# QMugs: Quantum Mechanical Properties of Drug-like Molecules

Clemens Isert  Department of Chemistry and Applied Biosciences, RETHINK, ETH
Zurich, 8093 Zurich, Switzerland.  Kenneth Atz  Department of Chemistry and
Applied Biosciences, RETHINK, ETH Zurich, 8093 Zurich, Switzerland.  José
Jiménez-Luna  Department of Chemistry and Applied Biosciences, RETHINK, ETH
Zurich, 8093 Zurich, Switzerland.  Department of Medicinal Chemistry,
Boehringer Ingelheim Pharma GmbH & Co. KG, Birkendorfer Straße 65, 88397
Biberach an der Riss, Germany.  corresponding authors: Gisbert Schneider
(gisbert@ethz.ch), José Jiménez-Luna<EMAIL_ADDRESS> Gisbert
Schneider  Department of Chemistry and Applied Biosciences, RETHINK, ETH
Zurich, 8093 Zurich, Switzerland.  ETH Singapore SEC Ltd, 1 CREATE Way,
$\\#$06-01 CREATE Tower, Singapore 138602, Singapore.  corresponding authors:
Gisbert Schneider (gisbert@ethz.ch), José Jiménez-Luna
<EMAIL_ADDRESS>
###### Abstract

Machine learning approaches in drug discovery, as well as in other areas of
the chemical sciences, benefit from curated datasets of physical molecular
properties. However, there is a lack of sufficiently large data collections
that include first-principle quantum chemical information on bioactive
molecules, such as single-point electronic properties, quantum mechanical wave
functions and density-functional theory (DFT) matrices. The open-access QMugs
(Quantum-Mechanical Properties of Drug-like Molecules) dataset fills this
void. The QMugs collection comprises quantum mechanical properties of more
than 665k biologically and pharmacologically relevant molecules extracted from
the ChEMBL database, totaling $\sim$2M conformers. QMugs contains optimized
molecular geometries and thermodynamic data obtained via the semi-empirical
method GFN2-xTB. Atomic and molecular properties (e.g., partial charges,
energies, and rotational constants) are provided on both the GFN2-xTB and on
the DFT ($\omega$B97X-D/def2-SVP) levels of theory. QMugs also comprises the
respective quantum mechanical wave functions, including DFT density and
orbital matrices, totaling over 7 terabytes of uncompressed data. This dataset
is intended to facilitate the development of models that learn from molecular
data on different levels of theory while also providing insight into the
corresponding relationships between molecular structure and biological
activity.

## Background & Summary

Machine learning methodologies are increasingly becoming well-established
tools in many chemistry-related disciplines, such as drug discovery [1],
material science [2], and physical chemistry [3]. In recent years, significant
progress has been made in quantum-based machine learning (QML) methods [4],
which aim to accurately and computationally inexpensively predict the
governing properties of atomistic systems, such as energies and forces [5, 6,
7, 8, 9, 10, 11, 12], dipole moments [13], wave functions [14, 15] and
electron densities [16, 17]. Despite the success and promise surrounding the
applicability of such approaches, several challenges remain for QML. Arguably,
one of the most important challenges is the increasing need for curated,
comprehensive datasets. [13] While several options, such as the QM9 [18] or
ANI-1 [19] sets have paved the way for the development of current-generation
QML methods [5, 6, 7, 8, 20, 21, 22], the computational cost entailed in their
generation limits both the scope of the explored chemical space (e.g.,
molecule size, atom-type diversity), and prospective modeling applicability
[13, 23].

There has been a recent surge in interest in the delta-learning
($\Delta$-learning) of chemical properties, which aims to use a machine
learning model to predict a physically relevant quantity, such as those
generated by density-functional theory (DFT) by utilizing information
extracted with a computationally cheaper method [24, 21] (e.g., semi-empirical
approaches such as GFN2-xTB [25, 26, 27, 28, 29] and PM6 [30]). Datasets that
enable this type of learning are scarce and could promote the development of
accurate models at potentially a fraction of the computational cost of more
precise alternatives [31]. Furthermore, datasets that provide three-
dimensional conformational data, for a wide variety of chemical space, at
levels of theory higher than classical force fields [32, 33], could boost the
performance of machine learning methods in predicting properties from
ensembles as well as generative models of conformations. Relevant examples
include the PubChemQC-PM6 [22] and GEOM [33] datasets, which include molecules
with properties computed using different semi-empirical levels of theory.
Finally, there is a clear potential to open up new lines of research by
combining biological annotations (e.g., from molecular databases such as
ChEMBL [34]), and additional QM-derived physical information.

This work introduces QMugs (Quantum-Mechanical Properties of Drug-like
Molecules), a data collection of over $665$k curated molecular structures
extracted from the ChEMBL database, with accompanying computed quantum
mechanical properties. Different levels of theory were combined in these
calculations. Per compound, three conformers were generated, and their
geometries were optimized using the semi-empirical GFN2-xTB method [25, 26,
27, 28, 29], whereas a comprehensive array of quantum properties was computed
at the DFT level of theory using the $\omega$B97X-D functional [35] and the
def2-SVP Karlsruhe basis set [36]. The data collection presented herein is put
in the context of other sets that also feature DFT-level properties. A
descriptive evaluation against QM9 [18], PubchemQC [37], and the ANI-1 [19]
datasets is provided in Figure 1B, as well as in Table 1. As previously
reported for the ChEMBL database [38], most of the considered drug-like
molecules in this study fall within the rod-disk axis in the principal moments
of inertia plot [39] (Figure 1A). Furthermore, the vast majority of the
included compounds ($\sim 641$k, $96.3\%$) were previously unreported in other
DFT data collections, while also providing equivalent information at
additional levels of theory, namely GFN2-xTB. With an average of $30.6$ and a
maximum of $100$ heavy atoms per compound (Table 1 & Figure 2), QMugs also
features molecular samples that are considerably larger than those provided by
other datasets. To the best of our knowledge, this work is the first to
provide a large and diverse dataset of quantum mechanical wave functions
represented as local bases of atomic orbitals (i.e., DFT density and orbital
matrices). Single-point properties as well as wave functions were computed
with the Psi4 software suite [40] for all the conformers ($\sim 2.0$M) present
in the database, totaling over $7$ terabytes of supplied quantum mechanical
data.

Overall, the utility of the presented dataset is fourfold: (i) it will provide
researchers with the largest-to-date dataset to either directly predict the
quantum chemical properties, or learn a property mapping between two popular
quantum mechanical levels of theory (i.e., GFN2-xTB and
$\omega$B97X-D/def2-SVP); (ii) it will facilitate the development of novel
machine-learning methodologies for the generation of molecular conformations
and molecular property predictions via their ensembles; (iii) it will enable
the development of novel deep learning frameworks for the prediction of the
quantum mechanical wave function in a local basis of atomic orbitals from
which all ground-state properties as well as electron densities can be
derived; and (iv) it will enable research towards the exploration of QML
methods quantum featurization in the context of pharmacologically relevant,
annotated biological data.

## Methods

Molecules were extracted from the ChEMBL database [34] (version 27).
Conformers were generated using RDKit [41] and GFN2-xTB [25, 26, 28, 29]. DFT
($\omega$B97X-D/def2-SVP) calculations were carried out via Psi4 [40]. A
similar approach was adopted in a previous study on transition-metal
complexes. [13] An overview of the data processing pipeline is given in Figure
3, while individual steps are described in more detail in the following
subsections.

In chemical terminology, the term “conformation” refers to any arrangement of
atoms in space, whereas “conformer” refers to a conformation that is a local
minimum on the potential energy surface of the molecule. [42] In the analyses
that follow, the term “conformation” is loosely used to refer to both, unless
explicitly mentioned otherwise.

### Data extraction and SMILES processing

Single-protein targets with assay information for at least $10$ compounds with
unique internal identifiers were extracted from the ChEMBL database. Several
activity and annotation filters were subsequently applied to these compounds
(see ESI for a detailed query description). This procedure resulted in
$685,917$ molecules with unique external identifiers (ChEMBL-IDs), represented
by their Simplified Molecular Input Line Entry Specification (SMILES) [43].
Molecules were neutralized, and salts and solvents were removed using the
ChEMBL Structure Pipeline package [44, 45]. For compounds consisting of
multiple separate fragments after this "washing" procedure, all except the one
with the highest number of heavy atoms were discarded. Additionally, molecules
containing fewer than $3$ or more than $100$ heavy atoms, as well as radical
species and molecules with a net charge different from zero after the
attempted neutralization, were removed. Atom types included in the QMugs
dataset are hydrogen, carbon, nitrogen, oxygen, fluorine, phosphorus, sulfur,
chloride, bromide, and iodine.

### Conformer generation and optimization

With the procedure described herein, a compromise between efficient molecular
conformational search and practical computational expense considerations was
sought.

The RDKit [41] implementation of the Experimental-Torsion Knowledge Distance
Geometry (ETKDG) method [46] was used to generate up to $100$ conformers for
each molecule, with a maximum of $1000$ embedding attempts and an initial
coordinate assignment using distance-matrix eigenvalues and default settings
(boxSizeMult=2.0, force-field tolerance=1e-3) . Upon no successful conformer
generation, it was re-attempted via random assignment of the starting
coordinates. The resulting conformers were further minimized using the Merck
molecular force field [47] (MMFF94s) for a maximum of $1000$ iterations, with
default settings (nonBondedThresh=100.0). The lowest-energy conformer
(according to the selected force field) for each structure was then used as a
starting point for meta-dynamics (MTD) simulations. Stereocenters that were
previously undefined in the SMILES extraction procedure were assigned in this
conformer generation process.

For each generated conformer, an MTD simulation was performed with the xTB
software package [28] for a duration of $5$ ps with time steps of $1$ fs, at a
temperature of $300$ K. The biasing root-mean-square deviation (RMSD)
potential used for all MTD simulations is given by
$E_{\mathrm{bias}}^{\mathrm{RMSD}}=\sum_{i=1}^{N}k_{i}\exp(\alpha\Delta^{2}_{i}$),
where $N$ is the number of reference structures, $k_{i}$ the pushing strength,
$\Delta_{i}$ the collective variable (i.e., the RMSD between structure $i$ and
a reference structure), and $\alpha$ the width of the Gaussian potential used
in the RMSD criterion. Simulations were carried out with
$\alpha=$$1.2\text{\,}\mathrm{\text{ $a_{0}$ }}$-1 and
$k_{i}=$$0.2\text{\,}\mathrm{m\text{ $E_{\textup{ h }}$ }}$ with snapshots
taken every $50$ fs, resulting in $100$ conformations stored with their
corresponding energies. To obtain conformationally diverse samples, these
structures were subsequently clustered into three groups via the $k$-means
[48] algorithm, as implemented in the scikit-learn [49] (version 0.23.1)
Python package using the pairwise RMSD of the aligned structures as molecular
features. The conformation with the lowest-energy value from each cluster was
then selected for further processing. The three resulting conformations for
each molecule were then optimized using the GFN2-xTB [25, 26, 28, 29] method
using energy and gradient convergence criteria of $5\times 10^{-6}$
$\mathrm{\text{ $E_{\textup{ h }}$ }}$ and $1\times 10^{-3}$ $\mathrm{\text{
$E_{\textup{ h }}$ }}$ $\alpha^{-1}$, respectively, and the approximate normal
coordinate rational function optimizer (ANCopt). Harmonic frequencies,
entropies, enthalpies and heat capacities at $298.15$ K were extracted at the
end of the geometry optimization process. Structures for which vibrational
frequencies with imaginary wave numbers were obtained — indicative of failure
to reach energy minima — were subjected to additional optimizations until no
significant ones remained, up to a maximum of $100$ attempts.

### Quantum mechanical calculations

Single-point electronic calculations were performed for the optimized
geometries using the $\omega$B97X-D quantum functional and the def2-SVP basis
set as implemented in the open-source quantum-chemistry software suite Psi4
[40]. Single-point properties such as formation and orbital energies, dipole
moments, rotational constants, partial charges, bond orders, valence numbers,
as well as wave functions including $\alpha$ and $\beta$ DFT-density matrices,
orbital matrices, and the atomic-orbital-to-symmetry-orbital transformer
matrix were obtained. For practical reasons, $52$ structures whose DFT
calculations required computational resources that exceeded empirically
determined limits, or for which calculations were unsuccessful, were discarded
(see ESI for details).

## Data Records

All computed molecular structures, as well as their corresponding properties
and wave functions are accessible through the ETH Library Collection service
(Data Citation 1) [50].

### Format specification

A summary.csv comma-separated file contains computed molecular-level
properties and additional annotations. A compressed tarball file
(structures.tar.gz) of $\sim 7$ gigabytes (GB) contains plain MDL structure-
data files [51] (SDFs) with embedded atomic and molecular properties, grouped
in sub-directories according to their respective ChEMBL identifiers. These
SDFs include single-point electronic properties calculated on the GFN2-xTB and
$\omega$B97X-D/def-SVP levels of theory, as described in Table 2. A second
compressed tarball file (vibspectra.tar.gz, $\sim 3$ GB) contains vibrational
spectra.

Wave function files, including additional properties such as DFT-density and
orbital matrices, as described in Table 3, are split into $100$ compressed
tarballs (wfns_xx.tar.gz) of $\sim 50$ GB each for easier management and
downloading. These are supplied as NumPy [52] (.npy) binary files, which can
be read using the Psi4 software package. Molecules (with all conformers
grouped together) were assigned at random to the tarballs to enable easy use
of subsets of the QMugs dataset without having to download all the files. The
assignment of ChEMBL identifiers to tarballs is described in a
tarball_assignment.csv file.

## Technical Validation

### Optimized geometry sanity checks

Four consecutive geometry checks were performed to filter out structures for
which the geometry optimization procedure converged to unrealistic
conformations. To determine suitable thresholds for removing a structure from
our dataset, the generated geometries were compared to experimental reference
values and to DFT-optimized geometries extracted from the PubChemQC dataset
[37]. Specifically, we investigated (i) the deviation of bond lengths from
experimental reference values, (ii) isomorphism between the initial molecular
graphs and those obtained after geometry optimization, (iii) linearity of
triple bonds, and (iv) planarity of aromatic rings. Structures were removed
from the dataset if they failed any test of these tests. In total, $10,986$
($0.55$%) conformations were discarded from the dataset. Each test is briefly
described in the following subsections, with further technical details
reported in the ESI.

#### Deviation of bond lengths from experimental reference values

Bond lengths in the optimized structures were compared to average experimental
reference values for bonds of the same bond type (single, double, triple, or
aromatic) and between the same atoms. Reference values were obtained from the
Computational Chemistry Comparison and Benchmark DataBase (CCCBDB) [53], and
the largest absolute bond-length deviation from reference values was recorded
per molecule. Bonds for which no reference value was available ($0.75\%$) were
omitted. The same analysis was carried out for molecules from the PubChemQC
dataset containing the same atom types as QMugs, in order to obtain a
comparable set with respect to the present atom types. The PubChemQC set
($3,834,382$ conformations with reference bond lengths) showed a deviation of
$0.06$ $\pm$ $0.04\text{\,}\mathrm{\SIUnitSymbolAngstrom}$ (median $\pm$ $1$
standard deviation), whereas the QMugs dataset ($2,004,003$ conformations with
reference bond lengths) showed a deviation of $0.07$ $\pm$
$0.03\text{\,}\mathrm{\SIUnitSymbolAngstrom}$. Based on the observed
distribution of bond-length deviations from experimental reference values
(Figure S1 in ESI) and manual investigation of example structures,
$0.2\text{\,}\mathrm{\SIUnitSymbolAngstrom}$ was determined to be a suitable
threshold for a conformation to be removed from the dataset, which included
$6,131$ ($0.31$%) examples.

#### Molecular graph isomorphism

It was investigated whether atom connectivity could be reconstructed after
removing bond information from the generated SDFs. To this end, molecular
graphs constructed exclusively from atom positions and types were compared to
those obtained using the original atom connectivity (see ESI for details).
$1,568$ ($0.08$%) conformations for which the resulting molecular graphs were
non-isomorphic failed this test.

#### Deviation of triple bonds from linear geometry

The deviation of triple bonds from their ideal linear geometry was examined.
In this investigation, ring triple bonds were not considered owing to
routinely-occurring deviations from linear geometry in systems with high ring
strain [54]. The largest deviation from a
$180\text{\,}\mathrm{\SIUnitSymbolDegree}$ (linear) bond angle was recorded
for each molecule containing at least one non-ring triple bond. The same
analysis was performed on the PubChemQC dataset [37]. Triple-bond-containing
molecules from PubchemQC and QMugs ($273,320$ and $165,101$ samples,
respectively) show deviations of $1.38$ $\pm$
$1.46\text{\,}\mathrm{\SIUnitSymbolDegree}$ (median $\pm$ $1$ standard
deviation), and $1.46$ $\pm$ $2.13\text{\,}\mathrm{\SIUnitSymbolDegree}$,
respectively. Based on the observed distribution of triple bond angles from a
linear geometry (Figure S2 in ESI) and manual inspection of structures, a
$10\text{\,}\mathrm{\SIUnitSymbolDegree}$ deviation was identified as a
suitable threshold. $1,147$ ($0.06$%) conformations failed this test.

#### Deviation of aromatic rings from planar geometry

The planarity of carbon-containing aromatic rings was also investigated. For
each molecule containing aromatic carbon atoms, the largest dihedral angle
between the two planes spanned by each aromatic carbon atom and its three
neighbors was recorded (see ESI for details). The same analysis was performed
on the PubChemQC dataset [37]. Molecules from PubchemQC and QMugs ($2,391,589$
and $1,950,929$ conformations with aromatic carbons, respectively) showed
median dihedral angles ( $\pm$ $1$ standard deviation) of $1.70$ $\pm$
$1.85\text{\,}\mathrm{\SIUnitSymbolDegree}$ and $2.99$ $\pm$
$2.20\text{\,}\mathrm{\SIUnitSymbolDegree}$, respectively. Based on the
observed distribution of dihedral angles from planar geometries (Figure S3 in
ESI) and manual inspection of structures, $2,769$ ($0.14$%) conformations with
aromatic carbon dihedral angles above
$15\text{\,}\mathrm{\SIUnitSymbolDegree}$ were discarded.

### Further geometrical assessment

The changes in the molecular geometries along the applied pipeline were
examined in order to evaluate the effects of the applied steps. Figure 4A
shows the mean pairwise RMSD of atom positions between the conformations of
each molecule at different steps along the pipeline. Conformations sampled
during MTD simulations show a mean pairwise RMSD of $2.40$ $\pm$ $0.52$
$\mathrm{\SIUnitSymbolAngstrom}$ (median $\pm$ $1$ standard deviation). The
$k$-means clustering procedure accomplishes the envisaged task of sampling
conformations with higher geometric diversity ($2.67$ $\pm$ $0.74$
$\mathrm{\SIUnitSymbolAngstrom}$). During the geometry optimization process,
conformational diversity decreases ($2.48$ $\pm$ $0.86$
$\mathrm{\SIUnitSymbolAngstrom}$). Unsurprisingly, for some molecules
featuring rigid structures, conformations tend to converge toward the same
energy minimum ($0.09$ % of molecules show a mean pairwise RMSD $<$
$0.01\text{\,}\mathrm{\SIUnitSymbolAngstrom}$ between their optimized
conformers).

The degree to which the molecular geometries changed during the final
optimization step was further analyzed. Molecules with initially more diverse
conformations (higher mean pairwise RMSD of pre-optimized conformations) were
shown to undergo a greater chance in atom positions (mean RMSD of pre- vs.
post-optimized conformations) during optimization with the GFN2-xTB method
(Figure 4B). The observed heteroscedastic behavior of these two properties
indicates that while the mean RMSD of pre- vs. post-optimized conformations
tends to increase with higher mean pairwise RMSD of pre-optimized
conformations, its variance also increases.

Finally, the suitability of GFN2-xTB as a lower-cost surrogate for DFT-level
geometry optimization (Figure 4C) was confirmed. $500$ randomly-chosen
structures prior to semi-empirical geometry optimization from the QMugs
dataset were further subjected to DFT-level geometry optimization
($\omega$B97X-D/def2-SVP), discarding structures that could not be converged
in 100 iterations or with the computational resources described in the ESI.
The RMSDs between the structures independently optimized at both levels of
theory were then measured. The pairs of structures show RMSDs of $0.47$ $\pm$
$0.63$ $\mathrm{\SIUnitSymbolAngstrom}$ (median $\pm$ $1$ standard deviation),
indicating that the chosen semi-empirical method obtains similar geometries to
those obtained with more expensive first-principle calculations. Large RMSDs
in some example pairs (Figure 4C) could be interpreted as indicative of
convergence to distinct local minima.

For $2,067$ molecules, their individual conformations have different SMILES
describing two different (E)/(Z) isomers. Those structures are either
$\alpha$-$\beta$-unsaturated ketones, $\alpha$-$\beta$-unsaturated nitriles,
imine, or azo compounds, for which isomerization might be plausible [55, 56].
In part due to the applied washing procedure, $17,176$ molecules can be
represented with a SMILES string that is shared with at least one other
ChEMBL-ID.

### Validation of single-point properties

To validate the general agreement between the two methods employed in this
work, the correlation between a series of single-point properties computed on
both levels of theory was analyzed. Both global, molecular (Figure 5) and
local, atomic/bond properties (Figures 6 & 7) were considered. All single-
point molecular properties showed a high degree of correlation. Formation
energies $E_{\mathrm{Form}}$ (Figure 5A), which were obtained by subtracting
atomic energies $E_{\mathrm{Atom}}$ from total internal energies $U_{RT}$,
show a Pearson correlation coefficient (PCC) of $0.998$. Dipole moments $\mu$
and rotational constants $A$ (excl. $22$ small structures with very high
rotational constants; Figure 5B,C) display PCCs of $0.969$ and $0.999$,
respectively. Orbital energies, namely the energies for highest occupied
(HOMO) $E_{\mathrm{HOMO}}$ and lowest unoccupied molecular orbitals (LUMO)
$E_{\mathrm{LUMO}}$ and HOMO-LUMO gaps $E_{\mathrm{Gap}}$ show PCCs of
$0.769$, $0.924$ and $0.830$, respectively (Figures 5 D, E & F). The observed
PCCs for all six single-point molecular properties indicate good agreement
between the two methods. Atom-type-specific partial charges for the 10 atom-
types in QMugs (Figure 6, ESI Table 1) as well as the 15 most abundant
covalent bonds orders (Figure 7, ESI Table 2) also show high correlations
between the two methods used herein. Regarding partial charges, $7$ out of the
$10$ atom types considered in QMugs were observed to have PCCs $>0.8$, with
the remaining carbon, nitrogen, and oxygen atom-types resulting in lower PCCs
of $0.574$, $0.124$, and $0.274$, respectively. Regarding bond orders, $10$
out of the $15$ show PCCs $>0.9$ and $14$ out of $15$ display PCCs $>0.75$
(see ESI Table 2 for additional metrics). Notably only carbon-fluorine bonds
display a larger discrepancy between both levels of theory, with an observed
PCC of $0.153$. The observed correlations in both molecular and atomic single-
point properties between GFN2-xTB and $\omega$B97X-D/def2-SVP confirm the
suitability of the former method as a computationally affordable starting
point for $\Delta$-learning of DFT-level properties.

## Usage Notes

All data files can be accessed via any modern web browser, and can be
programmatically downloaded using the provided instructions in the archive’s
readme. The provided SDFs can be processed using standard cheminformatics
software (e.g., RDKit [41], KNIME [57]), and wave function files using the
Psi4 [40] software package or directly using Numpy [52].

## Code availability

All analyses were supported by the Python programming language (version 3.7.7)
and its scientific software stack [52]. Molecular conformations were generated
using RDKit [41] (version 2020.03.3) and GFN2-xTB [25, 26, 27, 28, 29]
(version 6.3.1). All quantum mechanical calculations were carried out with
Psi4 [40] (version 1.3.2). Molecular structure visualizations were created
using PyMol [58] (version 2.3.5) and ChemDraw (version 19.1.1.32). The rclone
(https://rclone.org) WebDAV client was used for all data uploading purposes.

## Acknowledgements

We thank Dr. Jan A. Hiss and Dr. Petra Schneider for their valuable insights
and discussions. Special thanks to Andreas la Roi from the ETH Research
Collection team and to the ETH cluster team for their technical support
throughout this project. This work was financially supported by the ETH
RETHINK initiative, the Swiss National Science Foundation (grant no.
205321_182176), and Boehringer Ingelheim Pharma GmbH & Co. KG. C.I.
acknowledges support from the Scholarship Fund of the Swiss Chemical Industry.

## Author contributions statement

Clemens Isert: Methodology, Formal Analyses, Writing. Kenneth Atz:
Methodology, Formal Analyses, Writing. José Jiménez-Luna: Conceptualization,
Methodology, Writing. Gisbert Schneider: Supervision, Writing.

## Competing interests

G.S. is a cofounder of inSili.com LLC, Zurich, and a consultant to the
pharmaceutical industry.

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## Figures & Tables

Figure 1: A) Principal-moments-of-inertia plot [39] for molecules in the QMugs
dataset. $NPR_{x}$ = $x$-th normalized principal moment, $I_{x}$ = $x$-th
smallest principal moment of inertia. B) Venn diagram showing overlap between
QMugs and other well-known datasets with DFT-level computed properties: QM9
[18], PubChemQC [37], and ANI-1 [19]. Overlap was computed based on the
uniqueness of the InChI representations of the contained molecules. Numbers do
not add up to those reported in Table 1 because of InChI strings that occur
multiple times. Figure 2: Distribution of properties for the molecules
contained in the QMugs dataset. Figure 3: Overview of the data generation
process. Molecules were extracted from the ChEMBL database, standardized, and
filtered, and starting conformers were generated using the RDKit software
package. Metadynamics (MTD) simulations were performed using the GFN2-xTB
semi-empirical method to generate three diverse conformations before final
geometry optimization. Molecules that did not pass a series of geometric
sanity checks were removed. DFT-level properties ($\omega$B97X-D/def2-SVP)
were computed using the Psi4 software. Figure 4:  (A) Distributions of mean
pairwise RMSD of atom positions between conformations of each molecule in the
QMugs dataset at different stages along the pipeline. While the $k$-means
sampling process selects conformations that are, on average, more
geometrically diverse than the average pair of structures generated by MTD
simulations, geometry optimization reduces the geometrical diversity between
the optimized conformers. (B) Change in atom positions during geometry
optimization vs. mean pairwise RMSD of conformations before optimization.
Molecules with initially more diverse conformations displayed a greater change
in atom positions than those with initially less diverse conformations. (C)
Distribution of RMSD of structures prior to and after optimization with the
semi-empirical GFN2-xTB method, and of structures optimized with the same
approach vs. with $\omega$B97X-D/def2-SVP. The structures of three molecules
with varying differences between the two methods are shown as illustrative
examples (black and gray correspond to GFN2-xTB and $\omega$B97X-D/def2-SVP-
optimized structures, respectively). For illustrative purposes, the example
molecules are aligned on their substructures.  Figure 5: Comparison of
molecular properties computed at the two levels of theory considered herein
(GFN2-xTB, $\omega$B97X-D/def2-SVP) for the molecules contained in QMugs. The
molecular formation energy $E_{\mathrm{form}}$ in (A) was calculated by
subtracting the atomic $U_{\mathrm{Atom}}$ contributions from the total
molecular energies $U_{RT}$. Only the rotational constants $A$ are shown in
(C) as their $B$ and $C$ counterparts showed highly similar values. $22$
conformations of small molecules show very large rotational constants and are
not shown. RMSE and PCC for rotational constant $A$ are $845.834$ cm-1 and
$0.091$ respectively, if those structures are included. Abbreviations: RMSE,
root mean squared error; PCC, Pearson’s correlation coefficient. Figure 6:
Atom-type-specific partial charge correlations (GFN2-xTB,
$\omega$B97X-D/def2-SVP) for the QMugs dataset (see ESI Table 1 for additional
metrics) Figure 7: Comparison of Wiberg bond orders between GFN2-xTB and
$\omega$B97X-D/def2-SVP for the 15 most frequently occurring bond types in the
QMugs dataset. The latter level of theory uses Löwdin-orthogonalization. See
ESI Table 2 for additional metrics. For bond types which occurred $>1$M times
in the dataset, a randomly chosen sample of $1$M bonds is plotted. Table 1:
Descriptive statistics of the dataset reported herein in the context of other
DFT-level molecular datasets and the information provided by each. The number
of molecules for PubChemQC corresponds to that available on the website of the
project. [59] Heavy atom averages are weighted by the number of conformations.

Dataset |  Unique compounds |  Total conformations |  Heavy atoms max (mean) |  Method |  $\Delta$-learning possible |  Wave functions
---|---|---|---|---|---|---
QM9 |  $133,885$ |  $133,885$ |  $9$ ($8.8$) |  B3LYP/6-31G(2df,p) |  ✗ |  ✗
ANI-1 |  $57,462$ |  $22,057,374$ |  $8$ ($7.1$) |  $\omega$B97X/6-31G(d) |  ✗ |  ✗
PubChemQC |  $3,982,436$ |  $3,982,436$ |  $51$ ($14.1$) |  B3LYP/6-31G(d) |  ✗ |  ✗
QMugs |  $665,911$ |  $1,992,984$ |  $100$ ($30.6$) |  GFN2-xTB + $\omega$B97X-D/def2-SVP |  ✓ |  ✓

Table 2: Calculated properties as stored in the SDFs of the QMugs data collection. Abbreviations: a.u., atomic units; vib., vibrational; rot., rotational; transl., translational. Properties that enable $\Delta$ machine learning are labelled with $\blacklozenge$. Property | Symbol | Unit | Key |  $\Delta$-ML
---|---|---|---|---
ChEMBL identifier | - | - | CHEMBL_ID |
Conformer identifier | - | - | CONF_ID |
Total energy | $U_{RT}$ | $\mathrm{\text{ $E_{\textup{ h }}$ }}$ | GFN2:TOTAL_ENERGY | $\blacklozenge$
Internal atomic energy | $E_{\mathrm{Atom}}$ | $\mathrm{\text{ $E_{\textup{ h }}$ }}$ | GFN2:ATOMIC_ENERGY |
Formation energy | $E_{\mathrm{Form}}$ | $\mathrm{\text{ $E_{\textup{ h }}$ }}$ | GFN2:FORMATION_ENERGY | $\blacklozenge$
Total enthalpy | $H_{RT}$ | $\mathrm{\text{ $E_{\textup{ h }}$ }}$ | GFN2:TOTAL_ENTHALPY |
Total free energy | $G_{RT}$ | $\mathrm{\text{ $E_{\textup{ h }}$ }}$ | GFN2:TOTAL_FREE_ENERGY |
Dipole ($x$, $y$, $z$, total) | $\mu$ | D | GFN2:DIPOLE | $\blacklozenge$
Quadrupole ($xx$, $xy$, $yy$, $xz$, $yz$, $zz$) | $Q_{ij}$ |  D $\mathrm{\SIUnitSymbolAngstrom}$ | GFN2:QUADRUPOLE |
Rotational constants ($A$, $B$, $C$) |  $A$, $B$, $C$ |  $\mathrm{cm}$-1 | GFN2:ROT_RONSTANTS | $\blacklozenge$
Enthalpy (vib., rot., transl., total) | $\Delta H$ |  cal mol-1 | GFN2:ENTHALPY |
Heat capacity (vib., rot., transl., total) | $C_{V}$ |  cal K-1 mol-1 | GFN2:HEAT_CAPACITY |
Entropy (vib., rot., transl., and total) | $\Delta S$ |  cal K-1 mol-1 | GFN2:ENTROPY |
HOMO energy | $E_{\mathrm{HOMO}}$ | $\mathrm{\text{ $E_{\textup{ h }}$ }}$ | GFN2:HOMO_ENERGY | $\blacklozenge$
LUMO energy | $E_{\mathrm{LUMO}}$ | $\mathrm{\text{ $E_{\textup{ h }}$ }}$ | GFN2:LUMO_ENERGY | $\blacklozenge$
HOMO-LUMO gap | $E_{\mathrm{Gap}}$ | $\mathrm{\text{ $E_{\textup{ h }}$ }}$ | GFN2:HOMO_LUMO_GAP | $\blacklozenge$
Fermi level | $E_{\mathrm{Fermi}}$ | $\mathrm{\text{ $E_{\textup{ h }}$ }}$ | GFN2:FERMI_LEVEL |
Mulliken partial charges | $\delta_{M}$ | $\mathrm{\text{ $e$ }}$ | GFN2:MULLIKEN_CHARGES | $\blacklozenge$
Covalent coordination number | $N_{\textrm{coord}}$ | - | GFN2:COVALENT_COORDINATION_NUMBER |
Molecular dispersion coefficient | $C_{6}$ | a.u. | GFN2:DISPERSION_COEFFICIENT_MOLECULAR |
Atomic dispersion coefficients | $C_{6}$ | a.u. | GFN2:DISPERSION_COEFFICIENT_ATOMIC |
Molecular polarizability | $\alpha(0)$ | a.u. | GFN2:POLARIZABILITY_MOLECULAR |
Atomic polarizabilities | $\alpha(0)$ | a.u. | GFN2:POLARIZABILITY_ATOMIC |
Wiberg bond orders | $M_{AB}$ | - | GFN2:WIBERG_BOND_ORDER | $\blacklozenge$
Total Wiberg bond orders | $\sum_{A(A\neq B)}M_{AB}$ | - | GFN2:TOTAL_WIBERG_BOND_ORDER | $\blacklozenge$
Total energy | $U_{RT}$ | $\mathrm{\text{ $E_{\textup{ h }}$ }}$ | DFT:TOTAL_ENERGY | $\blacklozenge$
Total internal atomic energy | $E_{\mathrm{Atom}}$ | $\mathrm{\text{ $E_{\textup{ h }}$ }}$ | DFT:ATOMIC_ENERGY |
Formation energy | $E_{\mathrm{Form}}$ | $\mathrm{\text{ $E_{\textup{ h }}$ }}$ | DFT:FORMATION_ENERGY | $\blacklozenge$
Electrostatic potential | $V_{ESP}$ | $\mathrm{V}$ | DFT:ESP_AT_NUCLEI |
Löwdin partial charges | $\delta_{L}$ | $\mathrm{\text{ $e$ }}$ | DFT:LOWDIN_CHARGES |
Mulliken partial charges | $\delta_{M}$ | $\mathrm{\text{ $e$ }}$ | DFT:MULLIKEN_CHARGES | $\blacklozenge$
Rotational constants ($A$, $B$, $C$) |  $A$, $B$, $C$ |  $\mathrm{cm}$-1 | DFT:ROT_CONSTANTS | $\blacklozenge$
Dipole ($x$, $y$, $z$, total) | $\mu$ | D | DFT:DIPOLE |
Exchange correlation energy | $\hat{V}_{eN}$ | $\mathrm{\text{ $E_{\textup{ h }}$ }}$ | DFT:XC_ENERGY |
Nuclear repulsion energy | $\hat{V}_{eN}$ | $\mathrm{\text{ $E_{\textup{ h }}$ }}$ | DFT:NUCLEAR_REPULSION_ENERGY |
One-electron energy | $\hat{T}_{e}$ | $\mathrm{\text{ $E_{\textup{ h }}$ }}$ | DFT:ONE_ELECTRON_ENERGY |
Two-electron energy | $\hat{V}_{ee}$ | $\mathrm{\text{ $E_{\textup{ h }}$ }}$ | DFT:TWO_ELECTRON_ENERGY |
HOMO energy | $E_{\mathrm{HOMO}}$ | $\mathrm{\text{ $E_{\textup{ h }}$ }}$ | DFT:HOMO_ENERGY | $\blacklozenge$
LUMO energy | $E_{\mathrm{LUMO}}$ | $\mathrm{\text{ $E_{\textup{ h }}$ }}$ | DFT:LUMO_ENERGY | $\blacklozenge$
HOMO-LUMO gap | $E_{\mathrm{Gap}}$ | $\mathrm{\text{ $E_{\textup{ h }}$ }}$ | DFT:HOMO_LUMO_GAP | $\blacklozenge$
Mayer bond orders | $M_{AB}$ | - | DFT:MAYER_BOND_ORDER |
Wiberg-Löwdin bond orders | $W_{AB}$ | - | DFT:WIBERG_LOWDIN_BOND_ORDER | $\blacklozenge$
Total Mayer bond orders | $\sum_{A(A\neq B)}M_{AB}$ | - | DFT:TOTAL_MAYER_BOND_ORDER |
Total Wiberg-Löwdin bond orders | $\sum_{A(A\neq B)}W_{AB}$ | - | DFT:TOTAL_WIBERG_LOWDIN_BOND_ORDER | $\blacklozenge$
Table 3: Calculated molecular properties stored in the wave function files provided in the QMugs data collection. Mayer and Wiberg-Löwdin bond orders included here represent a superset of the bond orders in the SDFs which additionally comprise bond orders for non-covalent bonds. Property | Symbol | Key
---|---|---
Alpha density matrix | $\mathrm{D}_{\alpha}$ | matrix, Ca
Beta density matrix | $\mathrm{D}_{\beta}$ | matrix, Cb
Alpha orbitals | $\mathrm{C}_{\alpha}$ | matrix, Da
Beta orbitals | $\mathrm{C}_{\beta}$ | matrix, Db
Atomic-orbital-to-symmetry-orbital transformer | $\mathrm{C}_{\mathrm{AOTOSO}}$ | matrix, aotoso
Mayer bond orders | $M_{AB}$ | MAYER_INDICES
Wiberg-Löwdin bond orders | $W_{AB}$ | WIBERG_LOWDIN_INDICES