Quantities characterizing the electron dynamics

We begin with the spin-free reduced density matrix p (r, r t) is just the electron density. In chemical applications, it is often expanded in atomic orbitals XaA (r) as 

By extracting from 7 (r t) only the product terms between the atomic orbitals belonging two different atoms, say A and B, one can define bond-order density in such a way 

The positiveness of its spatial distribution represents a formation of chemical bonding. The familiar bond-order itself is recovered by integrating Bab (r,t) over r. 

It is well known that even the spin singlet ground-state can have a biradical character due to electron correlation. To monitor the distribution of such radicals, or unpaired electrons, at a given site r, we use the so-called unpaired electron density [417], which is defined as 

Chemical Theory Beyond the Born-Oppenheimer Paradigm 

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