# Why do the number of orbitals in successive subshells always increase by 2? **URL:** \ **Category:** Factual Questions\ **Created:** [August 1, 2019, 1:31pm UTC](https://boards.straightdope.com/t/why-do-the-number-of-orbitals-in-successive-subshells-always-increase-by-2/837994 "2019-08-01T13:31:13Z")\ **Posts on this page:** 9\ **Page:** 1
**Author:** ![KidCharlemagne](https://avatars.discourse-cdn.com/v4/letter/k/ed8c4c/32.png) [@KidCharlemagne](https://boards.straightdope.com/u/KidCharlemagne)\ **Post date:** [August 1, 2019, 1:31pm UTC](https://boards.straightdope.com/t/why-do-the-number-of-orbitals-in-successive-subshells-always-increase-by-2/837994/1 "2019-08-01T13:31:13Z")
S,P,D,F subshells allow for 1,3,5,7 orbitals respectively. Is there some obvious reason why successive subshells always allow for 2 more orbitals (besides obviously that it will allow for the total electrons to be 2N^2’ which I’m assuming is tautological)? Thanks in advance! ---
**Author:** ![leahcim](https://avatars.discourse-cdn.com/v4/letter/l/b4bc9f/32.png) [@leahcim](https://boards.straightdope.com/u/leahcim)\ **Post date:** [August 1, 2019, 3:14pm UTC](https://boards.straightdope.com/t/why-do-the-number-of-orbitals-in-successive-subshells-always-increase-by-2/837994/2 "2019-08-01T15:14:10Z")
This relates to the angular momentum of the orbital, which is governed by two quantum numbers, _l_ and _m_. Long story short, _l_ does go up by 1 each “level”, so the S orbitals are ones where _l=0_, the P orbitals are the ones where _l = 1_, &c. For each level, though, _m_ is constrained to be between _l_ and _-l_. So S orbitals have _m=0_, P orbitals have _m ∈ {-1, 0, 1}_, &c, which accounts for the “increasing by 2”, since every single increase in _l_ increases the number of possible values of _m_ by 2. (In the classical analogue, where angular momentum is a 3-vector, _l_ determines the length of the vector, and _m_ determines the value of one component of it, and the constraint between _l_ and _m_ is the constraint that a vector has to be at least as long as its longest component.) ---
**Author:** ![markn\_1](https://avatars.discourse-cdn.com/v4/letter/m/f9ae1b/32.png) [@markn\_1](https://boards.straightdope.com/u/markn_1)\ **Post date:** [August 1, 2019, 3:48pm UTC](https://boards.straightdope.com/t/why-do-the-number-of-orbitals-in-successive-subshells-always-increase-by-2/837994/3 "2019-08-01T15:48:52Z")
nm ---
**Author:** ![DPRK](https://avatars.discourse-cdn.com/v4/letter/d/4491bb/32.png) [@DPRK](https://boards.straightdope.com/u/DPRK)\ **Post date:** [August 1, 2019, 5:05pm UTC](https://boards.straightdope.com/t/why-do-the-number-of-orbitals-in-successive-subshells-always-increase-by-2/837994/4 "2019-08-01T17:05:10Z")
If the state of an electron is determined by the angular momentum, the problem boils down to enumerating the possible irreducible finite-dimensional representations of SU(2) (or its Lie algebra). The number l will essentially be the highest weight of the representation, and the weights that occur will be integers between -l and l, as stated above. The angular momentum is thus quantized. ---
**Author:** ![glowacks](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/glowacks/32/5548_2.png) [@glowacks](https://boards.straightdope.com/u/glowacks)\ **Post date:** [August 2, 2019, 12:00am UTC](https://boards.straightdope.com/t/why-do-the-number-of-orbitals-in-successive-subshells-always-increase-by-2/837994/5 "2019-08-02T00:00:18Z")
> [@KidCharlemagne](#): > > S,P,D,F subshells allow for 1,3,5,7 orbitals respectively. Is there some obvious reason why successive subshells always allow for 2 more orbitals (besides obviously that it will allow for the total electrons to be 2N^2’ which I’m assuming is tautological)? Thanks in advance! My chemistry teacher in high school claimed that the very simple formula for the number of electrons in each shell was clear proof that the universe was designed. Well, ok, maybe. There’s no simple reason, but its provable in quantum mechanics. The reasons are very complex and took physicists a long time to figure out, even once they knew about atoms and protons and electrons. As mentioned, it has to do with the various states of angular momentum that are possible for an electron in an atom, but why would take two college courses at least to explain. If you’re interested in learning, I suggest trying out MIT’s Open Course Ware. You probably can start with 804 (Quantum Physics; the 8 means Physics, by the way. This is the 4th course in Physics.), but learning 803 (Waves mechanics) helps a lot even if it’s just classical waves, because quantum theory is based on waves that have many similar principles to classical waves. 804 made a lot more sense after I went back and did 803. Knowing basic Electricity/Magnetism and Maxwell’s equations’ theoretical results helps a lot too even if you can’t work out any problems (I certainly can’t, but I know the physical meanings of all of it). You’ll learn a good deal about the energy levels in Hydrogen in 804, and a whole lot more in 805 if you’re willing to put up with learning a whole ton of math (I already knew the linear algebra required as a grad student in math). ---
**Author:** ![Saint\_Cad](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/saint_cad/32/18907_2.png) [@Saint\_Cad](https://boards.straightdope.com/u/Saint_Cad)\ **Post date:** [August 2, 2019, 1:12am UTC](https://boards.straightdope.com/t/why-do-the-number-of-orbitals-in-successive-subshells-always-increase-by-2/837994/6 "2019-08-02T01:12:42Z")
Is this discussion of possible angular momenta at all related to Pauli’s Exclusion Principle? ---
**Author:** ![g8rguy](https://avatars.discourse-cdn.com/v4/letter/g/7cd45c/32.png) [@g8rguy](https://boards.straightdope.com/u/g8rguy)\ **Post date:** [August 2, 2019, 3:53am UTC](https://boards.straightdope.com/t/why-do-the-number-of-orbitals-in-successive-subshells-always-increase-by-2/837994/7 "2019-08-02T03:53:23Z")
As a hand-wavy heuristic explanation, you can think of the subshell angular momentum like this: [ul][li]Angular momentum tells you about the shape of the orbital. You could describe shape via a factor like[/li]_x[sup]i[/sup] y[sup]j[/sup] z[sup]k[/sup]_ where _x_, _y_, and _z_ are meant to be the three directions. Since we’re only interested in shape and not in spatial extent, we can assume _x[sup]2[/sup] + y[sup]2[/sup] + z[sup]2[/sup]_ = 1. In terms of the kind of angular factor I’ve described above, the angular momentum is _l~i+j+k_. [li]But this isn’t quite right. We’re actually only interested in the _unique_ shape factors, and a lot of the ones generated by the procedure I’ve outlined above are redundant. The unique new shapes at angular momentum _l_ wind up being _2l+1_ in number, for reasons discussed earlier.[/li][/ul] So let’s make this concrete: [ul][li]For _l=0_ I just get one possibility: _x[sup]0[/sup] y[sup]0[/sup] z[sup]0[/sup]_ = 1. This is the shape of an _s_ orbital, which has no angular structure at all.[/li] [li]For _l=1_ I have three possibilities: _x_, _y_, and _z_. These are the shapes of the _p_ orbitals, and there are 3.[/li] [li]But then we come to _l=2_. If you work through it, you have 6 possibilities: _x[sup]2[/sup], y[sup]2[/sup], z[sup]2[/sup], xy, xz, yz_. But one combination of these things is[/li]_x[sup]2[/sup] + y[sup]2[/sup] + z[sup]2[/sup]_ = 1 and 1 is the shape factor for an _s_ orbital. In other words, there is one combination of these 6 potential “_d_” orbital shape factors which we already had. And that means we have to exclude it from our list of _d_ orbitals because the shape isn’t new. From the original list of 6 possibilities, we only have 5 new shapes. [li]Similarly, if you tried to do this with _f_ orbitals which have _l=3_, you should come up with 10 possibilities, but among these would be_x[sup]3[/sup] + x y[sup]2[/sup] + x z[sup]2[/sup] = x_[/li]_x[sup]2[/sup] y + y[sup]3[/sup] + y z[sup]2[/sup] = y_ _x[sup]2[/sup] z + y[sup]2[/sup] z + z[sup]3[/sup] = z_ and these have the same shape as the _p_ orbitals. You’re left with 7 new things from the original 10. [/ul] And so on. ---
**Author:** ![DPRK](https://avatars.discourse-cdn.com/v4/letter/d/4491bb/32.png) [@DPRK](https://boards.straightdope.com/u/DPRK)\ **Post date:** [August 2, 2019, 4:26am UTC](https://boards.straightdope.com/t/why-do-the-number-of-orbitals-in-successive-subshells-always-increase-by-2/837994/8 "2019-08-02T04:26:19Z")
> [@Saint\_Cad](#): > > Is this discussion of possible angular momenta at all related to Pauli’s Exclusion Principle? Well, electrons with different orbital angular momenta, or different electron spin (also a type of angular momentum) will not be excluded from being jammed into the same atomic orbital. Pauli originally stated his principle for electrons; maybe someone here is familiar with his original proof/derivation? ---
**Author:** ![DPRK](https://avatars.discourse-cdn.com/v4/letter/d/4491bb/32.png) [@DPRK](https://boards.straightdope.com/u/DPRK)\ **Post date:** [August 2, 2019, 1:07pm UTC](https://boards.straightdope.com/t/why-do-the-number-of-orbitals-in-successive-subshells-always-increase-by-2/837994/9 "2019-08-02T13:07:14Z")
Erratum: where I typed SU(2) above, read SO(3) (group of rotations in three dimensions). This is in fact a crucial point, since this rotational symmetry when analyzing angular momentum is precisely what (after some algebra) leads to spherical harmonics and the fact that the irreducible unitary representations have dimensions 1, 3, 5, … One can also realize these representations as homogeneous harmonic polynomials as explained by **g8rguy** : so the 3-dimensional representation will be spanned by x, y, and z; the 5-dimensional one will be spanned by x[sup]2[/sup]-y[sup]2[/sup], x[sup]2[/sup]-z[sup]2[/sup], xy, xz, and yz; etc. This mathematics may seem a bit ad hoc but spherical harmonics will in fact show up when solving the spherically symmetric Schrödinger’s equation.