Multi-Ion Interactions and Structures in Simple Metals

Abstract

The total energy of simple metals is calculated formally to all orders in the pseudopotential. The leading term (in the pseudopotential expansion) of the $n$-ion interaction is obtained from the $n$th-order terms and the asymptotic form for large separations is evaluated explicitly. The resulting $n$-ion interaction is proportional to $\frac{\left(\frac{E_F}{k_F}\right){\left(\frac{\lambda }{k_F}\right)}^n\left[cos{k}_F\right(l_1+l_2+\cdots +l_n\left)\right]}{l_1{l}_2\cdots l_n(l_1+l_2+\cdots +l_n)}$, where the $l_i$ are consecutive segments of a straight-line path connecting the $n$ ions and $\lambda$ is of the order of a pseudopotential form factor divided by the Fermi energy. This is to be summed over all continuous paths connecting the $n$ ions. The familiar two-body interaction proportional to $\frac{\left(cos2k_{F}r\right)}{r^3}$ is a special case. The three-body interaction is found to be strongest when the three ions form a straight line and are separated by nearest-neighbor distances. The assumption that the influence of $d$-state hybridization upon this interaction dominates the determination of structures leads to the correct distribution of cubic and hexagonal structures among the monovalent and divalent metals and to appropriate high and low axial ratios among the hexagonal structures.