Published: May 06, 2011 r2011 American Chemical Society 5461 dx.doi.org/10.1021/jp202489s | J. Phys. Chem. A 2011, 115, 5461–5466 ARTICLE pubs.acs.org/JPCA Crystal Orbital Hamilton Population (COHP) Analysis As Projected from Plane-Wave Basis Sets Volker L. Deringer, † Andrei L. Tchougr  eeff, †,‡ and Richard Dronskowski* ,† † Institute of Inorganic Chemistry, RWTH Aachen University, Landoltweg 1, D-52056 Aachen, Germany ‡ Poncelet Laboratory, Moscow Center for Continuous Mathematical Education, Independent University of Moscow, Bolshoi Vlasevsky Per. 11, 119002 Moscow, Russia I. INTRODUCTION There is a flourishing symbiosis between theoretical chemistry and physics when it comes to treating the solid state. In the 21st century, ab initio calculations not only guarantee a thorough un- derstanding of existing phenomena but are also of tremendous help in the prediction of fascinating new materials. One of the cornerstones of chemical theory, however, has always been the quest for simple, yet powerful models that can be easily visualized, 1 and such models are particularly valuable when dealing with ex- tended, three-dimensional structures, i.e., crystals. Here, the quan- tum-mechanical information is typically expressed in reciprocal space, which often poses a serious problem for both chemical in- tuition and imagination. To overcome such difficulties in the framework of density- functional theory (DFT), the crystal orbital Hamilton population (COHP) analysis was introduced in 1993, 2 a DFT successor of the familiar crystal orbital overlap population (COOP) concept 3 based on extended H€ uckel theory. 4 COHP is a partitioning of the band-structure energy in terms of orbital-pair contributions, and it is therefore based on a local basis (the so-called tight-binding approach) as is commonly used in chemistry and parts of physics as well. Given that the interaction between two orbitals (say, the μth and νth one), centered at neighboring atoms, is described by their Hamiltonian matrix element H μν = Æφ μ | ^ H|φ ν æ, the multi- plication with the corresponding densities-of-states matrix then easily serves as a quantitative measure of bonding strength because the product either lowers (bonding) or raises (antibonding) the band-structure energy. Thus, energy-resolved COHP(E) plots make bonding, nonbonding (no energetic effect), and antibond- ing contributions discernible at first glance, just like the earlier COOP(E) plots. Accordingly, COHP analysis has successfully answered numerous questions and furthermore made useful predictions in the “chemical” language of local, atom-centered orbitals 5 together with the underlying density-functional theory. Physics, on the other hand, has been following alternative pathways. Bloch’s theorem 6 suggests handling periodic systems quite differently, and the wave functions of the crystal are easily constructed in terms of plane waves that form an orthonormal and, in principle, complete description of the Hilbert space. In fact, plane waves appear as a natural (yet highly nonchemical!) choice for any crystalline system, and the price paid is obvious from the fact that the atomic nature of the material at hand is hidden in a plane-wave expansion; in addition, the atom’s nodal structure is totally removed by a numerically tractable pseudopotential ansatz. Today, an abundance of plane-wave electronic-structure codes is available, 7 and plane-wave calculations have become the method of choice for fast, yet reliable theoretical materials science. 8 To nonetheless apply chemical thinking, a couple of attempts have been carried out to reconstruct local quantities such as Mulliken charges from the results of plane-wave calculations. Already in 1995, S anchez-Portal et al. introduced a projection technique 9 that enabled studies on a broad range of different solids; 10 the idea is similar to the one presented in this work. To date, however, no attempts have been made public for re-for- mulating a COHP-like quantity as well. Given that such a method exists, insightful chemical models will be available even when relying upon nonchemical computational approaches, namely the state-of-the-art plane-wave codes. This paper is organized as follows. In section II, we describe the underlying theory and then develop the technique which we Received: March 16, 2011 Revised: April 21, 2011 ABSTRACT: Simple, yet predictive bonding models are essential achievements of chemistry. In the solid state, in particular, they often appear in the form of visual bonding indicators. Because the latter require the crystal orbitals to be constructed from local basis sets, the application of the most popular density-functional theory codes (namely, those based on plane waves and pseudopotentials) appears as being ill-fitted to retrieve the chemical bonding information. In this paper, we describe a way to re-extract Hamilton-weighted populations from plane-wave electronic- structure calculations to develop a tool analogous to the familiar crystal orbital Hamilton population (COHP) method. We derive the new technique, dubbed “projected COHP” (pCOHP), and demonstrate its viability using examples of covalent, ionic, and metallic crystals (diamond, GaAs, CsCl, and Na). For the first time, this chemical bonding information is directly extracted from the results of plane-wave calculations.