README.md

---
license: mit
library_name: skala
tags:
- chemistry
- density-functional-theory
- exchange-correlation-functional
- computational-chemistry
- quantum-chemistry
---

# Skala 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-1.0 exchange-correlation functional, as introduced
in [Accurate and scalable exchange-correlation with deep learning (arXiv v5),
Luise et al. 2025](https://arxiv.org/abs/2506.14665v5). 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, proton affinities, and elementary reactions, as well as a
small amount of publicly available high-accuracy data.

We demonstrate that combining a large-scale high-accuracy dataset with
our deep learning architecture yields a functional that predicts
atomization energies at chemical accuracy (1 kcal/mol), as measured on
the W4-17 benchmark set. On GMTKN55, which covers general main-group
thermochemistry, kinetics, and noncovalent interactions, the Skala-1.0
functional achieves a WTMAD-2 of 3.89 kcal/mol. This accuracy is
competitive with state-of-the-art range-separated hybrid functionals,
while only requiring a cost comparable to 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, Klaas Giesbertz, Giulia
    Luise, Jan Hermann, Megan Stanley, Paola Gori Giorgi, Rianne van den
    Berg, Sebastian Ehlert, 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-1.0 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-1.0 exchange-correlation functional.
3.  Evaluating the total energy of a molecule using an SCF calculation
    with the Skala-1.0 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-1.0
    functional so that other, more optimized open-source SCF
    codes—including GPU-enabled ones—can integrate it into their
    pipelines, for instance through GauXC.

## 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-1.0 functional
    in this code base.

## Risks and limitations

1.  Interpretation of results requires expertise in quantum chemistry.
2.  The Skala-1.0 functional is trained on atomization energies,
    conformers, proton affinities, ionization potentials, elementary
    reaction pathways, and non-covalent interactions, as well as a small
    amount of electron affinities and total energies of atoms. 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-1.0 functional has been trained on data containing the
    following elements: H–Ar, Br, Kr, I, Xe. It has been tested on data
    containing H–Ca, Ge–Kr, Sn–I, 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
    ($\sim$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.
3.  Skala-1.0 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:

- 99% of MSR-ACC:TAE ($\sim$78k reactions) containing
  atomization energies. This data was generated in collaboration with
  Prof. Amir Karton, University of New England, using 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).
- 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, 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 a two-point
  scheme of [Karton et. al 2006][karton2006]
- Four 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
- 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 2 or level 3.
- 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 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 (arXiv v5), Luise et al. 2025](https://arxiv.org/abs/2506.14665v5).

#### Speeds, sizes, times

Training on the dataset described above took approximately 36 hours for
500k steps on an [NC A100 v4 series
VM](https://learn.microsoft.com/en-us/azure/virtual-machines/sizes/gpu-accelerated/nca100v4-series?tabs=sizebasic)
with 4 NVIDIA A100 GPUs (80 GB each), 96 CPU cores, 880 GB RAM, and a
256 GB disk.

The model checkpoints have $\sim$276k 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.  GMTKN55. A diverse and highly accurate dataset of general main-group
    thermochemistry, kinetics, and noncovalent
    interactions from [Goerigk et. al 2017][goerigk2017]
3.  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]
4.  The dipole benchmark dataset from [Hait et. al 2018][hait2018]
5.  Conformer search benchmark dataset of 22 molecules spanning 24 to
    176 atoms, used for cost-scaling analysis, from
    [Grimme et. al 2019][grimme2019]

These five benchmark types serve to measure different performance
aspects of the functional. Benchmarks 1 and 2 focus on the accuracy of
predicted reaction energies, while 3 evaluates geometry optimization and
convergence to reference equilibrium structures. Benchmark 4 measures
dipole moments, providing a proxy for the quality of the self-consistent
electron density produced by the SCF procedure. Finally, benchmark 5
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:
    $\text{MAE} = \frac{1}{N} \sum_{r=1}^N |\Delta E_r - \Delta E_r^\theta|$.
    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.  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-1.0 functional predicts atomization energies at
chemical accuracy ($\sim$1 kcal/mol MAE). On GMTKN55,
it achieves a WTMAD-2 of 3.89 kcal/mol, competitive with
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 GGA. On the
dipole prediction benchmark, the error in dipole moment predictions is
comparable to that of state-of-the-art range-separated hybrid
functionals.

Finally, the scaling results show that the Skala-1.0 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-1.0 in your research, please reference it including the
version number as follows:

> This work uses the Skala-1.0 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={2506.14665v5},
   archivePrefix={arXiv},
   primaryClass={physics.chem-ph},
   url={https://arxiv.org/abs/2506.14665v5}, 
}
```

## 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
[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