MBD-ML: Many-body dispersion from machine learning for molecules and materials

Evgeny Moerman; Adil Kabylda; Almaz Khabibrakhmanov; Alexandre Tkatchenko

physics.chem-phFeb 25, 2026

MBD-ML: Many-body dispersion from machine learning for molecules and materials

Authors: Evgeny Moerman, Adil Kabylda, Almaz Khabibrakhmanov, Alexandre Tkatchenko

Organizations: Department of Physics and Materials Science, University of Luxembourg, L-1511 Luxembourg City, Luxembourg

Abstract

Van der Waals (vdW) interactions are essential for describing molecules and materials, from drug design and catalysis to battery applications. These omnipresent interactions must also be accurately included in machine-learned force fields. The many-body dispersion (MBD) method stands out as one of the most accurate and transferable approaches to capture vdW interactions, requiring only atomic $C_6$ coefficients and polarizabilities as input. We present MBD-ML, a pretrained message passing neural network that predicts these atomic properties directly from atomic structures. Through seamless integration with libMBD, our method enables the immediate calculation of MBD-inclusive total energies, forces, and stress tensors. By eliminating the need for intermediate electronic structure calculations, MBD-ML offers a practical and streamlined tool that simplifies the incorporation of state-of-the-art vdW interactions into any electronic structure code, as well as empirical and machine-learned force fields.

Figures & tables

Figure 1: Overview of the training and the validation of the MBD-ML model: (top left) MBD-ML was trained on the QCML molecular data set spanning almost the entire periodic system (top right; Reproduced from Ref. 16 under CC BY 4.0). (middle) The model has been integrated into the libMBD python interface, pymbd, to allow direct calculation of MBD properties from the atomic structure. (bottom) The model was validated on both molecules and molecular crystals in predicting fundamental properties including total energy and force contributions and – in the case of organic crystals – to perform geometry optimizations and recover the energy ranking of different polymorphs

	QCML a ( $N{=}83226$ )		DES370k a,b ( $N{=}309130$ )		OMOL25 b ( $N{=}825$ )		OMC25 ( $N{=}200$ )		OMAT24 ( $N{=}203$ )
	MAE	RMSE	MAE	RMSE	MAE	RMSE	MAE	RMSE	MAE	RMSE
Ratio Predictions
$\alpha_{0}^{\mathrm{r}}$	0.013	0.020	0.014	0.025	0.023	0.033	0.025	0.031	0.247	0.429
$C_{6}^{\mathrm{r}}$	0.014	0.023	0.013	0.022	0.021	0.030	0.017	0.022	0.269	0.895
MBD Properties
$E_{\mathrm{MBD}}$ (meV/atom)	0.158	0.228	0.117	0.165	0.198	0.226	0.713	0.841	–	–

Table 1: Performance of MBD-ML models on ratios and MBD properties for different test sets in terms of RMSE and MAE values. Errors for

\alpha_{0}^{\mathrm{r}}

and

C_{6}^{\mathrm{r}}

are unitless.

N

denotes the number of systems in each test set. a Excluding negatively charged molecules (due to insufficient accuracy of ab initio treatment; see QCML discussion in the Model validation section). b Excluding alkali and alkaline earth metals (due to under-representation in the training set; see DES370k discussion in the Model validation section)

Panel of Figure 2 — Figure 2: Performance of PBE0+MBD-ML in predicting the $C_{6}$ and $\alpha_{0}$ ratios and the MBD contribution to the total energy and atomic forces of the QCML test set

Panel of Figure 3 — Figure 3: Performance of PBE0+MBD-ML in predicting the $C_{6}$ and $\alpha_{0}$ ratios and the MBD contribution to the total energy, atomic forces and stress tensor components in the OMC25 test set

Panel of Figure 4 — Figure 4: Comparison of MBD-ML and DFT-D performance on predicting vdW force contributions compared to MBD-NL. Polar representation of atomic force deviations computed from the MBD-ML ( a ), D3 ( b ) and D4 method ( c ). The radial coordinate specifies the relative magnitude deviation of the atomic force vector to the MBD-NL reference in percent. The angular coordinate corresponds to the angular deviation from the reference in degrees. The point colour encodes the absolute magnitude of the MBD-NL atomic force As a guide to the eye the radial $100\%$ mark is highlighted in red

Figure 5: Structural comparison of polymorphs relaxed with PBE and PBE+MBD-ML compared to the PBE+MBD-NL relaxed structures: (top) RMSD and maximum atomic displacement and (bottom) ratio of the unit cell volume of the relaxed structures with respect to the PBE+MBD-NL-relaxed structure. Blue crosses represent cases where the PBE geometry optimization did not converge. The RMSD and volume ratio is shown for every polymorph of each molecular crystal. ANP refers to 2-amino-5-nitropyrimidine. Numerical values are tabulated in Table S4

Molecule		$\Delta E_{\mathrm{MBD-NL}}$ $(\mathrm{kJ/mol})$	$\Delta E_{\mathrm{MBD-ML}}$ $(\mathrm{kJ/mol})$
Mixed Interactions
Aspirin	I $\rightarrow$ II	-0.262	-0.291
Hydrogen Bonded
Maleic acid	I $\rightarrow$ II	0.670	0.467
ANP	I $\rightarrow$ II	0.183	-0.292
	I $\rightarrow$ III	0.237	0.261

Table 2: Polymorph energy differences (kJ/mol per molecule) from PBE0+MBD-NL (

\Delta E_{\mathrm{MBD-NL}}

) and PBE0+MBD-ML (

\Delta E_{\mathrm{MBD-ML}}

). Roman numerals indicate the polymorphic forms between which the energy difference is computed (e.g., I

\rightarrow

II denotes

E_{\mathrm{II}}-E_{\mathrm{I}}

). Systems are grouped by dominant intermolecular interaction. ANP refers to 2-amino-5-nitropyrimidine. Incorrect rankings likely due to incomplete cancellation of total energy errors are in bold. a Structures were taken from Ref. 46

Figure 6: Computational scaling of the calculation of the

a_{0}^{\mathrm{r}}

and

C_{6}^{\mathrm{r}}

ratio calculation by the MBD-ML model and of the MBD property calculation via libMBD

Appendix figures & tables16 assets

Supplementary material from the paper’s appendix.

Appendix

Figure S1 : Convergence of ( a ) total energies and ( b ) MBD-NL dispersion energies with respect to the cut_pot parameter for two example anions. Total energies are plotted as the differences relative to the default 4 Å setting (marked by a vertical dashed line).

C 4 H 7				C 2 NH 4
$Q$	$E_{\rm tot}$	EA	HOMO	$Q$	$E_{\rm tot}$	EA	HOMO
-2	-4252.28	-2.81	2.79	-2	-3621.08	-3.36	3.29
-1	-4255.09	0.96	-1.00	-1	-3624.44	1.61	-1.58
0	-4254.13		-9.15	0	-3622.84		-9.07
C 5 N 2 H 9				C 6 O 2 H 9
$Q$	$E_{\rm tot}$	EA	HOMO	$Q$	$E_{\rm tot}$	EA	HOMO

Appendix

Table S1 : Total energies, electron affinities and HOMO levels (all in eV) for various charged states of the four considered molecules. The first row in each case corresponds to the charged state included in QCML.

Figure S2 : Boxplots of $\alpha_{0}^{r}$ (left) and $C_{6}^{r}$ (right) with varying cut_pot for hydrogen atoms in the two negatively charged molecules: ( a ) $[\rm C_{4}H_{7}]^{2-}$ and ( b ) $[\rm C_{5}N_{2}H_{9}]^{3-}$ (the structures shown in the insets). The inset in the upper left panel illustrates the $\alpha_{0}^{r}$ ratios in pentane molecule for cut_pot = 4 Å and 8 Å.

Figure S3 : The same as Figure S2 but for ( a ) $[\rm C_{2}NH_{4}]^{2-}$ and ( b ) $[\rm C_{6}O_{2}H_{9}]^{3-}$ (the structures shown in the insets).

Panel of Figure 13 — Figure S4 : Performance of PBE0+MBD-ML in predicting the $C_{6}$ and $\alpha_{0}$ ratios of the QCML test set including all negatively charged molecules

Panel of Figure 14 — Figure S5 : Performance of PBE0+MBD-ML in predicting the MBD contribution to the total energy and atomic forces of the QCML test set including all negatively charged molecules

Panel of Figure 15 — Figure S6 : Performance of PBE0+MBD-ML in predicting the $C_{6}$ and $\alpha_{0}$ ratios and the MBD contribution to the total energy and atomic forces of the OMol25 test set including alkali and alkaline earth elements

Panel of Figure 16 — Figure S7 : Performance of PBE0+MBD-ML in predicting the $C_{6}$ and $\alpha_{0}$ ratios and the MBD contribution to the total energy and atomic forces of the OMol25 test set excluding alkali and alkaline earth elements

Panel of Figure 17 — Figure S8 : Performance of PBE0+MBD-ML in predicting the $C_{6}$ and $\alpha_{0}$ ratios and the MBD contribution to the total energy and atomic forces of the DES370k test set including alkali and alkaline earth elements

Panel of Figure 18 — Figure S9 : Performance of PBE0+MBD-ML in predicting the $C_{6}$ and $\alpha_{0}$ ratios and the MBD contribution to the total energy and atomic forces of the DES370k test set excluding alkali and alkaline earth elements

Figure S10 : Sensitivity of the MBD-NL energy to the introduction of uniform random noise to $a_{0}^{\mathrm{r}}$ and $C_{6}^{\mathrm{r}}$

Figure S11 : Sensitivity of the MBD-NL energy to the introduction of a systematic bias to $a_{0}^{\mathrm{r}}$ and $C_{6}^{\mathrm{r}}$

Figure S12 : Distributions of vdW force deviations from the MBD-ML, D3 and D4 method compared to MBD-NL. The distributions are represented by violin plots. Black crosses correspond to the mean value, the red line to the median. The vertical edges of the boxes are the quartiles, i.e the 25-th and 75-th percentile of the deviation distribution. The whiskers correspond to the minimum and the

95\%

deviation

	MBD-ML	D3	D4
Magnitude Deviation (%)
Mean	2.6	28.2	39.5
Median	1.8	18.2	35.4
95 th Percentile	7.7	60.7	80.8
Angular Deviation (°)
Mean	1.7	17.4	19.8

Appendix

Table S2 : Mean, median, and 95 th percentile deviations for relative magnitude and angular force predictions of MBD-ML, D3 and D4 compared to MBD-NL on the OMol25 test set

Molecule	Polymorphs	interaction type
aspirin	I (ACSALA07) II (ACSALA17)	mixed
maleic acid	I (MALIAC12) II (MALIAC13)	hb
2-amino-5-nitropyrimidine	I (PUPBAD01) II (PUPBAD02) III (PUPBAD)	hb
maleic hydrazide	I (MALEHY10) II (MALEHY01) III (MALEHY12)	hb
pentacene	I (PENCEN) II (PENCEN04)	vdW
coronene	I (CORONE04 a ) II (N/A a )	vdW

Appendix

Table S3 : Molecular crystal polymorphs from which geometry optimizations were performed. Along with the roman numeral denoting the polymorph the CSD reference code and the type of interaction predominantly found in the system is provided: mixed, van-der-Waals (vdW) or hydrogen-bonded (hb)

		RMSD (Å)		Max. Disp. (Å)		$\Delta V$ (%)
Molecule	Polymorph	PBE	PBE+MBD-ML	PBE	PBE+MBD-ML	PBE	PBE+MBD-ML
Mixed Interactions
Aspirin	I	0.177	0.002	0.248	0.003	26.00	0.09
	II	–	0.007	–	0.018	–	0.22
Hydrogen Bonded
Maleic acid	I	0.035	0.002	0.060	0.003	24.16	0.29

Appendix

Table S4 : Structural comparison of polymorphs relaxed with PBE and PBE+MBD-ML relative to the PBE+MBD-NL-relaxed reference structures. RMSD and maximum atomic displacement (Max. Disp.) quantify the deviation of atomic positions, while

\Delta V

gives the relative change in unit cell volume. Cases where the PBE geometry optimization did not converge are marked with “–”. ANP refers to 2-amino-5-nitropyrimidine

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