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As everyone knows, the atomic orbital can be classified as $s, p_z, p_x, p_y, d_{z^2},d_{xz},d_{yz},d_{xy},d_{x^2-y^2}$ and so on. I want to know the meaning of $z^2,x^2-y^2$ and so on. Maybe this is a fundamental question, but I'm not familiar with chemistry.


I have some ideas about this:

  1. For example, we consider the $p$ orbital: $p_z$ is symmetric about $z$ axis; $p_x$ is symmetric about $x$ axis; $p_y$ is symmetric about $y$ axis. But what's the meaning of $z^2$ or $x^2 -y^2$?
  2. We consider the symmetry of atomic orbital with group theory. The change of orbital under the symmetry operation is the same as the orbital subscript.
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  • $\begingroup$ I have deleted my post, if I get correct relevance I will repost. But, you can notice that d_xy is symmetric about x+y axis. $\endgroup$ – Immortal Player Mar 25 '14 at 0:46
  • $\begingroup$ @GODPARTICLE Do you mean x+y axis is x=y axis? $\endgroup$ – Ben Mar 25 '14 at 0:53
  • $\begingroup$ Yes. The vector x+y will be $45^0$ with respect to both x and y axis. $\endgroup$ – Immortal Player Mar 25 '14 at 5:11
  • $\begingroup$ But it does not work for all (╯_╰) $\endgroup$ – Ben Mar 25 '14 at 11:28
  • $\begingroup$ It does for all xy, yz, zx involved subscripts. $\endgroup$ – Immortal Player Mar 25 '14 at 11:31
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For each azimutal quantum number $l$, the magnetic quantum number $m$ ranges from $-l$ to $+l$. Those give you the number of atomic orbitals "subsripts" you should obtain. Those are then expressed using spherical harmonics. If you look at the table of spherical harmonics at the $l=1$, you will find that they contain the cartesian axes as $x-y$, $z$ and $x+y$. Therefore it is advisable to form the linear combination out of them, so you get $x$, $y$, $z$. If you go to higher angular momentum, things get more complicated, but there is some tradition what linear combinations to use.

To answer your second question, all this formulas are not some inevitable result you obtain by juggling the numbers. Contrary, it is carefully crafted in a way that the formulas are so simple.

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  • $\begingroup$ 1)What do you mean "you will find that they contain the cartesian axes as $x−y$, $z$ and $x+y$."? 2)I want know why the subscripts are written as $z$, $z^2$, $x^2-y^2$ and so on respectively. 3)You say "it is carefully crafted in a way that the formulas are so simple.", and I'm not clear about "the formulas". Can you explain it more specificly? $\endgroup$ – Ben Mar 24 '14 at 9:57
  • $\begingroup$ 1. Please have a careful look at the table of spherical harmonics at l=1, i.e. p-orbitals in cartesian coordinates and try to understand, what I have written. $\endgroup$ – ssavec Mar 24 '14 at 14:25
  • $\begingroup$ I know how to combine spherical to obtain $p_x$, $p_y$...But I do not know why choose x, y, z and so on as subscripts. $\endgroup$ – Ben Mar 24 '14 at 14:36
  • $\begingroup$ The orientation of the orbitals. For $p$, one is oriented in the x direction one is oriented in the y direction and one is one the z direction. $\endgroup$ – 1110101001 Mar 25 '14 at 4:17
  • $\begingroup$ Bach: We chose the orientation for immense convenience. It is great, if you expand some equation and magically 95% of the terms disappear just because you have chosen good coordinate system. And the labels $p_x$,... just show, which axes and in which power appear in the spherical harmonic expression. $\endgroup$ – ssavec Mar 25 '14 at 8:03

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