Toward a general dispersion-inclusive density-functional model for materials: delocalized metallic fluctuations, semiempirical polarizability, ionic references, and universal range separation in MBD-NL-type methods

Semilocal density-functional theory (DFT) combined with the many-body dispersion (MBD) framework has become a standard route to London dispersion (van der Waals) forces in molecules and materials. The MBD-NL model of Hermann and Tkatchenko (Phys. Rev. Lett. 124, 146401, 2020) unified atom-based many-body dispersion with nonlocal vdW functionals by parametrizing coupled quantum oscillators from the Vydrov–Van Voorhis (VV) polarizability functional, normalized to free-atom and jellium reference data, and obtained accurate energies across molecular, ionic, metallic, and hybrid systems. Hermann and Tkatchenko identified four open questions concerning MBD-NL: (1) the underlying polarizability functional is semiempirical; (2) the vdW parameters are normalized only to neutral free atoms, and normalization to ionic reference states is not covered; (3) the model does not capture truly delocalized electronic fluctuations in conductors and small-gap layered compounds, the largest remaining theoretical gap; (4) the range-separation parameter coupling the semilocal functional to the dispersion correction is empirical and functional-dependent. The 2020–2026 literature shows substantial progress on metallic fluctuations via the MBD+C scheme (Dobson and Ambrosetti, 2023) for idealized low-dimensional metals, and machine-learned surrogates (MBD-ML, 2026) that bypass but do not solve the polarizability-functional problem. A precise open core remains: one seamlessly coupled DFT+dispersion model, free of functional-specific fitted parameters, valid across insulating, ionic, and metallic electronic structures.

Background

Kohn–Sham density-functional theory (DFT) with semilocal exchange–correlation functionals such as PBE (the Perdew–Burke–Ernzerhof generalized-gradient-approximation functional) is the workhorse of first-principles materials simulation, but semilocal functionals miss the long-range electron correlation that produces London dispersion (van der Waals, vdW) forces. Three families of corrections dominate practice: pairwise atom-based schemes (e.g., Tkatchenko–Scheffler, which partitions the electron density among atoms via Hirshfeld partitioning — a prescription that assigns each atom a share of the total density — and scales free-atom dispersion coefficients accordingly); nonlocal vdW density functionals such as VV10, which add a nonlocal correlation term — a double integral over all pairs of points $\mathbf{r}$ and $\mathbf{r}'$ of a kernel built from the density and its gradient; and many-body dispersion (MBD) models, in which each atom is coarse-grained into a quantum harmonic oscillator (a fluctuating dipole), all oscillators are coupled through the dipole–dipole interaction with short-range damping, and the dispersion energy is the change in zero-point energy of the coupled modes — a dipole-level random-phase approximation that captures many-body screening and beyond-pairwise additivity.

The MBD-NL model of Hermann and Tkatchenko (2020) unified the atomic and nonlocal-functional approaches: the oscillator polarizabilities are obtained from the VV polarizability functional $\alpha_{\mathrm{VV}}[n](\mathbf{r},iu)$, a semiempirical local approximation to the microscopic density response derived by Vydrov and Van Voorhis (2010); the atomic parameters are normalized to exact neutral free-atom polarizabilities and $C_6$ coefficients; the uniform-electron-gas (jellium — the standard idealized model of a simple metal) limit is matched to VV10, which is what lets the model describe metals and their surfaces reasonably; and a nonlocal range-separation (damping) function with a single empirical parameter $\beta$ couples the dispersion correction to a given semilocal exchange–correlation functional. The result performed well on standard benchmarks: S66 molecular dimers, the X23 organic molecular-crystal set, 64 hard solids including metals and ionic crystals, and hybrid organic/metal interfaces, surpassing the earlier range-separated self-consistently screened variant MBD@rsSCS, which overbinds metals and ionic solids.

Hermann and Tkatchenko discuss several open questions regarding MBD-NL. The VV polarizability functional is semiempirical and uneven in accuracy; a more rigorous functional — for example at the meta-GGA level, i.e., depending also on the kinetic-energy density, or otherwise nonlocal — could substantially improve predictive accuracy but requires advances in the general theory of polarizability functionals. The source also raises, as an alternative possibility, normalizing the vdW parameters not only to neutral free atoms but also to ionic reference states; the fractionally ionic approach of Gould et al. (2016) is the earlier treatment of polarizability and van der Waals many-body dispersion for ionically bonded systems. The source identifies as the largest remaining theoretical gap that MBD-NL does not capture truly nonlocal, fully delocalized electronic fluctuations: in conductors (Dobson's 'type-C nonadditivity', the anomalous power-law decay of dispersion between metallic bodies arising from gapless collective charge fluctuations) and in interacting small-gap systems such as the transition-metal dichalcogenides (TMDCs, layered compounds like MoS$_2$ with small band gaps, 23 of the 26 layered materials in the standard '26' benchmark, which MBD-NL underbinds by about 21% while MBD@rsSCS and VV10 overbind). Incorporating such a mechanism would enable treatment of long-range interactions between fully metallic bodies and improve accuracy for small-gap compounds. Finally, the range-separation parameter $\beta$ is fitted per exchange–correlation functional; semilocal functionals differ substantially in their mid-range behavior (PBE and PBE0 — the latter a hybrid functional mixing a fraction of exact exchange into PBE — used in the source, are exceptions), and a seamless, universal, transferable coupling between semilocal functionals and vdW corrections (the subject of Hermann and Tkatchenko's 2018 balancing analysis) remains an open challenge. These strands share one objective: a single seamlessly coupled DFT+dispersion model that treats weak interactions across all electronic structures of materials — molecular, ionic, metallic, and hybrid — without functional-specific empiricism.

Problem Statement

Advance the MBD-NL framework — or an equivalent seamlessly coupled semilocal-DFT + many-body-dispersion model that addresses the same open questions — toward the general theory of van der Waals interactions in materials that its authors identified as incomplete:

1. Polarizability functional. Replace the semiempirical VV polarizability functional underlying the coarse-graining with a more rigorous polarizability density functional — for example a meta-GGA-level or otherwise nonlocal functional using additional density information — improving predictive accuracy.

2. Ionic normalization. Extend the normalization of the dispersion parameters beyond neutral free-atom references to ionic reference states, so that charged fragments in strongly ionic or charged environments are described consistently.

3. Delocalized electronic fluctuations (the largest remaining theoretical gap). Incorporate a mechanism that captures truly nonlocal, fully delocalized electronic fluctuations — the type-C collective charge response of conductors and of interacting small-gap systems such as transition-metal dichalcogenides — enabling the model to treat long-range dispersion between fully metallic bodies with the correct asymptotic behavior and to improve interlayer binding in small-gap layered compounds.

4. Universal range separation. Devise a seamless range-separation scheme that couples semilocal exchange–correlation functionals and dispersion corrections in a universal, transferable way — valid across functionals with different mid-range behavior — removing the empirical, functional-dependent damping parameter $\beta$.

These strands jointly serve one objective: a single dispersion-inclusive density-functional model, valid simultaneously for molecular, ionic, metallic, and hybrid systems and free of functional-specific fitted parameters. Items (1) and (2) concern the same limitation of the model — a semiempirical polarizability functional whose parameters are normalized only to neutral free atoms — and the source offers (2) as an alternative to it, whereas (3) and (4) are independent requirements. The construction or determination each strand asks for concerns MBD-NL or an equivalent seamlessly coupled semilocal-DFT + many-body-dispersion model, reaching beyond the earlier scheme in which the idea was formulated (as with ionic normalization beyond the earlier MBD@rsSCS scheme).

References

1. Jan Hermann; Alexandre Tkatchenko. Density Functional Model for van der Waals Interactions: Unifying Many-Body Atomic Approaches with Nonlocal Functionals. Physical Review Letters 124, 146401 (2020). DOI: 10.1103/PhysRevLett.124.146401 (arXiv:1910.03073). https://doi.org/10.1103/PhysRevLett.124.146401
2. John F. Dobson; Alberto Ambrosetti. MBD + C: How to Incorporate Metallic Character into Atom-Based Dispersion Energy Schemes. Journal of Chemical Theory and Computation 19, 6434–6451 (2023). DOI: 10.1021/acs.jctc.3c00353 (arXiv:2308.11855). https://doi.org/10.1021/acs.jctc.3c00353
3. Evgeny Moerman; Adil Kabylda; Almaz Khabibrakhmanov; Alexandre Tkatchenko. MBD-ML: Many-Body Dispersion from Machine Learning for Molecules and Materials. PRX Intelligence 1, 013003 (2026). DOI: 10.1103/pv86-l9h7 (arXiv:2602.22086). https://doi.org/10.1103/pv86-l9h7
4. Tim Gould; Sébastien Lebègue; János G. Ángyán; Tomáš Bučko. A Fractionally Ionic Approach to Polarizability and van der Waals Many-Body Dispersion Calculations. Journal of Chemical Theory and Computation 12, 5920–5930 (2016). DOI: 10.1021/acs.jctc.6b00925. https://doi.org/10.1021/acs.jctc.6b00925
5. John F. Dobson. Beyond pairwise additivity in London dispersion interactions. International Journal of Quantum Chemistry 114, 1157–1161 (2014). DOI: 10.1002/qua.24635. https://doi.org/10.1002/qua.24635
6. Oleg A. Vydrov; Troy Van Voorhis. Dispersion interactions from a local polarizability model. Physical Review A 81, 062708 (2010). DOI: 10.1103/PhysRevA.81.062708. https://doi.org/10.1103/PhysRevA.81.062708
7. Oleg A. Vydrov; Troy Van Voorhis. Nonlocal van der Waals density functional: The simpler the better. The Journal of Chemical Physics 133, 244103 (2010). DOI: 10.1063/1.3521275. https://doi.org/10.1063/1.3521275
8. Jan Hermann; Alexandre Tkatchenko. Electronic Exchange and Correlation in van der Waals Systems: Balancing Semilocal and Nonlocal Energy Contributions. Journal of Chemical Theory and Computation 14, 1361–1369 (2018). DOI: 10.1021/acs.jctc.7b01172. https://doi.org/10.1021/acs.jctc.7b01172
9. John F. Dobson. Towards efficient description of type-C London dispersion forces between low-dimensional metallic nanostructures. Electronic Structure 3, 044001 (2021). DOI: 10.1088/2516-1075/ac3807. https://doi.org/10.1088/2516-1075/ac3807
10. S. Pal; John F. Dobson. Do 1-dimensional metals prefer to form even-numbered van der Waals clusters? Physical Chemistry Chemical Physics 28, 10339–10349 (2026). DOI: 10.1039/d5cp04534g. https://doi.org/10.1039/d5cp04534g
11. Minho Kim; Won June Kim; Timothy Gould; Eok Kyun Lee; Sébastien Lebègue; Hyungjun Kim. uMBD: A Materials-Ready Dispersion Correction That Uniformly Treats Metallic, Ionic, and van der Waals Bonding. Journal of the American Chemical Society 142, 2346–2354 (2020). DOI: 10.1021/jacs.9b11589. https://doi.org/10.1021/jacs.9b11589
12. Matteo Gori; Philip Kurian; Alexandre Tkatchenko. Second quantization of many-body dispersion interactions for chemical and biological systems. Nature Communications 14, 8218 (2023). DOI: 10.1038/s41467-023-43785-z. https://doi.org/10.1038/s41467-023-43785-z
13. Heikki Muhli; Tapio Ala-Nissila; Miguel A. Caro. Atom-wise formulation of the many-body dispersion problem for linear-scaling van der Waals corrections. Physical Review B 111, 054103 (2025). DOI: 10.1103/PhysRevB.111.054103. https://doi.org/10.1103/PhysRevB.111.054103
14. Almaz Khabibrakhmanov; Dmitry V. Fedorov; Alberto Ambrosetti; Jason Crain; Katharine L. C. Hunt; Erin R. Johnson; Kenneth D. Jordan; Szabolcs Góger; Matteo Gori; Mohammad Reza Karimpour; Reinhard J. Maurer; Mainak Sadhukhan; Martin Stöhr; Alexandre Tkatchenko. Accurate noncovalent interactions in atomistic systems via quantum Drude oscillators. Journal of Chemical Physics 163, 151001 (2025). DOI: 10.1063/5.0281913. https://doi.org/10.1063/5.0281913
15. Elisabeth Keller; Volker Blum; Karsten Reuter; Johannes T. Margraf. Exploring atom-pairwise and many-body dispersion corrections for the BEEF-vdW functional. Journal of Chemical Physics 162, 074111 (2025). DOI: 10.1063/5.0248728. https://doi.org/10.1063/5.0248728