It is well-known that the Embedded Atom Method (EAM) predicts positive Cauchy pressures for cubic metals if physically-motivated embedding functions are used. Supris-ingly, even if the angular character of the covalent bonding is included within an orthorgonal or non-orthorgonal Tight-Binding (TB) description, the Cauchy pressure for most elemental and binary metallic systems remains positive. We describe the results of a detailed breakdown of the different contributions to the Cauchy pressure within the Harris-Foulkes approximation (HFA) to density functional theory. We show that negative values of the Cauchy pressure for both elemental transition metals such as Ir and binary intermetallics such as Ti3Al, TiAl and TiAl3 are well reproduced by the HFA. We argue that the negative Cauchy pressure (NCP) arises namely from the environment dependence of the local TB orbitals which leads to both environment-dependent bonding integrals and overlap repulsion. We discuss a particular functional form for overlap repulsion which leads to NCP and compare it with different fitting schemes proposed recently in TB theory.