We consider a multiatomic system where the nuclei are assumed to be point charges at fixed positions. Particles interact via Coulomb potential and electrons have pseudo–relativistic kinetic energy. We prove the van der Waals-London law, which states that the interaction energy between neutral atoms decays as the sixth power of the distance $|D|$ between the atoms. In the many atom case, we rigorously compute all the terms in the binding energy up to the order $|D|^{−9}$ with error term of order

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... $\mathcal{O}(|D|^{−10})$. This yields the first proof of the famous Axilrod–Teller–Muto three–body correction to the van der Waals–London interaction, which plays an important role in atom physics. As intermediate steps we prove exponential decay of eigenfunctions of multiparticle Schrödinger operators with permutation symmetry imposed by the Pauli principle, and new estimates of the localization error.