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 centre. (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 centre
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 centre, 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.
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 onwards, the shapes become more complex.
This orbital has two main lobes either side of a central torus.
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. 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.
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.
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).
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.
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).
Above: with m = 0
Above: with m = 1
Above: with m = 2
Article updated: 15th Feb 2016
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:
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:
2p orbital with m = 0
2p orbital with m = 1
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 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 apparently avoids directionality by resulting in three
dumbbell-shaped orbitals at right-angles to one-another. Although a linear superposition of stationary
states is also a solution of Schrodinger's wave equation, it is no longer 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.

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.
Personally, I favour the complex solutions as being the more likely eigenstates, however, no direct
measurements of the shapes of p-orbitals have so far been possible. 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. See the section on
quantum