MadSci Network: Chemistry

Re: Are there more atomic orbitals other than s, p, d, f?

Date: Sun Mar 4 04:32:07 2001
Posted By: Duje Bonacci, Grad student,
Area of science: Chemistry
ID: 977128298.Ch

Hallo, Richard!

Before I attempt to answer your question, I have to apologise to you for
the delay...

Now, to orbitals! First let's consider the physical system we will be

We shall take one nucleus, which is of course positively charged, but we
are not interested in the magnitude of that charge - it may be a single
proton with charge +e, or a more complex nucleus such as sodium (charge
+11e), or whatever... Along with the nucleus, which can for practical
purposes be considered a point particle, with no spatial extension, we
shall take a handful of electrons, each charged -e.

From these ingredients we start building an ion - a compound system of the
certain number of electrons, bound to the nucleus by the electromagnetic
First step in this construction will of course be to combine just one
electron with the nucleus. This system is simple enough that you can
actually solve the differential equation for it exactly. The solutions that
you get effectively describe the distribution of electron charge around the
nucleus. Among themselves, these solutions differ in a certain number of
parameters that we call 'quantum numbers'. These are basically just ordinary
integers, and for our convenience we label them N, L, m.

What do they mean? Well, as I said, basically they are just labels, but they
are also related to some physical properties of the whole nucleus-electron
system (actually, if they were not, they would be utterly useless - what
are labels that have no relation to anything measurable good for?!). 

One of the most important physical properties of such a system is its
energy. The electron is negatively charged. The nucleus is positively charged.
Opposites atract. So if you want to dismember the electron-nucleus system,
you have to make an effort. The 'energy' of this system is a physical quantity
that measures this effort. When you do the calculations, it turns out that the
energy of the e.-nuc. system is determined by just one of our labels.
Energy being such an important quantity, we call this particular label 'THE
PRINCIPAL QUANTUM NUMBER', and use symbol N for it.

Other labels that appear in the solution of the e.-nuc. system we call the
m). In this simple system they have no impact on the energy, and it turns
out that once you pick N (and hence the energy), you can choose between N
different values of L, ranging from 0 to N-1. One physical property of the
system that L is related to is the average distance of electron charge from
the nucleus - larger L corresponds to the electron charge beeing smeared
out farther away from the nucleus.

m I shall not discuss, as it is almost irelevant for the rest of the story.

Now to answer the part of your question. In the simple system described
above, there indeed are N 'orbitals' (different values of L) for each
prncipal quantum number N. Historically, and due to their spectroscopic
appearance, the lowest four orbitals are labeled by letters:
s(harp)       - L=0
p(rincipal)   - L=1
d(iffuse)     - L=2
f(undamental) - L=3 
Higher orbitals have no such poetic names attached, but they are also
labeled by letters in alphabetic order following f:
g - L=4
h - L=5

Why aren't these higher orbitals mentioned so offten? Well the reason is
simple: they exist in a simple system we discussed earlier, the e.-nuc.,
but such simple systems are extremely rare - they are almost never observed
in nature. The more common systems are the ones 'built' from the nucleus
and a larger number of electrons. 

Let's now consider what happens if we want to complicate our initial simple
sistem with a single additional electron. Again, each of the electrons is
attracted by the nucleus, so no problem there. BUT the two electrons repell
each other, so their interaction adds to the energy of the whole new system
(2e.-nuc.). This interaction becomes even more important when you add more
electrons. In atoms with large number of electrons, the interaction among
electrons destroys the simple bookkeeping we had in the case of just one
electron, and the notion of the 'orbital' with L higher then 3 (i.e.
f-orbital) becomes obsolete - it simply does not exist any more. I guess
you will agree that this non-existance is a pretty good reason not to
mention them at all...

Hope I have been helpfull...


Dan Berger adds:
Duje's answer is excellent, but Duje is a physicist. In chemistry we are more
comfortable working with approximate or even "unreal" models like orbitals
in atoms with lots of electrons. Chemists will, for example, talk about 
f-orbitals in europium or uranium atoms and whether they participate in bonding 
in compounds of these elements, even though the more philosophical chemists will 
admit that f-orbitals don't "exist" but are instead useful approximations. 
Again, I remind you that the orbitals we talk about only "exist" in the hydrogen
atom or in other one-electron atoms. BUT...

The first sighting of a "g-orbital" as having any importance even in an 
approximate description of an atom will not come until we synthesize elements 
with atomic numbers greater than 120. Since 118 is as high an atomic number as 
anyone has even CLAIMED to have made (see WebElements), there's no reason to 
bother with g or h or i or ... orbitals except for fun. 

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