Acknowledgment The authors would like to thank John B. Hart of Xavier's Physics Department for his successful implementation of the algorithm on a Hewlett-Packard HP-41CX programmable calculator. Literature Cited These same quantities can be used to fmd b2, which is an unbiased estimate for a2when all ti's are the same.
if C coefficients are estimated, there are N data points, and (FF) = Y wm. .. ' he estimates for the variances of the coefficients are
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Visualization of Electron Clouds In Atoms and Molecules
6.2 = P[(HH)(KK)- (HK)qID
John E. Douglas
Eastern Washington University
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Cases having just two terms (F, = aG, pH, €2 can be treated as special cases in this analysis. In such a situation, = (KK) = 0. K, = 0 for all r and thus (FK) = (GK) = (HK) Merely setting (KK) = 1 if (KK) = 0 yields the appropriate solution. Similarly, if the original model has only one term q), setting (HH)= (KK) = 1 will give the (F, = aG, solution. The above calculations are shown in algorithm form in Table 2. Steps 10 and 12 are mathematically equivalent to the corresponding calculations above and are to be preferred since they minimize round-off error. When the data gives an extremely good fit, it can happen that the number of signifcant digits (SD) used by the computer or calculator is insufficient to calculate a meaningful variance. When this occurs either $2 < 0 or log(FFIb2) > (SD - 1). The problem can be avoided by using the alternate method of calculating the variance indicated in Table 2, step 12, by asterisks. In any case, the highest precision available with the computer or calculator should be used. The algorithm has been tested with VAX, IBM-PC, Apple-11, and Commodore-64 computers using the same BASIC program.
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Welghtlng When all the error is assumed to be in the dependent variable, the weighting function for each point should be:
In the ~reviousdiscussion it was assumed that ui,the uncertainty in Y,wasthe same for all Y. If it is not, then the global weight (1/(6F/613') should be multiplied by ( l / ~ , ~all) ,eraluated at the point in question, i n order to calculate the The global part of the weight can be weighting factor W;. included explicitly for particular cases (6),but for reasons of generality, the procedure described here approximates the 6FIGY term as:
For situations where error exists in the independent varia b l e ( ~or ) where the eouation cannot be arraneed in a multimany line& form, iterative techniques are needed. eouations of chemical interest that are not linear in the dependent varinhle(s) are nonetheless multilinear in form. If the error is essentially all in the dependent variahle, these can conveniently be treated analytically, without iteration, by the technique described.
ow ever,
42
Journal of Chemical Education
Cheney. WA 89004
Visualization of the electron orbital concept continues to challenge and intrigue chemical educators. The concept is crucial both to the nonscientist who is taking a liberal arts course in chemistry and should understand the basic structure of matter and to the serious chemistry student who is exploring the sophisticated nuances of atomic and molecular orbitals. The chemical educator deals with both of these groups. Twenty-five years ago Ogryzlo and Porter ( I ) pointed out that "The ideal model of an atomic or molecular orbital would be a cloud-like structure showing the probability of findine the electron at all noints in soace. . .".Admittine the d i f f i z t y of preparing sucha model;they presented a gethod for comoutine contours and nre~arinesolid models. Bordass and ~ k n e t i ( 2 )Olcott , (3);and ~trGtweiserand Owens ( 4 ) were amone three- the first to use comouter-eenerated dimensional contour diagrams to represent atomic and molecular orbitals. The author (5) and Raughman (6)have used computer generated numerical grids asthe basis of student exercises for plotting orbital contours. A recent issue of this Journal contained two articles on the subject, one by Brenneman (7)that presented contours for the angular functions of all orbitals through the g type, and one by Leihl(8) that presented a computer program for plotting two- and threedimensional contour diaerams. However, these traditional presentations of orbitals as distribution functions and contour surfaces are abstract and sometimes inaccurately simplified. The viewer has difficulty. visualizing the true nature of the electron cloud, especially just how diffuse or dense it actually is in different regions of the atom or molecule. Dot-density diagrams are a more realistic form of presentation but have been utilized only to a limited extent in some texts and hy a few instructors. Frequently these are qualitative only and are not designed for interactive use by the student. Today, inexpensive microcomputers with powerful graphics capabilities enable the 25-year-old ideal of Ogryzlo and Porter to become a reality in the classroom with the easy production of accurate dot-density orbital diagrams either for demonstrations or as student exercises. One rudimentary computer program is available through a national exchange (9).This paper presents computer routines that can quickly nroduce accurate. manhic . reoresentations of electron clouds for horh atomic and molecular orbitals. with all parameters easilv adiustable. 'l'hev are suitable for all levels of instruction.-simple plots ma; be produced in a minute or so, suitable for a live class demonstration. Longer times me re-
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wired for more complete representations. Dependinp on the facilities and time available; the plots may be-reproduced in printed form, shown as overhead proiections, proiected as 35 mm slides, or storedon disk for i n i t k t recallon
