The charge density ρ, introduced in a previous chapter of this manual, is a scalar-valued function of the three cartesian coordinates x, y, z in ℝ3, i.e. ρ ≡ ρ(x,y,z). It can also be seen as a scalar-valued function of a vector variable in ℝ3, i.e. the position vector r with components x, y, z: ρ ≡ ρ(r). The electrostatic properties of a charge distribution of N nuclei are reported in Table 1. The same properties for a single nucleus are reported in Table 2.
| property | definition |
|---|---|
| charge | Q = N ∑ i=1 Zi |
| dipole | p = N ∑ i=1 qi ri − Q r0 |
| quadrupole | Θ = N ∑ i=1 qi ri ri† − p r0† − r0 p† + Q r0 r0† |
| electric potential | Φ(r) = N ∑ i=1 qi |r − ri| |
| electric field | E(r) = N ∑ i=1 qi r − ri |r − ri|3 |
| electric field gradient | Hαβ(r) = N ∑ i=1 qi [ 3 (rα − ri,α) (rβ − ri,β) |r − ri|5 − δαβ |r − ri|3 ] |
| electrostatic energy density | uE(r) = E2(r) 8 π |
| electrostatic energy | UE = 1 2 N ∑ i=1 qi N ∑ i=1 qj rij |
| Maxwell stress tensor | σαβ(r) = 1 4 π ( Eα(r) Eβ(r) − 1 2 δαβ E2(r) ) |