Comments (1)
Given that it is practical to compute the number of nodal regions but not the number of nodes, we could
- compute the number of nodal regions
- compute the number of nodes in a spherical/cubic system that has that number of nodal regions
- use this function
$\text{number of nodes}(\text{number of nodal regions})$ as a proxy variable. Since a given number of
nodes has many different number of nodal regions, one needs to use an average.
For spherical systems we have:
# nodes | # regions |
---|---|
0 | 1 |
1 | 2 |
2 | (3+3 * 4+4 * 4+1 * 3z^2)/9 |
3 | (4+3 * 6+4 * 8+6z^2+6 * 6+1z^3 * 4)/16 |
4 | (5+3 * 8+4 * 12+9z^2+6 * 12+1 * 8z^3+8 * 8+5z^4)/25 |
- | - |
We would need to build a spline for this function and use its inverse to compute the number of nodes as a function of the number of regions.
For Cartesian systems, we have:
# nodes | # regions |
---|---|
0 | 1 |
1 | 2 |
2 | (3 * 3 + 4 * 3)/10 |
3 | (3 * 4 + 6 * 6 + 8)/11 |
- | - |
James pointed out that the formula for the cube (or any rectangular prism) is that the number of regions is for quantum numbers
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