README.md

---
license: mit
library_name: skala
tags:
- chemistry
- density-functional-theory
- exchange-correlation-functional
- computational-chemistry
- quantum-chemistry
datasets:
- microsoft/msr-acc-tae25
---

# 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 that 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 2.8 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 and GPU4PySCF](https://arxiv.org/abs/2603.14155),
    which runs the functional on CPU and GPU.
    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 these 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
    due to 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) calculations. The basis
    sets used for H and He were aug-cc-pV(Q+d)Z and aug-cc-pV(5+d),
    while for the remaining elements B--Ar the basis sets 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
    $Z^{-3}$ expression, while the Hartree--Fock energy was extrapolated
    using the two-point scheme of [Karton 2006][karton2006].

**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 filtered 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, input density and derived meta-GGA features were
computed from density matrices of converged B3LYP SCF calculations
(def2-QZVP and ma-def2-QZVP basis sets) using a modified version of
PySCF.

### Training procedure

#### Preprocessing

The training datapoints are preprocessed as follows.

- For each molecule, the density and derived meta-GGA features are
  computed from the density matrix of a converged B3LYP SCF calculation
  using a def2-QZVP or ma-def2-QZVP basis set in a modified version of
  PySCF.
- Density fitting was not applied.
- The density features were evaluated on an atom-centered integration
  grid of level 1.
- The radial quadrature was performed with Treutler-Ahlrichs,
  Gauss-Chebyshev, Delley, or Mura-Knowles schemes 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; all grid points were retained for
  training.

#### Training hyperparameters

The training hyperparameter settings are detailed in the supplementary
material of [Accurate and scalable exchange-correlation with deep
learning, Luise et al. 2025](https://arxiv.org/abs/2506.14492). This
repository only includes the code to evaluate the provided checkpoints,
not the training code.

#### Speeds, sizes, times

The training of the functional on the dataset described above took
approximately 48 hours for 1M steps on an [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 GPUs (80 GB each), 96 CPU cores, 880 GB RAM, and a 6
TB disk.

The model checkpoints have ~385k 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 [Chan et al. 2023][chan2023],
    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 and bond
    lengths from [Grimme et al. 2015][grimme2015];
    (c) LMGB35, equilibrium structures and bond lengths
    from [Grimme et al. 2015][grimme2015]; and
    (d) W4-11-GEOM, equilibrium structures, bond
    lengths, and bond angles from [Karton et al. 2011][karton2011].
5.  The dipole benchmark dataset from [Hait et al. 2018][hait2018]
6.  Conformer search benchmark dataset of 22 molecules spanning 24 to
    176 atoms, used for cost-scaling analysis, from
    [Grimme et al. 2019][grimme2019]

These six benchmark types serve to measure different performance aspects
of the functional. Benchmarks 1 and 2 focus on the accuracy of predicted
reaction energies. Benchmark 3 evaluates general main-group
thermochemistry, kinetics, and noncovalent interactions. Benchmark 4
evaluates geometry optimization and convergence to reference equilibrium
structures. Benchmark 5 measures dipole moments, providing a proxy for
the quality of the self-consistent electron density produced by the SCF
procedure. Finally, benchmark 6 assesses computational cost scaling with
respect to system size.

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.
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.  For the dipole benchmark, we follow the metrics defined in
    [Hait et al. 2018][hait2018]. For molecules
    (indexed by *i*) for which only the reference magnitude of the
    dipole moment $\mu_i^{\text{ref}} = |{\vec\mu}_i^{\text{ref}}|$ is
    provided, the error is defined as
    $\text{Error}_i = \frac{\mu_i^\theta - \mu_i^\text{ref}}{\max(\mu_i^\text{ref}, 1D)} \times 100\%$,
    where $\mu_i^{\theta} = |{\vec\mu}_i^{\theta}|$ is the predicted
    magnitude and *D* denotes the unit of Debye. For molecules for which
    the reference dipole vector $\vec{\mu}_i^\text{ref}$ is also
    available, we instead compute
    $\text{Error}_i = \frac{|\vec{\mu}_i^\theta - \vec{\mu}_i^\text{ref}|}{\max(\mu_i^\text{ref}, 1D)} \times 100\%$.
    The RMSE is then
    $\text{RMSE} = \sqrt{\frac{1}{N} \sum_{i=1}^N \text{Error}_i^2}$.
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

On W4-17, the Skala-1.1 functional predicts atomization energies at
chemical accuracy (~1 kcal/mol MAE). On GMTKN55, which covers general
main-group thermochemistry, kinetics, and noncovalent interactions, it
achieves a WTMAD-2 of 2.8 kcal/mol, surpassing state-of-the-art
range-separated hybrid functionals while only requiring runtimes typical
of semi-local DFT.

On the geometry optimization benchmarks, the functional converges to
reference equilibrium structures with errors comparable to a
range-separated hybrid functional. On the dipole prediction benchmark,
the error in dipole moment predictions is better than that of
state-of-the-art range-separated hybrid functionals.

Finally, the scaling results show that the Skala-1.1 functional exhibits
the asymptotic scaling behavior of a meta-GGA functional, with an
approximate prefactor of 3 relative to r2SCAN.

## License

> MIT License
> 
> Copyright (c) Microsoft Corporation.
> 
> Permission is hereby granted, free of charge, to any person obtaining a copy
> of this software and associated documentation files (the "Software"), to deal
> in the Software without restriction, including without limitation the rights
> to use, copy, modify, merge, publish, distribute, sublicense, and/or sell
> copies of the Software, and to permit persons to whom the Software is
> furnished to do so, subject to the following conditions:
> 
> The above copyright notice and this permission notice shall be included in all
> copies or substantial portions of the Software.
> 
> THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR
> IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY,
> FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT. IN NO EVENT SHALL THE
> AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER
> LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING FROM,
> OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS IN THE
> SOFTWARE.

## 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 Stefano Battaglia and Gregor N. C. Simm and P. Bernát Szabó 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