PHYSICAL REVIEW B VOLUME 43, NUMBER 3 15 JANUARY 1991-II First-principles calculations of defect-induced lattice relaxation in ionic systems Koblar Jackson, Mark R. Pederson, and Barry M. Klein Complex Systems Theory Branch, Condensed Matter and Radiation Sciences Division, naval Research Laboratory, 8'ashington, D. C. 20375-5000 (Received 27 June 1990; revised manuscript received 17 September 1990) Local-density-approximation methods are used to investigate defect-induced lattice relaxation in LiCl:Cu+ and for the F center in MgO. Calculations are carried out on large finite clusters of atoms using a linear combination of atomic orbitals {LCAO) — type basis set, and defect-related properties are deduced from a comparison of results for related pure and defect clusters. The outer atomic shells of these clusters are allowed to relax, to respond to effects of truncating the clusters at finite sizes. We show that, with this relaxation, the calculated electronic and structural properties of the pure clusters are good approximations to the corresponding bulk properties. Using the finite-cluster procedure, we find essentially no relaxation of the defect near-neighbor ions in LiCl:Cu+, but an in- ward relaxation of 1.3% of the near-neighbor ions around the F center in MgO. I. INTRODUCTION An accurate description of the properties of point de- fects in solids should include the relaxation of bulk crys- tal atoms around the defect, since the formation energy of the defect and the positions of defect-related levels with respect to host crystal bands can both be strongly affected by relaxation. However, experimental measure- ments of defect-induced lattice relaxation are difficult to make, so that precise data are largely lacking. Calculat- ing the relaxation is also difficult. Most methods for treating the defect problem can allow for relaxation of host atoms in principle, but in practice, defect-induced lattice relaxation is often ignored. Many methods are formulated with the pure undistorted host crystal as a reference for the defect system, and allowing relaxation of the atoms near the defect makes the calculations ex- tremely complicated and costly. In this paper we discuss an approach to the defect- induced lattice relaxation problem based on free, finite atomic clusters. By effectively treating only the region of the solid in the immediate vicinity of the defect, the finite-cluster approach can be used to investigate defect- induced relaxation effects directly. Conceptually, our ap- proach is based on the fact that the local atomic environ- ment in the interior of a sufficiently large cluster is indis- tinguishable from that of the bulk, so that cluster esti- mates of bulk properties extracted from the central re- gion of a cluster can be expected to converge to the bulk values in the large cluster limit. It is not known how large a cluster must be treated in order to ensure full con- vergence to bulk properties, and we do not set out in this work to address this question. Instead we apply the clus- ter formalism in a pragmatic way to determine the feasi- bility of the approach for estimating bulk properties, par- ticularly those relating to point defects, using moderately sized clusters. The equilibrium atomic positions in the finite cluster are somewhat shifted from the corresponding equilibrium positions in the bulk solid due to stresses introduced by truncating the cluster at a finite size. Allowing the atom- ic shells in the cluster to relax from the ideal bulk lattice positions counteracts these truncation effects and pro- vides a more physically meaningful boundary for the clusters. Accordingly, we expect that estimates of bulk properties taken from relaxed clusters converge to the bulk values faster than those taken from unrelaxed clus- ters. This point can be made more general. Allowing the cluster to relax to its full variational ground state gives the cluster its greatest opportunity to respond to trunca- tion effects, and we expect that by optimizing the cluster energies with respect to all structural and electronic de- grees of freedom, we will obtain the fastest and smoothest convergence to the bulk values. The defect-related properties reported here are ob- tained from a comparison of calculations of pure clusters and the corresponding "defect" clusters, containing a sin- gle point defect. We have used this approach to investi- gate lattice relaxation around the Cu+ substitutional im- purity in LiC1, and the F center in MgO. These represent prototypical defect systems, and have been studied in the past using various methods. ' The paper is organized as follows: in the next section we provide a brief overview of our computational ap- proach, and discuss in more detail the use of finite clus- ters to obtain estimates of bulk properties. In Sec. III we present our results, for LiC1:Cu+ and for the MgO F center. We include in this section a comparison with re- sults obtained in previous studies. In the final section we summarize, offering our conclusions regarding this work and suggesting possible future directions. II. THEORY Our calculations are based on the Hohenberg-Kohn- Sham local-density approximation (LDA), which we use to study the structural and electronic properties of finite atomic clusters. In the LDA, the ground-state energy (in 43 2364 1991 The American Physical Society