ORBITAL for Web Browsers

It was written by Dr. Bruce W. Tattershall, Lecturer in Chemistry at Newcastle University, England.

What is it for?

• ORBITAL is an aid to teaching and learning about molecular orbital theory in chemistry courses at an elementary undergraduate level

What does it do?

• ORBITAL shows Atomic Orbitals or Molecular Orbitals on a computer screen, as coloured contour plots
• It provides simplified mathematical functions which are of the same form as the space-dependent parts of some of the one-electron wavefunctions for an atom
• Atomic functions can be shown, or functions can be added to make MOs for a diatomic or a linear triatomic molecule
• The program is a tool for students to use
• It was originally devised for use in a drylab situation in which students were taught interactively by a demonstrator while they carried out loosely specified exercises using ORBITAL
• It can be used in private study by students, but probably should be accompanied by a tutorial or a quiz provided to fit in with the course they are studying, suggesting work to do using ORBITAL
• Teachers possibly could show it as a lecture demonstration
• The student controls what the program does
• There is no tutorial or programmed learning nature to ORBITAL, beyond what is provided in this help facility

Realism

• Although the functions have been simplified by removal of almost all of the combinations of constants which look confusing in textbooks, the resulting plots are very similar to those obtained with properly calculated AOs
• The forms of some of the functions used are illustrated in a table available online
• This emphasizes that understanding the algebraic form of AOs is within the grasp of almost all university chemistry students
• Commercially available modelling packages may produce more accurate surfaces (or possibly contour plots) for orbitals, but the algebraic form of the functions used is generally hidden from the user
• If used for teaching or learning, the learning objectives are different:  these packages are not meant to promote understanding about how MO theory works, whereas ORBITAL sets out to do this
• Ready-made rotatable models of atomic orbitals or of diatomic molecular orbitals are among the excellent web publications of Dr. P. Falstad, but these also list the orbitals by name, rather than giving the functions used

Recognising the AOs

• Students select the atomic wavefunctions by their algebraic form, rather than by their name
• This exercises students' understanding of the algebraic nature of natural orbitals, nodes, etc.
• Wavefunctions are given in the Cartesian coordinate system, rather than in polar coordinates
• The author finds that students find it much easier to understand and assimilate these functions, and it is anticipated that they will be taught about them in this form
• Molecular modelling using MO methods, as used by chemists, employs Cartesian coordinates
• Polar coordinates were used in the middle years of the 20th century, before the computer era, when students learned to solve Schroedinger's Equation for the hydrogen atom by hand

Plane of the plot

• Atomic orbitals are plotted in the xy plane:  thus 3d(xy) and 3d(x²-y²) orbitals are provided, but 3d(z²) is not
• The molecular axis must therefore be the x axis, not the z axis which is the convention
• The x axis is horizontal across the screen, y is vertical down the screen (and z, which is not used, comes out of the screen towards the user)
• The nuclei lie in the plane of the plot, and are represented by black dots

Colour of the contours

• The contour bands are coloured with warm colours green to purple to represent positive values of the functions, and with cold colours cyan to deep blue to represent negative values

Nodes

• Values near zero are coloured neutral mid grey, so nodal surfaces intersecting the plane of the plot are easily recognised
• Students should exercise their knowledge of both AOs and MOs by identifying and counting the nodes in the plots
• The idea that the wavefunction has zero value at a node is of infinitely little importance to chemistry, because the nodal surface is infinitely thin
• What is of supreme importance is that the wavefunction changes sign at the node:  see the help item on animation
• Nodes are not shown as infinitely thin in ORBITAL, because the student needs to see them
• The coloured bands represent equal ranges of the value of the wavefunction, and the grey band spans zero symmetrically (except in the 1s orbital), making it wide enough to see easily

Making MOs

• MOs can be made by adding one AO at a time from each atom
• Sufficient AOs are provided to make σ bonds from s or p orbitals, or π bonds from p or d orbitals

Coefficients

• Students enter coefficients for each AO
• By varying the signs of these coefficients, bonding or antibonding MOs can be made
• Students thus learn to use the rule that the sign of the product of two coefficients must match the sign of the overlap integral of these two AOs, in order to produce a bonding contribution to the MO
• They should set out to produce a bonding or antibonding combination in each case, and confirm by counting the nodes in the MO that they have obtained the intended outcome
• By varying the magnitudes of the coefficients, polarised MOs can be produced
• Students should decide in advance which nucleus in the plot is to represent which nucleus in the molecule
• By examining the shape and colouring of the lobes, they should confirm that the polarisation shown agrees with their assumptions about relative electronegativity of the atoms, in both the bonding and the antibonding MOs
• Antibonding MOs are polarised in the opposite direction to a corresponding bonding MO

One overlap at a time

• Because understanding bonding or antibonding contributions is a main learning objective in using the program, use of more than one AO at a time per atom, e.g. for s and p orbital mixing in σ MOs, is not supported

Distance between the atoms

• The distance between the atoms can be varied, showing the progression from almost pure AOs to highly bonding or antibonding MOs, as the overlap integral becomes bigger in magnitude

Triatomic molecules

• The important ideas which can be explored with the linear triatomic molecule are:
• σ and π MOs each delocalised over all three atoms
• the non-mixing of s and p on the central atoms, because they require different combinations of signs of coefficients of the ligand orbitals
• there are π MOs which are neither bonding nor antibonding:  they are non-bonding by symmetry
• ORBITAL's triatomic molecule does not bend
• Understanding what happens to MOs if the molecule bent is an interesting and important topic, but it is more advanced than is usually covered in courses at the level which ORBITAL is meant to support
• As the symmetry is lowered, more mixing of AOs into MOs takes place, and the situation becomes rapidly more complicated
• The simple approach of ORBITAL, in which only one orbital at a time per atom is considered, becomes too limiting

What are the molecules?

• Since ORBITAL does not set out to make a realistic plot for a particular molecule, this is a matter for the imagination of the student or their teacher
• Diatomic molecules could be H₂, F₂, O₂, NO, or HF
• Triatomic molecules could be CO₂ or N₂O
• The triatomic combination could be part of a larger molecule:
• two trans ligands surrounding an octahedrally coordinated metal
• the metal-carbon-oxygen sequence of carbonyl complex, in which the d-π* π-bonding can be depicted
• Rotatable ab initio models of MOs, for comparison with the ORBITAL plots, are available for CO and for CO₂
• While a 3p(x) orbital is provided to support learning about inner lobes, 2s is not provided
• Because atoms do not approach sufficiently for major overlap of the inner lobes of their AOs, it is suggested that 1s and 2p AOs are used in constructing MOs, even where 2s, 3s, or 3p orbitals are involved in 'reality'
• This gives MOs which are of the correct form in the bonding region, but which are oversimplified in the core regions near the nuclei
• This approach is often taken in elementary textbooks to avoid unnecessary confusion

Animation

• Our ideas of σ and π bonding, or of bonding, non-bonding and antibonding orbitals, which have carried over from small inorganic molecules to the most symmetrical of organic molecules, are all about nodes, i.e. the sign of values of the wavefunctions at different points in space
• Students often need to work hard to lose the concept of electrons in atoms as particles, which is very harmful to their understanding of chemistry, and to replace it with the concept of electrons as wave phenomena
• What is being plotted in ORBITAL, and in elementary textbooks, is the space-dependent part of the wavefunction, i.e. a snapshot at some instant in time of a parameter which varies in a wavelike manner
• Lobes on opposite sides of a nodal surface have opposite signs because the wave phenomenon is out of phase in these two regions
• It does not matter which region contains positive and which contains negative values, because a snapshot at a different instant could have shown the signs reversed
• This is illustrated by the animation provided in ORBITAL, in which the space-dependent wavefunction set up by the student is multiplied by a sinusoidal time-dependent function
• This causes values of the total wavefunction, and hence the colours and extents of the contour bands to change periodically, from the maximum values shown in the static plot, down through zero, and up to the maximum values but with the opposite signs
• Students are advised to turn down the speed of the animation until they fully understand what they are seeing
• The animation may have some reality for single-electron atoms in a steady state:  for polyelectronic atoms, the time-dependence is very much more complex
• The animation is provided mostly to encourage students to think about the meaning of the static space-dependent plots, which are similar to those shown in most elementary textbooks on the subject, but which are frequently misunderstood by beginners

History of ORBITAL

• ORBITAL has been used for teaching first year Chemistry students the basics of LCAO for every Honours intake year since 1976
• The present author wrote the ancestor of ORBITAL in 1976 in Hewlett-Packard Basic
• Students accessed a time-shared computer via a 10-characters per second teletype and a telephone linkage
• The program printed different upper-case letters to represent contour values, with white paper in between to aid visibility
• A plot took about 12 minutes to print
• Students changed function definitions by editing the program before running it
• The idea came from L.J. Soltzberg, J. Chem. Educ., 1972, 49, 357-361
• By 1985 the program had been translated into various dialects of Pascal for use in stand-alone microcomputers, but was still being used in an edit-compile-run mode
• Monochrome screen output still used letters to represent contour values, but was a lot faster than teletype output
• In 1993, Dr. Chris M. Smith assisted in translating ORBITAL into Prospero Pascal, and used its colour graphics capability to produce coloured contours similar to the present version
• A sequential output-input conversation was used to set up all required parameters
• The general nature of the previous versions, in which students could enter any function (in Pascal), was sacrificed in favour of selection from a menu of predefined functions in a pre-compiled program
• In 1999, the ProPascal version became unusable because of purchase of PCs with non-compatible graphics adapters
• In 2003, a decision was made to translate to a stand-alone Windows application, using ClearWin+ Fortran, because of the greater ease of using interactive controls and particularly of implementing user-determined animation
• In 2018 a translation was made to use web browser-based HTML controls and the JavaScript language

Introduction Contents

Using ORBITAL

• ORBITAL loads with an atomic orbital already selected
• This may be changed using the drop-down function list for atom 2, or the X Range may be changed
• Any change except setting the Animation on, results first of all in the plot being greyed, to show that it no longer corresponds to the selected settings
• Most details of the plot can still be seen, to help judge what new settings, e.g. Range, are required
• Once the plot has been greyed, the controls respond quickly, so that e.g. the type-in box for Range keeps up with movement of the slider below it
• Small buttons may be provided by the browser in use, at the side of the type-in boxes:  these are 'spin wheels':  if they are kept pressed with the left mouse button, the value is stepped by pre-set amounts
• When the plot is greyed, the Redraw Plot button at the top right of the display is ungreyed
• Press Redraw Plot to remake the plot using the new settings
• X Range controls the square xy region to be plotted
• A higher value makes the orbital seem further away
• Beware of using too low a value, i.e. zooming in too far, as this could result in essential features of the orbital being outside of the field of view
• The same units are used for Range as for the interatomic Separations, but because the supplied functions are simplified, these are arbitrary:  they do not correspond to Ångstrom units nor to Bohr radii
• Animation may be stopped by clicking the Stop button
• Because animation stops at an arbitrary magnitude, the plot is left greyed and Redraw Plot should be pressed

Atomic Orbitals

• Atomic wavefunctions are plotted if any one of the three atoms has an algebraic function selected and the other two have None
• The coefficient typein boxes are greyed, and even if they are set already, they have no effect
• The interatomic Separations have no effect

Molecular Orbitals

• Molecular wavefunctions are plotted if any two, or all three of the three atoms have algebraic functions selected
• The bond direction is horizontal, i.e. along the X axis
• Use 2p(x) for σ bonding or antibonding functions and 2p(y) for π bonding or antibonding functions
• Left separation is between atoms 1 and 2, and right separation between 2 and 3
• If functions are selected only for atoms 1 and 3, the sum of the two separations may be used to select a much larger interatomic distance:  e.g. use a separation of at least 20 units, made in this way, to display σ bonding using 3p(x) functions
• Separations of 3 units may be suitable for π bonds using p orbitals, or for bonds using the 1s orbital
• A separation of 3 units is too short for σ bonds using 2p orbitals:  separation 5 units is suitable
• The 3p(x) function is supplied mainly for display as an atomic orbital (use a large X range)
• Remember to increase X Range to include the separation selected as well as enough range for the lobes themselves
• Selecting too large an X Range can easily be corrected, but selection of a too small range may lead to wrong conclusions
• Coefficients may be typed in, or, if 'spin-wheels' are provided, these may be used to adjust the current value
• The range of these spin wheels is less than may be typed in to the boxes
• For polarised bonds, if coefficients are set to above 1 for some atom(s) and below 1 for the other(s), the range of the spin wheels is sufficient to represent any reasonable degree of polarisation
• Clicking to the left of a coefficient allows a minus sign to be inserted, but some browsers may not provide a minus sign for numeric input
• Because of this, a ± button is provided for each coefficient
• This conveniently reverses the sign of the coefficient, and should work in all browsers

Modification

• The code of ORBITAL for Web Browsers is entirely client-based, so may be used offline, if the available browser allows that
• This means that the program is essentially open-source, since a user can view and save all of it, using the View Source facility of the browser
• A user with sufficient programming skills could modify the code, e.g. to change the list of functions provided
• If ORBITAL is republished online, modified or not, it would be reasonable to acknowledge its original author

Feedback

Thanks very much.

Bruce Tattershall
Chemistry in the School of Natural and Environmental Sciences
University of Newcastle
Newcastle upon Tyne
England

Email:       Bruce.Tattershall@ncl.ac.uk
Website:   http://www.staff.ncl.ac.uk/bruce.tattershall/