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.

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 (

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.

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.

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.

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

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.

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.

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):

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

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

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.

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).

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.

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

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

definite surface. Instead we

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).

definite surface. Instead we

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.

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.

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.

Comment on this article!

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).

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

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:

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

measurement for more information on the shapes of atomic orbitals.

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

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

measurement for more information on the shapes of atomic orbitals.