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 wit- nessed 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 in- organic, valence bond ideas concerning the nature of excited molecnlar states made their exit in the years 1945-55. But with the accelerated interest in in- organic spectroscopy by inorganic chemists com- mencing 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

Andrew D. Liehr Bell Telephone Laboratories, Inc.

Murray Hill, NewJersey

--

Presented at the Symposium an Ligand Field Theory, 140th Meeting of the ACS, Chicago, September, 1961.

A comparison of theories

Molecular Orbital, Valence

Bond, and Ligand Field

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. In 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 dynam- ics 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 usu- ally introduce too much ionicity into the ground elec- tronic 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 sym- metry 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 sym- metry designations of the primary atom and ligand atom orbitals prior to compound formation; that is, for symmetrically disposed reactants at infinity. Then

. .. Figure 1. Molecular orbitals for a linear triotomic transition metal com- pound ( D d . 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 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 quan- tum 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 anti- bonding orbitals. will lie. For example, the cs+* antibonding orbital lies higher than the s.* antibonding orbital, as the primary atom c.+ orbital, nds2, is direc- ted 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.

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 angu- lar 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

PeIMAW ATOM MOLECULAR ORBlTAL5 LlCAND MOLECULAR ORBITALS OF THE COMPOUND ORBITALS

Figure 2. Molecular orbitdr for o square planar transition metal complex IDd Observe that the o-bond structure is well approximated by the volence bond hybridsr'p'ff, ond notr'p2d'orpzd2alone.

PtC14-2.2 The placement of the bonding and anti- bonding orbitals is again accomplished by qualitative principles and may be subject to reordering. The location of the a@* orbital, which is primarily nd,,

%The molecular 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 2r ln 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

136 / Journal of Chemical Education

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 ( 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

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).

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 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.

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 -~e l l e r theorem 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 LIEHR, A. D., Adu. Chem. Phys. 4, 000 (1961); BALLHAUSEN, C. J., Progr. Inorg. Chem., 2,251 (1960).

Figure 4. The nuclear dirplocements dlowed a rquors p l m m complex. Intensity enhancement is provided only by the odd (01 coordinoter, and John- Tellw forces only by the wen (gl81, ond Bag madinmter

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.

Figure 5. Molecular orbitals for a trigand bypyramid geometry (DIJ. See that the <-bond rtrvcture is nicely approximated by 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) VAN VLECK, J. H., J. Chem. Phys. 3 , 803 (1935). (3) VAN VLECK, J. H., AND A., SHERXAN, Rev. Mod. Phys. 7 ,

lfi7 f l D R 6 ) - -. \ - - - - , . (4) M c G ~ ~ v i w , B. R., J. Phys. Chem. 60, 71 (1956). (5) KIMBALL, G. E., J. Chem. Phys. 8 , 188 (1940). (6) E~SENSTEIN, J. C., J. C h . Phys. 25, 142 (1956). (7) COULSON, C. A,, "Valence," Clarendon Press, Oxford, 1952. (8) HOUGEN, J. T.. LEROI, G. E., AND JAMES, T . C., J. Chem.

Phys. 34, 1670 (1961).

+BONDS ".80NDI PLIR3

PrtlMAW ATOM MOLECULAR ORBITALS LICI\ND MOLECULAR 01181TAL5 OF THE COMPOUND 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 rhown in Figure 6(b) for the nd in + 1 ) s in + i l p type .tom. Perceive thot the .-bond rtrvctvre is neatly opproximated by the valence bond hy- brid. 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(*) to the unfilled antibonding ta" is, rl orbital produces the 'A?, 'E, ITlr and '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 con- sidered ithis borrowing is usually inperceptiblel. The stater ' E , ITx, and IT2 ore, of COUIS~, John-Teller active (1 1).

(9) HEXTER, R. M., private communication, February 1961. (10) FENSKE, R. F., MARTIN, D. S., AND RUEDENBERG, K., p"-

"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).

' L o N C <- . "... - . ON.. v.,.., . .

PRIMARY ATOM MOLECULIIQ OFlBlTALS LIGAND MOLECULAR OFtB1TALS OF THE COMPOUNO 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 + 1 lp type otom. Discern that the o-bond structure i s readily opproximated by the valence bond hy- brids rLpsd2 in agreement with the usual notion. An electronic excitation from the filled nonbinding h,(d tothe unfilled ontibonding f&) gives rise to the electronic states 'AI., LEm,%u, and of which only the 'TI. i s electronic- ally accessible from on 'A1. ground state. The 'E" and ITnu $totes ore vi- bronimliy dowed, but the 'A,. date is strictly forbidden [if first hormonic vibranicinteractionsare discounted). The'E,.'T~,,and 'Tz.stotesallow John- Teller antics il 1).

138 / Journal o f Chemicd Edvcufion

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,

Figure 8. Molesvlar orbitals for an eight coordinated cubic environs (Oh) about a central tramition metal atom. Mark thot the c-bond rtruchm is easily approrirnded by the valence bond hybrids (to obtoin eight equivalently directed valence orbitdr o a-bonding f-type orbital must be included).

S u a ~ ~ o , S., 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., 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., 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 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. 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.

GRIFFITH, J. S., "The Theory of Transition Metal Ions," Csm- bridge University Press, London and New York, 1961.

J@RGENSEN, C. K., "Ab~orption Speotra and Chemical Bonding in Complexes," Pergamon Press, Ltd., London and New York, 1961.

Volume 39, Number 3, March 1962 / 139