Add model card for Skala-1.1

README.md

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+---
+license: mit
+library_name: skala
+tags:
+- chemistry
+- density-functional-theory
+- exchange-correlation-functional
+- computational-chemistry
+- quantum-chemistry
+---
+
+# Skala 1.1 model
+
+## Model details
+
+In pursuit of the universal functional for density functional theory
+(DFT), the OneDFT team from Microsoft Research AI for Science has
+developed the Skala-1.1 exchange-correlation functional, as introduced
+in [Accurate and scalable exchange-correlation with deep learning, Luise
+et al. 2025](https://arxiv.org/abs/2506.14665). This approach departs
+from the traditional route of incorporating increasingly expensive
+hand-designed non-local features from Jacob\'s ladder into functional
+forms to improve their accuracy. Instead, we employ a deep learning
+approach with a scalable neural network that uses only inexpensive input
+features to learn the necessary non-local representations.
+
+The functional is based on a neural network architecture that takes as
+input features on a 3D grid describing the electron density and derived
+meta-generalized-gradient (meta-GGA) quantities. The architecture
+performs scalable non-local message-passing on the integration grid via
+a second, coarser grid, combined with shared local layers that enable
+representation learning of both local and non-local features. These
+representations are then used to predict the exchange-correlation energy
+in an end-to-end data-driven manner.
+
+To facilitate this learning, the model is trained on a dataset of
+unprecedented size, containing highly accurate energy labels from
+coupled cluster theory. The largest subset focuses on atomization
+energies and was generated in collaboration with the University of New
+England. This subset is released as part of the Microsoft Research
+Accurate Chemistry Collection (MSR-ACC, [Accurate Chemistry Collection:
+Coupled cluster atomization energies for broad chemical space, Ehlert et
+al. 2025](https://arxiv.org/abs/2506.14492v5)). To broaden coverage of
+other types of chemistry, the training dataset is further complemented
+with in-house generated datasets covering conformers, ionization
+potentials, electron affinities, proton affinities, noncovalent
+interactions, distorted equilibrium geometries, and elementary
+reactions, as well as a small amount of publicly available high-accuracy
+data.
+
+We demonstrate departure from the historical trade-off between accuracy
+and efficiency is enabled by learning non-local representations of
+electronic structure directly from data, bypassing the need for
+increasingly costly hand-engineered features. The Skala-1.1 functional
+surpasses state-of-the-art hybrid functionals in accuracy across the
+main-group chemistry benchmark set GMTKN55, which covers general
+main-group thermochemistry, kinetics, and noncovalent interactions, with
+an error of 2.72 kcal/mol, while retaining the lower computational cost
+characteristic of semi-local DFT. With this work, we demonstrate the
+viability of our approach toward the universal density functional across
+all of chemistry.
+
+Users of this model are expected to have a basic understanding of the
+field of quantum chemistry and density functional theory.
+
+Developed by
+:   Chin-Wei Huang, Deniz Gunceler, Derk Kooi, Gregor Simm, Klaas
+    Giesbertz, Giulia Luise, Jan Hermann, Megan Stanley, Paola Gori
+    Giorgi, P. Bernát Szabó, Rianne van den Berg, Sebastian Ehlert,
+    Stefano Battaglia, Stephanie Lanius, Thijs Vogels, Wessel Bruinsma
+
+Shared by
+:   Microsoft Research AI for Science
+
+Model type
+:   Neural Network Density Functional Theory Exchange Correlation
+    Functional
+
+License
+:   MIT
+
+## Direct intended uses
+
+1.  The Skala-1.1 functional is shared with the research community to
+    facilitate reproduction of the evaluations presented in our paper.
+2.  Evaluating reaction energy differences by computing the total energy
+    of all compounds in a reaction using a self-consistent field (SCF)
+    calculation with the Skala-1.1 exchange-correlation functional.
+3.  Evaluating the total energy of a molecule using an SCF calculation
+    with the Skala-1.1 exchange-correlation functional. Note that, as
+    with all density functionals, energy differences are predicted much
+    more reliably than total energies of individual molecules.
+4.  The SCF implementation provided uses PySCF, which runs the
+    functional on CPU. We also provide a traced version of the Skala-1.1
+    functional so that other, more optimized open-source SCF
+    codes—including GPU-enabled ones—can integrate it into their
+    pipelines, for instance through GauXC. A compatible fork of GauXC is
+    included in this repository.
+
+## Out-of-scope uses
+
+1.  Evaluating the functional with a single pass given a fixed density
+    as input is not the intended way to evaluate the model. The model\'s
+    predictions should always be made by using it as part of an SCF
+    procedure.
+2.  We do not include a training pipeline for the Skala-1.1 functional
+    in this code base.
+
+## Risks and limitations
+
+1.  Interpretation of results requires expertise in quantum chemistry.
+2.  The Skala-1.1 functional is trained on atomization energies,
+    conformers, proton affinities, ionization potentials, electron
+    affinities, elementary reaction pathways, distorted equilibrium
+    geometries, and non-covalent interactions, as well as a small amount
+    of total energies of atoms and transition metal atoms and dimer
+    properties. We have benchmarked performance on W4-17 for atomization
+    energies and on GMTKN55, which covers general main-group
+    thermochemistry, kinetics, and noncovalent interactions, to provide
+    an indication of generalization beyond the training set. We have
+    also evaluated robustness on dipole moment predictions and geometry
+    optimization.
+3.  The Skala-1.1 functional has been trained on data containing the
+    following elements: H–Xe. It has been tested on data containing
+    H–Xe, Pb, and Bi.
+4.  Given points 2 and 3 above, this is not a production model. We
+    advise testing the functional further before applying it to your
+    research and welcome any feedback.
+
+## Recommendations
+
+1.  In our PySCF-based SCF implementation, the largest system tested
+    contained 180 atoms using the def2-TZVP basis set
+    (~5000 orbitals) on [Eadsv5
+    series](https://learn.microsoft.com/en-us/azure/virtual-machines/sizes/memory-optimized/eadsv5-series?tabs=sizebasic)
+    virtual machines. Larger systems may run out of memory.
+2.  For implementations optimized for memory, speed, or GPU support, we
+    recommend integrating the functional with other open-source SCF
+    packages, for instance through GauXC. A compatible fork of GauXC is
+    included in this repository.
+3.  Skala-1.1 will also be available through [Azure AI
+    Foundry](https://labs.ai.azure.com/projects/skala/), where it is
+    coupled with Microsoft\'s GPU-accelerated [Accelerated
+    DFT](https://arxiv.org/abs/2406.11185) application.
+
+## Training details
+
+### Training data
+
+The following data is included in our training set:
+
+**MSR-ACC**
+:   99% of MSR-ACC/TAE25 (~78k reactions) containing atomization
+    energies for up to five non-hydrogen atoms. This data was generated
+    in collaboration with Prof. Amir Karton, University of New England,
+    with the W1-F12 composite protocol based on CCSD(T) and is released
+    as part of the [Microsoft Research Accurate Chemistry
+    Collection](https://arxiv.org/abs/2506.14492v5) (MSR-ACC).
+    Additionally the MSR-ACC subsets for larger TAEs (up to 9
+    non-hydrogen atoms), conformers, ionization potentials, electron
+    affinities, proton affinities, reaction paths and distorted
+    equilibrium structures were included. The labels for this data sets
+    are obtained with the W1w method and are part of the currently
+    unpublished subsets of the MSR-ACC.
+
+**Atomic Data**
+:   Total energies, electron affinities and ionization potentials (up to
+    triple ionization) for atoms, from H to Ar (excluding Li and Be
+    because of basis set constraints).This data was produced in-house
+    with CCSD(T) by extrapolating to the complete basis set limit from
+    quadruple zeta (QZ) and pentuple zeta (5Z) basis set
+    calculations.The basis sets used for H and He were aug-cc-pV(Q+d)Z,
+    aug-cc-pV(5+d), while for the remaining elements B-Ar the basis sets
+    used were aug-cc-pCVQZ and aug-cc-pCV5Z. All basis sets were obtained
+    from the [Basis Set Exchange
+    (BSE)](https://www.basissetexchange.org/). Extrapolation of the
+    correlation energy was performed by fitting a simple Z\^(-3)
+    expression, while extrapolation of the Hartree-Fock energy was
+    performed using a two-point extrapolation.
+
+**Transition metal properties**
+:   Additional data for transition metal atoms and dimers, including
+    ionization potentials, spin splittings and dissociation energies.
+    The reference energies were obtained from literature.
+
+**NCI-Atlas**
+
+:   Five datasets from the [NCI-Atlas collection of non-covalent
+    interactions](http://www.nciatlas.org/):
+
+    - [D442x10](http://www.nciatlas.org/D442x10.html), dissociation
+      curves for dispersion bound van-der-Waals complexes
+    - [SH250x10](http://www.nciatlas.org/SH250.html), dissociation
+      curves for sigma-hole bound van-der-Waals complexes
+    - [R739x5](http://www.nciatlas.org/R739.html), compressed
+      van-der-Waals complexes
+    - [HB300SPXx10](http://www.nciatlas.org/HB300SPX.html), dissociation
+      curves for hydrogen bound van-der-Waals complexes
+    - [IHB100x10](http://www.nciatlas.org/IHB100.html), dissociation
+      curves for ionic hydrogen bound van-der-Waals complexes
+
+**GDB9**
+:   The graph data base with up to non-hydrogen atoms computed at
+    W1-F12 level of theory from [Karton 2025][karton2025].
+
+**BH9**
+:   Reactions and barrier heights from [Prasad et. al 2021][prasad2021]
+    The data set was filted for systems with up to ten
+    non-hydrogen atoms.
+
+**NCIBLIND**
+:   Data set of non-covalent dissociation curves from [Taylor et. al 2016][taylor2016].
+
+**Water2510**
+:   Data set of the potential energy surface of the water
+    dimer from [Smith et al. 2016][smith2016]. The data set was fully relabeled with W1w.
+
+**DES370k**
+:   Subset with CCSD(T)/dCBS(aug-cc-pVQZ) non-covalent interaction
+    energies from [Donchev et al. 2021][donchev2021].
+
+**MB2061**
+:   Dataset containing decomposition energies of artificial
+    molecules from [Gasevic et al. 2025][gasevic2025].
+
+**W4-CC**
+:   Containing atomization energies of carbon
+    clusters from [Karton et al. 2009][karton2009].
+
+For all training data we have created input density and derived meta-GGA
+features using density matrices of converged SCF calculations with the
+B3LYP functional (def2-QZVP and ma-def2-QZVP basis set) using a modified
+version of the PySCF software package.
+
+### Training procedure
+
+#### Preprocessing
+
+The training datapoints are preprocessed as follows.
+
+- For each molecule the density and derived meta-GGA features are
+  computed based on the density matrix of converged SCF calculations
+  with the B3LYP functional using a def2-QZVP or ma-def2-QZVP basis set
+  using a modified version of the PySCF software package.
+- Density fitting was not applied for the SCF calculation.
+- The density features were evaluated on an atom centered integration
+  grid of level 1.
+- The radial integral was performed with the Treutler-Ahlrichs,
+  Gauss-Chebychev, Delley, or Mura-Knowles based on Bragg atomic radii
+  with Treutler based radii adjustment.
+- The space-partitioning was performed with Becke partition and
+  Treutler-Ahlrichs radii adjustment, Stratmann-Scuseria-Frisch (SSF)
+  partition scheme, and Laqua-Kussmann-Ochsenfeld (LKO) partition
+  scheme.
+- The angular grid points were pruned using the NWChem scheme.
+- No density based cutoff was applied and all grid points were retained
+  for training.
+
+#### Training hyperparameters
+
+The training hyperparameter settings are detailed in the supplementary
+of [Accurate and scalable exchange-correlation with deep learning, Luise
+et al. 2025](https://arxiv.org/abs/2506.14492v6). This repository only
+includes the code to evaluate the checkpoints provided, not the training
+code.
+
+#### Speeds, sizes, times
+
+The training of our functional using the training dataset as detailed in
+the section \"Training data\" took approximately 48h for 1M training
+steps on a [ND A100 v4 series
+VM](https://learn.microsoft.com/en-us/azure/virtual-machines/sizes/gpu-accelerated/ndasra100v4-series?tabs=sizebasic)
+with 8 NVIDIA A100 GPU with 80 GB memory each, 96 CPU cores, 880 GB RAM,
+and a 6 TB disk.
+
+The model checkpoints have ~385,217 trainable parameters.
+
+## Evaluation
+
+### Testing data, factors, and metrics
+
+We have evaluated our functional on several different benchmark sets:
+
+1.  W4-17. A diverse and highly accurate dataset of atomization
+    energies from [Karton et. al 2017][karton2017]
+2.  Transition metal data sets including
+    MOR41 from [Dohm et. al 2018][dohm2018],
+    ROST61 from [Maurer et. al 2021][maurer2021],
+    MOBH35 from [Semidalas et. al 2022][semidalas2022],
+    3dTMV from [Neugebauer et. al 2023][neugebauer2023],
+    CuAgAu83 from [Chan 2019][chan2019],
+    DAPd from [Author et. al 2020][dapd2020],
+    3d4dIPSS, TMB11, and TMD10 from [Liang et. al 2025][liang2025]
+3.  GMTKN55. A diverse and highly accurate dataset of general main group
+    thermochemistry, kinetics and noncovalent
+    interactions from [Goerigk et. al 2017][goerigk2017]
+4.  Geometry optimization datasets: (a) CCse21, equilibrium structures,
+    bond lengths and bond angles from [Piccardo et. al 2015][piccardo2015]
+    (b) HMGB11, equilibrium structures, bond
+    lengths from [Grimme et. al 2015][grimme2015]
+    (c) LMGB35, equilibrium structures, bond lengths,
+    and from [Grimme et. al 2015][grimme2015]
+    (d) W4-11-GEOM, equilibrium structures, bond
+    lengths and bond angles.
+5.  The Dipole benchmark dataset from [Hait et al. 2018][hait2018]
+6.  Conformer search benchmark dataset of 22 molecules spanning
+    molecular size from 24 to 176 atoms for cost scaling from
+    [Grimme et al. 2019][grimme2019]
+
+The evaluation of our model using the 5 different types of benchmarks as
+defined above serve to measure different performance aspects of our
+functional. For example, 1 and 2 focus on the accuracy of predicted
+reaction energies, and 3 focuses on the ability of our functional to
+perform geometry optimization and to converge to the right equilibrium
+molecular structure. Furthermore, 4 measures the dipole moment of the
+molecules in the benchmark set, which provides a measure for the quality
+of the self-consistent electron density that a converged SCF procedure
+produces with our model. Finally, 5 determines the speed of employing
+SCF with our model and compares its scaling behavior with respect to
+system size with the scaling of traditional functionals.
+
+The metrics for the different benchmark sets are:
+
+1.  Mean Absolute Error (MAE) in kcal/mol for reactions in W4-17
+    $MAE = \frac{1}{N} \sum_{r=1}^N |\Delta E_r - \Delta E_r^\theta|$.
+    Here *N* is the number of reactions in W4-17, *r* is the index
+    denoting reactions in W4-17, $\Delta E_r$ is the energy difference
+    of reaction r as calculated by a high-accuracy method from the W4
+    family (CCSDT(Q)/CBS to CCSDTQ56/CBS), and $\Delta E_r^\theta$ is
+    the prediction of the reaction energy difference using SCF
+    calculations with our functional, and
+2.  Weighted total mean absolute deviations 2 (WTMAD-2) in kcal/mol for
+    the GMTKN55 benchmark set
+    $\text{WTMAD-2} = \frac1{\sum^{55}_{i=1} N_i} \sum_{i=1}^{55} N_i \frac{56.84\text{ kcal/mol}}{\overline{|\Delta E|}_i} \text{MAE}_i$
+    Here $N_i$ is the number of reactions in subset *i*,
+    $\overline{|\Delta E|}_i$ is the average energy difference in subset
+    *i* in kcal/mol and $\text{MAE}_i$ is the mean absolute error in
+    kcal/mol for subset *i*.
+3.  For the geometry benchmark sets that report bond lengths, we measure
+    the absolute error in bond lengths in Angstrom, averaged over the
+    number of bonds and the number of equilibrium structures in the
+    dataset. For the benchmark that also contains bond angles, we report
+    the absolute error of the angles, averaged over the number of bonds
+    and equilibrium structures in the dataset.
+4.  We follow the metrics defined in `hait2018`{.interpreted-text
+    role="footcite"}. We measure the Root Mean Squared Error (RMSE) of
+    the dipole moment with respect to reference values provided by the
+    benchmark dataset. For those molecules (indexed with *i*) for which
+    only the reference magnitude of the dipole moment
+    $\mu_i^{\text{ref}} = |{\vec\mu}_i^{\text{ref}}|$ is provided, we
+    measure the RMSE of the predicted magnitude of the dipole moment
+    $\mu_i^{\theta} = |{\vec\mu}_i^{\theta}|$ is available, the error is
+    defined as
+    $\text{Error}_i = \frac{\mu_i^\theta - \mu_i^\text{ref}}{\max(\mu_i^\text{ref}, 1D)} \times 100\%$.
+    Here *D* denotes the unit of Debye. For those molecules for which
+    the reference value of the dipole vector $\vec{\mu}_i^\text{ref}$ is
+    also available we instead compute
+    $\text{Error}_i = \frac{|\vec{\mu}_i^\theta - \vec{\mu}_i^\text{ref}|}{max(\mu_i^\text{ref}, 1D)} \times 100\%$.
+    Using these errors we compute the RMSE as follows:
+    $\text{RMSE} = \sqrt{\frac{1}{N} \sum_{i=1}^N \text{Error}_i^2}$
+5.  We fit a power law of the form
+    $C(M) = \left(\frac{n(M)}{A}\right)^k$ to the 22 data points of the
+    test set where *C(M)* and *n(M)* are the computational cost and
+    number of atoms of molecule *M*, respectively, and *A* and *k* are
+    fitted parameters. We report the scaling power *k* as the main
+    metric.
+
+### Evaluation results
+
+We demonstrate that the combination of a large-scale high-accuracy
+dataset combined with our deep learning architecture produces the Skala
+functional that predicts atomization energies at chemical accuracy (1
+kcal/mol), as measured on the public benchmark set W4-17. On the public
+benchmark set GMTKN55, which covers general-main group thermochemistry,
+kinetics and noncovalent interactions, our model makes predictions
+around 2.72 kcal/mol. This accuracy is better than state-of-the-art
+range-separated hybrid functionals while only requiring runtimes typical
+of semi-local DFT.
+
+On the geometry optimization benchmarks we demonstrate that we can
+converge to the reference equilibrium structure with an error that is
+comparable to a range-separated hybrid functional. On the dipole
+prediction benchmark test we demonstrate that the error of our dipole
+moment prediction with respect to reference values is better than
+state-of-the-art range-separated hybrid functionals.
+
+Finally, our scaling results demonstrate that our functional shows the
+asymptotic scaling behavior of a metaGGA functional, with an approximate
+prefactor of 3 compared to the r2SCAN.
+
+## License
+
+:::: dropdown
+MIT License
+
+::: {.literalinclude lines="3-"}
+../../LICENSE.txt
+:::
+::::
+
+## Citation
+
+When using Skala-1.1 in your research, please reference it including the
+version number as follows:
+
+> This work uses the Skala-1.1 functional.
+
+``` bibtex
+@misc{luise2025,
+   title={Accurate and scalable exchange-correlation with deep learning}, 
+   author={Giulia Luise and Chin-Wei Huang and Thijs Vogels and Derk P. Kooi and Sebastian Ehlert and Stephanie Lanius and Klaas J. H. Giesbertz and Amir Karton and Deniz Gunceler and Megan Stanley and Wessel P. Bruinsma and Lin Huang and Xinran Wei and José Garrido Torres and Abylay Katbashev and Rodrigo Chavez Zavaleta and Bálint Máté and Sékou-Oumar Kaba and Roberto Sordillo and Yingrong Chen and David B. Williams-Young and Christopher M. Bishop and Jan Hermann and Rianne van den Berg and Paola Gori-Giorgi},
+   year={2025},
+   eprint={2506.14665},
+   archivePrefix={arXiv},
+   primaryClass={physics.chem-ph},
+   url={https://arxiv.org/abs/2506.14665},
+}
+```
+
+## Model card contact
+
+- Rianne van den Berg, <rvandenberg@microsoft.com>
+- Paola Gori-Giorgi, <pgorigiorgi@microsoft.com>
+- Jan Hermann, <jan.hermann@microsoft.com>
+- Sebastian Ehlert, <sehlert@microsoft.com>
+
+[karton2006]: https://doi.org/10.1007/s00214-005-0028-6
+[karton2009]: https://doi.org/10.1080/00268970802708959
+[karton2011]: https://doi.org/10.1016/j.cplett.2011.05.007
+[karton2017]: https://doi.org/10.1002/jcc.24854
+[goerigk2017]: https://doi.org/10.1039/C7CP04913G
+[piccardo2015]: https://doi.org/10.1021/jp511432m
+[grimme2015]: https://doi.org/10.1063/1.4927476
+[hait2018]: https://doi.org/10.1021/acs.jctc.7b01252
+[grimme2019]: https://doi.org/10.1021/acs.jctc.9b00143
+[dohm2018]: https://doi.org/10.1021/acs.jctc.7b01183
+[maurer2021]: https://doi.org/10.1021/acs.jctc.1c00659
+[semidalas2022]: https://doi.org/10.1021/acs.jctc.1c01126
+[neugebauer2023]: https://doi.org/10.1021/acs.jctc.3c00617
+[chan2019]: https://doi.org/10.1021/acs.jpca.9b03976
+[chan2023]: https://doi.org/10.1021/acs.jctc.3c01060
+[liang2025]: https://doi.org/10.1021/acs.jctc.5c01380
+[smith2016]: https://doi.org/10.1021/acs.jpclett.6b00780
+[taylor2016]: https://doi.org/10.1063/1.4961095
+[donchev2021]: https://doi.org/10.1038/s41597-021-00833-x
+[prasad2021]: https://doi.org/10.1021/acs.jctc.1c00694
+[gasevic2025]: https://doi.org/10.1021/acs.jcim.5c01364
+[karton2025]: https://doi.org/10.1016/j.cplett.2025.142030