This spreadsheet is designed to show that atomic orbital wavefunctions for hydrogen-like atoms can be calculated and visualized on a personal computer. It enables the user to calculate values of the wavefunction at points on an x-y grid in a slice of 3-dimensional space which may include the nucleus or may be offset from the nucleus by a displacement in the z direction. It also uses a very crude visualization technique.
To start, you should download the software to your computer. The spreadsheet is written in EXCEL 4.0 for the Macintosh format, and has been binhexed (saved as a .hqx file) for ease of network transmission, but hopefully it can be read out and used on any Windows or Macintosh computer just by clicking here and then opening the downloaded file "wf98.xls" using Microsoft EXCEL (you must have EXCEL vs 4 for Mac or EXCEL '95 or later for Windows). If your browser has difficulty downloading and opening this file, you can try downloading this compressed version. Please let me know if neither type of download works on your computer.
When you open the spreadsheet, you should see something like the picture below (what you see will be slightly different since some updating of the software has occurred, and the green writing won't be on what you open with EXCEL). The spreadsheet is showing a 3dz2 orbital for a C5+ ion (a hydrogen-like atom with Z = 6 and one electron). (The green writing are anotations that you won't see.) The z-axis for the 3dz2 orbital is the vertical axis. However, because the spreadsheet calls the vertical axis the y axis, this view of the 3dz2 orbital is called a "3dy2" orbital by the spreadsheet.
The y values have units of Å-3/2 so that the y2 (probability density) values have units of Å-3. In order to convert these to a probability of finding the electron in a given small volume, you must multiply the y2 value by the volume. For instance, to calculate the probability of finding the electron in the spreadsheet cell with the circled white "2" at x = 0.4 Å, y = 1.0 Å and z = 0 Å (all values are shown for z = 0 because of the value of "z(slice)"), multiply y2 = 0.2 Å-3 by the volume of the spreadsheet cell (each cell is 0.2 Å by 0.2 Å; assume that the square is in fact a cube, 0.2 Å on a side, so that its volume is 0.008 Å3). Thus the probability is 0.002, which means that the electron is in this cell about 1/500th of the time. Even in the most probable cells (at x = z = 0 and y = +/- 0.6 Å), the electron is present only 1 % of the time.
The distance from the nucleus (=( x2 + y2 + z2)1/2) is usually represented by r. To simplify the formulae for y, we "scale" the r by dividing it by rb which is the radius predicted by Bohr for the orbit of the one electron atom with the same Z and n as the orbital we are considering. Thus
r' = r / rb = r / [(n/Z) * 0.529 Å]
Similarly, the x, y and z values can be scaled by the same factor:
x' = x / rb = x / [(n/Z) * 0.529 Å], etc.
Values of r' are calculated in cells AY40:BS60 of the spreadsheet. Above and to the left, values of x' and y' are calculated. The value of z' is in cell AW8 (it will be zero as long as z(slice), cell A5, is zero).
The formulae for calculating y for orbitals of one-electron atoms can be represented as a product of a radial function and one or more nodal functions:
y = (radial fall-off) * (nodal function(s))
The "radial fall-off" is a function only of the distance from the nucleus. It is calculated in cells AY70:BS90 according to the formula:
radial fall-off = (π-1/2 rb-3/2) e-r'
Note that the radial fall-off function depends only on the distance from the nucleus, and on n (the principal quantum number) and Z (the charge on the nucleus). For instance, it will be the same function for any 3s, 3p or 3d orbital of a C5+ ion.
For any orbital with a principal quantum number of n, there are (n-1) nodal surfaces where the y = y2 = 0. Each of these surfaces can be described by the equation (nodal function) = 0. Multiplying the nodal functions together with a suitable constant gives the "nodal function(s)" part of the wavefunction. The constant part is shown in red below and is needed so all probabilities add up to one.
Here is a table of the nodal functions for the wavefunctions with n = 1, 2, or 3.
|
Orbital |
Nodal surfaces (planes unless noted) |
Nodal function(s) |
||
|
1s |
none |
(use nodal function(s) = 1) |
||
|
2s |
r'- 1 = 0 (a sphere) |
(r' - 1) |
||
|
2px |
x' = 0 |
x' |
||
|
2py |
y' = 0 |
y' |
||
|
2pz |
z' = 0 |
z' |
||
|
3s |
|
(r' - 2.37)(r' - 0.63) * (2/3) = (2/3) r'2 - 2 r' + 1 |
||
|
3px |
|
x' (r' - 2) * (2/3)1/2 |
||
|
3py |
|
y' (r' - 2) * (2/3)1/2 |
||
|
3pz |
|
z' (r' - 2) * (2/3)1/2 |
||
|
3dxy |
|
x' y' * (2/3)1/2 |
||
|
3dxz |
|
x' z' * (2/3)1/2 |
||
|
3dyz |
|
y' z' * (2/3)1/2 |
||
|
3dx2-y2 |
|
(x' - y')(x' + y') * (1/6)1/2 = (x'2 - y'2) * (1/6)1/2 |
||
|
3dz2 |
|
(31/2z' - r')(31/2z' + r') * (1/18)1/2 = (3z'2 - r'2) * (1/18)1/2 |
The nodal functions are calculated in cells AY10:BS30. Above this area are 15 cells from which formulae can be copied and pasted into AY10:BS30.
These are done in cells AY100:BS120 and AY130:BS150, respectively. The display of these functions in the upper left hand corner of the EXCEL spreadsheet makes use of cell formatting commands available in EXCEL.
By the way, the spreadsheet is protected to prevent users from accidentally changing a formula which shouldn't be changed. However, if you wish, you may "Unprotect Document" to make changes to the format, etc.
These workbooks are available either in Excel 98 format, or in an older format: stuffed and binhexed EXCEL 4.0 workbooks. The latter workbooks are optimized for smaller screensizes (480x640 pixels).
This workbook illustrates the formation of sigma bonding and antibonding orbitals by constructive and destructive overlap of orbitals on neighboring atoms.
The first two worksheets in the workbook (named "atom A" and "atom B") are "snapshots" of the wf98.xls worksheet for 1s orbitals of hydrogen (Z=1). The only difference is that the x scale on atom A goes from -2.5 to +1.5, while on atom B it goes from -1.5 to +2.5. This has the effect of offseting the nucleus by 1 Å along x when comparing the wavefunction of atom A with that of atom B. Thus, it is as if the two atoms are 1 Å apart.
The third worksheet ("sigma bond") has been modified from a normal wf98.xls sheet because the psi values are now generated by adding the psi values from atom A and from atom B. Thus the psi for the sigma bonding molecular orbital = psi(atom A) + psi(atom B). The yellow vertical stripes indicate the position of the two nucleii. Note that most of the electron density is concentrated in the region between the atoms.
The fourth worksheet ("sigma antibond") is similar to the previous worksheet, except that the formula for the antibonding molecular orbital = psi(atom A) - psi(atom B). Note that this makes the sign of the molecular orbital wavefunction negative on the side of atom B, and also creates a nodal plane between the atoms. Most of the electron density is concentrated on the left and right sides of the molecule (rather than in the region between the nucleii).
Strictly speaking, the molecular orbitals generated in this way as bonding and antibonding orbitals should be multiplied by constants so that the sum of the probabilities of finding the electrons (psi squared values) will add up to 1. The constant for the bonding orbital will be < 1 and for the antibonding orbital will be > 1. This spreadsheet has not included the constant in the calculations, so the probabilities for the antibonding orbital look smaller than for the bonding orbital. The constants were not included because there is no easy way to find the constant; it is found by adding up the psi^2 values for each "box" in the simulation (including simulations with z at values other than 0) and then correcting the psi values so that the next time this sum is made, the result will be 1.
The workbook can be modified to show pi bonding by the following sequence. I suggest changing the atomic number to 2 because this is an estimate for the effective atomic charge in a carbon or nitrogen atom.
This is simple enough to do, but you may also just download the PiBondWorkbook.xls (Excel 98 format).
This workbook shows how hybrid orbitals are constructed by adding together atomic orbitals. The particular example illustrated is the construction of three sp2 hybrid orbitals from the 2s, 2px and 2py orbitals. (These are easier to illustrate than sp3 orbitals because the lobes all lie in the xy plane, so that the z=0 cross-sections contain the regions of highest electron probability).
The first three worksheets within the workbook are "snapshots" of wf98.xls worksheets for the 2s, 2px and 2py orbitals. The three worksheets that follow generate the wavefunctions for the hybrid orbitals using the following formulae (0.57 is really sqrt(1/3); 0.82 is really sqrt(2/3), and 0.41 is really sqrt(1/6)):
Notice in which direction the hybrid orbitals extend the furthest away from the nucleus (reminder: the nucleus is at (0,0,0)).
Screen snapshots of this EXCEL spreadsheet can be viewed even if you don't have EXCEL on your computer by clicking here.