Imagine that we find very special 3D cameras on sale. We get a few hundred of them and give them to several groups of students for a weekend project. Each student gets a camera and just one hydrogen atom in a special box. Their task is to take as many photographs of their hydrogen atom as possible. Well, you know that in the real world this would not be possible for many reasons, but there are few limits to our imagination. We collect all the digital pictures taken and add them together in Photoshop 3D in such a way that all nuclei are perfectly aligned at the point where the Cartesian axes intersect. In our special pictures, the nucleus is invisible, but electrons show up as dots. Our congregate image (left below) represents a measure of the probability of finding the electron in different volumes of space around the nucleus of the groundstate hydrogen atom. You may easily discern the probability density pattern. It has spherical symmetry, with high density toward the center (nucleus) and low and diminishing densities at the outer edges of our observation box. By taking thousands of pictures of thousands of electrons in thousands of individual positions in thousand of atoms, we have learned about the probability of finding one electron in any “space” (unit volume cube) around the nucleus. Interestingly, collections of pictures from students from some groups showed different patterns (above on the right). It turns out that these groups were given boxes with an extra energy supply (batteries included!) that kept the hydrogen atoms in various excited states. The groundstate distribution (a) is repeated to match the size of the others, and all are shown in crosssections to reveal the internal details. Some of these probability patterns (b, d) are still spherical, although they have larger spreads than that observed for the ground state hydrogen atom (a). They also have some white spaces within their interiors, illustrating regions where electrons are not allowed to be (nodes). Other probability distributions (c, e, f) are a bit more complicated. They have “directionality” and more complex node patterns. These pictures are our first exposure to the different shapes and sizes of hydrogen orbitals. All orbitals with spherical symmetry (a, b, d) are stype orbitals: 1s, 2s, and 3s with quantum numbers n and l (=0) defined. The dumbbell shapes (c, e) are ptype orbitals, 2p and 3p, with nodes at the nucleus (l=1). They are representatives of a 3piece set in each case. The last picture (f) is a 3d orbital (l=2) that belongs to a 5member family of even more diverse profiles. In hydrogen atom, orbitals with the same n are degenerate, so, for example 3s, 3p, and 3d, all have the same energy. What that means is that in our imaginary experiment we would have great difficulty sorting the pictures from the students given boxes with n = 3 hydrogen atoms (where a total of 9 orbitals, i.e. different electron distributions, have the same energy!). In reality, the shapes, sizes and directions of orbitals are obtained by solving the Schrödinger equation. The applications of the Schrödinger equation, HΨ = EΨ, require advanced calculus, but its solutions and some important basic ideas may be often presented graphically in a qualitative but useful manner. At the first glance the equation looks quite simple, but the “quantum devil” is in the detail. It all starts with de Broglie’s idea that electrons in atoms are described by standing waves. The mathematical expressions of these waves are called wavefunctions and are usually represented by a Greek Ψ (psi). The waves cannot be measured directly (they have no physical meaning), but their squares (Ψ^{2}) represent the probability of an electron being in a given tiny volume of space (probability density). The wavefunctions for an atom (or a molecule, or any quantum system) are found by solving the Schrödinger equation. In that equation, “H” hides a set of mathematical operators (mathematical instructions on how to manipulate the wavefunction) that calculate all energies of the system including all electrostatic interactions between all nuclei and electrons and the kinetic energies of all electrons. The results of the calculations are the energies of the quantum systems, E, and the wavefunctions describing them.
The function shown above (right) is two dimensional; for each point [x] we have a value of the wavefunction as shown on z axis [Ψ(x)]. If we spin our function around the z axis we can recreate the water’s surface. That surface is threedimensional; for each point [x,y] in the xy plane the function has a value expressed on z axis [Ψ(x,y)]. To transition to atomic wavefunctions we have to add another dimension. For each point in space [x,y,z] around the nucleus we have to specify the value of our wavefunction [Ψ(x,y,z)]. Unfortunately, displaying things in four dimensions is rather difficult. We will rely on various crosssections to reduce dimensionality (as we did in our waterwave example here) or use other graphical “tricks” to simplify the presentation. The probability of finding an electron in a miniscule volume ("cube") of space around a point [x,y,z] is equal to the square of the value of the wavefunction at this point. We need to use these tiny volumes (dxdydz cubes, for those mathematically inclined) as points themselves have no volume. This probability density (probability per unit volume) is also called electron density and can be visualized as a fraction of an electron in that minuscule volume cube. Notice that squared values are never negative, as probability must be, even if the wavefunctions may be negative in certain regions of space. Electron density is a directly observable quantity. Many modern techniques allow us to probe it for atoms and molecules. Now we are ready to explore some specific solutions of the Schrödinger equation for the hydrogen atom. All plots have the same horizontal scale, and, within sets, the same vertical scale. You may click on any of the plots to enlarge them for sidebyside comparisons.. The distance from the nucleus, r, is measured in atomic units (bohr, a_{o}) and in Å.. The functions are plotted in both directions along the horizontal axis in a way analogous to our water wave. The specific direction in space or the sign of r are arbitrary and irrelevant for orbitals with spherical symmetry. 
The total probability of finding an electron within the orbital must be unity (orbitals are normalized), that is all probability densities for all volume units added up must equal 1. As you may have noticed, the wavefunctions and electron density functions stretch to infinity, even if with extremely low values of electron density at lager r values.. To facilitate graphical presentations of orbitals, an arbitrary cutoff point is chosen in such a way that the probability of finding the electron inside the demarcated volume is 90%. That cutoff point corresponds to a very low value of electron density, let's say 0.002 au (anyway, what’s 10% difference between friends?). For orbitals, it represents an artificial isosurface (surface with all values of electron density equal to 0.002 au, in our example) that serves as an imaginary border (“skin”). This isosurface makes the outside orbital layer smooth in our orbital representations, instead of being fuzzy, as shown in our electron "pictures" on the top of this page. It also makes our 4D problem disappear. Now, we have to show only [x,y,z] points at which the electron probability has the set value (0.002 au, for our example), i.e. we are dealing with 3D representations. We have sacrificed all the details of the internal electron density distribution, but we gain in simplicity of presentation that is important for our qualitative (nonmathematical) models. The only other information we will need is the algebraic sign of the underlying wavefunction. We will display it on the surface of our orbitals with color or other graphical marks. Since we do not have a way to experimentally distinguish the algebraic signs of the wavefunctions (we only observe the squared values) the labels are assigned arbitrarily and only aim to show if the signs are the same or opposite for different orbitals or different orbital lobes. That information will be important when orbitals start to overlap and form bonds. We would need to know then if they are going to interfere (like water waves) in a constructive or destructive way. But that is a subject for another page. Let's now look at our s orbitals in 3D! 
On this scale, the 3s orbital would take the width of page and a half of your monitor. Its interior would be even more complex (what is the maximum number of internal spheres that could be present inside the 3s orbital?), but the beauty of our simplified presentation is that all s orbitals may be drawn as uniformly colored spheres of various sizes. The p orbitals come in sets of three, all having the same overall shape but each pointing in a different direction in space. We call them p_{x}, p_{y}, and p_{z} to indicate their relative spatial orientation. Examples of wavefunctions and radial probability for p_{z }orbitals are shown below in a way analogous to that used for s orbitals. In this case the horizontal axis is chosen to be the z axis. Individual p orbitals do not have spherical symmetry, so the choice of direction is important. The p_{x} and p_{y} orbitals look the same, except are perpendicular to p_{z} and to each other. The functions shown here are accurate, but only along the z axis, not in all directions as was the case for swavefunctions. 

Let's start reviewing 3D attributes of p orbitals with the familiar . with opposite algebraic signs of the wavefunction. We need to scale our models down a little as compared to s orbitals (above) since we are dealing with larger orbitals with quantum numbers of 2 or 3. Here, we align the z axis to match the plots above, even if z axis is usually given vertical orientation (directions in space are chosen arbitrarily), We can turn the surface into a or make it . to explore the interior, but there is nothing there of interest. To spruce it up we add colors by drawing a or plot. All of these pictures tell us why a quick drawing of an "8" is a fair symbolic representation of ptype orbitals. The other two 2p orbitals (p_{x} and p_{y}) look the same, but point to different directions in space. We can visualize all of them at once by using different colors for the lobes to make them easier to distinguish from each other. The familiar is first, then we have and, finally all in mesh representation so we can look through them. You should give them the spin to gain a 3D appreciation of their relative spatial orientation. Let's add them up. Start with and then look at all . As you may notice, all three p orbitals as a group have again spherical symmetry. This is true for all sets of subshells orbitals (p, d, and f), although in organic chemistry we concentrate on s and p orbitals only. Finally we can have some fun with 3p_{z} orbital.
We will just list the possible display modes for your
enjoyment:
We have learned all we wanted to ever know about hydrogen orbitals and their shapes. The simple summary is that they are very similar on the outside, spherical for s orbitals, or dumbbellshaped for p orbitals. Gratifyingly, atoms other than hydrogen have very similar orbitals with two small, but important differences. The atoms have varying nuclear charges and since the electrons shield each other, the given electrons experience different effective nuclear charges. In general, as the result, the orbitals shrink (as compared to the corresponding hydrogen orbitals) and have lower energies than those in the hydrogen atom. These important differences affect bonding properties of the orbitals. 

Molecular Gallery  Last updated 08/31/12  Copyright 19972013 
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