PAMoC Tutorials

Properties of the nuclear charge density

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. ρρ(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: ρρ(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.

Table 1 - Electrostatic properties of a distribution of N nuclei.
property definition
charge Q = N i=1 Zi
dipole p = N i=1 qi riQ r0
quadrupole Θ = N i=1 qi ri rip r0r0 p + Q r0 r0
electric potential Φ(r) = N i=1 qi / |rri|
electric field E(r) = N i=1 qi rri / |rri|3
electric field gradient Hαβ(r) = N i=1 qi [ 3 (rαri,α) (rβri,β) / |rri|5 δαβ / |rri|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) )