Atomic orbitals: 5g 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 (5 for the 5g orbitals)
k = various constants

**Table of equations for the 5g orbitals.**
Function	Equation
Radial wave function, R_5g	= (1/900√70) × ρ⁴ × Z^3/2 × e^-ρ/2
Angular wave functions:
Y_{5g_z⁴}	= √(9/64) × (35z⁴ - 30z²r² + 3r⁴)/r⁴ × (1/4π)^1/2
Y_{5g_z³y}	= √(45/8) × yz(7z² - 3r²)/r⁴ × (1/4π)^1/2
Y_{5g_z³x}	= √(45/8) × xz(7z² - 3r²)/r⁴ × (1/4π)^1/2
Y_{5g_z²xy}	= √(45/16) × 2xy(7z² - r²)/r⁴ × (1/4π)^1/2
Y_{5g_{z²(x² - y²)}}	= √(45/16) × (x²-y²)(7z² - r²)/r⁴ × (1/4π)^1/2
Y_{5g_zy³}	= √(315/8) × yz(3x² - y²)/r⁴ × (1/4π)^1/2
Y_{5g_zx³}	= √(315/8) × xz(x² - 3y²)/r⁴ × (1/4π)^1/2
Y_{5g_xy(x²-y²)}	= √(315/64) × 4xy(x² - y²)/r⁴ × (1/4π)^1/2
Y_{5g_{(x⁴ + y⁴)}}	= √(315/64) × (x⁴ + y⁴ - 6x²y²)/r⁴ × (1/4π)^1/2
Wave functions:
ψ_{5g_z⁴}	= R_5g × Y_{5g_z⁴}
ψ_{5g_z³y}	= R_5g × Y_{5g_z³y}
ψ_{5g_z³x}	= R_5g × Y_{5g_z³x}
ψ_{5g_z²xy}	= R_5g × Y_{5g_z²xy}
ψ_{5g_{z²(x² - y²)}}	= R_5g × Y_{5g_{z²(x² - y²)}}
ψ_{5g_zy³}	= R_5g × Y_{5g_zy³}
ψ_{5g_zx³}	= R_5g × Y_{5g_zx³}
ψ_{5g_xy(x²-y²)}	= R_5g × Y_{5g_xy(x²-y²)}
ψ_{5g_{(x⁴ + y⁴)}}	= R_5g × Y_{5g_{(x⁴ + y⁴)}}
Electron density	= ψ_5g²
Radial distribution function	= r²R_5g²

The radial equations for all the 5g orbitals are the same. The real angular functions differ for each and these are listed above.

For s-orbitals the radial distribution function is given by 4πr²ψ², but for non-spherical orbitals (where the orbital angular momentum quantum number l > 0) the expression is as above. See D.F. Shriver and P.W. Atkins, Inorganic Chemistry, 3rd edition, Oxford, 1999, page 15.