Talk:Atomic orbital

Re : d orbitals and above
I made the edit adding a warning to some of the links relating to orbitals - I was not logged in at the time so my user name appears as a number. Also there is a picture at the top of the article that also includes an incorrect d orbital etc - but I do not know how to edit it. Unfortunately I cannot find a link to an image of a set of 5 more correct d orbitals on the web - sorry. HappyVR 18:28, 11 February 2006 (UTC)

We need to pick a page to explain electron shells in general: Electron shell, Orbitals, Atomic orbital, Electron configuration are all valid candidates, and none of them explain the basic ideas yet

- I agree but I am not a chemist, I am more a general knowlegist. Also I am only 16 and this subject is not one i know well. - fonzy That's too bad because you're starting out at the intermediate level of fuzzy thinking.


 * Electron shell, Orbitals, Atomic orbital, Electron configuration and possibly periodic table block should all be merged into a single article, preferably by someone who knows what they are talking about. The Anome


 * I agree with this remark. This is in fact very difficult to write easy to read articles on all these topics separately.  They are so strongly linked with each other! Splitting all those entries provides the reader the not untrue impression of a big mess very difficult to edit even by specialists. --131.220.68.177 07:40, 27 July 2005 (UTC)


 * Perhaps they should be introduced in Basics of quantum mechanics. The problem is, the only good way to learn chemistry is to learn through a series of simpler outdated models then add details and new discoveries along the way. All the encyclopedia articles just jump into the currently most accepted models, which are very intimidating and complex.

I think there needs to be a different page for each part of this. Atomic orbitals, their own page, Electron configuration, their own page, and etc. It would help separate different parts of the Atomic structure. I'm doing research on one specific part: Atomic Orbitals, and when all of the information is meshed, it keeps it from being allowed to be expanded upon. I don't know, just putting in my two cents.--- JulieRaven

I agree with User:JulieRaven - the articles need to be kept separate - they are all quite long as it is - although there is quite a lot of duplication between them I think this to be expected as they all cover closely related topics. As for a place to start the explanation of electron shells - how about with the first experimental evidence for them - I think this was Balmer/Lyman lines in the sun's spectra - this leads to (amongst others) Bohr's model of the atom (first example of a theory involving quantisation?) and eventually to quantum theory. I think it is important to include the experimental data/physical phenomana in an explanation of a model that attempts to explain it - if these lines in the spectrum hadn't been found we wouldn't have this article in the first place...HappyVR 20:49, 12 February 2006 (UTC)

It is interesting to note how the mathematicians have grabbed the need to subdivide the 3 dimensional space around the nucleus of an atom and used it to develop an elaborate concept of spacial occupation and probability values. From the Bohr concept the first thing that was abandoned was the Physical principle that the Conical orbit concept required that the angular momentum related to the motion be a constant throughout the distance of the path. Instead we have a definition that the level of lost free energy is a constant and changes in accordance with calculated probability of location properties. Next we have a decision that it is a spherical volume of space that is to be subdivided up into certain discrete quantities related to a desired mathematical series such that the filling of the volume coincides with the end of the series, and when that results in overlapping of the spacial subdivisions we have exceptions to the rule. And at the end we run into so many permutation possibilities that cant be differentiated that the explanation lacks cohesion and intelligibility. And this is all based on the idea that an electrons are unattached particles moving in association with an atomic nucleus and having the property of being able to interact and exchange energy with it. I surely hope that this elaboration of mathematical concept leads to a better understanding of the physical and chemical activities of the atom. But there are alternate methods of conception of how the electron interacts with the atomic nucleus and I hope they are likewise explored.WFPM (talk) 19:20, 21 April 2010 (UTC)

Removing content
I've pulled the following content from the article, because I think that at least one of the following conditions applies: (a) It's duplicated at Molecular orbital; (b) It's false. --Smack 03:56, 14 Oct 2004 (UTC)

''In the quantum-chemical treatment of molecules, it is usually necessary to express the solutions as linear combinations of one-electron functions which are centered on the nuclei of the constituent atoms of the molecule. These functions are referred to as atomic orbitals even though they may not actually be solutions of the Schrödinger equation for those atoms taken in isolation. This method is referred to as the linear combination of atomic orbitals molecular orbital method (LCAO MO method).''

The orbitals used in the LCAO method are usually either exponentially decreasing from the atomic center (radial component of the form $$r=e^{-kx}$$, referred to as Slater-type orbitals) or decreasing as a Gaussian function from the atomic center (radial component of the form $$r=e^{-kx^2}$$, referred to as Gaussian orbitals), though other forms have been used.


 * More comment: agreed it doesn't belong in atomic orbitals (a) but it is accurate (not b!). I'm gearing up to revise the various items on orbitals and chemical bonding.... --Ian 10:37, 29 Jan 2005 (UTC)

Anon edit
There was an anonymous edit correction (possibly?). I am no expert in this topic, so please check if the minor edit was factual. -- AllyUnion (talk) 10:28, 10 Dec 2004 (UTC)


 * Worry not. I have this page watchlisted.  That was a genuine mistake on my part.  I'm no expert myself, but it'll take a smarter-than-average vandal to get one past me.  --Smack 01:45, 11 Dec 2004 (UTC)

Link suggestions
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orbitals, shape and energy
I've amplified the introduction to say what an orbital is, and is not, and linked to the electron configuration and periodic table pages. The former page duplicates quite a bit of this one, but why not? Having the same material in several places may not be efficient storage wise, but can help the reader. Similarly, I reckon its helpful to have large topics slpit into smaller pages, so wouldn't want to consolidate everything together. I've also clarified the description of the p and d atomic orbitals, and their relation to the orbital's energy - it isn't the shape that determines it, but the detail of the probability density in the radial direction.--Ian 10:02, 30 Jan 2005 (UTC)

Possible links for demonstrating the quantum atom
Below is a link to a video produced by astronauts at the international space station that shows wave patterns in a very large sphere of water (something not attainable on Earth). It reminded me of the various orbitals within an atom. It would be particularly useful for demonstrating the electron wave patterns of hydrogen.

http://www.youtube.com/watch?v=zaHLwla2WiI

Notice that at 0 & 180 deg. to the initial energy pulse, the oscillations resembled a p-orbital along that axis, but the globule retains the basic (lower energy) spherical shape throughout the course of its vibration, and eventually comes to rest back at that ground state (you can almost pick out the other two p-orbitals as well, but they are not well resolved. Also, the multitude of concentric rings running up and down the axis of the initial pulse seem to me to resemble one of the possible d-orbitals (dSubscript zSuperscript 2) and f-orbitals (I think it is the fSubscript x)Superscript 3Subscript - 3/5xrSuperscript 2). These patterns are all simultaneously present, oscillating from one orbital type to another and back again, or even appearing and vanishing again. I found it very interesting to see the one lobe of the p-orbital mimetic appearing in one direction, and then vanishing again as the lobe in the opposite direction appears.

These vibrational patterns appear to me to be analogous to an atom absorbing a photon, kicking an electron to a higher orbital, followed by the atom relaxing back to its ground state as it remits the photon and that orbital shape dissapears (although the possibility for the orbital to reappear is always present when the next photon interacts with it).

It would be interesting to see the result of using a more energentic pulse to initiate the globual's vibration. I suspect that at higher energy, a smaller water droplet would be ejected from the larger globual, similar to what is observed with the photoelectric effect, where only photons with sufficient energy can kick electrons from a metal plate to complete a circuit.

Many students find the entire subject to be confusing in the extreme, and seem to fixate on the idea that these orbitals are solid, tangible objects hiding under the surface of the atom like steel girders in a building). Perhaps this video could help (assuming no one here finds serious fault with my interpretation).

There are some other very interesting videos on cymatics (the study of wave behavior) given below:

1. http://www.youtube.com/watch?v=8ik6RgdoIMw (The sequence from ~ 0:43 sec to 0:50 sec reminds me of benzene, with its alternating pi-bonds, and equivalent resonance structures.)

2. http://video.google.com/videoplay?docid=2795869048702157810&q=cymatics&hl=en

3. http://video.google.com/videoplay?docid=7253148167375317006&q=cymatics&hl=en

Since I am not certain that this wiki page is the best location for any this material, I felt that placing it here for further discussion would be best.

Thanks, DF 72.48.34.162 04:21, 4 December 2006 (UTC)

Although mathematically, the s orbitals have a maximum probability at the origin, is it possible that in reality, this is not the case if you factor in the spin angular momentum of the electron? Regards Geoff e j (talk) 05:57, 28 February 2011 (UTC) 220.101.3.150 (talk) 01:59, 9 March 2011 (UTC)

Relation to the Heisenberg-Bohr-Sommerfeld Picture and Hydrogen SO(4)
The discussion regarding the shapes of orbitals is not complete until the Heisenberg picture view of the Hydrogen atom (and more generally: the Kepler problem) is included. The two sets of parameters that determine an orbit's shape and size in the classical Kepler problem (angular momentum and eccentricity/direction of closest approach) have magnitudes that assume perfectly well-defined eigenvalues in the quantum version of the Kepler problem for each orbital state; hence the shape of the orbital is, itself, well-defined. The uncertaintly actually applies to the orientation of the orbit, not its shape or size.

In the quantized problem the two vectors generate a constrained version of SO(4) (hence the term "Hydrogen SO(4)"); or E(3) for parabolic orbits, or SO(3,1) for hyperbolic orbits. Both the eccentricity e and the magnitude of the angular momentum |L| assume eigenvalues in the orbital states. The orbital shapes resulting from the eigenvalues both supersede and refine those that had historically been associated with the Bohr-Sommerfeld orbits; namely that the major axis (a) is proportional to the square of the energy number n; the semilatus rectum is proportional to 1 + l(l + 1) (in contrast to Bohr-Sommerfeld's l squared), where l is the angular momentum quantum number. The terms in the correction 1 + l to the Bohr-Sommerfeld figure arise, respectively, from the uncertainty in the direction of the closest approach, and in the components of the angular momentum.

More generally, there should be a discussion concerning the Heisenberg picture and Hydrogen SO(4) -- either directly here, or provided by a link. -- Mark, 23 October 2006 ' —The preceding unsigned comment was added by 129.89.32.142 (talk • contribs). '
 * Hi Mark, the way you describe it here is probably too much for a high-school student (which probably use the wikipedia more to increase their knowledge than people with a degree in quantum mechanics), but I do agree, a good and complete picture is necessary. Could you write either a paragraph about this subject, or indeed maybe a new articles, e.g. something like atomic orbital shape and molecular orbital shape?  I will be glad to help you with polishing, comments, but I am not really good in quantum mechanics, but hey, maybe I can learn something from it!
 * Could you please sign}} by typing four tildes at the end of your contribution (i.e. ~ ). This converts to your signature automatically upon saving.  Thanks!  --[[User:Beetstra|Dirk Beetstra T  C 20:19, 23 October 2006 (UTC)

Edits by User:147.231
I'm distressed by these modifications. I don't have the expertise to corroborate or dispute their factual correctness, but one thing is clear: they're inaccessible to anyone without extensive training in mathematics and quantum mechanics. Wikipedia is not a physics reference text; it's supposed to be written so as to be readable by as many people as possible. The corrections should be written into a special section, and the remainder of the article be left as it was. --Smack (talk) 02:31, 15 July 2005 (UTC)


 * I agree that we should go back to the simpler first paragraph. Talking about slater determinants right up front is way out of line.  People who read this may not even know what a wavefunction is.  But the rest of the changes look good, and the info in the new first paragraph can be moved down to another section.  Pfalstad 03:28, 15 July 2005 (UTC)


 * The problem with the older first paragraph was that is was simply false. Moreover an atomic orbital is a wave function.  Thus it is quite difficult to discuss about an atomic orbital without referring to the concept of wavefunction. Of course one can make it more pedagogic but pay attention not to write anything false. --131.220.68.177 12:01, 25 July 2005 (UTC)

Hydrogen-like atoms
I believe the section Hydrogen-like atoms should be moved to the article Hydrogen atom or to a new article which could be "derivation of the hydrogen atom formulae" which solution would be preferred since the article Hydrogen atom is very good as it stands and should not be expanded anymore.

Atomic orbitals are quite different from the eigenfunctions of the hydrogen atom. The hydrogen atoms are one electron atoms only and this is a big conceptual difference. Of course one can use the Hydrogen atom eigenfunctions as atomic orbitals but one does not have to. In practice one often does not! --131.220.68.177 08:23, 26 July 2005 (UTC)


 * Two important points:
 * Hydrogen wavefunctions are indeed different from the wavefunctions seen in more-complex atoms, but they're an important special case, and should IMHO be retained here.
 * Hydrogen wavefunctions are not unique to hydrogen. Ions such as $$He^+$$ or $$Li^{+2}$$ may be oddities that occur primarily in physics textbooks, but they do exist. --Smack (talk) 02:15, 27 July 2005 (UTC)


 * What does IMHO mean?


 * Of course they exist but they are usually discussed together with the hydrogen atom wavefunctions. They belong to some remark in the hydrogen atom article.


 * Could someone tell be why the proof of the formulae of the hydrogen atom are found in this article and not in hydrogen atom.--131.220.68.177 07:33, 27 July 2005 (UTC)


 * Sorry for the neologism. See Internet slang.
 * The partial derivation (it's neither complete nor a proof) is found here because, as I said above, these things are not unique to hydrogen. They're also found (in a somewhat more general form) in one-electron ions such as $$He^+$$.  Though these ions are exceeedingly rare here on earth, they probably occur in larger quantities in space, and at any rate, the natural occurrence of a substance is of little theoretical consequence. --Smack (talk) 18:14, 27 July 2005 (UTC)


 * OK I found that by typing IMHO on wikipedia.


 * The reason why this result can be found in almost all textbooks (and also in the hydrogen atom article by the way) is because the atomic orbitals used is practice are often Slater-type orbitals but with some Z different from the actual charge of the nuclei.


 * If it is nor complete nor a proof why must it be here? I definitively think these details belong somewhere else. --131.220.44.10 08:47, 28 July 2005 (UTC)


 * So, if hydrogen-like orbitals (or Slater-type, or one-electron, or whatever you want to call them) are used in LCAO, isn't that by itself justification enough to talk about them in a general article?
 * Why do you insist on calling this derivation a 'proof'? A proof always starts with the thing to be proven.  Even if it is not used explicitly as a starting point in a series of transformations, the end result is always given beforehand.
 * More to the point, why must all derivations given on the wiki be complete? You still haven't given me a reason, but I can give you a list of reasons to the contrary:
 * Most people don't care about all of the messy mathematical details, and shouldn't be forced to scroll through them.
 * Many people who do care about the mathematical details probably already own at least one physics reference text that outlines the derivation in sufficient detail.
 * But, you say to #1, we can put the full derivation on a separate page. To that I reply that people like you and I have more valuable things to do with our wiki-time than to code pages and pages of TeX. --Smack (talk) 00:58, 30 July 2005 (UTC)

Well I agree with you we don't need it so suppress it because what this paragraph is all about is explaining people a bright example of separation of variables which is summarized in the hydrogen atom article and can be found in any low level text book! Do you really think someone interested in atomic orbitals wants to know something about how to derive their mathematical formula in the very special case of hydrogen-like atom - information which can be found easily on wikipedia anyway --81.209.204.11 08:16, 7 August 2005 (UTC)


 * Sorry, what do you mean by 'it' and by 'this paragraph'?
 * Honestly, I don't care much more than a rat's thighbone about whether or not a particular bit of information is contained somewhere on the wiki. I often like to ask myself: should this information be contained here; does it really belong in that other place where someone has decided to put it; and how likely is it that someone will come here looking for it and not be able to find it?  In my opinion, there are two good places to put that derivation: here, or in a separate article called Hydrogen-like orbital or something to that effect. --Smack (talk) 15:27, 7 August 2005 (UTC)


 * Sorry I agree my English was not that bright! It and this paragraph both mean the Hydrogen-like atoms paragraph.  Thus I see we begin to agree : we could put this info in a new article named Hydrogen-like atom -- hydrogen-like orbital is equivalent to [[Slater-type

orbital]]. I would suggest you to merge this with hydrogen atom but this is just my opinion.--81.209.204.4 15:39, 7 August 2005 (UTC)

sorry i have a ques. and am not finding it`s answer.so am posting it here.In a p-subshell we have 2 electrons in each orbital which are accomodated by 2 lobes in each orbital and between 2 lobes there is a zero probability region called node.the ques. that arises in my mind is why there is zero probability if electrons can oscillate between two lobes they have to pass through node then there must be some probability of finding electrons there.Raje80887 (talk) 10:04, 5 July 2008 (UTC)

the giguere periodic table
The giguere periodic table is a very simple table. It classifys its elements according to what orbital the electrons end in. Oh and I do'nt appreciate what you talked about about the p or s block it just not right

Circular reference - bad for newbs
The top of this article states "A less formal description of the electrons in atoms can be found at Electron configuration.", wehereas the Electron configuration article states "The discussion below presumes knowledge of material contained at Atomic orbital."!

Basically, both articles are stating that if you are new to the subject you should read the other one first. Can this please be rectified by someone who understands the subject. I expect the best solution is to rewrite the Electron Configuration article to work as a standalone introduction for newbies, and for this article to be a more technical treatment of the subject, but other solutions are possible. --HappyDog 01:29, 21 December 2005 (UTC)


 * I'd like to see the two articles merged into this one. - Aug-11-06 —The preceding unsigned comment was added by 12.33.19.11 (talk • contribs).


 * Nope, no merge, see discussion on Electronic configuration. --Dirk Beetstra T  C 16:08, 11 August 2006 (UTC)

d orbitals and above
I'm still concerned that the pictures associated with this text as well as some of the links (appear to) show incorrect (ie non degenerate) d orbitals - I can't see the point of including an image for visualisation purposes if the image gives the wrong impression. I refer to [electon_orbitals.png] and also the external link "the orbitron" and "David Manthey's Orbital Viewer renders orbitals with n ≤ 30" also gives incorrect d orbitals - I can't check the other links at the moment. Previously I added a warning to the external links but that was removed.HappyVR 19:37, 17 June 2006 (UTC)

Orbital Definition
So, is an orbital a "region in which an electron may be found", or is is a "mathematical description"? There are geometric and a methematical definitions. This needs to be clearer or we are looking at a confusing start to an inherently difficult topic. Dr Thermo 19:44, 12 June 2007 (UTC)
 * How about "In physics and chemistry, an atomic orbital is a mathematical description of the region in which an electron may be found around a single atom." (emph mine for diff)? Baccyak4H (Yak!) 20:10, 12 June 2007 (UTC)
 * Yeah that sounds way better. Amit 21:15, 25 July 2007 (UTC)

History of Orbital Definition
While presenting at the 1979 Sanibel Symposium, I had the occasion to ask Per-Olov Löwdin for his definition of an orbital. Without hesitation, he told me that it was first used by Robert S. Mulliken "in 1925 as the English translation of Schroedinger's use of the German word, 'Eigenfunktion'." Mulliken had been working in Germany in 1925 with many of the founders of Quantum Mechanics and particularly in 1927 with Friedrich Hund on the beginnings of molecular orbital theory. Mulliken presented his work at that time in Physical Review. Löwdin told me to look there for the first use of a hydrogen "orbital" in English. He also agreed that a suitable definition of an orbital would be "a mathematical function that describes the wave-like behavior of an electron". I will edit the definition to reflect this usage of the term as a mathematical function, but also connect it to the more general description of an orbital as a "region of space" that can be calculated from the function. The advent of graphing calculators has enabled even high school students to grasp the definition of an orbital as a mathematical function. I invite educators to have their students graph the "simplified' 1s orbital, y = exp(-abs(Z.x)), where Z=1,2,3(Atomic Number). This simple exercise can display the effects of electronegativity and the fact that atoms and isoelectronic ions "decrease" their size in going to the right in a row in the periodic chart. The bonding and antibonding sigma MO can also be displayed by y = exp(-abs(Zx+1) + or - exp(-abs(Zx-1) . After graphing functions, students can grasp the idea of "combining" functions (atomic orbitals) to get other functions (molecular orbitals), and what is meant by the + and - signs or the red/blue, white/gray colors of the orbitals. Laburke21:13, 25 October 2007 (UTC)

Why integer solutions?
The third paragraph of the "Connection to uncertainty relation" section doesn't seem to tell me why an electron must be in a specific set of states. It basically says there are several laws that apply and more-or-less leaves it at that. (The Bohr model page similarly fails to explain why there are discrete quanta.) How is a reader to understand why an electron can't be at n=1.5? I think this article needs to explain why the wave state collapses to the states at n=1, 2, 3, .... Thanks.&mdash;RJH (talk) 16:25, 31 January 2008 (UTC)

Apparent contradiction in discussion of limitations in an electron's location
The "Connection to uncertainty relation" section states "...this does not mean that the electron could be anywhere in the universe" and concludes with the sentence "An electron's location... stops at the nucleus and before the next n-sphere orbital begins." The first sentence of the "Shapes of orbitals" section states "... a given electron, regardless of which orbital it occupies, can at any moment be found at any distance from the nucleus and in any direction due to the uncertainty principle." This apparently contradictory text will surely confuse an unknowledgeable reader. It certainly confuses me! The text of both sections needs to be amended by a subject matter expert to clear up this seeming contradiction. Ross Fraser (talk) 17:56, 18 March 2008 (UTC)
 * I've tried to do some of this. I eliminated the discussion of phase-space, which while correct is confusing because phase space is not a place! It's a mathematical graph of position and momentum, and although it has voxels of minimal size, these are voxels in a 6-d graph (3 dimensions of momentum, three of space), NOT volume cells in 3-space. So mentioning this as a way to limit particles in space (3-space) is bound to confuse the reader. The basic reason Heisenberg implies a limitation to how closely you may localize bound particles in space, is that particles are waves, and the only way you can localize a wave is to localize a wave packet. Such packets need spreads in wavelength (which means momentum, in quantum mechanics). For low energy/momentum packets, the localization of the particle and wave-packet becomes bad, and thus the packet is large, and that's why the electron can't be found at any location smaller than a certain range of distances from the nucleus. But this is a property of all kinds of waves, as Bohr pointed out. The Heisenberg relationship just gives the scale of localization, once particles are connected with wave packets. Born supplies the last connection, with how you decide the connection of particle and wave: the complex congugate of the wave in any volume gives the probability density of finding the particle within that volume. S  B Harris 02:02, 3 May 2008 (UTC)

Electron orbital images
I have created a new set of orbital images at much higher resolution than the pre-existing: please review, comment & include if appropriate. I am happy to fulfil requests for further orbitals to be generated (within reason). Dhatfield (talk) 15:24, 5 October 2008 (UTC)


 * They look very nice!
 * Could you make them have a consistent scaling scheme for different n? The existing ones are all the same size, which makes the nodes and shapes clearly visible for small n (vs your 1s--just a few pixels in the table), but isn't really "correct" and obscures their relative radial extents and lobe positions. Some readers have asked about that. The new ones have s scaled (1s is smaller sphere than 2s, etc) but the other l appear all the same size (6p the same distance from nucleus (or even smaller?) than 2p). Either way is could be useful (constant image size vs scaled for n) but should pick one of those two for all.
 * Two graphical thoughts:
 * Consider switching to clear background. Doing so would increase reusability of the images.
 * Reduce whitespace/margins around the actual orbitals. The table is by its nature very wide, would be good to avoid more side-scrolling than necessary.
 * Could you also generate an XYZ coordinate-axes image for the orientation of these renderings? That would make it easier to correlate the m numbering with the more common "px" naming system. Wouldn't be good to have it on all the orbitals themselves, but useful on its own as part of the explanation of the table.
 * More general throught: should this table, which is over 100K just for the thumb images and quite wide, be moved to a separate Gallery of atomic orbitals page? Then we could even have two sets of images: one that are "same visual size", another that are "constant scale". DMacks (talk) 16:04, 5 October 2008 (UTC)
 * I'm not familiar with the "px" naming system. I studied orbitals in 1994 - I'm just a renderer :)
 * Axes embedded in the images are not possible but I'll try and generate an example axis - all orbitals are rendered from a consistent viewpoint and it's pretty easy to figure out what it is from the d m=0 orbital.
 * One of the problems with 'scale' when discussing orbitals is that the electron probability density that is selected to represent the 'surface' that is rendered is arbitrarily chosen. Hence, it is commonly chosen to represent the maximum amount of detail regarding the structure of the orbital. However, the selection of this cutoff level affects both the 'size' of the orbital and the visibility of certain structures (eg. the 'rings' in the d m=o orbitals). Do you have a recommendation on this? Dhatfield (talk)

I would actually vote for a constant size depiction, since they size of any given orbital is semi-arbitrary anyway, and can be scaled up or down depending on the Z (charge) of the nucleus. That said, I'm curious as to why the f-orbital images here are so small-- isn't the fact that they must be at least at the 4 primary quantum number more or less mean that they're going to be larger than the 1,2, and 3 orbitals of all types? Again, though, I don't think understanding would be severely hurt to show all orbitals as large as the available box space available to contain them allows. This shows detail of smaller inner nodes (which is not seen in books too often-- I wonder how many chem students have thought about the extra nodes in 3p and 4p orbitals? Another issue I bring up, is the matter than these are pictures of the Ψ function, and not the ||Ψ||2, and thus they are a bit more bulbous than the Ψ2 solutions we often see which are more physical because they show the actual volumetric density of the electron. Have you considered using a set of these, instead? Yes, we lose the alternating colors of the lobes between nodes (since this is phase and is squared out), but on the other hand, do we really need it? The lobes are separated by nodes anyway, so it's clear where they are. You could even put in "false color" representations of adjoining lobes (as now), with the explanation that the color now doesn't represent anything at all, but is put in for contrast. What do you think? Anyway, nice job! S B Harris 00:44, 15 December 2008 (UTC)
 * Or you know, have the shape follow the squared wavefunction, keep the colors, then explain that they relate to the phase/sign of the unsquared wavefunction.Headbomb {{{sup|ταλκ}}κοντριβς – WP Physics} 19:34, 16 January 2009 (UTC);

Notation Errors?
I agree, these are great pictures! I especially like the fact that the z-axis is consistent, which does not appear to be the case in the present diagram.

And for what it's worth, my two cents of input: I also agree with the note above that the table is too wide. The pictures works fine on my Mac at home, but on my PC in the lab where I work I can't see the final column. I'm not sure how I feel about the proposal to square the functions. They're certainly prettier with the two colors, but I guess SBHarris is right above about that being a little bit non-physical. However, I don't think squaring $$\psi$$ can change the shape of the orbital. If $$|\psi|$$ is constant on a given contour surface, then $$|\psi|^2$$ will also be constant, and thus contour lines for the two functions will have the same shape (see Levine, Ira "Quantum Chemistry" p. 151). Dhatfield, do you know the probability of finding an electron within the contours you rendered? That information would be nice.

My biggest concern is with the notation used to identify the orbitals, both in the current article's picture and in the proposed revision. There seem to be two systems of orbitals being conflated here. If you identify orbitals by angular momentum number $$l$$ and magnetic quantum number $$m$$, you are, at least from a physicist's point of view, choosing orbitals that are eigenstates of the projection of momentum along the z axis. These are NOT the same as the "real hydrogenlike wave functions" that chemists use. I come from a physics background, so maybe I don't fully understand the chemists' notation, but something should be mentioned to clarify this difference. I recommend using a notation that specifies absolute value of $$m$$, such as in

http://www.sccj.net/publications/JCCJ/v5n3/a81/text.html

Alternatively, real hydrogenlike orbitals are frequently identified by their relation to Cartesian axes ($$p_z, p_x, p_y, d_{z^2}, d_{yz}, d_{xz}, d_{x^2-y^2}, d_{xy},$$ etc.) Is this the $$p^x$$ system mentioned above? Csmallw (talk) 18:46, 16 January 2009 (UTC)


 * If you really want your mind blown, there are some really good links to applets in the article, such as http://www.falstad.com/qmatom/ which has phase animation, and also http://www.orbitals.com/orb/orbtable.htm which has about every one of 30 orbital combos, including many not seen above. Some of the other comments: I think somebody earlier commented that that nodes and antinodes in the various lobes are not due to the radial part of the wave equation, but that's wrong-- of course they are. They come from the Laguerre polynomial zeros, and are the most prominant feature of the radial solutions at higher energies; these are just like higher vibrational modes in a pipe with one end closed and one open (and of course distorted here from the 1/r potential, too). But otherwise, the same idea. Every radial function gets more nodes and antinodes as the energy goes up (n= higher principle quantum numbers), simply due to the shorter wavelengths of the faster particle "confined" in the potential well. As to the angular $$|\psi|^2$$ looking no different from the $$|\psi|$$'s I can't see how that can be. The simplest ones I can easily check by "hand" are the angular part functions for 2p along the Z axis, which depend only on cos(theta) (theta = the angle from the z axis). You can see that the simple psi function is a sort of double squashed pumpkin, but if it's not squared it goes from z=1 at theta = 0, and falling off to 1/sqrt(2)=.707 at 45 degress off the z axis, then back to zero as you get to the x-y plane. But if you square it, it gets to be a near double sphere with z= 0.5 at 45 degrees, then back to z=0 on the x-y plane. Clearly, a different shape. And add in the fact that with psi^2 you're really doing an integral of [psi^2]dV from the origin to your point of interest, if you're looking at surfaces denoting "90% of the electron probability is inside THIS". That integration should give you again something slightly different than the surface of psi^2 which is the probability density itself, which you can really only probe by taking slices of it and looking at the shading or something, which is not the same thing as looking at the summed-up function which can be respresented by looking at the other thing. So I suppose we really three different spacially dependent function to consider, here. S  B Harris 08:18, 17 January 2009 (UTC)


 * "And add in the fact that with psi^2..." Huh? You've lost me after this point. The note and reference to Levine above applies if the method of rendering orbitals is done with contours (Specifically, p. 151 says: "if $$|\psi|^2$$ is constant on a given surface, $$|\psi|$$ is also constant on that surface; the contour surfaces for $$|\psi|^2$$ and $$|\psi|$$ are identical.") I see your greater point, though. You can depict an orbital's phase, or you can assign an electron's probability density within a contour, but it doesn't really make sense to try doing both at once. So I'll agree. It would be better to plot $$|\psi|^2$$ if possible.
 * I hadn't looked at the external links until just now. Trippy :) Csmallw (talk) 23:14, 19 January 2009 (UTC)
 * I was talking about trying to sum up total probability of finding an electron "inside" a given surface (in the volume bounded by that surface), which is an integral (from zero to the surface) of psi^2dV, not the psi^2 at any given surface. So it's not the same function. But even the comment about the contours of psi and psi^2 are wrong, so far as I can see. Cos(theta) = z for a polar coordinate system where theta is the angle from the z axis, is NOT the same shaped function as Cos^2(theta) = z. So I have no idea what the author is talking about. The psi function of theta is not shaped like the psi^2 function on theta in this case. That's what *I* mean. I don't know what HE means. S  B Harris 00:46, 20 January 2009 (UTC)
 * I think that you are confusing a contour surface with a graph of solid angle. Csmallw (talk) 20:23, 20 January 2009 (UTC)
 * You can't graph a solid angle per se. I do not know how you would even construct a function of a solid angle, exept to specify surfaces of constant radius as the output of your "function." But anyway, atomic orbital functions are not functions of solid-angles, but functions of 2 different very normal angles, and a radius (in polar coordinates). You can graph the function of a standard angle, and a particular contour surface is merely the surface defined by a particual value of that function, with all 3 variables allowed to vary in every way that gives that function value. The (constant) contour of a function of 3 variables will be a 3-D surface. That's what we're plotting here. Something that shows you many contours would be nested surfaces, and to get the full function of 3 dimensions, you need a 4th graphical device, like a color, or a see-through proxy like density where varies from point to point as a marker for higher value of the function, and you can sort of sense that in a 3-D graph, as a thicker "fog", whose thickness varies from point to point in space. Anyway, the contour of the square of one of these functions just is not going to look like the non-square. Graph it as a curve in 2-D and see-- don't take my word for it. Graph the contour for the 2-D polar equation R = cos(theta) where theta is the angle from z. You get the mushroom with no stem (rotate it around the z axis to extend to 3-D polar). Now the contour requires that F be held constant, so you need some other radial function of R which drops off with distance from the origin, so you have a constant surface F = contant = R(r)*F(theta). Otherwise the surface of constant psi dosen't extend out in a lobe farther in one direction than another. It has to be some combination of radial and angular functions that gives you a constant surface which extends in a lobe in any direction. Anyway, Cos^2(z) gives you a function which is a near circle in 2-D and a near hollow sphere (tangent to the x-y plane) in 3-D. S  B Harris 01:35, 21 January 2009 (UTC)
 * Oh, boy. I'll have a look, but maybe we ought to just agree to disagree for the time being. Csmallw (talk) 08:59, 21 January 2009 (UTC)

It is important to remind that $m$ of an orbital is not necessarily certain. Surely s and pz have $m = 0$, but $m$ is uncertain for px and py, let alone more intricate d and f orbitals in the real form. Surely the “doughnut” p orbitals for $m = ±1$ are well-defined, but the $m$ basis is important for atomic physics and is less important for chemistry where “dumbbell” p orbitals are predominantly considered. Incnis Mrsi (talk) 18:16, 10 August 2019 (UTC)

Revision
I modified the table and went ahead and put it into the article (better off there than here). I also made an attempt to rectify issues of notation (physics vs. chem). The rest of the article could use an overhaul with respect to this issue, though. Csmallw (talk) 09:23, 21 January 2009 (UTC)
 * Thanks for including the table in the article and improving the notation. A number of important questions have been raised on this talk page regarding these images and I am flattered by the compliments, but I'm really not knowledgeable enough to take this project further. The excellent little application used to generate the images can be found at http://www.orbitals.com/orb/ov.htm (as noted in each image's description) and I would ask that someone more knowledgeable than myself trawl through the details of the configuration of these orbitals to address the community's concerns & preferences. Dhatfield (talk) 18:53, 22 February 2009 (UTC)

P = Principal?
I'm relatively new to quantum mechanics here so I might be wrong, but the article says p stands for principal, shouldn't it be principle? Psycholian (talk) 23:06, 18 February 2009 (UTC)
 * No, it is "principal" as in "first in rank, authority, importance" to quote my dictionary. p, s, d and f come from "sharp", "principal", "diffuse" and "fundamental" which were names given to various series of lines in the alkali metal spectra. -- Bduke   (Discussion)  23:44, 18 February 2009 (UTC)
 * Ah I see, thank you, just wondering. Psycholian (talk) 08:26, 22 February 2009 (UTC)
 * Remember, the principal is not actually your pal (a palpable lie), which is one more reason it's bad to think of school positions as the main use of the spelling "principal." Rather the principal is named because he or she is the principal teacher. "Principle" is used for maxims or laws, but "principal" is actually a quite common word in other contexts, including the spectrographic one here. S  B Harris 08:04, 9 October 2010 (UTC)

Didactics: Orbital shapes and the atoms that have them
SUMMARY: can we add the chemical symbols of the atoms that have each shape in the orbital shape table?

A didactic suggestion. The text states The orbital table shows all orbital configurations for the real hydrogen-like wave functions up to 7s, and therefore covers the simple electronic configuration for all elements in the periodic table up to radium. So I take it each cell is a given element. If the chemical symbol for that element were added to the table, then I for one would find this chart a key bridge between the world of physics (orbitals) and chemistry (bonds; the shape of molecules). Thanks for all this community has slogged through on the way to a "B" rating. Perhaps this suggestion is a sign that the page has matured -- both correct and accessible. --jerry Jerry-va (talk) 13:13, 31 May 2009 (UTC)

Merge from electron cloud
The article on electron cloud was never good, and the only thing saving it was one cite where Feynman uses the phrase, and the fact that an atoms' electron cloud isn't really the same as its atomic orbitals or even the sum of them. But the last is sort of getting at the right idea-- there is a sense in which for complex atoms the total electron cloud is ALMOST a sum of 1-electron orbitals for hydrogenic atoms out of the Schroedinger equation: if that weren't so, we wouldn't have the periodicity of the periodic table, which can be seen in the Schrodinger hydrogenic solution. That fact is a remarkable thing, when you think about it! So, in order to get rid of the awful electron cloud Wiki (bring it up if you don't believe me) which ahs been sitting there with no improvement for a year with a merge tag, I've re-written the LEAD of this article, to make it clear that there is an electron cloud, and that it is "sort of" composed of atomic orbitals, but not exactly. It's an appoximation. And that about does it for the electron cloud--- and now it can be done-away-with. So I was bold and redirected it to here. I left the TALK page. You can look at the history for what it was like. Feel free to rewrite this LEAD, but as you do, remember how it got to where it is, and what we're trying to do. Our best "picture" of the "electron cloud" of a complex atom, bad though it is, is a bunch of filled-in atomic orbitals ala Shroedinger, for 1-electron atoms. So it's in THIS article, if anywhere, that we have the chance to give a picture of what all these things look like, and how they go together. S B Harris 06:36, 16 June 2009 (UTC)
 * It's hard for me to be impartial about the electron cloud article, as I re-wrote a fair bit of it, and liked it, but apparently that left me in the minority (maybe of one or two). What is interesting about the electron cloud is really the unpredictability of electron. For me it is not the physics implications but instead those for philosophy and free will. To have free will, the future cannot be pre-ordained. I later found the schroedinger himself (What_Is_Life%3F) had thought about this but, unlike myself, did not believe the randomness was related to will. In any case, I believe this is the only area where physics has made headway on the "I think therefore I am" topic, ie the future cannot be calculated from the present. A significant advance on this in the realm of physics would dwarf all physics achievements to date.

I was glad to see that Feynman found simple intuitive ways of understanding things, including negotiating the math. His simplicity does not survive on a wiki though, as there is an academic force that has not embraced the value of making things simple, and it is easier to complicate topics than simplify them (Feynman spent about 10hrs to prepare each 1hr lecture). For example, look at the wiki article on Feynman diagrams, which have a fairly simple explanation in his Book QED (quantum electro dynamics). Hilbert has a similar stature in math. I was hoping that interested folks, especially the upcoming generation would achieve greater progress by knowing what is important, and the value of simple language and notation to achieve more with the limited time we all have. The term electron cloud was really a visual depiction for a first brush with the concepts needed to explain the double slit experiment, and understand shroedingers model of the atom. JeffTowers (talk) 17:02, 3 September 2010 (UTC)

New discussion on atmosphere analogy
I am concerned by this wording: "Despite the obvious analogy to planets revolving around the Sun, electrons cannot be described as solid particles. In addition, atomic orbitals do not closely resemble a planet's elliptical path in ordinary atoms. A more accurate analogy might be that of a large and often oddly-shaped "atmosphere" (the electron), distributed around a relatively tiny planet (the atomic nucleus)." Actually, that's a terrible analogy because it leads to wrong physics. Firstly, electrons ARE point charges, as far as anyone can tell, and so a solid particle is not a bad description. If you treat an electron as an extended charge distribution given by the ground state of the hydrogen atom and apply Gauss's Law you will see that the nucleus is almost entirely shielded; if this were the case hydrogen would not form molecules. I guess I would say that there is a cloud which tells you where you are likely to find the point-particle electron, and that's just fundmentally not like a satellite in orbit or a smeared out cloud of charge. Happy to discuss.Shrikeangel (talk) 16:01, 9 August 2011 (UTC)
 * I agree that the atmosphere analogy is incorrect and should be removed. However a solid particle is not a very good description either, and should probably not be mentioned either. Instead we should just use the usual description in physics books, which is a point particle whose probability of being found at each point is given by the wave function.Dirac66 (talk) 23:20, 10 August 2011 (UTC)

Atomic orbital material
message to User talk:71.217.73.148

I see that you were adding a lot of informtion to atomic orbital that was simply mathematized versions of what is already discussed in the main article. I've moved it to an old version of spin-orbital, see here. But the math probably isn't appropriate here, but in the Schroedinger equation wiki. So don't be disheartened. Use the talk page.

To other editors-- this editor was obviously adding material in good faith, but has been bitten and treated a bit shabbily, unless I've missed something. What he was doing certain wasn't vandalism.

S B Harris 01:46, 14 November 2010 (UTC)

Merger proposal
There is a lengthy section in this article which is about the development of atomic models (ancient Greeks --> Dalton --> Rutherford --> Bohr), and not about the modern atomic orbital model at all. It would be better to move this into the Atomic theory article, which discusses the historical development of atomic theory, and keep this article on subject. Djr32 (talk) 18:17, 29 December 2008 (UTC)
 * Done. I think there's still too much about the Bohr model in this article, might have another go at focusing the article on topic at some stage in the future.  Djr32 (talk) 21:45, 13 January 2009 (UTC)

Merger proposal: Atomic orbital and Atomic orbital model
Does anyone have any views on merging the articles Atomic orbital and Atomic orbital model? I don't think that either of them work alone, and a combined article would seem more coherent. Djr32 (talk) 22:21, 30 December 2010 (UTC)


 * The solution is deletion, as there's not much to combine. These two articles have been evoloving in parallel, and the atomic orbital article (the OTHER one) is by far the most complete and detailed, and I don't think this one atomic orbital model has anything at all that the much longer one doesn't. This one should (again) just be deleted, much as it pains me to suggest it to the people who've put in the separate work. If they can find a few bits that aren't mentioned in the other, they can put them in (so it's a "merge" in the sense that a bit survives from the much smaller article). But after a quick read of this article, I can't find anything that qualifies, save perhaps for some new references. S  B Harris 23:05, 30 December 2010 (UTC)


 * I agree with Sbharris. We don't need two articles on this topic, Atomic orbital is much more complete (42K vs. 9K), and also Atomic orbital is a better title. (Yes, atomic orbitals are a model, but that can be said of the title subject of many articles.) Dirac66 (talk) 23:38, 30 December 2010 (UTC)


 * I agree that the atomic orbital article is much more comprehensive, and any merger would probably want to add what material we want to keep from this article to that one. I suggested keeping "model" in the title because I would see the article as being about the modern QM model of the atom, rather than being about the mathematical solution to Schrodinger's equation for a central potential.  (It also keeps the pattern Plum pudding model, Rutherford model, Bohr model, Atomic orbital model.)  Likewise I would keep text from the atomic orbital model article that describes the model, though actually the model is probably described better in the atomic orbital article than in the atomic orbital model article!  Djr32 (talk) 00:05, 31 December 2010 (UTC)
 * Whatever we name the result, our primary job is to find anything in atomic orbital model which is salvagable at all! That is, different or better than in the longer atomic orbital article. As for adding "model," to the name of THAT article, atomic orbitals are a feature of a larger quantum mechanical model of nature, but that article is called quantum mechanics not quantum mechanics model. Also, we do not have special relativity model, general relativity model, and so on. With precident for both systems of naming, I personally prefer the shorter one. Bohr model is better because it's old and outdated, and doesn't have an alternate name anyway. If atomic orbitals or quantum mechanics become old and outdated, that will be a better time to re-open this discussion. Not putting in "model" should not be taken to mean we're under the impression that this stuff is True, i.e., the Last Answer. Or that it isn't (in some sense) still a model for nature, instead of Nature Herself.  S  B Harris 01:06, 31 December 2010 (UTC)

I agree with the essential gist of the above. We only need one article. It should be called atomic orbital, so atomic orbital model should become a redirect. Technically this would be a merge, but I too see little content to merge. -- Bduke   (Discussion)  01:17, 31 December 2010 (UTC)


 * OK, I've started this process off. I have tended towards keeping everything from both articles, on the grounds that it's a lot easier to remove stuff that we don't want later, rather than to remember where it was and find it again.  Opinions welcome.  Djr32 (talk) 19:28, 8 January 2011 (UTC)


 * I've duplicated the material here (which is probably where the "merge" suggestion should initially have gone, since it tends to disappear when the other article gets redirected). Speaking personally I have to say: "thanks for doing a messy but needed job". I would have deleted all of the other article, but I see that the parts of atomic orbital model that survive now, serve as overall segement introductions in THIS article, which is complicated enough that it can use them anyway, and isn't hurt. I just got through reading this article from start to finish, and it's fine. BTW, I'm going to add a bit to the drum vibration section, which is interesting. Notice that the first set of modes (u0x) are the only ones where the center moves, exactly like the s orbital set, which are the only ones with a center antinode. None of the other drum modes have a moving center, and the same is true for the atom-- these are all counterparts to orbitals with angular momentum. In these images, the electron is mostly "likely" to be in the places where the surfaces are moving the most. The analogy is closer with a 2-D surface (a drum) than with full spherical harmonics on a sphere. This is strange, but appears to be due to the fact that this allows the drum surface displacement in the third-dimension, to correspond to the electron radial coordinate (and thus energy). Thus, the process of modeling energy for bits of a 2D surface, as 3D displacement, works to show energy intuitively as a 3D coordinate. S  B Harris 23:11, 8 January 2011 (UTC)

Language?
I'm confused by:


 * For elements with high atomic number Z, the effects of relativity become more pronounced, and especially so for s electrons, which move at relativistic velocities as they penetrate the screening electrons near the core of high Z atoms. This relativistic increase in momentum for high speed electrons causes...

as I understand it, the idea that the electrons have speeds is not the QM view; indeed, ealier the article says "The electrons do not orbit the nucleus in the sense of a planet orbiting the sun, but instead exist as standing waves." If they had speeds, they would be accelerating electrical charges and they would radiate William M. Connolley (talk) 09:48, 16 January 2011 (UTC)


 * We can calculate the average kinetic energy of the electron. When we talk loosely of the speed of electrons we are talking about what one derives from that. The main relativistic effect is on the the core 1s electrons in heavy atoms and then the effect of that on the other electrons. The statement you give does need some modification, but not for the reasons you give. -- Bduke   (Discussion)  10:21, 16 January 2011 (UTC)


 * Also, although an electron does not have a uniquely defined velocity in quantum mechanics, it does have a distribution of possible velocities with associated probailities. In this sense velocity (or better momentum) is analogous to position, for which again an electron does not have a uniquely defined position, but rather a distribution of possible positions.
 * Relativistic effect are then consequences of the probability that electrons have relativistic speeds becomes significant, for the inner electrons of heavier atoms. However this idea is probably above the level of this article. Dirac66 (talk) 17:28, 16 January 2011 (UTC)

Connection to uncertainty relation - delete dubious paragraph?
Today's edits by Spiral5800 call attention to the last paragraph of this section which begins "Although Heisenberg used infinite sets ...". The whole paragraph is still quite confusing and has now accumulated two "citation needed" and one "dubious - discuss" tags. My opinion is that it adds little to the article and that we should just delete this paragraph. Comments? Dirac66 (talk) 15:13, 11 April 2011 (UTC)


 * I actually agree with this sentiment.  Lacking the addition of quality citations, it should go.  Spiral5800 (talk) 11:04, 13 April 2011 (UTC)


 * It appears this has been taken care of. Though I consider the matter of Heisenberg's work implying that there is a non-zero probability of an atom's electrons being found at arbitrarily large distances from its nucleus to be interesting, here - in the context it was mentioned - citations were needed and it was more an interesting factoid than a necessary issue for this article.  Spiral5800 (talk) 11:17, 13 April 2011 (UTC)

Inline citations
In lieu of putting up the template noting a need for more inline citations in this article, I instead am bringing up this issue here. For an article of this length and detail, I find myself longing for more inline citations through which I can fact-check various claims. I also noticed that in addition to a significantly noticible lack of inline citations, this article has only 15 cited references at the end. Not only do we need more inline citations, but more sources as well.

The template is obtrusive and because I otherwise find this to be a rather high quality article, I decided against the template in favor of hopefully fruitful discussion and eventual resolution here. Thanks! Spiral5800 (talk) 11:22, 13 April 2011 (UTC)

Orbital Energy chart
I'm confused about this chart in the orbital energy section:

Why is the 8p subshell followed by the 6g subshell? Shouldn't it be followed by the 9s? Or am I missing something? The pattern displayed for the first 20 subshells would seem to imply that it should be:

Is the chart given in the article correct? If so, why is the pattern broken after 8p? XinaNicole (talk) 01:27, 25 April 2011 (UTC)
 * Yes, your version is correct. Thank you for pointing out this error. The paragraph preceding this table says that there are 24 subshells, so I suspect that an earlier version of the table stopped at 24, and someone came along and continued the table by filling the wrong boxes. So I have just deleted the values after 24, since only 19 are needed to describe all currently known chemical elements. Dirac66 (talk) 02:38, 25 April 2011 (UTC)

Number of electrons in the n=3 state
In the last paragraph of the section "The Bohr Atom," shouldn't it be "and the n=3 state can hold 18 electrons"?

JKW (talk) 13:08, 15 May 2011 (UTC)
 * It can hold 18 but it only holds 8 in argon. I'll reword this sentence and eliminate "can hold". Dirac66 (talk) 14:13, 15 May 2011 (UTC)


 * Yes. Rule for the maximal number of electrons with principal quantum number n, is 2n2. Thus for n=3 you get 2*3^2 = 18. For n=4, it's 2*4^2 = 32. S  B Harris 18:21, 15 May 2011 (UTC)

Orbitals meaning and use.
Fortunately for Wikipedia, the contributions to the Discussion section do not seem to get scrambled as the articles do. So, here is the definition and explanation of an atomic orbital from someone who has published in the field of Quantum Chemistry since 1974 and has taught what orbitals are and how to use them in university courses at the introductory, intermediate, and advanced undergraduate and graduate levels (and even taught high school students with the aid of a graphing calculator) since 1980.

1) When you mention the word orbital to someone in the field of Quantum Chemistry, that person usually thinks you mean a mathematical function that describes the wavelike behavior of one particle, usually an electron.

2) When you say Atomic Orbital, one of two things is implied. Either you are talking about:

a) (Exact case, used rarely and only in one system) a mathematical function that describes the wavelike behavior of the electron in a hydrogen atom or ion containing only one electron (He+, Li2+, etc.) These are exact in that the forms of the functions (1s, 2s, 2px, etc.) provide the correct (non-relativistic) energy levels and these forms can be found without resorting to any mathematical approximations. These forms are called solutions to the Schrödinger equation.

b) (Approximate case, the usual case) a mathematical function that provides a starting point to approximate the wavelike behavior of two or more'' electrons in an atom, molecule, polymer, or crystal. The term, atomic orbital model, implies the use of approximate (i)methods and (ii)functions.

i) (Approximate methods) Since the time of Isaac Newton, physicists and astronomers have not found a way through calculus to describe the motions of the sun and two or more planets exactly (by consulting integral tables, solving differential equations and such). They resort to numerical techniques, which usually consist of putting more and more bits of the puzzle together to finally get close to the whole picture. Three of these techniques in Quantum Chemistry are configuration interaction(CI), perturbation theory, and density functional theory. Let's take CI as an example. The true but known wave function for an atom is approximated as a weighted sum of the ground electron configuration (1s22s2 for the Be atom) and excited configurations (e.g. 1s22s12px1, 1s22s02px2, etc.) The more excited configurations that are used, the better the approximation becomes and the more the approximate energy approaches the true energy of the atom. What is meant by a weighted sum of configurations is that they can each be assigned a coefficient (ci) and a best set of coefficients can be found through calculus (dE/dc = 0). The percentage that each configuration makes to the overall wave function can be indicated by the value of each weighting coefficient.

Here's an illustration using the Be atom.

In high school and first year chemistry (US), one learns that the electronic configuration of the Be atom is

ΨBe = 1s22s2    (resulting E = -14.566764 atomic units, STO-3G)

Then, in third year chemistry (US), one learns about the variational principle and LCAO techniques. Using the CI method for the Be atom, a wave function with six participating electron configurations atom might be

ΨBe = (90%)1s22s2 + (3%)1s22s02px22py02pz0 + (3%)1s22s02px02py22pz0 + (3%)1s22s02px02py02pz2 + (0%)1s22s12px12py02pz0 + (<1%)1s22s03s2  (E = -14.613493 atomic units, or 0.046729 a.u. more stable, STO-3G/CISD)

The percentages are derived from the weighting coefficients that are found for each configuration. The ground configuration contributes the most, 90%, to the wave function but not 100%. In modern Quantum Chemistry, wave functions with only one configuration are rarely acceptable in research. Thus, an atomic orbital is a mathematical function that describes the wavelike behavior of one electron and this function is used as a start in approximating the wavelike behavior of each electron in a multi-electron system.

ii)(Approximate functions) The functions shown in this article are the exact functions found by solving an equation for the hydrogen atom but they are almost never used for many electron atoms or molecules. In practice the functions in the atomic orbital model only resemble these functions. The reasons for this are that (1) the form of a hydrogen orbital makes solving the equations exceedingly difficult when there are several functions (orbitals) on the atom, (2) the hydrogen orbital can itself be approximated by a weighted sum of simpler functions (LCAO). The two most popular approximate orbitals are the Slater-type orbital and the gaussian orbital.

The 2s orbital in an abbreviated form for these three types is:

2s(H like) = NZ (2 - r) e-   ... Z = atomic number, NZ = normalization constant

2s(Slater type) = Nζ e-  ... ζ = effective atomic number, Nζ = normalization constant

2s(Gaussian type) = Nα e- α r•r ... α = effective atomic number, Nα = normalization constant

If you look in a first year chemistry text, you will probably see the 1s, 2s, 3s, and 4s orbitals plotted and you will see that each has (n-1) nodes. These come from a polynomial in front of the e function, (2 - r) for 2s. J. C. Slater dropped this polynomial but linear combinations of STOs can approximate the H orbital. Likewise, the similar gaussian functions, which contain an r squared term in the exponent, can be combined to form an STO.

Please forgive this old professor for giving you this Exercise. A trick question in chemistry exams is, "Does the atomic radius increase, decrease, or remain the same when going to the right in a row for the Representative Elements in the Periodic Table? Most students who haven't memorized the answer usually pick increase. Some pick remain the same because there are the same number of electrons and protons in each atom.

Plot 2s (x= -5 to 5) for a C atom using a gaussian, Ψ2s,C = Nα e- α x•x ... α = 6, Nα = 2.732

Plot 2s for a F atom using a gaussian, Ψ2s,F = Fα e- α x•x ... α = 9, Nα = 3.703

The area under each curve corresponds to the electron density. An orbital is defined as the mathematical function for one electron. So the total areas are equal. What is different is that the density for a 2s electron in F is closer to the nucleus than in C. This is found for all the orbitals. Therefore, the F atom is smaller than the C atom. The nuclear charge is one greater in F, but all the electrons are described by the same type of orbitals, a mathematical function that describes the wavelike behavior of ONE electron at a time.

BTW, if you want an index for electronegativity, you can use ζ or α.

The rational basis and the conceptual basis for almost every electronic property of substances and reactions in Chemistry are built on the wavelike nature of electrons. Functions no more difficult than sine or gaussian functions can be used to 'picture' electronegativity, bonding, conjugation, hyperconjugation, aromaticity, symmetry in pericyclic reactions, etc. In fact, the idea of vibrations on a string often suffice and can be taught in first year chemistry. That's the reason for my breaking Wikipedia rules and writing a very long Discussion section. I am sick and tired of having students arrive in mid-level chemistry courses who are burdened with the intellectual dead-end concepts of regions, shapes, and clouds. Thank you for your patience. Again BTW, that was a great idea to put in the moving images of vibrations on a drum head! Do you know what you get when you do it for vibrations on a sphere? (It rhymes with 1s, 2s, 2px, etc.) Slán, Laburke (talk) 02:50, 16 May 2011 (UTC)
 * Yes, the talk page is a record of a discussion and normally preserves individual contributions intact, whereas the article itself is a collective effort. However the object of the discussion is to improve the article, so my reaction to your long contribution is to ask which points we should add to this article. Where do we go from here?
 * My own opinion is that the most important point you have made is that a single configuration is only an approximation to the exact wavefunction. I would suggest adding a section at the end to mention more exact methods such as CI, explain briefly the CI function which you have given for beryllium as an example (though with the correct coefficients from some published calculation), and link to other articles on CI, DFT etc. Other opinions? Dirac66 (talk) 14:21, 16 May 2011 (UTC)

This is a most interesting account that is valuable here. However, like a lot of experts (and I am one too, I suppose) you miss the obvious. You say "Three of these techniques in Quantum Chemistry are .." but you miss Hartree-Fock theory, which is generally the reference function for CI and PTn methods and in itself gives useful results and simple pictures for students. Orbitals in both atoms and molecules are almost always, and certainly best seen as, Hartree-Fock orbitals which are optimised to give the lowest energy when each is in the mean field of electrons in all the other occupied orbitals. It is important to recognise that orbitals are generally approximations and obtained by improvement just as you describe for CI. They have no obvious form, except for the one-electron atoms and of course for symmetry derived parts such as the angular part of atomic Hartree-Fock orbitals. The CI and PTn approximations simply can not be understood without understanding the Hartree-Fock approximation. DFT is best understood by students as a form of Hartree-Fock theory, although of course it is not. An article on atomic orbitals should focus on Hartree-Fock along of course with the simple pictures of Hydrogen-like orbitals. -- Bduke   (Discussion)  22:26, 16 May 2011 (UTC)


 * Yes, even though it is integral to Orbital Theory, I skipped HF Theory, Fock matrices, etc. for brevity since some of the theory was already discussed in the body of the article. Laburke (talk) 03:43, 30 October 2011 (UTC)


 * As Brian Duke and Laburke, I consider myself also somewhat of an expert and wrote an encyclopedic article about the AO, found here. --P.wormer (talk) 05:47, 17 May 2011 (UTC)
 * It is a good article, but I do not believe AOs in atoms these days are the same as basis functions. I think AOs in atoms are Hartree-Fock orbitals built from basis functions. The term LCAO is misleading as it really goes back to when only minimum basis sets were used and these were considered to be AOs. I think the term is less used now and that is a good thing. Basis sets are just a tool to get one-electron orbitals of various kinds in MO and VB theory. These one-electron orbitals may have some meaning within the context of the theory used, but basis functions have no meaning in themselves. -- Bduke    (Discussion)  06:47, 17 May 2011 (UTC)


 * Brian, when you write "It is a good article, but I do not believe AOs in atoms these days are the same as basis functions", which article are you referring to? The present WP article? I completely agree that in the case of atoms, AOs are often Hartree-Fock orbitals (and always if we see LDA as a form of HF). In practice atomic HF orbitals are either linear combinations of STO's or numerical tables. I also agree with the rest of your statement. It seems to me that this article is in complete agreement with your views, or do I overlook something in your arguments? If I do, I may adapt my article on AOs. BW, Paul. --P.wormer (talk) 07:31, 17 May 2011 (UTC)
 * I was referring to the citizendium article where, to me, the section headed "AO basis sets" totally confuses the difference between atomic orbitals and basis functions. I think we use basis functions, not atomic orbitals, whether they are atom-centred or otherwise. -- Bduke   (Discussion)  09:17, 17 May 2011 (UTC)


 * Thank you Brian, I will adapt it. I agree with what your are saying, you have in mind, among others, bond-centered orbitals and floating Gaussians, I presume? Both are basis functions that are not centered on atoms. However, you don't deny that atom-centered-one-electron functions (to avoid the term AO) are used frequently as basis functions, do you? If you do, I'm misunderstanding  your meaning of "basis function". Paul. --P.wormer (talk) 09:53, 17 May 2011 (UTC)
 * I am clearly not making myself clear. Of course I do not deny that atom-centered-one-electron functions are used frequently as basis functions. No, I did not have bond-centered orbitals and floating Gaussians in mind. My point was that these are basis functions. They are not atomic orbitals. An atomic orbital in an atom is a Hartree-Fock orbital. There is no other way to define an AO. These Hartree-Fock orbital can be built in many different ways including from basis functions, but basis functions are not atomic orbitals - they are merely one tool to build atomic orbitals. I think it is important to keep that distinction clearly in mind. I am not so sure about the relevance to this article however. -- Bduke   (Discussion)  11:42, 17 May 2011 (UTC)
 * I think this article should not go into computational methods too deeply. It would be enough to just state that atomic orbitals can be computed by the Hartree-Fock method, with a link for further explanations. There is no need to mention basis functions at all in this article which discusses the concept. Readers interested in the HF method can follow the link to learn more. Dirac66 (talk) 12:15, 17 May 2011 (UTC)
 * Brian, I now see your point, but follow you only halfway. I changed my CZ article going partly your way, but not completely. As I see it, your definition was the only one in the pre-computer era. The definition of atomic orbital has changed since then. Nowadays, quantum chemists call arbitrary one-electron atomic-centered functions also "atomic orbitals". These functions obviously lack a well-defined energy (are not eigenfunctions of an energy operator). I can appreciate that you stick to the original meaning, but as I perceive it, the modern, more sloppy, definition is now used generally. (BTW I'm misusing this forum for improvement of my CZ article, I wouldn't touch the WP article). --P.wormer (talk) 12:19, 17 May 2011 (UTC)
 * I think it did change for a while, but I think the most common usage now is to call them basis functions. Let me give an example. If we are calculating the hydrogen molecule and want to make a really good calculation at Hartree-Fock or CI or MP2 or whatever, we start with s orbitals that can be used to make a pretty good stab at getting the 1s orbital on the hydrogen atom right. We then add p, d, f, g, h, etc basis functions to polarize the molecular orbitals we are using and get them to approach as close as we can the best possible molecular orbitals (if we are doing Hartree-Fock, we want to approach the Hartree-Fock limit). It makes no sense to say we are mixing in h atomic orbitals and people do not say this. They talk of basis functions. -- Bduke   (Discussion)  23:05, 17 May 2011 (UTC)

Can we begin at the beginning? When chemists put a standard electron configuration for an element, what do those numbers represent? Do they mean to suggest that each of those electrons in a subshell occupies a mere Slatter determinant?!? That they sit in things that are no more real than a basis function, like the cartesian axes are basis functions for vectors? And any old set of basis functions could be used instead? Or do these electrons reside (in pairs of different orientation) in sets of Hartree-Fock orbitals which now just-so happen to be called 3d something-or-other, as though they WERE sets of hydrogenic atomic orbitals that make up a subshell (even though they aren't)? What DO those numbers mean, in an atom's electron configuration notation? IOW, are they (2p, 3d, 4f, etc) the first two quantum-numbers of the one-electron Hartree-Fock wavefuctions, and we're now hijacked Schroedinger hydrogenic notation for them-- but forgot to tell chem students? Inquiring minds want to know. S B Harris 17:52, 17 May 2011 (UTC)
 * I think you have it correct, although I would probably use different language. We are not hijacked Schroedinger hydrogenic orbitals, as will already have explained that these do not apply strictly in atoms with more than one electron. The s, p, d, f part of the nomenclature means that the angular part is a s, p, d or f spherical harmonic. The number are just numbers but, yes, we do start at 2 for p, 3 for d and so on to match the hydrogen case. The electrons sit in orbitals that arise only within the one-electron or Hartree-Fock approximation, so in one sense they are not real, but I think they are more real than the basis functions which are used (as just one way) to build the atomic orbitals. Using the one-electron orbitals with configuration interaction can give a good result but the electron configuration picture has mostly disappeared. Electron configurations miss electron correlation - the way electrons keep out of each others way - and we are increasingly understanding that getting electron correlation right is really important. -- Bduke   (Discussion)  23:05, 17 May 2011 (UTC)
 * Okay. So my simpleminded view of a single boron atom floating in vacuum in ground state, is that there are 2 electrons (spin up, spin down) with energy quantum number 1 and no angular momentum, and 2 more with energy quantum number 2 and no angular momentum, and one last electron with energy quantum 2 but 1 unit of angular momentum. So I think of these orbitals as being as real as in the 1 electron hydrogen atom: 1s22s22p. No? Yes? If the "real" orbitals for each of these electrons are not hydrogenic, what DO they look like? Sort of hydrogenic? In beryllium the 4 electrons really don't have any angular momentum, as the bonding electron chemical state has a big effect on the rate of electron-capture beta decay of Be-7, which makes sense as it happens only with electrons that are in s states that penetrate the nucleus. So there is something about the s-state of these that is "real." Why can't we think of the electrons in Be as all being in hydrogenic s-type spherical orbitals? Even though there be 4 of them? S  B Harris 01:01, 18 May 2011 (UTC)
 * They are not hydrogenic, because the radial part of the the wave functions are not identical to the form in the hydrogen atom, but they are similar. In the H atom there is a neat analytical form to the orbitals, but for B and Be orbitals have no such analytical form and we have to find the form numerically. The real wave function is not as you describe it because it is not a product of one-electron functions. It is more complicated. The thing about the S state that is real is that the electron density at the nucleus is not zero. The approximation (1s)**2(2s)**2 for Be points to that fact as s orbitals are non-zero at the nucleus but that does not make the s orbitals real. The ground state of Be is an S state but the excited states are very close. You can think of that as similar to saying that the energy of the p orbitals is very close to that of the 2s orbital. -- Bduke   (Discussion)  02:44, 18 May 2011 (UTC)


 * @Bduke. Brian, you are saying that "AO" in the meaning of "atom-centered basis function" is disappearing from the literature? You could be right. Although, the latest book I have on the subject, Frank Jensen, Introduction to Computational Chemistry Wiley, (1999), states on page 65:


 * Each MO is expanded in terms of the basis functions, conventionally called atomic orbitals (MO-LCAO, Linear Combination of Atomic Orbitals), although they are generally not solutions to the atomic HF problem.


 * Jensen then continues to use the term AO as a synonym for basis function. IMHO, even if the term is getting out of fashion, an encyclopedia still ought to mention it, if only for historical reasons. There is whole lot of printed stuff out there where the abbreviation MO-LCAO-HF is used for the method in which a HF-MO is written as a linear combination of "AOs". --P.wormer (talk) 09:46, 18 May 2011 (UTC)
 * OK, it is used, but the sooner it goes out of fashion the better. It makes no sense as a g polarisation function is not a AO. I think the term LCAO is used a lot less now than it was when Jensen wrote his book and when it is used it is more about arm-waving MOs from minimum basis functions. I think it is time to close this discussion or take it to email. -- Bduke   (Discussion)  10:46, 18 May 2011 (UTC)

If you would like to close this discussion, please allow me. I wanted to point out that in English an orbital is generally understood to be a mathematical function that describes the wavelike behavior of one electron. The mathematical function depends on the coordinates of one electron.

I have found that students understand many concepts in Chemistry in a qualitative, but still important way if they can use sine functions on paper or exponential functions on a graphing calculator (sometimes the calculator is not even needed). These functions are orbitals.

Thanks to computers, we are going beyond orbitals when we want to calculate chemical properties very precisely. There are even some reactions that can not be adequately described by orbitals alone. Without explicitly naming any mathematical technique, this is how I would like them to see where orbitals historically fit into our calculations: 1) If we have the mathematical function that describes the wavelike behavior of the particles in a system (say, the nuclei and electrons of an atom, molecule, polymer, crystal, etc.), we can calculate the energy and any other physical property of that system. 2) However, the mathematics presently available to us will not let us find the exact form of this function if there are two or more electrons in the system, and so the energy of this system can not be calculated exactly. 3) We have been developing mathematical methods over the last seventy years to take advantage of the development of computers and are getting energies and properties closer to experiment (and in some cases more precise than experiment). Many of these approximate methods start with an atomic orbital for each electron in an atom or molecule and then apply these new techniques to get a mathematical function that describes the wavelike behavior of all the electrons in the system. The atomic orbitals usually all have similar forms but they are all mathematical functions that depend on the coordinates of one electron. 4) Several decades ago we started using a technique that combines atomic orbital functions for each atom in a molecule gave us another type of orbital, the molecular orbital, which, being an orbital, is a mathematical function which describes the wavelike behavior of one electron in a molecule. 5) Although the energy and other chemical properties of a molecule can be calculated using molecular orbitals, they too can be used as starting points for better mathematical functions that describe the wavelike behavior of all the electrons in a molecule together and not one-by-one as in orbitals. Laburke (talk) 20:17, 23 May 2011 (UTC)

Spherical or cartesian coords for wave functions-- it doesn't matter to the graph or most places (just makes the radially symmetric equation easier to solve)
Recently an editor has changed a lot of nice simple |ψ|s into  |ψ(r,θ,φ)|s and I don't see that this helps at all, except to make it uglier and more complicated in notation. After all, these are the very same graphs as if you used (say) |ψ(x,y,z)|s or any other spacial coordinate choice. Why don't you just change them back to the simpler form? Are you trying to show a spacial dependence? Just say they're being graphed in terms of spacial dependence and then you don't confuse the reader (by implication) into thinking that perhaps choice of coordinate systems makes some kind of difference. S B Harris 19:44, 1 November 2011 (UTC)


 * Dear Dr Sb, the ugly truth is that there were no parameters for the functions until I put them in. There is no better way of pointing out to beginners the fact that an atomic orbital, Ψ, is a mathematical function, thus Ψ, that depends on the coordinates of ONE electron in the atom, thus Ψ(x,y,z) than to put it right up there in the titles of the graphics. It doesn't matter which coordinate system is used, ultimately, and the text mentions which coordinate systems are used for atoms and molecules. Ease of solving the Schrödinger Equation (Hartree-Fock equations for us, Brian) is what determines the use of ugly spherical coordinates for isolated atoms, (very ugly) cylindrical for diatomics and linear triatomics, or (pretty) cartesians for molecules, or even (Halloween scary) complex variables for atoms in certain fields. BTW, I don't mind your using my name or user-name instead of "an editor". Laburke (talk) 02:32, 2 November 2011 (UTC)

"Vandalism" vs "good-faith edits"
Earlier today the edit summary "Revert vandalism" was used to describe the removal of the sentence "Atomic orbitals are the regions or volumes in space in which the probability of finding electrons is highest" and its replacement by the correct mathematical definition of an orbital as a one-electron function. While I do agree with the edit, I think the word "vandalism" should be reserved for edits which are totally irrelevant to the article, or so obviously incorrect that the error must be deliberate. In this case the sentence removed is a simplistic description which is often repeated or implied by teachers at an elementary level, so it is most probably an honest attempt which should be described as a "good-faith edit" or even an "imprecise description" since it has some merit.

I have taught both theoretical (quantum) chemistry and first-year chemistry. I try to tell students that an orbital is a mathematical one-electron function. But in first year when their eyes glaze over and they ask what the orbital drawings mean, the usual answer is that they show the region in space in which the probability of finding electrons is highest for a given quantum state. Yes I know, the drawing is not the orbital, but at a first-year (or high-school) level this is not very clear. Dirac66 (talk) 19:47, 13 January 2012 (UTC)
 * I agree that the wording that was reverted was not "vandalism". However, the editor did include vandalism in one of his edits at that time, which he then undid when he inserted the wording that was reverted today, so I understand the ease by which one might apply that label to the whole set of edits. DMacks (talk) 20:21, 13 January 2012 (UTC)

What I don't like is that by calling it an "orbital", they're trying to relate it in some sneaky way to a legitimate conic orbit, which has a fixed amount of angular momentum as well as additional amount of exchangeable free to kinetic energy. Can you imagine a motion path of an electron around an atomic nucleus that has a constant amount of lost free energy but with a zero to some amount of angular momentum? The only thing I can think of is a tethered orbit, where the electron is moving around in a circle at a fixed distance from the center of attraction.WFPM (talk) 18:52, 13 April 2012 (UTC)

Real orbitals
Are you sure the definitions of p_x and p_y are corrrect? I think they should be switched, such that p_x=-1/sqrt{2}*(p_1-p_{-1}) and p_y=i/sqrt{2}(p_1+p_{-1}). — Preceding unsigned comment added by Mariussimonsen (talk • contribs) 08:00, 16 April 2013 (UTC)
 * No, they are correct. The relation of cartesian to spherical coordinates is z = r cos θ, x = r sin θ cos Φ, y = r sin θ cos Φ. Neglecting constant factors, the complex orbitals are p1 = r sin θ eiΦ and p-1 = r sin θ e-iΦ. So px = (1/√2)(p1 + p-1) = r sin θ [eiΦ + e-iΦ] = r sin θ cos Φ = x, as required for a px orbital. Similarly py = (1/√2i)(p1 - p-1) = r sin θ [eiΦ - e-iΦ] = r sin θ sin Φ = y. Dirac66 (talk) 18:39, 16 April 2013 (UTC)
 * There were some changes made to this notation in the Wikipedia text on Aug. 24th, 2013. I have just reverted the notation back to its original form so as to be consistent with your math, Dirac66, and also to be consistent with the reference "Levine," but I would appreciate it if someone could also take the time to check for consistency with the other two references. If there is a different convention most commonly used, perhaps we should change the Wikipedia text to that instead. Csmallw (talk) 18:42, 25 January 2014 (UTC)
 * Thanks, I hadn't noticed the changes made in August. The possibility of switching p_x and p_y is not a matter of convention. The p_x function correctly defined is large on the x-axis and zero on the y-axis, so it is not reasonable to call it p_y and I doubt that any published book really does so.
 * I did check the cited Theochem article by Blanco et al., which does not in fact include the definitions of p_x and p_y explicitly. It does contain more general equations which lead to these definitions, but that requires algebra which we cannot check, and some Wikipedia editor has probably made an error judging by the result. I don't have access to the book by Chisholm.
 * There is however another alternate convention which can be mentioned. The phase of any orbital is purely conventional, so it is acceptable to multiply either p_x or p_y or both by (-1), and some books can be found which do this for one or both. I will not object if someone wants to point this out in the article. As long as the orbitals are not switched with each other. Dirac66 (talk) 21:17, 25 January 2014 (UTC)
 * I have Chisholm. What you want me to check? -- Bduke   (Discussion)  23:24, 25 January 2014 (UTC)
 * Please check that Chisholm does not actually claim that
 * $$p_x = \frac{1}{\sqrt{2}} \left(p_{-1} - p_1 \right) $$
 * $$p_y = \frac{i}{\sqrt{2}} \left( p_{-1} + p_1 \right) $$
 * These two equations were in this article from August until they were corrected yesterday by Csmallw. The article cited Chisholm (as well as the Theochem article which I have already checked), but it is difficult to believe that Chisholm would have switched p_x with p_y. Dirac66 (talk) 03:16, 26 January 2014 (UTC)
 * I have only just got around to checking Chisholm. He does not explicitly give the equations we have, but our equations are consistent with his equation 4.15 on page 47, but that is much more general. Chisholm is a very advanced text and I do not think it an appropriate reference for this article, so I will remove it. The reference to Levine is not clear about which edition it refers to. The correct equations are given on pages 133 and 134 of the 4th edition, which I have, but that is 1991, not the 2000 edition. Could someone update the Levine reference to the latest edition? -- Bduke    (Discussion)  02:00, 31 March 2015 (UTC)
 * Done. Dirac66 (talk) 23:56, 1 April 2015 (UTC)

I think that the previous relation was right. The sign of the spherical harmonics associated to p(1) and p(-1) are opposite. So if you add them, a sin(phi) will appear corresponding to p(y). Thisis in agreement with the real spherical harmonics page. As it is said in the text about spherical harmonics different conventions exists. — Preceding unsigned comment added by 193.50.159.70 (talk) 16:47, 26 March 2015 (UTC)
 * Hm. OK, there are two different sign conventions for the Ylm and the article now follows books such as Levine, but is inconsistent with the Spherical harmonics article. Since both conventions are found in reliable sources, Wikipedia should not say that one is correct and the other wrong, but rather adopt a neutral point of view (WP:NPOV). In principle this means giving both sign conventions and their consequences (and some sources). It would probably be confusing to put both in the main text however. I suggest putting one sign convention in the text with its consequences and sources, and adding a footnote to point out that some authors use the other convention with its consequences and sources. Dirac66 (talk) 23:32, 26 March 2015 (UTC)
 * I have now inserted a mention of the alternate sign convention in the text (not a footnote). I hope that what I have written is correct, as the phase conventions are confusing. The Spherical harmonic article actually seems to have more than two - the original Laplace version with just eimφ, the version with a phase factor (-1)m, the Condon-Shortley version with (-1)m for m > 0 only, and on the talk page a Racah convention with (-1)l+m. In this article (Atomic orbital), I have only mentioned the version (-1)m for all m which we discussed above. Dirac66 (talk) 03:09, 4 October 2015 (UTC)

Ok, I have made what may be considered a major revision to the Real atomic orbitals section. It is possible I overreached, if so please let me know, educate me on my editing approach, and feel free to revert. I tried to keep all that I could from the previous version. I am trying to more clearly and explicitly explain the relationship between the real and complex spherical harmonics. I would say I have done 3 things to the section. (1) I spell out the mathematical relationship very clearly, making clear link to the corresponding relationship for spherical harmonics which is already illustrated in Table of spherical harmonics, (2) I switch the section to using the Condon-Shortley phase predominantly because this is consistent with the Wikipedia pages on spherical harmonics (and frankly I prefer this convention since it's predominant in my personal literature-base) and (3) I give some exposition and add a talbe about how the Cartesian expansion of the real spherical harmonic is used to generate the common "atomic orbital" nomenclature for different orbitals. I could see this third thing I do being split out into a separate subsection.

I'm a new editor, so like I said, please educate me if I've done something wrong here. I've tried to modify this section to add clarity, but perhaps I take an approach that is too pedagogical and not encyclopedic enough. I may also be too lengthy in some of my explanations or lack references in important places. I also think there's a risk that this exact explanation I've given doesn't appear in the literature and is something that I have put together from looking at various Wikipedia pages and references there-in. If this explanation does not exist in the literature I wish it did and that's why I'm including it here, but maybe my efforts would be better spent somehow getting this kind of explanation into the peer-reviewed literature.

Anyways, I hope folks here can help me figure out how to fit these contributions onto this page in a nice way. The relationship between these two descriptions of hydrogen orbitals confused me for a long time but I've finally found clarity that I would like to share. Thanks for any help! Twistar48 (talk) 01:43, 22 February 2022 (UTC)

The "Qualitative understanding of shapes" section
In the current article these lines don't make sense to me:

''If such a particle were gravitationally bound to the Earth it would not escape, but would pursue a series of passes in which it always slowed at some maximal distance into space, but had its maximal velocity at the Earth's center (this "orbit" would have an orbital eccentricity of 1.0). If such a particle also had a wave nature, it would have the highest probability of being located where its velocity and momentum were highest, which would be at the Earth's core.''

Generally speaking this is wrong. The particle would have the highest probability of being found where its velocity is lowest. This is because the particle spends a higher fraction of its time in locations where its velocity is lowest. The above either needs to be completely removed or better explained. — Preceding unsigned comment added by O. Harris (talk • contribs) 08:18, 27 May 2013 (UTC)
 * Of course, this was a rubbish, I [ fixed it]. I can remember that I heard some presentation about such limits of Keplerian orbits, but IMHO it is next to a trivial fact about cofocal ellipses. Incnis Mrsi (talk) 09:11, 27 May 2013 (UTC)

Problem with "Bohr's Atom" section
At the end of Bohr's Atom section there is following statement: "Ultimately, the relationship between orbital occupation and chemical behavior was resolved by the discovery of modern quantum mechanics and the Pauli Exclusion Principle. Namely in helium, all n=1 states are fully occupied; the same for n=1 and n=2 in neon."

But in next inert gas Ar n=3 is not fully occupied, so as for Kr, Xe and Rn n=4,5 and 6 are not fully occupied. Instead, n=3 is fully occupied for Zn and n=4 is fully occupied in Yb. Therefore, this is a poor basis for the statement which implies that all noble gases are characterized by closed electron shells. Drova (talk) 02:47, 3 November 2013 (UTC)
 * Yes, I think you were correct. I have now rewritten this paragraph to specify which subshells are full in argon. Dirac66 (talk) 18:04, 3 November 2013 (UTC)

Sockpuppets
user:AWomanWithAPlan is clearly formerly blocked IP user 103.10.64.20, see, and has now reverted 3 times a change two editors want. I've had it. S B Harris 06:20, 17 October 2014 (UTC)

User:Pandaron is also a sockpuppet. AManWithNoPlan (talk) 14:12, 17 October 2014 (UTC)

Omission of J
Why is J omitted? 108.65.83.125 (talk) 18:52, 27 October 2016 (UTC)
 * Ah, at last, a reasonable question. Because some languages do not distinguish between 'i' and 'j'. See Spectroscopic notation. Double sharp (talk) 06:33, 28 October 2016 (UTC)
 * I have now included your answer in the article, citing the same source as in Spectroscopic notation. Dirac66 (talk) 21:50, 29 October 2016 (UTC)

lobes as tangent
I have concerns about two statements:
 * " The three p-orbitals for n = 2 have the form of two ellipsoids with a point of tangency at the nucleus (the two-lobed shape is sometimes referred to as a "dumbbell"—there are two lobes pointing in opposite directions from each other)."
 * "Four of the five d-orbitals for n = 3 look similar, each with four pear-shaped lobes, each lobe tangent at right angles to two others, and the centers of all four lying in one plane."

If one is plotting the spherical harmonic and only giving the magnitude of that function in a certain direction (that is, plotting Y as a distance for each θ and φ), then these are accurate. But this is deceptive, since Y contributes to a magnitude of ψ at each point in space.

It's more common (and probably more correct) to plot an isosurface, all the values of ψ at a triple (x,y,z) or (r,θ,φ). These isosurfaces cannot be tangent to each other (or to the nodal plane), since for all any arbitrary value of |ψ| > 0, the distance of the closest point of an isosurface is always going to be at a positive, non-zero distance away from zero. This misconception is presented in a number of Journal of Chemical Education and articles and quantum chemistry textbooks. Olin (talk) 15:33, 22 November 2016 (UTC)

How to Explain Muons sub-s orbital?
I find it interesting that there can be no orbital below s and yet Muon orbitals are below s.

Anyone care to explain this?


 * A Muon is not an electron so the Fermi exclusion principle does not apply between muons and electrons. The muon forms its own tight orbital, irrespective of the electrons.  From the electrons point of view what they see is a new "nucleus" with an atomic number = #protons - #muons, and the atom chemically behaves as if it had this new atomic number. Cloudswrest (talk) 17:57, 24 April 2018 (UTC)

Definitions of px and py: I think they are reversed.
The current article has

px = (p_1 + p_-1)/Sqrt[2] when I simplify this, I get

Exp[- constant r] y.

And for

py = (p_1 - p_-1)/i Sqrt[2]

I get

Exp[- constant r] x.

Thus, I think they are likely backwards in the article. — Preceding unsigned comment added by WCraigCarter (talk • contribs) 09:11, 13 July 2018 (UTC)

Black hole central region
The black hole's central region is so degenerate that it looks like the atomic orbitals, and has different energy levels (both the singularity and the ringularity violate Planckian shrinkage limits; the Big Bang exploded [it didn't shrink in more] because nothing can violate the Planckian threshold of degeneracy = maximal degeneracy = precosmic state; actually the black hole's central region less degenerate than the "precosmic state"). But we need more on the specifics. — Preceding unsigned comment added by 2a02:587:4114:20de:49:ffd3:fa6f:fc35 (talk) 00:37, 17 July 2021 (UTC)

Suggestions for page improvements:
I recently made a draft page you can see here: https://en.wikipedia.org/wiki/Draft:Gallery_of_Atomic_Orbitals which was rejected. I think much of the content that appeared there would clarify some topics in this article. I think this article along with hydrogen atom and hydrogen-like atom are overcluttered and share a lot of redundant information. This is unfortunate and makes me not want to put in more material to clutter further, but perhaps the way of Wikipedia is to jam in too much information, sometimes in bad form, and then clean it up after the fact.. or don't.. I don't know.

In any case I propose to make the following adjustments: (1) Reformat the existing table of spherical harmonics to match that on the draft page. This is a more logical sorting that showcases that patterns between orbitals much better. (2) include a second table of complex orbitals next to the real orbitals for comparison. (3) ensure these tables follow the Condon-Shortley phase for consistency with the Table of spherical harmonics. (4) general cleanup of the text to be more concise in explaining the geometry of orbitals, (5) Include the formula for the solution to the Schrodinger equation for hydrogen like atoms on this page.

Regarding point (5), I know that that equation already appears on Hydrogen-like atom and hydrogen atom, but I don't understand why it appears there and not here. If that equation can't appear on this page then I don't think visualizations of those orbitals should appear so prolifically on this page. Regarding point (3) above, unfortunately the existing table of real spherical harmonics, while nice, and of very high quality, does not follow the Condon-Shortley phase as far as I can tell. Unfortunately this will spoil a little bit the comparison between real and complex orbitals, so long term it would be best to generate new images in the table which do follow that phase convention.

Folks should let me know if they think any of these updates would be "Wikipedia worthy" or valuable otherwise and I can get to work on them. Especially let me know what should be done about the inclusion of the Hydrogen orbital equation.. my main point there is that it is silly to have this page with so many orbitals visualized but without the equation. It makes the pictures difficult to talk about. My opinion is that this is the page that should have the equation and visualizations, and if that is too much for one page then it should be broken out into a specific page like the draft page I already made, not smattered across three largely redundant pages. — Preceding unsigned comment added by Twistar48 (talk • contribs) 03:31, 23 February 2022 (UTC)

Also, make it easier for laypeople to understand. Azbookmobile (talk) 00:00, 29 August 2022 (UTC)

Minor error in "Real orbitals"
At the end of the first paragraph are three equations for (real)psi(n,l,m). The lower right-hand corner reads "for m < 0". Common sense dictates this should be "for m > 0". Then again, common sense only plays a minor role in quantum physics. Could somebody more knowledgeable than me make the edit, or explain why we have two possible equations for m 0? Thanks. 2600:6C5E:517F:E91D:2A:BF1A:6A3:1A5D (talk) 02:26, 15 February 2023 (UTC)


 * OK, I made the change for you and added a useful link. I notice the equations in question use absolute values for $$m$$. This is kind of odd, since we already broke out the $$m<0$$ case, so why not get rid of the $$|m|$$ and just use $$-m$$ or $$m$$ depending on $$m<0$$ or not. But like you I'm gonna leave that for others to consider. --JohnH~enwiki (talk) 19:00, 23 February 2023 (UTC)
 * Yes, $$m<0$$ was a typo. Thank you for catching this and thank you @JohnH~enwiki for correcting it.
 * It was a style choice to use $$|m|$$ instead of $$-m$$. The idea is to make the top and bottom equations look more similar. In particular, using this form, we can simply say that the two real orbitals for $$|m|>0$$ correspond directly to the real and imaginary parts of the complex orbital with $$m = |m|$$. That simpler than saying one of the real orbitals is the imaginary part of the complex orbital with negative $$m$$ and the other is the real part of the complex orbital with positive $$m$$. I think you could alternatively absorb this difference between $$\pm m$$ into some sort of overall sign difference between the equations as a different option. But I found this the simplest, though I admit this is just my opinion and other opinions are valid on this point. Twistar48 (talk) 05:20, 6 August 2023 (UTC)
 * So, what's the number one root of all origins of the first defined element? What came first, 0 or 1? Who defined the origin of what is defined as our origin of all man as defined by fossils discovered most recently or simply stayed as the oldest logic following binary logic as defined I'm the CTMU defined by Chris Langan who is for a lack of insight the best life to live in the free world? 108.147.10.69 (talk) 10:50, 16 November 2023 (UTC)

"Shapes of orbitals" comment/question
This section (Shapes of orbitals) says "there is a non-zero probability of finding the electron (almost) anywhere in space". [1] Does this apply to all of an atom's electrons, or just those in the valence shell?

I'm going to suppose that holds for all electrons. Consider, for example, a (stable, non-radioactive) potassium atom, K. Suppose an electron in the 1s orbital (radius 3.175 pm) wanders off in the general direction of the 2s orbital (radius 14.82 pm). [2] Can it move very close to the 2s orbital, in apparent violation of the Pauli exclusion principle, because it "really" belongs to the 1s orbital? [3] If not, how close can it get to the 2s orbital before the exclusion principle takes over? [4] How would that "critical distance" be calculated, and does it depend on the position of the 2s electron with the same spin as the wandering 1s electron? Thanks for any help. JohnH~enwiki (talk) 23:11, 13 March 2023 (UTC)


 * @JohnH~enwiki
 * There is no critical cutoff distance, but the probability of finding an electron close to another electron decreases rapidly as the distance between them approaches zero. Each orbital actually occupies all of space so that each electron can be found anywhere, but the chance of finding two electrons in a given volume simultaneously decreases with the size of the volume considered. This is called electron correlation and must be included in accurate atomic or molecular structure calculations. Dirac66 (talk) 11:48, 16 May 2023 (UTC)
 * Asking "Where is the electron?" in an orbital is like asking "Where is the spark?" in an overcharged capacitor *before* the dielectric breaks down. The orbital IS the electron, unless you force its point existence by looking for it. Cloudswrest (talk) 00:44, 17 May 2023 (UTC)