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THE PROBABILISTIC INTERPRETATION OF ATOMIC ORBITALS (Part 2)

In this second part of the experiment, you will be examining the wave functions in more detail to reinforce your understanding of material from Cotton, Wilkinson and Gaus or Huheey, Chapter 2. The applet below allows you to display slices of constant z-coordinate, which you can set, through a number of different orbitals.

The image which is displayed illustrates the other way of interpreting the wave function, that is, treating the electron as a delocalized "cloud" whose density is related to y. The brightness of the colour that you see is related (logarithmically, for practical reasons) to the value of y2. The brightest regions correspond to high "electron density". The colour used in the different regions is determined by the sign of y: red is positive, blue is negative. The nucleus of the atom is shown as a yellow spot. Remember, PLEASE, that this sign is COMPLETELY unrelated to the sign of the charge on the electron.

Using the applet

You can select an orbital to view by clicking on the appropriate button as in part 1. The calculation of the 40 000 points comprising the image (200x200 pixels) is quite a chore for the computer so be patient!

Once an orbital has been chosen, you can select a z-section by entering a value for z in the little box at the bottom right of the applet. To do this, click in the box, type your choice of z and press "enter" (or its equivalent) ONCE on the keyboard. You MUST press "enter" or the z value will not be registered by the applet, and the calculation will not start.

If you are just experimenting, to get a sense of the three-dimensional shape of an orbital, your choice of z-coordinate(s) should be guided by the scale shown below the image. Imagine the three- dimensional orbitals (which have been scaled to appear a reasonable size) centered in a cube which has faces which are the size of the image. (The questions below give more specific values of z to use.)

Once you have displayed the section you want, you can find out y, (radial) and (angular) for any point in the section by moving the cursor (mouse pointer) to it and then clicking the left mouse button. The program gives a continuous read-out of the cartesian and polar coordinates to help you while you are positioning the mouse. You will need a steady hand to position the cursor on a specific pixel!

N.B. If you scroll the applet off the screen and back again, the panel which shows the mouse position etc may disappear. It will reappear when you select an orbital or a z value. In general, the y values shown will not be valid until after you click somewhere on the contour plot.

The relation between the cartesian and polar coordinate systems is shown in the figure below.

Diagram showing the polar cordinate system



Execise 3: On the three-dimensional character of the orbitals

Display the section of the 3s orbital at z = 0. Sketch what you see. By using the mouse find and report the radii of the nodes. Now change z by 50 pm intervals up to 500. Where do the nodes go, and why?

Execise 4: On the shape of nodal surface of the 3dz2 orbital

Display the section of the 3dz2 orbital at z = 0. Sketch and describe what you see. Now change z by 100 pm intervals up to 1000. Find, using the mouse, the approximate theta (q) values for the node. How does this relate to what you know of the three-dimensional shape of the 3dz2 orbital. There is a picture of it in Fig 2-6 (p43) in Cotton, Wilkinson and Gaus or Fig 11.3 (p 396) in Huheey.

Exercise 5: On the radial part of the y  functions

Display the section of the 3dxy orbital at z = 0. Sketch what you see. By trial and error find and report the coordinates of the maximum and minimum values of y. (Just do the antinode in the quadrant with x and y both positive. Figure the other three out from the symmetry of the orbital.) How does the value of r at the maximum compare with the value you expect from Fig 2.7 (p45) in Cotton, Wilkinson and Gaus or Figure 2.1 (p 12) in Huheey.

Execise 6: On the three-dimensional shape of the 3dxz orbital and polar coordinates

Investigate a series of sections through the 3dxz orbital. Change z by 50 pm intervals up to 200 and by 100 thereafter up to 1000. Find, by trial and error with the mouse, the coordinates of the maximum value of y on each section. Determine r for the overall maximum from a graph of y vs r. Compare it with the value from question 5 - why should it be the same?

Exercise 7: Left over from part 1!

The selection of orbitals available in the applet which you used in the first part of this experiment did not include the 2pz, 3dxz or 3dyz. Why?

Exercise 8: On the radial probability functions

Select the 2px orbital with z = 0. Sketch what you see. Using the mouse to read off the values from the diagram keeping y = 0 and varying x, plot a graph of 4.p.r2.y2 vs r. What is the relation of this graph to the histogram for the same orbital from the first part of the dry lab?

Execise 9: On the radial and angular parts of the y  functions

Repeat the procedure in Exercise 8 but read off y values along the line x = y (ie at 45o) between the axes. Compare this plot to the one you got from Exercise 8, in particular, where are the maxima and how do the values relate in general to those you found for question 8. Are your results consistent with the functions for the radial and angular parts of y as illustrated in the text?

POSTSCRIPT

Remember that you can return to the applets in this, and indeed any of the dry labs, any time you like, to help you in your studies.