After giving a concise overview of the current knowledge in the field of quantum mechanical bonding indicators for molecules and solids, we show how to obtain energy-resolved visualization of chemical bonding in solids by means of density-functional electronic structure calculations. On the basis of a band structure energy partitioning scheme, i.e. rewriting the band structure energy as a sum of orbital pair contributions, we derive what is to be defined as crystal orbital Hamilton populations (COHP). In particular, a COHP( epsilon ) diagram indicates bonding, nonbonding, and antibonding energy regions within a specified energy range while an energy integral of a COHP gives access to the contribution of an atom or a chemical bond to the distribution of one-particle energies. A further decomposition into specific atomic orbitals or symmetry-adapted linear combinations of atomic orbitals (hybrids) can easily be performed by using a projector technique involving unitary transformations. Because of its structural simplicity and the availability of reliable thermodynamic data, we investigate the bonding within crystalline silicon (diamond phase) first. As a basis set, both body-centered-cubic screened and diamond screened atomic-centered tight-binding linear muffin-tin orbitals (TB-LMTOs) are used throughout. The shape of COHP versus energy diagrams and the significance of COHP subcontributions (s-s, sp3-sp3) are analyzed. Specifically, the difference between the COHP energy integral (one-particle bond energy) and the experimental bond energy is critically examined. While absolute values for the one-particle bond energy show a high basis set dependence due to changing on-site (crystal field) COHP terms, the shape of off-site (bonding) COHP functions, elucidating the local bonding principle within an extended structure, remains nearly unaffected.
 

Journal of Physical Chemistry, 97 8617-24, 1993.