Arnold C. Wahl
and Uldis Blukis' Chemistry Division Argonne Notional Laboratory Argonne, Illinois 60439
Educational Film Loops on Atomic and Molecular Structure
T h e teaching of quantum mechanics and its bearing on atomic and molecular structure poses a problem a t the introductory level. Torn between a rigorous presentation which bewilders his students, and a primitive approach which is understood but often misleading, the teacher is usually forced to an unhappy compromise. One of the main reasons for the lack of success in teaching this material, we feel, has been the dearth of good teaching aids a t the introductory level which are necessary to overcome the students' preconditioned classical thinking. A further difficulty in accurately introducing atomic and molecular structure to the beginning student is the rapid advance which has been made in the field during the past ten years. The material described in this paper represents an effort to provide visual material intended to supplement a general chemistry or physical chemistry course ( 1 ) . These film loops are based on very recent advanced calculations (8-6) on the systems involved. It should be stressed that the pictures of atomic and molecular orbitals used in these films are not "artists conceptions" hut actual contour diagrams (7) of the appropriite electron density obtained from these recent (2-5) calculations. The display techniques have relied heavily for their accuracy on the use of computer-controlled automatic plotting devices ( 7 ) , and particular care has been taken not to introduce erroneous concepts in the name of simplicity. I n what follows, we describe six films ( 1 ) dealing with fundamental principles of atomic and molecular structure. The film titles are (1) Electrons and Electron Densities (2) Orbitals and the Building-Up Principle of Atomic Structure (3) The Covalent, Bond: Forming the Hydrogen Molecule (4) The Ionic Band: Forming the Lithium Fluoride Molecule (5) Repulsion of Noble Gwes: The Helium-Helium Interaction (6) Chemical Bondingin Polystomic System: The Formation of the Water Molecule.
In the second section of this paper we give a summary of the material presented in each film and a brief outline of the actual sequence of material in the film. I n the third section, we discuss the interpretation of probable electron density. I n the fourth section, we present preliminary student
under the aauspwesof the I:.S. .\tumtc i.:nrrgv Commi*siml.
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11210.
H, C h a r g e C l o u d
2
- 3 Dimensional
Dimensional Contour
D e n s i t y Plot
Diagram
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Figure 1. Three different ways o f representing the probable electron density ( 0 ) of the hydrogen atom; ( I ) or a diffuse "charge cloud." 121 or on x.yp graph o f electron denlity in on x - y plane p a e i n g through the nucleus. and I31 as a two dimensional contour diagram in tho x - y plane. The innermost contour line corresponds to an electron density of 2 5 e-/bohrJ and each rvccerrive outer one decreases b y 0 factor of two down t o 1/2" = 4.9 X iO-4/bbhr3. FILM LOOP
II
ATOMIC O R B I T ~ L S AND THE BUILDING UP PRINCIPLE OF ATOMIC STRUCTURE SAMPLE FRAME Sequence for building up the carbon atom. This process is performed for each atom in the series H Ne.
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ATOM is
2s
ATOMIC ORBITALS ZPr ~ P x
TOTAL
SEOUENCE I. In this film each orbital is displayed as electrons ore added. 2. Then shell density ir merged into total column to build up atomic density. in the case d the 2 p orbital merging is rpherieolly averoged (symbolized by rototing orbital os merging occurs1 Figure 2. Disploy o f the orbital 9tructvrs o f the carbon atom. In Film II there are such diagrams given in sequence for the series H through Ne. For the carbon d a m 1s. 2s. and total diagrams the innermost contour correrponds to m probable electron density of 1.0 e - bohr3, whde the 2p, and 2p, innermost contours correspond to ,225 e-/bohra. h;h successive outer rontow decreases b y a factor of 2 down 104.9 X 10-'s-/bohr8.
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reaction to the film content and discuss various ways of incorporating them into the chemistry curricula. Film Content
I n this section we present a brief outline of each film with accompanying sample diagrams. Film No.
I
Film I, which is intended to be an introductory one in this series on atomic and molecular structure, has two main parts. Part I illustrates the difference between classical and quantum mechanics and does so by starting with the prediction of the future position of a marble after its position and velocity has been measured. We then illustrate that the prediction agrees with observation. We then perform the same (idealized) experiment on electrons and find that our prediction does not come {rue in every event, and that the final position of the electron must be expressed in terms of probabilities. In Part I1 of the film, we turn to the electron structure of the simplest atom, H, display its probability charge cloud as a sphere of varying darkness, take a planar cross section through this charge cloud, plot the resulting probable electron density as a three dimensional (p-x-y graph) into a contour diagram (7) (Fig. 1) and, finally reduce the three-dimensional graph to a two-dimensional contour map (p map). Film No. I1
In Film 11, relying upon the introductory material on probable electron densities presented in Film I. the buildine-uo principle of atomic structure is illus: trated. The film proceeds sequentially from H to Ale. For each atom, the contour maps of probable electron density (p maps) for every occupied orbital appear and are
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&
FlLM LOOP m THE COVALENT BOND: FORMATION OF THE HIOROGEN MOLECULE
then merged into a p map for the total probable density. I n order to facilitate comparison between corresponding p maps of different atoms, two adjacent atoms are retained on the screen simultaneously (Fig, Z), the one whose orbital structure has just been completed and the new one in progress. Films No. Ill-V
These three films deal with the diatomic systems, H?, LiF and Hez. The films illustrate a covalent, an ionic, and a repulsive interaction, respectively. Each film begins with the question "How does the energy of the diatomic system change with internuclear distance?" To answer this question, we show the drawing of a potential curve coincident with the convergence of two symbolic atoms. Next, the p maps are built up contour by contour for the separated atoms. The question is then asked "How does the probable electron density change as the two atoms approach one another?" The atoms are then continuously brought together with a display of their changing probable electron density contours at selected internuclear distances. During the above sequences, the potential curve is continuously displayed so that the relationship between the changes in the energy and the probable electron density can be seen. I n Films I11 and IV, the atoms move closer together than their equilibrium separation and are then allowed to return to the equilibrium position. Several steps are shown in the top portions of Figures 3 4 . At the end of Film V, the differencesin probable electron density changes between the stable system Hp and the repulsive system Hen are compared. Film No. VI
In Film VI, H%Ois formed from OH and H. Changes in the electron density associated with the variation in the 0-H distance and the FlLM L W P P THE REPULSION BETWEEN NOBLE GASES: HE-HE INTERACTION
Figure 3. (left) Displays of tho beginning ond end diagrams in the formation of the HZ molecule. The innermost contour corresponds to o probable electron denrib of .25 e-/bohr3 ond eochsvcce~siveouter contour decreases b y a foctor of 2 down to 4.9 X 10-4 d-/bohr8. Figure 4. (center) Dirployr of three sequences in the formation of the highly ionic system LiF from the 1; and Ruorine atoms. Note change from Li and F atoms to Lit and F- atoms d l o distance of bohrr where fhe ionic conflgvrotion becomer more stoble. In 011 diagrams the innermost contour corresponds to a probable electron denrily of 1.0 eK/bohra and each succerrive outer contour decreases b y a factor of 2 down to 4.9 X 1 0-4 ee-/bahr3. Figurq 5. (right) Disployr of the beginning ond end diagrams in the repulsive inrermdion of t w Helium ommr. Note haw os atoms move together ths electron charge i s squeezed from between the nuclei to the ends of the molecule. (A consequence of the Pouli Exciurion Principle.) In d l diagrams the iQnermort contour corresponds to 0 probable electron density or 1.0 e-/bohrJ and each wccessive outer contow decreases b y o factor of 2 down to 4.9 X lo-' e-/ bohr5.
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H-0-H
bond angle are displayed (see Fig. 6). FILM LOOP V l T ~ Formotion L of the WOIW Malcsule fmm "and OH
Figure 6. Dirplays of eledmn density contwr diagrams for tho O H molssule and the H atom and the Rnol diagrams for the HsO molecvlo in i h equilibrium configuration. In tho two left hand diagrams the innermost contour corresponds to a probable electron density of 1.0 e-/bohr5, in the .ight hand diagram O.25e-/bohra and each rvccesive oulor contour dsreases b y o fador of 2 down to 4.9 X 10-'=-/bohva.
Significance of Probable Electron Density
The detailed interpretation of the p maps (7) is discussed in the teacher's manual (1) which accompanies the films. This section is a summary of the major themes in the manual. An analysis of the electron experiment in Film I, with help of the uncertainty principle, brings out the physical reasons why descriptions of low energy electrons must use probabilities, e.g., the probable electron density p. The relation between p and the Nelectron wave function rC. is (9)
Here x,, sf are the space (xi = x,y4,) and spin coordinates, respectively, of the ith electron. rC. is assumed to be real. What do the p maps in these film strips tell us? Future use and developments will provide many detailed answers to this question (8). However, even now it is clear that p maps contain a wealth of information about molecules and atoms. The p maps (a) Give a quantum mechanicdly correct quantitative and compact picture of probable electron density p. One can see how the size of atomic orbitals decreases with increasing nuclear charge. Yet, the orbital demities add up to give atoms of nearly constant size (Film 11). Relations between p of different nuclear configurations and different molecules are easy to see, e.g., the effects of Pauli's principle on the p of B and He$ (cf., Figs. 4-5). (b) Show that for all common operators F which are sums of one electron operators f (except those containing dierentid operators), the expectation value 7 is directly related to p (9)
'For example, the nucles.relectron attraction operator V,. for an atom with charge Z is V,. = -Ze'ZriP, where e is the charge of an electron and ri is the distance between the nucleus and elert,ron i. 'We are preparing written materials for student self-instruction.
where the integral is over all space.
Therefore different expectation values for, e.g., the multipole moments of different configurations or molecules can he related to changes in their p maps.% It is intriguing to relate the size, shape, and number of nodes of the atomic orbitals and their respective energies to the total atomic electron density and energy. In respect to Films.111-VI particularly interesting are such semiquantitative relations between the bond energy and p. When two H atoms (or an Li and an F atom) approach each other a stable bond results because the electron-nuclear attraction energy p,, is sufficiently lowered to overcome the increased internuclear (V,,) and inter-electronic (pa,) potential, as well as the electron kinetic (T) energies. In Hez the lowering of F,, is not sufficient to cause a stable bond. Since p,, is calculable by eqn. (3, changes in the p,, can be readily related to differences in p maps. For ground states the repulsive terms Teeand T also dependupon p only implicitly (10). However, the p maps show that the volume occupied by the electrons decreases when the molecule forms. This makes it at least plausible that Fa, and T should incrmse upon bonding. Instructional Uses of the Film Loops
These film strips may be used in a variety of ways: (a) their contents are largely self-explanatory; therefore, a no-stop showing is possible or they may he used with (b) stop-and-go showing with interspersed interpretations of changes in p maps; (c) repeated stopand-go showings with different levels or aspects of interpretation interspersed; (d) gradual introduction to concepts needed for understanding the films, followed by either (b) or ( c ) ; (e) self-instr~ction.~With this last method students could compare p maps quantitatively which, in turn, could lead to estimates of charge shifted and of changes in 7,The materials used in the films have been shown to physical and general chemistry students as well as to our colleagues. The observations which were made during the showings and written student answers to subsequent questions form the basis of the conclusions: these films can teach something constructive to all groups; explicit training in the reading and interpreting of the p maps is useful for everyone; and for most general chemistry students, the number of new concepts to be retained and understood is high enough to make a gradual introduction to the films advisable. Some of the conceptual difficulties which general chemistry students had were the following (a) Difference between probable electron and mass densities (b) Differences and relations between probable electron density and total charge ( c ) Relating charge shifts to energy changes ( d ) Interpreting potential energy curves (e) Difference between s.force and the related potential energy (f) Tendency to vkualize energy levels as localized in particular regions of the molecule, e.g., higti energy levels in high potential energy regions
As can be seen, most of these difficulties should not present serious obstacles, hut they do suggest strongly that preparation before showing the films is advisable. We are currently further assessing student and Volume 45, Number 12, December 1968
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faculty reaction to this material by actual demonstration and are incorporating the results of these trial showings in the student manual which will accompany the films. a Acknowledgments
We are very grateful to Dr. Max Matheson, Director of the Chemistry Division; to Dr. Louis W. Dini, and to Dr. John Bradish of the Office of College and University Cooperation for their enthusiastic support and encouragement. We wish to thank Professors D. Howery and G. Mennitt of Brooklyn College for their help in showing the electron density maps to classes and for useful comments and criticisms. We are also deeply indebted to Professors Schrage and Lippincott for their early and enthusiastic encouragement to produce these films. Literature Cited (1) WAHL,A. C., AND BLUKIS,U., "Atoms to Molecules," McGraw-Hill Book Co., Inc., New York, 1968.
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(2) DAS, G., AND WAHL,A. C., J. Chem. Phys., 44, 87 (1966).
Hydrogen molecule wave functions. (3) WAHL,A. C., AND BLUKIS,U., Lithium fluoride wave fonc-
tions. Unpublished cdculrttions: See A. C. Wahl, et al., Int'l. J . Qwmturn Chem., IS,123 (1967). (4) GILBERT, T. L., A N D WAHL,A. C., J. Chem. Phys., 47, 3425 (1967). Helium-helium wave functions. (5) BAGUS,P. S., AND GILBERT, T. L., Argonne National Laboratory unpublished calculations. First row atomic wave functions. (6) J A N I ~ Z E W ~ J., K IBLUKIS, ~T. U.,AND WAHL,A. C., to be submitted to J. Chem. Phys.; Multiconfiguration self-consistent field wsvefunctiom for the water molecule. (7) WAHL,A. C., Sciace, 159,962 (1966). (8) The fallowing references contain aiternative interpretations, not discussed in this paper: BADER,R. F.W., KEAVENY, I., A N D C n m , P. E., J . Chem. Phys., 47, 338 (1967) (References t,o earlier work of these authors will be found in this paper). RANSIL,B. J., AND SIN.AI,J. J., J . Chem. Phys., 46,4050 (1967). P. O., Phys. Rev., 97, 1474 (1955); MCWEENY, R., (9) LOWDIN, Proe. Roy. Soc., (London), AZ32, 114 (19.5); and RUEDENBERG, K., Reu. Mod. Phys., 34, 326 (1962). P., AND KOHN,W., Phys. Rev., 136, B864 (10) HOHENBERG, (1964).
