Stephen 1. Holmgren' and J a m e s S. Evans2 Lawrence University Appleton, Wisconsin 5491I
Accurate contours for spa Hybrid Orbitals Some computer-assisfed exercises
Undergraduates are expected to learn the fundamental quantum concepts of the orhital approximation in elementary chemistry courses and to enrich their command of these concepts in subsequent courses, hut they often fail to achieve proficiency in visualizing atomic and molecular orhitals. Even though textbooks are improving in the quality and accuracy of illustrations, some students develop serious misconceptions about orbitals, probably because of inadequate opportunity to manipulate these abstract models and gain familiarity with them through practice as is possible with other abstract models in chemistry (e.g., the calculations for chemical equilibria). Osterheld (1) has outlined a sequence of wave function calculations suitable for unner-division courses. hut still lacking are experiments a i d homework problems dealing with orbital shapes to help students having limited mathemitical sophistication overcome their difficulties and acquire a good grasp of these important concepts. An attractive approach toward meeting this need has become possible through the availability of timesharing computers, the development of interfaces for computer control of ordinary X-Y plotters, and the declining cost of video graphics terminals3 This paper describes an interactive Basic computer program that enables students to perform exercises concerned with hybridized orhitals of the spa family. A timesharing computer terminal with attached plotter produces more accurate plots than can he obtained using line printers, teletypes, and the noninteractive programs previously .described (2-5); furthermore. onlv two or three minutes are required to draw a typic& orbital (at a transmission rate of i 0 characters per second). Using this program, individual students or small study goups c a n explore, visualize, and gain familiarity with several fundamental concepts of quantum chemistry, including the shapes of orhitals, the overlap of orbitals to form chemical bonds, and the origin of definite bond angles. Definition and Properties of spa Hybrid Orbitals The normalized wave functions for the 2s and 2p hydrogen-like atomic orbitals in spherical polar coordinates
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'Mr. Holmgren, now a graduate student at Haward University, leveloped as an independent project the program for calculating the plotting hybrid orbitals during his sophomore and junior years.
2Persons wishing to obtain a copy of the program, comprising approximately 500 Basic statements, may communicate with Dr. Evans concerning the availability of listings, paper tapes, or mag: netic tapes. 3The Lawrence University timesharing system has several hard-wired remote terminals to a PDP-11/20 computer equipped with disk and tape drives; a TSP-212 plotter can he attached at any of these terminals. The orbital plotting program has also been adapted for the PDP-10 timesharing system at the University of Oregon, where Dr. Evans spent a sabbatical leave in 197273. Different versions of the progrim bperate with TSP-212 plotter interfaces or Tektronix 4010 video graphics terminals and either PDP-10or PDP-11computers.
where a. is the Bohr radius (0.529 A) and Z' is the effective nuclear charge after accounting for the screening effects of electrons in other orbitals. Suitable approximate values for Z', adapted from the results of self-consistent field calculations (6). for four common elements are: C, 3.15; N, 3.80; 0 , 4.45; F, 5.10. The advantages of using different radial wave functions for 2s and 2p orbitals, rather than Slater functions, have been discussed thoroughly elsewhere (7,8). ?he normalized wave function for an spa hyhrid orbital isdefined as a linear combination of & $ and dzp
where the magnitude of a determines the degree of hybridization. Since the 2s and 2p wave functions are orthogonal, the electron probability distribution for J',p= is (1 a)-' (+z,2 a+zpZ); thus the 2s and 2p contributions to the probability are i n the ratio l:a indicated in the notation spm.The functions J.z, and J.,@ are both cylindrically symmetric about the axis specified by 0 = 0; hence eqns. (2) and (3) describe the variation of these functions in polar coordinates in any plane containing the symmetry axis. Another spa hyhrid orhital with its symmetry axis rotated by an angle /3 is easily calculated by changing the angular dependence in eqn. (2) to cos (@- @).For these two orhitals to he orthogonal, the angle /3 and the degree of hybridization a must satisfy the relationship
+
+
cos 6
=
- lla
(4)
Coulson (9) has outlined a derivation of eqn. (4), which is the basis for the familiar 180", 120", and 109"28' bonding angles for sp, spZ, and spa orbitals that are employed in discussions of molecular geometry. Computer-Plotted Contour Diagrams Of the many schemes devised for picturing atomic wave functions, contours of constant electron probability per unit volume, i.e., constant b2, offer the best combination of quantitative significance and graphic clarity in twodimensional representations (10, 11). Of course the computation of several contours for a function such as J.,,. (r, @)definedby eqns. (I)-@) is excessively tedious by the. manual methods outlined previously (1, 12). Even with a computer, efficient calculation of smooth curves requires rather intricate programming. The general approach of such programming will he outlined in a later section. The quality of contour plots attainable with a computer-driven X-Y plotter is illustrated in Figures 1-3. Figure 1 shows the change in shape of hybrid orbitals for carbon as the amount of p character in the wave function increases. Figure 2 demonstrates both the side-to-side and end-to-end configurations for the overlap of p orhitals, showing that the latter arrangement results in a somewhat greater total electron density in the bonding region. Figure 3 shows that the ~ o l a r i t vin a carbon-fluorine bond originates from the differing nuclear charges. In these ild orhitals. it isrelativelv lustrations of s u ~ e r i m ~ o s eatomic easy to envisionthe contour shapes of the molecular orbitals that would be formed. Volume 51, Number 3, March 1974
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189
Hybrid Orbitals for Carbon
5) 6)
7, 81
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I#(
Figure 1. Four spo hybrid orbitals with contour valuer = 0.15. 0.3. and 0.6 A-3/2, showing the effect of increasing amountsofp character.
sigma
proper angles three sp2 orbitals on nitrogen, suitable for forming the sigma bonds tooxygen in the planar nitrate ion. Demonstrate the sigma and pi types of bonds between car-. bon atoms in the ethylene molecule, which has an internuclear distance of 1.34.k. The contour values printed by the program correspond to the magnitude of in units of A-stz, while the squares of these values give ILZ in units of kSalong the contours. Find the aooroximate electron density in the laree lobe of a .. . (iC2) . rnrhm .p3 orhltal a t adctanceuf tJ.7 .\ Iron) thr nurlrus Draw the slgrna urbitol* for t h r rnrhon skelrron of benzene. What rhnrnarz are newssay for humrole inniX3He,. curne. rimes rnllwt "inorgan~rbenzene"? I t the H ('-H bond angles r118"l in ethylene are taken t o mdirara rhnt each carhm a t o m c m t r h r c s t w o sp' urbltals for overlapping with two hydrogen atoms, then one unhybridized p orbital and one sp2 hybrid orbital are "left over" at each carbon atom, i.e., a total of 'h of an s orbital and % of a p orbital, or an stp ratio of 1 5 . Illustrate the "bent hand" (also called tau bond, or valence bond) description of the double bond in ethylene hy using two sp5 orbitals oriented at an angle of cos-I (-'h)with respect toeach other.
+
Since the interactive style of the program facilitates exploratory learning on an individual basis, each student can be encouraged to devise and work on a special project as well as to complete some of these standard exercises. Textbooks abound with highly schematic bonding diagrams that can he redrawn more accurately with the program, or with verbal assertions that can be tested. After acquiring this firsthand experience with computer-drawn orbital diagrams, students should be better prepared to analyze the subtleties of more advanced orhital representations (13). Description of the Program
Figure 2. Two orientations for the overlap of p orbitals at the same internuclear distance (simulating two carbon atoms 1.34 A apart).
The program has major sections for calculating contours, plotting contours, and communicating with the user. Since h2will be constant if 111.1 is constant, a contour can he mathematically specified by either of two equations, 111.1 = K or 11.2 = K2; however, the former equation leads to distinctly simpler programming. The method for calculating these contours essentially involves searching for the several distances r, where I$,* (r,O)I for a gwen 0 has the particular value K, followed by similar searches a t enough additional values of the angle 0 to give a smooth contour in a polar-coordinate plot of r versus 0. To see how this method proceeds, consider the general shapes of 11. versus r for constant 0 shown in the left half of Figure 4, where two different contour levels K and K' are drawn to
Figure 3. Overlap of sp3 orbitals of carbon (left) and fluorine (right). showing that differing nuclear charges give rise to bond polarity. The = 0 . 3 and 0.6 kW2. contour values are
I+I
Suggested Exercises ~k~features of hybrid orbitals can be readily demonstrated with the computer program described in the next section. The exercises utilizing that program are listed below approximately in order of increasing conceptual difficulty. The first few exercises illustrate fundamental concepts, and are suitable for students a t any level; subsequent exercises cover more specialized applications discussed in organic, inorganic, and physical chemistry courses. 1) Find out how the features of spa orbitals change as n is
made larger. 2) Find out how the features of sp0 orbitals change as Z' is made larger. 3) Demonstrate the two ways, side-to-side and end-to-end, in which twop orbitals can overlap. 4) Two spe hybrid orbitals with the same value of rr have the quantum property called orthogonality if one is oriented at an angle 4 with respect to the other, where cos 4 = -l/a. What is the value of 6 for a = 1, 2, 3? Superimpose at the 190 / Journal of Chemical Education
Figure 4. Mapping technique used in the construction of the contour lines, illustrated for one lobe of the 2p orbital of hydrogen.
YHAT EFFECTIVE N U C L E l R CHARGE V H A T AMOUNT OF P C H A R O C T E R 7 3 HOW M A N Y C O N T O U R S ? 2 APPROX. P S I M 1 X = 1.4755 A T R C H O O S E T O U R OWN C O N T O U R S ? Y E S PLEASE ENTER S M e L L E S T F I R S T I.3 7.6
User O~tionsin the Hybrid Orbital Plotting Program
C O N T O U R S USED 0.3 0.6 COMPARE O R M A X S C A L E ICOMPI\RE OPTIONS2 SCALEITRANS~ROTITE~PLOTTOXIS~MOREOFINI
larger valueeof %'. Ir L-
YOUR C H O l C E ?SCALE REDUCE S C I L E BY F A C T O R 01 1 . 4 YOUR C H O I C E ?TRANS MAXIMUM TRANSLATION 1.11164 X-TRANS 1 1-.69 Y-TRANS I0 YOUR C H O I C E ? R O T A T E DEGREES OF R O T A T I O N 1 ,180 YOUR C H O I C E ?PLOT
I I O I L ~ ~ I ItheI C slrr I 0~1 t ~ h - ,plot . ~ by ~ r~p m~f ~ y t n~g
a (acwr IrulhanunLty,rurhns11.4.
-
-
SCALE nducrr t h e phyairal a w of the plot hy any numericnl i , . u r , x w h a*?.-or 2. M r h r *.vr..4urbttalacm be plotted on t h e %me pace. With
DIRECTION
TRANS moves the plot any desired,distanee (in A1 in the x or y directions. The mgram computes the maxrmum p w h l e motion for the current S C A ~ E=sting.
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A X I S d r a w a 1.m 2 .ilung, w w r d d i n r t l y knrsth rbr origin, vavnlly the nuclrus to ~ ~ l i h m t the ( . ~ ~ l ~ y wdlr s w l of thr plot. Stnrp the Inm. l # , m uf rhu marktnp varlcr wlrh the 'i('ALE . ~ n d'L'IlANS wtlmgx a re.
YOUR C H O I C E ?SCALE REDUCE SCALE BY FACTOR O F 7.4 YOUR C H O I C E ? A X I S WISH TO S C A L E O R T R A N S F I R S T ?NO
lllinds. i x . , \ m
m c a * Ih" u a r w r l l r s r o r h l n ~ cthen,.
Y O U R C H O I C E ?MORE
VHAT EFFECTIVE NUCLEAR C H A R G E 7 5 . 1 8 WHAT AMOUNT OF P CHARACTEH 7 3 HOW MANY CONTOURS 7 2 APPRO%. P S I MAX 3.R6946 A T R --n.025 CHOOSE Y O U R OWN CONTOURS ,YES P L E A S E ENTER SMALLEST F I R S T 7.3
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7.6
C O N T O U R S USED
YOUR C H O I C E ?SCALE REDUCE SCALE BY FACTOR OF ?.4 YOUR C H O I C E ?TRANS MAXIMUM THaNSLATION 1.40645 X-TRaNS = 7 . 6 9 Y-TRANS i ?:A YOUR C H O l C E 7PLOT
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I N X OR Y D I R E C T I O N
YOUR C H O I C E ? F l u 1
Figure 5. Interactive dialog involving all of the user options. This particular dialog resulted from drawing Figure 3.
intersect the curves. Note that there may be up to four intersections of each contour level with the II. curves. The problem of locating these intersections is equivalent to finding the roots of eqn. (5). In effect, the horizontal axis is moved up or down K units
to the proper contour level, so that the desired roots ri become the zeroes of the new function f(r). Newton-Raphson and interval-halving techniques search for a zero in each region where a root can exist. The radial distance for each zero is stored in a doubly-subscripted array; for instance, R2(J,I) contains the root in region 2 for contour level I a t angle J. Of course roots 3 and 4, or roots 1 and 2, or both pairs, may not exist at some angles because the magnitude of II. in those regions is everywhere less than the contour value. Although the calculations for the contours are best handled in polar coordinates because of the form of eqns. (1)-(3), these data must be converted to cartesian coordinates in order to provide driving signals to the X - Y recorder. Figure 4 indicates how the r; values a t three different angles correspond to seven numbered points on a car-
the current set of contour data and transfers mntml hack to the beginning of the program where new contours may be ealevlated for a different type of orbital.
MORE erases
FIN1 terminates the pmgrsm.
tesian plot for the positive lobe of a p orbital. The orhital is symmetric with respect to reflection in the horizontal axis; therefore, for every point (x,y) calculated along the contour, a second point fx,-y) can he obtained very simply from the symmetry relationship. Finally, the X - Y recorder will draw the contour plot when the arrays of points in cartesian coordinates are read out in the proper sequence to drive the pen around each contour curve for both the positive and negative lobes of the orbital. The options available to the user of the program are illustrated by the sample dialog in Figure 5 (for the preparation of Figure 3). The routing section of the program requests the additional parameters needed for each user option; these options enable the user to scale, translate, or rotate an orbital before plotting it, or to plot another orbital of the same kind without having to generate new data. Further information about these options is presented in the table. Acknowledgment
The Lawrence University timesharing computer system was partially funded by grant GJ-832 from theNationa1 Science Foundation and a Title VI grant from the US. Office of Education. T h e senior author thanks the University of Oregon for computer time for program development. Literature Cited (11 Osterheld: R.K.. J.CHEMEDUC..44.286(19671. (2) Bsder. M.. J:CHEM.EDUC.,M,175 (19711. (3) Craig, N. C., ShexrU, D. D., Carletan. T. S., and Aebrman, M. N., EDUC.. 48.310 11971). (4) Peterson. D.L..andFul1er.M. E..J.CHEM.EDUC.,48.314 (19711. ( 5 ) Soltrberg. L.J.. J. CHEM.EDUC..49.357 (1972). (61 Clemcnti. E., sndRaimondi, D.L.,J C h e m Phys.. 58,2686(19631. 171 Moffitt. W. E.. andCoulion.C.A..Phii. M a e . 9%.6-4 ,1967)
J. CHEM.
New York. 1468.
(121 Ogvdo,E. A. end Porter. 0.B.. J. CHEM. EDUC.. 40.256 (1963). (I31 Streituie~er,Jr., Andrew, and O m s , Peter H.."Orbits1 and Eloctmn Density Diagrams: An Application of Computer Gmphiea." The Mscmillan Company, New York,
1973.
Volume 51, Number 3, March 1974
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