First-principles interatomic potentials for transition-metal aluminides: Theory and trends across the $3d$ series

In this paper the first-principles generalized pseudopotential theory (GPT) of transition-metal interatomic potentials [J. A. Moriarty, Phys. Rev. B 38, 3199 (1988)] is extended to $\mathrm{AB}$ binary compounds and alloys. For general transition-metal (TM) systems, the GPT total-energy functional involves a volume term, central-force pair potentials, and angular-force multi-ion potentials, which are both volume $\left(\Omega \right)$ and concentration $\mathrm{}\left(x\right)\mathrm{}$ dependent and include all $sp,$ $sp-d,$ and $d-d$ interactions within local density-functional quantum mechanics. The formalism is developed here in detail for intermetallic systems where $A$ is a simple metal and $B$ is a transition metal and applied to the prominent special case of the transition-metal aluminides ${\mathrm{TM}}_x{\mathrm{Al}}_{1-x},$ where $sp-d$ hybridization is especially important. Emphasis is given to the aluminum-rich $3d$ binary systems for $x<\mathrm{}0.30,$ which appear to be well described at the pair-potential level without angular forces and for which the present GPT potentials can be used directly in atomistic simulations. Volume terms and pair potentials for all of the $3d$ aluminides have been calculated and their behavior with atomic number, $\Omega ,$ and $x$ is elaborated through illustrative applications to the cohesive and structural trends across the $3d$ series. More extensive applications to the Co-Al and Ni-Al phase diagrams will be given elsewhere.