Density-functional calculation for K lattices in condensed phase and quantum-chemical model for the cohesive energy of heavy alkali metals N. H. March and Angel Rubio Departamento de Fı ´sica Teo ´rica, Universidad de Valladolid, E-47011 Valladolid, Spain ~Received 4 December 1996; revised manuscript received 11 July 1997! Following a summary of deductions from experiment on the bonding of Rb and Cs in very different liquid and solid environments, energy calculations based on density-functional theory ~DFT! are presented on ordered chains of K as a function of nearest-neighbor distances. At a given bond length, and sufficiently low density, a regime occurs in which modest zig-zag behavior is found to stabilize the original linear chains. As for Rb and Cs, we conclude that K may exhibit low coordination in highly expanded forms. The previous ab initio results on lattices of K atoms for coordination numbers z 5 8, 4, and 2 are analyzed by means of a quantum-chemical model in which a nearest-neighbor Heisenberg Hamiltonian is characterized by free-space K 2 dimer potential energy curves. The satisfactory accord between the two different treatments has prompted us to present results also for Rb and Cs lattices for five different coordination numbers for which DFT calculations are not currently available. The relevance to experiments on expanded fluid Cs and to zig-zag chains of Cs on semiconductor substrates is briefly referred to. @S0163-1829~97!02445-4# I. INTRODUCTION In pioneering work on the measurement of liquid structure factors S ( q ) using neutron diffraction, Hensel et al. 1,2 have studied liquid metallic Rb and Cs in a number of thermody- namic states along the liquid-vapor coexistence curve to- wards the critical point. Their findings were that the high coordination numbers just above freezing, compatible with the local coordination in the hot bcc solids, were progres- sively lowered as these heavy alkali liquid metals were highly expanded. At the same time, it was found that the nearest-neighbor distance remained largely intact, as evi- denced by the position of the main peak in S ( q ) remaining at almost constant value of q max . These findings suggested the study of different phases ~with different coordination num- bers! for expanded alkali metals for a fixed nearest-neighbor distance ~see below!. One of us 3 noted that the coordination number data of Hensel et al., 1,2 when plotted against mass density d , could be fitted by d 5az 1b , ~1! where a 5230 and b 5280, both in kg/m 3 . For a low- density state of Cs described by Winter and Hensel, 2 with atomic number density r 50.004 16 Å 23 and temperature T 51923 K, pair potentials f ( r ) were extracted from the mea- sured S ( q ) by Ascough and March, 4 following the proposal of Johnson and March. 5 One of these potentials 4 ~dashed curve! is compared with a theoretical pair potential obtained by Arai and Yokoyama 6 in Fig. 1 ~continuous line!. This potential has a sharp minimum at 5.6 Å followed by a repul- sive region out to ; 9Å ~with the maximun at ;8.5 Å! and a second minima at ;9.4 Å. We shall return to consequences of these pair interactions for expanded liquid metal Cs be- low. Turning from these liquid metals, with short-range order ~SRO! characterized by S ( q ), we next consider a quite dif- ferent type of study made by Whitman et al. 7,8 In their work, Cs was deposited on semiconductor crystal surfaces, and as a result data has, subsequent to the experiments of Hensel et al. on SRO, become available on expanded Cs structures with long-range order ~LRO!. While, as emphasized by Free- man and March, 9 this data is at least partially about Cs in interaction with the semiconducting substrate, there is never- theless again an important message about chemical bonding in highly expanded Cs, but this time with LRO. Specifically, Whitman et al. 7 have measured the structural properties of Cs adsorbed on GaAs~110! and InSb~110! surfaces at room temperature using scanning tunneling microscopy. Their work establishes that Cs initially forms long, one- dimensional zig-zag chains on both these surfaces. To take one example, 7,9 their Fig. 1~a! shows a large-area image of Cs chains on GaAs~110! that includes chains more than 1000 Å in length. Their experiments conclusively demonstrate, in addition, that the chains tend to be separated by some tens of nm and have no LRO along the @001# direction; thus estab- lishing that they are truly one-dimensional structures. 7 In their Fig. 1~b!, showing the higher-resolution image, the Cs structures are revealed as zig-zag chains of single atoms in registry with the substrate ~110! surface. In the present con- text, it is important to stress that the Cs-Cs nearest-neighbor distance here is 6.9 Å, to be compared with the correspond- ing distance of 5.2 Å in bulk Cs. On the InSb~110! surface, the formation of Cs zig-zag chains was again revealed by the experiments of Whitman et al., 7 but because of InSb having the greatest spacing of the III-V semiconductors, the Cs nearest-neighbor distance is now 8.0 Å. Thus we have a low- ering of the coordination of Cs induced by the semiconductor substrate. This is to be compared with the previously dis- cussed lowering of the coordination in an expanded Cs liquid taken along the liquid-vapor coexistence curve toward the critical point. In this last case, it seems to be established beyond reasonable doubt that the building block of the ex- panded liquid metals Cs and Rb is a chemical bond with a rather constant length of ;5.4–5.7 Å with coordination number between 2 and 3. PHYSICAL REVIEW B 1 DECEMBER 1997-I VOLUME 56, NUMBER 21 56 0163-1829/97/56~21!/13865~7!/$10.00 13 865 © 1997 The American Physical Society