How to picture atomic/electron orbitals

[dolimitless]

I don’t understand why the arrangement of electrons around an atom (the electron cloud) orientates itself around the nucleus in different shapes i.e, spherical, dumbbell shape, etc? I know electrons have wavelike properties but I can’t picture in my head why, after two electrons, why they orientate themselves in the structures we see represented in orbitals and subshells.

I know in the lowest-energy state hydrogen atom, the electrons are most likely to be found within a sphere around the nucleus of an atom. In a higher energy state atoms, the shapes become lobes and rings, obviously due to the interaction of the “quantum effects” between the different atomic particles (that’s what quantum mechanics is all about I guess.) Can someone explain in words or through imagery the interaction of these quantum effects that orientate these electrons in such structures? I do not know the math involved or have much experience in quantum mechanics, but can someone maybe perhaps explain it to me in laymen’s terms? Is it the wave structure of more than two electrons combining that cause the dumbbell shapes? This is bugging me so bad. I am a chemistry major and it seems to me that if I don’t understand this or picture it in my head, then learning atomic structure in general will seem useless to me. I know trying to understand or picture the effects of quantum mechanics is a futile process, but I need to know!

 

[malawi_glenn]

The shapes just come from solving the Schrödinger equation, basic differential equation. You are chemist major, you don't need layman terms, since layman terms is never even close to be correct.

There is no "interaction", except for the interaction of the electron and the nucleus itself. So the shape of the hydrogen energy levels just comes from mathematics of the schrödinger equation.

In more electron atoms, electron-electron effects must be taken into account, but that is another story. The dumbell shapes, sub-shells in the hydrogen atom just comes from math.

First, try to understand how the hydrogen atom works, then you can move on to several electron atoms. Try Atkins book in physical chemistry, it explain things well and from first principles to chemists.

 

[alxm]

Well as malawi_glenn points out, it's not something easy to understand. Ultimately it's what the solutions look like, and solutions to partial differential equations like the S.E. aren't easy to predict in general.

You're essentially looking at wave-like functions in three dimensions. And you can deduce some things by the shape of the orbitals. For instance, the higher orbitals (p,d,f) correspond to higher angular momentum, so the electrons have a more curved trajectory, hence the increasingly tighter 'dumbbells'.

Also, like a simple standing wave, more nodes means higher frequency means more energy. So a p-orbital which has a single nodal plane must, by simple inspection, have higher energy than an s orbital that has none, and lower energy than a dx orbital which has two.

 

[Forestman]

That is a really good question.

 

[alxm]

I just noticed you wrote:

after two electrons, why they orientate themselves in the structures we see represented in orbitals and subshells.

An electron will move in the orbitals you see regardless of how many other electrons there are. In fact, most of the time, orbitals that are drawn are the orbitals of a single electron system (hydrogenlike orbitals), which can alternately be viewed as the orbitals for electrons that are completely non-interacting.

Since electrons are fermions, only two (with opposite spin states) can occupy any single spatial orbital, due to the Pauli exclusion principle.

So it's not an effect of electron-electron interactions in themselves. (which do affect the shape of the orbitals, but not much).

 

[genneth]

The picture I have in my head is to treat electrons as essentially fuzzy gas. The orbitals are simply density profiles of these gas clouds. DO NOT think of the electrons as particles with specific position or velocity. They are simply clouds of gas, sitting in some wells.

Now, this cloud will settle into the lowest energy state it can manage, but there are some constraints. One is that they can't occupy the same well. This is the exclusion principle, and entirely quantum --- no reasonable way to explain it classically. Two is that due to quantisation of angular momentum (or the fact that wavefunctions need to come back to themselves after going about the nucleus), these clouds need to form density modulations about an axis through the origin. Lastly, the potential due to the nucleus and the effective potential due to the angular momentum makes little wells in the radial direction (need to do maths at this point).

There are various other effects I've entirely ignored there, but that should be enough to estimate a hydrogenic atom.

 

[dolimitless]

I just noticed you wrote:



An electron will move in the orbitals you see regardless of how many other electrons there are. In fact, most of the time, orbitals that are drawn are the orbitals of a single electron system (hydrogenlike orbitals), which can alternately be viewed as the orbitals for electrons that are completely non-interacting.

Since electrons are fermions, only two (with opposite spin states) can occupy any single spatial orbital, due to the Pauli exclusion principle.

So it's not an effect of electron-electron interactions in themselves. (which do affect the shape of the orbitals, but not much).

I don't get this. How are all the orbitals representations just of an single electron system? Why is it in my chemistry textbook and when people talk about orbitals it shows that in order to go on to say, the 3s sublevel, you have to fill two electrons for 1s shell, then the 6 electrons for the three p orbitals, etc?

 

[alxm]

I don't get this. How are all the orbitals representations just of an single electron system? Why is it in my chemistry textbook and when people talk about orbitals it shows that in order to go on to say, the 3s sublevel, you have to fill two electrons for 1s shell, then the 6 electrons for the three p orbitals, etc?

The orbitals are higher-energy states, so a single electron will naturally 'want' to be in the lowest orbital. But the other ones are still "there" as excited states.

So as you add electrons, the ground state (HOMO - Highest Occupied Molecular Orbital) becomes increasingly higher, since the exclusion principle demands only two electrons can occupy any single orbital.

(If electrons were bosons, OTOH, most of them would just stay in the lowest, 1s, orbital)

 

[DaveC426913]

I grant that there's no classical counterpart of the shapes of orbitals but are the orbital lobes kind of harmonic-like?

For example: if I have a circular semi-rigid plastic hoop, I can easily twist it into a figure 8. It's also possible to twist it even harder into a cloverleaf (some sunshades and camping tents do this). No matter how I twist, the results will always make harmonic standing "waves" that are an integer of the length of the hoop (i.e. I can't make 3 1/2 lobes).

Again, I know it's not a good analogy but it is sort of like that?

 

[malawi_glenn]

I don't get this. How are all the orbitals representations just of an single electron system? Why is it in my chemistry textbook and when people talk about orbitals it shows that in order to go on to say, the 3s sublevel, you have to fill two electrons for 1s shell, then the 6 electrons for the three p orbitals, etc?

Yes the additional electrons must be put somewhere else, but the shape of the places where they must go is almost exactly the same as for the single particle case.

There are two things that contributes to the atomic spectra and shapes of the orbitals.

1) Pauli principle

2) Electromagnetic force between electrons

the Pauli principle is the thing that says that only one electrons can be in the same state, that is why one must put the 3rd electron in another shell than the 1s shell. And so on.

But the shape of the shells are the same, if one neglects the electromagnetic force between electrons, which will disturb those shapes a little bit, but that is not important in the first treatment of the atom, this you will learn more about in higher courses in atomic physics and structure.

 

[Piscuit]

Yes the additional electrons must be put somewhere else, but the shape of the places where they must go is almost exactly the same as for the single particle case.

There are two things that contributes to the atomic spectra and shapes of the orbitals.

1) Pauli principle

2) Electromagnetic force between electrons

the Pauli principle is the thing that says that only one electrons can be in the same state, that is why one must put the 3rd electron in another shell than the 1s shell. And so on.

But the shape of the shells are the same, if one neglects the electromagnetic force between electrons, which will disturb those shapes a little bit, but that is not important in the first treatment of the atom, this you will learn more about in higher courses in atomic physics and structure.

Okay, I BELIEVE that I just read that the shapes of these electrons orbit will be altered slightly because of the electromagnetic forces between them. Is this because electrons are negative, and thusly they want to be farther away from each other and may be forced to change the orbital shape slightly? I used way too many commas in that sentence.

 

[malawi_glenn]

they will disturb the shapes of the hydrogentic levels since just because they are charged, the rest you wrote one does not have to bother about - how they do it, the fact that there will an interaction between the electrons due to their charge is enough to consider.

But the important thing for you to take with you is that the orbitals have this "funny" shape even in the single electron case - as a consequence of the math.

 

[alxm]

Okay, I BELIEVE that I just read that the shapes of these electrons orbit will be altered slightly because of the electromagnetic forces between them. Is this because electrons are negative, and thusly they want to be farther away from each other and may be forced to change the orbital shape slightly? I used way too many commas in that sentence.

Yes, electrons avoid each other, but since this effect is fairly uniformly distributed, it doesn't change the shape much. The more significant concern tends to be what it does to the energy.

Also, the Pauli principle, or rather the underlying anti-symmetry conditions of fermions, has a real impact on the orbitals (other than just the requirement that you only have two electrons per orbital). But this is also a fairly uniform effect.

But you have to watch out to think too much in terms of classical charged particles. Electrons in an atom don't actually repel each other as much as one might think (thanks to correlated motion). To give a concrete example:

The electronic energy of Hydrogen is -0.5 a.u. For He⁺ it's -2.0 a.u. How much energy an electron loses by joining He⁺? Well you might think that the charge of the existing electron would cancel half the charge of the nucleus, so it should be about the same as for Hydrogen ~0.5. But it's actually 0.903 a.u. - almost twice that! So electrons don't shield each other from the nucleus as much as you might think.

 

[feynmann]

The shapes just come from solving the Schrödinger equation, basic differential equation. You are chemist major, you don't need layman terms, since layman terms is never even close to be correct.

There is no "interaction", except for the interaction of the electron and the nucleus itself. So the shape of the hydrogen energy levels just comes from mathematics of the schrödinger equation.

In more electron atoms, electron-electron effects must be taken into account, but that is another story. The dumbell shapes, sub-shells in the hydrogen atom just comes from math.

First, try to understand how the hydrogen atom works, then you can move on to several electron atoms. Try Atkins book in physical chemistry, it explain things well and from first principles to chemists.

Yes, we know the shapes just come from solving the Schrödinger equation, basic differential equation. But what is it? It can't be electron itself, since electron can't have fractional charge.

 

[isabelle]

Yes, we know the shapes just come from solving the Schrödinger equation, basic differential equation. But what is it? It can't be electron itself, since electron can't have fractional charge.

The basic explanation is that the value of the wavefunction at a particular point is the probability density for finding the electron to be in a particular volume around that point. In other words, let $\psi(\vec{r})$ be the value of the wavefunction for an electron at the point in space $\vec{r}$, and let $\Delta V$ be a very small volume centered around the point $\vec{r}$, then $|\psi(\vec{r})|^2 \Delta V$ is the probability that the electron will be within the volume $\Delta V$. If you want to find the probability of being in a large volume (like the left half of space for example) then you could integrate $|\psi(\vec{r})|^2 * \Delta V$ over that space.

 

[feynmann]

The basic explanation is that the value of the wavefunction at a particular point is the probability density for finding the electron to be in a particular volume around that point. In other words, let $\psi(\vec{r})$ be the value of the wavefunction for an electron at the point in space $\vec{r}$, and let $\Delta V$ be a very small volume centered around the point $\vec{r}$, then $|\psi(\vec{r})|^2 \Delta V$ is the probability that the electron will be within the volume $\Delta V$. If you want to find the probability of being in a large volume (like the left half of space for example) then you could integrate $|\psi(\vec{r})|^2 * \Delta V$ over that space.

Isn't this the so-called Copenhagen interpretation? Why is the Copenhagen interpretation being taken for granted here?

 

[malawi_glenn]

Isn't this the so-called Copenhagen interpretation? Why is the Copenhagen interpretation being taken for granted here?

So then you already knew what wavefuntion is? Why did you then ask?

 

[alxm]

Isn't this the so-called Copenhagen interpretation? Why is the Copenhagen interpretation being taken for granted here?

No, it is not.

 

[conway]

The shape of the orbitals doesn't have all that much to do with quantum mechanics. They're more about how differential equations work in general.

Each level of energy describes a more involved way in which the orbital can be distorted. The simplest case, the s orbital, is spherically symmetric.

The next group of orbitals, the p orbitals, are just those shapes which can mix with the s orbital to distort it in the simplest possible way: a lateral displacement. A combination of the s orbital with a p orbital gives it a lateral shift in the x, y or z direction. You can mix and match to force the displacement towards any desired direction.

The next group of orbitals, the d orbitals, are simply the shapes which can mix with the s orbital to distort the spherical symmetry into an ellipsoid. By using different combinations of the 5 d orbitals, you can force the ellipsoid to have any desired orientation.

Chemists and physicists tend to draw the orbitals differently. The chemists like to draw the p orbitals as 3 similar dumbells lined up on the x, y, and z axis. The physicists, to create symmetry about the z axis, take the combinations x+iy and x-iy to create p orbitals with unit spin clockwise or counterclockwise. A similar but more complex game is played with the d orbitals.

To the OP: it's nice that you think you need to understand this stuff to get anywhere. But in fact people generally get pretty far without understanding it in the slightest.

 

[feynmann]

So then you already knew what wavefuntion is? Why did you then ask?

No, I do Not know what wavefunction really is and I think Bohr's version of quantum mechanics was deeply flawed. I'm still searching...

Here is Gell-Mann comment: 'Bohr brainwashed a whole generation of physicists into believing that the problem had been solved', lamented the Nobel Prize-winning physicist Murray Gell-Mann

 

[alxm]

To the OP: it's nice that you think you need to understand this stuff to get anywhere. But in fact people generally get pretty far without understanding it in the slightest.

Depends on your major! You do need to understand orbitals to get anywhere with chemistry these days. Which is why basic orbital theory/MO theory is first-year chemistry undergraduate stuff. They learn what the solutions look like long before they know what the actual Schrödinger equation looks like!

Chemistry and quantum physics are beautifully integrated. (Okay being a chemical physicist I'm very biased here) But it's really one of the most profound 'paradigm shifts' around. You can't really overstate the impact of QM on chemistry. Yet, people's, even some physicists, perceptions of chemistry is largely colored from high-school chemistry, which tends to be all 19th century 'pre-quantum' chemistry.