Relativistic and Nonrelativistic Electron Densities

The spherically averaged electron density is an important quantity [481] as it can be defined even for an atom in a molecule. It follows from the general definition of the electron ground-state density p(r), which is to be calculated from the total electronic ground state wave function To, Eq. (4.9). For a representation in spherical coordinates r = (r, qj, j ) we define a radial electron density D(r) such that 

Comparing this definition with the integration of p r) over all space, Eq. (4.10) 

In order to define a density which addresses both an atom in a general molecule and a spherically symmetric atom, we need to define a spherically averaged density p r) that yields the total (electronic) charge in a spherical shell Pq with inner radius r and thickness dr, i.e., D(r)dr, divided by the volume of this shell (47rr dr), 

= (Yo p, Yo) = (apYoKYo) with the excitation operator tpq of second quantization 

in the case of atoms, the angular degrees of freedom are integrated out analytically, we switch from spinor indices p,cj to shell indices i,j and arrive 

---