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Atomic orbitals: 3s equations

The symbols used in the following are:

  • r = radius expressed in atomic units (1 Bohr radius = 52.9 pm)
  • π = 3.14159 approximately
  • e = 2.71828 approximately
  • Z = effective nuclear charge for that orbital in that atom.
  • ρ = 2Zr/n where n is the principal quantum number (3 for the 3s orbital)
Table of equations for the 3s orbital.
Function Equation
Radial wave function, R3s = (1/9√3) × (6 - 6ρ + ρ2) × Z3/2 × e-ρ/2
Angular wave function, Y3s = 1 × (1/4π)1/2
Wave function, ψ3s = R3s × Y3s
Electron density = ψ3s2
Radial distribution function = 4πr2ψ3s2

The origin of the spherical nodes becomes clearer upon examining the wave equation, which includes a (6 - 6ρ + ρ2) term. When (6 - 6ρ + ρ2) = 0, then we must have nodes. Since for the 3s orbital ρ = 2Zr/3 (Z = effective nuclear charge, r = radius in atomic units), and we can solve the zero values for the quadratic to give ρ = 3 + √3 or 3 - √3 then the nodes are at the radii r = 3(3 + √3)/2Z and 3(3 - √3)/2Z atomic units.

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