provoc~tive opinion The Theoretical Emperor Is Wearing the Proper Clothing! A Detailed Defense of Teaching - Quantum Chemical Ideas in Undergraduate Chemistry Courses Clyde K. Edmiston University of Wyoming. Laramie. WY 82071 This title is adapted from the Opinion article ( I ) of R. T. Sanderson, "Is the Theoretical Emperor Wearing Any Clothing?", in which he again tries to make the case for teaching less qualitative quantum chemistry and for the teaching of his approach to calculation of chemical binding effects. Here it is argued that students in chemistry should he learning elementary atomic and molecular orhital ideas about to the extent that is present in most texts today. Not only are these ideas well-grounded in quantum theory, which everyone including Sanderson accepts as the correct basic theory, but they also give the intelligent students insights into many properties of molecules that are otherwise not understandable a t all. There are well-known and very good reasons why experimental chemists and physicists early accepted and used the qualitative orbital insights of Pauling, Mulliken, Slater, Hund, Lennard-Jones, Huckel, Bethe, Coulson, and others. Furthermore, increasingly accurate ab initio and semiempirical molecular orhitals (MO's) and configuration interaction (CI) calculations of recent years have shown that fairly accurate results (2) can he obtained for many small molecules besides Hz. New methods for calculation (26) and transformation (2c) of molecular orhital energy integrals has made this possible. Also there has been much progress in developing better and more visualizahle ways of understanding even complex wavefunctions, such as localized molecular orhitals (LMO's) (3),electron-pair correlations theories (4). and density-functional theohes (50).Chemical bonding is incrrasinglyk n g better understood rfi),within theorbirulnpproaches,asare also the reasons why molecules have the shapes that they have (7). The self-consistent field (SCF: Hartree-Fock) MO's are essentially those qualitatively described in general chemistry texts (for small molecules) and their localized version (LMO's) are very visualizable for students ( 8 ) . The SCF approximation has been successful for calculation of many properties of molecules, even for molecular binding energies when (1)isodesmic reactions (2) are treated, or (2) low-order perturbation theory corrections are "tacked on" as in the independent-electron-pairs-approximation and its extensions (2a). In fact the true nonrelativistic energy of a molecule with N electrons can he written as
where hbis the kinetic energy and nuclear-electronic attraction energy for the electron in th_eith MO, NR is the nuclearnuclear repulsion energy, and J; = Jjj- Kjj e;j. Here Jjj
+
and K;j are, respectively, the coulombic and exchange repulsions between the electrons in the ith and jth MO's (spin MO's) and eij is the corresponding electron-pair-correlation correction. Thus J;jcan he thought of as an "effective electron repulsion" between the electron in spin orhital "i" and the electron in spin orhital "j."This energy formula holds whether localized MO's (LMO's) are used or delocalized "spectroscopic" MO's are used. I am not advocating that this formula and its meaning be taught, necessarily, in the general chemistry course, though it should be given in physical chemistry texts, hut it illustrates that SCF theory and its extensions are not so unvisualizahle as some critics claim. It is not beyond the abilities of even first-year chemistry students to visualize that the total electrons' energy would be the sum of their kinetic energies, their attraction by the various nuclei, and their mutual repulsions (Jij's). Since increasingly accurate properties are now being calculated for small molecules using the SCF theory and its extensions (with CI), we would do a disservice to students not to give them some qualitative insight into the nature of molecular orhitals and also some insight into the valence-bond-theory ideas of "resonance structures", etc. Students can easily visualize LMOs corresponding to bonds, lone pairs, and inner shells, and then visualize forming the delocalized spectroscopic MO's as simple linear combinations of these LMO's. That would be a good approach for future writers of chemistry texts. Furthermore the connections to valencebond theory are much more easily visualized in terms of the hybrid atomic orhitals (HAO's), which are the principal components of agiven LMO, as is well known (9). It would be unfortunate if future writers of general, organic, inorganic, and physical chemistry texts took Sanderson's advice and wrote very little about molecular-orbital and valence-bondtheow ideas: verv little about orhitals. We need to i m ~ r o v e rhe presentntim, n i sugpe51ed abow, bur nbt to stop doing so. Elernenraw and inorvanic chemistrv text; do discusr the general ideas bf electronegativities ocatoms and how this leads to unequal sharing of electrons in polar covalent bonds. Usually the electronegativities are those of Pauling and Mulliken, or refinements. The polarization of most sigma-honding LMO's toward the more electronegative atom is an important concept for students, as is also the resulting reverse polarization of some pi-MO's in some cases, for example, in the CO molecule, with the two effects contributing to a nearly zero dipole moment of the molecule. Except for homonuclear diatomic molecules, delocalized (canonical: spectroscopic) MO's are somewhat difficult for students to visualize, so perhaps other CMO's should not he introduced in the "freshman chemistry'' course, unless most Volume 65 Number 3 March 1988
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of the students have had a quite good introduction to descriptive chemistry in their high school course (8). Even the introduction of LMO's. and the HAO's which vrinci~allv compose them, should always be carefully tied ti howthey "ex~lain"molecular stabilities or instabilities, homologous series of molecules, nearly transferable bond energies, etc. We must becareful to exulain that it is not known how murh A 0 hybridization in LMO'S causes a molecular geometry, and how much i t is the result of that geometry (7n. Also we should point out that we do not yet know what is the basic "physics" behind the success of very simple models like the VSEI'H one. such as the relative importance of nuclear repulsion, "Pauli forces", electrostatic interactions of orbital rharee"rloud;i". etc. 1 7 ~ )In. fact studentscan be ~hullenced by tGe knowledge that "cry many of the theoretical ideas must vet be sorted out and examined using better experime& data and wavefunctions yet to be calculated. " ~ o d e l building" is really only getting started in modern-day quantum chemistry (7), and students should be given some awareness of theereat excitement in this endeavor. Whether they themselveswill build these "models" or will only use them in the years ahead, they need to feel the excitement of human minds' attempt to "make sense of" the physical world. Even if chemistry were of no practical importance whatsoever for making "better things for better living", the beautiful insight given us by quantum theory, statistical mechanics, and thermodynamics should be taught a t every level for which the students are mathematically ready. Total Failures of the Elec(ronegatlv1ty Approach Along with qualitatiue discussions of localized MO's (and sometimes of delocalized MO's) it would he wonderful if we could present a reliable, easily explainable, and simple method for the calculation of molecular binding energies, as Sandersou claims his approach provides. However, his approach totally fails to describe the bonding or antibonding of some molecules and molecular complexes, although he has found many for which i t works fairly well for this single property. Although an "electronegativity equalization theorem" has been rigorously derived from "density functional (Hohenherg-Kohn theorem) theory" by P a n et al. @a), apparently it is quite misapplied in Sanderson's approach or else his theory would not totally fail in some cases as shown below. Consider cyclobutadiene(CaH4), which is well known from experiment and from good quantum calculations (2) to he unstable relative to two C2Hz (acetylene) molecules. Benzene (CsHs) ismore stable than three C2Hz molecules by A& = -143 kcallmol. Sanderson's approach necessarily calculates that aEg for 2C2H2 CIHq is exactly two-thirds of A&, if we assume the same bond lengths (for C-C and C-H bonds) in both cyclic compounds. If Sanderson's approach were correct, then CaH4 should he even more stable than this for its optimum hond lengths. The sigma electrons' strain energy for C4Ha should not differ greatly from that in cyclic ClHs and cyclic CaH6 (-20-25 kcallmol) (2). As is well known, molecular orbital theory easily and immediately explains this situation with the "(4n 2) rule" for aromatic stabilities. Similarly Sauderson (Ic) calculates AHf = -120.0 kcallmol for B3N3H6in very good agreement with the experimentalvalue of -122.3 kcallmol, by assuming that the B-N hond of the cyclic compound (analogous to CsHd is half a BN' and half a B-N" bond. Therefore he must calculate for the cyclic BzN2Ha that AHr 2 -80 kcallmol, and yet this molecule is not found and is unstable for much the same reasons as is C4Ha. Since by its nature the Sanderson approach does not account for molecular orbitals, or even for the reasonance of valence-bond-theory "structures", there is no way the approach can be modified to give the correct energies for both C4H4and CsHs. Furthermore, organic and inorganic chemists are familiar with many more examples
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Comparison of lsodesmic Reaction Energles: Experimental vs Calculateda Molecule H9C-CH2-CH2-CH3 HJC-CHI-CHZ-OH H3C-CH2-C-H H3C-CH=CH-CH3 H3CC%H H3C-CH4He H3C-CHZ-NHI HJC-CHI-OH HnC-NH-CHJ HIC-0-CH, O=CH-CH, O=CH-NH2 O=CH-OH C4H-CHI-CH, O4H-0-CH, O*(CH& =(OH)-CHI
Experimental
Sanderson
5.43 8.52 10.01 10.79 7.46 5.27 3.53 5.93 4.47 5.87 9.68 29.20 32.80 13.06 35.66 19.51 22.91
3.2 2.6 9.3 4.3 2.3 3.1 -7.8 1.6 -8.1 -2.0 1.9 17.0 27.6 5.3 25.2 2.6 6.9
Benson 4.84 8.13 9.49 10.72 7.07 5.36 4.21 5.71 5.22 5.20 10.79 31.11 34.11 13.48 39.01 20.54 24.28
SCF 2.20 6.87 10.14 6.89 8.95 3.91 3.58 5.39 2.48 2.33 10.98 25.10 30.11 12.63 29.53 21.16 21.84
A l l quamities in units of kcallmol.
where resonance stabilization (delocalization-stabilization) is a very important factor. Comparison with Other Theoretical Approaches lsodestnic reartions maintain the same numbers of bonds ofearhrype,e.g.. forO=C-0 CH4 * 2(H?C-OJ we ha\'e two C=O and four C-H bonds on each side of the reaction equation. In these reactions the bond can he thought of as changing environment. If a theory of bonds is valid, it should describe these slight changes in the "bonds". Even for molecules where Sanderson's method calculates fairly accurate total binding energies, the isodesmic reaction energies are not given well as can be seen from the table. The linearcombination-of-atomic-orbitals (LCAO) molecular-orbital (MO) self-consistent-field (SCF) method does better (2). The table shows the "large molecule" formed from smaller "pattern molecules" in an isodesmic reaction, for example, 2CzHs CzHa - 2CHa H3C-CH-CH-CH3. Even though the "molecular mechanics'' approach (10) makes use of much experimental data, when delocalized electrons are involved. it is found necessarv to carrv out approximate SCF calchations for those ele&ons. sanderson's less parameterized approach has less chance of avoiding these quantum delocalization-of-electrons effects. The "lone- air bond-weakeninn effect" (LPBWE) exists in many m o ~ c u l e swith N, 0 , and F atoms, according to Sanderson's calculations. The extent to which he invokes this effect is fairly unsystematic. He gives a series of 14 observations (rules) concerning this LPBW effect, after which he states: "A consistent interpretation of all these observations, in terms of atomic fundamentals, bas not yet heen develoned" (OD 75-79 of ref lb). Even these rules do not allow one to know h a t for CFa (and for other molecules as well) one must use an average of the "fully weakened" (CF')and "half weakened" (C-F") bond energies ((108.1 + 126.9)/2 = 117.5 kcallmol) to net a value close to the "experimental" C-F bond energy o r 117.0 kcallmol. No reason is given why there should be 50% of each type of bond energy present. Sanderson does this sort of arbitrary selecting and1 or averaging for many molecules in order to get fair agreement with experiment.
+
+
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Conclusions Theoretical chemists have made great efforts to decompose the total molecular binding energy into bond energies that are more or less transferable between molecules expect-
ed to have similar "bonds". Thev have not succeeded, and it appears to be impossible in principle. However, it has been known for about 50vears that this is approximately true (91, due often to canceliations of variousefffects in many cases. Many of the ideas Sanderson discusses closely parallel those of the localized MO discussions of molecular stabilities. Various "population analysis" techniques, like that of Mullikeu ( l l ) ,attribute partial charges to atoms, which implies some "ionic" contril)utions to binding. However, there are also nonbonded van der Waals-like interactions u.ith~nthe molecule, inductive effects, dclocalizarion effects, and orthogonalitv constraints between the l.hlO's, through which the environment of a bond affect its contiibution to the total molecular binding energy. Binding effects result from the positive overlap of atomic orbitals to form bonding LMO's, and antibonding effects result from negative overlap, causing MO nodes in "bond regions". T o ignore the teaching of these most basic ideas would deprive students of much important understanding. There needs to be much more work done to decompose total molecular bindine (and antibindine) energies into .'bond components" (bond energies) u,irh i6eproper currpctiuns for the bond's environment. The molecular-mechan~cs method (10) works in this direction, using MO calculations and experimental parameters as a guide. Furthermore, we need to understand better the basic bond energy itself, and LMO's and associated electron pair correlations (eii's) can contribute much to this. Ruedenberg's analysis (12) of the important energy components of binding in simple molecules has exnanded the earlier understandine. includine that due to ~erl;n'suse (13) of the ~ e l l m a n n - ~ G n m atheorem n (14). A study of these approaches shows that Sanderson's approach is indeed a gross oversimplification. There are manv nrofessional chemists, to sav nothing of students, who wishfully hope that they might he able t o understand the pro~ertiesof molecules without having to face either the conckpts or elementary mathematics necessary for a qualitative "feel" for quantum theory. All of us would like to avoid the huge quantum calculations sometimes necessary in order to predict theoretically some of these properties, even for quite small molecules. But nature is made so that electrons obey quantum theory, and there is not another known quantitative theory, even for a single property, for all molecules. Benson and co-workers (15) have had much success at predicting the binding energy components associated with certain eroups of atoms (usuallv one atom) existing in a certain iype'of environment within the given molecde by using total experimental binding energies of related molecules and "least-squares fitting". The types of molecules treated this way is limited by lack of experimental data and lack of homologous series of similar molecules (except for certain organic molecules). Nevertheless, their success is a motivation for quantum chemists to try to set up simplified computational schemes (based on LMO's) in which a "standard LMO" is perturbed more or less correctly in different environments, giving approximately correct estimations of "bond-energy fluctuation" due to environmental change. Physical chemistry texts properly show students how to use soectrosco~icconstants for gas-phase molecules and functions to calculate ah&l"te entropies, heat capacities, and e n t h a l ~ ychanges accurately. The Debye thebry of specific heatsworks very well for many simplesolids. Simple Huckel theory allows the understanding of many otherwise mysterious spectra of pi-electron systems. The heat capacity of gaseous Hz (o- and p-Hz, o- and p-Dz, and HD) can be totally calculated from Schrodinger's equation and statistical mechanics. These are all triumphs of quantum theory prior to the advent of large digital computers, and there are many more such "hack of an envelope" suc-
cesses. The student wholearns of these successes can not but marvel at the power of quantum theory. I t is not a t all fair to pick out the single property of molecular "energy surfaces", for which accurate results are coming more slowly, as a basis to attack the teaching of quantum chemical ideas and therefore also statistical mechanical ideas. As long as these ideas and equations are taught at the mathematical level of the students, they can undkrstand them and, i t is hoped, appreciate the great insights they can gain. How much purely descriptive chemistry we should ask students to "ro6memorize" will alwavs be a touchy subject among teachers of chemistry. ldeaily the students would have a real "thirst" for learning both facts and ideas and equations that systematize those facts. Some quantum chemical ideas presented to undergraduate students are oversimplifications, so that they may be more easilv The teaching of Benson's additivity rules, . grasped. - . particularly in organic and phyiical chemistry courses (15), is the proper alternative to teaching Sanderson's flawed approach, aithough a few calculations using electronegativity ideas are useful if the pitfalls are pointed out. Also the students should be warned that averaging the homonuclear covalent bond energies to get a heteronuclear covalent hondenerev estimate..E..-. (hv . whatever averaee is used) is a lea^ of f z h " without theoretical foundatik whatever. ~ h e h combining this E, with an ionic binding contribution, Ei, by either Sanderson's or Pauling's method, is another such "leap of faith", and so on for the manv other assum~tionsof these methods: Transistors, lasers, nuclear magnetic resonance, other spectroscopies (IR, UV, Mossbauer, microwave, etc.) have all had tremendous impacts on chemistry and chemical research. All these are easily understood in principle using a little qualitative MO theory, or its analogues (e.g., vibrational normal modes) in terms of quantum ideas. The future holds even more innovations from "quantum devices" (e.g., Josephson junctions, quantum switching devices), superconductivity, superfluidity, new spectroscopies such as Compton scatterina. etc.. which chemists who use them should understand i&&iple. I t is dangerous to chemical research for chemists to use devices that they donot understand even in a qualitative way. Certainly even qualitative quantum chemical ideas are difficult for undergraduate chemistry students to grasp, hut the rewards for them in future life as chemists, engineers, physicists, biologists, etc., make it well worth the effort. The problem is not whether we should teach these ideas, but how we can teach them so the student appreciates how very beautiful and powerful they are for explaining everything in his world, given some effort. I view Sanderson's approach as an unneeded digression from what should be taught, even if he does think that he has explained whv CO? is a gas and SiO? is a solid. w e are on the verge of extremely accurate quantum calculations of atomization energies for small molecules. Using an asymptotic convergence formula for ah initio e,'s, calculated with pair-natural orbitals (PNO's) (16), the energies of Ne and H 2 0 have been estimated to within about 1%of the EC (the correlation energy). This i s a n accuracy of greater than about 2 kcalhol. This might be somewhat fortuitous, but nevertheless i t appears to be sound, and no empirical parameters are involved.
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Literature Clted 1. ( a ) Sanderson, R.T. J. Chrm. Educ. 1986.63, MS:(b) Sanderaon, R. T.Polor Cooalonra: Academic: New York, 1983: (e) Sanderaon, R. T. Chorniml Bonds ond Bond Ensrgy; Aesdemie: New York. 1967. 2. (a) See for example, Hehre, W. J.: Radom. L.: Schleyer, P, v. R.;Pople, J. A. Ab inifio Moleeulor Orbital Theory; Wiley: New York, 1986; see p. 371 for eydobutadiene renults. Clark. T H o d b o o k olCornputotiona1 Chemistry. Wiley: New York, 1985: lbl B o w S. F. Pror. ROY.Soc. (London1 1950,AZO0,542: ("1Tan., K.;Edmiston,C. i c h & P ~ W 1970,5i, . 997. 3. (s)Daudel,R.:Lcroy,G.: Peeton,D.;Ssna, M. Quantum Charnistry;Wiley: Near York. 1983: p. 210;Edmiston. C.; Ruedonborg, K. Rau. Mod.Phys. l963,34,457;J. Chern.
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Phyr 1965.43 (nl, 101. S9:: qmnrum Theory olrlroma, M o l ~ c n r b .ond rhv .%Id ~ b Hund. l t P 1Ih)s.k 0 .FA,A ~ r s d ~NOS m ~ 1 ~ 1 k1966:~26: . 1931.7i.l.~5:Hell,C.G:Lcnnard.June3.J h r RO Soc tLonorn 1951.1N*I. 357. Hurley. A. C.. Eleetrnn Correlation in Smoll Maloeulea; Academic London, 1976; Schaefer H. F., 111. Eledronir Strwlvre of Atom and Molrcule~.A Surwy of Rigornu Quantum Meehonicol Reaulfs; Addiaon-Wealcy: Reading, MA, 1972; Sehaefer,H. F., 111. TheDeuelopment of AbinifioMefhd inMol~eulorE l ~ ~ t m n i c Structure Thmry; Oxford, Clarendon: New York. 1981; Sinanoglu, 0. J. Chsm. Phya. 1962.36, 706, Hurley, A. C.; Lennard-Jonas. J. E.: Pople, J. A. Proc. Roy Soc. (London) 1953,A220,446:Edmiston.C.J. Chem.Phw 1963,39,2394:Edrniston.C.: Krsusn,M. J,Chem.Phys. 1965,42,1119; J. Chem.Phys. 1966,45,1833; J Chem. Phya. 1968.49.192; J.Chsm. Phrr. 1970,52,3419. ( s ) P a ~ r , R C . I n J. k QuonfumChem. 1984,26,687:Parr.R.G.:Donnelly,R.A.;Iaw, M.: Pslke, W. E. J. Cham. Phya 1978, 68. 3801; lb) Parr.R G.:Bsrtolotti, L.J . J. Am. Chom. Sm. 1982,104.3Wl. Ruedenberg, K. Locolizofion and Delocolizofion in Bvontum Chemistry, Chalvet, 0.; Dsudel. R.: Malrieu, S.. Ms.; Raidel: DordrechfiHolland, 1976; Val. I, p. 225: Edmiston, C.; Lindner, P. Int. J.Qvanrum Chem. 1973,7,309. (a) Gimarc, B. "Mokculor Sfrueturn ondBanding the Quolitoti~MolsculorOrbifol Approach: Academic: New York, 1979: (b) Pearson, R. 0. Symmetry Ruler for Ch~mieolReoliaw: Wiley: New Yark, 1976; (c) Wmdward, R. B.: Hoffmann. R. The Consemtion of Orbital Symmetry: Academic: New Yo*, 1970: (dl Edmiston, State: Londm. I'
4.
5. 6. 7.
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C .Jaww.J . R a r t l ~ m . J.I Cnom P h , , 19%Rd,G90:.l? J Am Cnem Sw 1986. IN, 359l:tF Jarnc J W # . s r n . \ l . I l h l # t t l e . . l . t d m # a f ~ n . C J Chrm Ph,r 1973, 59. ~UtO 8. Seethefollo~ingBxB,whiihdiigood job of iittaluali"ggumffm fffffpts: Porterh i d , W. W. Concepts ofchemlstry; Norton: New Yor*; Dickerson. R. E.; Gray, H. B.;Haight,G.P., Jr. ChomieolPlineiples.2ndad.:Benjamin:MenloPark,CA, 1975: P i m e n ~ IG. . C.;Spratley, R.D. Understanding Chemistry; Holden-Dsy;SsnFrmeisco. 1971; Levin., I. N. Physical Chmwtry, 2nd ed.; McGraw-Hill: New York, 1983. 9.Slam, J. C. Qvontum TheoryofMntter; MeGraw-Hill: NmYork, 1 9 6 6 ; ~55CSMI. ~. 10. Burkert, U.;Allinger, N. L. Molecular Mechanics: ACS Monograph 177: American Chemical S ~ i e t Washington, y DC 198C 11. (a) Daudel, R.; Leroy,0.; Petera, D.; Sana, M.Quonfum Chem*.fry Wile?: NewYmk, 1983: p 137. 12. Ruedenberg. K. Reu.Mod. Phys. 1962,34,326; J.Phys. Chem. 1964,68,1028; Wileon, C. W.:Gddard, W.A. Theor Chim. Aefo 1972.26,195. 13. Berlin, T. Cham.Phys. 1958.19,208. 14. See Lovine. I. N. Quantum Chemistry, 3rd ed.; AUyo and Bacon: Boaton, 198%p 404, for a dinmadon and ~orefandeuioaniandn~refnoran.dseuioaniandn~refnoran.dseuioaniandn~refnoran.dseuioaniandn~refnoran.dseuioaniandn~refnoran.dseuioaniandn.refna.s 15. Benron, S. W. Thermochrmicol Kinetics, 2nd ed.: Wiley 1976; Herndon, W. C. J. Chem. Edue. 1974,51,10. 16. PeterPon,G.A.:Yea.A.K.:Bennett,A. J.Chem.Phys.lM5,R7,5105.5129;aeeref4f-h lor mrly usem of PNO'a, then called "pneudo~naturalorbitals".
