# Atomic orbitals: 7g 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 (7 for the 7g orbitals)
Table of equations for the 7g orbitals.
Function Equation
Radial wave function, R7g = (1/17640√154) × (8 – ρ)ρ4 × Z3/2 × e-ρ/2
Angular wave functions:
Y7gz4 = √(9/64) × (35z4 - 30z2r2 + 3r4)/r4 × (1/4π)1/2
Y7gz3y = √(45/8) × yz(7z2 - 3r2)/r4 × (1/4π)1/2
Y7gz3x = √(45/8) × xz(7z2 - 3r2)/r4 × (1/4π)1/2
Y7gz2xy = √(45/16) × 2xy(7z2 - r2)/r4 × (1/4π)1/2
Y7gz2(x2 - y2) = √(45/16) × (x2-y2)(7z2 - r2)/r4 × (1/4π)1/2
Y7gzy3 = √(315/8) × yz(3x2 - y2)/r4 × (1/4π)1/2
Y7gzx3 = √(315/8) × xz(x2 - 3y2)/r4 × (1/4π)1/2
Y7gxy(x2-y2) = √(315/64) × 4xy(x2 - y2)/r4 × (1/4π)1/2
Y7g(x4 + y4) = √(315/64) × (x4 + y4 - 6x2y2)/r4 × (1/4π)1/2
Wave functions:
ψ7gz4 = R7g × Y7gz4
ψ7gz3y = R7g × Y7gz3y
ψ7gz3x = R7g × Y7gz3x
ψ7gz2xy = R7g × Y7gz2xy
ψ7gz2(x2 - y2) = R7g × Y7gz2(x2 - y2)
ψ7gzy3 = R7g × Y7gzy3
ψ7gzx3 = R7g × Y7gzx3
ψ7gxy(x2-y2) = R7g × Y7gxy(x2-y2)
ψ7g(x4 + y4) = R7g × Y7g(x4 + y4)
Electron density = ψ7g2