Mathematical plots of hydrogen atomic orbitals
Above: left, the radial wave function for a 1s (100) atomic orbital of hydrogen plotted as a function of distance from the atomic center. (This function has been normalised to ensure that the integral sum
of all the probabilities is equal to 1). The Greek symbol rho (
p) indicates distance from the center
along a radius in units of the
Bohr radius (the atomic radius for hydrogen in the ground state as on
the left gives the square of the radial wave function, which gives us the
probability density function
(PDF)
. The PDF tells us the probability of encountering the electron at a given distance from the
atomic centre in any single measurement (or equivalently the proportion of measurements in which
the electron is encountered at that position in an ensemble of measurements carried out on a large
number of atoms prepared in the same quantum state). Notice that the electron is most likely to be
found at one Bohr radius from the center, in approximate agreement with the classical atomic model.

The panel on the right shows a mathematical plot of the hydrogen orbital in 2D in greyscale (top) and
rgb colour (bottom, with blue indicating a lower probability, red a higher probability). To achieve this,
the radial probability function is multiplied by a function called a
spherical harmonic, which tells us how the radial distribution has to rotated about each axis to generate the 2D and 3D plots. The 2D plot is essentially a cross-section of the 1s orbital. Also shown is our 3D model. S-orbitals are also called s-waves.
Above: left, radial wave functions and 2D plots of the 2s (200) hydrogen atomic orbital. Right, top -
greyscale computed 2D probability density distribution and our 3D model, bottom right. Note that
s-orbitals (1s, 2s, 3s, ...) are spherically symmetric, so that the radial probability distribution applies
along any radius of the atom - it is simply rotated about two of the axes. The 3D model this time shows
the natural fuzziness in the PDF of the orbital (the model for the 1s orbital omitted this and constructed
a hard smooth surface). All orbitals are in reality similarly diffuse.
When we get to the 2p orbital (210, i.e. n=2,
l=1, m=0 orbital) the radial probability function
no longer tells us all we need to know about
the geometry. The spherical harmonic causes
an angular deviation from a spherical shape.
The 3s orbital is spherically symmetric like the 1s
and 2s orbitals. However, notice that the peak of
the 3s PDF (bottom left) is further out from the
atomic centre at the origin than for the 2s orbital.
This trend continues - the orbitals get larger as
the first quantum number, the principle quantum
number (n) increases. This quantum number n
corresponds to the energy of the s-orbital, so
more energetic orbitals are larger and focused
further from the atomic centre.

Notice also that the PDF of the 1s orbital has no
zero outside the centre (but gradually decays to
zero after reaching a peak) but that the 2s orbital
has one such zero and the 3s has two such
zeros. These zeros form thin spherical shells at
which the electron is never found and they are
called
nodal surfaces. The 2s nodal surface is
visible as a blue ring in-between two red rings
9where the electron is most likely to be found) in
the 2D colour plot, which is a section across the
orbital. There are n-1 nodal surfaces for an
s-orbital.
Above the 3d (320, n=3, l=2, m=0) orbital. From the 3d orbital onward, the shapes become more complex.
This orbital has two main lobes either side of a central torus. D-orbitals are also called d-waves.
In general a hydrogen atomic orbital has
n-l-1 nodal surfaces (l=0 for an s-orbital
giving n-1 nodal surfaces). Since, for a
p-orbital, l=1, there are n-2 nodal surfaces.
Orbitals for which n=l-1 have no nodal
surface and just a single peak. For the 3p
orbital there is one nodal surface (n=3, l=1,
so n-l-1 = 1) giving rise to the first minimum
in the PDF and separating the main lobes
from the smaller lobes toward the centre.

The 2D plots for these atomic orbitals were generated using software written by Bot for Windows using visual C# for the .NET framework. Below is a screen shot of this software, called OrbPlotter (I eventually changed the name of the WinForm from its default 'Form1'), which has just finished plotting a 2s orbital in greyscale. This program used a very early version of C# and is slow to execute due to the limitations of the class library available at the time, and/or my understanding of it: the way the code handles the bitmaps is inefficient. I suspect if I rewrote the code on a more modern platform I could speed it up). Note that the intensity (darkness) of the plot can be preset as desired (if we plotted them all on relative scales then some orbitals would come out pale and diffuse):

The shapes of the hydrogen atom atomic orbitals are given by solving Schroedinger's wave equation for an electron trapped inside a Coulomb potential well. The Coulomb force is the force due to the electric attraction between two electric charges, in this case the attractive force between the negatively charged electron and the positively charged nucleus (in the hydrogen atom the nucleus contains only a single positively charged proton). A potential energy well is a force field that keeps a particle in place, rather like a water well in which water has to be raised against gravity to lift it out (the physical water well is also a gravitational potential well due to the gravitational force field of the Earth). In a similar manner, work must be done (and energy supplied) to an electron to pull it away from the nucleus.

According to the Schroedinger equation, a particle behaves like a wave (as indeed they do) and so the solutions to the equation, which tell us the behaviour of the particle, are waves, called
wave functions. Essentially, a particle in an energy well is like a water wave trapped in a harbour - it bounces about between opposite 'walls' and interferes with its own reflection to establish a stationary wave which appears not to be travelling from one wall to the next but simply moves up and down. The walls in the case of the atom are provided by our Coulomb force-field - when the electron flies too far from the nucleus it is pulled back toward it.

Now, on Cronodon we don't normally include much in the way of maths (unless it's something unusual), since this can be found in standard textbooks for those who want to understand such technical things. However, Bot has produced a pdf explaining the solution of the Schrodinger equation for both the atomic nucleus and the electron in the hydrogen atom.
Download the Schrodinger pdf. Alternatively, you can simply follow the general gist of the method with the following background information:

Here, the h with the bar across it (h-bar) is Planck's constant (h) divided by 2 x pi. This constant determines the scale of quantisation, or the graininess of energy, since energy and momentum are always found in multiples of Planck's constant. V is the force field, in this case the Coulomb potential, m is the mass of the particle (the electron mass in this case) and E is the energy of the system. A particle not confined by a force field is called a free particle and is also described by the TDSWE. The TISWE is a second-order linear partial differential equation and can be solved by a mathematical technique called separation of variables. Essentially this separates out three solutions, one for each variable, the variables here being spatial coordinates. The best method uses spherical polar coordinates, in which each point is specified by its radial distance (r) from the origin, its azimuthal angle (phi) from the positive x-axis, and its zenith angle (theta) with the positive z-axis.

Above: the position of the point P (on the surface of a sphere of radius r) can be described by the standard Cartesian coordinates x, y and z. However, for problems involving spherical symmetry the maths gets much easier if one uses spherical polars (these are also more intuitive for such problems once you are used to them).

Separation of variables then gives us a solution (wave function) consisting of three functions multiplied together: a radial function for r and an angular function for theta and an angular function for phi. The two angular functions together form the
spherical harmonics and the radial function is as plotted in the figures above. The 2D plots used the radial function multiplied by the angular function for theta.

Finally the functions have to be normalised, that is they must sum (by integration) to one. This is because the functions describe the probability of finding the electron at a particular location in space, and since the electron must be somewhere the probabilities must add to one! Both the radial functions and the spherical harmonics must be normalised to give accurate probabilities. The 2D plots only illustrate the shape of the orbital and so normalisation is not important for this, since normalisation does not alter the shape only the scale on the axes of the graph. To obtain radial plots with the right shape again normalisation is not required, however, to make the vertical axis read correct probabilities the radial functions have been normalised. Here, let Bot add a note of caution! Many textbooks appear to state the incorrect normalisation. Only by plotting the functions does this become obvious and other authour's doing similar plots have found the same problem. Without resorting to calculating each one directly (quite a bit of work) Bot found two quick methods that gave (apparently) correct results (although one of these methods, which uses recurrence formulae, appeared to be only accurate for s and p orbitals). The method given in The Picture Book of Quantum Mechanics (Brandt and Dahmen, Springer-Verlag) seems to be correct. Normalisation is also well explained in materials produced by the Open University. Nevertheless the intensity of each plot has been adjusted for ease of visualisation: d and f orbitals are diffuse and pale when plotted on the same greyscale as the s and p orbitals.

Controversy! The Shapes of Orbitals

The shapes of the hydrogen atomic orbitals which we have derived here are called complex orbitals because the wave functions are complex (involving the i, the square-root of -1)! This isn't particularly a problem, since what is actually observable is the square of the wavefunction (this gives us the probability wave) which removes i (i x i = -1). However, these solutions do impose an apparent geometric direction to the atom, and in the absence of external magnetic fields (or in a spherically symmetric field) we might expect it to have an overall spherical symmetry. For example, consider the two 2p-orbitals shown below, which differ in magnetic quantum number m:

In this case, the radial PDF is plotted along the long-axis (the vertical; axis) of the 2p orbital shown in
the 3D model above and so shows the probability density along one of the two lobes of the 'dumbbell'
shape. There are two other p-orbitals - the 211 and 212 orbitals (not shown). These three 2p orbitals
have very different shapes, however they are easily perturbed by neighbouring atoms in which case
they average out to form three 2p orbitals which have the same dumbbell shape but directed along a
different perpendicular axis in each case (hence these are labelled the 2px, 2py and 2pz orbitals). P-orbitals are also called p-waves.
Note that atomic orbitals have no
visualise an orbital with the
darkest regions indicating the
highest probability of finding the
electron at that position. We can
thus visualise say the total area
in which the electron is 90% or
99% likely to be found simply by
altering the intensity of the plot.
Two different intensity scales
have been used here (not
normalised).
(m=0) orbital.

Right: the 3d (m=2) orbital.
Note the two inner nodal planes (spheres) where the rings
left). The electron density also drops off toward zero in the
centre of the atom. Note, however, that the probability of
finding the electron is only nominally zero in these regions
at mathematical 'points' as shown in the radial plot. Hence,
we should not think of the electron being absent near
these regions - raising the intensity of the plot will narrow
the white rings visualised in the plot as this area is really a
very pale shade of grey where there is a low but
nevertheless definite probability of finding the electron.
Above: with m = 3

Note that these orbitals are radially symmetric
about the vertical axis: rotating each image around
the vertical by 180 degrees will generate the 3D
shape (some of the apparent 'lobes' are actually
toroidal belts). I made these plots quite intense
(dark). The f-orbitals are very diffuse and have been darkened in these images. F-orbitals are also called f-waves.
Above: with m = 0
Above: with m = 1
Above: with m = 2

2p orbital with m = 0

2p orbital with m = 1

On the face of it these orbitals look like those we are used to: two dumbbell shaped orbitals
perpendicular to one-another. However, to obtain their actual 3D shapes we have to rotate them about
the vertical axis, so the m = 0 orbital gives us the usual dumbbell shape, whilst the one with m = 1 (and
similarly the third with m = -1) gives us an approximately torus-shaped orbital! Note, however, that these
orbitals have the same energy (in the absence of an external magnetic field) and when superposed we
have a sphere! This is illustrated below:

With the 2p orbitals having equivalent energies there is no reason to assume that a single electron would be confined to one or the other of these orbitals. In this case there would be no preferred spatial direction (caveat: we would have to check that normalisation gives a spherically symmetric distribution of probabilities, which we have not done here).

The alternative solutions, typically shown in chemistry books, are called real orbitals. This does not mean that they are any more physical, but that the appearance of i (the square root of minus one) in the wave functions is avoided by taking linear superpositions of the stationary states (or at least the spherical harmonics) such that the solutions are mathematically real. This aims to avoid directionality by resulting in three dumbbell-shaped orbitals at right-angles to one-another. A similar device is used to construct 'real' d-orbitals.

Although a linear superposition of stationary states is also a solution of Schrodinger's wave equation, it is no longer generally a stationary state (indeed, this is how we obtain motion in quantum mechanics: by superposing stationary states) and following measurement, the electron must be found in a stationary state. Specific combinations of the stationary states (eigenstates) may also be eigenstates, however, such as the triplet state in electron spin which is an entangled state of two electrons consisting of a linear combination of spin up and spin down which is an eigenstate of certain measurements of spin on the electron pair. The complete set of eigenstates, whether simple or mixed, must all be independent of one another, so the choice is not arbitrary.

The rational given for using these real solutions is that they are superpositions of different states with the same orbital quantum number, l, but different magnetic quantum numbers, ml. Since these energy levels are degenerate then a linear combination of them is also a solution to Schrodinger's equation with the same eigenvalue (the same energy) and is mathematically allowed. The motivation is to make the 3 orbitals geometrically equivalent, so there is no preferred axis. However, what we end up with is a new basis using Cartesian axes (and hence 3 preferred axes in any particular atom) rather than a basis in spherical polars. Does this matter?

Of course, if a directional external magnetic field is applied to the atom, then the degeneracy of the p-orbitals is lifted (they have different energies) and the atom would indeed have a 'preferred direction' in that the energies of the eigenstates would change as some orbitals become energetically favoured (perhaps the spherical symmetry of its electron orbitals would be broken and the new symmetry would align with the magnetic field in some way to also change the shape of the set of eignestates?). A change in the magnetic field must impose a change in the eigenstates resulting from measurement (even though any set of eigenstates can be constructed from any other mathematically since each set is 'complete'). If instead of Schrodinger's equation we use Dirac's equation, which factors in relativistic effects, then the degeneracy of the p and d-orbitals is in fact lifted ( the fine structure of the atomic spectra due to spin-orbit interaction: the electron reacts to the internal magnetic field of the atom). The orbitals of chemistry become less good approximations.

Personally, I favor the complex solutions as being the more likely eigenstates, or some linear combination of them that gives the isolated atom spherical symmetry. To be more exact, however, we would need to use Dirac's equation, or more accurate still quantum electrodynamics and consider any external fields around the atom, however weak. However, no direct measurements of the shapes of p-orbitals have so far been possible, though some measurements have been made on other atomic orbitals (see quantum measurement) and these confirm theoretical expectations. It should also be remembered that while the wavefunction describing the orbital itself has no physical meaning, its square does give the probability distribution of the electron and is, potentially, obesrvable. Squaring also removes the complex terms to give real solutions. In the end we need empirical observation to verify our theories. The real orbitals perhaps remain a more convenient model of atomic orbitals when considering how atoms bond together to form molecules.

Additionally, an important requirement for each subshell is spherical symmetry. The complex p-orbitals of physics add to form complete spherical shells. The 3 p-orbitals of chemistry do not seem to do this exactly, but may perhaps better describe the contribution orbitals make to covalent bonding (in bonding orbitals may combine in linear combinations to achieve the observed molecular geometry). It seems to me that the p-orbitals of chemistry are convenient approximations, useful to chemists in analysing molecular structure.

Another issue arises with the d-orbital solutions: there are six solutions on geometric grounds but there are only 5 independent solutions to Schrodinger's wave equation. The basic shape is 4-lobes with two nodal planes at right-angles passing through the nucleus. Three orbitals: dxz, dyz and dxy, have the same shape but with different orientations in space. The fourth orbital, dx2-y2 is also the same shape and identical to the dxy orbital but rotated through 45 degrees. Two more geometric possibilities exist: dy2-z2 and dz2-x2, but there are only five independent eigenstates and these latter three solutions are not all independent (if we superpose them then they tend to cancel out one-another). The fifth, therefore, has a very different shape since it is the superposition of the remaining two (this is OK as long as the superposition is an eigenstate):

Ψdz2 = 1/√3(Ψdz2-x2 - Ψdy2-z2)

Some texts view the choice of the 5 as arbitrary, to the extent that one of the six gemetric possibilities can be dropped. However, the exact solutions should once again add to give exact spherical symmetry. Certainly, the actual state the electrons are in will in general be a superpositions of these eigenstates (which form a basis rather like the x, y and x axes do when we plot a graph) but a (strong) measurement will collapse the system into an eigenstate so the eigenstates do have physical meaning: their square describes the probability distribution of the electric charge around the nucleus following a measurement, so there choice is surely not arbitrary. Other arguments state that the choice of basis eigenstates (eigenbasis) is arbitrary, which is true to an extent, for example we have a choice when plotting a graph of what axes to use, but this choice must have no effect on the measurable physics of a system.

It is worth stating that certain features of the shapes of atomic orbitals can be indirectly verified by experiments without measuring the shape directly. For example, the observed lowering of energy in the 4s orbitals can be explained by the fact that their electron clouds pass through the nucleus, whereas in p and d orbitals they do not (p and d orbitals have nodes at the nucleus). Thus an electron in an s-orbital is more likely to be found near the nucleus, lowering its average energy. The d-orbitals are also very spread-out and diffuse which explains their very weak tendency to form molecular bonds (so-called delta bonds are very weak and make a very small contribution to molecular bonding). The energy values of atomic orbitals can be measured with precision and since the shape of the wavefunction impacts upon these energies, the shape is evidently physical and not arbitrary, even when it can not be directly measured.

When describing the energies of the orbitals in an atom there are other factors that need to be taken into account, apart from shell number and angular momentum (subshell number). The angular momentum of the electron in hydrogen interacts with the spin of the electron (both motions generate magnetic fields that interact). The s-orbitals have zero angular momentum and are not effected, but higher orbitals are. The 2p orbital, for example, is split into two closely-spaced energy levels. These energy levels are hard to distinguish as the energy separation between them is very small (microwave range) and Doppler broadening of the spectral lines 'blurs' the distinction between them. This is called spectral fine structure. The spin-orbit interaction responsible can also be explained as relativistic effects and relativistic corrections predict them.

If we take a hydrogen atom, for example, then its 1s energy level is split into two very closely-spaced energy levels, which require high resolution to distinguish as they are so close together. This is called spectral hyperfine structure. The energy absorbed or emitted by transitions within the pair of closely-spaced 1s energy levels (the 1s doublet) gives the 21 cm (radio-frequency) hydrogen line of importance in radio-astronomy. This splitting is due to magnetic (spin) interactions between the electron and the proton. The 2p energy levels produced by fine structure splitting in the hydrogen atom are each split further into a pair of energy levels in the hyperfine structure, giving a total of four 2p energy levels (for 3 electrons). What effect do these and other corrections have on orbital shape? (Other inaccuracies may stem from the fact that the Schrodinger equation is a linear approximation: what effects do non-linear terms have?).

In the end I think the only way to settle these disputes is by empirical measurement. Others have argued that the wavefunctions themselves are just mathematical tools that have no physical meaning as they can never be measured: however, the squares of these wavefunctions can potentially be measured and in a few cases such measurements have recently been carried out. It is true that properties we associate with matter, such as color and texture, are only manifest in large collections of atoms, and no individual atom can be said to have a 'color' but individual atoms nevertheless have a range of measurable properties.

See the section on quantum measurement for more information on the shapes of atomic orbitals and the possibility of measuring them.

Do Orbitals Exist?

A number of scientists have claimed that orbitals do not exist, that they are simply mathematical tools and so one can choose any set of such orbitals. First of all, even on mathematical grounds we are not free to choose any linear combination of the eigenfunctions (since any function can be represented in this way as the eigenbasis is complete): we can only choose a set of orbitals that form a complete set of basis eigenstates and which correspond to stationary states of the system and our choice of basis should not change the observable physics anymore than a choice of axes would. (We would not conclude that the three dimensions of space and every spatial entity therein do not exist simply because our choice of axes is arbitrary!). Many linear combinations describe motion or entangled states and we are certainly not free to choose them without radically altering observable physics.

The predictions of quantum mechanics are clear: the square of the wavefunction gives us the probability of observing the position of the electron in a region of space if we measure its position within a specified degree of accuracy. (It is possible to extract other information, of course, such as momentum and energy, but we are not concerned with these observables in this argument). This is a prediction of a real system that can, at least in principle, be tested. Furthermore, after a measurement the system is predicted to be in one of the eigenstates (e.g. 1s, 2p, etc.). However, in between measurements there is nothing to stop the system evolving to be in a superposition of states, e.g. one-third of each of: 1s + 3d + 4s. Indeed, such superpositions can account for motion in bound systems. Physicists would agree that orbitals do not consist of electrons following well-defined trajectories as planets do around the Sun (since this would violate Heisenberg's uncertainty principle). Nevertheless they do make observable physical predictions which are not arbitrary.

Physics essentially makes no other claims on the matter, except that a large number of measurements performed on an ensemble of identical systems, or the same system in which sufficient time elapses between each measurement, would recreate the pattern of the orbitals, and also that this pattern effects the readily observable energy levels and other observable properties such as electric radius. In a similar way, photons striking a screen reconstruct the wave pattern predicted in a two-slit diffraction experiment. In what way could we argue that the wave describing the pattern does not exist? Indeed, we could use any other mathematical wave as long as it gave the same observable outcome but we would require our method to be consistent and to work in other situations to qualify as a physical theory. The observable outcome is the only definite reality. Of course, the model may not be completely correct, but as long as it makes testable predictions and those predictions match experiment then it can not be simply rejected on philosophical grounds. Physicists generally take a modest viewpoint: what can be empirically observed is the only 'reality', the other underlying mechanisms of the models are a means to an end. The best models make the most reliable predictions in the widest possible number of scenarios without introducing unnecessary complexity.

While it is true that the Schrodinger model contains inaccuracies (and can be improved upon using Dirac's equation or even more so using quantum electrodynamics (QED)) this does not invalidate the model any more (or less) than general relativity invalidates the Newtonian concept of planetary orbits. It is also true that in multi-electron atoms the electrons become indistinguishable, but the physics is still able (within computational limitations) to predict the probability of finding electron charge in any given region of space. Orbitals have a certain lack of individuality. We could then certainly argue that we are free to use any set of atomic orbitals that reconstruct the observable outcome, but the choice should be consistent and work for the single-electron hydrogen atom too. If our methods are inconsistent, for example, if we use a set of atomic orbitals that accurately predicts molecular shapes but fail to describe isolated atoms then our set is merely a convenient tool rather than a physical theory. Scientists are free to do this, but they can not readily reject physical theories without first subjecting them to rigorous empirical tests. We may be unable to visualize what an electron is, and model it mathematically, but we are nevertheless quite certain that our concept of the electron corresponds to some observable aspect of cosmic reality. (Again i am restricting 'reality' to mean that which can be empirically observed and nothing more).

Finally, a number of experiments have observed atomic orbitals, or rather observed the electron distributions that these models predict, in both atoms and molecules and these predictions match the theory surprisingly well. More observations are, however, needed to fully test the model. Of course, there is the issue of what is an orbital - to what extent does a mathematical model mirror reality, but to dismiss orbitals is to dismiss the whole of physics, since the entire subject relies on theoretical models, and indeed chemistry and biology too. This last point is well made by Labarca and Lombardi 2010 (Why atomic orbitals do not exist? Foundations of Chemistry 12(2): 149-157). Indeed, if we dismiss atomic orbitals then we must dismiss molecular orbitals and molecules too. I think it is better to think of atomic orbitals as a prediction made by a model and a prediction that can, at least in principle, be tested (and indeed has been tested in a number of ways albeit incompletely) rather than to be so sure as to dismiss them altogether! Atomic force microscopes and other developments are beginning to allow us to measure the position of electron charge in atoms and molecules, as predicted by quantum mechanics.

Conclusion

The controversy over atomic orbitals can be summarized as follows: chemistry adopts a set of atomic orbitals that form a useful tool for constructing approximate molecular orbitals and so tend to view the choice of set as arbitrary. For these purposes an arbitrary set is fine, but this should not be confused with observable physics. Nevertheless, measurements on the shapes of molecular orbitals have been performed more readily than those on atoms and the shapes confirm the predictions of molecular orbital theory. If the molecular orbitals obtained from atomic orbitals match observation, then the atomic orbitals must have some reality. Physics realizes that the shapes of the atomic orbitals correspond to a measurable observable, though actual measurements so far have been limited by practical constraints, those that have been made support the theory. Physics also realizes that orbitals derived from Schrodinger's model are an approximation: Dirac's relativistic theory is more exact and further refinements can be made. The actual form of the probability clouds will also depend on the environment the atom finds itself in.

Article updated: 15th Feb 2016, 22 Aug 2020, 23 Aug 2020, 15 Sep 2020, 25 Sep 2020