the red of the HF frequency, corresponding to its increased bond length. Note also that the frequencies of H2S change little from the monomer. New intermolecular freauencies are present on the formation of the complex, commonly called stretch. bend. and shear. these are small, and mav be observable tumicrokave s p e r t ~ o ~ c u ~ iWe s t i are . only aware of m e exoerimental studv on this comolex ( 5 ) ,this predicts 129 cm-"for the intermo~cularstretch, and 320 cm? for the bend. This calculation demonstrates that the microcomputer has to be considered not only as a teaching aid, but also as a significant aid to research scientists (even to quantum chemists who are number crunchers). These calculations on H&HF and the monomers took a total of 400 hours on the IBM PCIAT. (Note that with one of the available accelerator boards inserted, this would be reduced by a factor of five.) Their cost on a mainframe would he considerable, probably in excess of the cost of the PCIAT! Further information on MICROMOL is available from the authors.
cuts through the center of the hydrogen atom with both distance and polar angle. One type of two-dimensional mapping draws the contours of the orbital. An orbital's contour lines are similar to the eeoeraohic contours of a hillside. Geographic contours are &c& parallel to the earth's surface that pick out lines of equal altitude. Orbital contours are slices parallel to the cross section that pick out lines of equal value of the wave function for the orbital. A second type of two-dimensional mapping produces a pseudo-three-dimensional plot of the orbital. Using the cross section as a base plane, the values of the wave function for the orbital at different locations are mapped as a distance above or below the base plane. The procedure produces a picture of the "hill" through which the contour cuts were made. The algorithm to produce the two-dimensional plots proceeds as follows. Load the (x, y ) array of scaled values representing the solution to the Schroedinger equation for the orbital. Step through the points in the array sequentially. For contour plots In a cell whose corners are (r,y), (r + 1,y), (r,y + 1) and (x + 1, v + 1-, ) * ~
Orbital Plots of the Hydrogen Atom
If a predetermined contour value is neither greater nor less than the orbital value at any corner, then draw a point. For pseudo-three-dimensional plots Draw straight line segments from the orbital value at ( x , y) to the value at ( x + 1, y). Project the picture onto a two-dimensionalviewing plane.
Michael Llebl Mount Michael Benedictine High School Elkhorn, NE 68022 Orbitals give coherence to many aspects of chemical behavior because they describe the electron density of regions ahout a nucleus. For examnle orbitals helo exolain theaeometry of chemical bonds. A; a result, orbitals &e now f;ndamental to the study of chemistry. Unfortunately, the mathematical methods required to generate orhitals are not simple. For that reason, introducing students to the concept of orbitals is prohlematic. On the other hand, when students can visualize orhitals, their imoortance and utility becomes apparent. The software described in this articleenables a 48K Apple I1 with a single disk drive to plot the orhitals of the hydrogen atom in one, two, or three dimensions. The software works in the following manner. Solutions to the Schroedinger equation for the 1s orbital through all the 3d orbitals have been stored as arrays of binary values on disk. When an orbital is selected. the aoorooriate solution is .. . loaded from disk into memory. Then a drawing routine is called to trace out a oicture of the orbital. Five Woes of plots are available. ~ a d i a plots i show the orbital's variation-with distance from the nucleus. I'olar plots centered on the nucleus show the orbital's variation wkh polar angle. Two-dimensional plots of orhitals map the variation of cross-sectional
Three-dimensional plots yield the most complete view of the orbital. These plots draw contour lines that encompass the regions about the nucleus with high electron density. The algorithm to produce these plots is as follows. Load the (x, y, z) array of scaled values representing the solution to the Schroedinger equation for the orbital. Step through the points in the array sequentially. In a cell whose corners are ( x , y, z), (X 1, y, z), (x, y + 1, z ) and
+
(x + L Y + L d
If a predetermined contour value is neither greater nor less than the orbital value at any corner, then draw a point. In a cell whose corners are ( x , y, z ) , (x + 1, y, z ) , ( x , y, z + 1) and (x+ l,Y,L+l)
Ifapredeterrninedcontour value is neifhergreatrr nor l e s ~ t h n n the orbital value at any corner, then drawn point. Project the picture onto a two-dimensionalviewing plane
T o aid in visualizing the orbitals, both the three-dimensional plots and the pseudo-three-dimensional plots can be rotated
r
--
RFlDIFlL FUNCTION
I Figure 2. Radial plot of the 2p orbital.
SPHERICFlL HFlRMONIC FUNCTION
POLAR PLOT
Figure 3. Polar plot 01 the p, orbital Volume 65
Number 1 January 1988
23
The plots shown in these figures are all screen dumps of images generated bv the programs. The software supports an opt& that enablrbany image to he swed onto a spare disk. Herause the three-dimensional pluts and the pseudo-thrrrdimensional plots can be rotatedahout two axes, any one of a large number of images can be saved. If auser decides to save an orhital plot, the software prompts the user for a file name for the image. The file name is limited to the letters A through Z and to alength of six characters. Once the name is entered, the high-resolution image is saved as a file heginning at decimal address 16384 (hexadecimal 4000) with a decimal length of 8192 (hexadecimal 2000). If an orhital plot has been saved as a file with file name, ORBTL, the file can be retrieved from disk by the command BLOAD ORBTL. The file will he loaded into page 2 of the Apple II's highresolution graphics. Any utility capable of a screen dump of the high-resolution graphics pages can produce hard copy of the orhital nlots. The softkare fits onto two disks. The programs are integrated and entirely menu driven. Menu programs are coded in BASIC. Machine language routines generate the plots. The use of machine language routines for driving the graphics makes the software fast. No picture takes more than 20 seconds to execute. A short set of instructions for the use of the software is included as a text file on the second disk. Binary data files store the values of the wave functions for the arrays. These arrays could be refigured to study hybrid orbitals or to study orbitals in transition from one state to another during photon emission/absorptiou. The software should he useful to students who are struggling to understand the various ways of representing orhitals. I t could he especially effective for teachers in classrooms equipped with large screen or projection displays. The software with instructions is available from the author for $15.
axORBITAL Figure 4. Contour plot of the 2p,wbital.
Visualizing Boltzmann-Like Distributions
Figure 5. PseudMhreedimensional plot of the 2p,
orbital
2 p ~ ORBITAL
Figure 6. Three dimensional plat of the 2p, orbital
about two axes. Examples of the radial plot, the polar angle plot, the contour plot, the pseudo-three-dimensional plot and the three-dimensional plot for the 2p, orhital are shown in Figures 2 through 6, respectively. 24
Journal of Chemical Education
Fred M. Hornack University of North Carolina at Wilmington Wilmington, NC 28403
Some elementary aspects of statistical mechanics can be pictured with the aid of a micro~omputer.~ Suppose that all of the molecules in an iodine crvstal are in the same vibrational energy level. If molecules can exchange energy, this unlikely situation willnot last very long. Amolecule can emit a photon and drop to a lower level while another molecule absorbs the photon and advances to a higher level. A continuation of the haphazard interactions leads to more probable enerev distributions. Likewise, when a finite amount of energyUflowsinto a system in finite time, i t cannot instantaneously develop the most probable arrangement of molecules in energy levels. After the flow stops, the system continues to redistribute energy. The interactive programs herein described address these concepts graphically and can attract the interest of students in both introductory and advanced chemistrv courses. The BASIC code was devel&ed on the'l'andy i000and should trawfer withont modification to other MS-DOSsystems such as the lBhl I'C. Input to the internally documrnted program HOI.'l'% consists t,f a oarameter that afiects the speed of the action and the initiai populations of 21 energy levels, which are numbered from zero to 20. Each level can accommodate up to 76 units. The dynamic display, in which energy is randomly exchanged in an adiabatic system, illustrates the drive toward more probable distributions and the attainment of
The implementation was suggested by Charles R. Ward.
