Open PaulWAyers opened 1 month ago
Given that it is practical to compute the number of nodal regions but not the number of nodes, we could
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 $n_x, n_y, n_z$ = $(n_x+1)(n_y+1)(n_z+1)$. Then one can fill in the rest of this table relatively simply (just looping through all the ways to add up 3 integers to form $n_x + n_y + n_z = \text{number of nodes} - 3$
The idea is to leverage QTAIM technology to get the number of nodes in orbitals.
This is complicated (more than I thought) as pointed out by @FarnazH . So this is mostly storage of an ideas/discussions for future consideration.
Isosurface-Driven Approach
gbasis
,grid
, and/orcugbasis
.@FarnazH pointed out that this is more complicated, because nodal surfaces can (and often do) interesect. Intersecting nodal surfaces are connected, however.
By identifying isosurfaces, we can identify closed isosurfaces (do not contain any boundary points) and also open isosurfaces (that stretch to infinity, or the boundary of the region). Closed isosurfaces always count; I do not think they can intersect except at a catastrophe point, where the number of nodal surfaces changes.
For the open isosurfaces, one stores the points at the boundary, and defines a new graph that contains only the boundary points. If this graph has just one cycle, then the open isosurface does not interesect any others. If two open isosurfaces intersect, it seems there are twelve cycles. For three or more interesecting open surfaces it depends on the topology of the intersection.
One can also characterize the number of regions. The number of regions is obtained by:
gbasis
,grid
, and/orcugbasis
.In one dimension, the number of nodal regions is one greater than the number of nodes. In two dimensions, there is already the potential for intersecting nodes. So, for example, four nodal regions could correspond to four interesecting nodes (like a typical $3d$ orbital) or three nonintersecting nodes (like a $3p$ or $4s$ orbital). Three nodal surfaces can give anywhere from 4 nodal regions to 8 nodal regions, depending on how (and whether) the nodal surfaces intersect. For example, a typical $4d$ orbital has 8 nodal regions. Four nodal surfaces can give anywhere from 5 nodal regions to 16 nodal regions. In three dimensions, it should be similar: for $n$ nodes you can have anywhere between $n+1$ and $2^n$ nodal regions.
Trying to count them is hard. There is probably a formula in terms of the number of cycles at infinity, the number of connected nodal surfaces, and the number of nodal regions. It may be possible to make a complete taxonomy up to a relatively small number of nodes (~5), but beyond that one needs to use one of the asymptotic formulas that we have cooked up elsewhere. There would be better ways to do this, but it would require a detailed analysis of the way the nodal hyperplanes interested.
Old Bad Idea
I thought that by constructing the function, $d(\mathbf{r})$ that was the distance to the nearest nodal surface we would have a function that would have one maximum for every nodal region. This doesn't work because there could be dumbell-shaped regions, which have two maximum for $d(\mathbf{r})$.