Pseudopairwise potential and lattice dynamics of rare-gas crystals

We considered the potential U = Ur+ Uc of a system of rare-gas atoms, where Uc is the Van der Waals potential and Ur is the repulsion potential determined by the permutation symmetry of the many-electron wave function Ψ. Hartree-Fock calculations of Ur are cumbersome and unreliable. In view of this, in calculating Ur, we made allowance for the permutation symmetry in a different way, i.e., by using the nodal surface Σ, of the function Ψ generated by the symmetry. In some important particular cases the form of the surface Σ is found from general considerations. Thus, for a pair of rare-gas atoms A and B the surface Σ is close to a plane normal to the segment AB and dividing it in a ratio which does not depend on its length. For the wave function of an atom in a lattice Σ is a polyhedron formed by such planes. The many-particle repulsion potential, obtained as the difference of the energy of atoms with and without the nodal surface, is approximated (with an estimation of the error) by the sum of pair-wise potentials U(2) (Ref. 1). The potential U(2), which (unlike the true pairwise potential) depends on the lattice constant, is called pseudopairwise; it contains a parameter β which determines the position of the nodal surface. The other unknown parameter of the potential U = Ur + Uc is the coefficient C in the principal term of the expansion of the Van der Waals potential (determining the other terms as well2). The calculations for an anharmonic lattice described by a pseudopairwise potential were performed by a variational method with subsequent estimation and correction of the systematic error. With an optimal choice of β and C all the calculated functions (phonon dispersion law, specific heat, thermal expansion, and compressibility over a wide range of temperatures and pressures) for Ne, Ar, and Kr crystals practically coincide with experimental data. The values of β so determined, virtually the same for various rare-gas atoms, can also be used in calculations for closed-shell ions. The values of C somewhat exceed the data obtained by nonrelativistic calculations3 (the discrepancy increases with an increase in the atomic number). For comparision with the pseudopairwise potential, in much the same way we obtained the true pairwise (molecular) potential; the difference between them is governed by a law and amounts to several per cent.

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