edited by: Eric S. Proskauer and Leonard W. Fine

I Quantum Mechanics and Chemical C. J. Ballhausen

University of Copenhagen Copenhagen, Denmark

I Bonding in Inorganic Complexes

I 11. Valency and inorganic metal complexes

The history of chemistry has taught us that whenever a theory cannot put numbers to findings, but shrouds itself in verbal mist, i t is time to abandon it.

Valency and Inorganic Metal Complexes

The Valence Bond Method

In his hook nuhlished in 1927 Sidewick summed UD the electronic interpretation of coordination as follows In the first dace it is clear that the links which ioin the units of a coordination comolex to the central atom are covalent. This is really self-evident if our'theorv is true. since Werner showed that thev were ~ ~~~ ~

nc.t i w n w c l , and a ctwlrnt h n k i r themly altrrnativr whwh *re have ndmittcd: I m r 11 13 ~ ~ t i t h l ~ ~ h e d ~t,nclu>i\.ely by I I W t ~ n i l l 1estsof PO- mctrtral and opt~<ni w,rnrrlim. Hcncr the maxlmum cwrrl~natlm number is the maximum covalency number: an atom with a cwrdi- nation number of six is one which can form six covalencies, and so have s valeney group of twelve shared electrons.

The working out of these ideas in a quantum mechanical language was done by Pauling (27). In his very important paper of 1931 he showed

that many more results of chemical significance can beobtained from the quantum mechanical equations, permitting the formulation of an extensive and powerful set of rules for the electron-pair bond supplementing those of Lewis.

We shall here concentrate on Pauling's work on the tran- sition metal coordination compounds (27, 28). The basic problem to solve was which metal orhitals should he used to form the Heitler-London covalent bonds with the ligands. Pauling showed that by using a single 3d eigenfunction, the 4.7, and two 4p eigenfunctions, four equivalent strong honds can be formed, and these lie in a plane, directed toward the corners of a square. Using a (3d)2(4s)'(4p)3 hybridization, six equivalent eigenfunctions can he formed directed toward the corners of a reeular octahedron. A (3d)3(4s)' hvhridization would lead to k u r equivalent strong honds directed toward the corners of a tetrahedron. These hvhrid orbitals could then he used in the formation of covalent k~ectron pair honds. The

electrons which formed the honds were supplied from lone pairs of electrons on the ligands.

The numbers and types of metal orbitals which had to he used in the construction of the sp2d (square planar) sp3d2 (octahedral) and sd3 (tetrahedral) hybrids appeared to be something of a mystery. Using group theory this was, however, cleared up by Van Vleck (29) in 1935. Concentrating on what we now call u-bonding between the metal atom and the ligands Van Vleck showed that for the

directional properties. . .to be correlated with one particular attached atom . . . the Pauling-Slater central functions must have the same transformation properties as do those wave-functions of the attached atoms before line& combinations of the latter are taken.

An exam~le (29) mav clarifv this. Usine the modern lan- . . . . "

guage of group theory, the six ligand u bonds in an octahe- drallv coordinated metal com~lex sDan the irreducible ren- resetkations al,, e, and t',. he metal orbitals span (s): a&, (P,, P,, P,): t ~ , , (daz2-+ dr2-y2): ep, and (d,,, d,,, d,,): tz,. By comparison we see that the dtz, orhitals are entirely u non-bondina. In this way, the metal d-orbitals which form octahedral hybrids are seen to be dX9-,1 and d3+4.

A quantum mechanical calculation of the 12-electron va- lencebond wave function which can he written down for an octahedral complex was a t that time an impossible under- taking. Pauling's bonding scheme remained therefore a pos- tulate, unsupported by numerical calculations. However, modern molecular orbital calculations have shown that the picture is essentially correct. The greatest element of weakness in Pauline's hondina theorv was ~ o i n t e d out hv Van Vleck and ~herman.(26), namely that ligand electrons to a large extent had to he absorbed by the central metal ion. For instance, for an octahedral complex six ligand electrons. had to he trans- ferred to the central atom. Many years later Pauling took account of this critique and formulated (30) the

About the Editors. . . Erie S. Proskauer received his doctoral degree in physical

chemistry from the University of Leipzig. His early uzork as a writer, editor, and student of theliteratureof chemistry led him into publishing and a life-long relationship with the creation of scientific'lileratore that has already spanned 50 years. He was co-fuunder of Interscience Publishers and, eventually, director and senior vice president af the Wiley Interscience Division of .John Wiley and Sons. In semi-retirement now, his contributions to the literature of chemistry continue.

Leonard W. Fine received his doctoral training in organic chemistry at the University of Maryland. He is currently Pmfessor of ('herniatry at Housatcmic College (State of Cooneeticut) and lecturer i n the General Studies Proerarn at Columbia Universitv.

About the Author. . . Carl J. Hallhausen. Pn,tessur 01' ('hemistry in the Institute f o r I'hysicsl ('hemistry at the Ilniversity nI'(hpenhsgen, has earned a n international repu- Iaticm thrcurh his scientific researrh, irhcdarly writing, and concerns I'or chemical ed- ucation.

This is the second part of Carl Ballhausen's contribution to our understandine of the

His three textbrr,ks and active interest in the work of the Division ~I'Chemical Education suggest a growing amcern among younger chemists fiw a~mmunicating chemistry, education technology, and matters that traditionally helmged to the history of science.

quantum mechanical ideas as applied to inorganic ampiexes." The first part appeared in the preceding (April) issue of this Journal; the third part follows next. month

294 / Journal of Chemical Education

ion isdetermined entirely by thenumber of unpaired electrons, being equal to N, = 2 m .

Hans Bethe reminiscing. Courtesy of Goudsmit collection.

postulate of the essentiil electrical neutrality of atoms: namely, that the electronic structure of substances is such as to cause each atom to have essentially zero resultant electric charge.

In an octahedral complex [M3+Ls] this could be achieved with the use of onlv three metal orhitals, resonatinn among the six positions. ~ a c h bond therefore would he 5036 ionic and 50% covalent. A true valence bond wave function is of course a linear combination of functions representing the different possible phases through which the system resonates. Yet again, it must be said that the numerical testing of Pauling's ideas was out of the question.

Assumine a saturated valence hond scheme the next oues- tion nhich ;ad tr, he answered was where macmmmodatk the s~lrplu-. metal electrons. In his 1931 paper I'auling makes a distinction 11t.tween "nn c.lt,ctn)n-pair l~cmd structure" and an "ionic structure." In an rlrctnn-pair hond scheme fur an oc- tahedral metal complex three 3d orbitals (the tz, orhitals) are emptv, and thus can be used to house the metal electrons. ~ o k e v e r , when more than six metal electrons were present more orbitals were needed. Either the electrons had to occupy thetwo 3d (em) orhitals. which therefore could not he used f i r bonding p"rposes, or they had to be accommodated in the hieh-lvine 4d orbitals. The last situation was thought to lead " . - to very unstable complexes. Consequently the problem of where to house the "surplus" electrons could he met only by assuming an "ionic structure" for the complex. As to the actual type of honding utilized in the complexes Pauling suggested that in many cases this could he inferred from a knowledge of the magnetic moments of the complexes.

Let L and S he the quantum numbers corresponding to the total anplar orbital momentum and the total spin momentum of the electrons. Provided the interaction between L and S is small, so that the multiplet separation is small compared with kT, the magnetic moment of an atom is given by Van Vleck's formula (39)

Pauling now reaches the conclusion that

the perturbing effect of the atoms or molecules surrounding a mag- netic atom destroys the contribution of the total momentum to the magnetic moment,. . . the magnetic moment of a molecule or complex

Using this criterion he observed, for instance, that the complex Fe(Hz0)P with six 3d electrons had a magnetic moment of 5.25, corresponding roughly to S = 2. Fe(CN)Q-on the other hand had S = 0. Hence the honding in Fe(H20)%+ was "ionic" but in Fe(CN)Q-it was "covalent". Boldly, Pauling further used measurements of magnetic susceptibilities to infer molecular structures. The Ni(CN)i- complex is dia- magnetic, and provided we use one 3d orbital to form four square planar covalent bonds, the eight metal electrons can onlv be housed in the remaining four 3d orbitals. all beine pai;ed up in the spins. From thiHpauling predicted that the Ni(CN)!- com~lex should be olanar. No X-rav data on this . . . ion were available when pauling made his but the fact was soon established.

With very little effort and with virtually no knowledge of theoh it was therefore possible for inorganic chemists to make magnetic measurements and to classify complexes as "ionic" or "covalent." Many molecular structures were further pre- dicted on the basis of susceptibility measurements. Most of this work has been shown by time to he unreliable. As we shall see next Van Vleck soon proved that the "magnetic criterion" was not of much value, and Pauline also realized this later (30). In 1948 he said

We conclude accordingly that the magnetic criterion distinguishes, not between essentially covalent bonds and essentially ionic bonds, but between strong covalent bonds, using good hybrid bond orbitals and with the possibilities ofunsynchronized ionic-covalent resonance, and weak covalent bonds. usine Door bond orbitals. and with the ne- ,. . rrsslry h r iym hnmmtmn of r h s rowlent phase* cat the hcnndq. I n n ccmplex ufihr f l r i r art the stnhlhty otthr wmpler ,-. due in Inrger part t o the t ~ m l thtmirlvei and in smaller parr t u thr nrmnlr elrr- trons, and in a complex of the second sort the situation is reversed.

In his 1948 statement Pauling tried to conceal by a cir- cumlocution that his original simple theory was unable to deal with the accumulated experimental evidence. The history of chemistry has taught us that whenever a theory cannot put numbers to its findings, but shrouds itself in a verbal mist, it is time to abandon it. Let us. however. also reflect for a minute that in 1931 Pauling gave a hasicall; correct explanation of the bonding in organic transition metal comolexes and also - gave an accurate description of the electronic configurations of the mound states in those complexes where one only needs to inciude the 3d (t2#) atomic orbitals. Had Pauling aiso as a matter of course utilized the two 4d (e,) orhitals as recipients for the metal electrons, and used theenergy separation be- tween the 3d (tw) and 4d (ex) orbitals as a parameter, the history of transition metal chemistry would undoubtedly have been advanced many years. Evidently Pauling had not read the two-year-old paper by Bethe (16); had he done so, the idea should have been obvious. I t would have provided the natural explanation of the variations in the magnetic moments of the complexions without any ad hoc explanations. However, notwithstanding all "bonding theories" the time was evidentlv not ripe to realrze that the complexes are just not thinly dis- guised metal ions, hut true molecular entities.

The Crystal Field and Magnetic Susceptibilities

Using "the new buantum mechanics" the formulas for the magnetic susceptibility of a molecule had been worked out by Van Vleck (31) in 1928. A new result was obtained from these, namely that molecules without a spin may still he slightly paramagnetic. In addition to the usual diamagnetic term proportional to - 2; ( r : ) , there is also for a molecule a posi- tive, so-called, high frequency term. I t is also called the Van Vleck term after its discoverer. The reason for its oresence is that for a molecule ( n I LIn) may be zero hut (nlL>ln) never is. The sum of the diamagnetic term and the high frequency term is invariant of which point we take as the origin of ri.

The cases of most interest are, however, those where the

Volume 56, Number 5, May 1979 / 295

The work of Bethe had advanced the model based on the idea that in a complex the electrons associated with the central atom experience an electric field coming from the surrounding ligands.

molecules possess spin magnetism. The work of Bethe (16) had advanced the model based on the idea that in a complex the t.lectrtmn a h t ~ r i n t t ~ l wt11 the central alum experience an electric field coming irom [he surroundinr l i~and i . The sear 1932 saw Van ~ ~ e c k and his coworkers busil; engaged in in- vestigating how the "crystal field" of the ligands would in- fluence the paramagnet& susceptibility of the central tran- sition metal ion.

Van Vleck wrote (32)

I was m a ! rnm hhdiww iur m<.;t of 19:kI helawe I war in E I I ~ ~ I P un n (;ug~enh<im F v I I ~ I u ~ I ~ . . . . New the bt~inniti~: 0 1 m y frllornh~p !I h ; 4 1 a wall. n i l h Kmmer; 310#1!:s<)n>ei>itI1t d t l n ~ s in Ilolland. He told me of Bethe's wonderful group theory paper on the energy levels of magnetic ions in crystals, and also of his own work on the double degeneracy of odd ions. I feel I learned more in this one walk than in the whole rest of my fellowship. One can never tell when a turning point will arise in one's career in research.. . . When I returned to Wisconsin.. . I was kept busy finishing up my book on Electric and Magnetic Susceptibilities, as well as learning group theory so I would have a thorough understanding of Betheh paper. However, in the late summer of 1931, two physicists from Great Britain came to Madison on post-doctoral fellowships, William (now Lord) Penney, and Robert Schlaoo. and Isueeested thatthev work resoectivelvon the effect of .., "- a crystalline field on the paramagnetic suseeptib~lities of rare earth compounds and of salts containing certain ions of the iron group.

Let us consider the second paper by Schlapp and Penney (33). "Influence of Crvstalline Fields on the Susceptibilities of salts of paramagnetic Ions. . . ,Especially Ni, C ; and Co." As their starting point they took the well known spectroscopic term for the ground states of Ni" +("F), Cr" (("F), and CoZ+ (,'F). The electric field of the crystal is then able to break down the relatively weak coupling between orbit and spin, and the soin-orbit coupling may he treated as a perturbation on an unperturbed prohiem which neglects the spin. The unper- turbed problem is then the same for all the above three ions, since thky all have an F state as ground state. Notice that the troublesome llr12 terms, which in all other molecular prob- lems are a great bottle-neck, in this model are already incor- porated in the zero-order functions.

The crystalline potential in salts of the iron group is dom- inantly cubic, corresponding to a regular octahedral six-fold coordination of the metal ions. For d-electrons the most

H. A. Krarners enjoying a discussion. Courtesy of Goudsmit collection.

296 I Journal of Chemical Education

general crystalline potential with cubic symmetry was shown by Van Vleck (34) to be

For Ni2+ and Cr" such a cubic field will split the F state leaving a one-fold orbital degenerate state of A7a svmmetrv ." da the ground state prwidt-d IJ is pmitiw, hut in octahedral ( ' (I" wmr)lext:s ~t will l ~ e a three-idd orhirnl deaenerate 7'1,

state. NO; a Tlg state is also susceptible to spin-orbit co": pling. I t can carry orhital momentum and it can split into further components in a lower than cubic crystal field. None of these effects are operative in a A% state. The first triumph of crvstal field theorv was therefore that it exolained whv nickel salts are nearl; isotropic magnetically, w'hile those df Co2+ exhibit large anisotrooies even though the Ni2+ and Co2+ ions are both i n k states and are adjacentin the periodic table. As stated by Van Vleck (320

The article 1341 in which I published this result is my favorite of the various papers I've written as it involved only rather simple caleula- tions, and yet it gave consistency and rationality to the apparently irregular variations in magnetic hehaviour from ion to ion. . . . In fact I still have some reprints left of my favorite 1932 paper. In the post- war era its results are considered well known, and the calculation straightforward-hence no need of a reprint.

Provided the orhital contrihution to the magnetic moment is neglected, the magnet on number would he the spin-only value 2v'S(S + 1) .A one-fold orbital degenerate state cannot, in first order, carry any orhital momentum. However, the spin-orbit coupling may give rise to an orbital contribution to the magnetic moment of a one-fold degenerate state.

The spin magnetic moment of an electron is measured in Bohr magnetons, P. The ratio of the magnetic moment to the angular momentum is two times as great for the spin as for the orbital motion. The total angular momentum J of an atom is a constant; it is made up of orhital, L, and spin, S, contribu- tions from the electrons. Therefore J = L + S. The magnetic moment, P(L + 251, will not in general he a constant, but will in each state have an average value. This will be equal to its

J. H. Van Vleck on the steps of the National Academy of Sciences. Courtesy of the Goudsmit collection.

oroiection on the constant total angular momentum. The ratio, g, of the projection of the magnetic moment on the an- gular momentum to the angular momentum is thus

F r o m J - S = L d o w e g e t 2 J - S = J2+S2-L2.WithJ2,L2 and S2 being diagonal with values J ( J + I) , L(L + 1)and S(S + 1) we have for the Land6 g factor

With L = 0, we have g = 2, the spin-only value. Let the crystal field perturbation Hamiltonian be

Here A is the spin-orhit coupling conslnnt. X is positive in the firit half oian atomic shell, negative in thr lnst half. The or- l h l contrihutim to a onr-fold deeenerntc state will therefore - by second order perturbation theory be given as the expec- tation value (L) = AID. Identifying L with (L), wesee how Schlapp and Penney could explain why the deviations of the effective Land6 g factor should be positive for the (3dP system of Ni" and negative for the (3d)3 system of Cr3+. Fitting their formulas for the magnetic susceptibilities to the available experimental data they found 10 Dq = 12600 cm-' for Ni(H20)it and 10 Dq = 37,300 cm-' for Cr (HzO);'. The "modern" spectroscopic values are 8500 em-' and 17,400 cm-', respectively. Particularly the Cr3+ value is therefore off by more than a factor of two.

By doing an electrostatic calculation Gorter (35) had shown that provided D is positive for an octahedral arrangement of six ligands, it will he negative for a tetrahedral four coordi- nation. This had importance for the work of Jordahl(36) on the magnetic susceptibility of CuS0c5H20. In 1933 it was eenerallv suooosed that the Cu2+ ion was four-coordinated. ?'he magnetIEsusceptibility of CuS04.5HzO could, however, only be explained by Jordahl on the assumption of a positive D. The agreement between the calculated and observed mean susceotibilitv was therefore taken as an indication of an oc- tahedral arrangement around the CuZ+ ion. The X-ray proof that this was indeed so was given very shortly after (37).

For an orhital degenerate ground state one should not a uriori expect a "spin-only" formula for the magnetic sus- kt.ihili<v to he valid. As the measurements of magnetic --r

susceptibkities as a function of temperature accumul&ed i t became, however, apparent that the "spin-only" formula had a much wider applicability than expected. In cases where a cuhic crystal field would lead to orbital degeneracies it was, therefore, necessary to include crystal fields of lower sym- metrv than cuhic in order to do away with the orbital degen- eracies.

A hexacoordinated magnetic ion of course need not be lo- cated in th,: crystal at a pnint ha\,ing cuhic symmetry. Such a situation would naturally introduce lower than cuhic ele- mrnts in the field. Howe\w. as first oointedout hv Van Vlrrk - - - ~ ~ ~ ~ ~

(38) in 1939, the Jahn-~e l l e r effect will always Bee that the arraneement of the lieands is such that all orhital deaeneracy is abokhed.

- .

The Jahn-Teller effect! How many times since its discovery in 1937 has this theorem (39) (whichis not an effect a t all) n i t been invoked to explain trouhlesome molecular electronic manifestations. As t o its discovery Edward Teller records (40):

In the year 1934 both Landau and I were in the Institute of Niels Bohr at Copenhagen. We had many discussions. I told Landau of the work of one of my students, R. Renner, on degenerate electronic states in the linear Con molecule . . . (Landau) said that I should be very careful. In a degenerate electronic state the symmetry on which the degeneracy is based will in general he destroyed. I managed to con-

vince Landau that his doubts were unfounded (for a linear molecule) . . . . Avear later in London I asked myself the question whether an- other ekeeption to Landau's postulated statement might exist. . . . The question did not appear simple. I proceeded to discuss the problem with H. A. Jahn who, as I, was a refugee from a German university. We went through all possible symmetries and found that linear molecules constitute the only exception. In all other eases Landau's suspicion was verified.

Jahn (41) extended the theorem to cover the cases where there is both soin and orhital deeeneracv. A non-linear sym- ~~~~

metrical position of a polyatomk moleche cannot be stable if there is other soin deaeneracv than a Kramers spin doublet. The spin-orbit Eoupliig can consequently stabilize a non- linear confieuration which would he unstable if orbital de- generacy alone were present, provided the coupling is large and leaves as its lowest state a non-degenerate "double group" state or a Kramers doublet.

Jahn and Teller (39) showed that orbital electronic de- generacy and stability of the nuclear configuration are in- compatible unless all the atoms of the molecule lie on a straieht line. Their oroof was based on croup theory and i t .. .. . does not give an" recipe as to how to ralrulatr the magnitude of the instahilitv. The sirnolest molrculnr s\stem which should exhibit a Jahn'~e1ler configurational instability is the equi- lateral triangle configuration of Ha. For this three-electron system it was already possible in 1938 for Hirschfelder (42) to calculate the angle dependence of the energy by the varia- tional method. I t was found that the (unstable) Ha system had its lowest energy for a linear configuration.

The Jahn-Teller instability of the big octahedral complexes had, however, to be attacked by perturbation methods. The necessarvmathematical theorv was worked out bv Van Vleck (43) anda calculation made ofthe magnitude of the effect. As his oerturhation Hamiltonian Van Vleck took an expansion of (he crystal field potential in the octahedral vibrational symmetry coordinates eg and ~ 2 ~ . TO this he added a perma- nent trigonal field

12(m) # ( . 123Jav) 1 = - c 1 + - 'Q , at, ' 0 , a r k 0

For an E state, only the first term is active; the locus of mini- mum energy is a circle in the @, $'plane. This is the, by now well known, "Mexican hat" potential surface. For TI and Tz states, it was found that with the first term in %('I being dominant we would have a tetragonal distortion: with a dominims of the lasf two terms tht,distortion would hr nl~m:! a hod\.dingnnnl of theocwhedron. For instsncr, forTi(H10I.:' . - with a ground state of 2T2 the tetragonal splitting of Tg was calculated to he 315 em-', the trigonal splitting to be 550 cm-'.

The treatment Van Vleck gave the Jahn-Teller configura- tional instahilitv oroblem of octahedral comolexes is bv now the standard procedure. In 1939 it must have been a tour de force. Its significance was not understood until long after the war.

Literature Cited

I261 Jordahl.0 M.,Phys. Rsu., 45.87 (1934). I371 Beevers, C. A,, and Lipron, L, Pmc. Ruy. Soc. (London1 A146 570 (19341. (381 VanV1ock.J. H., J. Cham. Phyr, 7.61 119391. I391 Jshn, H. A.,sndTeller.E..Pmc. Ray. Soc lLondonI.A161.220 (19371. (40) Englrnsn. R., "TheJahn~Teller Effect in Moleculesand Crystals.An HinWrical Note."

Wilev-lnterscienee. New York. 1972.

+ + +

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