Molecular Orbital, Valence Bond, and Ligand Field

Home , Crystal Field Theory, Ligand field theory

A comparison of theories Andrew D. Liehr Bell Telephone Laboratories, Inc. Molecular Orbital, Valence Murray Hill, NewJersey Bond, and Ligand Field

Before jumping into the intricate details magnetic criterion. [A most recent example of such of the various theories of valency, let us first pause a a blow is the demonstration by McGarvey (4) that hit and try to obtain some historical insight into their the odd electron in square planar CU+=complexes is course of development, their successes, and their in a 3d-like orbital and not a 4p-like orbital.] At the failures. Only in this manner shall we be able to present time the valence bond theory of coordination assess with any assurance the future progress of the compounds is being systematically supplanted by a theory of the chemical bond. molecular orbital-crystal field amalgamation, which Soon after the announcement of the Schrodinger has been aptly dubbed ligand field theory. [Crystal equation for electronic motions, there were proposed field theory was originally an ionic theory of chemical and utilized three approximate means of formulating bonding until modified by Van Vleck. In its modified solutions of this equation as it applied to molecular form it was a highly simplified molecular orbital theory problems: (1) the valence bond technique of Heitler, of the nd, (n + l)s, (n + l)p, and nf electrons in which London, Slater, and Pauling; (2) the molecular orbital orbital energies were of the molecular orbital type, but technique of Hund, Bloch, Mulliken, Lennard-Jones, electron-electron electrostatic repulsion energies and and Hiickel; and (3) the crystal field technique of spin-orbit energies were of the atomic type. Ligand Bethe, Kramers, and Van Vleck. Each of these field theory is a molecular orbital theory of the nd, techniques had its limitations, its strong points, and (n + l)s, (n + l)p, and nf electrons in which both the its weak points. And the years 1930-45 saw a struggle individual orbital energies and the electron-electron among the three for pre-eminence in the minds of repulsion and spin-orbit energies- are of a molecular chemists and physicists, even though Van Vleck had type. I shown in 1935 that they were absolutely equivalent Why has this particular pattern of historical dynam- when carried to completion (2, 3). These years wit- ics evolved? The answer to this question is simple: nessed the adoption of the valence bond and molecular the valence bond theory, although it is by far and away orbital methods by the organic chemist, the valence the superior outlook for ground electronic states, bond method by the inorganic chemist, the molecular becomes hopelessly complex as a description of excited orbital method by the molecular and solid-state electronic states. The jungle of ionic valence bond chemist and physicist, and the crystal field method by struebures nverruns all attempts to describe electronic the magneto-physicist. Each of these adoptions had excitations by this technique. Molecular orbital and the same driving reason: most chemists and physicists crystal field theory on the other hand suffer from the of this era were primarily interested in the ground complementary deficiency: they provide adequate electronic states of chemical systems. pictures of the excited electronic states, hut not of With the end of the war, scientific interest began to the ground electronic state of most molecules (they usu- swing toward a concern over the excited electronic ally introduce too much ionicity into the ground elec- states of molecules. This precipitated a rapid fall tronic state). Therefore as long as ground electronic from favor of the valence bond theory, and a consequent state properties were the vogue, valence bond theory rise in esteem of the molecular orbital and crystal field shone, but when the properties of electronically excited theories. Thus in all fields of chemistry except in- states became the style its gleam was dulled. organic, valence bond ideas concerning the nature of With this historical perspective behind us, let us excited molecnlar states made their exit in the years now see how to extend the ligand field technique to 1945-55. But with the accelerated interest in in- encompass all inorganic compounds. In two frequently organic spectroscopy by inorganic chemists com- overlooked papers by Kimball (5) in 1940 and by mencing around 1955, even this last stronghold of Eisenstein (6) in 1956, there are tabulated the sym- valence bond concepts has begun to fall. Indeed, in metry classifications of the primary atom and ligand the particular area of inorganic chemistry called atom orbitals for most geometries of interest. With coordination chemistry, valence bond theory has these classifications written down once and for all, suffered its most grevious blows; even its descriptions it is a simple matter to construct a molecular orbital of the ground electronic charge distributions have energy level diagram, and with this diagram to predict proved false in many important cases, especially with the number and classification of the excited electronic regard to the much vaunted, but completely specious states. We shall demonstrate this fact explicitly for a few exemplary systems. -- Presented at the Symposium an Ligand Field Theory, 140th In Figure 1 we show the orbitals which are primarily Meeting of the ACS, Chicago, September, 1961. 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, of the ligand orbitals and the a.+* and T.* antihonding from Kimball's and Eisenstein's tables, the correct sym- orbitals are composed mostly of the primary atom metry designations of the primary atom and ligand nd+ and nd,,, orbitals, respectively. Hence, for atom orbitals prior to compound formation; that is, CuCI2 we expect the ground electronic state to be a for symmetrically disposed reactants at infinity. Then 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 actu- ally 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 (Dd. Note that the <-bond strudure is closely approximated by 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 ap- proach one another in a symmetrical fashion to pro- duce the ha1 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 quan- PeIMAW ATOM MOLECULAR ORBlTAL5 LlCAND MOLECULAR tum numbers). We can, of course, obtain only so ORBITALS OF THE COMPOUND ORBITALS many complete bonding and antibonding molecular Figure 2. Molecular orbitdr for o square planar transition metal complex orbitals of a given symmetry type as there are primary IDd Observe that the o-bond structure is well approximated by the atom and ligand atom orbitals of this same symmetry volence bond hybridsr'p'ff, ond notr'p2d'orpzd2alone. classification initially present. The ordering of the resultant molecular orbitals of the product molecule PtC14-2.2 The placement of the bonding and anti- is completely based on qualitative concepts: the more bonding orbitals is again accomplished by qualitative the primary atom and ligand orbitals are directed principles and may be subject to reordering. The toward one another, the deeper the consequent bonding location of the a@* orbital, which is primarily nd,, orbitals will lie and the higher the consequent antibonding orbitals. will lie. For example, the cs+* %Themolecular symbols corresponding to the atomic s, p, d, f, antibonding orbital lies higher than the s.* antibonding etc., designations for non-linear compounds are a, b, e, and 1 orbital, as the primary atom c.+ orbital, nds2,is direc- (an icosahedral molecule has the additional symbols y. and h, ted more strongly toward the ligand cg+ orbital than which are not to be confused with tho analogous atomic terms). The molecular symbol a corresponds to the atomic s (and the the primary atom s, orbital, nd,,,,,, is toward the linear molecule designation a)-it means that the molecular wave ligand T. orbital (the molecular axis is the z axis). function does not change sign under a rotation of 2rln about the Moreover, as the ligands have a greater affinity for molecular n-fold rotational axis of symmetry (e.g., the four-fold z their electrons than does the primary atom, the c.+ axis of PtCL-2): the symbol b means that it does. (In a very s, direct sense this is equivalent to saying that the molecular a type and bonding molecular orbitals are composed mostly orbitals have a component of angular momentum along the n-fold rotational axis of symmetry (the e axis) whose magnitude is a 'For the linear molecule the Greek letter symbols (G, r, 6, p multiple of n (e.g., 0 or n), and the b type a companmt which is etc., replace the stomic designations a, p, d, j, etc. Just as these an odd multiple of n12.) The symhals e and 1 mean that the Latter symbols denoted orbital angular moments of 0, 1, 2, 3, ete., molecular wave function is degenerate (just as the atomic states respectively, in the atom, the former now indicate the magnitude p, d, f, eto., am degenerate), with two-fold and three-fold do of the component of orbital angular moment 0, 1, 2, 3, etc., along generacy, respectively (the icosahedral g and h symbols indicate the internuclear axis (thez axis) of the moleoule. four-fold and five-fold degeneracy). Such degenerate sets of

136 / Journal of Chemical Education in character, is especially vague as it 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 (lo)]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 vi- bronic (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 plmm complex. Intensity enhancement is provided only by the odd (01 coordinoter, and John- Tellw forces only by the wen (gl81, ond Bag madinmter I Figure 3. The JohmTeller energy By now the procedures utilized to construct molecular surface characterirtic of a square orbitil energy level diagrams are apparent, and so my planar system (i 11. Nuclear motions on this surface ore essentially one purpose in presenting this lecture has been accomplished. dimensional. 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 displacements which cause these Jahn-Teller motions stimulate a fervent interest in the science of inorganic and intensity enhancement are given in Figure 4.3 spectroscopy. I eagerly look forward to the day when Figures 5-8 depict the molecular energy level dia the idem of Mnlliken (1) and Van Vleck (2, 3) which grams (in the absence of spin orbit coupling) for the were here outlined will be universally accepted. 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 resultant 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). 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, "per- mutational momentum") about some dold 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.

J A Jahn-Teller molecule is one which exists in a degenerate (or nearly degenerate) electronic state. Such a molecule, acoording to the theorem of Jahu and Teller (PTOC.Roy. SOC.,161A, 220 (1937)), may experience eoulombic forces which tend to destroy this degeneracy by distortion of the nuclear framework. The

~ahn-~ellertheorem 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 Proy. Inonorg. Chem., 4, 000 (1962). In addition, see BALLHAUSEN,C. J., Adv. Chem. Phys., 4, 000 (1961). A similarly graphic-type discussion of the closely re- lated intensity problem has also been given elsewhere. See Figure 5. Molecular orbitals for a trigand bypyramid geometry (DIJ. LIEHR,A. D., Adu. Chem. Phys. 4, 000 (1961); BALLHAUSEN, See that the <-bond rtrvcture is nicely approximated by the valence bond C. J., Progr. Inorg. Chem., 2,251 (1960). hybridsr'padJ, and not r1p3d'or r'pWalone.

Volume 39, Number 3, March 1962 / 137 Literature Cited (9) HEXTER,R. M., private communication, February 1961. (10) FENSKE,R. F., MARTIN,D. S., AND RUEDENBERG,K., p"- (1) MULLIKEN,R. S., Phys. Rar. 40, 55 (1932). "ate eammuni<:i~tion,May 1961, also 140th Meeting of (2) VANVLECK, J. H., J. Chem. Phys. 3, 803 (1935). ACS, September, 1960. (3) VAN VLECK,J. H., AND A., SHERXAN,Rev. Mod. Phys. 7, (11) LIEHR,A. D., XVIIIth International Congress of Pure and lfi7 flDR6) - -. \ - - -- ,. Applied Chemistry, Montreal, Canada, August 6-12, (4) McG~~viw,B. R., J. Phys. Chem. 60, 71 (1956). 1961. 'This reference pertains to the general theory of (5) KIMBALL,G. E., J. Chem. Phys. 8, 188 (1940). Jahn-Teller and non-Jahn-Teller energy surfaces. (6) E~SENSTEIN,J. C., J. Ch.Phys. 25, 142 (1956). (7) COULSON,C. A,, "Valence," Clarendon Press, Oxford, 1952. Recent Qualitative Discurrions (8) HOUGEN,J. T.. LEROI,G. E., AND JAMES,T. C., J. Chem. Phys. 34, 1670 (1961). 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).

+BONDS ".80NDI PLIR3

PrtlMAW ATOM MOLECULAR ORBITALS LICI\ND MOLECULAR 01181TAL5 OF THE COMPOUND ORBiTAL5 Figure 6(0)

' LoNC Figure 61bl. Molecular orbitols far a tetrahedral arrangement ITd) <-. "... -.ON.. v.,.., about a cectrml nr np nd type atom ore shown in Figure 6i.I. The same ir .. rhown in Figure 6(b) for the nd in + 1)s in + ilp type .tom. Perceive PRIMARY ATOM MOLECULIIQ OFlBlTALS LIGAND MOLECULAR thot the .-bond rtrvctvre is neatly opproximated by the valence bond hy- OFtB1TALS OF THE COMPOUNO OeBITALJ brid. r'#da, and not +pa or r'dJ alone. The electronic excitations from the Figure 7[b). Molecvlor orbitals for an oetohedrol disposition (Oh) filled nonbonding h irl orbital to the unfilled ontibonding e*(r) arbitd giver about 0 central nr np nd type otom ore rhown in Figure 7(4. The some rise to the low lying single excited stater ITt and LTs, while that from fl(*) is rhown in Figure 7ibl for the nd (n + 1 lr in + 1 lp type otom. Discern to the unfilled antibonding ta" is, rl orbital produces the 'A?, 'E, ITlr and that the o-bond structure is readily opproximated by the valence bond hy- 'T* states. As the ground electronis state is 'A1 and the transition dipole brids rLpsd2in agreement with the usual notion. An electronic excitation restor, el: transforms ashonly the electronic jumpslA1 -IT2 ore allowed. from the filled nonbinding h,(d tothe unfilled ontibonding f&) gives rise to TheLAl -'E,'Tr hops are made vibronically ollowed via intensity theft from the electronic states 'AI., LEm,%u, and of which only the 'TI. is electronic- the aIlowedlT~ state by the and n nuclear modes. ThelAz state is strictly ally accessible from on 'A1. ground state. The 'E" and ITnu $totes ore vi- forbidden unless intensity borrowing by the nuclear first harmonics ore con- bronimliy dowed, but the 'A,. date is strictly forbidden [if first hormonic sidered ithis borrowing is usually inperceptiblel. The stater 'E, ITx, and vibranicinteractionsare discounted). The'E,.'T~,,and 'Tz.stotesallow John- IT2 ore, of COUIS~, John-Teller active (1 1). Teller antics il 1).

138 / Journal of Chemicd Edvcufion SUTTON,L. E., J. CHEM.EDUC., 37,498 (1960). Sua~~o,S., J. Appl. Phys. Suppl., in press (1962). LIEHR,A. D., Bell Syrt. l'ech. J., 39,1617 (1960). PEARSON,R. G., Chem. Eng. News, 37, 72 (June 29, 1959); J. Recent Reviews CHEM.EDUC., 38,164 (1961). MANCH,W., AND FERNELIUS,W. C., J. CHEM.EDUC., 38, 192 NYHOLM,R. S., ORGEL,L. E., AND J@ROENSEN,C. K., in Reports (1961). of the 10th Solvay Conference, Bruxelles, May, 1956. LEWIS,J., AND NYHOLM,R. S., Chem. Eng. News, 39, 102 (Dec. 4, MOFFITT,W. E., AND BALLHAUSEN,C. J., in "Annual Reviews of Physical Chemistry," Val. 7, Annual Reviews, Inc., Pdo 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., AND 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., in "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 Co- ordination 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. Faday Soe., No. 26, 1958. ORGEL,L. E., "An Introduction to Transition-Metal Chemistry: Ligand Field Theory," John Wiley & Sons, Inc., New York, 1960. GRIFFITH,J. S., "The Theory of Transition Metal Ions," Csm- bridge University Press, London and New York, 1961. J@RGENSEN,C. K., "Ab~orptionSpeotra and Chemical Bonding Figure 8. Molesvlar orbitals for an eight coordinated cubic environs (Oh) in Complexes," Pergamon Press, Ltd., London and New York, about a central tramition metal atom. Mark thot the c-bond rtruchm is easily approrirnded by the valence bond hybrids (to obtoin eight 1961. equivalently directed valence orbitdr o a-bonding f-type orbital must be included).

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