edited by Leonard W. Fine and Eric S. Prwkauer
C. J. Ballhausen University of Copenhagen Copenhagen, Denmark
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Quantum Mechanics and Chemical Bonding in Inorganic Complexes Ill. The spread of the ideas
Valency and Inorganic Complexes (continued) The Molecular Orbital Method
In 1935,I wrote a paper (44) outlining how crystalline potential theorv could in nrincinle. . . usine.. the Mulliken tme .. annroach. .. .be een.. r r n l m 4 m u ~ 1 1 3 is 1 nt,urallrd l~gandlirld theory i n ahirn eleitnms fron, thr pararl>agneticrati 4. In the high spin case we lose orhital energy, but gain "exchange energy" because the electrons can have parallel spins. In the case of the low spin we gain orhital
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Here the a wave function of the attached ligand i is denoted by # j , with ligands 1,4 located on the x axis; 2,5 on the y ; and 3,6 on the z. The values of a, P, and y are determined by solving a secular equation. The lower roots of the three quaThis is the third and concluding part of Carl Bellhausen's contribution to our understanding of the "influence and development of quantum mechanical ideas as applied to inorganic complexes." The first part, which dealt with the static concepts of bonding and the dynamic concepts of valency, appeared in the April issue; the second part which initiated the discussion of valency and inorganicmetal complexes completed in this Dart. aoneared in the Mav issue.
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energy, but lose "exchange energy" because the electrons are forced to pair-up. It is therefore a balancing of the different energies which determines the number of unpaired spins and not whether we have "covalent" or "ionic" bonding. Indeed. we can pass from pure "ionic" bonding corresponding to the crystal field model to full covalent bonding by letting a,8, and y move from 0 to We can therefore conclude that the calculations hased on the crystal field model of magnetic susceptibilities (44)
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retain practically as much significance with the molecular orbital method as with the crystal field model, except that the size of the crvstalline ~otentialis not to be taken tooliterallv. The s~littinesof a few volts hay now relate to the magnitude of t6e ~undresonance inteeralsrather than of the crvstalline ootential. but remain comoathe (high spin). Apart from admiring the beauty of the paper one other thine strikes a modern reader with amazement. Van Vleck's paper was received by the J o u ~ n aof l Chemical Physics on October 7, 1935. It was published in the Decemher issue, 1935.
Leslie Orgel
The Excited States
All considered the ereatest service of the crvstal field thenrv derives from its ahili6 to handle the excited states of the (3d)" complexes. This is a unique featwe of the theory-neither the valence bond method nor the molecular orbital theory can hope to do as well within an order of magnitude. The reasons are, as we now know, that for many octahedral complexes of the first transition e- r o -u ~metals, the t 2 , molecular orbitals are to a very good approximation given byinre 3d atomic orbitals and that the e; excited orbitals only are slightly contaminated with ligand functions. In the crystal field calculations the handling of the electron-electron repulsion terms can therefore be taken over from the theory of atomic spectroscopy. The first calculation of excited electronic states of an inorganic complex was performed by Finkelstein and Van Vleck (45) in 1940. Empirically the excited terms of the (3d)3 system of Cr" were fairlv well known. The atomic states are characterized in a L-s scheme. Ignoring the spin-orbit coupling the ~ositionsof all excited molecular states were evaluated for C~(H~O);+ using a perturbation treatment on the complete L-S basis set. I t had long been well known (46) that the Cr(H2O)p complex both has a broad continuous absorption hand with a maximum a t 17.500 cm-' and some sharb lines centered a t e and lines were c&efully investigated 15,000 cm-I. ~ h & bands a t the temperature of liquid hydrogen by Spedding and Nutting (47) in 1934. The positions of the broad bands were found to he very sensitive to variations in temperature, and their edges drew in several hundred Angstroms between room and liquid hydrogen temperature. Measurements of the Zeeman effect in the sharp line group revealed that the Zeeman pattern consisted of unshifted lines and of lines displaced by f28H, where 8 is the Bohr magneton, H the magnetic field. From an analysis of the splitting pattern Van Vleck had proven conclusively (48) that the sharp lines terminated in an excited state having S = %. The lines were therefore
looked odd a t first sight. The two lowest doublet atomic states are 2G (15,200 cm-I) and 2P (19,400 cm-1, actually 14,200 cm-I). Assuming a "weak field" perturbation, that is retaining L as a "good quantum number" the energy of the ground state is 4A2,(F)(-12 Dq). For the excited doublet states we get 2 T ~ g ( P ) ( 0Dq) and 2A~p(G)(-2 Dq), 2E(G)(-2/7 Dq), 2T1,(G)(-Dq), 2T2g(G)('3!7 Dq). The lowest "inter-system combination" was therefore in this scheme to be expected a t E('AlgG) - E(4AzgF)= 15,200 cm-' + 10 Dq. From magnetic susceptibility measurements 10 Dq for Cr(H2O)Zt was known to he some 15,000 cm-I. Therefore one thing stands out clearly from the calculation that without including the effect of matrix elements non-diagonal in L, no account can be given of the doublet states. With full configuration interaction included the quantum number L loses, on the other hand, all meaning. Taking a value of 10 Dq = 15,000 em-', and using a full configurational mixing, Finkelstein and Van Vleck calculated the lowest spin doublet state 2E, to he 18,200 cm-' above the ground state. Thr dicrrcpancy may beduetorhr fart that our assumed rnlus 1500 .. . miry he a little low. There arr no adequate determ~. n a t i o n a d Dq ava;lal,lr forrhnme nlum.ond it ia nercuiarv for u s to base our estimates on salts not merely of different chemical composition, but also of different valence.. . The doublets are brought to within the DroDer distance 14.900cm-' ofthe basis ouartet ifwe take Dq about i820 cm-l. r m - ' of 1)q
The first identification of an excited state in an inorganic complex had been made. Finkelstein and Van Vleck must, however, have been too accustomed to the sharp line spectra of atomic spectroscopy to consider the claims of broad featureless bands. The opening line of Finkelstein and Van Vleck's paper is Usually the spectra of solids are characterized by continuous bands rather than discrete lines. They then make the observation that
or the ~ a h n - ~ e l effect. L r As the lines are inter-systemcombinations, as well as contrarv to the Laoorte rule. thevare hiehlv forbidden.. .
They could be due to dipde transitions caused"bicoupling with crystal vibrations, to magnetic dipole radiation, or quadrupale radiation. The tricky question of the intensity source in the crystal field bands and lines indeed already had been treated by Van Vleck (49) in 1937. The presence of a spin-doublet in C r ( H 2 0 ) ca t 15,000cm-I 358 I Journal of Chemical Education
tion. Had Finkelstein and Van Vleck identified the broad continuous band with the maximum a t 17,500cm-' as the looked-for line, their calculation would immediately have given Dq = 1750 cm-', and the first spin allowed ligand field transition would have been identified. This had instead to wait ten years.
Paramagnetic Rotation Working in Gottinpen with Max Born in 1928 Rosenfeld ( 4 9 ) had &hlished the quantum mechanical theory fur the natural optical activity of rhiral m~~lrculrs.'I'he theorv for the rotation of the plane df polarization in a magnetic fieid (Faraday effect) had likewise been treated by Rosenfeld (50) for free paramagnetic atoms or ions. In t h e case of the multiplet widths Au small compared to kTlh and with the frequency of the incident lieht heine far from anv ahsorntion line. the " magnetic rotation can he expressed as a sum of two parts. These show different dependence on frequency, and are respectively independent of and inversely proportional to the absolute temperature. They are referred to as the "diamagnetic" and the "paramagnetic" terms. The diamagnetic term arises from the splittings produced by the magnetic field in the absorption frequencies for right and left circularly polarized light. The paramagnetic part arises in case the distribution of the atoms in the ground manifold depends on temperature through the Boltzmann factors. The paramagnetic rotation of the crystal tysonite (a mixed fluoride of cerium and other rare earth elements) had been treated in 1929 by Kramers (51). Without prior knowledge of Bethe's paper (16), he introduced a trigonal crystal field and derived a formula which could account for the nhenomenon. In an extremely important paper published in i930, Kramers developed the general theory of the paramagnetic Faraday effect in a crystal, and proved the so-called Kramers' degeneracy mentioned above (52): all electronic levels in molecules containing an odd number of electrons must remain a t least two-fold degenerate provided that no magnetic field is Dresent. Upon Van Vleck's instigation and under his direction Robert Serher (53) in 1932 worked out the Faradav effect for molecules. The formula contained, of course, the well-known diamagnetic A terms and the ~aramaeneticC terms. However. for molecules, the diamagnktic B terms were introduced: These have the same frequency de~endenceas the C terms. and stem from the mixingof the zeroth order molecular states by the perturbing magnetic field. Due to lack of information on the excited states Van Vleck and co-workers (54,55) could only try to establish a proportionality between the paramagnetic susceptibility and the "paramagnetic" term in the Faraday effect. Primarily the interest before the war was therefore centered on thme mound state spliti~ngiwhich were compnruhle to k T The first caleffect in a cumolex had to wait c u l s t i o ~01'3 ~ ~nal(nelo-wticdl until 1965 when ~tep'hens (56) considered the allowed "charge-transfer" transitions found in Fe(CN)i-.
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The Spread ol the Ideas Our general view of the influence and developments of quantum mechanical ideas as aoalied to inoreanic comolexes . have now reached the years 193940. This seems a very suitable point to pause for a moment and take stock of the situation.For onething World War I1 broke out and curtailed so much scientific work. The year 1939 also marks the first issue of L. Pauling's hook: "The Nature of the Chemical Bond." I t constitutes in essence a summing up of seven papers by Pauling, published between 1931 and 1933. Pauling's aim was t o popularize the structural aspects of chemistry. The valence bond method and the concept of resonance were his primary tools. To this, in the case of inorganic com~lexes.came the maenetic criterion for bond tvoe. After ~ the war, most of this material found its way into the general textbooks of chemistry. The hook inspired and exerted a profound influence on a generation of chemists. The immense success of "The Nature of the Chemical Bond" (virtuallv unaltered third edition, 1960) is prohahly due to the' fact that to a generation untrained in the formalism of quantum mechanics it gave easy, understandable answers to very difficult questions. Very few-if any-chemists seem to have taken notice of
rhe crystal-field mdccular orbital theory developed by Van \'lrck nnd his collnhordtors. Only the physics drpartmcnts at Harvard (Van Vleck): in Leiden. Holland IKramersl:. . and at the Impt&l College, London ~I'rnnw),cm;lrl h i m ~ctivitivs in the lirld. From Hollnn(l cam(:some maenetic measurements hy Sirgcrt (;i7) and Polder (58)considerid the paramagnetic anisorropy ~ L s o m eC d ' salts. Prnnes (.??I showed in 1DItl that the apparent irregularities of theheats of formation of the divalent ions of the transition group can be exolained if one corrects for the crystal field siahilization of the ground state. But from 1942 there was silence. In Germany most of the senior scientists had been driven out by the Hitler regime before the war. After the war all major German university towns were com~letelvbombed out and the scientific llbrt&s destroyed. ll&r ;hew working w n diriuns F. F.llse wrote his thesis in lD4G under thedirection of H. Hartmann. The place was Frankfurt am Main. The thesis dealt with a point-charge crystal field calculation of octahedral (3d)' and (3d)2systems. Shortly after having written his thesis Ilse died, and its contents were not published (60.61) until 1951. It is clear that Ilse had no access to the relevant literature. Most of his calculations of the state energies had indeed been done by Bethe in 1929 and by Siegert (57) in 1937. What was new, and what made history, was his identification of the hroad band found in T i ( H 2 0 ) pa t 17,500cm-' as a transition 2Ee Ilse further between the crystal field levels 2Tz, pointed out that the transition could only occur if a molecular skeleton vibration of rl, or 12, was simultaneously excited. The two hroad "visible" hands of V(H20)Et were also identified as transitions between the crystal field split components Q 3T1, of the atomic ground state (3dI2 3F viz. 3Tlg ~ T and "AQ. We know now that the assignment of the second band in V(H20)F is wrong; the correct one being3Tlp 3Tlg ( 3 P ) . With these two papers the identification of the spin-allowed crystal field bands had thus heeun. I%'ith~,ut bring awnre of the papers by ilse and Hnrtmann, 0rgt:l (621w~rkingin the I'niversity of Oxti,rd HIS,,identified the, l~rnsdfeatureless bands of transition metal complexes with transitions beween the crystal field levels. In his ID52 nauer he uses a strong field basisset for his calculations, that is ;set quantized after the occupation numbers n and m in ( t ~ ) " The well known similarity between the spectra of Cr(NH3);' and Co(NHs)i+ found in this way its natural explanation. The possible effects of a-bonding between the ligands and the t2, were also pointed out. Most important, for the (3dP confieuration Oreel drew a correlation diaeram which show& how the crystal field energy levels behaved as a function of the crystal field strength, 10 Do. Orgel further indicates that the unusual stereochemistry of Cu2+with four planar and two more distant neighbors may be connected with the Jahn-Teller effect, as calculated by Van Vleck (43). The effect on the heats of hydration of the crystal field stabilization, first discussed by Penney (59). was also rrdiscovercd. All of these themes werr to he worked over again many rimes by sul~sequentworkers. In dupnn, \I. Kotani (63) had in ID4Dcilculated the magnetic moments oicmnplt:x ions having the electronic contigurations ir,,)rc. 1 5 n 5 5. The effective number of Hohr magnetons were expressed in close form using only the parameter x = A/kT, where A is the one electron spin-orbit -coupling parameter. Now in 1954, inspired by the work of Ilse and Hartmann, but unaware of the paper by Orgel (621, Y. Tanahe and S. Sugano (64) puhlished the complete matrices to calculate all energy levels for the (d)n, 15 n 5 9, octahedral complexes. They also depicted all the full correlation diagrams. Their calculation was made using the strong field basis sets, without including the spin-orbit coupling. In the beginning of the fifties, five centers were actively engaged in crystal field research, namely the German school in Frankfurt am Main, a Danish school in Copenhagen, the
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Volume 56, Number 6, June 1979 1 359
Japanese school of Tanahe and Sugano, Orgel who published three important papers from California Institute of Technology and a group of physicists in the Clarendon laboratory in Oxford working on paramagnetic resonance. Soon a sixth group, headed by W. Moffitt at Harvard, was going to be active. Oreel both calculated and oictured some correlation diagrams (I?:;),gave n quantitative treatment of "the magnetic criterion ot'thr bond tvoe" of comolexes 661 and oointed out that the breadth of theipectral hands (67) depended upon the quantity d(AE)ld(Dq). He therefore explained the sharp bands as due to transitions inside the same (tQ)n(ef)melectronic configuration and the broad hands to trans~tionsin which an electron is transferred from the t 2 , shell to thee, shell. The English nhvsicists were narticularlv interested in the measurements and calculations of the paramagnetic resonance in complexes. A magnetic field is applied to the paramagnetic ions, so that the ground state undergoes a Zeeman effect. At the same time the ions are subjected to a high-frequency magnetic field, introducing transitions between the Zeeman levels. The splittings are given as g@H = hhu. The experimental results are expressed using a spin Hamiltonian. The idea of a spin Hamiltonian is to construct an operator containin a polynomial in the components of the spin-operaton &, and 9, which when operating on a molecular state gives us the Zeeman energies. The perturhation procedure to do this was developed by M. H. L. Pryce (68) in 1949. The Clarendon Lahoratorv . eroun - . now discovered that fittine" the data to the perturhation formulae it was necessary to use a delocalized description of the electron wavefunction. In oarticular Stevens (69) introduced n honding between the tz, metal- and ligand-ort~itals. Theg liictor is calculated using the operator JH.tL + 2s). I t is only in centro-symmetric systems that L commutes with the Hamiltonian. Furthermore, in a molecule L is not associated with any particular center. Stevens now defined the orbital reduction factor k,,, by k,,,(d,, llzId,,) = (1.2 1 h lyz) where I, is the 2 component of the angular momentum and 1x2) is the molecular ?r-orbital transforming as xz. With f, being the fraction of the electron which spends its time on the lieands one can show that k,, = 1- '/.fT.k-, - .. .. ,.. is thus a measure orthe delocalization of the k e t i c n electron. The synthesis of the magnetic features. the delocalization of the maanetic electrons and the low lying excited "crystal field" state; were finally considered by Owen (70). The proposal of the "sandwich" structure for bis-cyclopentadienyl iron(I1) in 1952 immediately produced a number of honding schemes. The years 1952 and 1953 saw three qualitative electronic descriptions by E. Ruch and E. 0. Fischer (71) in Germany and by J. J. JaffB (72) and Dunitz and Orgel (73) in the U S . The more detailed examination of the electronic structure of ferrocene was, however, given hy Moffitt (74) in 1954. Moffitt's purpose in his paper was twofold:
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On the other hand, it is hoped to present a plausible and useful account of the electronic structures attributable to bis-cyclopentadienyl compounds of the transition metals. . .And on the other hand, these systems are beautifully symmetrical.They therefore also offer an opportunity to illustrate in a simple manner the principles by means of which symmetry arguments are used to elucidate electronic . ..In oarticular. the useof erouo orooerties . . .. . theorv in theresolution ot pmhlems wlth high symmetry. . may Ire illustrartd in a stralght fonvnrd f:uhion,and that this may aid the experimentalist in deciding for himseli the rcl;,tive merits of pruposed rlrrtrunir structure
