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In molecular orbital theory, the term molecular orbital extents was mentioned, although I can't really picture it. Does molecular orbital extent refer to the spatial distribution of electrons in a molecule?

When I calculate molecular orbital extents with $\ce{H2O}$, the following table is provided.

  Orbital extents (a.u.):
        MO          <x^2>          <y^2>          <z^2>          <r^2>
   0 A1  0   0.0177335990   0.0177345751   0.0314235271   0.0668917013
   1 A1  1   0.5165227329   0.7453494022   0.5948204224   1.8566925575
   2 B2  0   0.4227875842   1.8458686046   0.6860187318   2.9546749206
   3 A1  2   0.4563255952   0.6542639495   1.4243758427   2.5349653874
   4 B1  0   1.3086927738   0.4362309246   0.4500334749   2.1949571734
   5 A1  3   3.2682125957   5.9000542358   4.8761186497  14.0443854813
   6 B2  1   2.8796959432  10.2335899558   4.1395581510  17.2528440501
   7 B2  2   1.2848895351   4.1475258156   2.3869042266   7.8193195772
   8 A1  4   2.2181684269   3.7911494454   2.5513702157   8.5606880879
   9 B1  1   2.8880726336   0.9626908779   0.9764934282   4.8272569397
  10 A1  5   1.0167358798   1.0037029230   2.9703269612   4.9907657641
  11 B2  3   1.1288038413   3.4979906480   1.5412110158   6.1680055051
  12 A1  6   1.5715312779   1.8762037282   1.5440486064   4.9917836126
  13 A1  7   1.0293258031   3.4022681941   1.8912304400   6.3228244372
  14 B2  4   0.8946420761   3.9039643575   1.9319712799   6.7305777135
  15 B1  2   0.8650504801   0.2883501600   0.3021527104   1.4555533504
  16 A1  8   0.2888876394   0.3407682209   0.8571143170   1.4867701773
  17 B2  5   0.2835432585   0.9003917411   0.3971506399   1.5810856395
  18 A1  9   0.0413406458   0.0433229266   0.0554579364   0.1401215088

However, I am not sure what it means. If someone is familiar with this topic, please teach me.

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    $\begingroup$ I don't know which program you are using, but it might be worth seeing if you can visualise the orbitals somehow (in, Avogadro, ChimeraX, Jmol, or any other program you have access to) $\endgroup$
    – isolated matrix
    Commented Jul 14, 2023 at 7:03
  • $\begingroup$ I see, I should visualize it! $\endgroup$
    – user250756
    Commented Jul 14, 2023 at 7:04
  • $\begingroup$ For something like water, though, you can just google 'water molecular orbitals' and look at the images... it took me less than 10 seconds to find this: chem.libretexts.org/@api/deki/files/77758/… $\endgroup$
    – isolated matrix
    Commented Jul 14, 2023 at 7:08
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    $\begingroup$ I recommend using Avogadro. It's open source, and reasonably user friendly. $\endgroup$
    – Tensor
    Commented Jul 14, 2023 at 13:22
  • 2
    $\begingroup$ What is the origin of the table? $\endgroup$
    – Buck Thorn ♦
    Commented Jul 14, 2023 at 15:25

1 Answer 1

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It looks like $<x^2>$ relates to the variance of the electron distribution, i.e. a measure how "big" the molecular orbital is in each direction. The final column seems to be the sum of the variances in x, y and z.

The first MO is almost exactly equal to the inner atomic orbital of the oxygen atom, and has the smallest extent. (Strangely, it is not spherical with equal variances in the three directions.)

The header says that values are in units of a.u., yet you would expect the square of a length if the values were the variance. On the other hand, if the extent were defined as the square root of the variance, I would not expect the values to add up to the last column.

I'm not sure orbital extent is a widely used technical term. I found one paper that talks about orbital variance, though:

The orbital variance (or spread) of orbital | p〉is a measure of the spatial extent of that orbital and thus of the locality of the orbital.

Source: https://pure.au.dk/ws/files/38005281/JChemPhys_134_194104.pdf

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  • $\begingroup$ Isn't the last column the sum of the other three just because, $<r^2> = \int \psi^*\hat{r}^2\psi \; \mathrm{d}\tau = \int \psi^*(\hat{x}^2 + \hat{y}^2 + \hat{z}^2)\psi \; \mathrm{d}\tau = <x^2> + <y^2> + <z^2>$? $\endgroup$
    – Metal Storm
    Commented Jul 14, 2023 at 13:23
  • $\begingroup$ @MetalStorm Yes, that makes sense. But then (a.u.) does not make sense unless it is meant to say "appropriate atomic units", which is a0 squared in this case, I guess. $\endgroup$
    – Karsten ♦
    Commented Jul 14, 2023 at 13:27
  • $\begingroup$ Yes, that was my same question... $\endgroup$
    – Metal Storm
    Commented Jul 14, 2023 at 13:29

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