Organic Chemistry at Penn State: Hydrogen Orbitals

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 ground-state 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.

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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 ground-state distribution (a) is repeated to match the size of the others, and all are shown in cross-sections 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 s-type orbitals: 1s, 2s, and 3s with quantum numbers n and l (=0) defined. The dumbbell shapes (c, e) are p-type orbitals, 2p and 3p, with nodes at the nucleus (l=1). They are representatives of a 3-piece set in each case. The last picture (f) is a 3d orbital (l=2) that belongs to a 5-member 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 (Ψ²) 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.

Here is our standing wave illustration: tall in the middle, then dropping below the level of the undisturbed water surface (serving as our reference plane), to climb again, and drop again in a series of diminishing concentric hills and valleys. If we produce a cross-section trace of the water’s surface on graph paper, with the height of the water’s surface represented on the vertical axis (z) and the separation from the center on the horizontal axis (x), we get a mathematical function that has positive algebraic values in the upper part of the plot and negative values below the x axis.

two dimensional representation of the wave

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 three-dimensional; 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 cross-sections to reduce dimensionality (as we did in our water-wave 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 side-by-side 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.

1s	2s	3s
1s wavefunction has no nodes and drops off rapidly with r. The top part of the plot was cut off to show all wavefunctions on the same scale.	2s wavefunction has one node and spreads farther out from the nucleous. It has regions with opposite algebraic sign of the wavefuntion..	3s wavefunction has two nodes (i.e three separate region with different algebraic sign) and stretches far from the nucleus.	These wavefunctions have the highest value at the origin (nucleus) and they drop off equally in all directions as r grows. The wavefunctions for 2s and 3s extend further from the center than 1s, have regions of positive and negative values, and have nodes in spots where the wavefunctions change their sign.
			The orbital cross-sections are aligned with the corresponding electron density 2D plots (below). The circular (i.e. spherical in 3D) nodes are readily apparent.
1s electron density profile is quite compact; beyond 3 Å. the values are very close to zero.	2s electron density is more spread out with a node around 2 Å.	3s electron density has two nodes and stretches beyond 10 Å.	The squares of the wavefunctions represent electron density, i.e, the probability of finding an electron in a tiny volume of space around each point of atomic territory. These are given as fractional numbers (probabilities) per unit volume (in atomic units). The squares of the wavefunctions are always positive (or zero). as probabilities must be.
The peaks in radial probability plots correspond to the most probable distance for the electron. For 1s it is 1 bohr (a_o= 0.529 Å).	The 2s radial probability has two distinct peaks with the larger (the most probable) at ca. 3 Å.	For 3s orbital the most probable distance of the electron is about 7 Å, but the orbital stretches well beyond 10 Å.	In the radial probability plots all electron density of tiny cubes at a given r are added up to provide the total probability of a given separation between the nucleus and the electron. They demonstrate well the increasing spread of electron density with n and normalization of atomic orbitals (the total area under each plots is equal unity).

Since the s orbitals have spherical symmetry, i.e. they stretch equally from the nucleus in all directions of space, we may ask what the most probable separation distance is between the electron and the nucleus. Looking at the electron density plots, an intuitive answer might be that electron should “statistically” be at (or very, very near) the proton. That is not correct, however. Although the electron density is high at or near the nucleus, that volume of space is very small. Electron density is expressed as probability per volume element (dxdydz), i.e. a small “cube” of space. The number of such cubes is very small at the center, but increases with r. At a given r, the probability function is simply [Ψ(r)]² multiplied by the surface area of the sphere with radius r (i.e. multiplied by 4πr²). In other words, we add up probabilities of all the cubes contained in a thin spherical shell ("onion layer") of radius r. The results of these calculations are shown by the last set of functions above called radial probability. Close to the nucleus we have high values of electron density, but few volume cubes. At larger distances the electron density drops off, but the number of volume cubes increases as our spherical shell gains in radius. As the result, the most probable distance for electron in the ground state is 0.529 Å, i.e. 1 bohr which is the atomic unit of distance and corresponds to radius of the Bohr orbit, a_o for n = 1, After all, Bohr was “almost” right! The most probable distance increases for the other s orbitals.

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 cut-off 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 (non-mathematical) 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!

We start by drawing the , or more precisely an isosurface encompassing 90% of electron density. Well... it is quite boring, just a sphere painted blue to represent one of the possible signs of the underlying wavefunction. Let's (arbitrarily) say it is positive to match the picture above. You may spin it or move it, but it looks the same from all sides. That's what spherical symmetry means. For all the artists out there, we can be more adventurous and try to use to indicate the curvature of the surface. And if we draw the lines more densely we get a presentation. It looks a bit like a cage for the electron. At this scale, the proton (the nucleus of hydrogen atom) would be smaller than one pixel at the crossing point of the coordinate axes.

If you play play with the models using your mouse, spinning them, moving them, zooming in or out, remember to press the "reset" button before moving on to the next model. The reset will return the model to the same size and orientation for easy comparisons.

Sometimes it is convenient to make the orbital skin translucent to see what's inside. This presentation technique is especially useful for molecular surfaces where we may want to peak inside to check on what atoms are participating in bonding and what is the molecular geometry. For now, let just make our orbital skin It is still quite monotonous, like its solid-color analogue. To add some accents we may instead show a of the orbital, replacing dot density (see above) with a color gradient. We can even take it for a spin..

On the other hand, for those who like hiking, maps are the preferred way to read topography of the terrain, or electron density (you may click the picture and use the mouse weel to zoom in for e closer look). One of the contour lines, when span around its axis generates back the orbital isosurface. Finally, we can go and show three mutually perpendicular cross-sections at the same time, just for fun.

The , is very similar, except twice bigger and it takes a bit longer to calculate and display (patience, young grasshoper). It is "painted" red to consistently show (see the plots above) that the "outside" algebraic sign of the 2s wavefunction is opposite to that on the "inside". We can barely see the axes unless we make it . If we do, we can immediately recognize the two "inside" spheres, the bigger one of the the same sign as the outside surface and the smaller one of the opposite algebraic sign, all matching the outside's isosurface absolute value. The existence of these three concentric isosurfaces may be confirmed by examination of the 2s-electron density plot above that shows that a line drawn at some low value of electron density (parallel to the horizontal axis) will cross the blue Ψ² function 3 times on each side of the origin. As before, to add some color we can paint to shows some internal values of the wavefunction.

2p	3p
2p wavefunction has a node at the nucleus.	3p wavefunction has a node at the the nucleous and one additional one spreading on both "sides" of the of the orbital.	The p-wavefunctions have nodes on the nucleus and opposite algebraic signs on the two sides of the axis. They have similar "spreads" to the s-function when compared to the 2s and 3s orbitals, respectively.
		The orbital cross-sections are aligned with the corresponding electron density 2D plots (below). The nodal region are clearly visible. Both orbitals have one nodal plane cutting through the nucleus, and 3p has an additional radial node.
2p electron density; spinning the function along the z axis would yield a dumbbell shape.	3p electron density; spinning the function along the z axis would yield a "double dumbbell" shape.	The squares of the wavefunctions (all positive) represent electron density. Spinning the functions around the z axis will give you an approximation of of 3D representations (see below).
2p radial probability; the values apply only along the z axis.	3p radial probability; the values are valid along the z axis only.	As before we can ask about the most probable distance for the electron by plotting radial probabilities. Because the p orbitals are not spherical the plotted data apply only along the z axis. Note that the position of the maxima are similar to those for 2s and 3s orbital radial probabilities..

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 p-type 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 dumbbell-shaped 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.