Acknowledgment The authors would like to thank John B. Hart of Xavier's Physics Department for his successful implementation of the algorithm on a HewlettPackard HP41CX 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

Visualization of Electron Clouds In Atoms and Molecules
6.2 = P[(HH)(KK) (HK)qID
John E. Douglas
Eastern Washington University
+
+
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 roundoff 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, IBMPC, Apple11, and Commodore64 computers using the same BASIC program.
+
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. Twentyfive years ago Ogryzlo and Porter ( I ) pointed out that "The ideal model of an atomic or molecular orbital would be a cloudlike 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 comoutereenerated 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. Dotdensity 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 25yearold ideal of Ogryzlo and Porter to become a reality in the classroom with the easy production of accurate dotdensity 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
.~
.u
wired for more complete representations. Dependinp on the facilities and time available; the plots may bereproduced in printed form, shown as overhead proiections, proiected as 35 mm slides, or storedon disk for i n i t k t recallon
Program ATORB plots hydrogenlike orbitals for which the user specifies the orbital type and the effective nuclear charge. The present program includes all functions with principal quantum numbers up to three. Higher functions and hybrid orbitals may easily be added. Program MORB is similar to ATORB, except that it allows for the weighted combination of atomic orhitals either on one center to illustrate hybrid orbitals, or on two centers to illustrate molecular orhitals. Slaterorhitals are used for simpler computation. multkentered The program could easily be modified to orbitals. The effective nuclear charge for atoms can be computed from the screening constants given by Burns (10). For the Slater orhitals used as basis functions in the molecular orbital plots the orbital exponents of Clementi and Raimondi (11) may he used, or else optimized orhital exponents and coefficients can be taken from SCF calculations in the literature. The program operates by randomly choosing an xy point in the d o t area. I t then computes the ratio of psisquare a t that location to the maximum value of psisquare found anywhere in that orbital, as determined by a preliminary calculation. This provides a probability a t the point normalized from zero to one. This normalized probability is com
pared to a random number between zero and one. If the point3sprohahility is greater than the random number,a dot is placed, i.e., one pixel is illuminated, otherwise not. The process continues until a predetermined number of dots have been placed. In addition to online viewing, the plot may he stored on disc for future display, or a hard copy may he made on the printer. The screen has been successfullv ohotoeraohed  . with 35mm camera to produce dide transparencies. The figures in this article were printed from the Atari ST "higbresolution" hlackandwhite mode with a screen resolu~ionof 640 X 400 pixels, using an E ~ s o n LX8009pin dotmatrix ~ r i n t er. Plots have alsobeen produced usinithe "mediumrksolution" color mode with 640 X 200 pixels. In this mode provision is made to plot successively darker colors each time any one pixel is hit again, thereby emphasizing the denser regions of the electron cloud. The highresolution blackandwhite mode is superior for producing hard copy on a dotmatrix printer. However, the color monitor provides a more dramatic visual image for online viewing.
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a
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~
ResuHs and Appllcatlons
In Figure 1are representations of all atomic orbital types of principal quantum number 1, 2, and 3. Each electron is represented by 1300 dots. Although some orbitals might be represented optimally by more or less dots, allare represented by the same number for the sake of comparable representation of electron density. Points to be emphasized to the student include the greater differences of orbitals with higher principal quantum number and how the electron density ismuch more spread out from they axis than in the common dumbbell form of representation. The effect of nuclear charge on the size and density of the electron cloud can be demonstrated by plotting the same orbital for differing values of z.
Flgure 1. Output of Program ATORB for atomic orbitals of principal quantum numbers 1. 2. and 3.
Volume 67
Number 1 January 1990
43
Orbital 1 x= 0 n=P: Type: S
Ex=1.61: CFI
Orbital 2 x= P n=Z: Type: PSIGHR Ex=1.57: C F l Orbital 3 X= 0
n=2; I y p r : PPI Ex=1.57: Co=I
!
. ... ,: . .
ma ~ t s ,nn
i
1 3 i I 8. 1 Figure 3. An sp3hybrid wbital produo&by wagram MORB.
2'
I
Figure 2. Component w b i i b fa Uw sodium amm, 1300 points representing each electron. mese can be overlaid to model Uw entire atom.
Figure 2 shows the component orhitals for an atom of sodium. Each electron in the orbital is again re~resentedby 13W dots. For example, the 1s and 2s oihitala iach contain 2600 dots, representing two electrons in each orhital: the 2p(xy) orbital contains 5200 dots representing four electrons. By using these plots as overlays on an overhead projector, the difference between core electrons and valence electrons is vividly shown. While constructing such visualizations. the user runs across manv interestine observations such as the fact that the outer lithium 2s orbital is visually the same size as that for the outer sodium 3s orhital, even though both the van der Wads and metallic radii are &eater for sodium. The use of Program MORB to illustrate the combination of atomic orbitals isillustrated in Figure 3, which shows one of the hvhrid so" orbitals. The shift of electron densitv into one lobe, resulting in stronger bonding, is evident. Program MORB can be used straightforwardly to visualize the qualitative features of molecular orhitals in homonuclear molecules, such as sigma and pi bonds. Beginning students often do not realize that this simple, quantitative application of MO theorv to heteronuclear molecules by the unweiehted combination of atomic orbitals leads to uhrealistic electron distributions between the two atoms. The program computes the dipole moment of the calculated distribution so that students can discover for themselves the errors resultine from Door a~oroximations. .. Accurate descriptions of molecular orhitals are obtained fromselfconsistent field molecular orhitalcalculationssurh as thosr for H F by Hansil (12) and for HCI by Neshit (13). These c o m ~ u t e dorbitals can easilv he visuali~edusine Program MORB. As an example, ~ i & r e4 shows the 3sigma orhital of H F from the calculations of Ransil. Thus Program MORB is useful for leading the student beyond qualitative molecular orhital theory and into the quantitative results of sophisticated MO computations. The Programs
The programs are available either in GFA Basic for the Atari S T and Amiga personal computers or in True Basic for Macintosh, IBM compatible, Amiga, and Atari S T comput44
Journal of Chemical Education
Figure 4. SCF 3sigma Mil fa HF as computed by Ransil(12).
ers. The GFA Basic version has the advantage of a faster running time. Program listings are available from the author; please specify which version is desired. A set of 39 hlackandwhite prints of representative orbitals, suitable for reproduction, either in printed form or as overhead transnarencies. is available from the author for $20. Included in the set are: a representative of each atomic orbital tvve with z=1: examoles showine the effect of nuclear charge; component atomic orbitalslfor constructing an overlay of the atoms and ions of lithium, sodium, fluorine, chlorine; hybrid orbitals; molecular orhitals of homonuclear molecules showing ss, p(sigma)p(sigma), and p(pi)p(pi) overlap; and the four SCF molecular orbitals for H F from the literature. The author invites inquiries about the availability of a set of 35mm transparencies depicting a similar group of orbitals. Literature Clted I ( , g n > 0 . F .a: 1'nncr.C B J C n e m E d r r 1965.4..2i(. 1 B,rdar*.(I T..Lc~.wu.,I\I'..I ('hen! Fu.i ll10.1:,611 3 (11nr1.R .I .I Chrm &:our 1972.49 61 I