Response Functions and Interatomic Forces

Abstract

If a tight-binding Hamiltonian H is perturbed by some displacement of the atoms, then the second order change in energy is given by:

$$\vartriangle {U^{\left( 2 \right)}} = \vartriangle U_{rep}^{\left( 2 \right)} - \left( {1/\pi } \right)\operatorname{Im} Tr\sum\limits_\sigma {\int {^{\varepsilon F}} } \left\{ {{G^\sigma }\left( \varepsilon \right)\vartriangle {H^{\left( 2 \right)}} + \frac{1}{2}{G^\sigma }\left( \varepsilon \right)\vartriangle {H^{\left( 1 \right)}}{G^\sigma }\left( \varepsilon \right)\vartriangle {H^{\left( 1 \right)}}} \right\}d\varepsilon$$

, where G^σ(ε) is the σ-spin Green function for the unperturbed lattice, and ∆H ⁽¹⁾ and ∆H ⁽²⁾ are the first and second order changes in the Hamiltonian whose matrix elements between sites and orbitals are obtained from the Slater-Koster two-centre integrals expanded as a Taylor series in the atomic displacements u _aα . Force constants ⌽ ^ab_αβ are the coefficients of −u _aα u _bβ in the above expansion of ∆U ⁽²⁾. The recursion method is used to calculate the matrix elements of G^σ(ε). The energy integrals of products of these are response functions, which should satisfy a sum rule over lattice sites to equal the density of states. By calculating the density of states directly and via the response function sum rule we have a useful test of the range and accuracy of the response functions which we have calculated to the tenth shell of neighbours in the bcc lattice ([333]a/2 and [115]a/2), using 13 – 15 exact levels of the continued fraction for the canonical d-band model. Results of this test are presented, followed by calculations of force constants. Features of the phonon spectra in bcc transition metals are explained.