Atomic orbitals: 7f 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 7f orbitals)
Function | Equation |
---|---|
Radial wave function, R7f | = (1/17640√42) × (720 - 270ρ + 30ρ2 - ρ3)ρ3 × Z3/2 × e-ρ/2 |
Angular wave functions (general set): | |
Y7fz3 | = √(7/4) × z(5z2 - 3r2)/r3 × (1/4π)1/2 |
Y7fyz2 | = √(42/16) × y(5z2 - r2)/r3 × (1/4π)1/2 |
Y7fxz2 | = √(42/16) × x(5z2 - r2)/r3 × (1/4π)1/2 |
Y7fxyz | = √(105/4) × 2xyz/r3 × (1/4π)1/2 |
Y7fz(x2-y2) | = √(105/4) × z(x2-y2)/r3 × (1/4π)1/2 |
Y7fy(3x2-y2) | = √(70/16) × y(3x2-y2)/r3 × (1/4π)1/2 |
Y7fx(x2-3y2) | = √(70/16) × x(x2-3y2)/r3 × (1/4π)1/2 |
Angular wave functions (cubic set): | |
Y7fy3 | = √(7/4) × y(5y2 - 3r2)/r3 × (1/4π)1/2 |
Y7fz3 | = √(7/4) × z(5z2 - 3r2)/r3 × (1/4π)1/2 |
Y7fx3 | = √(7/4) × x(5x2 - 3r2)/r3 × (1/4π)1/2 |
Y7fy(z2-x2) | = √(105/4) × y(z2-x2)/r3 × (1/4π)1/2 |
Y7fz(x2-y2) | = √(105/4) × z(x2-y2)/r3 × (1/4π)1/2 |
Y7fx(z2-y2) | = √(105/4) × x(z2-y2)/r3 × (1/4π)1/2 |
Y7fxyz | = √(105/4) × 2xyz/r3 × (1/4π)1/2 |
Wave functions (general set): | |
ψ7fz3 | = R7f × Y7fz3 |
ψ7fyz2 | = R7f × Y7fyz2 |
ψ7fxz2 | = R7f × Y7fxz2 |
ψ7fxyz | = R7f × Y7fxyz |
ψ7fz(x2-y2) | = R7f × Y7fz(x2-y2) |
ψ7fy(3x2-y2) | = R7f × Y7fy(3x2-y2) |
ψ7fx(x2-3y2) | = R7f × Y7fx(x2-3y2) |
Wave functions (cubic set): | |
ψ7fy3 | = R7f × Y7fy3 |
ψ7fz3 | = R7f × Y7fz3 |
ψ7fx3 | = R7f × Y7fx3 |
ψ7fy(z2-x2) | = R7f × Y7fy(z2-x2) |
ψ7fz(x2-y2) | = R7f × Y7fz(x2-y2) |
ψ7fx(z2-y2) | = R7f × Y7fx(z2-y2) |
ψ7fxyz | = R7f × Y7fxyz |
Electron density | = ψ7f2 |
Radial distribution function | = r2R7f2 |
For any atom, there are seven 7f orbitals. The f-orbitals are unusual in that there are two sets of orbitals in common use. The cubic set is appropriate to use if the atom is in a cubic environment. The general set is used at other times. Three of the orbitals are common to both sets. These are are the 7fxyz, 7fz3, and 7fz(x2-y2) orbitals.
The radial equations for all the 7f orbitals are the same. The real angular functions differ for each and these are listed above.
Each of the orbitals is named for the expression based upon x, y, and z in the angular wave function, but some abbreviated names are useful for simplicity. These are:
- 7fx3 used for 7fx(5x2 - 3r2)
- 7fy3 used for 7fy(5y2 - 3r2)
- 7fz3 used for 7fz(5z2 - 3r2)
- 7fxz2 used for 7fx(5z2 - r2)
- 7fyz2 used for 7fy(5z2 - r2)
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.
The OrbitronTM, a gallery of orbitals on the WWW: https://winter.group.shef.ac.uk/orbitron/
Copyright 2002-2023 Prof. Mark Winter [The University of Sheffield]. All rights reserved.