What are the axes in the structure of an atom?

Question

When learning about the structure of atoms, I learnt that there are orbitals oriented along certain axes. What does it mean to be oriented along the axes? What is the reference? Also, when learning coordination chemistry, I learn that the ligands approach from the +ve and -ve x, y, and z axes, and hence the energy of different orbitals changes due to repulsion. But why can't those approach between the axes? Does the atom align itself with respect to the approaching ligands?

Also, there should be no preference for z axis over the other two, right? Why is there a $d_{z^2}$ orbital, but no $d_{x^2}$ or $d_{y^2}$ orbitals, and instead a $d_{x^2-y^2}$ orbital?

Edit: Added another question

(This question may belong to chemistry SE. If so, please inform me, and I will remove it from here. I thought there is enough physics for it to belong here.)

Related: this question that asks why spins can only be up/down (as opposed to say, left/right). — jacob1729, Dec 02 '20 at 15:58
My question is not about spin of electron, but about the orbitals in the atom. (p orbital, d orbital, etc) — Manish S, Dec 02 '20 at 16:00
Hence, 'related' and not 'duplicate'. The $p$ orbitals form a vector space - you can rotate from $p_x,p_y,p_z$ to some $p_{x'},p_{y'},p_{z'}$ basis. — jacob1729, Dec 02 '20 at 16:08

Gert · Accepted Answer · 2020-12-02T16:15:33.083

2

What is the reference?

The reference is the reference. Just like in any other physics problem, you choose it, usually for maximum convenience (ease of use). There is no 'absolute' reference point. For the calculation of the orbitals of the hydrogen atom a spherical coordinate system is used.

But why can't those approach between the axes?

They do sometimes and the phenomenon is the cause of many coloured transition metal (d-block) complexes. For an introduction, see my post here (scroll down to A partial explanation of the colours of transition cation complexes) on the colour of $\text{Ti }+3$ complexes.

I hope this helps.

edited Dec 02 '20 at 16:15

answered Dec 02 '20 at 16:08

Gert

35,289

Why only sometimes? It should be equally likely in all directions, right? Correct me if I am wrong. – Manish S Dec 03 '20 at 01:32
"It should be equally likely in all directions, right?" What makes you think that? – Gert Dec 03 '20 at 12:45
Basic symmetry. – Manish S Dec 03 '20 at 12:48
Hmm... the $\text{d}$ orbitals are calculated from the TISE for the $H$ atom. Not sure where symmetry fits into all this? – Gert Dec 03 '20 at 13:01
What is TISE here? – Manish S Dec 03 '20 at 13:07
Time Independent Schodinger Equation: $\hat{H}\Psi=E\Psi$. – Gert Dec 03 '20 at 13:11

score 2 · Answer 2 · answered Dec 02 '20 at 16:34

Here is a semi-handwaving explanation.

Whenever we attempt to confine an electron in a small volume of space, we discover that the chances of finding it at any particular location within that space become nonuniform: there are some places where we are more likely to observe it than in other locations, and also some energy levels of the electron will become more likely than others. Furthermore, the smaller the space in which we try to confine it, the stronger these effects become. When we reach the size scale of an atom, the positional probability of the electron is highly nonuniform, and its energy levels have become discrete- which is to say that the electron can only hang out in certain regions, and we call those regions orbitals, and that hanging out somewhere inbetween two allowed orbitals is forbidden.

The electron itself resists being stacked into an orbital that already is occupied by another electron- unless we invert the spin of one of the electrons, in which case we can fit more than one of them into one orbital.

The lowest-energy orbitals look like spherical clouds, but as we try to put more and more electrons into position around a larger and larger nucleus, we discover that the electron's angular momentum is also quantized, and there are certain angular positions in space around the nucleus where those additional electrons are more likely to be observed than in others. These orbitals are not spherically symmetric- they stick out along certain preferred directions, and all higher-energy orbitals have to accommodate by avoidance all of those other orbitals.

So the facts that any given electron wants to occupy a state of minimum energy in the vicinity of a nucleus, that energy states or levels for electrons being confined by electrostatic attraction to the nucleus of an atom are discrete, that electrons cannot be stacked on top of each other unless their spins are opposite, and that their angular momentum is quantized, all give rise to the orbital structure as described in your course materials.

The positions of all those orbitals are relative to all the other orbitals- that is, there isn't anything in the nucleus itself that preferentially steers a given electron pair into two lobe-shaped volumes that stick out in opposite directions from one another; the nucleus appears to the electrons around it to be electrostatically spherical even though the locations of the allowed orbitals in space around the nucleus are not.

Thank you. But, this triggered a new question. I added it to the question. — Manish S, Dec 03 '20 at 01:34

score 1 · Answer 3 · answered Dec 03 '20 at 10:43

The shape of the orbitals are an artifact of the way Schrodinger's equation is solved. The first part of the standard solution is to pick a direction for the x,y,z axis and convert to polar coordinates.

The solutions we get reflect the chosen coordinate system. The distribution is actually isotropic if there is no preferred direction in space (no electric or magnetic field). The electrons are indistinguishable and in states that are "spherical" mixtures of orbitals (until a measurement is made/interaction happens). The picture is complicated by Pauli exclusion principle which constrains which mixtures are allowed at the same time. Statements such as "one electron is in the p_x orbital" are somewhat misleading and simplified, but much easier on high school Chemistry students.

What are the axes in the structure of an atom?

3 Answers3