R. Keith Osterheld Student Exercises University of Montana

Missoula, 59801 I in Wave Function

The introduction of reasoning based on wave functions at the more elementary levels in chemis- try has been accompanied by frequent misinterpretation of the graphs by which these functions are commonly presented. Adamson recently pointed out a reason for these misinterpretations when he called attention to "over literal acceptance of lobes as a kind of photograph of an atom" by students who do not do calculations with the wave functions (1). A number of elementary wave function calculations can be assigned profitably tounder- graduate students if the instructor offers some guidance to reduce the time required for the exercises.

It is assumed that the students have bcen introduced to the fundamental approach of quantum mechanics: $, the wave function, is a solution to the Schrodinger equation. For hydrogen and assumed hydrogen-like atoms, the complete wave function can be factored into two, $,.dl.l and $ I.,, with appropriate dependence on t.he variables:

The coordinated set of exercises described here has been used in a senior inorganic chemistry course and in a refresher course in chemical principles for high school teachers. Texts' for these courses commonly supply tables of normalized wave functions, separated into angular and radial components. Individual problems require from one to two hours of work. Since a fair amount of repetitive arithmetic is involved, several of these projects can profitably serve as examples for students in a computer programming course.

We have arranged that the student separately evalu- ate the angular and radial portions of the wave function. He then must bring these together to arrive a t a com- plete description. Many of the conceptual errors with wave functions arise from a failure to consider both parts of the wave function. In Exercises 8 and 10 we have chosen a graphical method of finding electron probability contour lines to give further emphasis to the interrelation of the angular and radial components of

'Useful discussions of the description of orbitals appear in the following: HARVEY, K. B., AND PORTER, G. B., i'Introduct,ion to Physical

Inorganic Chernist,ry," Addison-Wesley Publishing Ca., Read- ing, M~~ass., 1963, pp. 65-71.

PHILLIPS, C. S. G., AND WILLIAMS, R. 3. B., "Inorgsnie Chemie try," Oxford Cniversity Press, London, Vol. 1, pp. 13-21.

STREITWEISER, A,, "Molecular Orbital Theory for Organic Chemists," - ~~ John Wiley & Sons, Inc., New York, 1961, pp. 7-20.

WIBERG, K. B., "Physical Organic Chemistry," John Wiley & Sons, Ine., New York, 1964, pp. 1441 , 57-65.

COHEN, I., AND BUSTARD, T., J. CHEM. EDUC. 43,187 (1966). WAHL, A. C., Science, 151, 961 (1966).

Calculations

the wave function. Others might prefer a more direct method (2).

For simplicity we have chosen to deal only with or- bitals of hydrogen. One can obtain approximate de- scriptions of orbitals of other atoms by substituting for Z in the wave functions, the effective nuclear charge values determined by Slater's screening rules (8). A number of orbital descriptions derived on this basis appear in an article by Ogryzlo and Porter (4). Berry (5) has provided a very useful summary of current views on atomic orbitals.

In our use of these exercises we have provided the student with detailed instructions for the earlier prob- lems. For the later exercises an outline of the method is all that is given. For Exercises 2, 4 , and 7 we have specified for the student the points to be evaluated in order that smooth curves will result from a minimum of effort and in order to insure that useful values are amassed for combining wave function descriptions in later exercises.

Exercise 1. $.,, for an s Electra. The student is asked to evaluate fie,, for an s electron.

Exercise 2. Radial Dependence of $, J.%, and 4 ~ r ~ J . ~ for a 1 s Electron. The student is asked first to evaluate J., for a 1s electron for selected r values from the radial component of the wave function. The evaluation of $, for this and the later exercises can be done by conversion of the $,expression to logarithmic form, through the use of a log log slide rule, or through the use of a table of ex- ponential~. The student is then asked to calculate the values of J., q2, and 4 ~ r ~ J . ~ at each r value listed, recog- nizing that $ represents the complete wave function, J.,&,. Each of these quantities is to be graphed as a function of r.

Exercise 5. Electron Probability Contour Map for a 1s Electron. Using the results of Exercise 2, the student is asked to draw contours in the xz plane for electron grob- ability values of 1.50, 1.20, 0.90,0.60, and0.30 e-/AS.

Exercise 4. Radial Dependence of $, $2, and 4~r2$Z for 2s Electron. Values of J., and then of $,$%, and 4 ~ r ~ $ ~ are to be calculated for a 2s electron for selected r values. The dependence on r is to be graphed for $, $%, and 4 ~ r ~ $ ~ .

Exercise 5. Electron Probability Contour Map for a 2s Electron. Using the results of Exercise 4, the student is asked to draw contours in the xz plane for the successive encounters, moving out from the nucleus, with electron probability values of 0.0036, I$0018, 0.0000, 0.0018, 0.0036, 0.0036, and 0.0018 e-/Aa. Shading of the re- gions for which the probability is greater than 0.0036 e-/A3 emphasizes the pattern.

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Exercise 6. &., and for a p Electron. Since the angular component expression is simplest for a p, elec- tron, the student is asked to evaluate $,,, and $a,,2 for a p, electron for 8 values from 0" to 180" in 10" incre- ments. He is then asked to graph the $,,,, and values as a function of 8.

Exercise 7. Radial Dependence of $ and $? for a 2p Electron. $, for a 2 p electron is evaluated for selected r values. $ and $? are then evaluated for these distances either side of the nucleus along the axis on which the p orbital is oriented, e.g., along the z axis for a 2p, elec- tron, using $,,, values for 8 = 0" and 8 = 180" from Exercise 6. The student is asked to graph $and $% ver- sus r , either side of the nucleus.

Exercise 8. Electron Probability Contours for a %p, Electron. We have the student evaluate and graph $as a function of 1. for 10' intervals of 8,combining values for

and $, from Exercises 6 and 7. From these curves the distance-direction ( r , 8) combinations that give the following $ values are to be read: zt0.063, zt0.090, *0.126, ~'~0.155, and zt0.179. These values correspond to electron probability (f) values of 0.004, 0.008, 0.016,0.024, and 0.032 e-/A3, respectively. Contours in the xz plane are then drawn for these electron probabil- ity values. I t is desirable to develop the contours in this exercise from $ values, rather than $? values, in order that the values obtained for $ as a function of r in each direction can be used again in Exercise 10.

Exercise 9. Electron Probability Profile along the Bond A x i s for a n sp-Hybrid Electron. Normalized wave functions for the diagonal s p hybrids are:

along the bond axis of a hydrogen molecule ion on three bases, to allow comparison.

( A ) The electron probability profile for a pair of non-interacting hydrogen atoms based on averaging at each point the electron probabilities for a 1s electron of each hydrogen atom. This profile is obtained by evaluating + $2) for points along the line pass- ing through the nuclei of the atoms. $., and $B refer to the 1s wave functions of the respective hydrogen atoms.

( B ) The electron probability profile for an electron shared in a als bonding molecular orbital. Electron probability values for this profile are calculated from:

where S represents the integral of the "overlap term," $A$-

( C ) The electron probability profile for an electron shared in a u*ls anti-bonding molecular orbital. Values for this profile are calculated from

The accepted separation of the atoms in H2+ is 1.06 A. It is suggested that the student consider the nuclei to be 1.00 A apart and that he evaluate the electron proba- bility for each of these profiles for those points along the bond axis for which Exercise 2 provides values for both atoms. A value of $2 + $2 can be calculated for each location from the results of Exercise 2 and used in the evaluation of each of the three profiles. For profiles (B) and ( C ) , the value of 2 $& also must be calculated for each location from the results of Exercise 2. Profiles - . - ~ ~~~~ ~~ ~ ~~~ ~ ~~ ~~ ~ ~~

$a = (lid?) ($'>a + $'*A (B) and (C) could be calculated by evaluating q, and q,, then squaring these values, but this approach would

*Dl = (1142) ($2. - Ad. not bring out the influence of the overlap term. The The student is asked to evaluate $D, and J.o,2 for those value of the overlap integral, S , in the normalizing fac- positions on the bond axis (the z axis here as we have tors for qO2 and qS2 can be obtained (6, 7) for these used the 2p, orbital) for which he has already calculated molecular orbitals of the hydrogen molecule ion from: the $s, and hp values in Exercises 4 and 7. Both fi and $2 are to be graphed as a function of distance either side s = (& + + 1) e-piao

nf tihe I I I I P I ~ ~ S . . . . . . . . . - . . . -. . in which r is the separation of the atoms and aa is the

Exercise 10. Electron Probability Contours for a n s p ~~h~ radius, H y b d Eledron. Using J.2. and qzV values from Exer- Exercise 11, in conjunction with Exercise 9, helps the cise 4 and 8, the procedure of Exercise 9 is used to obtain student avoid the development of MO de- the ~rofile of $D, values along lines in the ~z plane de- scriptions (using AO's of two atoms) with the develop- scribed by the following 8 value combinations: 20'- ment of hybrid AO descriptions (using dissimilar ~ 0 ' ~ of 16O0, 40-140°, 6&120°, 70-110°, and 80-100". Each the same abm). pair of 0 values combines to describe a line passing through the nucleus, one 0 value applying onone side of Literature Cited the nucleus and the second value applying on the other side of the nucleus. Each of these profiles is graphed. (I ) *. W., J. CHEM. EDUC. 42, 140

From thefie graphs and the fi graph of Exercise 9 the ii; g::;,? J d ~ ~ ~ ; ~ ~ ~ $ ' , " 7 " ( ( ~ , " , " ~ ~ ; for more student is asked to read r , 8 combinations with which to source see C o u ~ s o ~ , C. A., "Valence," (2nd ed.), Oxford plot contour lines in the xz plane for $ values of 0.000, University Press, London, 1961, p. 40. +0.063. +0.090. +0.126, and +0.155. These come- (4) OGRYZLO, E. A., AND PORTER, G. B., J. CHEM. EDUC. 40,

256 (1963). to contours for *? Of O.OOO' 0.0041 0'008~ (5) BERRY, R. S., J. CHEM. Eu~c.43,283 (1966). This reference 0.016, and 0.024 e-/A3. includes an extensive annoted bibliography.

Exercise Electrrm Probability Profile along the (6) SLATER, J. C., "Quantum Theory of Molendes and Solids," Vol. 1, McGraw-Hill Book Co., New York, 1963, pp. 23-25.

Bond for lhe Ion. The student ( 7 ) WIBERG, K. B., "Physical Organic Chemislry," John Wiley is asked to evaluate the profile of electron probability & Sons, Ine., New York, 1964, pp. 59-61,

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