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A comparison of theories Andrew D. Liehr Bell Telephone Laboratories, Inc. Murray Hill, NewJersey Molecular Orbital, Valence Bond, and Ligand Field Before jumping into the intricate details of the various theories of valency, let us first pause a hit and try to obtain some historical insight into their course of development, their successes, and their failures. Only in this manner shall we be able to assess with any assurance the future progress of the theory of the chemical bond. Soon after the announcement of the Schrodinger equation for electronic motions, there were proposed and utilized three approximate means of formulating solutions of this equation as it applied to molecular problems: (1) the valence bond technique of Heitler, London, Slater, and Pauling; (2) the molecular orbital technique of Hund, Bloch, Mulliken, Lennard-Jones, and Hiickel; and (3) the crystal field technique of Bethe, Kramers, and Van Vleck. Each of these techniques had its limitations, its strong points, and its weak points. And the years 1930-45 saw a struggle among the three for pre-eminence in the minds of chemists and physicists, even though Van Vleck had shown in 1935 that they were absolutely equivalent when carried to completion (2,3). These years witnessed the adoption of the valence bond and molecular orbital methods by the organic chemist, the valence bond method by the inorganic chemist, the molecular orbital method by the molecular and solid-state chemist and physicist, and the crystal field method by the magneto-physicist. Each of these adoptions had the same driving reason: most chemists and physicists of this era were primarily interested in the ground electronic states of chemical systems. With the end of the war, scientific interest began to swing toward a concern over the excited electronic states of molecules. This precipitated a rapid fall from favor of the valence bond theory, and a consequent rise in esteem of the molecular orbital and crystal field theories. Thus in all fields of chemistry except inorganic, valence bond ideas concerning the nature of excited molecnlar states made their exit in the years 1945-55. But with the accelerated interest in inorganic spectroscopy by inorganic chemists commencing around 1955, even this last stronghold of valence bond concepts has begun to fall. Indeed, in the particular area of inorganic chemistry called coordination chemistry, valence bond theory has suffered its most grevious blows; even its descriptions of the ground electronic charge distributions have proved false in many important cases, especially with regard to the much vaunted, but completely specious -- Presented at the Symposium an Ligand Field Theory, 140th Meeting of the ACS, Chicago, September, 1961. magnetic criterion. [A most recent example of such a blow is the demonstration by McGarvey (4) that the odd electron in square planar CU+=complexes is in a 3d-like orbital and not a 4p-like orbital.] At the present time the valence bond theory of coordination compounds is being systematically supplanted by a molecular orbital-crystal field amalgamation, which has been aptly dubbed ligand field theory. [Crystal field theory was originally an ionic theory of chemical bonding until modified by Van Vleck. I n its modified form it was a highly simplified molecular orbital theory of the nd, (n l)s, (n l)p, and nf electrons in which orbital energies were of the molecular orbital type, but electron-electron electrostatic repulsion energies and spin-orbit energies were of the atomic type. Ligand field theory is a molecular orbital theory of the nd, (n l)s, (n l)p, and nf electrons in which both the individual orbital energies and the electron-electron repulsion and spin-orbit energies are of a molecular type. I Why has this particular pattern of historical dynamics evolved? The answer to this question is simple: the valence bond theory, although it is by far and away the superior outlook for ground electronic states, becomes hopelessly complex as a description of excited electronic states. The jungle of ionic valence bond struebures nverruns all attempts to describe electronic excitations by this technique. Molecular orbital and crystal field theory on the other hand suffer from the complementary deficiency: they provide adequate pictures of the excited electronic states, hut not of the ground electronic state of most molecules (they usually introduce too much ionicity into the ground electronic state). Therefore as long as ground electronic state properties were the vogue, valence bond theory shone, but when the properties of electronically excited states became the style its gleam was dulled. With this historical perspective behind us, let us now see how to extend the ligand field technique to encompass all inorganic compounds. In two frequently overlooked papers by Kimball (5) in 1940 and by Eisenstein (6) in 1956, there are tabulated the symmetry classifications of the primary atom and ligand atom orbitals for most geometries of interest. With these classifications written down once and for all, it is a simple matter to construct a molecular orbital energy level diagram, and with this diagram to predict the number and classification of the excited electronic states. We shall demonstrate this fact explicitly for a few exemplary systems. In Figure 1 we show the orbitals which are primarily involved in the bonding of a linear MX2 transition + + + + Volume 39, Number 3, March 1962 / 135 metal compound (e.g., CuC12). We have written down, from Kimball's and Eisenstein's tables, the correct symmetry designations of the primary atom and ligand atom orbitals prior to compound formation; that is, for symmetrically disposed reactants at infinity. Then of the ligand orbitals and the a.+* and T.* antihonding orbitals are composed mostly of the primary atom nd+ and nd,,, orbitals, respectively. Hence, for CuCI2 we expect the ground electronic state to be a 2Z,+ state [as there is one unpaired electron in the cg+* orbitarcapital letters denote the over-all state electronic distribution], a state with zero orbital angular momentum along the molecular axis, and the first two excited states to be the so-called nd excitations, TI, and 2Ax,which arise from the one electron jumps s.*-te,+* and . 8 c.+*, each. This is what is actually observed (8). As a second example, we present in Figure 2 the energy levels of a square planar complex such as - . . . Figure 1. Molecular orbitals for a linear triotomic transition metal compound ( D d . Note that the <-bond strudure is closely approximated b y the valence bond hybrids r'p'dl, and nots'p' or pld'alone. recalling that the symmetry quantum numbers,' c.+, c.+, s,,6,, etc., are exact quantum numbers at all internuclear distances, we allow the reactants to approach one another in a symmetrical fashion to produce the h a 1 molecule, and we combine only those primary atom and ligand atom molecular orbitals to form the complete over-all bonding, antibonding, and nonbonding molecular orbitals, which have the self-same symmetry designations (i.e., symmetry quantum numbers). We can, of course, obtain only so many complete bonding and antibonding molecular orbitals of a given symmetry type as there are primary atom and ligand atom orbitals of this same symmetry classification initially present. The ordering of the resultant molecular orbitals of the product molecule is completely based on qualitative concepts: the more the primary atom and ligand orbitals are directed toward one another, the deeper the consequent bonding orbitals will lie and the higher the consequent antibonding orbitals. will lie. For example, the cs+* antibonding orbital lies higher than the s.* antibonding orbital, as the primary atom c.+ orbital, nds2,is directed more strongly toward the ligand cg+ orbital than the primary atom s, orbital, nd,,,,,, is toward the ligand T. orbital (the molecular axis is the z axis). Moreover, as the ligands have a greater affinity for their electrons than does the primary atom, the c.+ and s,bonding molecular orbitals are composed mostly 'For the linear molecule the Greek letter symbols ( G , r , 6, p etc., replace the stomic designations a, p, d, j,etc. Just as these Latter symbols denoted orbital angular moments of 0,1,2, 3, ete., respectively, in the atom, the former now indicate the magnitude of the component of orbital angular moment 0, 1, 2, 3, etc., along the internuclear axis (thez axis) of the moleoule. 136 / Journal of Chemical Education P e I M A W ATOM ORBITALS Figure 2. MOLECULAR ORBlTAL5 OF T H E COMPOUND LlCAND MOLECULAR ORBITALS Molecular orbitdr for o square planar transition metal complex I D d Observe that the o-bond structure is well approximated b y the volence bond hybridsr'p'ff, ond notr'p2d'orpzd2alone. PtC14-2.2 The placement of the bonding and antibonding orbitals is again accomplished by qualitative principles and may be subject to reordering. The location of the a@* orbital, which is primarily nd,, % T h emolecular symbols corresponding to the atomic s, p, d , f, etc., designations for non-linear compounds are a, b, e , and 1 (an icosahedral molecule has the additional symbols y. and h, which are not to be confused with tho analogous atomic terms). The molecular symbol a corresponds to the atomic s (and the linear molecule designation a)-it means that the molecular wave function does not change sign under a rotation of 2 r l n about the molecular n-fold rotational axis of symmetry (e.g., the four-fold z axis of PtCL-2): the symbol b means that it does. (In a very direct sense this is equivalent to saying that the molecular a type orbitals have a component of angular momentum along the n-fold rotational axis of symmetry (the e axis) whose magnitude is a multiple of n (e.g., 0 or n), and the b type a companmt which is an odd multiple of n12.) The symhals e and 1 mean that the molecular wave function is degenerate (just as the atomic states p, d, f, eto., am degenerate), with two-fold and three-fold d o generacy, respectively (the icosahedral g and h symbols indicate four-fold and five-fold degeneracy). Such degenerate sets of in character, is especially vague as i t depends quite strongly upon axial perturbations (the z-axis is the four-fold axis) which are always present in solid or solution. With our assignment [similar to that of Fenske, Martin, and Ruedenberg ( l o ) ] we find the ground electronic state to be 'Al., and the first three excited nd states to be 'I?,., lA2., and 'E.. These transitions should be strongly enhanced due to vibronic (vibrational-electronic) intensity theft from the nearby en* and bl,* n-antibonding orbitals. The 'E, electronic state should exhibit a Jahn-Teller energy surface similar to that of Figure 3. The nuclear Figure 4. The nuclear dirplocements dlowed a rquors p l m m complex. Intensity enhancement is provided only by the odd (01 coordinoter, and JohnTellw forces only b y the wen (gl81, ond Bag madinmter I Figure 3. The JohmTeller energy surface characterirtic of a square planar system ( i 11. Nuclear motions on this surface ore essentially one dimensional. displacements which cause these Jahn-Teller motions and intensity enhancement are given in Figure 4.3 Figures 5-8 depict the molecular energy level d i a grams (in the absence of spin orbit coupling) for the trigonal bipyramid (PFk), the tetrahedron (SiFp, VCL, and MnOa-2), the octahedron (SF, RepB, and TiFe-a), and the cube (Ti+S:CaFd. Discussion of the result-, ant excited states is similar to that given previously. Note especially the number of Jahn-Teller states expected for these systems. Several such Jahn-Teller states have been recently observed for SiF4 (9). By now the procedures utilized to construct molecular orbitil energy level diagrams are apparent, and so my purpose in presenting this lecture has been accomplished. I sincerely hope that what I have said today will be of aid to those of vou who are concerned with the excited states of inorganic systems, and that it will stimulate a fervent interest in the science of inorganic spectroscopy. I eagerly look forward to the day when the idem of Mnlliken (1) and Van Vleck (2, 3) which were here outlined will be universally accepted. functions transform one function into the other under certain of the rotations and reflections permitted the overall molecular symmetry, and are identified in practioe in this way. This criterion for degeneracy is entirely equivalent to saying that such states have components of angular momentum (better yet, "permutational momentum") about some d o l d rotational axis of symmetry of the molecule which are not multiples of nI2. The subscripts "g" and "u" indicate that the wave functions are even or odd, respectively, under inversion in the center of symmetry; and the subscripts 1 and 2, etc., that they are even or odd under reflection in some given plane of symmetry. A Jahn-Teller molecule is one which exists in a degenerate (or nearly degenerate) electronic state. Such a molecule, acoording Roy. SOC.,161A, 220 to the theorem of Jahu and Teller (PTOC. (1937)), may experience eoulombic forces which tend to destroy this degeneracy by distortion of the nuclear framework. The J ~ a h n - ~ e l l theorem er and its consequences has been given by the author elsewhere. See LIEHR,A. D., Revs. Mod. Phys., 32, 436 (1960); "Annual Reviews of Physical Chemistry," Val.. 13, Annual Reviews, Ine., Palo Alto, California, 1962; Progr. Inorg. Chem., 3, 281 (1961); and P r o y . Inonorg. Chem., 4, 000 (1962). C. J., Adv. Chem. Phys., 4, 000 In addition, see BALLHAUSEN, (1961). A similarly graphic-type discussion of the closely related intensity problem has also been given elsewhere. See LIEHR,A. D., Adu. Chem. Phys. 4, 000 (1961); BALLHAUSEN, C. J., Progr. Inorg. Chem., 2,251 (1960). Figure 5. Molecular orbitals for a trigand bypyramid geometry (DIJ. See that the <-bond rtrvcture is nicely approximated b y the valence bond hybridsr'padJ, and not r1p3d'or r'pWalone. Volume 39, Number 3, March 1962 / 137 Literature Cited (1) MULLIKEN, R. S., Phys. Rar. 40, 55 (1932). (2) VANVLECK,J. H., J. Chem. Phys. 3 , 803 (1935). Rev. Mod. Phys. 7 , (3) VANVLECK,J. H., AND A., SHERXAN, -lfi7 -. f l-D-R--6 ,) . \ McG~~viw B., R., J. Phys. Chem. 60, 71 (1956). KIMBALL, G. E., J. Chem. Phys. 8 , 188 (1940). J. C., J. C h . Phys. 25, 142 (1956). E~SENSTEIN, COULSON, C. A,, "Valence," Clarendon Press, Oxford, 1952. HOUGEN, J. T.. LEROI,G. E., AND JAMES,T . C., J. Chem. Phys. 34, 1670 (1961). (4) (5) (6) (7) (8) +BONDS PrtlMAW ATOM 01181TAL5 MOLECULAR ORBITALS OF THE COMPOUND ".80NDI (9) HEXTER,R. M., private communication, February 1961. K., p"(10) FENSKE,R. F., MARTIN,D. S., AND RUEDENBERG, "ate eammuni<:i~tion,May 1961, also 140th Meeting of ACS, September, 1960. (11) LIEHR,A. D., XVIIIth International Congress of Pure and Applied Chemistry, Montreal, Canada, August 6-12, 1961. 'This reference pertains to the general theory of Jahn-Teller and non-Jahn-Teller energy surfaces. Recent Qualitative Discurrions NYHOLM, R. S., Biochem. Sac. Symposia, No. IS, 1 (1958); Record Chem. Progr., 19, 44 (1958); La Ricerea Seiatijea, Suppl., p. 3 (1958). KIMBALL, G. E., AND LOEBL,E. M., J. CHEM.EDUC.,36, 233 (1959). PLIR3 LICI\ND MOLECULAR ORBiTAL5 Figure 6(0) Figure 61bl. Molecular orbitols far a tetrahedral arrangement ITd) about a cectrml nr np nd type atom ore shown in Figure 6i.I. The same ir 1 ) s in i l p type .tom. rhown in Figure 6(b) for the nd in Perceive thot the .-bond rtrvctvre is neatly opproximated by the valence bond hybrid. r'#da, and not +pa or r'dJ alone. The electronic excitations from the filled nonbonding h irl orbital to the unfilled ontibonding e*(r) arbitd giver rise to the low lying single excited stater ITt and LTs, while that from fl(*) is, rl orbital produces the 'A?, 'E, ITlr and to the unfilled antibonding ta" 'T* states. As the ground electronis state i s 'A1 and the transition dipole restor, el: transforms a s h only the electronic jumpslA1 -IT2 ore allowed. TheLAl -'E,'Tr hops are made vibronically ollowed via intensity theft from the aIlowedlT~ state by the and n nuclear modes. ThelAz state is strictly forbidden unless intensity borrowing by the nuclear first harmonics ore considered ithis borrowing is usually inperceptiblel. The stater ' E , ITx, and IT2 ore, of COUIS~, John-Teller active (1 1). + 138 / + Journal o f Chemicd Edvcufion .. PRIMARY ATOM OFtB1TALS <-. MOLECULIIQ OFlBlTALS OF T H E COMPOUNO "...- '. ON. LoNC v.,.., L I G A N D MOLECULAR OeBITALJ Figure 7[b). Molecvlor orbitals for an oetohedrol disposition (Oh) about 0 central nr np nd type otom ore rhown in Figure 7(4. The some is rhown in Figure 7ibl for the nd (n 1 lr in 1lp type otom. Discern that the o-bond structure i s readily opproximated by the valence bond hybrids rLpsd2in agreement with the usual notion. An electronic excitation from the filled nonbinding h,(d tothe unfilled ontibonding f&) gives rise to the electronic states A ' I., LEm,%u,and of which only the T ' .I i s electronically accessible from on A ' 1. ground state. The 'E" and ITnu $totes ore vibronimliy dowed, but the A ' ,. date is strictly forbidden [if first hormonic vibranicinteractionsare discounted). The'E,.'T~,,and 'Tz.stotesallow JohnTeller antics il 1). + + SUTTON,L. E., J. CHEM.EDUC.,37,498 (1960). LIEHR,A. D., Bell Syrt. l'ech. J . , 39,1617 (1960). PEARSON, R. G., Chem. Eng. News, 37, 72 (June 29, 1959); J. CHEM.EDUC.,38,164 (1961). MANCH,W., AND FERNELIUS, W. C., J. CHEM.EDUC.,38, 192 (1961). LEWIS,J., AND NYHOLM, R. S., Chem. Eng. News, 39, 102 (Dec. 4, S u a ~ ~S., o ,J. Appl. Phys. Suppl., in press (1962). Recent Reviews NYHOLM, R. S., ORGEL,L. E., AND J@ROENSEN, C. K., in Reports of the 10th Solvay Conference, Bruxelles, May, 1956. MOFFITT,W. E., AND BALLHAUSEN, C. J., i n "Annual Reviews of Physical Chemistry," Val. 7, Annual Reviews, Inc., P d o Alto, California, 1956, p. 107. GRIFFITH,J. S., AND ORGEL,L.E., Quad,. Rev., 11,381 (1957). PRYCE,M. H. L., NuovoCimenlo Suppl. 3 (lo), 6,817 (1957). HARTMANN Elektroehem., ,~. 61,908 (1957). SUTTON, L. E., J . Inorg. Nud.Chem., 8,23 (1958). RuNCIMAN, W. A., Rep&. Progr. Phys., 21,30(1958). McCLunE, D. S., in "Solid State Physics," edited by SEITZ,F., AND TURNBULL, D., Academic Press, New York and London, Vol. 9, 1959, pp. 399525, GEORGE,P., A N D MCCLURE,D. S., in "Progress in Inorganic Chemistry," Vol. 1, edited by COTTON,F. A,, Interscience Publishers, Ino., New York, 1959, pp. 381-463. DUNITZ,J. D., AND ORGEL,L. E., i n "Advances in Inorganic Chemistry snd Radiochemistry," Vol. 2, edited by EMELELIS, H. J., and SHARPE, A. G., Academic Prcss, New York, 1960, pp. 1-60. BALLHAUSEN, C. J., in "Advances in the Chemistry of the Coordination Compounds," edited by KIRSCHNER, S., Mscmillan Co., New York, 1961, pp. 3-14. CARRINGTON, A., and LONGUET-HIGGINS, H. C., Qumt. Rev., 14, 427 (1960). Books "Ions of the Transition Elements," Disc. F a d a y Soe., No. 26, 1958. ORGEL,L. E., "An Introduction to Transition-Metal Chemistry: Ligand Field Theory," John Wiley & Sons, Inc., New York, 1960. Figure 8. Molesvlar orbitals for an eight coordinated cubic environs (Oh) about a central tramition metal atom. Mark thot the c-bond rtruchm is (to obtoin eight easily approrirnded by the valence bond hybrids equivalently directed valence orbitdr o a-bonding f-type orbital must b e included). GRIFFITH,J. S., "The Theory of Transition Metal Ions," Csmbridge University Press, London and New York, 1961. J@RGENSEN, C. K., "Ab~orptionSpeotra and Chemical Bonding in Complexes," Pergamon Press, Ltd., London and New York, 1961. Volume 39, Number 3, March 1962 / 139