Quantum Drude oscillators without the dipole approximation

Interactions between Drude oscillators (two opposite charges, one fixed, one mobile, interacting by a spring) are almost always modelled in the dipole approximation. This is because the Hamiltonian for describing Drude oscillators in this way is quadratic in the creation/annihilation operators (or equivalently, is quadratic in position and momentum) and can therefore be diagonalized exactly through a normal-mode transformation.

drudes.jpg

Second-order perturbation theory for the interaction of two such oscillators produces the expected –C6 R–6 dispersion interaction. At short distances this diverges, and the exact interaction energy ceases to be defined at all.

Therefore people use screening functions to fix up the short range. How much are the screening functions fixing up the divergence, and how much are they just compensating for the use of the dipole interaction Hamiltonian? Now we can find out, because Mainak has calculated the (almost) exact binding energy between Drude oscillators with the full Coulomb interaction.

Quantum mechanics of Drude oscillators with full Coulomb interaction
M. Sadhukhan and F. R. Manby, Phys. Rev. B 94, 115106 (2016) 10.1103/PhysRevB.94.115106

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