# Simple Explanation of orbitals

## The Question :

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Imagine I was your little brother, how would you explain an orbital to him? (assuming he knows what atoms electrons and neutrons are, and the rest of the basics till that point.)

I have been reading up on this and can’t make heads out tails of it.
As a side how do the different energy levels relate to them.

I hope I’m clear, I’m doing my best. I read several books but can’t get a clear bearing on this.

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Well, the first step is to stop thinking electrons as being very small balls that orbit around the nucleus in a circular path. This is known as the Bohr model. Despite this model being an excellent model at an introductory level, it doesn’t tell the entire story. There are several issues with the notion that electrons travel in a circular orbit around the nucleus, the main one being that they should eventually undergo orbital decay and hence eventually slow down and crash into the nucleus. Of course, this doesn’t happen. So how do electrons move around the nucleus?

This is where quantum theory came to the rescue in the early 20th century. It stated that electrons could not be treated as a classical particle and didn’t not have a definite position and momentum. So this basically means that we can’t exactly know an electron’s position and its momentum at the same time. This relationship is given by the Heisenberg Uncertainty Principle.

I won’t go into detail about it since to keep this explanation basic, but if you are interested check out this video which provides a really good explanation.

So basically what quantum theory tells us that we actually can’t specify where and what a path an electron is going to take. However the Bohr model assumes that you can precisely plot out where an electron is going to be, but we now know that due to the Heisenberg Uncertainty Principle you can’t precisely know the exact position and momentum of an electron. So we actually can’t construct an orbit and say that the electron is always going to follow that path.

Therefore, it makes it impossible to plot an orbit for an electron as how can you? If you don’t know where the electron is going to be next, there is no way you can predict its path.

So how can we get around this problem? Well, if we can’t draw the orbit for an electron the next best thing we can do is to plot an electron density probability orbital for it. Also known as atomic orbitals.

An orbital is a well defined region of space. Therefore an atomic orbital is a region of space that shows where the electron will be 95% of the time (usually we take 95% but it could be any number really such as 90% or 75%).

A simple way of thinking of orbitals is that imagine that you have a magic camera that can take a sequence of photos of an electron in a hydrogen atom. The electron appears as a dot. Now if we superimpose all of these images, you will see something like this:

Since we generally take atomic orbitals to be the region of space that shows where the electron will be 95% of the time, the very outer sphere of the above image is said to be the orbital of the electron.

Hopes this helps you understand orbitals. If you have any question, feel free to ask.

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Why don’t you try with analogies: at least, as a vary basic introduction, I very much appreciate the attempt done here by Goh et al.

An orbital is defined as a region in space in which there
is a high probability of finding an electron. An analogy for
an orbital is the possible location of a student based on his
or her timetable. For example, according to the timetable,
on Monday at 9.00 am, a student has to be in Lecture Room
1 for a chemistry lesson. One can say that, over a period of
weeks, there is a high probability that this student will be
in Lecture Room 1 at that time on that day. However, one
cannot be 100% sure that this student will be there because he or she may be absent from school on that day. One
also usually cannot predict exactly where the student will
sit in the lecture room, but one can say, with a high probability that this student will be within the confine of the
lecture room.
This analogy describes an orbital fairly well, because it
takes into consideration the probability and region of
space in the definition of an orbital.

I will definitely start with such a analogy which seems to give a quite simple, although thorough, description of how orbitals should be regarded.

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Ok. Forget everything about the orbitals for a second.

You have nuclei. You have electrons around them.

First step: electrons are not balls. Imagine them as being a smeared charge in space. Which is precisely what they are: smeared charge in space, with fuzzy borders, like an unfocused blob. The shape of this blob depends from many factors, but the biggest one is the position of the nuclei.

Now suppose you want to describe the shape of this blob, like if you want to 3d print it. Well an easy way would be to divide space in little cubes and say “charge” if you have some smear or “no charge” if you don’t have that smear. In practice, since it’s fuzzy, you would say “3.0 charge” in some cubes, “1.3 charge” in other cubes, “0.1 charge” in other cubes and so on. This is a perfectly legitimate way of describing what is called “charge density” in space.

The problem with this approach is that it’s rather inconvenient. It has poor accuracy and it scales badly. If you make the little cubes smaller, because you don’t like living in a minecraft world, you need a lot more cubes to do that.

So now there’s a smarter method to do so, and to explain it, I need you to understand Fourier decomposition. It’s not as hard as you think it sounds.

The problem is the following. You have a complex sound wave, like the one produced by blowing a whistle or singing a song, and it turns out that you can create any complex shape of this wave by summing simpler waves together: a careful choice of sines and cosines of certain frequencies (pitches), and with certain intensities (volumes). Play them all together, you get the initial wave back. This is exactly what you see in your spectrum display in your stereo Where each column is a different frequency, and the height of the column is the intensity of that particular frequency. It changes all the time because you are playing complex music, but try to play a uniform sound (e.g. a violin playing a single note) and you will see it stays the same throughout.

Now back to the orbitals.

The orbitals are the “sines and cosines” of our problem of describing that blob. We have a complex entity (our smeared blob) and we want to describe it by summing together “something”. It does not matter what “something” we use, but it turns out that 3 dimensional functions of that shape has a lot of nice properties which makes the problem much more compact.

Let’s make you a simple example. Suppose you have a spherical blob of charge. That’s probably described well by a single orbital of spherical shape (an s orbital), exactly like a sine wave from a tuning fork is described well by a single sine function.

Now add an electric field so that the electrons are pulled, and the smeared sphere is now more like an elongated egg. That one is not really described well by a sphere, is it? so you need to describe the lobe, meaning that you need an additional orbital (a p orbital) to add to the mix so that the result is egg-shaped, exactly like you need more than one sine wave to describe the sound of a violin.

That’s it, really. orbitals are just convenient 3d equivalents of the sine and cosine. We could well use anything else (and in fact we do, in some cases) and it would work as well, but with some potential disadvantages.

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Elucidating the atomic orbital structure of an atom is a fascinating story. It has just been solved within about the last hundred years. It wasn’t a linear journey, but a journey filled with all sorts of dead ends and side trips. The Greeks thought that the elements were air, fire, water and earth. Not understanding the atom alchemists spent zillions of hours and a countless amount of money trying to turn other substances into gold.

So the first significant step in solving atomic structure starts with Dmitri Mendeleev who is credited with the notion of a periodic table. This allowed for the discovery of additional elements which were needed to complete the table. However why the arrangement of the table worked was unknown.

About that same time other work was happening. Maxwell created his famous equations linking electric current and magnetism.

The discovery of the electron occurred around about the same time. But it wasn’t until about 1900 that the understanding of an electron was fairly complete.

Physicists still didn’t understand how to put an atom together. At this point many atomic models were floated to explain various phenomena. One such model was the plumb pudding model. The idea was that an atom was like a blob of pudding in which the electrons were suspended like plums in the pudding.

One big breakthrough came from Rutherford scattering about 1910. These experiments showed that the nucleus (positive charge) was located at the center of an atom and that it had a very small volume compared to the whole atom. This lead to the planetary model of the atom. The negatively charged electrons orbited the positively charged nucleus like the planets orbiting the sun. This didn’t explain other phenomena that were known like line spectra of hydrogen that Balmer had discovered before 1900. Such line spectra were related via Rydberg formula which related the principle quantum number $n$ to atomic structure.

$\frac{1}{\lambda} = R(\frac{1}{{n_1}^2} – \frac{1}{{n_2}^2})$

in 1913 the Bohr model was developed. This explained that the electrons were arranged in shells, and the filling of the shell structure could be linked back to the periodic table! The shell structure is basically tacked onto the planetary model by Arnold Sommerfeld and the notion was that for whatever reason the electrons couldn’t revolve around the atom’s nucleus in “any” orbit but that they had to occupy certain orbits which became known as shells. Subshells ($l$, $m$ and $s$) were added to Bohr’s model to fudge additional orbits to explain the fine spectroscopic structure of some elements. Like the Fraunhofer lines observed in the spectra of the sun. So at this point physicists and chemists had cataloged much of the behavior of atoms, but still were lacking a complete understanding of how atoms worked.

In the mid 1920’s Schrödinger deveoped his famous equation which completed the puzzle of the atoms structure into the more fundamental electrons and nucleus. From the Bohr model it was realized that electrons have to be in certain orbits because the orbits are quantized. That is to say that each orbital has a specific energy. What Schrödinger showed was that the orbits were not like the lanes at a track and field event that constrained the electrons, but that the electron orbit was distributed in a 3D cloud around the atom. Another aspect of Schrödinger’s work and others was that the electron had both wave characteristics and particle characteristics.

Using the Schrödinger wave equation the shape of orbitals could be calculated. Now this was in the 1930 time frame computers were still decades away. So these calculations were done by hand! The “problem” was that solving the equations was only possible for one electron. Using calculus it is nice to be able to solve equations so that the integrals can be easily calculated. However the three body problem doesn’t allow such a solution. With modern computers it is possible to calculate numerical solutions even though the integrals don’t exists.

So back to the question. An orbital is a mathematical function that describes an electron’s 3D path around the nucleus. Rather than an planetary like “orbit” the orbital is a probability function. The density of the orbit varies as a function of radius. Depending on the orbital the probability function also shows an orientation in 3D space. So a S orbital is spherical like in that there is no X-Y-Z preference. the P orbitals though not only have a radial aspect but they have spacial orientations. Thus the three P orbitals are like 3D rotations of a two-leafed rose in 2D space. So P orbitals have lobes oriented along +/- x-axis, the +/- y-axis, and the +/- z-axis.

A word of caution. The orbital representations are very useful for predicting chemical behavior, but they are not “real.” This obviously gets a bit crazy when four $sp^3$ molecular orbitals are formed each with 25% S character and 75% P character! It’s sort of like thinking that a unicorn is a hybrid of a horse and a rhinoceros.

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I don’t know enough physics to know whether this is on-target or not, but…

The idea of different vibrational modes is easy to demonstrate with a big bowl of water and your finger. Fill the bowl, and use your finger to excite different standing-wave patterns on the surface.

Start at the center of the bowl, move your finger slowly up and down, and you should be able to get a radially-symmetric standing wave that looks something like a two dimensional version of a 1S orbital. Once you find the fundamental frequency, double it, and you’ll get something like a 2S orbital. Move off-center, and you probably can excite a mode that looks like a 1P, and a 2P.

Unfortunately, you’ll have to stop there, because the higher-order modes on the two-dimensional surface of the water don’t bear much similarity to the higher order, three-dimensional orbitals.

But what you can show, is that standing waves in the confined space of the bowl are limited to only a few discrete possible kinds, and that for each kind, the frequency seems to always be an integer multiple of some fundamental frequency.

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Orbital is an one electron wave function. It describes where an electron (or pair, if needed) is to be expected, somehow ignoring all other electrons.

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An electron is both a particle and a wave.

Sound waves travelling in free air can have any frequency. But a vibrating disc can only resonate at certain frequencies. These correspond to the vibration modes of the disc. The animations at the bottom of https://en.wikipedia.org/wiki/Vibrations_of_a_circular_membrane show these vibration modes, and there is a one-to-one correspondence with orbitals of s,p and d symmetry (more complex orbital symmetries require 3 dimensions.)

Although this analogy is far from perfect, it shows the principles of a wavefunction. Note that in all cases except 1s, there is a part of the membrane that does not move at all. This is called a node. Nodes can be either planar or circular/spherical. In an orbital, a node is a place where the probability of finding an electron is zero.

The total number of zones where the probability of finding an electron is zero (this includes the zone of infinite distance from the atom) is given by the number in the orbital designation. The number of planar nodes is given by the letter.

Thus all orbitals with designation 3 have 3 zones where the probability of finding an electron is zero. 3s has 2 spherical nodes (the 3rd zone is infinite distance from the atom.) 3p has one planar and 1 spherical node. 3d has 2 planar nodes (giving it the familiar 4-lobed shape.) Note also that there is one 3d orbital with a conical node, but we will consider it as belonging to the same group as the planar notes.

Going back to the analogy of the circular disc, we see that it can vibrate in a number of discrete modes, and its motion can be described as a combination of those modes.

In an atom, the pauli exclusion principle dictates that all electrons must have different quantum numbers. Thus each orbital can contain up to 2 electrons, provided they have different spins. In this way the atom differs from the vibrating disc analogy, in that the disc can vibrate with any amplitude, but an orbital can only contain 0,1 or 2 electrons.

Each electron has a certain energy, depending on what orbital it is in. The energy of the electron depends also on its interaction with other electrons in other orbitals. Largely because of this interactions we find that the orbitals with more planar nodes are at higher energy than the ones with less planar nodes.

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Imagine I was your little brother, how would you explain an orbital to him? (assuming he knows what atoms electrons and neutrons are, and the rest of the basics till that point).

There’s a bit of a problem there, in that he probably won’t know what electrons are. There’s no electron model in the Standard Model. It’s described as a fundamental particle, but there’s no real description of how it’s created in gamma-gamma pair production or what it really is. You can find information about spinors and Dirac’s belt and the Dirac equation, which is a wave equation. But then people talk about point-particles and probability and everything gets confusing.

I have been reading up on this and can’t make heads out tails of it.

IMHO it’s easy when you do a bit of detective work. Have a look at the Wikipedia atomic orbitals article and note that electrons “exist as standing waves”. The moot point is that when you take an electron out of an atomic orbital it still exists as a standing wave. You made it along with a positron out of photons in gamma-gamma pair production. You can diffract electrons. Electron diffraction “refers to the wave nature of electrons”. The electron is a wave. A standing wave. Standing wave, standing field. The electron doesn’t orbit the nucleus like a planet, it’s more like an electromagnetic Saturn’s ring. It isn’t flat, it’s spherical, but that’s hard to imagine, so in describing it to your little brother, stick with a flat ring. To show it to him, give him a sparkler, and get him to rotate it round and round. This makes a ring-shaped image, which relates to the s orbital below:

Now get him to rotate the sparkler in a figure-of-eight fashion. This relates to the p orbital in either the x y or z orientation. Then get him to rotate the sparkler in a four-leaf-clover fashion. That relates to a d orbital, and so on.

As a side how do the different energy levels relate to them.

Because we’re dealing with electromagnetic waves rather than sparklers, and the energy levels are a bit like gears. See this picture by Kenneth Snelson: There’s always an integer number of wavelengths in the standing wave. The atom is a “quantum harmonic oscillator”. The electron wave can effectively change up a gear when it absorbs an E=hf photon wave. When it emits a photon, it effectively changes down a gear.

Of course electron orbitals aren’t quite as simple as that, but it should get your little brother off first base.

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I’d no more try to explain what “an orbital” “is” to my hypothetical little brother, than I would to a dog. IF there were some reason to explain them, then I’d have to know what it was. I noticed you didn’t even mention whether you were talking about atomic orbitals or molecular orbitals…
If pushed, I’d say that orbitals describe the shape of the volume of space in which the bound electron with a given set of quantum numbers is most likely to be, the shape of the space where that electron spends most of its time.