Nonadditive Three-Body Potential Models

Updated 5 February 2026

Nonadditive three-body potentials are interaction terms that capture irreducible energy contributions in three-particle systems, which cannot be described as a simple sum of pairwise interactions.
They arise from quantum fluctuations, induction, exchange, and resonance effects, making them essential for accurately modeling van der Waals forces, trimer binding, and noncovalent cooperativity.
Advanced computational methods such as ab initio calculations, symmetry-adapted perturbation theory, and machine learning are employed to rigorously quantify these nonadditive interactions.

A nonadditive three-body potential is an interaction term in the potential energy of a system of three particles that cannot be decomposed as a sum of pairwise (two-body) interactions. Such nonadditive terms are essential for the accurate modeling of many quantum, atomic, molecular, and condensed-matter phenomena—ranging from van der Waals interactions in noble gases, to trimer binding in ultracold Rydberg molecules, to noncovalent cooperativity in halogen-bonded complexes. Nonadditivity arises fundamentally from correlated quantum fluctuations, orbital or electronic structure effects, or, in field-theoretical settings, from genuine irreducible three-body processes.

1. Definition and General Formalism

Consider three particles at positions $\mathbf{r}_1$ , $\mathbf{r}_2$ , and $\mathbf{r}_3$ . The total potential energy $V_{\text{tot}}$ may be written in the many-body expansion: $V_{\text{tot}} = \sum_{i<j} V_2(\mathbf{r}_i, \mathbf{r}_j) + \sum_{i<j<k} V_3(\mathbf{r}_i, \mathbf{r}_j, \mathbf{r}_k) + \dots$ where $V_2$ are pairwise potentials and $V_3$ is the three-body (nonadditive) term. $V_3$ cannot be written as a sum of pairwise terms and encodes all interaction energy contributions that are not captured by the two-body terms alone. In quantum chemistry, the nonadditive three-body interaction energy is defined as: $E_{\text{int}}^{(3)} = E_{123} - (E_{12} + E_{13} + E_{23}) + (E_1 + E_2 + E_3)$ where $E_{ij}$ is the energy of pairs, and $E_{123}$ is the energy of the trimer (Ochieng et al., 23 Jul 2025, Hapka et al., 2017).

2. Physical Origins of Nonadditivity

Nonadditive three-body potentials arise from several physical mechanisms:

Triple-dipole (Axilrod–Teller–Muto, ATM) interactions: Quantum-mechanical correlation of instantaneous dipole fluctuations in three atoms leads to the classic ATM $R^{-9}$ three-body dispersion energy, with a well-defined angular factor $1 + 3\cos\alpha_1\cos\alpha_2\cos\alpha_3$ (Lang et al., 2023, Hellmann et al., 29 Jan 2026, Ibrahim et al., 3 Jun 2025).
Induction and exchange effects: When particles possess permanent or induced multipoles, nonadditive induction emerges; for example, in S–S–P systems, three-body terms appear already at second order in perturbation theory (Yan et al., 2016, Yan et al., 2017, Yan et al., 2021). Exchange nonadditivity is governed by Pauli exclusion and can be significant, especially in closed-shell and hydrogen-bonded systems (Hapka et al., 2017).
Resonance and degeneracy effects: For degenerate or nearly degenerate states, such as in mixtures of excited and ground state atoms or in hybrid atom-ion systems, nonadditive interactions can appear at lower perturbative order and be strongly enhanced (Yan et al., 2021).
Correlated scattering/field-theoretic mechanisms: Nonadditivity is essential in effective field theory, for example, in three-body terms generated by two-meson exchange in nuclear forces (Hadizadeh et al., 2011) or induced by multi-channel processes in Efimov physics (Kraats et al., 2022, Efremov et al., 2013).

3. Analytic Forms and Models

a. ATM and Beyond in Rare Gases

The canonical long-range nonadditive three-body interaction (for three isotropic, nonpolar atoms) is the Axilrod–Teller–Muto potential: $V_{3}^{\text{ATM}}(r_{12}, r_{23}, r_{31}; \theta_1, \theta_2, \theta_3) = C_9 \, \frac{1 + 3 \cos\theta_1 \cos\theta_2 \cos\theta_3}{r_{12}^3 r_{23}^3 r_{31}^3}$ with $C_9$ a triple-dipole constant (Hellmann et al., 29 Jan 2026, Lang et al., 2023, Ibrahim et al., 3 Jun 2025). Modern ab initio three-body potentials augment this with short-range exchange and induction corrections using high-rank, permutationally-symmetric expansions damped at short distances (Hellmann et al., 29 Jan 2026, Lang et al., 2023): $V_3(r_{12}, r_{23}, r_{31}) = V_{3}^{\text{ATM}} + \sum_{\ell_1 + \ell_2 + \ell_3 \le n} A_{\ell_1\ell_2\ell_3} P_{\ell_1}(\cos\theta_1)P_{\ell_2}(\cos\theta_2)P_{\ell_3}(\cos\theta_3) f(r_{ij})$ where $P_{\ell}(x)$ are Legendre polynomials, and $f(r_{ij})$ are damping functions (Hellmann et al., 29 Jan 2026).

b. Degenerate/Excited State Systems

For S–S–P or S–P–ion systems, the second-order nonadditive terms take the general form: $V_{3}^{(2)} = -\sum_{\rm cyc} \frac{C_{ij,jk}}{R_{ij}^3 R_{jk}^3}$ with angular factors dependent on internal angles and electronic state projections, often resulting in significant geometry-dependent nonadditivity (Yan et al., 2016, Yan et al., 2017, Yan et al., 2021).

c. Strongly Correlated/Few-Body Physics

In ultralong-range Rydberg trimers, the three-body term is not simply a correction but fundamentally determines binding. For $R_1 = R_2 = R$ ,

$E_{3\text{-body}}(R,\theta) = E_{\text{dim}}(R) [1 \pm (-\tfrac{1}{2} + \tfrac{3}{2}\cos^2\theta)]$

The resulting angular dependence leads to trimer energies that cannot be written as integer multiples of dimer energies, directly reflecting nonadditivity (Fey et al., 2018).

d. Machine Learning and Empirical Potentials

Nonadditive three-body contributions can be fitted directly from high-dimensional electronic structure data using fully general basis representations, e.g., B-splines over triangles subject to permutation symmetry: $E = \sum_{i<j} \phi_2(r_{ij}) + \sum_{i<j<k} \phi_3(r_{ij}, r_{ik}, r_{jk})$ This fully coupled potential captures all nonadditive correlations and can be trained efficiently via linear regression (Pozdnyakov et al., 2019).

4. Methods of Computation and Benchmarking

Symmetry-adapted perturbation theory (SAPT): Enables decomposition of $E^{(3)}$ into induction, exchange, and dispersion components, and quantifies the importance of each mechanism in a given system (Ochieng et al., 23 Jul 2025, Hapka et al., 2017).
Ab initio supermolecular and counterpoise techniques: For rare gases and small molecules, full configuration interaction or coupled cluster (CCSD(T), CCSDT, CCSDTQ) with large basis sets and supermolecule energy differences provides benchmarks for three-body energies (Lang et al., 2023, Hellmann et al., 29 Jan 2026, Ibrahim et al., 3 Jun 2025).
Path-integral Monte Carlo (PIMC): Used for direct evaluation of quantum virial and acoustic coefficients including the three-body term and its uncertainty (Lang et al., 2023, Hellmann et al., 29 Jan 2026).
Model Hamiltonians/Born–Oppenheimer approaches: Employed in ultracold Rydberg physics and Efimov/STM-type three-body systems to extract effective nonadditive three-body potentials (Fey et al., 2018, Kraats et al., 2022, Efremov et al., 2013).

5. Influence on Physical Properties

Accurate three-body potentials are crucial for:

Thermodynamic properties: Correct prediction of third and higher virial coefficients in noble gases and hydrogen, particularly under nonideal conditions (Hellmann et al., 29 Jan 2026, Lang et al., 2023, Ibrahim et al., 3 Jun 2025).
Spectroscopy and binding energies: Interpreting noninteger-multiplied line splittings and the existence of stable trimers in ultracold gases (Fey et al., 2018).
Elastic and structural properties: Accounting for observed low- and high-pressure equations of state, bulk modulus, and shear modulus in condensed helium and neon (Cazorla et al., 2015).
Noncovalent interaction cooperativity: Governing attractive/repulsive cooperative effects in halogen-bonded molecular clusters, often dominated by induction nonadditivity (Ochieng et al., 23 Jul 2025).
Defect and surface energetics in materials: Machine-learning potentials with explicit three-body terms afford significantly improved accuracy in modeling grain boundaries and large-cell MD (Pozdnyakov et al., 2019).

6. Angular and Geometric Dependence

The three-body potential is inextricably tied to the geometry of the three-particle cluster. The angular structure, already present in the simple ATM form ( $1 + 3 \cos\alpha_1\cos\alpha_2\cos\alpha_3$ ), becomes dramatically more complex in excited/degenerate systems or under the influence of field-mediated or quantum-gravitational fluctuations (Yan et al., 2017, Hu et al., 2022). Notably:

For Cs($6s$)-Cs( $nd$ ) Rydberg trimers, the nonadditive term depends nontrivially on the angle $\theta$ between the ground-state Cs atoms and is responsible for the observed spectroscopic signatures of nonadditivity (Fey et al., 2018).
For para-H $_2$ and rare gases in condensed phases, only a subset of compact triangles (e.g., equilateral in hcp solids) contribute dominantly due to the strong angular sensitivity of $V_3$ (Ibrahim et al., 3 Jun 2025).
The sign of the nonadditive term, and thus whether the three-body interaction is attractive or repulsive, can be tuned by the triangle's internal angles in quantum and even gravitational settings (Hu et al., 2022).

7. Impact, Open Problems, and Future Directions

Nonadditive three-body potentials have established indispensability across quantum chemistry, atomic, molecular, and optical physics, and materials modeling. Their accurate inclusion is essential for:

Achieving sub-percent-level uncertainties in virial coefficients required by metrology (Lang et al., 2023, Hellmann et al., 29 Jan 2026).
Correctly predicting trimer binding and quantum phenomena such as Borromean states and Efimov resonances (Kraats et al., 2022, Efremov et al., 2013, Fey et al., 2018).
Designing quantum simulators and exploring exotic few- and many-body phases in ultracold gases (Fey et al., 2018).
Advancing force-field and ML-potential development for large-scale simulations with chemical accuracy, especially as the three-body term strongly impacts defect and interface energetics (Pozdnyakov et al., 2019).

Challenges and frontiers include extending nonadditive potential modeling to include four-body and higher contributions (key at high densities and pressures), systematically capturing relativistic and quantum field-theoretical nonadditive effects, and constructing transferable models across chemical and physical regimes.

References

(Fey et al., 2018, Yan et al., 2021, Pozdnyakov et al., 2019, Efremov et al., 2013, Hellmann et al., 29 Jan 2026, Ibrahim et al., 3 Jun 2025, Hadizadeh et al., 2011, Yan et al., 2016, Cazorla et al., 2015, Hu et al., 2022, Ochieng et al., 23 Jul 2025, Kraats et al., 2022, Hapka et al., 2017, Pricoupenko, 2018, Lang et al., 2023, Yan et al., 2017)