Chemical Physics 142 (1990) 321-334 North-Holland zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA A STRUCTURAL JELLIUM MODEL OF CLUSTER ELECTRONIC STRUCTURE Zhenyang LIN, Tom SLEE and D.M.P. MINGOS zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLK Inorganic Chemistry Laboratory, South Park Road, Oxford OX1 3QR, UK Received 10 August 1989; in final form 6 November 1989 A structural jellium model has been developed which accounts for the electronic structures of clusters using a crystal-field perturbation. Although it is related to previous models proposed by Knight et al., this model allows predictions to be made concerning the gross stuctural features of metal clusters; something the original jellium model is incapable of doing. The zero- order potential we derive is of central-field form, depends on the geometry of the cluster, and has a general form that can produce an energy-level ordering similar to that of Knight’s rounded well, but can also accommodate other orderings. In particular, quali- tative arguments suggest that this potential produces different energy level orderings for clusters with a nucleus with large positive charge at the centre of the cluster, enabling the spherical jellium model to be applied to alkali metal clusters seeded with magne- sium and zinc. Analysis of the effects of the non-spherical perturbation on the spherical jellium shell structures leads to the conclusion that for a cluster with a closed-shell electronic structure a high symmetry arrangement which is approximately or precisely close packed will be preferred and provides a basis for rationalising those structures, which have been predicted using ab initio calculations, of clusters with incomplete shell electronic configurations. 1. Introduction Trends in the stabilities of bare clusters as a func- tion of their nuclearity have now been explored for many elements [ l-3 1. Mass spectral data, taken un- der appropriate conditions, have shown large abun- dances for specific number of metal atoms and these have been associated with particularly stable clusters. These “magic numbers” are particularly evident in the mass spectral data for alkali metal clusters [ 45 1, noble gases [ 6 1, and carbon [ 7 1. In this paper we are concerned with the alkali metals. In this group, mass spectra have been reported for Na [ 8 1, K [ 9 1, LiK, [lo] andLi,Na,,[ll]. Many theoretical studies of alkali metal clusters have been carried out [ 3 1. The simplest model of their electronic structure is the spherical jellium model of Knight et al. [ 8 1, which has successfully predicted several of the magic numbers for alkali metal clusters by associating these nuclearities with closed-shell electronic configurations. The assumption of spheri- cal symmetry leads to electronic shells, specified by a principal and an angular momentum quantum num- ber (n and I). Mass spectra [ 81 showed discontinui- ties at masses corresponding to N=2, 8, 20, 40, 58 0301-0104/90/$03.50 0 Elsevier Science Publishers B.V. (North-Holland) and 92 atoms for sodium and potassium [ 9 ] clusters generated in gas evaporation experiments, and Knight’s spherical jellium model correctly predicted high stabilities at these numbers, albeit along with others for which the mass spectrum showed no par- ticularly large peaks.. More accurate self-consistent jellium calculations have been carried out by Ekardt to investigate the effective potentials, energy levels and charge densities in comparison to the semi-infi- nite half-space [ 12,13 1. The spherical jellium model has attracted both in- terest and criticism [ 5 1. One of the observations that has cast some doubt on its usefulness is that alkali metal clusters seeded with Mg and Zn exhibit a large peak corresponding to the ten electron clusters KsMg and KsZn [ 5 1. Knight et al. [ 141 have suggested that these could be interpreted in jellium terms, and a modified jellium model, employing a smeared out background density of positive charge to produce the potential, has been constructed [ 15 1. The spherical jellium model has been extended to spheroidal clusters by Clemenger [ 16 1, an extension that encompasses even subtle features of the alkali metal mass spectral abundances. The spheroidal model allows a single degree of freedom in the cluster