Current local descriptors computed at r = (x, y, z)

• Electron density ρ(r).
• Components of electric dipole moment, - x ρ(r); - y ρ(r); - z ρ(r)
• Radial distribution, 4 π r² ρ(r) with (r² = x² + y²+ z²).
• Gradient of electron density ∇ρ(r) its norm |∇ρ(r) |.
• Eigenvalues of the Hessian matrix of electron density : λ₁(r) _, λ₂(r) and λ₃(r). 
• Laplacian of electron density  ∇²ρ(r) = λ_{1 +} λ₂ + λ₃ ; L(r) = - ∇²ρ(r).
• Shape function, ρ(r)/N, where N is the total number of electrons.
• Reduced Density Gradient (RDG), see J. Am. Chem. Soc., 132, 6498 (2010).

• τs(r) or τ(r) or G(r) is the exact kinetic energy density (KED) of the non-interacting electron system defined within the KS-DFT theory.

       
          where φ_i are the occupied molecular orbitals.

• Local Temperature (2/3) G(r)/ρ(r) in Hartree/kB. see PNAS, 81, 8028 (1984)
  
• KED per electron G(r)/ρ(r).
• Approximate Kirzhnits G(r),
  

        
        with τ_TF(r) is the Thomas-Fermi KED, exact KED of an uniform electron gas (non-interacting system). The degree of ELF localization is given with respect to theThomas-Fermi localized degree.

• Hamiltonian K(r) kinetic energy density

• Potential energy density (Virial field): V(r) = - [ G(r) + K(r) ] = 0.25 ∇²ρ(r) - 2 G(r)
• Ratio V(r)|/G(r). At the QTAIM bond critical bond, ratio<1 for a closed-shell interaction and a ratio >1 for a typical covalent bond. See J. Chem. Phys., 117, 5529 (2002).
• Ratio |V(r)|/(2G(r)). At the QTAIM bond critical bond, ratio> for a typical covalent bond.
• Total energy H(r) = G(r) + V(r) = - K(r)
• Weizsäker kinetic energy density τ_w(r)

• Electron Localization Function (ELF)

    
        ELF definition for HF/Kohn-Sham DFT. See Angew. Chem. Int. Ed. Engl., 31, 187 (1992). ELF reveals the atomic shell structure as well as the nature of intra- and intermolecular interactions.

• Approximate ELF  computed from the Kirzhnits approximation of G(r). See Chem. Phys. Lett., 351, 142 (2002).
• Ellipticity,

      
        where λ₁ and λ₂ are the lowest and the second lowest eigenvalues of the electron density, respectively. This descriptor reveals an asymmetric distribution of the electron density around the QTAIM bond critical bond.
        see J. Am. Chem. Soc., 105, 5061 (1983).

• Bond metallicity, ρ(r)/∇²ρ(r). Indicates the metallic interaction computed at the BCP. see J. Phys.: Condens. Matter, 14, 10251 (2002).
• Shannon Entropy Density, - ρ(r) ln(ρ(r)). See J. Chem. Phys., 126, 191107 (2007).
• Molecular Electrostatic Potential MESP, electron & nuclear parts of the MESP.