lity, the pressure and volume axes must be scaled appropriately, and the critical temperature region must be examined; compare Figures 1 and 2. Similarly, using the Henderson-Hasselbalch equation to describe the buffer region in a titration involving a weak acid or base requires a bit more work than one might expect in order to obtain reasonable results in the vicinity of the equivalence point. Although one of the points of this article is that spreadsheets for chemical calculations are quite easy to construct, the author has come to realize that a few detailed, welldocumented examples might be valuable to educators who are not yet familiar with spreadsheet preparation. The author will be happy to provide a documented disk containing example spreadsheets for gas laws, kinetics, titrations, multiple equilibria, linear regression of experimental data, molecular orbitals, and an electronic gradebook. The spreadsheets require either LOTUS 12-3 (Version 2 or later) or SYMPHONY running on an IBM PC or compatible computer. To order, please send a check for $10.00 made out to the author to help defray costs. When ordering, please specify whether the SYMPHONY or LOTUS 1-2-3 version of the spreadsheets is desired; otherwise the LOTUS 1-2-3 version will be sent.
Molecular Orbitals on a Spreadsheet
the molecular orbitals in less than five seconds after the internuclear distance parameter is changed. Thus the instructor can begin with the atoms widely separated and show that the orbitals are localized on the two atoms. Then the atoms can be "moved together" and the formation of the bonding and antibonding molecular orbitals can be seen. The LCAO-MO spreadsheet begins with the Bohr radius, a0 = 0.529 A,stored in cell Al, the normalization factor 1/fi in cell A2 and the recurring constant l/a~Tr112 in cell A3. Evaluating recurring constants once at the beginning of a routine is a standard programming practice that speeds execution.
The internuclear distance is specified as a variable parameter in cell C4, and is the only variable parameter in this spreadsheet. The two hydrogen nuclei are assumed to lie along the X axis with nucleus A at the origin and nucleus B located at a point on the positive X axis specified by the internuclear distance. Each row in the main body of the spreadsheet represents a point along the internuclear axis. The values for the various parts of the wave functions at that point are tabulated in that row. The first two columns of the (Continued on page A31 6)
J. Van Houten St. Michael's College Winooski, VT 05404
The preceding article (7) has documented some of the benefits of using spreadsheets for teaching chemistry. The variety of applications is virtually limitless and depends on the imagination and ingenuity of the user. Most of the examples in the literature (2-6) involve problems frequently encountered in general chemistry or in elementary physical chemistry. This article will attempt to show that spreadsheets can be used even to do theoretical calculations. Although the simplest possible case will be treated-sigma overlap between Is orbitals in a molecule of Hz-it should be obvious that the approach could be extended to more complicated situations. A portion of the spreadsheet used to treat sigma bonding in Hz is shown in Figure 3. The sigma bonding and antibonding molecular orbitals are obtained as an appropriately normalized linear combination of the Is atomic orbitals on the two hydrogen atoms-the familiar LCAO-MO approach. This technique is relatively easy to describe qualitatively, but it is difficult to illustrate quantitatively because the calculations are so tedious. Unfortunately, students must often be expected to accept on faith the instructor's assertions concerning the nature of the orbitals which are formed as two atoms approach one another. The LCAO-MO spreadsheet described here can be used in class to demonstrate bond formation in an H2 molecule "live". The spreadsheet will recalculate and replot
Volume 65
Number 12
December 1988
spreadsheet body consist of radial distances from the two nuclei at various points along the internuclear axis, the radial distance from nucleus A being in column A and the distance from nucleus B in column B. The negative signs in columns A and B merely indicate points to the left of each nucleus. For the radially symmetric s orbitals this sign has no significance, whereas for p, d, and f orbitals, which are not radially symmetric, the sign would affect the atomic orbital wave function. Columns C and D contain the values of the Is atomic orbital wave functions of the two atoms evaluated at the radial distances specified in columns A and B, respectively. Columns E and F contain the values of the sigma bonding and antibonding molecular orbital wave functions, respectively. Thus the contents of any cell in column E is simply the normalized sum of the contents of the cells in the corresponding row in columns C and D. Similarly the cells in column F contain the normalized difference between the values of cells C and D in the same row. By constructing a spreadsheet in this manner the problem has been reduced to a series of relatively simple steps. An examination of the functional form of the spreadsheet contents, Figure 4, shows the similarity between the spreadsheet entries and the explicit mathematical functions representing the orbitals. Let us focus on row 10, the first row in the main body of the spreadsheet. The X coordinate shown in cell A10 corresponds to the radial distance from nucleus A. The contents of cell B10 correspond to the radial distance from nucleus B at that X coordinate, the distance being computed by taking the difference between the X coordinate in cell A10 and the internuclear distance in cell C4, hence the expression +A10-C$4. The dollar signs are used to indicate cell addresses that are unchanged when the formula is later copied to generate other rows of the spreadsheet. Now, the Is orbital wave function for the hydrogen atom (2 = 1) is
Recalling that an is stored in cell Al, the recurring constant l/apTr1/2is stored in cell A3, and the radial distance from atom A is in cell A10, t h e spreadsheet function +$A$3*EXP(- @ABS(AlO)/$A$l) in cell C10 represents the I s atomic orbital wave function for atom A. Copying the function from cell C10 into cell Dl0 yields the Is atomic orbital wave function for atom B. The explicit expressions in column D have been omitted to save space in Figure 4. The form of the functions in column D is similar to the functions in column C with the terms A10, A l l , . . . replaced by B10, B l l , .. Finally, cell El0 contains the sum of the contents of cells C10 and Dl0 normalized by multiplying by the factor I/@ from cell A2 to generate the bonding molecular orbital wave function. The antibonding molecular orbital wave function is obtained in a similar fashion in cell F10 by taking the normalized difference between the wave function values in cells C10 and D10.
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Figure 4. Functional form of spreadsheet.
Figure 5. Molecular orbitals in a H2 molecule with a 0.800-A internuclear separation.
Once the formulas in row 10 have been written, the remainder of the spreadsheet is generated simply by copying row 10 into the rows below. It is not necessary to rewrite the expressions. In order to yield smooth curves, the spreadsheet used to generate the plots in Figure 5 contained many more rows than the examples in the tables. That complete spreadsheet as well as others to simulate a variety of other chemical problems (see ref 7) are available on a disk from the author. To order, please send a check for $10.00 made out to the author to help defray costs. The spreadsheets require either LOTUS 12-3 (Version 2 or later) or SYMPHONY running on an IBM PC or compatible computer. When ordering, please specify whether the SYMPHONY or LOTUS 1-2-3 version of the spreadsheets is desired; otherwise, the LOTUS 1-2-3 version will be sent.
Literature Cited' 1. Ouchi, G. I. Personal Computers for Scientists; American Chemical Society: Washington,DC, 1987; Chapter 5.
2. Breneman, G. L. J. Chem. Educ. 1986,63,321. 3. Levkov, J. S. J. Chem. Educ. 1987.64.31.
4. 5. 6. 7.
Ibrahim, S. I. J. Chem. Educ. 1986,63,322. Miller, V. R. J. Chem. Educ. 1987,64,793. Coe, D. A. J. Chem. Educ. 1987,64,496. Van Houten, J. J. Chem. Educ. 1988, 65, preceding article in this column.
Presented in part at the Symposium on Computer-Assisted Instruction in Chemistry, cosponsored by the Divisions of Chemical Education and Computers in Chemistry, 192 ACS National Meeting, Anaheim, CA, September 1986, Abstract CHED 36.
