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| Atomic orbitals |
| Level 1 1s |
| Level 2 2s 2p |
| Level 3 3s 3p 3d |
| Level 4 4s 4p 4d 4f |
| Level 5 5s 5p 5d 5f 5g |
| Level 6 6s 6p 6d 6f 6g |
| Level 7 7s 7p 7d 7f 7g |
| Hybrid orbitals |
| 2s+2p hybrids sp sp2 sp3 |
| 3s+3p+3d hybrids dsp3 d2sp3 |
| Molecular orbitals |
| H2, dihydrogen σ* σ |
| N2, dinitrogen σp* πx* πy* σp πx πy σs* σs |
| Introduction | Wave function | Electron density | Dots! | Radial distribution | Equations |
|---|---|---|---|---|---|
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 (5 for the 7g orbitals)
- k = various constants
| Function | Equation |
|---|---|
| Radial wave function, R7g | = (1/900√70) × (= - =ρ + =ρ2)ρ4 × Z3/2 × e-ρ/2 |
| Angular wave functions: | |
| Y7gz4 | = k × (35z4 - 30z2r2 + 3r4)/r4 × (1/4π)1/2 |
| Y7gz3x | = k × xz(4z2 - 3x2 - 3y2)/r4 × (1/4π)1/2 |
| Y7gz3y | = k × yz(4z2 - 3x2 - 3y2)/r4 × (1/4π)1/2 |
| Y7gz2xy | = k × xy(6z2 - x2 - y2)/r4 × (1/4π)1/2 |
| Y7gz2(x2 - y2) | = k × (x2-y2)(6z2 - x2 - y2)/r4 × (1/4π)1/2 |
| Y7gzx3 | = k × xz(x2 - 3y2)/r4 × (1/4π)1/2 |
| Y7gzy3 | = k × yz(3x2 - y2)/r4 × (1/4π)1/2 |
| Y7gxy(x2-y2) | = k × xy(x2 - y2)/r4 × (1/4π)1/2 |
| Y7gx4 + y4 | = k × (x4 + y4 - 6x2y2)/r4 × (1/4π)1/2 |
| Wave functions: | |
| ψ7gz4 | = R7g × Y7gz4 |
| ψ7gz3x | = R7g × Y7gz3x |
| ψ7gz3y | = R7g × Y7gz3y |
| ψ7gz2xy | = R7g × Y7gz2xy |
| ψ7gz2(x2 - y2) | = R7g × Y7gz2(x2 - y2) |
| ψ7gzx3 | = R7g × Y7gzx3 |
| ψ7gzy3 | = R7g × Y7gzy3 |
| Electron density | = ψ7g2 |
| Radial distribution function | = r2R7g2 |
The radial equations for all the 7g 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.
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