized | best variational | -0.00877159(1) | VMC (MP-NQS) Liu et al. (2026) | |
| rect-3x3x2 cell, N = 36, rs = 86.9, unpolarized | best variational | -0.00876228(1) | VMC (MP-NQS) Liu et al. (2026) | |
| rect-3x3x2 cell, N = 36, rs = 87, unpolarized | best variational | -0.008753017(8) | VMC (MP-NQS) Liu et al. (2026) | |
| rect-3x3x2 cell, N = 36, rs = 87.1, unpolarized | best variational | -0.008743822(8) | VMC (MP-NQS) Liu et al. (2026) | |
| sc cell, N = 7, rs = 1, polarized | best variational | 1.120358(3) | VMC (WAP-net) Wilson et al. (2023) | |
| best projected | 1.120400(5) | FN-DMC (iterative BF) Wilson et al. (2023) | ||
| sc cell, N = 7, rs = 10, polarized | best variational | -0.0531436(2) | VMC (WAP-net) Wilson et al. (2023) | |
| best projected | -0.0531380(5) | FN-DMC (iterative BF) Wilson et al. (2023) | ||
| sc cell, N = 7, rs = 100, polarized | best variational | -0.00774654(1) | VMC (WAP-net) Wilson et al. (2023) | |
| best projected | -0.00774625(5) | FN-DMC (iterative BF) Wilson et al. (2023) | ||
| sc cell, N = 7, rs = 2, polarized | best variational | 0.110850(1) | VMC (WAP-net) Wilson et al. (2023) | |
| best projected | 0.110879(2) | FN-DMC (iterative BF) Wilson et al. (2023) | ||
| sc cell, N = 7, rs = 20, polarized | best variational | -0.0322689(1) | VMC (WAP-net) Wilson et al. (2023) | |
| best projected | -0.032266(2) | FN-DMC (iterative BF) Wilson et al. (2023) | ||
| sc cell, N = 7, rs = 5, polarized | best variational | -0.0664225(5) | VMC (WAP-net) Wilson et al. (2023) | |
| best projected | -0.0664055(5) | FN-DMC (iterative BF) Wilson et al. (2023) | ||
| sc cell, N = 7, rs = 50, polarized | best variational | -0.01466089(1) | VMC (WAP-net) Wilson et al. (2023) | |
| best projected | -0.0146604(1) | FN-DMC (iterative BF) Wilson et al. (2023) | ||
| sc cell, N = 14, rs = 0.1, unpolarized | best projected | 106.90015 ○ | CCD Butler et al. (2024) | butler-julie/HEG-SRE |
| best extrapolated | 106.89994(1) | CCSDT Spencer & Thom (2016) | ||
| best approximate | 106.87643 ○ | dRPA Shepherd et al. (2014) | ||
| sc cell, N = 14, rs = 0.2, unpolarized | best extrapolated | 25.4050004 ○ | CCD Shepherd et al. (2014) | |
| best approximate | 25.3821433 ○ | dRPA Shepherd et al. (2014) | ||
| sc cell, N = 14, rs = 0.3, unpolarized | best extrapolated | 10.6954209 ○ | CCD Shepherd et al. (2014) | |
| best approximate | 10.6732066 ○ | dRPA Shepherd et al. (2014) | ||
| sc cell, N = 14, rs = 0.4, unpolarized | best extrapolated | 5.6765448 ○ | CCD Shepherd et al. (2014) | |
| best approximate | 5.6547591 ○ | dRPA Shepherd et al. (2014) | ||
| sc cell, N = 14, rs = 0.5, unpolarized | best variational | 3.412684(4) | VMC (FermiNet) Cassella et al. (2023) | uses FermiNet |
| best projected | 3.41253(4) | ph-AFQMC (RHF) Lee et al. (2019) | uses PAUXY | |
| best extrapolated | 3.41226(2) | FCIQMC Shepherd et al. (2012) | ||
| best approximate | 3.3916087 ○ | dRPA Shepherd et al. (2014) | ||
| sc cell, N = 14, rs = 0.6, unpolarized | best extrapolated | 2.2154921 ○ | CCD Shepherd et al. (2014) | |
| best approximate | 2.1945636 ○ | dRPA Shepherd et al. (2014) | ||
| sc cell, N = 14, rs = 0.7, unpolarized | best extrapolated | 1.5130365 ○ | CCD Shepherd et al. (2014) | |
| best approximate | 1.4923937 ○ | dRPA Shepherd et al. (2014) | ||
| sc cell, N = 14, rs = 0.8, unpolarized | best extrapolated | 1.0698103 ○ | CCD Shepherd et al. (2014) | |
| best approximate | 1.0493818 ○ | dRPA Shepherd et al. (2014) | ||
| sc cell, N = 14, rs = 0.9, unpolarized | best extrapolated | 0.7746685 ○ | CCD Shepherd et al. (2014) | |
| best approximate | 0.7534542 ○ | rxmCCD Shepherd et al. (2014) | ||
| sc cell, N = 14, rs = 1, unpolarized | best variational | 0.568851(1) | VMC (FermiNet) Liu et al. (2026) | |
| best projected | 0.56869(6) | ph-AFQMC (RHF) Lee et al. (2019) | uses QMCPACK | |
| best extrapolated | 0.56850(3) | FCIQMC Shepherd et al. (2012) | ||
| best approximate | 0.5464629 ○ | rxmCCD Shepherd et al. (2014) | ||
| sc cell, N = 14, rs = 10, unpolarized | best variational | -0.0552126(6) | VMC (MP-NQS) Pescia et al. (2024) | uses NetKet 3 |
| best projected | -0.0552180(5) | FN-DMC (iterative BF) Wilson et al. (2023) | ||
| best extrapolated | -0.05289 ○ | DCD Liao et al. (2021) | ||
| best approximate | -0.0791573 ○ | rxmCCD Shepherd et al. (2014) | ||
| sc cell, N = 14, rs = 100, unpolarized | best variational | -0.00773024(1) | VMC (WAP-net) Wilson et al. (2023) | |
| best projected | -0.00773430(2) | FN-DMC (iterative BF) Wilson et al. (2023) | ||
| best extrapolated | -0.0064603 ○ | CCD Shepherd et al. (2014) | ||
| best approximate | -0.0104603 ○ | rxmCCD Shepherd et al. (2014) | ||
| sc cell, N = 14, rs = 15, unpolarized | best variational | -0.0408619(1) | VMC (FermiNet) Liu et al. (2026) | |
| best approximate | -0.0580244 ○ | rxmCCD Shepherd et al. (2014) | ||
| sc cell, N = 14, rs = 2, unpolarized | best variational | -0.0084275(7) | VMC (FermiNet) Cassella et al. (2023) | uses FermiNet |
| best projected | -0.00853(6) | ph-AFQMC (RHF) Lee et al. (2019) | uses QMCPACK | |
| best extrapolated | -0.00873(3) | FCIQMC Shepherd et al. (2012) | ||
| best approximate | -0.0478896 ○ | rxmCCD Shepherd et al. (2014) | ||
| sc cell, N = 14, rs = 20, unpolarized | best variational | -0.0324553(2) | VMC (MP-NQS) Pescia et al. (2024) | uses NetKet 3 |
| best projected | -0.0324578(2) | FN-DMC (iterative BF) Wilson et al. (2023) | ||
| best extrapolated | -0.03035 ○ | DCD Liao et al. (2021) | ||
| best approximate | -0.0457024 ○ | rxmCCD Shepherd et al. (2014) | ||
| sc cell, N = 14, rs = 3, unpolarized | best projected | -0.07352(4) | ph-AFQMC (RHF) Lee et al. (2019) | uses QMCPACK |
| best extrapolated | -0.07432(2) | CCSDTQ5 Neufeld & Thom (2017) | uses HANDE-QMC | |
| best approximate | -0.1179136 ○ | rxmCCD Shepherd et al. (2014) | ||
| sc cell, N = 14, rs = 30, unpolarized | best extrapolated | -0.0203291 ○ | CCD Shepherd et al. (2014) | |
| best approximate | -0.0321148 ○ | rxmCCD Shepherd et al. (2014) | ||
| sc cell, N = 14, rs = 4, unpolarized | best extrapolated | -0.0792519 ○ | CCD Shepherd et al. (2014) | |
| best approximate | -0.1240376 ○ | rxmCCD Shepherd et al. (2014) | ||
| sc cell, N = 14, rs = 40, unpolarized | best extrapolated | -0.0155875 ○ | CCD Shepherd et al. (2014) | |
| best approximate | -0.0247303 ○ | rxmCCD Shepherd et al. (2014) | ||
| sc cell, N = 14, rs = 5, unpolarized | best variational | -0.0798630(2) | VMC (FermiNet) Liu et al. (2026) | |
| best projected | -0.079842(1) | FN-DMC (iterative BF) Wilson et al. (2023) | ||
| best extrapolated | -0.07997(7) | FCIQMC Shepherd et al. (2012) | ||
| best approximate | -0.1175392 ○ | rxmCCD Shepherd et al. (2014) | ||
| sc cell, N = 14, rs = 50, unpolarized | best variational | -0.01462631(6) | VMC (MP-NQS) Pescia et al. (2024) | uses NetKet 3 |
| best projected | -0.0146287(1) | FN-DMC (iterative BF) Wilson et al. (2023) | ||
| best extrapolated | -0.01323 ○ | DCD Liao et al. (2021) | ||
| best approximate | -0.0201249 ○ | rxmCCD Shepherd et al. (2014) | ||
| sc cell, N = 14, rs = 6, unpolarized | best extrapolated | -0.0706647 ○ | CCD Shepherd et al. (2014) | |
| best approximate | -0.1085933 ○ | rxmCCD Shepherd et al. (2014) | ||
| sc cell, N = 14, rs = 60, unpolarized | best extrapolated | -0.0106188 ○ | CCD Shepherd et al. (2014) | |
| best approximate | -0.0169759 ○ | rxmCCD Shepherd et al. (2014) | ||
| sc cell, N = 14, rs = 7, unpolarized | best extrapolated | -0.0651069 ○ | CCD Shepherd et al. (2014) | |
| best approximate | -0.099964 ○ | rxmCCD Shepherd et al. (2014) | ||
| sc cell, N = 14, rs = 70, unpolarized | best extrapolated | -0.0091195 ○ | CCD Shepherd et al. (2014) | |
| best approximate | -0.0146909 ○ | rxmCCD Shepherd et al. (2014) | ||
| sc cell, N = 14, rs = 8, unpolarized | best extrapolated | -0.0599259 ○ | CCD Shepherd et al. (2014) | |
| best approximate | -0.0920688 ○ | rxmCCD Shepherd et al. (2014) | ||
| sc cell, N = 14, rs = 80, unpolarized | best extrapolated | -0.0080403 ○ | CCD Shepherd et al. (2014) | |
| best approximate | -0.0129689 ○ | rxmCCD Shepherd et al. (2014) | ||
| sc cell, N = 14, rs = 9, unpolarized | best extrapolated | -0.0553862 ○ | CCD Shepherd et al. (2014) | |
| best approximate | -0.0851719 ○ | rxmCCD Shepherd et al. (2014) | ||
| sc cell, N = 14, rs = 90, unpolarized | best extrapolated | -0.0071484 ○ | CCD Shepherd et al. (2014) | |
| best approximate | -0.0115769 ○ | rxmCCD Shepherd et al. (2014) | ||
| sc cell, N = 19, rs = 1, polarized | best variational | 1.046242(3) | VMC (WAP-net) Wilson et al. (2023) | |
| best projected | 1.0462250(5) | FN-DMC (iterative BF) Wilson et al. (2023) | ||
| sc cell, N = 19, rs = 10, polarized | best variational | -0.0528624(2) | VMC (MP-NQS) Pescia et al. (2024) | uses NetKet 3 |
| best projected | -0.0528652(3) | FN-DMC (iterative BF) Wilson et al. (2023) | ||
| sc cell, N = 19, rs = 100, polarized | best variational | -0.00771899(1) | VMC (iterative BF) Wilson et al. (2023) | |
| best projected | -0.00772142(1) | FN-DMC (iterative BF) Wilson et al. (2023) | ||
| sc cell, N = 19, rs = 2, polarized | best variational | 0.096303(1) | VMC (WAP-net) Wilson et al. (2023) | |
| best projected | 0.096302(2) | FN-DMC (iterative BF) Wilson et al. (2023) | ||
| sc cell, N = 19, rs = 20, polarized | best variational | -0.03201095(5) | VMC (WAP-net) Wilson et al. (2023) | |
| best projected | -0.0320167(1) | FN-DMC (iterative BF) Wilson et al. (2023) | ||
| sc cell, N = 19, rs = 5, polarized | best variational | -0.0672510(2) | VMC (WAP-net) Wilson et al. (2023) | |
| best projected | -0.0672540(5) | FN-DMC (iterative BF) Wilson et al. (2023) | ||
| sc cell, N = 19, rs = 50, polarized | best variational | -0.01456685(3) | VMC (MP-NQS) Pescia et al. (2024) | uses NetKet 3 |
| best projected | -0.01456920(4) | FN-DMC (iterative BF) Wilson et al. (2023) | ||
| sc cell, N = 27, rs = 1, polarized | best variational | 1.051771(4) | VMC (MP-NQS) Pescia et al. (2024) | uses NetKet 3 |
| best projected | 1.05200(3) | FN-DMC (SJ) Pescia et al. (2024) | ||
| sc cell, N = 27, rs = 10, polarized | best variational | -0.0526015(2) | VMC (MP-NQS) Pescia et al. (2024) | uses NetKet 3 |
| best projected | -0.052540(6) | FN-DMC (SJ) Pescia et al. (2024) | ||
| sc cell, N = 27, rs = 2, polarized | best variational | 0.098460(2) | VMC (MP-NQS) Pescia et al. (2024) | uses NetKet 3 |
| best projected | 0.09866(2) | FN-DMC (SJ) Pescia et al. (2024) | ||
| sc cell, N = 27, rs = 30, polarized | best variational | -0.0228184(1) | VMC (MP-NQS) Pescia et al. (2024) | uses NetKet 3 |
| best projected | -0.0228080(8) | FN-DMC (SJ) Pescia et al. (2024) | ||
| sc cell, N = 27, rs = 5, polarized | best variational | -0.0665523(6) | VMC (MP-NQS) Pescia et al. (2024) | uses NetKet 3 |
| best projected | -0.066429(4) | FN-DMC (SJ) Pescia et al. (2024) | ||
| sc cell, N = 27, rs = 50, polarized | best variational | -0.01457976(5) | VMC (MP-NQS) Pescia et al. (2024) | uses NetKet 3 |
| best projected | -0.0145752(6) | FN-DMC (SJ) Pescia et al. (2024) | ||
| sc cell, N = 27, rs = 70, polarized | best variational | -0.01074986(3) | VMC (MP-NQS) Pescia et al. (2024) | uses NetKet 3 |
| best projected | -0.010749(2) | FN-DMC (SJ) Pescia et al. (2024) | ||
| sc cell, N = 27, rs = 90, polarized | best variational | -0.00853156(2) | VMC (MP-NQS) Pescia et al. (2024) | uses NetKet 3 |
| best projected | -0.0085314(5) | FN-DMC (SJ) Pescia et al. (2024) | ||
| sc cell, N = 38, rs = 0.5, unpolarized | best projected | 3.2080745 ○ | CCD Butler et al. (2024) | butler-julie/HEG-SRE |
| best extrapolated | 3.20691(3) | FCIQMC Shepherd et al. (2012) | ||
| sc cell, N = 38, rs = 1, unpolarized | best extrapolated | 0.52478(3) | FCIQMC Shepherd et al. (2012) | |
| sc cell, N = 38, rs = 2, unpolarized | best extrapolated | -0.01422(5) | FCIQMC Shepherd et al. (2012) | |
| sc cell, N = 54, rs = 0.5, unpolarized | best variational | 3.221226(2) | VMC (DeepSolid) Li et al. (2022) | bytedance/DeepSolid |
| best projected | 3.22087(2) | ph-AFQMC (RHF) Lee et al. (2019) | uses QMCPACK | |
| best extrapolated | 3.2202(2) | FCIQMC Shepherd et al. (2012) | ||
| best approximate | 3.2119035 ○ | rxmCCD Shepherd et al. (2014) | ||
| sc cell, N = 54, rs = 1, unpolarized | best variational | 0.52973(1) | VMC (MP-NQS) Pescia et al. (2024) | uses NetKet 3 |
| best projected | 0.52967(2) | ph-AFQMC (RHF) Lee et al. (2019) | uses QMCPACK | |
| best extrapolated | 0.52991(6) | FCIQMC Shepherd et al. (2012) | ||
| best approximate | 0.5147818 ○ | rxmCCD Shepherd et al. (2014) | ||
| sc cell, N = 54, rs = 10, unpolarized | best variational | -0.054448(1) | VMC (MP-NQS) Pescia et al. (2024) | uses NetKet 3 |
| best projected | -0.054443(2) | FN-DMC (SJB) López Ríos et al. (2006) | uses CASINO | |
| best extrapolated | -0.05157 ○ | DCD Liao et al. (2021) | ||
| best approximate | -0.0728389 ○ | rxmCCD Shepherd et al. (2014) | ||
| sc cell, N = 54, rs = 100, unpolarized | best variational | -0.00769203(5) | VMC (MP-NQS) Pescia et al. (2024) | uses NetKet 3 |
| best projected | -0.007674(3) | FN-DMC (SJ) Pescia et al. (2024) | ||
| sc cell, N = 54, rs = 15, unpolarized | best extrapolated | -0.0363669 ○ | CCD Shepherd et al. (2014) | |
| best approximate | -0.0359224 ○ | mCCD Shepherd et al. (2014) | ||
| sc cell, N = 54, rs = 2, unpolarized | best variational | -0.014046(8) | VMC (MP-NQS) Pescia et al. (2024) | uses NetKet 3 |
| best projected | -0.01429(3) | ph-AFQMC (RHF) Lee et al. (2019) | uses QMCPACK | |
| best extrapolated | -0.01286 ○ | DCD Liao et al. (2021) | ||
| best approximate | -0.0363623 ○ | rxmCCD Shepherd et al. (2014) | ||
| sc cell, N = 54, rs = 20, unpolarized | best variational | -0.0320524(5) | VMC (MP-NQS) Pescia et al. (2024) | uses NetKet 3 |
| best projected | -0.032047(2) | FN-DMC (SJB) López Ríos et al. (2006) | uses CASINO | |
| best extrapolated | -0.02925 ○ | DCD Liao et al. (2021) | ||
| best approximate | -0.0429947 ○ | rxmCCD Shepherd et al. (2014) | ||
| sc cell, N = 54, rs = 3, unpolarized | best approximate | -0.0706229 ○ | mCCD Shepherd et al. (2014) | |
| sc cell, N = 54, rs = 4, unpolarized | best approximate | -0.0774134 ○ | mCCD Shepherd et al. (2014) | |
| sc cell, N = 54, rs = 5, unpolarized | best variational | -0.079090(2) | VMC (MP-NQS) Pescia et al. (2024) | uses NetKet 3 |
| best projected | -0.079036(3) | FN-DMC (SJB) López Ríos et al. (2006) | uses CASINO | |
| best extrapolated | -0.07655 ○ | DCD Liao et al. (2021) | ||
| best approximate | -0.104002 ○ | rxmCCD Shepherd et al. (2014) | ||
| sc cell, N = 54, rs = 50, unpolarized | best variational | -0.0145015(1) | VMC (MP-NQS) Pescia et al. (2024) | uses NetKet 3 |
| best projected | -0.01387(2) | FN-DMC (SJ) Pescia et al. (2024) | ||
| best extrapolated | -0.01267 ○ | DCD Liao et al. (2021) | ||
| best approximate | -0.01281 ○ | TC-DCD Liao et al. (2021) | ||
| sc cell, N = 54, rs = 7.5, unpolarized | best approximate | -0.0602626 ○ | mCCD Shepherd et al. (2014) | |
| sc cell, N = 66, rs = 1, unpolarized | best variational | 0.5748(3) | VMC (SJ) Kwon et al. (1998) | |
| best projected | 0.57290(3) | FN-DMC Fantoni (2013) | ||
| sc cell, N = 66, rs = 10, unpolarized | best variational | -0.05318(1) | VMC (SJ) Kwon et al. (1998) | |
| best projected | -0.053728(3) | FN-DMC Fantoni (2013) | ||
| sc cell, N = 66, rs = 2, unpolarized | best projected | -0.00208(4) | FN-DMC Fantoni (2013) | |
| sc cell, N = 66, rs = 20, unpolarized | best variational | -0.031515(5) | VMC (SJ) Kwon et al. (1998) | |
| sc cell, N = 66, rs = 5, unpolarized | best variational | -0.07583(2) | VMC (SJ) Kwon et al. (1998) | |
| best projected | -0.076676(2) | FN-DMC Fantoni (2013) | ||
| sc cell, N = 114, rs = 0.5, unpolarized | best projected | 3.48453(8) | ph-AFQMC (RHF) Lee et al. (2019) | uses QMCPACK |
| sc cell, N = 114, rs = 1, unpolarized | best variational | 0.6039(3) | VMC (SJ) Kwon et al. (1998) | |
| best projected | 0.59877(6) | ph-AFQMC (RHF) Lee et al. (2019) | uses QMCPACK | |
| sc cell, N = 114, rs = 10, unpolarized | best variational | -0.05276(1) | VMC (SJ) Kwon et al. (1998) | |
| sc cell, N = 114, rs = 2, unpolarized | best projected | 0.00487(6) | ph-AFQMC (RHF) Lee et al. (2019) | uses QMCPACK |
| sc cell, N = 114, rs = 20, unpolarized | best variational | -0.031390(5) | VMC (SJ) Kwon et al. (1998) | |
| sc cell, N = 114, rs = 5, unpolarized | best variational | -0.07433(1) | VMC (SJ) Kwon et al. (1998) | |
| sc cell, N = 128, rs = 110, unpolarized | best variational | -0.0071121(2) | VMC (MP-NQS) Pescia et al. (2024) | uses NetKet 3 |
| best projected | -0.007009518(1) | FN-DMC (SJ) Pescia et al. (2024) | ||
| sc cell, N = 128, rs = 200, unpolarized | best variational | -0.0040615(1) | VMC (MP-NQS) Pescia et al. (2024) | uses NetKet 3 |
| best projected | -0.0039923(7) | FN-DMC (SJ) Pescia et al. (2024) | ||
| sc cell, N = 162, rs = 1, unpolarized | best variational | 0.5581(2) | VMC (SJ) Kwon et al. (1998) | |
| sc cell, N = 162, rs = 10, unpolarized | best variational | -0.053210(5) | VMC (SJ) Kwon et al. (1998) | |
| sc cell, N = 162, rs = 20, unpolarized | best variational | -0.031350(5) | VMC (SJ) Kwon et al. (1998) | |
| sc cell, N = 162, rs = 5, unpolarized | best variational | -0.07619(1) | VMC (SJ) Kwon et al. (1998) | |
| sc cell, N = 246, rs = 1, unpolarized | best variational | 0.5969(2) | VMC (SJ) Kwon et al. (1998) | |
| sc cell, N = 246, rs = 10, unpolarized | best variational | -0.052740(5) | VMC (SJ) Kwon et al. (1998) | |
| sc cell, N = 246, rs = 20, unpolarized | best variational | -0.031375(5) | VMC (SJ) Kwon et al. (1998) | |
| sc cell, N = 246, rs = 5, unpolarized | best variational | -0.07443(2) | VMC (SJ) Kwon et al. (1998) |
E/N (Ha)
| instance | value | energy | method | code |
|---|---|---|---|---|
| square cell, N = 21, rs = 10, polarized | best variational | -0.08492(4) | VMC (SJ) Tanatar & Ceperley (1989) | |
| square cell, N = 21, rs = 20, polarized | best variational | -0.04634(2) | VMC (SJ) Tanatar & Ceperley (1989) | |
| square cell, N = 21, rs = 30, polarized | best variational | -0.031965(5) | VMC (SJ) Tanatar & Ceperley (1989) | |
| square cell, N = 21, rs = 40, polarized | best variational | -0.024434(5) | VMC (SJ) Tanatar & Ceperley (1989) | |
| square cell, N = 21, rs = 5, polarized | best variational | -0.1451(1) | VMC (SJ) Tanatar & Ceperley (1989) | |
| square cell, N = 21, rs = 75, polarized | best variational | -0.013457(1) | VMC (SJ) Tanatar & Ceperley (1989) | |
| square cell, N = 26, rs = 1, unpolarized | best variational | -0.1850(5) | VMC (SJ) Tanatar & Ceperley (1989) | |
| square cell, N = 26, rs = 10, unpolarized | best variational | -0.08433(2) | VMC (SJ) Tanatar & Ceperley (1989) | |
| square cell, N = 26, rs = 20, unpolarized | best variational | -0.04587(1) | VMC (SJ) Tanatar & Ceperley (1989) | |
| square cell, N = 26, rs = 30, unpolarized | best variational | -0.03166(1) | VMC (SJ) Tanatar & Ceperley (1989) | |
| square cell, N = 26, rs = 5, unpolarized | best variational | -0.14685(5) | VMC (SJ) Tanatar & Ceperley (1989) | |
| square cell, N = 26, rs = 50, unpolarized | best variational | -0.019659(4) | VMC (SJ) Tanatar & Ceperley (1989) | |
| square cell, N = 37, rs = 10, polarized | best variational | -0.08440(4) | VMC (SJ) Tanatar & Ceperley (1989) | |
| square cell, N = 37, rs = 20, polarized | best variational | -0.04614(1) | VMC (SJ) Tanatar & Ceperley (1989) | |
| square cell, N = 37, rs = 30, polarized | best variational | -0.0318870(5) | VMC (SJ) Tanatar & Ceperley (1989) | |
| square cell, N = 37, rs = 40, polarized | best variational | -0.024379(4) | VMC (SJ) Tanatar & Ceperley (1989) | |
| square cell, N = 37, rs = 5, polarized | best variational | -0.14373(4) | VMC (NCT) Xie et al. (2023) | fermiflow/CoulombGas |
| square cell, N = 37, rs = 75, polarized | best variational | -0.0134380(5) | VMC (SJ) Tanatar & Ceperley (1989) | |
| square cell, N = 42, rs = 1, unpolarized | best variational | -0.2225(3) | VMC (SJ) Tanatar & Ceperley (1989) | |
| square cell, N = 42, rs = 10, unpolarized | best variational | -0.08450(2) | VMC (SJ) Tanatar & Ceperley (1989) | |
| square cell, N = 42, rs = 20, unpolarized | best variational | -0.04588(1) | VMC (SJ) Tanatar & Ceperley (1989) | |
| square cell, N = 42, rs = 30, unpolarized | best variational | -0.031675(5) | VMC (SJ) Tanatar & Ceperley (1989) | |
| square cell, N = 42, rs = 5, unpolarized | best variational | -0.14800(5) | VMC (SJ) Tanatar & Ceperley (1989) | |
| square cell, N = 42, rs = 50, unpolarized | best variational | -0.019654(3) | VMC (SJ) Tanatar & Ceperley (1989) | |
| square cell, N = 57, rs = 10, polarized | best variational | -0.08390(5) | VMC (SJ) Tanatar & Ceperley (1989) | |
| best projected | -0.08427(3) | FN-GFMC (SJ) Tanatar & Ceperley (1989) | ||
| square cell, N = 57, rs = 20, polarized | best variational | -0.04607(3) | VMC (SJ) Tanatar & Ceperley (1989) | |
| best projected | -0.04618(1) | FN-GFMC (SJ) Tanatar & Ceperley (1989) | ||
| square cell, N = 57, rs = 30, polarized | best variational | -0.031825(5) | VMC (SJ) Tanatar & Ceperley (1989) | |
| best projected | -0.031913(4) | FN-GFMC (SJ) Tanatar & Ceperley (1989) | ||
| square cell, N = 57, rs = 40, polarized | best variational | -0.024360(5) | VMC (SJ) Tanatar & Ceperley (1989) | |
| best projected | -0.024420(1) | FN-GFMC (SJ) Tanatar & Ceperley (1989) | ||
| square cell, N = 57, rs = 5, polarized | best variational | -0.14225(5) | VMC (SJ) Tanatar & Ceperley (1989) | |
| best projected | -0.14291(5) | FN-GFMC (SJ) Tanatar & Ceperley (1989) | ||
| square cell, N = 57, rs = 75, polarized | best variational | -0.0134275(5) | VMC (SJ) Tanatar & Ceperley (1989) | |
| best projected | -0.013473(2) | FN-GFMC (SJ) Tanatar & Ceperley (1989) | ||
| square cell, N = 58, rs = 1, unpolarized | best variational | -0.1953(1) | VMC (SJ) Tanatar & Ceperley (1989) | |
| best projected | -0.2046(3) | FN-GFMC (SJ) Tanatar & Ceperley (1989) | ||
| square cell, N = 58, rs = 10, unpolarized | best variational | -0.08426(2) | VMC (SJ) Tanatar & Ceperley (1989) | |
| best projected | -0.08552(4) | FN-GFMC (SJ) Tanatar & Ceperley (1989) | ||
| square cell, N = 58, rs = 20, unpolarized | best variational | -0.04582(2) | VMC (SJ) Tanatar & Ceperley (1989) | |
| best projected | -0.04636(1) | FN-GFMC (SJ) Tanatar & Ceperley (1989) | ||
| square cell, N = 58, rs = 30, unpolarized | best variational | -0.031655(3) | VMC (SJ) Tanatar & Ceperley (1989) | |
| best projected | -0.031967(3) | FN-GFMC (SJ) Tanatar & Ceperley (1989) | ||
| square cell, N = 58, rs = 5, unpolarized | best variational | -0.1470(1) | VMC (SJ) Tanatar & Ceperley (1989) | |
| best projected | -0.14990(5) | FN-GFMC (SJ) Tanatar & Ceperley (1989) | ||
| square cell, N = 58, rs = 50, unpolarized | best variational | -0.019651(3) | VMC (SJ) Tanatar & Ceperley (1989) | |
| best projected | -0.019763(3) | FN-GFMC (SJ) Tanatar & Ceperley (1989) | ||
| square cell, N = 69, rs = 10, polarized | best variational | -0.08412(3) | VMC (SJ) Tanatar & Ceperley (1989) | |
| square cell, N = 69, rs = 20, polarized | best variational | -0.046085(5) | VMC (SJ) Tanatar & Ceperley (1989) | |
| square cell, N = 69, rs = 30, polarized | best variational | -0.031852(4) | VMC (SJ) Tanatar & Ceperley (1989) | |
| square cell, N = 69, rs = 40, polarized | best variational | -0.024375(2) | VMC (SJ) Tanatar & Ceperley (1989) | |
| square cell, N = 69, rs = 5, polarized | best variational | -0.14310(5) | VMC (SJ) Tanatar & Ceperley (1989) | |
| square cell, N = 69, rs = 75, polarized | best variational | -0.0134305(5) | VMC (SJ) Tanatar & Ceperley (1989) | |
| square cell, N = 74, rs = 1, unpolarized | best variational | -0.2078(4) | VMC (SJ) Tanatar & Ceperley (1989) | |
| square cell, N = 74, rs = 10, unpolarized | best variational | -0.08429(2) | VMC (SJ) Tanatar & Ceperley (1989) | |
| square cell, N = 74, rs = 20, unpolarized | best variational | -0.045845(5) | VMC (SJ) Tanatar & Ceperley (1989) | |
| square cell, N = 74, rs = 30, unpolarized | best variational | -0.031662(4) | VMC (SJ) Tanatar & Ceperley (1989) | |
| square cell, N = 74, rs = 5, unpolarized | best variational | -0.14735(5) | VMC (SJ) Tanatar & Ceperley (1989) | |
| square cell, N = 74, rs = 50, unpolarized | best variational | -0.019651(2) | VMC (SJ) Tanatar & Ceperley (1989) | |
| square cell, N = 90, rs = 30, unpolarized | best variational | -0.0317373(2) | VMC (SJ) Azadi et al. (2024) | uses CASINO |
| best projected | -0.031919(1) | FN-DMC Azadi et al. (2024) | uses CASINO | |
| square cell, N = 113, rs = 10, polarized | best variational | -0.08396(2) | VMC (SJ) Tanatar & Ceperley (1989) | |
| square cell, N = 113, rs = 20, polarized | best variational | -0.046030(5) | VMC (SJ) Tanatar & Ceperley (1989) | |
| square cell, N = 113, rs = 30, polarized | best variational | -0.031821(3) | VMC (SJ) Tanatar & Ceperley (1989) | |
| square cell, N = 113, rs = 40, polarized | best variational | -0.024356(2) | VMC (SJ) Tanatar & Ceperley (1989) | |
| square cell, N = 113, rs = 5, polarized | best variational | -0.14215(5) | VMC (SJ) Tanatar & Ceperley (1989) | |
| square cell, N = 113, rs = 75, polarized | best variational | -0.0134245(5) | VMC (SJ) Tanatar & Ceperley (1989) | |
| square cell, N = 114, rs = 1, unpolarized | best variational | -0.1983(4) | VMC (SJ) Tanatar & Ceperley (1989) | |
| square cell, N = 114, rs = 10, unpolarized | best variational | -0.08423(1) | VMC (SJ) Tanatar & Ceperley (1989) | |
| square cell, N = 114, rs = 20, unpolarized | best variational | -0.045825(5) | VMC (SJ) Tanatar & Ceperley (1989) | |
| square cell, N = 114, rs = 30, unpolarized | best variational | -0.031647(4) | VMC (SJ) Tanatar & Ceperley (1989) | |
| square cell, N = 114, rs = 5, unpolarized | best variational | -0.14685(5) | VMC (SJ) Tanatar & Ceperley (1989) | |
| square cell, N = 114, rs = 50, unpolarized | best variational | -0.019648(2) | VMC (SJ) Tanatar & Ceperley (1989) | |
| triangular cell, N = 16, rs = 30, polarized | best variational | -0.031990687(3) | VMC (Fermi Sets) Li et al. (2026) | |
| best projected | -0.03199636(7) | FP-DMC (Fermi Sets) Li et al. (2026) | ||
| triangular cell, N = 91, rs = 30, polarized | best variational | -0.03193763(3) | VMC (self-attention) Gaggioli et al. (2026) | |
| best projected | -0.0319370(1) | FN-DMC (SJB) Gaggioli et al. (2026) | ||
| triangular cell, N = 91, rs = 40, polarized | best variational | -0.02443814(2) | VMC (self-attention) Gaggioli et al. (2026) | |
| best projected | -0.0244377(2) | FN-DMC (SJB) Gaggioli et al. (2026) | ||
| triangular cell, N = 169, rs = 30, polarized | best variational | -0.03192077(3) | VMC (self-attention) Gaggioli et al. (2026) | |
| best projected | -0.0319197(1) | FN-DMC (SJB) Gaggioli et al. (2026) |
E/N (Ha)
| instance | value | energy | method | code |
|---|---|---|---|---|
| H4 chain, open, R = 1.51178, STO-3G | exact | -0.5418901 | FCI QMBL check, checks/hchain-fci/ | QMBL code |
| best variational | -0.5418875 ○ (+2.6e-6) | VQE Li et al. (2023) | tencent-quantum-lab/TenCirChem | |
| H4 chain, open, R = 1.8, STO-6G | exact | -0.547595 | FCI Li et al. (2023) | |
| best variational | -0.547595 ○ (+<1e-6) | VMC (RBM) Li et al. (2023) | ||
| H4 chain, open, R = 3.6, STO-6G | exact | -0.481645 | FCI Li et al. (2023) | |
| best variational | -0.481645 ○ (+<1e-6) | VMC (RBM) Li et al. (2023) | ||
| H6 chain, open, R = 1.51178, STO-3G | exact | -0.5340686 | FCI QMBL check, checks/hchain-fci/ | QMBL code |
| best variational | -0.5340233 ○ (+4.5e-5) | VQE Li et al. (2023) | tencent-quantum-lab/TenCirChem | |
| H6 chain, open, R = 1.8, STO-6G | exact | -0.5444567 | FCI Li et al. (2023) | |
| best variational | -0.5444567 ○ (+<1e-7) | VMC (RBM) Li et al. (2023) | ||
| best approximate | -0.5444567 ○ (+<1e-7) | CCSD(T) Li et al. (2023) | ||
| H6 chain, open, R = 2, STO-3G | exact | -0.5362833 | ED Kowalski & Bauman (2023) | |
| best variational | -0.5362167 ○ (+6.7e-5) | QFlow Kowalski & Bauman (2023) | ||
| best projected | -0.5363333 ○ (−5.0e-5) | CCSDT Kowalski & Bauman (2023) | ||
| H6 chain, open, R = 2.83459, STO-6G | exact | -0.5033663 | FCI Stair et al. (2020) | |
| H6 chain, open, R = 3, cc-pVDZ | exact | -0.5265167 | ED Kowalski & Bauman (2023) | |
| best projected | -0.5265167 ○ (+<1e-7) | CCSDTQ Kowalski & Bauman (2023) | ||
| best approximate | -0.5264833 ○ (+3.3e-5) | DUCC-QFlow Kowalski & Bauman (2023) | ||
| H6 chain, open, R = 3, STO-3G | exact | -0.4929333 | ED Kowalski & Bauman (2023) | |
| best variational | -0.4920167 ○ (+9.2e-4) | QFlow Kowalski & Bauman (2023) | ||
| best projected | -0.4929 ○ (+3.3e-5) | CCSDTQ Kowalski & Bauman (2023) | ||
| H6 chain, open, R = 3.6, STO-6G | exact | -0.4817533 | FCI Li et al. (2023) | |
| best variational | -0.4817517 ○ (+1.7e-6) | VMC (RBM) Li et al. (2023) | ||
| H8 chain, open, R = 1.51178, STO-3G | exact | -0.5304239 | FCI QMBL check, checks/hchain-fci/ | QMBL code |
| best variational | -0.530315 ○ (+1.1e-4) | VQE Li et al. (2023) | tencent-quantum-lab/TenCirChem | |
| H8 chain, open, R = 1.8, STO-6G | exact | -0.543135 | FCI Li et al. (2023) | |
| best variational | -0.5431337 ○ (+1.3e-6) | VMC (RBM) Li et al. (2023) | ||
| best approximate | -0.543125 ○ (+1.0e-5) | CCSD(T) Li et al. (2023) | ||
| H8 chain, open, R = 2, STO-3G | exact | -0.53575 | ED Kowalski & Bauman (2023) | |
| best variational | -0.5355875 ○ (+1.6e-4) | QFlow Kowalski & Bauman (2023) | ||
| best projected | -0.53575 ○ (+<1e-5) | CCSDTQ Kowalski & Bauman (2023) | ||
| H8 chain, open, R = 2.83459, STO-6G | exact | -0.503519 | FCI Stair et al. (2020) | |
| H8 chain, open, R = 3, STO-3G | exact | -0.4930875 | ED Kowalski & Bauman (2023) | |
| best variational | -0.491525 ○ (+1.6e-3) | QFlow Kowalski & Bauman (2023) | ||
| best projected | -0.4929875 ○ (+1.0e-4) | CCSDTQ Kowalski & Bauman (2023) | ||
| H8 chain, open, R = 3.6, STO-6G | exact | -0.4818225 | FCI Li et al. (2023) | |
| best variational | -0.4818188 ○ (+3.7e-6) | VMC (RBM) Li et al. (2023) | ||
| H10 chain, open, R = 1, aug-cc-pVQZ | best variational | -0.41566 ○ | HF Motta et al. (2020) | |
| best approximate | -0.43918 ○ | MP2 Motta et al. (2020) | ||
| H10 chain, open, R = 1, aug-cc-pVTZ | best variational | -0.41495 ○ | HF Motta et al. (2020) | |
| best projected | -0.44012 ○ | CCSD Motta et al. (2020) | ||
| best approximate | -0.43752 ○ | MP2 Motta et al. (2020) | ||
| H10 chain, open, R = 1, cc-pV5Z | best variational | -0.4155 ○ | HF Motta et al. (2020) | |
| best projected | -0.44236(8) | ph-AFQMC Motta et al. (2017) | ||
| best approximate | -0.43927 ○ | MP2 Motta et al. (2020) | ||
| H10 chain, open, R = 1, cc-pVDZ | best variational | -0.42186 ○ | DMRG Motta et al. (2017) | |
| best projected | -0.4220847 ○ | DMET Motta et al. (2017) | ||
| best approximate | -0.4217751 ○ | RCCSD(T) Motta et al. (2017) | ||
| H10 chain, open, R = 1, cc-pVQZ | best variational | -0.41505 ○ | HF Motta et al. (2020) | |
| best projected | -0.44160(8) | ph-AFQMC Motta et al. (2017) | ||
| best approximate | -0.4413939 ○ | RCCSD(T) Motta et al. (2017) | ||
| H10 chain, open, R = 1, cc-pVTZ | best variational | -0.4395 ○ | DMRG Motta et al. (2017) | |
| best projected | -0.43969(6) | ph-AFQMC Motta et al. (2017) | ||
| best approximate | -0.4394551 ○ | RCCSD(T) Motta et al. (2017) | ||
| H10 chain, open, R = 1, basis-free | best variational | -0.441499(9) | VMC (JAGP) Motta et al. (2017) | |
| best projected | -0.442430(7) | LR-DMC (AGP) Motta et al. (2017) | ||
| best extrapolated | -0.4432(3) | MRCI+Q Motta et al. (2017) | ||
| best approximate | -0.4427(2) | RCCSD(T) Motta et al. (2017) | ||
| H10 chain, open, R = 1, STO-6G | exact | -0.3824389 | FCI Motta et al. (2017) | |
| best variational | -0.3824389 ○ (−<1e-7) | MRCI Motta et al. (2017) | ||
| best projected | -0.38248(3) (−4.1e-5) | ph-AFQMC Motta et al. (2017) | ||
| best approximate | -0.382432 ○ (+6.9e-6) | RCCSD(T) Motta et al. (2017) | ||
| H10 chain, open, R = 1.2, cc-pV5Z | best variational | -0.4884091 ○ | HF Motta et al. (2017) | |
| best projected | -0.51451(6) | ph-AFQMC Motta et al. (2017) | ||
| H10 chain, open, R = 1.2, cc-pVDZ | best variational | -0.50271 ○ | DMRG Motta et al. (2017) | |
| best projected | -0.5029843 ○ | DMET Motta et al. (2017) | ||
| best approximate | -0.506634 ○ | GW Motta et al. (2017) | ||
| H10 chain, open, R = 1.2, cc-pVQZ | best variational | -0.4882713 ○ | HF Motta et al. (2017) | |
| best projected | -0.51419(8) | ph-AFQMC Motta et al. (2017) | ||
| best approximate | -0.5140214 ○ | RCCSD(T) Motta et al. (2017) | ||
| H10 chain, open, R = 1.2, cc-pVTZ | best variational | -0.51302 ○ | DMRG Motta et al. (2017) | |
| best projected | -0.51315(9) | ph-AFQMC Motta et al. (2017) | ||
| best approximate | -0.5129509 ○ | RCCSD(T) Motta et al. (2017) | ||
| H10 chain, open, R = 1.2, basis-free | best variational | -0.513646(9) | VMC (JAGP) Motta et al. (2017) | |
| best projected | -0.514432(6) | LR-DMC (AGP) Motta et al. (2017) | ||
| best extrapolated | -0.5153(2) | MRCI+Q Motta et al. (2017) | ||
| best approximate | -0.5149(2) | RCCSD(T) Motta et al. (2017) | ||
| H10 chain, open, R = 1.2, STO-6G | exact | -0.4766378 | FCI Motta et al. (2017) | |
| best variational | -0.4766378 ○ (−<1e-7) | DMRG Motta et al. (2017) | ||
| best projected | -0.476638 ○ (−2.3e-7) | CCSDT Low et al. (2019) | ||
| best approximate | -0.4766261 ○ (+1.2e-5) | RCCSD(T) Motta et al. (2017) | ||
| H10 chain, open, R = 1.4, cc-pV5Z | best variational | -0.5228716 ○ | HF Motta et al. (2017) | |
| best projected | -0.54871(9) | ph-AFQMC Motta et al. (2017) | ||
| H10 chain, open, R = 1.4, cc-pVDZ | best variational | -0.54087 ○ | DMRG Motta et al. (2017) | |
| best projected | -0.5411661 ○ | DMET Motta et al. (2017) | ||
| best approximate | -0.54465 ○ | GW Motta et al. (2017) | ||
| H10 chain, open, R = 1.4, cc-pVQZ | best variational | -0.547832 ○ | MRCI Motta et al. (2017) | |
| best projected | -0.54855(5) | ph-AFQMC Motta et al. (2017) | ||
| best approximate | -0.5483736 ○ | RCCSD(T) Motta et al. (2017) | ||
| H10 chain, open, R = 1.4, cc-pVTZ | best variational | -0.54769 ○ | DMRG Motta et al. (2017) | |
| best projected | -0.5477564 ○ | MRCI+Q Motta et al. (2017) | ||
| best approximate | -0.547569 ○ | RCCSD(T) Motta et al. (2017) | ||
| H10 chain, open, R = 1.4, basis-free | best variational | -0.548046(7) | VMC (JAGP) Motta et al. (2017) | |
| best projected | -0.548748(7) | LR-DMC (LDA determinant) Motta et al. (2017) | ||
| best extrapolated | -0.5493(2) | MRCI+Q Motta et al. (2017) | ||
| best approximate | -0.5491(1) | RCCSD(T) Motta et al. (2017) | ||
| H10 chain, open, R = 1.4, STO-6G | exact | -0.5205094 | FCI Motta et al. (2017) | |
| best variational | -0.5205094 ○ (−<1e-7) | MRCI Motta et al. (2017) | ||
| best projected | -0.52051(5) (−5.9e-7) | ph-AFQMC Motta et al. (2017) | ||
| best approximate | -0.520492 ○ (+1.7e-5) | RCCSD(T) Motta et al. (2017) | ||
| H10 chain, open, R = 1.41729, STO-6G | exact | -0.522856 | FCI Stair & Evangelista (2020) | |
| H10 chain, open, R = 1.51178, STO-3G | exact | -0.5283552 | FCI QMBL check, checks/hchain-fci/ | QMBL code |
| best variational | -0.528221 ○ (+1.3e-4) | VQE Li et al. (2023) | tencent-quantum-lab/TenCirChem | |
| H10 chain, open, R = 1.6, cc-pV5Z | best variational | -0.5368359 ○ | HF Motta et al. (2017) | |
| best projected | -0.5629322 ○ | MRCI+Q Motta et al. (2017) | ||
| H10 chain, open, R = 1.6, cc-pVDZ | best variational | -0.55702 ○ | DMRG Motta et al. (2017) | |
| best projected | -0.5573602 ○ | DMET Motta et al. (2017) | ||
| best approximate | -0.56045 ○ | GW Motta et al. (2017) | ||
| H10 chain, open, R = 1.6, cc-pVQZ | best variational | -0.5620252 ○ | MRCI Motta et al. (2017) | |
| best projected | -0.5627104 ○ | MRCI+Q Motta et al. (2017) | ||
| best approximate | -0.562514 ○ | RCCSD(T) Motta et al. (2017) | ||
| H10 chain, open, R = 1.6, cc-pVTZ | best variational | -0.56191 ○ | DMRG Motta et al. (2017) | |
| best projected | -0.5620(1) | ph-AFQMC Motta et al. (2017) | ||
| best approximate | -0.5617783 ○ | RCCSD(T) Motta et al. (2017) | ||
| H10 chain, open, R = 1.6, basis-free | best variational | -0.562237(5) | VMC (JAGP) Motta et al. (2017) | |
| best projected | -0.562900(7) | LR-DMC (LDA determinant) Motta et al. (2017) | ||
| best extrapolated | -0.56325(8) | MRCI+Q Motta et al. (2017) | ||
| best approximate | -0.56311(9) | RCCSD(T) Motta et al. (2017) | ||
| H10 chain, open, R = 1.6, STO-6G | exact | -0.5384361 | FCI Motta et al. (2017) | |
| best variational | -0.5384361 ○ (+<1e-7) | MRCI Motta et al. (2017) | ||
| best projected | -0.5384361 ○ (+<1e-7) | MRCI+Q Motta et al. (2017) | ||
| best approximate | -0.5384141 ○ (+2.2e-5) | RCCSD(T) Motta et al. (2017) | ||
| H10 chain, open, R = 1.8, cc-pV5Z | best variational | -0.5401556 ○ | HF Motta et al. (2017) | |
| best projected | -0.566277 ○ | MRCI+Q Motta et al. (2017) | ||
| H10 chain, open, R = 1.8, cc-pVDZ | best variational | -0.56141 ○ | DMRG Motta et al. (2017) | |
| best projected | -0.5618118 ○ | DMET Motta et al. (2017) | ||
| best approximate | -0.564291 ○ | GW Motta et al. (2017) | ||
| H10 chain, open, R = 1.8, cc-pVQZ | best variational | -0.5654171 ○ | MRCI Motta et al. (2017) | |
| best projected | -0.5660681 ○ | MRCI+Q Motta et al. (2017) | ||
| best approximate | -0.5658519 ○ | RCCSD(T) Motta et al. (2017) | ||
| H10 chain, open, R = 1.8, cc-pVTZ | best variational | -0.56523 ○ | DMRG Motta et al. (2017) | |
| best projected | -0.5653303 ○ | MRCI+Q Motta et al. (2017) | ||
| best approximate | -0.5651061 ○ | RCCSD(T) Motta et al. (2017) | ||
| H10 chain, open, R = 1.8, basis-free | best variational | -0.565634(5) | VMC (JAGP) Motta et al. (2017) | |
| best projected | -0.566255(7) | LR-DMC (LDA determinant) Motta et al. (2017) | ||
| best extrapolated | -0.5667(1) | SBDMRG Motta et al. (2017) | ||
| best approximate | -0.56638(4) | RCCSD(T) Motta et al. (2017) | ||
| H10 chain, open, R = 1.8, STO-6G | exact | -0.5424385 | FCI Motta et al. (2017) | |
| best variational | -0.5424385 ○ (−<1e-7) | MRCI Motta et al. (2017) | ||
| best projected | -0.542439 ○ (−4.6e-7) | CCSDTQ Low et al. (2019) | ||
| best approximate | -0.542419 ○ (+2.0e-5) | CCSD(T) Li et al. (2023) | ||
| H10 chain, open, R = 1.88973, STO-6G | exact | -0.5415393 | FCI Stair & Evangelista (2020) | |
| H10 chain, open, R = 2, cc-pV5Z | best variational | -0.538217 ○ | HF Motta et al. (2017) | |
| best projected | -0.5638526 ○ | MRCI+Q Motta et al. (2017) | ||
| H10 chain, open, R = 2, cc-pVDZ | best variational | -0.55949 ○ | DMRG Motta et al. (2017) | |
| best projected | -0.5599357 ○ | DMET Motta et al. (2017) | ||
| best approximate | -0.561647 ○ | GW Motta et al. (2017) | ||
| H10 chain, open, R = 2, cc-pVQZ | best variational | -0.563033 ○ | MRCI Motta et al. (2017) | |
| best projected | -0.5636456 ○ | MRCI+Q Motta et al. (2017) | ||
| best approximate | -0.5634155 ○ | RCCSD(T) Motta et al. (2017) | ||
| H10 chain, open, R = 2, cc-pVTZ | best variational | -0.56282 ○ | DMRG Motta et al. (2017) | |
| best projected | -0.5628926 ○ | MRCI+Q Motta et al. (2017) | ||
| best approximate | -0.562658 ○ | RCCSD(T) Motta et al. (2017) | ||
| H10 chain, open, R = 2, basis-free | best variational | -0.563249(4) | VMC (JAGP) Motta et al. (2017) | |
| best projected | -0.563852(7) | LR-DMC (LDA determinant) Motta et al. (2017) | ||
| best extrapolated | -0.5643(1) | SBDMRG Motta et al. (2017) | ||
| best approximate | -0.56391(1) | RCCSD(T) Motta et al. (2017) | ||
| H10 chain, open, R = 2, STO-6G | exact | -0.5389626 | FCI Motta et al. (2017) | |
| best variational | -0.5389626 ○ (−<1e-7) | DMRG Motta et al. (2017) | ||
| best projected | -0.538963 ○ (−4.1e-7) | CCSDTQ Low et al. (2019) | ||
| best approximate | -0.5380497 ○ (+9.1e-4) | UCCSD(T) Motta et al. (2017) | ||
| H10 chain, open, R = 2.4, cc-pV5Z | best variational | -0.5281156 ○ | HF Motta et al. (2017) | |
| best projected | -0.5516514 ○ | MRCI+Q Motta et al. (2017) | ||
| H10 chain, open, R = 2.4, cc-pVDZ | best variational | -0.54761 ○ | DMRG Motta et al. (2017) | |
| best projected | -0.5480704 ○ | DMET Motta et al. (2017) | ||
| best approximate | -0.547728 ○ | GW Motta et al. (2017) | ||
| H10 chain, open, R = 2.4, cc-pVQZ | best variational | -0.5509118 ○ | MRCI Motta et al. (2017) | |
| best projected | -0.5514268 ○ | MRCI+Q Motta et al. (2017) | ||
| best approximate | -0.5512171 ○ | RCCSD(T) Motta et al. (2017) | ||
| H10 chain, open, R = 2.4, cc-pVTZ | best variational | -0.55059 ○ | DMRG Motta et al. (2017) | |
| best projected | -0.5506631 ○ | MRCI+Q Motta et al. (2017) | ||
| best approximate | -0.550459 ○ | RCCSD(T) Motta et al. (2017) | ||
| H10 chain, open, R = 2.4, basis-free | best variational | -0.551106(4) | VMC (JAGP) Motta et al. (2017) | |
| best projected | -0.551699(8) | LR-DMC (LDA determinant) Motta et al. (2017) | ||
| best extrapolated | -0.5520(1) | SBDMRG Motta et al. (2017) | ||
| best approximate | -0.55167(3) | RCCSD(T) Motta et al. (2017) | ||
| H10 chain, open, R = 2.4, STO-6G | exact | -0.5227936 | FCI Motta et al. (2017) | |
| best variational | -0.5227936 ○ (+<1e-7) | MRCI Motta et al. (2017) | ||
| best projected | -0.522794 ○ (−3.5e-7) | CCSDTQ Low et al. (2019) | ||
| best approximate | -0.5207534 ○ (+2.0e-3) | UCCSD(T) Motta et al. (2017) | ||
| H10 chain, open, R = 2.8, cc-pV5Z | best variational | -0.5174129 ○ | HF Motta et al. (2017) | |
| best projected | -0.53749(7) | ph-AFQMC Motta et al. (2017) | ||
| H10 chain, open, R = 2.8, cc-pVDZ | best variational | -0.53344 ○ | DMRG Motta et al. (2017) | |
| best projected | -0.5347972 ○ | SEET Motta et al. (2017) | ||
| best approximate | -0.5336142 ○ | RCCSD(T) Motta et al. (2017) | ||
| H10 chain, open, R = 2.8, cc-pVQZ | best variational | -0.5367131 ○ | MRCI Motta et al. (2017) | |
| best projected | -0.5370991 ○ | MRCI+Q Motta et al. (2017) | ||
| best approximate | -0.5370925 ○ | RCCSD(T) Motta et al. (2017) | ||
| H10 chain, open, R = 2.8, cc-pVTZ | best variational | -0.53626 ○ | DMRG Motta et al. (2017) | |
| best projected | -0.53633(5) | ph-AFQMC Motta et al. (2017) | ||
| best approximate | -0.5363253 ○ | RCCSD(T) Motta et al. (2017) | ||
| H10 chain, open, R = 2.8, basis-free | best variational | -0.536832(3) | VMC (JAGP) Motta et al. (2017) | |
| best projected | -0.537424(9) | LR-DMC (LDA determinant) Motta et al. (2017) | ||
| best extrapolated | -0.5377(1) | SBDMRG Motta et al. (2017) | ||
| best approximate | -0.53749(6) | RCCSD(T) Motta et al. (2017) | ||
| H10 chain, open, R = 2.8, STO-6G | exact | -0.5050245 | FCI Motta et al. (2017) | |
| best variational | -0.5050245 ○ (−<1e-7) | MRCI Motta et al. (2017) | ||
| best projected | -0.50507(5) (−4.6e-5) | ph-AFQMC Motta et al. (2017) | ||
| best approximate | -0.5029597 ○ (+2.1e-3) | UCCSD(T) Motta et al. (2017) | ||
| H10 chain, open, R = 2.83459, STO-6G | exact | -0.5036293 | FCI Stair & Evangelista (2020) | |
| H10 chain, open, R = 3.2, cc-pV5Z | best variational | -0.5096605 ○ | HF Motta et al. (2017) | |
| best projected | -0.52525(5) | ph-AFQMC Motta et al. (2017) | ||
| H10 chain, open, R = 3.2, cc-pVDZ | best variational | -0.52125 ○ | DMRG Motta et al. (2017) | |
| best projected | -0.5234726 ○ | SEET Motta et al. (2017) | ||
| best approximate | -0.5228438 ○ | RCCSD(T) Motta et al. (2017) | ||
| H10 chain, open, R = 3.2, cc-pVQZ | best variational | -0.5243413 ○ | MRCI Motta et al. (2017) | |
| best projected | -0.52488(5) | ph-AFQMC Motta et al. (2017) | ||
| best approximate | -0.5254869 ○ | RCCSD(T) Motta et al. (2017) | ||
| H10 chain, open, R = 3.2, cc-pVTZ | best variational | -0.52382 ○ | DMRG Motta et al. (2017) | |
| best projected | -0.52417(5) | ph-AFQMC Motta et al. (2017) | ||
| best approximate | -0.5248344 ○ | RCCSD(T) Motta et al. (2017) | ||
| H10 chain, open, R = 3.2, basis-free | best variational | -0.524373(4) | VMC (JAGP) Motta et al. (2017) | |
| best projected | -0.524987(7) | LR-DMC (AGP) Motta et al. (2017) | ||
| best extrapolated | -0.52512(6) | DMET Motta et al. (2017) | ||
| best approximate | -0.5257(1) | RCCSD(T) Motta et al. (2017) | ||
| H10 chain, open, R = 3.2, STO-6G | exact | -0.4910383 | FCI Motta et al. (2017) | |
| best variational | -0.4910383 ○ (−<1e-7) | MRCI Motta et al. (2017) | ||
| best projected | -0.4910383 ○ (−<1e-7) | MRCI+Q Motta et al. (2017) | ||
| best approximate | -0.4895218 ○ (+1.5e-3) | UCCSD(T) Motta et al. (2017) | ||
| H10 chain, open, R = 3.6, cc-pV5Z | best variational | -0.5050291 ○ | HF Motta et al. (2017) | |
| best projected | -0.51584(8) | ph-AFQMC Motta et al. (2017) | ||
| H10 chain, open, R = 3.6, cc-pVDZ | best variational | -0.51238 ○ | DMRG Motta et al. (2017) | |
| best projected | -0.51287(3) | ph-AFQMC Motta et al. (2017) | ||
| best approximate | -0.5119131 ○ | PC-NEVPT2 Motta et al. (2017) | ||
| H10 chain, open, R = 3.6, cc-pVQZ | best variational | -0.5150225 ○ | MRCI Motta et al. (2017) | |
| best projected | -0.51580(5) | ph-AFQMC Motta et al. (2017) | ||
| best approximate | -0.514026 ○ | UCCSD(T) Motta et al. (2017) | ||
| H10 chain, open, R = 3.6, cc-pVTZ | best variational | -0.51448 ○ | DMRG Motta et al. (2017) | |
| best projected | -0.51507(3) | ph-AFQMC Motta et al. (2017) | ||
| best approximate | -0.513382 ○ | PC-NEVPT2 Motta et al. (2017) | ||
| H10 chain, open, R = 3.6, basis-free | best variational | -0.514968(3) | VMC (JAGP) Motta et al. (2017) | |
| best projected | -0.51558(1) | LR-DMC (LDA determinant) Motta et al. (2017) | ||
| best extrapolated | -0.51612(9) | DMET Motta et al. (2017) | ||
| best approximate | -0.5143(1) | UCCSD(T) Motta et al. (2017) | ||
| H10 chain, open, R = 3.6, STO-6G | exact | -0.4818701 | FCI Motta et al. (2017) | |
| best variational | -0.4818701 ○ (+<1e-7) | MRCI Motta et al. (2017) | ||
| best projected | -0.4818701 ○ (+<1e-7) | MRCI+Q Motta et al. (2017) | ||
| best approximate | -0.4809595 ○ (+9.1e-4) | UCCSD(T) Motta et al. (2017) | ||
| H10 chain, open, R = 3.77945, STO-6G | exact | -0.4790989 | FCI Stair & Evangelista (2020) | |
| H12 chain, open, R = 1.51178, STO-3G | exact | -0.5270349 | FCI QMBL check, checks/hchain-fci/ | QMBL code |
| best variational | -0.5268575 ○ (+1.8e-4) | VQE Li et al. (2023) | tencent-quantum-lab/TenCirChem | |
| H12 chain, open, R = 1.8, STO-6G | exact | -0.5420192 | FCI Li et al. (2023) | |
| best variational | -0.5420125 ○ (+6.7e-6) | VMC (RBM) Li et al. (2023) | ||
| best approximate | -0.5419892 ○ (+3.0e-5) | CCSD(T) Li et al. (2023) | ||
| H12 chain, open, R = 1.88973, STO-3G | exact | -0.5377347 | ED Huang et al. (2026) | |
| H12 chain, open, R = 3.6, STO-6G | exact | -0.4819042 | FCI Li et al. (2023) | |
| best variational | -0.4819025 ○ (+1.7e-6) | VMC (RBM) Li et al. (2023) | ||
| H14 chain, open, R = 1.51178, STO-3G | exact | -0.5261238 | FCI QMBL check, checks/hchain-fci/ | QMBL code |
| best variational | -0.525915 ○ (+2.1e-4) | VQE Li et al. (2023) | tencent-quantum-lab/TenCirChem | |
| H16 chain, open, R = 1.51178, STO-3G | exact | -0.5254592 | FCI QMBL check, checks/hchain-fci/ | QMBL code |
| best variational | -0.5252269 ○ (+2.3e-4) | VQE Li et al. (2023) | tencent-quantum-lab/TenCirChem | |
| H20 chain, open, R = 3.6, STO-6G | best projected | -0.481978(1) | FCIQMC Guther et al. (2020) | ghb24/NECI_STABLE |
| H22 chain, open, R = 1, STO-6G | best projected | -0.357541(9) | ph-AFQMC Motta et al. (2017) | |
| H22 chain, open, R = 1.4, STO-6G | best projected | -0.51444(1) | ph-AFQMC Motta et al. (2017) | |
| H22 chain, open, R = 1.8, STO-6G | best projected | -0.54115(2) | ph-AFQMC Motta et al. (2017) | |
| H22 chain, open, R = 2.8, STO-6G | best projected | -0.50541(2) | ph-AFQMC Motta et al. (2017) | |
| H30 chain, open, R = 1, cc-pV5Z | best variational | -0.3969058 ○ | HF Motta et al. (2017) | |
| H30 chain, open, R = 1, cc-pVDZ | best variational | -0.3937633 ○ | DMRG Motta et al. (2017) | |
| best projected | -0.39379(3) | ph-AFQMC Motta et al. (2017) | ||
| best approximate | -0.3936663 ○ | RCCSD(T) Motta et al. (2017) | ||
| H30 chain, open, R = 1, cc-pVQZ | best variational | -0.3948131 ○ | HF Motta et al. (2017) | |
| best projected | -0.42184(7) | ph-AFQMC Motta et al. (2017) | ||
| best approximate | -0.421883 ○ | RCCSD(T) Motta et al. (2017) | ||
| H30 chain, open, R = 1, cc-pVTZ | best variational | -0.3898388 ○ | HF Motta et al. (2017) | |
| best projected | -0.41760(3) | ph-AFQMC Motta et al. (2017) | ||
| best approximate | -0.41743 ○ | RCCSD(T) Motta et al. (2017) | ||
| H30 chain, open, R = 1, basis-free | best variational | -0.42157(2) | VMC (SJ) Motta et al. (2017) | |
| best projected | -0.42510(3) | LR-DMC (LDA determinant) Motta et al. (2017) | ||
| H30 chain, open, R = 1, STO-6G | best variational | -0.3518877 ○ | DMRG Motta et al. (2017) | |
| best projected | -0.3528238 ○ | DMET Motta et al. (2017) | ||
| best approximate | -0.35206 ○ | GW Motta et al. (2017) | ||
| H30 chain, open, R = 1.2, cc-pV5Z | best variational | -0.4739482 ○ | HF Motta et al. (2017) | |
| H30 chain, open, R = 1.2, cc-pVDZ | best variational | -0.4891083 ○ | DMRG Motta et al. (2017) | |
| best projected | -0.48927(3) | ph-AFQMC Motta et al. (2017) | ||
| best approximate | -0.489002 ○ | RCCSD(T) Motta et al. (2017) | ||
| H30 chain, open, R = 1.2, cc-pVQZ | best variational | -0.4736828 ○ | HF Motta et al. (2017) | |
| best projected | -0.50143(5) | ph-AFQMC Motta et al. (2017) | ||
| best approximate | -0.5011587 ○ | RCCSD(T) Motta et al. (2017) | ||
| H30 chain, open, R = 1.2, cc-pVTZ | best variational | -0.4731441 ○ | HF Motta et al. (2017) | |
| best projected | -0.50030(3) | ph-AFQMC Motta et al. (2017) | ||
| best approximate | -0.4999623 ○ | RCCSD(T) Motta et al. (2017) | ||
| H30 chain, open, R = 1.2, basis-free | best variational | -0.50020(1) | VMC (SJ) Motta et al. (2017) | |
| best projected | -0.50177(2) | LR-DMC (LDA determinant) Motta et al. (2017) | ||
| H30 chain, open, R = 1.2, STO-6G | best variational | -0.461805 ○ | DMRG Motta et al. (2017) | |
| best projected | -0.4627503 ○ | DMET Motta et al. (2017) | ||
| best approximate | -0.4617621 ○ | RCCSD(T) Motta et al. (2017) | ||
| H30 chain, open, R = 1.4, cc-pV5Z | best variational | -0.5155844 ○ | HF Motta et al. (2017) | |
| H30 chain, open, R = 1.4, cc-pVDZ | best variational | -0.5344613 ○ | DMRG Motta et al. (2017) | |
| best projected | -0.5354635 ○ | DMET Motta et al. (2017) | ||
| best approximate | -0.5343077 ○ | RCCSD(T) Motta et al. (2017) | ||
| H30 chain, open, R = 1.4, cc-pVQZ | best variational | -0.5154895 ○ | HF Motta et al. (2017) | |
| best projected | -0.54253(5) | ph-AFQMC Motta et al. (2017) | ||
| best approximate | -0.542323 ○ | RCCSD(T) Motta et al. (2017) | ||
| H30 chain, open, R = 1.4, cc-pVTZ | best variational | -0.5152041 ○ | HF Motta et al. (2017) | |
| best projected | -0.54186(3) | ph-AFQMC Motta et al. (2017) | ||
| best approximate | -0.5414947 ○ | RCCSD(T) Motta et al. (2017) | ||
| H30 chain, open, R = 1.4, basis-free | best variational | -0.54142(1) | VMC (SJ) Motta et al. (2017) | |
| best projected | -0.54276(1) | LR-DMC (LDA determinant) Motta et al. (2017) | ||
| H30 chain, open, R = 1.4, STO-6G | best variational | -0.5132063 ○ | DMRG Motta et al. (2017) | |
| best projected | -0.5140977 ○ | DMET Motta et al. (2017) | ||
| best approximate | -0.5131409 ○ | RCCSD(T) Motta et al. (2017) | ||
| H30 chain, open, R = 1.6, cc-pV5Z | best variational | -0.5337486 ○ | HF Motta et al. (2017) | |
| H30 chain, open, R = 1.6, cc-pVDZ | best variational | -0.5542583 ○ | DMRG Motta et al. (2017) | |
| best projected | -0.55440(3) | ph-AFQMC Motta et al. (2017) | ||
| best approximate | -0.5540527 ○ | RCCSD(T) Motta et al. (2017) | ||
| H30 chain, open, R = 1.6, cc-pVQZ | best variational | -0.5336765 ○ | HF Motta et al. (2017) | |
| best projected | -0.55999(7) | ph-AFQMC Motta et al. (2017) | ||
| best approximate | -0.5599073 ○ | RCCSD(T) Motta et al. (2017) | ||
| H30 chain, open, R = 1.6, cc-pVTZ | best variational | -0.5334553 ○ | HF Motta et al. (2017) | |
| best projected | -0.55942(3) | ph-AFQMC Motta et al. (2017) | ||
| best approximate | -0.559162 ○ | RCCSD(T) Motta et al. (2017) | ||
| H30 chain, open, R = 1.6, basis-free | best variational | -0.559065(9) | VMC (SJ) Motta et al. (2017) | |
| best projected | -0.56038(1) | LR-DMC (LDA determinant) Motta et al. (2017) | ||
| H30 chain, open, R = 1.6, STO-6G | best variational | -0.534978 ○ | DMRG Motta et al. (2017) | |
| best projected | -0.5357627 ○ | DMET Motta et al. (2017) | ||
| best approximate | -0.534889 ○ | RCCSD(T) Motta et al. (2017) | ||
| H30 chain, open, R = 1.8, cc-pV5Z | best variational | -0.5394784 ○ | HF Motta et al. (2017) | |
| H30 chain, open, R = 1.8, cc-pVDZ | best variational | -0.560521 ○ | DMRG Motta et al. (2017) | |
| best projected | -0.5616303 ○ | DMET Motta et al. (2017) | ||
| best approximate | -0.560279 ○ | RCCSD(T) Motta et al. (2017) | ||
| H30 chain, open, R = 1.8, cc-pVQZ | best variational | -0.5394138 ○ | HF Motta et al. (2017) | |
| best projected | -0.56530(5) | ph-AFQMC Motta et al. (2017) | ||
| best approximate | -0.564994 ○ | RCCSD(T) Motta et al. (2017) | ||
| H30 chain, open, R = 1.8, cc-pVTZ | best variational | -0.5392194 ○ | HF Motta et al. (2017) | |
| best projected | -0.56451(3) | ph-AFQMC Motta et al. (2017) | ||
| best approximate | -0.564239 ○ | RCCSD(T) Motta et al. (2017) | ||
| H30 chain, open, R = 1.8, basis-free | best variational | -0.564180(9) | VMC (SJ) Motta et al. (2017) | |
| best projected | -0.56550(1) | LR-DMC (LDA determinant) Motta et al. (2017) | ||
| H30 chain, open, R = 1.8, STO-6G | best variational | -0.5409897 ○ | DMRG Motta et al. (2017) | |
| best projected | -0.5416268 ○ | DMET Motta et al. (2017) | ||
| best approximate | -0.5408894 ○ | RCCSD(T) Motta et al. (2017) | ||
| H30 chain, open, R = 2, cc-pV5Z | best variational | -0.5386818 ○ | HF Motta et al. (2017) | |
| H30 chain, open, R = 2, cc-pVDZ | best variational | -0.5595467 ○ | DMRG Motta et al. (2017) | |
| best projected | -0.55961(2) | ph-AFQMC Motta et al. (2017) | ||
| best approximate | -0.55928 ○ | RCCSD(T) Motta et al. (2017) | ||
| H30 chain, open, R = 2, cc-pVQZ | best variational | -0.5386235 ○ | HF Motta et al. (2017) | |
| best projected | -0.56379(4) | ph-AFQMC Motta et al. (2017) | ||
| best approximate | -0.5634263 ○ | RCCSD(T) Motta et al. (2017) | ||
| H30 chain, open, R = 2, cc-pVTZ | best variational | -0.5384493 ○ | HF Motta et al. (2017) | |
| best projected | -0.56295(3) | ph-AFQMC Motta et al. (2017) | ||
| best approximate | -0.5626683 ○ | RCCSD(T) Motta et al. (2017) | ||
| H30 chain, open, R = 2, basis-free | best variational | -0.562636(9) | VMC (SJ) Motta et al. (2017) | |
| best projected | -0.56403(1) | LR-DMC (LDA determinant) Motta et al. (2017) | ||
| H30 chain, open, R = 2, STO-6G | best variational | -0.5385523 ○ | DMRG Motta et al. (2017) | |
| best projected | -0.5390239 ○ | DMET Motta et al. (2017) | ||
| best approximate | -0.5384874 ○ | RCCSD(T) Motta et al. (2017) | ||
| H30 chain, open, R = 2.4, cc-pV5Z | best variational | -0.5292216 ○ | HF Motta et al. (2017) | |
| H30 chain, open, R = 2.4, cc-pVDZ | best variational | -0.5482932 ○ | DMRG Motta et al. (2017) | |
| best projected | -0.5493137 ○ | DMET Motta et al. (2017) | ||
| best approximate | -0.5480643 ○ | RCCSD(T) Motta et al. (2017) | ||
| H30 chain, open, R = 2.4, cc-pVQZ | best variational | -0.5291749 ○ | HF Motta et al. (2017) | |
| best projected | -0.55215(4) | ph-AFQMC Motta et al. (2017) | ||
| best approximate | -0.5517193 ○ | RCCSD(T) Motta et al. (2017) | ||
| H30 chain, open, R = 2.4, cc-pVTZ | best variational | -0.529024 ○ | HF Motta et al. (2017) | |
| best projected | -0.55142(2) | ph-AFQMC Motta et al. (2017) | ||
| best approximate | -0.5509443 ○ | RCCSD(T) Motta et al. (2017) | ||
| H30 chain, open, R = 2.4, basis-free | best variational | -0.550876(9) | VMC (SJ) Motta et al. (2017) | |
| best projected | -0.55248(2) | LR-DMC (LDA determinant) Motta et al. (2017) | ||
| H30 chain, open, R = 2.4, STO-6G | best variational | -0.5231183 ○ | DMRG Motta et al. (2017) | |
| best projected | -0.523325 ○ | DMET Motta et al. (2017) | ||
| best approximate | -0.5239338 ○ | RCCSD(T) Motta et al. (2017) | ||
| H30 chain, open, R = 2.8, cc-pV5Z | best variational | -0.5183556 ○ | HF Motta et al. (2017) | |
| H30 chain, open, R = 2.8, cc-pVDZ | best variational | -0.5340762 ○ | DMRG Motta et al. (2017) | |
| best projected | -0.5347214 ○ | SEET Motta et al. (2017) | ||
| best approximate | -0.5327924 ○ | UCCSD(T) Motta et al. (2017) | ||
| H30 chain, open, R = 2.8, cc-pVQZ | best variational | -0.5183116 ○ | HF Motta et al. (2017) | |
| best projected | -0.53788(4) | ph-AFQMC Motta et al. (2017) | ||
| H30 chain, open, R = 2.8, cc-pVTZ | best variational | -0.5181643 ○ | HF Motta et al. (2017) | |
| best projected | -0.53718(3) | ph-AFQMC Motta et al. (2017) | ||
| H30 chain, open, R = 2.8, basis-free | best variational | -0.536326(9) | VMC (SJ) Motta et al. (2017) | |
| best projected | -0.53821(2) | LR-DMC (LDA determinant) Motta et al. (2017) | ||
| H30 chain, open, R = 2.8, STO-6G | best variational | -0.5054037 ○ | DMRG Motta et al. (2017) | |
| best projected | -0.5055314 ○ | DMET Motta et al. (2017) | ||
| best approximate | -0.5037865 ○ | UCCSD(T) Motta et al. (2017) | ||
| H30 chain, open, R = 3.2, cc-pV5Z | best variational | -0.5102735 ○ | HF Motta et al. (2017) | |
| H30 chain, open, R = 3.2, cc-pVDZ | best variational | -0.5216948 ○ | DMRG Motta et al. (2017) | |
| best projected | -0.5240882 ○ | SEET Motta et al. (2017) | ||
| best approximate | -0.5205262 ○ | UCCSD(T) Motta et al. (2017) | ||
| H30 chain, open, R = 3.2, cc-pVQZ | best variational | -0.5102287 ○ | HF Motta et al. (2017) | |
| best projected | -0.52512(7) | ph-AFQMC Motta et al. (2017) | ||
| H30 chain, open, R = 3.2, cc-pVTZ | best variational | -0.5100848 ○ | HF Motta et al. (2017) | |
| best projected | -0.52443(4) | ph-AFQMC Motta et al. (2017) | ||
| H30 chain, open, R = 3.2, basis-free | best variational | -0.523410(9) | VMC (SJ) Motta et al. (2017) | |
| best projected | -0.52555(2) | LR-DMC (LDA determinant) Motta et al. (2017) | ||
| H30 chain, open, R = 3.2, STO-6G | best variational | -0.4913033 ○ | DMRG Motta et al. (2017) | |
| best projected | -0.4914582 ○ | SEET Motta et al. (2017) | ||
| best approximate | -0.4900449 ○ | UCCSD(T) Motta et al. (2017) | ||
| H30 chain, open, R = 3.6, cc-pV5Z | best variational | -0.5053753 ○ | HF Motta et al. (2017) | |
| H30 chain, open, R = 3.6, cc-pVDZ | best variational | -0.512651 ○ | DMRG Motta et al. (2017) | |
| best projected | -0.5128862 ○ | DMET Motta et al. (2017) | ||
| best approximate | -0.511785 ○ | UCCSD(T) Motta et al. (2017) | ||
| H30 chain, open, R = 3.6, cc-pVQZ | best variational | -0.5053287 ○ | HF Motta et al. (2017) | |
| best projected | -0.5162413 ○ | DMET Motta et al. (2017) | ||
| H30 chain, open, R = 3.6, cc-pVTZ | best variational | -0.5051879 ○ | HF Motta et al. (2017) | |
| best projected | -0.5154642 ○ | DMET Motta et al. (2017) | ||
| H30 chain, open, R = 3.6, basis-free | best variational | -0.513577(9) | VMC (SJ) Motta et al. (2017) | |
| best projected | -0.51601(3) | LR-DMC (LDA determinant) Motta et al. (2017) | ||
| H30 chain, open, R = 3.6, STO-6G | best variational | -0.4820203 ○ | DMRG Motta et al. (2017) | |
| best projected | -0.4828246 ○ | SEET Motta et al. (2017) | ||
| best approximate | -0.4812442 ○ | UCCSD(T) Motta et al. (2017) | ||
| H50 chain, open, R = 1, cc-pV5Z | best variational | -0.3930718 ○ | HF Motta et al. (2017) | |
| H50 chain, open, R = 1, cc-pVDZ | best variational | -0.388254 ○ | DMRG Motta et al. (2017) | |
| best projected | -0.38822(2) | ph-AFQMC Motta et al. (2017) | ||
| best approximate | -0.3882394 ○ | RCCSD(T) Motta et al. (2017) | ||
| H50 chain, open, R = 1, cc-pVQZ | best variational | -0.3909264 ○ | HF Motta et al. (2017) | |
| best projected | -0.4169616 ○ | RCCSD Motta et al. (2017) | ||
| best approximate | -0.418261 ○ | RCCSD(T) Motta et al. (2017) | ||
| H50 chain, open, R = 1, cc-pVTZ | best variational | -0.3867182 ○ | HF Motta et al. (2017) | |
| best projected | -0.41352(4) | ph-AFQMC Motta et al. (2017) | ||
| best approximate | -0.4133358 ○ | RCCSD(T) Motta et al. (2017) | ||
| H50 chain, open, R = 1, basis-free | best variational | -0.41856(3) | VMC (SJ) Motta et al. (2017) | |
| best projected | -0.42197(3) | LR-DMC (LDA determinant) Motta et al. (2017) | ||
| H50 chain, open, R = 1, STO-6G | best variational | -0.3456814 ○ | LDMRG Hachmann et al. (2006) | |
| best projected | -0.3466732 ○ | DMET Motta et al. (2017) | ||
| best approximate | -0.345728 ○ | GW Motta et al. (2017) | ||
| H50 chain, open, R = 1.2, cc-pV5Z | best variational | -0.4724546 ○ | HF Motta et al. (2017) | |
| H50 chain, open, R = 1.2, cc-pVDZ | best variational | -0.486182 ○ | DMRG Motta et al. (2017) | |
| best projected | -0.48637(1) | ph-AFQMC Motta et al. (2017) | ||
| best approximate | -0.4860532 ○ | RCCSD(T) Motta et al. (2017) | ||
| H50 chain, open, R = 1.2, cc-pVQZ | best variational | -0.471144 ○ | HF Motta et al. (2017) | |
| best projected | -0.49888(3) | ph-AFQMC Motta et al. (2017) | ||
| best approximate | -0.4983418 ○ | RCCSD(T) Motta et al. (2017) | ||
| H50 chain, open, R = 1.2, cc-pVTZ | best variational | -0.4700515 ○ | HF Motta et al. (2017) | |
| best projected | -0.49746(4) | ph-AFQMC Motta et al. (2017) | ||
| best approximate | -0.4971284 ○ | RCCSD(T) Motta et al. (2017) | ||
| H50 chain, open, R = 1.2, basis-free | best variational | -0.49748(1) | VMC (SJ) Motta et al. (2017) | |
| best projected | -0.49913(4) | LR-DMC (LDA determinant) Motta et al. (2017) | ||
| H50 chain, open, R = 1.2, STO-6G | best variational | -0.458953 ○ | LDMRG Hachmann et al. (2006) | |
| best projected | -0.4599375 ○ | DMET Motta et al. (2017) | ||
| best approximate | -0.4588914 ○ | CCSD(T) Hachmann et al. (2006) | ||
| H50 chain, open, R = 1.4, cc-pV5Z | best variational | -0.5142633 ○ | HF Motta et al. (2017) | |
| H50 chain, open, R = 1.4, cc-pVDZ | best variational | -0.5331854 ○ | DMRG Motta et al. (2017) | |
| best projected | -0.5342916 ○ | DMET Motta et al. (2017) | ||
| best approximate | -0.5330286 ○ | RCCSD(T) Motta et al. (2017) | ||
| H50 chain, open, R = 1.4, cc-pVQZ | best variational | -0.5141705 ○ | HF Motta et al. (2017) | |
| best projected | -0.5401476 ○ | RCCSD Motta et al. (2017) | ||
| best approximate | -0.5411192 ○ | RCCSD(T) Motta et al. (2017) | ||
| H50 chain, open, R = 1.4, cc-pVTZ | best variational | -0.5138868 ○ | HF Motta et al. (2017) | |
| best projected | -0.54063(3) | ph-AFQMC Motta et al. (2017) | ||
| best approximate | -0.540299 ○ | RCCSD(T) Motta et al. (2017) | ||
| H50 chain, open, R = 1.4, basis-free | best variational | -0.53996(2) | VMC (SJ) Motta et al. (2017) | |
| best projected | -0.54153(2) | LR-DMC (LDA determinant) Motta et al. (2017) | ||
| H50 chain, open, R = 1.4, STO-6G | best variational | -0.5118756 ○ | LDMRG Hachmann et al. (2006) | |
| best projected | -0.5127902 ○ | DMET Motta et al. (2017) | ||
| best approximate | -0.5117824 ○ | CCSD(T) Hachmann et al. (2006) | ||
| H50 chain, open, R = 1.6, cc-pV5Z | best variational | -0.5332496 ○ | HF Motta et al. (2017) | |
| H50 chain, open, R = 1.6, cc-pVDZ | best variational | -0.5307409 ○ | HF Motta et al. (2017) | |
| best projected | -0.5548843 ○ | DMET Motta et al. (2017) | ||
| best approximate | -0.5535398 ○ | RCCSD(T) Motta et al. (2017) | ||
| H50 chain, open, R = 1.6, cc-pVQZ | best variational | -0.5331783 ○ | HF Motta et al. (2017) | |
| best projected | -0.558395 ○ | RCCSD Motta et al. (2017) | ||
| best approximate | -0.5594386 ○ | RCCSD(T) Motta et al. (2017) | ||
| H50 chain, open, R = 1.6, cc-pVTZ | best variational | -0.5329572 ○ | HF Motta et al. (2017) | |
| best projected | -0.55895(3) | ph-AFQMC Motta et al. (2017) | ||
| best approximate | -0.5587018 ○ | RCCSD(T) Motta et al. (2017) | ||
| H50 chain, open, R = 1.6, basis-free | best variational | -0.55836(1) | VMC (SJ) Motta et al. (2017) | |
| best projected | -0.55992(2) | LR-DMC (LDA determinant) Motta et al. (2017) | ||
| H50 chain, open, R = 1.6, STO-6G | best variational | -0.5343888 ○ | LDMRG Hachmann et al. (2006) | |
| best projected | -0.5351831 ○ | DMET Motta et al. (2017) | ||
| best approximate | -0.5342628 ○ | CCSD(T) Hachmann et al. (2006) | ||
| H50 chain, open, R = 1.8, cc-pV5Z | best variational | -0.5393799 ○ | HF Motta et al. (2017) | |
| H50 chain, open, R = 1.8, cc-pVDZ | best variational | -0.560418 ○ | DMRG Motta et al. (2017) | |
| best projected | -0.5615634 ○ | DMET Motta et al. (2017) | ||
| best approximate | -0.5601304 ○ | RCCSD(T) Motta et al. (2017) | ||
| H50 chain, open, R = 1.8, cc-pVQZ | best variational | -0.5393155 ○ | HF Motta et al. (2017) | |
| best projected | -0.56515(3) | ph-AFQMC Motta et al. (2017) | ||
| best approximate | -0.564877 ○ | RCCSD(T) Motta et al. (2017) | ||
| H50 chain, open, R = 1.8, cc-pVTZ | best variational | -0.5391208 ○ | HF Motta et al. (2017) | |
| best projected | -0.56451(4) | ph-AFQMC Motta et al. (2017) | ||
| best approximate | -0.5641164 ○ | RCCSD(T) Motta et al. (2017) | ||
| H50 chain, open, R = 1.8, basis-free | best variational | -0.56381(1) | VMC (SJ) Motta et al. (2017) | |
| best projected | -0.56543(2) | LR-DMC (LDA determinant) Motta et al. (2017) | ||
| H50 chain, open, R = 1.8, STO-6G | best variational | -0.540773 ○ | LDMRG Hachmann et al. (2006) | |
| best projected | -0.5414071 ○ | DMET Motta et al. (2017) | ||
| best approximate | -0.540629 ○ | CCSD(T) Hachmann et al. (2006) | ||
| H50 chain, open, R = 2, cc-pV5Z | best variational | -0.5387831 ○ | HF Motta et al. (2017) | |
| H50 chain, open, R = 2, cc-pVDZ | best variational | -0.5376519 ○ | HF Motta et al. (2017) | |
| best projected | -0.560789 ○ | DMET Motta et al. (2017) | ||
| best approximate | -0.5593002 ○ | RCCSD(T) Motta et al. (2017) | ||
| H50 chain, open, R = 2, cc-pVQZ | best variational | -0.5387247 ○ | HF Motta et al. (2017) | |
| best projected | -0.56380(3) | ph-AFQMC Motta et al. (2017) | ||
| best approximate | -0.5634712 ○ | RCCSD(T) Motta et al. (2017) | ||
| H50 chain, open, R = 2, cc-pVTZ | best variational | -0.5385504 ○ | HF Motta et al. (2017) | |
| best projected | -0.56316(5) | ph-AFQMC Motta et al. (2017) | ||
| best approximate | -0.5626926 ○ | RCCSD(T) Motta et al. (2017) | ||
| H50 chain, open, R = 2, basis-free | best variational | -0.56242(1) | VMC (SJ) Motta et al. (2017) | |
| best projected | -0.56409(2) | LR-DMC (LDA determinant) Motta et al. (2017) | ||
| H50 chain, open, R = 2, STO-6G | best variational | -0.5385218 ○ | DMRG Motta et al. (2017) | |
| best projected | -0.5389803 ○ | DMET Motta et al. (2017) | ||
| best approximate | -0.538418 ○ | CCSD(T) Hachmann et al. (2006) | ||
| H50 chain, open, R = 2.4, cc-pV5Z | best variational | -0.5294429 ○ | HF Motta et al. (2017) | |
| H50 chain, open, R = 2.4, cc-pVDZ | best variational | -0.5484576 ○ | DMRG Motta et al. (2017) | |
| best projected | -0.549566 ○ | DMET Motta et al. (2017) | ||
| best approximate | -0.5481804 ○ | RCCSD(T) Motta et al. (2017) | ||
| H50 chain, open, R = 2.4, cc-pVQZ | best variational | -0.5293962 ○ | HF Motta et al. (2017) | |
| best projected | -0.5505126 ○ | RCCSD Motta et al. (2017) | ||
| best approximate | -0.5518596 ○ | RCCSD(T) Motta et al. (2017) | ||
| H50 chain, open, R = 2.4, cc-pVTZ | best variational | -0.5292454 ○ | HF Motta et al. (2017) | |
| best projected | -0.55145(4) | ph-AFQMC Motta et al. (2017) | ||
| best approximate | -0.5510824 ○ | RCCSD(T) Motta et al. (2017) | ||
| H50 chain, open, R = 2.4, basis-free | best variational | -0.55049(1) | VMC (SJ) Motta et al. (2017) | |
| best projected | -0.55262(3) | LR-DMC (LDA determinant) Motta et al. (2017) | ||
| H50 chain, open, R = 2.4, STO-6G | best variational | -0.5232114 ○ | LDMRG Hachmann et al. (2006) | |
| best projected | -0.5234051 ○ | DMET Motta et al. (2017) | ||
| best approximate | -0.521806 ○ | UCCSD(T) Motta et al. (2017) | ||
| H50 chain, open, R = 2.8, cc-pV5Z | best variational | -0.518544 ○ | HF Motta et al. (2017) | |
| H50 chain, open, R = 2.8, cc-pVDZ | best variational | -0.51777 ○ | HF Motta et al. (2017) | |
| best projected | -0.5352079 ○ | DMET Motta et al. (2017) | ||
| best approximate | -0.5329892 ○ | UCCSD(T) Motta et al. (2017) | ||
| H50 chain, open, R = 2.8, cc-pVQZ | best variational | -0.5185002 ○ | HF Motta et al. (2017) | |
| best projected | -0.53798(3) | ph-AFQMC Motta et al. (2017) | ||
| H50 chain, open, R = 2.8, cc-pVTZ | best variational | -0.5183529 ○ | HF Motta et al. (2017) | |
| best projected | -0.53742(2) | ph-AFQMC Motta et al. (2017) | ||
| H50 chain, open, R = 2.8, basis-free | best variational | -0.53563(1) | VMC (SJ) Motta et al. (2017) | |
| best projected | -0.53834(3) | LR-DMC (LDA determinant) Motta et al. (2017) | ||
| H50 chain, open, R = 2.8, STO-6G | best variational | -0.505496 ○ | LDMRG Hachmann et al. (2006) | |
| best projected | -0.5056322 ○ | DMET Motta et al. (2017) | ||
| best approximate | -0.503953 ○ | UCCSD(T) Motta et al. (2017) | ||
| H50 chain, open, R = 3.2, cc-pV5Z | best variational | -0.5103961 ○ | HF Motta et al. (2017) | |
| H50 chain, open, R = 3.2, cc-pVDZ | best variational | -0.5217914 ○ | DMRG Motta et al. (2017) | |
| best projected | -0.5226823 ○ | DMET Motta et al. (2017) | ||
| best approximate | -0.5206612 ○ | UCCSD(T) Motta et al. (2017) | ||
| H50 chain, open, R = 3.2, cc-pVQZ | best variational | -0.5103515 ○ | HF Motta et al. (2017) | |
| H50 chain, open, R = 3.2, cc-pVTZ | best variational | -0.5102075 ○ | HF Motta et al. (2017) | |
| best projected | -0.52450(4) | ph-AFQMC Motta et al. (2017) | ||
| H50 chain, open, R = 3.2, basis-free | best variational | -0.52221(2) | VMC (SJ) Motta et al. (2017) | |
| best projected | -0.52570(4) | LR-DMC (LDA determinant) Motta et al. (2017) | ||
| H50 chain, open, R = 3.2, STO-6G | best variational | -0.4913656 ○ | DMRG Motta et al. (2017) | |
| best projected | -0.4916637 ○ | SEET Motta et al. (2017) | ||
| best approximate | -0.4901495 ○ | UCCSD(T) Motta et al. (2017) | ||
| H50 chain, open, R = 3.6, cc-pV5Z | best variational | -0.5054446 ○ | HF Motta et al. (2017) | |
| H50 chain, open, R = 3.6, cc-pVDZ | best variational | -0.5046794 ○ | HF Motta et al. (2017) | |
| best projected | -0.5134833 ○ | DMET Motta et al. (2017) | ||
| best approximate | -0.5118654 ○ | UCCSD(T) Motta et al. (2017) | ||
| H50 chain, open, R = 3.6, cc-pVQZ | best variational | -0.5053981 ○ | HF Motta et al. (2017) | |
| H50 chain, open, R = 3.6, cc-pVTZ | best variational | -0.5052571 ○ | HF Motta et al. (2017) | |
| best projected | -0.5156009 ○ | DMET Motta et al. (2017) | ||
| H50 chain, open, R = 3.6, basis-free | best variational | -0.51192(1) | VMC (SJ) Motta et al. (2017) | |
| best projected | -0.51616(7) | LR-DMC (LDA determinant) Motta et al. (2017) | ||
| H50 chain, open, R = 3.6, STO-6G | best variational | -0.4820554 ○ | LDMRG Hachmann et al. (2006) | |
| best projected | -0.4831221 ○ | SEET Motta et al. (2017) | ||
| best approximate | -0.4813012 ○ | UCCSD(T) Motta et al. (2017) | ||
| H50 chain, open, R = 4.2, STO-6G | best variational | -0.4749942 ○ | LDMRG Hachmann et al. (2006) | |
| best approximate | -0.4302988 ○ | MP2 Hachmann et al. (2006) | ||
| H70 chain, open, R = 1, STO-6G | best projected | -0.34303(1) | ph-AFQMC Motta et al. (2017) | |
| H70 chain, open, R = 1.4, STO-6G | best projected | -0.511254(9) | ph-AFQMC Motta et al. (2017) | |
| H70 chain, open, R = 1.8, STO-6G | best projected | -0.54073(1) | ph-AFQMC Motta et al. (2017) | |
| H70 chain, open, R = 2.8, STO-6G | best projected | -0.50560(1) | ph-AFQMC Motta et al. (2017) | |
| H82 chain, open, R = 1, basis-free | best variational | -0.41649(2) | VMC (SJ) Motta et al. (2017) | |
| best projected | -0.42018(3) | LR-DMC (LDA determinant) Motta et al. (2017) | ||
| H82 chain, open, R = 1.2, basis-free | best variational | -0.49544(1) | VMC (SJ) Motta et al. (2017) | |
| best projected | -0.49896(7) | LR-DMC (LDA determinant) Motta et al. (2017) | ||
| H102 chain, open, R = 1, basis-free | best variational | -0.41616(2) | VMC (SJ) Motta et al. (2017) | |
| best projected | -0.41967(4) | LR-DMC (LDA determinant) Motta et al. (2017) | ||
| H102 chain, open, R = 1, STO-6G | best projected | -0.34087(1) | ph-AFQMC Motta et al. (2017) | |
| H102 chain, open, R = 1.2, basis-free | best variational | -0.49576(1) | VMC (SJ) Motta et al. (2017) | |
| best projected | -0.49844(4) | LR-DMC (LDA determinant) Motta et al. (2017) | ||
| H102 chain, open, R = 1.4, STO-6G | best projected | -0.51125(1) | ph-AFQMC Motta et al. (2017) | |
| H102 chain, open, R = 1.8, basis-free | best variational | -0.563316(7) | VMC (SJ) Motta et al. (2017) | |
| best projected | -0.56538(1) | LR-DMC (LDA determinant) Motta et al. (2017) | ||
| H102 chain, open, R = 1.8, STO-6G | best projected | -0.54069(2) | ph-AFQMC Motta et al. (2017) | |
| H102 chain, open, R = 2.8, STO-6G | best projected | -0.50565(2) | ph-AFQMC Motta et al. (2017) |
E/N (Ha)
| instance | value | energy | method | code |
|---|---|---|---|---|
| H4 sheet (2x2), open, R = 2.4, basis-free | best variational | -0.52710(7) | VMC (JAGP) Genovese et al. (2019) | |
| best projected | -0.52812(5) | LR-DMC (JAGP) Genovese et al. (2019) |
Details
Every row has a bound_type that says what kind of number the energy is
(RULES.md §4).
Bold marks the record. RULES.md §6 says how it is chosen.
○ marks a row where no error bar and no energy variance was found in the source.
† marks a sampled energy with a variance but no error bar. Both rows stay in rank
order. Error metrics defines the markers.
The code column links the code that produced a result, where its authors published it: the
place to start when you want to rerun it. For a row QMBL computed, "rerun (QMBL code)" links the
command, the commit and the package versions it ran with. Where the authors published no code
of their own, a muted "uses …" names the library they used, if their paper says. An empty
cell means that no source read so far names the code or a library.
How the links were found.
Rules: RULES.md. Units, conventions, error metrics, provenance and defects:
DATA.md.
Citing
T. Zerweck and J. Carrasquilla, QMBL: the Quantum Many-Body Leaderboard,
arXiv:2610.22387 (2026).
To cite the data itself, use the Zenodo concept DOI, which resolves to the latest release: 10.5281/zenodo.22753734.
Contributing
To add an energy, use the new result form.
To add an instance QMBL does not have yet, a lattice model or an electron-gas cell, use the
new instance form.
To correct a row or comment on it, click the pencil at the end of the row on
qmbl.org/instances; it opens the
about-a-row form with the row filled in. Be specific and technical
(RULES.md §10). A sampled energy without an
error bar is marked †. Send the error bar so the row can be compared with its
neighbours.
CONTRIBUTING.md gives the details: what happens after you submit, how
to change the code, and how AI systems submit.
Contributors
Everyone whose issue or pull request leads to a change is listed here.
License
Apache-2.0. The data tables are freely reusable with attribution.
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