Richard J. Olcoll Southwestern at Memphis Memphis, Tennessee 38112
Visualization of Molecular Orbitals Formaldehyde
O n e of the objectives of current chemical education is the development in the student of an understanding of the behavior of electrons in molecules. Unfortunately, the abstract wave-functions necessarily used to describe this behavior carry the triple burden (especially for the mathematically unsophisticated) of obscure derivation, interpretation outside of normal experimental frameworks, and difficult visualization. In freshman courses, the derivation is usually skipped and the molecular orbitals (MO's) presented ex cathedra (although we have had some success with an axiomatic approach based on symmetry). The wave-particle dilemma may be met head-on by appeal to Bohr's Principle of Complementarity (I), or else by total suppression of the idea of "the electron" as anything but a measure of electrical charge. I n the latter case the function W is interpreted as a distribution of electrical charge density, rather than a positional probability distribution. In either approach, there remains the necessity of communicating the shapes of the distributions corresponding to the MO's of a molecule. Direct presentation of the mathematical functions (if available) is inadequate, and indeed unwise for most freshman classes, so one must perforce turn to models and diagrams. The literature (8) and the market-place (3) are replete with models of various types of atomic orbitals, but thcre are as yet no really satisfactory solid molecular orbital models. Gymer (4) has described a fluid flow analog, and the tcchnique described by NIcClcllan (5) could be extended to mol~cularsystems at a considerable investment of time and effort. A number of computer programs have been described (6) which produce grids of symbols which may be hand-connectpd to give contour maps of the (z,y,*) or (x,y,W) surface for simple molecules. Such a map of the valence-level (3a1,4al,5a,,lbl,2bl)electron density in the plane of the formaldehyde molecule is shown in Figure 6. The next logical step is an isometric projection of a portion of the surface, its detailed topography delineated by loci of *(x,y) [or W(x,y)] a t constant x or y. Such a representation of atomic orbitals has appeared recently in THIS JOURNAL (7). This type of diagram carries thc same information as the contoured representation, but in a manner much bcttcr suited to apperception by the untrained eye. Assuming the availability of a computer to produce properly arranged tables of function value versus x and y, the amount of labor 'An IBM 1627 plotter, connected to an IBM 1130 computing system with a 16K memory. 'The Angstrom unit is not part of the SI dimension system, so molecule-scale distances must now be stated in picometers (1.00 A = I00 pm).
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involved in preparing the projection is about the same as for a clean contour diagram. Furthermore, we have found that students readily graduate to the more easily sketched contour plots after some familiarization with the projections. Formaldehyde, H&O, presents a number of points of interest for discussions of bonding. The C2. symmetry elements and the separation of the various atomic orbitals into symmetry-compatible sets are readily demonstrated even without formal group theory. The Lewis diagram (used in our course as a bookkeeping device) is H.
implying the existence of two CH bonds, a CO a-bond, a CO rr-bond, and two "lone pairs" (either non-bonding or "canceled" anti-bonding electron pairs) on the oxygen atom. The disparity in C and O electronegativities suggests that the double bond should show polarization toward the oxygen. According to the valence shell electron pair repulsion (VSEPR) theory of molecular structure (8, 9),the two lone pairs should be in the molecular plane, a t slightly less than 120' away from the CO bond. In order to illustrate these points we havc used a computer-driven automatic plotter' to generate "solid" representations of the wave-functions and charge densities of formaldehyde. We chose to illustrate the all-elect,ron, self-consistent field, linear combination of Slater-type atomic orbitals (SCF-LCSTAO) solutions provided by Aung, Pitzer, and Chan (10). The MO energy level diagram is presented in Figure 1. The origin was placed at C, the z-axis along the CO bond, and the x-axis in the molecular plane. The published functions are normalized including differential overlap, so thc charge density a t (x,y,z) is simply the occupation number of the orbital times the square of the function value, in electrons per unit volume. Net densities for combinations of orbitals are sums of orbital densities. In the program (written in Basic Fortran IV), the desired function is evaluated along lines of constant z (12.52 pm apart)2a t points separated by 7.33 pm in x or y. Computing time was considerably reduced by setting up and referring to a table of cxp(-[r) over the range 0 < p < 10.00 a t 0.10-unit intervals. A typical plot required about 5 min of computer time, of which some three minutes were taken up by the plotter. The calculation time could be further reduced with a small loss in accuracy by obtaining square roots from a pre-computed t,able. The resulting maps of charge density are given in Figures 3,4, 5, and 7.
Figure 2. Binding (clear) ond antbbinding lhakhedl regions ~ b w t otoms A and 8.
In studying the effects of particular MO charge distributions, we first consider the atoms pairwise. If charge is concentrated between two nuclei (either onaxis for a a-MO or off-axis for a s- or 6-MO), then the distribution is binding for those nuclei. On the other hand, if charge is kept sparse between two atoms, and in particular if an M 0 has a node in that region, then the distribution is anti-binding f o ~those nuclei. I n the molecule as a whole a distribution is bonding or antibonding (note the change in vowel) depending on the net effect of all the pairwise interactions. There is a third possibility: if charge is evenly divided between binding and anti-binding regions (for instance, if the atoms are far apart and show virtually no interaction), then the two influencesare balanced and we have a nonbonding distribution. This description is a verbal transcription of Mulliken's population analysis equations (12).
Figure 1.
MO energy level diagram for formaldehyde.
Before we consider the figures in detail, a few words are in order regarding their use. Our classroom discussion of bonding in molecules is based upon the Hellmann-Feynman electrostatic theorem (If). Put baldly, the content of the theorem is that the business of quantum mechanics is the derivation of electron charge distributions; once these are determined the forces on the nuclei can be calculated by classical electrostatics. On this basis one can classify volume elements near a pair of atoms as either binding or anti-binding (Fig. 2 ) . Electronic charge in the binding region holds the atoms to each other through two effects: simultaneous attraction of the two nuclei to the negative charge, and reduction of the internuclear repulsion by this interposed charge. (The two are of course inseparable, but, we have found that emphasis of both reduces the difficulty of later work, viz. VSEPR structure problems.) Charge in.the anti-binding region has. the opposite consequence, for two reasons. First, charge here represents a loss of charge from the binding region and thus a reduction in the two binding effects. For the second reason, suppose that electronic charge is concentrated to the left of atom A. This charge will attract both nuclei, but since electrostatic force follom an inverse-square law atom A will be more strongly pulled to the left than will atom B. Thus, cbarge in the anti-binding region not only abstains from binding interactions but actively pulls the two atoms apart.
Figure 3.
Charge density distribution%in the bl (in-plane
system.
Turning to Figures 3-7: the b, MO's are derivcd from s,"CO), s,*(CO), and a*(H2). These tlirec orbitals con~bineto produce a situation very much like the three MO-four electron bonding proposed for the rare-gas fluorides (IS). Thc diagrams in Figure 3 shorn vividly that lbl is bonding, 2bl nonbonding (or nearly so), and 3bl anti-bonding. Note the continuous ridges connecting all atoms in lbl and compare with the internuclcar nodes in 3bl. The sharp dips a t the hydrogen atoms are t,he nodes in the 1s charge density at each nucleus. The 2bl MO is of special interest. It is thc highest filled MO (Fig. 1) and thus controlling in react,ions\ ~ i t h electrophillic reagent,^; notice the heavy loading of the oxygen atom. The two most stable al RilO's (Fig. 4 ) arc essentially pure lso and i s , atomic orbitals, as cxpected from the large cnergy difference betwccn these orbitals and all others involved in the molecule (Fig. 1A). Thr diagram for 3al displays a problem that arises ~vhent,hr 1\10 includes a significant contribution from 2s at,omic orbitals: charge attributable to such a function is concentrated close to the nuclcus so that linear plots of !P2 on a reasonable scale show littlc of the bonding structure. The accompanying plot of log(!?+) c o v m Volume 49, Number 9, Sepfember 1972
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al+bl Figure 5.
Figure 4.
Charge dmrity distributions in lhe ol[o)system.
the rangr down to (W),., and emphasizes the low densities a t the expense of the peaks. The log plot shows charge in the binding rcgions between all thc nuclei, hut the orbital is predominantly the oxygen 2s function. I n contrast to the oxygen-polarized 3al MO, 4a1 has most of its charge on the CH2 end of the molecule. In 4al we see thc first appearance of an antibinding region in a bonding MO: the node between carbon and oxygen is quite plain in the log plot but not in the linear one. The 5al MO iv essentially a CO a-bond, again heavily favoring the oxygen. The last two al 110's are anti-bonding, shown hoth by their positive energies and their node structures. Notice the similarity between 7al and the a* MO of carbon monoxide. At the inception of this work, we had the notion that, even though neither thc al nor the bl MO's were likely to concentrate charge at the 120' lone pair positions, the two sets together might well overlap to produce the same effect. Figure 6 shows the results of summing various sets of charge densities in the occupied in-plane orbitals. The lal and 2al contributions have been omitted for reasons cited above. The (aI bl) view provides inconclusive evidence on the matter of localized lone pairs: the faint wings visible in the loci to the rlght of the oxygen atom may or may not be a t the '(proper" angles. A vertical viewing angle is necessary to settle the question and for this we return to the con-
al
Net in-plane shwge density distributions.
tour grid format (Fig. 6). This plot was calculated with the same algorithm as the others but on an IBM 1620 computing system (40K memory). Interval symbols were chosen to maximize contrast between neighbors and provide "automatic contours." Close examination of the region near the 120" line shows no evidence of an accumulation of charge relative to the 90" and 180" regions. The disparity between these results and thc predictions of the VSEPR theory is due to a shortcoming of the SCF calculation (and all other one-electron MO theories). Strict Hiickel formalisms explicitly consider only the attractive interactions between electrons and nuclei. SCF theory adds to these forces the interelectronic repulsions, but in an approximate form: the clectron "in" a particular MO is considered to move in the summed, time-averaged field set up by all the other electrons in "their" MO's. More sophisticated theories must attempt to take quantitative account of the instantaneous set of forces acting on each electron, resulting in a correlated motion of a11 of them. VSEPR theory and Linnett's "doubiequartet" proposal (14) hoth concentrate in a qualitative manner on this spatial correlation of electron positions, and thus appear very different from "ordinary" MO descriptions. I n the case of formaldehyde, the averaging proccss inhercnt in the SCF calculation of reference (10) has had the effect of obliterating the lone pairs. Two other features of Figurcs 5 and 6 are north comment. The contour diagram shows definite evidence of a direct interaction between the two hydrogen atoms, again reminiscent of the "H, plus CO" description of this molecule. I n Figure 5, comparison of the distribu-
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+
bl). Symboh rhonge at Figure 6. Grid plot eqdvaknt of Figure 5(01 regular intervals of increasing charge density in the order /+ElJMNTY38. 9'.
.1 . . . . . . . . . . . I .
;...p
T '""pq
+',
.w.J
..............
lb2 Figure 7.
.......
2b;
....
0 I.. ..
..-..
'
...-
log la,+b2)
Net out-of-plone r-orbitals.
tions with and without the bl contribution shows the latter to be surprisingly important in establishing the CH bonding. The two out-of-plane s-orbitals, shown in Figure 7, appear roughly as expectcd. Notice in particular the distinct internuclear nodal plane in 2b2, and its considerable extension beyond thc carbon atom. Similar extension in the analogous s * orbital of carbon monoxide facilitates thc metal-to-ligand baclc-bonding in transition-metal carbonyl compounds (15). The last diagram, showing the charge density sum of t,he occupied a, and b2 orbitals, is notable for the smooth variation of density along the y-axis. The charge distributions of the al and s, systems merge with no evidence of the "wings" shown in many texts. I n summary, we have presented a novel form of MO charge density depiction and used it to analyze the SCF-LCSTAO wave-functions of the formaldehyde molecule. We have confirmed the MO-type allocation
inferred from the Lewis diagram. We have also showed that there is no reason to cxpect Hiickel or SCF MO theories to yield the localized lonc-pair charge concentrations predicted by the VSEPR and doublequartet theories. Acknowledgment
This work was supported by a grant from Southwestern at Memphis. Literature Cited (1) K A u e w m ~ ,W.. "Quantum Chemistry," Aoademia Press, Inc.. New York, 1957, p. 166. (2) cf. baok isaues of J. C ~ E M Eonc. . (3) Gonnon, A. J., J. CXEM.&DUO., 47, 30 (1970). . G.. J. C ~ E MEouc.. . 46,493 (1969). (4) G r ~ e nR. (5) M C C L ~ L G AA. N .L., J. CHEM.Eouc.. 47, 761 (1970). (6) la) R e l ~ ~ R. n . C. A N D HOVBE.J. E., JR.. J. CREW.EDVC..45. 465 (1968); ( b ) BADE%M. J. CXDU.EDUC.,48, 175 (1971); (c) C R A I ~ . N. C.. SHERETZ.D. D . , CARLTON. T. S., A N D ACKERMAN. M. N . , J . C x e r . Eouc.. 48, 310 (1971): (d) D. L. Perensoa A N D F U L L E ~ M. E.. J. C m x . Eouc.. 48. 314 119711. (7) ~ono*s,'w. T.. A N D LINNETT. J. w., J.'CHEX.EDUO., 47, 872 (1970). (a) A s this material was being nrepared for submission, similar maps of states of the Hz moleoule were given by DEWAX. M. J. S., m o KELEIAN,J.. J. CABM.E ~ ~ c . , 4 8 , 4 9(1971). 4 (8) D A Y , Jn.. C . M., A N D SELIIS, J.. " T h e o m i d Inorganic Chemistry," (2nd ed.), Reinhold Book Corp.. New York. 1969, up. 320R. (9) G T L L B ~ PRI ~. J.. ; . J. Cwem. Eouc., 47, 18 (1970). (10) AUNG.S., PITZER, R . M.. A N D CHAN,6. I., . I . Cham. Phus.. 45, 3457 (1966). (11) P m ~ n . F. L.. "Elementary Quantum Chemistry," MoGraxv-Hill Book Co., New York, 1968, p. 474. (12) Murmren, R. S., J . Chem. P h y s . , 23, 1833 (1955) and subsewent pspera. (13) R u n o L ~R . . E., . I . Amar. Chem. Soc., 85, 112 (1963) (14) L u m n . W. F..J. C a ~ m Eouc.. . 44,206,268 (1967). (15) Reference (81,pp. 4448.
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