# Calculation of the vdW interactions without DFT

I am looking for proxy methods of calculating vdW interactions similar to the DFT-D2, DFT-D3, and DFT-D4 methods which require only atomic positions and/or properties of the atoms. It would be good to identify relative speeds of methods as well, but the ideal methods should be relatively low cost (more than classical potentials, less than DFT).

An easy answer to this is DFT-D2, DFT-D3, and DFT-D4 but I am looking for less well-known alternatives. If it can run using ASE or LAMMPS as a backend that is best, but I am willing to look at stand-alone tools or even just papers which describe the form.

• The van der Waals interaction between two atoms is often given by Eq. 4 or Eq. 5 of this. In both cases, the interaction potential is given by the function V(R) = C6/R^6 in which R is the distance between the two atoms (so you only need the atomic positions). C6 can be calculated based on the formula given in [cotd] Jan 19 at 21:13
• this question. Is this the type of thing you're seeking? If so, I can write a more self-contained answer that explains this. Jan 19 at 21:15
• That is close, but essentially DFT-D2. This page for lammps covers it to some degree, but I am curious about other attempts at the same thing. There are definitely other ways to go about this since there are more versions of D2, D3, D4 but its unclear if these are the only available tools. Jan 20 at 1:01
• Something like this? Jan 20 at 6:18
• In that case, I would edit the question to say that. ‘Interactions’ doesn’t say very much at all about what you want. Mar 6 at 16:58

Recently I came across a paper that might be what you are looking for. It's an universal parametrization of the vdW interactions depending on the static dipole polarizability of the atoms and the $$C_{6}$$ dispersion coefficient. The potential is derived from a quantum Drude oscillator model and was applied for noble gas dimers, Group 2 dimers and molecular dimers, although I think it only works for the long-range behavior.

Anyways, the work is also from Tkatchenko and gives some nice results for the systems mentioned. A very brief explanation is as follows: the vdW potential is written in the form $$V(r) = \left(\frac{1}{2}+\frac{2C_{8}}{3C_{6}R_{e}^{2}}+\frac{5C_{10}}{6C_{6}R_{e}^{4}}\right)\frac{q^{2}}{r}e^{-\mu\omega r^{2}/2\hbar} - \sum_{n=3}^{5}\frac{C_{2n}}{r^{2n}}$$ where $$R_{e}$$ is the equilibrium distance (obtained from a relation to the dipole polarizability and the fine-structure constant) and $$q,\mu,\omega$$ are the parameters of the Drude oscillator. No DFT calculation is required for this, since it's a classical model potential (derived from quantum-mechanical relations) depending on the chemical nature of the system, and the results are compared to CCSD(T) values.

All details can be found in J. Chem. Theory Comput. 2023, 19, 7895-7907. The calculation of the oscillator's parameters is described in the supporting information. Also, in Refs. 53, 54 and 106 of that paper there are another approaches to dispersion interactions using a machine learning approach.

• That looks like much more than just a vdW potential. The term that has a decaying exponential in it, is not even a long-range (vdW) potential, and it describes the Coulomb potential of the inner wall towards the united atom limit (r->0), rather than the vdW limit (r->$\infty$). After the term with the exponential in it, you have three dispersion terms (C6, C8, C10). If you want just the vdW interaction, why not just write $V(r) = -C_6/r^6$ and use any answer from here to calculate the $C_6$ without DFT? Jan 23 at 22:47

Symmetry-adapted perturbation theory (SAPT) provides a means of directly computing the noncovalent interaction between two molecules, that is, the interaction energy is determined without computing the total energy of the monomers or dimer. In addition, SAPT provides a decomposition of the interaction energy into physically meaningful components: i.e., electrostatic, exchange, induction, and dispersion terms.

Here, the Hamiltonian is written as a sum of the usual monomer Fock operators, $$F$$, the fluctuation potential of each monomer, $$W$$, and the interaction potential, $$V$$. The monomer Fock operators, $$F_a$$ and $$F_b$$, are treated as the zeroth-order Hamiltonian and the interaction energy is evaluated through a perturbative expansion of , $$V$$, $$W_a$$ and $$W_b$$. Through first-order in $$V$$, electrostatic and exchange interactions are included; induction and dispersion first appear at second-order in $$V$$. For a complete description of SAPT, the reader is referred to the excellent review by Jeziorski, Moszynski, and Szalewicz. (1, 2)

• Would you please check the following: mattermodeling.stackexchange.com/help/how-to-answer on how to provide a good answer, it is encouraged that users provide enough explanation in their answers. When providing links, even if that link might potentialy have the answer to a question, it might not be accessible later on. Thus, please pay attention to that and try to edit your answer accordingly, unless it might be removed. Feb 23 at 13:44
• @JaafarMehrez done Feb 23 at 14:23
• While this link may answer the question, it is better to include the essential parts of the answer here and provide the link for reference. Link-only answers can become invalid if the linked page changes. - From Review Feb 27 at 17:23
• @HemanthHaridas done Mar 3 at 16:15