# What's the crystal orbital Hamiltonian population?

What's the crystal orbital Hamiltonian population (COHP) or the crystal orbital overlap population (COOP)? What's the relation between the bandstructure or density of states (DOS) and COHP/COOP? Why we can read the information about bonding or antibonding information from COHP and COOP?

A clear physical picture will be appreciated.

There is a great description given of COOP in Roald Hoffman's Solids and Surfaces: A Chemist's View of Bonding in Extended Structures. It is sometimes referred to as the Overlap Population Weighted Density of States (OPWDOS) which makes the meaning a bit clearer: its the density of states, weighted by the overlap for a given bond ($$c_ic_jS_{ij}$$). This results in a signed DOS, as the overlap population can be positive or negative. This quantity is similar (though not exactly equivalent) to the molecular concept of bond order, where a positive COOP denotes a favorable/bonding interaction and a negative COOP indicates an unfavorable/antibonding interaction. We can see an example of this below from Hoffman's book, giving an example for a simple hydrogen chain.