Harry B. Gray
Columbia
University N e w York
I I
Moletular Orbital Theory for Transition Metal Complexes
At present there are three different theories which may be used to describe the electronic structures of transition metal complexes. These are the valence bond theory (1, $), the crystal field theory (3-13, 20), and the molecular orital theory (6-9, 14-30). The merits of the valence bond and the crystal field theories as applied to transition metal complexes have been outlined in great detail. It is the purpose of this paper to demonstrate the utility (hopefully the superiority) of simple molecular-orbital-theory language in discussing the spectral, magnetic, and bonding properties of transition metal complexes. First of all let us remind ourselves of the weak and strong points of the valence bond and crystal field theories in dealing with transition metal complexes. Valence Bond Description of Metal Complexes
The first step is to determine the metal orbitals that are available for bonding with the ligands. For a first-row transition metal complex, this is accomplished by finding the total number of electrons in excess of eighteen (the Ar configuration) for the metal ion under consideration. This information is combined with the known number of unpaired electrons in the metal complex, giving the orbitals not occupied by metal ion electrons, and hence available for the ligands. The ligands are assigned to empty metal orbitals in the order 3d, 4 s , 4 p , and 4d (if necessary).
orbitals
Figure 1.
Valence bond orbital diagram for NilNHn)s'f.
The Ni(NH&%+complex will be used as an example. The NiZ+ ion has a 3d8 configuration. Further, Ni(NHB)s2+has two unpaired electrons. Thus, since all the 3d metal orbitals are occupied (three orbitals have two electrons each, two orbitals have one electron each), 4s, 4 p , and 4d metal orbitals are used for valence bonding with the six ligands, as shown by the conventional box diagram in Figure 1. This spadZvalence bonding correctly predicts an octahedral structure for Ni(NHA2+. unfortunately, this success in predicting structure is countered by the failure (at least up to the present moment) of the valence bond theory in the matter of understanding the electronic transitions responsible for the near infrared, visible, and ultraviolet spectral 2 / Journal of Chemical Education
bands exhibited by transition metal complexes. Thus the three band electronic spectrum of Ni(NHJsZ+ is a mystery to the valence bond advocates. Crystal Field Description of Metal Complexes
The bonding in transition metal complexes is considered to be electrostatic (ion-ion or ion-dipole), and, if a point charge (or a point charge-point dipole) model is used, the relative energies of the d orbitals of the central metal ion can be calculated. This leads to the now standard diagram for an octahedral complex with the d,. - and d,. (eg)more unstable than the dm, d,,, and d,, (ko)orbitals (see Fig. 2). The idea is that electrons will stay away from the point charges (or point dipoles) as much as possible; thus the result is that the nonaxially directed d,, d,,, and d , orbitals are more stable. The separation between the e, and t9# orbitals is called
,.
A.
I
- 1 dxz.d,,d,, Figure 2.
,2s
SpiiUing of the metal d orbitols in on odahedral crystal fleld.
The crystal field theory offers a simple explanation for the colors of metal complexes; that is, absorption bands arise when electrons undergo the tz0 + ep transition. For calculative purposes, there are two limiting cases: A is much larger than the interelectronic repulsions, the so-called strong field limit, and A is smaller than the interelectronic repulsions, or the weak-field limit. As an example, a calculation will be made for Ni(NH8)s2+. I n the strong-field limit we assign individual electrons to the k, and e, orbitals. Thus, the ground state for Ni(NH3)e2+has the configuration (t2,)6(e,)2. Excited states arise when electrons jump from t2gto ev Each (hQ)m(e,)"electronic configuration will have an orbital energy equal to m(-2/5A) ~ L ( ~ / ~ AThe ) . states and orbital energies are summarized below:
+
Electronic eonficurat,ion Ground state Excited states
Orbital enerm -. -
(L)Ye,)" ( W ( e 2
8
A A
'16
A
-
State
desimation SA2, a%g,
ITS.
I n the weak-field limit, we consider the splitting of the states of the free ion as a result of the octahedral field of 6 NHa groups. The 3ds configuration of the free NiZ+ ion has a ground state 3F and an excited state 3P. An
octahedral field splits 3Finto 3A20,3T,v.and 3T2,states; aPdoes not split up, giving a single ST1, state. There are two conditions,
+
+
C ( ~ A Z , ) 3 ~ ( & T 2 ~ 3e(JT10(F)) ) = 0
(1)
E ( ~ T , . ( P )= ) AE(SF - T )
(2)
Since the 3Az,and 3T2, states are unique to 3F,we may take their energies from the strong-field values of -*/,A and -'/,A, respectively, Solving (1), we obtain c ( ~ T ~ , ( F )= ] 3/6A. The comhined weak-field and strong-field energy level diagram for octahedral Xi2+ complexes is given below :
The crystal fields experimentally encountered are usually somewhere between the weak- and strong-field limits. In such an intermediate field the energies of 3T1,(F) and aTl,(P) may be calculated from the equation, a/h
A
-
X =
X
0
(3)
AE(aF - 3P) -
where the X mixing element may be obtained from the strong-field limit. Multiplying out equation (3) gives 2
- C('/SA+ AE) -k
a / 5 A ( A E )- X' = 0
a calculation yields the wrong ordering of the tz, and e, levels (83). (2) It is impossible to understand the spectrochemical series in terms of any ionic model. Why, for example, does NH3 give larger A's than F-? Why does CN- give much larger A's than F-? (3) There are a number of experiments which directly demonstrate that some covalent bonding does exist in transition metal complexes; also, there are many other experiments which are best interpreted using an assumption of covalent bonding. Let us summarize the important experiments now. Electron Spin Resononce (ESR)
Perhaps the most direct evidence that electrons in transition metal complexes are in molecular orbitals is provided by ESR. The ESR spectrum of IrCls2-, the first such example reported, shows hyperfine splitting due to the six chlorine nuclei, clearly indicating that the I r d orbital electrons are to some extent delocalized over the six chlorines (24-26). There are now numerous examples of this type. The ESR spectrum of the interesting [(NH3)6Co02Co(NH3)~]5+ complex shows eleven lines, indicating that the unpaired electron is in a molecular orbital with equal probability on each cobalt (27, 28). The ESR spectra of MnFz, FeF?, and CoF2 show that the electrons are delocalized over the Auorines (29). The three line ESR spectrum exhibited by all metal nitrosyls (measured to date) is due to the nitrogen (I = 1) splitting, and indicates considerable covalency of the A[-NO bond (3&33). The ESR in benzene solution spectrum of Fe(N0) [S2CN(CH3)2]2 is shown in Figure 3.
(4)
I n the strong-field limit, we drop the terms in AE, and, since the product of the roots of equation (4) is -4/,5 A2, we have -XP = - 4 / 1 5 ~ 1 (5)
x
-
(6)
= +%/,A
-
-
For Ni(NH3)62+,the three spin-allowed bands ohserved (81) are assigned as the three transitions, 3A2. 3T20, aA2. 3T1,(P), and 3An, 3T10(P). With A = 10,750 cm-' and AE(,F - SP)= 15,836 cm-1 (88), equation (3) yields the predicted energies of the 3A2, 3T1U(P)and 3A2, 3T10(P)transitior~s. The quantitative agreement between theory and experiment is summarized below:
-
Transition t--'L
=A*' "AI"
-
Predicted e!oergy
Observed energy
(cm-')
(em-')
-'
Figure 3.
The
ESR rpedrum
of
FelNOIlDTCh in acetone
soh;
DTC
S*CNlCHslz.
The spectra of many other octahedral metal complexes also have been closely calculated using the methods outlined above, and it is clear that the crystal field theory is successful in dealing with the d d spectral bands in these cases. I n spite of this success, a variety of criticisms may be directed a t the ionic model. These are the following: (1) If, instead of point charges, one uses a better ionic model with allowances for the actual size of the charge cloud (but carefully neglects all covalent terms), alas,
-
Nuclear Mogneiic Resonance (NMR)
It is possible to convert 19Fchemical shift data for a number of transition metal fluoride complexes into information about the fraction of unpaired electron spins in fluorine 2s, 2 p , and 2 p , orbitals (3457). The abbreviations f,, f,, andf,, are used to denote the fractions in 29, 2p. and 2 p , orbitals, respectively. The results: for the complexes KMnFa and KNiF,, f, = Volume 41, Number 1, Jonuory 1964
/
3
0.5%; however, f,, and f,, are much larger, approximately 4-5%. This means that the electrons in h, and e, orbitals mainly based on the central metal spend a non-negligible portion of their time around the fluorine nuclei. We will see later that the e, orbitals are r-type, and the tz, orbitals are rr-type. It is significant that for the Mn2+complex, with the ( t ~ ) ~ ( econfiguration, ,)~ the measurable quantity f,, f,, is nearly zero (0.2%), exposing the fact that the T and c metal unpaired electrons spend about equal time in fluorine orbitals. Consistently, KNiF3, with unpaired electrons only in thee, orbitals ((h,)"e,)2), has fnr - f,, = 5%, and K2NaCrS (Lz,)~, hasf,, - f,, = -5%.
These results are given in Table 2, along with the Ni2+ comparison mentioned above. Finally, the fact that further reductions in interelectronic-repulsion parameters are observed when eolid complexes are subjected to high pressures may be interpreted as due to increased covalency in the "squeezed" solids (46). Table 2. Comparison of the Values of Spin-Orbital Coupling Constants in Free Ions and in Complexes
Complex
Free ion € (em-')
Complex E (cm-')
Reference
Nudeor Quadrupole Resonance (NQR)
It is possible to relate shifts in the pure NQR spectra of the halide ions in various molecules to a so-called percentage ionic character (58). Such analyses require that reasonable assumptions be made concerning the orbitals mainly involved in bonding. A number of octahedral and square planar halide complexes of Pd2+ and Pt2+have recently been analyzed (59, LO), and the results show considerable covalency in the Pd-X and Pt-X bonds. These results are summarized in Table 1. Toble 1. Partial Ionic Character of Metal-Halide Bonds as Determined by Nuclear Quodrupole Rerononce of the Halogens"
Complex
Yo Ionic ehmaoter
PdCla2PdBr2PtCW PtBrasPtIa2PdBr2PtBr?
of M-X 43 37 44 38 30 60 57
From ref. 40. Reduction in Interelectronic Repulsions and Spin-Orbital Coupling Constants
It is experimentally observed that the interelectronicrepulsion parameters and spin-orbital coupling constants (t's) always are reduced in going from free to complexed metal ions. For example, the 3F - 3P term interval for free Ni2+ is 15,836 cm-', but for Ni(H20)62+,this same interval has a "reduced" value of 14,500 em-'. Note also that the calculation of the 3Az, 3T1p(P)band maximum for Ni(NH3)s2+would have been improved if a "reduced" 3F- 3Pterm interval had been used (see above). Similarly, the free ion Z: for NiZ+is -324 em-', and for N ~ ( H Z O ) ~Z:~ +=, -270 em-' (41). These characteristic reductions are attributed to covalent bond formation in the conlplex ion. The idea is that as the valence electronic cloud around the metal expands, both interelectronic repulsions and spin-orbital coupling constants decrease (6-8,4% @). I n accord with these ideas, complexes in which there exists substantial r-bonding show large reductions in spin-orbital coupling constants. Typical examples are the dl metal oxycations, in which the complex E is only 2055% of the free ion value (44, 45).
-
4
/
Journol of Chemicol Education
Spectrol Intensities of Tetrohedrol Complexes
The unusually large absorption band intensities observed for the tetrahedral ions C u C l P and CoClk2have been interpreted as due to considerable covalency in the metal-ligand bonds in these cases (47, 48). Recently it was shown that decreasing interelectronic repulsions (attributed to covalency) in a wide variety of tetrahedral Co2+ complexes is accompanied by a corresponding increase in the intensity of a particular d d absorption band (increases in intensities of d-d transitions in tetrahedral complexes also attributed to increasing covalency) (49). This is excellent indirect evidence of covalency in metal-ligand bonds. The indirect evidence of covalency as outlined above may possibly be capable of interpretation withim the framework of crystal field theory. An attempt has been made to blame the reduction of the Ni2+ aF - 3P term interval in Ni(H20)62+ on configuration interaction effects (50). Only one excited configuration, 3d74s, was considered in this case. It is not inconceivable that a calculation mixing configurations such as 3d" 4p2and 3p43d"+ with 3dn might lead to a correlation of increasing intensities with decreasing interelectronic repulsions and spin-orbital coupling constants. Mognetic Susceptibility
It has been pointed out that the measured orbital contributions to the bulk magnetic susceptibilities of metal complexes faU short of predictions based on an ionic model (51). Thus, "orbital reduction factors" are invoked, the explanation being that covalency decreases the orbital contribution to the magnetic moment (recall the reduction in spin-orbital coupling constants). Recently the "ring current" diamagnetism of square planar Ni(CN),2- was estimated a t -98 X 10-e cgs (63). Such a "ring current" is presumably due to electron delocalization in an extensive rr-molecularorbital which includes the Ni2+4p, orbital, and the four filled CN- r-bonding orbitals perpendicular to the plane. In summary, the experimental evidence now available overwhelmingly insists that some covalent bonding is nresent in metal comdexes. Thus we must consider the electrostatic model an extreme approximation of electronic structure in these cases. It remains to discuss the merits of the molecular orbital theory, an approach A~
which retains the best feature of the crystal field theory (that is, the explanation of the spectra of complexes) while allowing for the covalent bonding which experiment forces upon us. Molecular Orbital Theory
The general idea is that molecular orbitals are formed by mak'mg suitable lmear combinations of atomic orbitals. This is the so-called LCAO-MO (linear combination of atomic orbitals-molecular orbital) method. The method as usually may be divided into . applied .. three steps: Step 1. Pick out the valence atomic orbitals for the atoms in the molecule under consideration. The valence orbitals of an atom are the orbitals which have received electrons since encountering an inert gas (looking backward from the atom's position in the periodic table), and also any others which are in the stability range of the orbitals which will receive electrons before reaching the next inert gas. Step Z. Construct the proper linear cornhinations of the valence atomic orbitals for the molecule. This requires a knowledge of the molecular symmetry. Step 3. Evaluate the energies of the single electron molecular orbitals and obtain the values of the atomic orbital coefficients. A sample calculation of the Hzmolecule serves as an introduction to both the method and the jargon of the trade. For the molecule, H. - Ha:
+
which means that a molecular orbital with energy q j3 is more stable than the combining valence atomic orbitals. Such a molecular orbital is said to he bmding. The molecular orbital with energy q - 6 is less stable than the combining valence atomic orbitals. Such a molecular orbital is said to be antibmding. Further, q, and dzare a molecular orbitals; that is, they are symmetric about a line joining the two nuclei. Thus, 4, is a-bonding, abbreviated 2 in Figure 4 , and TZis a-antibonding, or a*. The difference in the orbital energies of 2 a n d a* is 28, which is relatcd to the longest wavelength electronic transition in the HZ molecule.
,-
Figure 4.
%
4-8
Molecvlor orbit01 energy level diagram for the Ha molecule.
Pictures of the 4, and qz functions are helpful in showing the overlap of atomic orbitals in the molecular orbitals.
Step 1. The valence orbital of H is Is. Step 8. The correct liieer combinations for the two la orbitals must be of the form
Y.(MO) = a is,
+ b lsa
(7) Step S. We rant. the solution of the Schrodinger equation, HY. = EY., which gives the best energy. Let us solve for the energy, E: E = JY.H*d, (8) if Y. is red and normalized; that is, if
By inspection, la1 = Ibl , and the two correct molecular orbitals, neglecting overlap, are
An electron in the a-bonding orbital (41) is found most of the time in the "overlap region" between the H. and Hb nuclei. On the other hand, an electron in the a-antibonding orbital (qZ)will not be found a t all in the maximum overlap region halfway between H. and Ha. Thus develops the overlap criterion of the stability of a bonding molecular orbital. Finally, remember that no orbitals are thrown away in the process of making molecular orbitals; that is, the total number of molecular orbitals must equal the total number of atomic orbitals started with. Octahedral Complexes
The energies of Y.Iand %from equation (8) are E,=q+IB
(12)
&=q-B
(13)
where the notation is
and @ =
exchange integral
=
J(ls.)H(lsa)dr
(15)
I n most cases it is very useful to construct a diagmm which shows the relative orbital energies. Such an energy level scheme for the Hz case is shown in Figure 4 . The orbital energies (q's) of the combining atoms are given in the outside columns. The middle column shows the single electron molecular orbital energies for the molecule. The exchange integral 3j is negative,
Let us proceed to apply the molecular orbital method to an MLBoctahedral transition metal complex, including both a- and a-bonding. Step 1 . For a first-row transition metal, the valence orbitals are 3d, 4s, and 4p. The ligands may have a and a valence orbitals. The atomic orbitals which correspond to the a and a valence orbitals of course vary from ligand to ligand. For example, the fluorine a valence orbital is composed of 2s and 2p, atomic orbitals, and the a valence orbitals are 2p, and 2p,. Step 2. The molecular orbitals for a metal complex all have the general form
+ CL@(L)
Y.(MO) = CU@(M)
(16)
where the values possible for Cnnand CL are restricted by conditions of normalization and orthogonality. The @(M)and @(L) functions are the proper metal (M) and ligand (L) orbital combinations for the molecular orbital under consideration. Volume 41, Number I, January
I964
/
5
For an octahedral complex (Oh symmetry), a convenient coordinate system is shown in Figure 5 . The metal a valence orbitals are 3d,, - ,,, 342, 48, 4,,, 4p,, and 4p,. These orbitals may be further classified according to symmetry, as follows:
of the plus lobes, functioning like a cigar band. The proper ligand u-orbital comhination for 3d,. is easily written down if x 2 y2 z2 is substituted for rZ in 3za - r2. This gives 22%- (xa y2). Thus the proper ligand combination is Z(US ~ 6 ) (al a? a3
+ +
+
4.
+
+ + +
The metal orbitals which can be used for =-bonding are 3d,,, 3d,, 3d,, 4p,, 4pV, and 4p,. The symmetry classification is the following:
For students without any knowledge of group theory, the representations t,,, e,, and a,, may be regarded as (3dzv, 3da,, 3 d d ko (4P=,4P",4 ~ ~ ) tl" labels for the different classes of u-orbitals in an octahedral complex. I t is a relatively simple matter to Notice that the 4 p orbitals (1,") are involved in both perform a few symmetry operations in 0"and show that a- and T-bonding. The proper tl.(s) and to(s) ligand the (4p,, 4p,, 4 p J , (3& - ,., 3d,J, and 4 s orbitals combinations are written down in the same way as for behave differently. A recent article in THIS JOURNAL the a-combinations. discusses the meaning of such symmetry symbols in The &,(a) and t,,(s) ligand combinations only acsome detail, and can be consulted for further clarificacount for six of the twelve ligand T-orbitals. The tion (53). The symmetry labels of the metal and ligand remaining six are accounted for by the tl,(a) and &,(a) orbitals for almost all geometries encountered in cocombinations, which are made up of the same a-orbitals ordination chemistry are given in another recent aras k,(r) and tl,(a), respectively, but with every other ticle in THIS JOURNAL (17). sign in the linear combination changed. For example, the ligand combmation for the 3d,, is s,, T,* T,, TZ,, and the corresponding t,,(a) function is - ?r,% s,, - T,,. The t,,(s) and tl.(s) ligand orbitals have no metal orbital counterparts, and therefore they are nm-bonding with respect to the metal complex. A summary of all the a and s metal and ligand orbitals for an octahedral complex is given in Table 3.
+
Table 3.
Representations
Figure 5. complex.
Coordinate syrfem for a- ond w-bonding in on MLa octahedral
We now proceed to find the linear combinations of ligand u-orbitals which may bond with the t,., e,, and a,, metal orbitals. This is done by writing down (by inspection) the linear combination of a-orbitals which has the same symmetry properties as the metal orbital in question. For example, the linear comhination of ligand a-orbitals which goes with the 4 p , metal orbital has a plus sign in the plus x direction and a minus sign in the minus x direction. This is the combiiation al - u3. Similarly, u2 - a4 and as - as are obtained to bond with 4 p , and 4p,, respectively. The 4 s function has the same sign in all directions. Therefore the correct ligand combination is a, an US as. The 3dZn function has four a. lobes with alternating plus and minus signs; this matches with theligand combmation a, - u2 aa - u4. The ligand a-orbital combination for 3 4 is the only one which poses any difficulty. The analytic function for the 3d,, is proportional to 3,= - r2. The orbital is directed along the z axis, with plus lobes (z2) in both the plus and minus z directions; in addition, the -r2 gives rise to a minus "blob" which wraps up the middle
+ + +
+ +
-,.
+
6
/
Journal o f Chemical Education
+
+
+
Proper Metal and Ligand Orbital Combinations for Octahedral Complexes
Metal Orbitals
Ligsnd Orbital Combinations
Step 3. It remains to find the energy of each $(MO) molecular orbital, and obtain along with it the CM and CL coefficients (see equation (16)). In order to carry out this step, we must have values for the metal and ligand orbital coulomb integrals, and, more important, for the various a and a metal-ligand orbital exchange integrals. There are a number of approximate methods which may be used to estimate molecular orbital enzrgies. I n particular, methods devised by Roothaan (54) and Wolfsberg and Helmholz (55) have been used in calculating molecular orbital energies for transition metal complexes. Mention may be made of a successful calculation (44) of VO(HSO)~~+, using a self-consistent Wolfsberg-Helmholz method. Reasonable approximations for the various coluomb and exchange integrals may be summarized in three ''rules" (52) : 1. The order of coulomb energies is taken to be a(L), d L ) , 3d, 4 s , 4 p . 2. The amount of mixing of atomic orbitals in the molecular orbitals is proportional to atomic orbital overlap and inversely proportional to their coulomb energy difference. 3. Other things being approximately equal, 4 molecular orbitals are more stable than ah molecular orbitals and a* molecular orbitals correspondingly less stable than a * molecular orbitals. Using the above rules, it is possible in some cases to decide the relative energies of the single electron molecular orbitals. Certainly in high symmetry cases when the total number of levels is small, confidence may he placed in the energy level scheme arrived at. The relative molecular orbital energy ordering ex~ e c t e dfor a general octahedral complex is shown in kigure 6. We have yet to consider metal complexes which contain lieands which have a-svstems of their own. A good example of such a ligand is CK-, which has filled
Figure 7. Molecular orbital energy level diagram for octohedrol metal complexes containing ligonds whish hove sbond relatively stable s* orbitals.
Mefal
Molecular
orbifals
Ligand
orbitalS
-
Figure 8. Molecular orbital energy level diogrom for rquore planar m e t d comp1exer.
Figvre 6. Molecular orbital energy level diagram for octahedral metd complexer
aB and empty a * orbitals. Interaction of metal d, electrons with ligand a * orhitals stabilizes the molecular orbital mainly based on the metal. Such interaction is conveniently called a-back-bonding (or rr-back-donation), and such ligands are referred to as r-acceptor ligands. To be very effective, the a* level of the ligand should be more stable than the 4 p metal orbital. The relative molecular orbital energies for octahedra1 complexes containing ligands with accessible 4 and a* levels are shown in Figure 7.
Volume 41, Number 1, January 1964
/
7
Square Planor Complexes
.
I n Dlh symmetry, the metal orbitals which may be used for u-bonding are 3d,4al,), 4s(a1,), 3d, _ ,2(bl,), and 4p,, 4p,(eu). Those which may a-bond are 3d,, 3d,&,), 3dr,(bd, 4p8(a2J, and 4p=, 4pv(e,). The 4p,, 4p, orbitals are involved in both u- and d o n d i n g . The molecular orbital energy level scheme which has been estimated for square planar metal complexes is shown in Figure 8 (52). There are four separate d (levels (e,(a*), al,(u*) bz,(s*), and bl,(a*)), and therefore three different orbital parameters, A,, A2, and Az. The full d orbital splitting will be designated A; thus A1
+ A1 +
A3
= A.
Tetrahedral Complexes
For the symmetry Td, the only pure metal u-orbital is 4s (ad. The 3&,, 3d,,, 3d&) and 4pZ,4p,, 4p&d orbitals may be used in both u- and a-bonding. The 3dZ8 3d,. (e) are pure a-orbitals (56). A generally accepted energy level scheme is shown in Figure 9 (47,57-59).
_
Metal
cirbifals
Molecular orbitals
Ligand
orbtfals
C4J, such as L.(NI-0) metal oxycations (44) arid L,(M-NO) metal nitrosyls (n = 4 or 5) (31). Spectral Properties of Transition Metal Complexes d-d Bands
The Magnitude of A. Electronic transitions within the t2,(aY)and e,(u*) orbitals (for the octahedral case) are the d-d transitions of the crystal field theory (vide supra); these transitions are responsible for the colors of many transition metal complexes. The claseic example is the Ti(H,O)2+ complex, with the electronic configuration [t2,(a*)]. The purple color of Ti(H20)s3+is due to light absorption (centered about 20,300 cm-') accompanying the transition ffp(a*)+ %(a*) (64). For the same ligand, the difference in energy (A) between the most stable and most unstable d levels in transition metal complexes varies with the geometry of the complex, with the charge on the central metal ion, and with the n of the d valence orbitals. For the same metal ion, A is different for each different ligand; the systematic ordering of ligands in terms of experimentally observed A values is known as the spectrochemical series. We will consider separately all of the factors which affect A. Effect of Geometry. For the important complex geometries, the value of A for the same ligand follows the order square planar
Figure 9. Molecular orbital energy level diagram for tetrahedral metal com~lexes.
There may be zero to five a-bonds in a tetrahedral complex, depending on the occupancy of the Z and a * orbitals. Maximum a-bonding occurs when all the # l e v e l s are filled and all the s * levels are empty. For example, the Mn04- complex has ten electrons in rborbitals (six in h(sb)and four in e(ab))and therefore five a-bonds. For complexes containing a-acceptor ligands, the e(a*) and tz(u*, a*) are not strongly antibonding (the e level may possibly be bonding in extreme back-donation cases such as Fe(CO)a2-), and even for dlocases the number of a-bonds is closer to five. Other Geometries
Molecular orbital energy level schemes have been estimated for linear (Dm,,) (60), trigonal bipyramidal (Dan)(61), and, for some of the levels, for Archimedean antiprismatic (DI,) (68, 63) metal complexes. There are also level schemes available for complexes containing one unusually strong metal-ligand link (effectively 8' / Journol o f Chemicol Educofion
> octahedral > tetrahedral
For example, square planar N i ( C N ) p has A = 35,500 cm-' (5t) and octahedral Fe(CN)64-has A = 33,800 cm-' (65); planar PdC142- has A = 26,800 cm-I (5t) and octahedral R h C l p has A = 20,300 cm-1 (66). Comparing octahedral and tetrahedral complexes, ~~MnCls4- has A = 7500 cm-' (66) while C O C ~ has A = 3100 cm-' (67). Metal d orbitals participate in strong u-bonding in square planar (d,. - ,S and octahedral (dzt- ,*, dJ complexes. Thus there is a substantial energy difference (A) between the very unstable u* and the relatively stable a* molecular orbitals. I n tetrahedral complexes, however, the metal & d orbitals need not he involved in such strong a-bonding, since the p valence orbitals probably take care of most of the & u-bonding. It follows that there is a rather small A between the two sets of weakly antibonding orbitals. Effect of Charge a the Central Metal Ion. For metal complexes containing ligands which do not have appreciable a-acceptor ability, A is known to increase as the positive charge on the metal ion increases. As an example, V(H20)s2+has A = 11,800 cm-', while V(Hz0)e3+ has A = 17,850 cm-I (68). This effect may be interpreted as an increase in u-bonding strength with increasing positive charge on the metal ion, making e,(a*) relatively more unstable than &,(a*). Although there is not enough information to make any sweeping generalization, complexes with good a-acceptor ligands do not appear to show appreciable increases in A with increasing positive charge on the metal ion. For example, both Fe(CN)E4- and Fe(CN)E3- have A's in the 34,000-35,000 cm-I range (65). Since A is the relative energy separation of a* and a * levels, the expected increase in A due to an increase in Fe-CN
a-bonding in going from Fez+to Fez+ may be cancelled by the corresponding decrease in metal-to-ligand s-bonding. Effect o j Sd, 4d, and 5d Valence Orbitals. For metal complexes with the same ligand, A increases on changing from 3d to 5d valence orhitals. For most metal complexes there is a regular increase in A in the order 3d < 4d < 5d. For example, the square planar
complexes have A, = 14,300 cm-I for Ni2+, A, = 18,200 cm-' for Pd2+,and A1 = 21,850 em-' for P t 2 + (fig). Again, an increase in a-handing strength in the order 3d < 4d < :id may be responsible. However, some octahedral complexes with good s-acceptor ligands have A about the same for 3d and 4d, and only slightly 1are;er for 5d. Example: F ~ ( C N ) R and ~Ru(CNls4- have A = 34,000 cm-I, hut OS(CN),~have has A > 3,5,000 em-'; Cr(CO)s and MO(CO)~ A = 34,150 cm-'; W(CO)c has A = 34,570 em-' (65). For such complexes, the a and s effects must cancel in the 3d to 4d step, but the U-strength of the 5d metal valence valence orbital probably slightly outweighs any changes in metal-ligand s-bonding. The Spectcrohemical Series. The orderly variation of A upon changing the ligand in metal complexes is the spectrochemical series. I n order or decreasing A, the spectrochemical series of the most important ligands is
The molecular orbital theory defines A as an energy difference between a* and a * orbitals. Thus, any quantitative theory of the spectrochemial series must consider the effect of different ligands on the stability of both the a* and T* molecular orbitals (70). I t is encouraging that one successful molecular orbital calculation of A has been made, in the case of KNiF,, (NiF, octahedra in the crystal) (71). This calculation lends support to the molecular orbital idea that A is not due to ionic effects, but to covalent effects; that is, d-orbital a-bonding ie stronger than d-orbital =-bonding, and therefore a* is more unstahle than s * (see Rule 2). A semiquantitative explanation of the spectrochemical series is possible by considering the effect of different ligands on the stability of the a * molecular orbitals mainly based on the metal. In doing this we assume that most ligands have similar a-bonding abilities, while possessing widely ranging =donor and s-acceptor properties. The metal d(a*) molecular orbitals are most stable with ligands which have appreciable a-acceptor properties; they are least stable with good sdonor ligands. Thus A is largest for strong s-acceptor ligands and smallest for strong a-donor ligands. I n the middle are ligands which lead to little or no =-interaction. We thus have the correlation -A
decreases in the order
-
Strong r-acceptor > Weak r-acceptor> Small r-interaction > Weak r-donor > Strong =donor
Of course the very strongly u-bonding ligands will always have large A values, regardless of s-interactions. For this reason -CH3 and -H are high in the spectrochemical series (73). Intensities. The absorption hands arising from transitions involving molecular orbitals which consist mainly of metal d orbitals are weak, and can be identified by their rather small molar extinction coefficients (r's). For spin-allowed transitions, a large fraction of experimentally observed r values for the different geometries are found within the following limits: Geometry
r range
Octahedral Square planar Tetrahedral
1-150 5-250
5C-7.50
The mechanism (or mechanisms) by which the d-d transitions gain intensity is still largely an unsolved matter. For octahedral and square planar complexes, which have a center of symmetry, the transitions are parity forbidden (even even). I n these cases a coupling of vibratioual and electronic motions may be blamed for the intensity (the so-called vibronic mechanism) (48). Slightly distorted octahedral and square planar metal complexes may have a fractional part of a d d transition "allowed," and this "static distortion" mechanism may be responsible for some intensity in many cases. The d-d absorption bands exhibited by tetrahedra1 complexes are much more intense than those exhibited by their centrosymmetric cousins, presumably because a greater fraction of these transitions is allowed. The greater the ligand orbital mixing in molecular orbitals which remain mainly localized on the metal, the larger the predicted intensity of any given d-d type transition (in the absence of a symmetry center) (48). Recall that the relatively large absorption intensities of tetrahedral complexes were taken as evidence for covalent bonding (47,W.
-
Charge Transfer Bonds
Energies. Electronic transihns involving molecular orbitals mainly localized on different atoms are called charge transfer transitions. The charge transfer absorption bands which result from such transitions usually (but not always) occur a t higher energies than the d d bands, and thus are found mainly in the ultraviolet region of the spectrum. The charge transfer processes are of two types: the transfer of an electron from an orbital mainly localized on the ligand(s) to an orhital mainly localized on the metal, abbreviated ligand metal, or L M; and the transfer of an electron in the opposite direction, that is, from an orbital mainly based on the metal to an orbital situated on the ligand(s), abbreviated metal ligand, or M -t L. The ligand metal type of charge transfer is exhibited prominently by complexes containing such ligands as 02-,C1-, Br-, and I-. For complexes of do metal ions, there are of course no d d bands and A can only be estimated from the charge transfer maxima. A good example is illnOl-, colored purple due to low energy 02- Mn7+ charge transfer (47, 55, 57-69). The first three bands in Rho4-have maxima at 18,320, 32,210, and 44,000 em-'. These bands may be assigned
-
-
-
-
-
Volume
41, Number I, Jonuory 1964 / 9
to the transitions tl(a) -+ e(a*), &(a" + e(a*), and tl(a) -+ &(u*,a*), respectively (see Fig. 9). Thus A = 25,700 cm-' is obtained for Mn04-. A recent molecular orbital calculation including all metaboxygen and and oxygen-oxygen interactions yielded A = 23,000 cm-1 forMn0~~--(7~). The metal + ligand type of charge transfer occurs in comnlexes containine ligands which oossess relatively staGe but empty orbit& (r-accepto; ligands). cornplexes containing NO, CO, and CN- have absorption bands which are most probably due to metal -+ ligand charge transfer (31, 66, 73). The Fe(CNjB3-complex, which with the electronic configuration [t2,(a*)15has an empty place in the &,(a*) level, exhibits a very rich charge transfer spectrum, indicating that both metal -+ ligand and ligand + metal transitions occur (74). It is possible t o identify charge transfer bands as ligand -+ metal or metal -t ligand by observing energy shifts in a number of analogous complexes. Ligand + metal charge transfer should occur a t lowest energy when there is a large positive charge on the central metal ion. Thus, the first charge transfer band maximum in 140,"- complexes is a t 18,320 cm-' for Mn04(Mn7+) and a t 26,810 cm-1 for CrOna- (Cra+), consistent with a ligand + metal assignment (47, 57-59). On the other hand, metal -t ligand charge transfer should occur a t lowest energy for the smallest positive charge on the metal. I n d6 M(CKjs4- complexes, the first charge transfer maximum appears a t 45,870 cm-' for Fe(CNle4 (Fez+) (73), and a t 49,500 cnl-I for Co(CN)P(Coa+)(65), providing evidence for metal + ligand transitions. Intensities. Many charge transfer transitions are strongly allowed, and the resulting absorption hands are very intense, with 6 values usually in the 500-15,000 range. Part of the identification of absorption bands as d-d or charge transfer is based on the r values of the band maxima. Transitions are very likely d-d with r < 200 and very likely charge transfer with c > 1000; in the 200-1000 range it is usually difficult to tell whether the band is a d-d band with large c or a weak charge transfer band. Molecular Structures and Magnetic Properties of Transition Metal Complexes
We must consider the relative importance of steric and electronic effects before discussing the molecular shapes of first-row transition metal complexes. Steric Effects. The mutual repulsions of the ligands: The ligands in metal complexes repel each other due t o coulomb and van der Waals forces. This repulsion is greatest for large one-atom ions such as I- and Br-, which a t close range experience substantial coulomb and van der Waals repulsions. It is smallest for small molecules such as H 2 0 and NH8, with only weak dipole-dipole interactions. I n the intermediate repulsion range are larger molecules and two- and threeatom ions. Mutual Ligand-Ligand Repulsion Greatest Simple ions, Verv Iarm 'molecules I-, Br~~
10
/
Least molecule ions,
small molecules
> large molecules > CN; SCN-, P(CsH& > NHz, HIO
lournd of Chemical Educofion
Electronic Effects. Electronic distribution in the 2 and u* molecular orbitals: This is most important for complexes containing ligands which give large A values (so-called strong-field ligands). The overwhelming majority of transition metal complexes have one of three structures: octahedral, tetrahedral, or square planar. The octahedral structure, which utilizes a.ll the good u-orbitals available (3d,., 3dZ2-,., 4s, 4p,, 4p,, 4pJ, is the preferred structure unless there is a compelling steric or electronic reason to abandon it. It is convenient in a structural discussion to divide the ligands into three classes: (1) Non-strong-field ligands with largest mutual repulsion-complexes containing such ligands will prefer a tetrahedral structure, thus minimizing the repulsions. (2) Strong-field ligands-when A is large, the first six d electrons are accommodated in the ho (a*) level, giving low-spin octahedral complexes. For a d7 metal ion, however, one electron would be forced into a very unstable e, (a*) orbital for the octahedral geometry. In order to stabilize this electron, a reduction in the number of ligands must take place. We are left with an ML6 complex with an unpaired electron in a potential a-orbital (say, the metal dd). Thus d7 metal ions with strong-field ligands such as CO and CN- form dimeric complexes with metal-metal a-bonds. Examples are Mnz(CO)lo and CO~(CN)IO-' (75). The d8 metal ions with strong-field ligands have a square planar structure, when there is a formal positive charge on the central metal. An example is Ni(CN)r2-. The square planar energy level scheme (Fig. 8) accommodates all electrons in d8 metal ions in relatively stable orbitals. When there is zero or negative formal charge on the metal, d8 metal complexes prefer a trigonal bipyramidal structure, which emphasizes the metal-to-ligand a-bonding. Examples are Fe(C0)6 and hln(CO)s-. For do metal ions, both octahedral and square planar structures force one or more electrons in very unstable u* orbitals. I n order t o prevent this we must orient the ligands so that d orbitals are not required for strong u-bonding. Again, with the one unpaired electron in a potential d u-orbital, dimeric complexes with metalmetal bonds are observed. Complexes containing dl0 metal ions are tetrahedral, thus enabling all the electrons to occupy relatively stable orbitals. (3) Non-strong-field ligands with intermediate or small mutual repulsions-complexes containing such ligands and with d1-d8 metal ions usually have no compelling steric or electronic reason to abandon the octahedral structure. The decision a t d9 is difficult, and both square planar (or distorted square planar) and octahedral (or distorted octahedral) structures are observed. Again, complexes with dL0metal ions are usually tetrahedral. Electronic effects are mori important in determining structure for second- and third-row transition metal complexes. There are two reasons for this: the A values are generally larger, and the mutual repulsions of the ligands decrease since metal-ligand bond distances are greater. Thus, structures usually follow the strong-field pattern discussed above, which is octahedral for d'-d@,square planar or trigonal hi-pyramidal for
d8, and tetrahedral for dlo. Metal-metal bonds are common for the d' and d9 configurations. Good examples of a change in structure from first-row t o second- and third-row transition metal ions are the complexes of Xi2+,Pd2+,and Pt2+(ds)with the ligands Br- and I- (largest ligand-ligand repulsions and smallest A's). The i'X&- complexes are tetrahedral (steric preference), but the PdX4%-and P t X P complexes are square planar (electronic preference). Magnetic properties of interest include bulk magnetic susceptibility and electron spin resonance. Magnetic susceptibility measurements are used to determine the number of uupaired electrons in a complex. Precise measurements also allow an estimate of the orbital contribution to the magnetic moment. This information is of great value in determining structure. The numbers of unpaired electrons in different d" metal complexes are indicated in Table 4. Table 4.
Structures and Magnetic Moments of First-row Transition Metal Comolexes
-
Electronic Number of unpaired electronseonfigurntion of -Octahedral-Square planar--Tetrahedralthe,metal High- LoyHi& Loy- High- Lo!". lon spm spm spn spm spin spm
Literature Cited
PAULING, L., "The Nature of the Chemical Bond," Cornell University Press, 1960. PAULING, L., J . CHEM.EDUC.,39, 461 (1962). BETHE,H., Ann. Physik, 3 , 133 (1929). GRIEFITH, J. S., "The Theory of Transition Metal Ions," Cambridge University Press, 1961. MCCLURE,I). S., "Solid State Physics," Vol. 9 , F. SEITZ AND I). TURNBULL, editors, Academic Press, New York, 1959.
OROEL,L. E., "An Introduction to Transition-Element Chemistry: Ligand Field Theory," Methuen, London, 1960.
J~RGENSEN, C. K., "Absorption Spectra and Chemical Bonding in Complexes," Pergamon, New York and London, 1961. BALLHAUSEN, C. J., "Introduction to Ligand Field Theory," McGrav-Hill, New York, 1962. J~RGIENSEN, C. K., "Solid State Physics," Vol. 13, F. SEITZ A N D D. TURNBULL, Editors, Academic Press, 1962. BASOLO,F., AND PEARSON, R. G., "Mechanisms of Inorganic Reactions," Wiley, Piew York, 1958. DUNN,T. M., "Modern Coordination Chemistry," J. LEWISAND R. G. WILKINS,editom, Interscience, New York, 1960, chap. 4 . PEARSON, R. G., Record Chem. Progress, 23, 53 (1962). DUNN,T. M., Pme. 7th ICCC, Stockholm, 1962; plenary lecl.ure published in "Coordination Chemistry," Butterworths, London, 1963, p. 1. VANVLECK,J . H., J . Chem. Phys., 3 , 803 (1935). VANVLECK,J. H., AND SHERMAN, A., Rev. Mod. Phys., 7 , 167 (19%).
. .
" For 1-5 unpaired electrons spin-only magnetic moments' (in
Bohr magnetons) are: 1
=
1.73; 2 = 2.83; 3
=
3.88; 4 = 4.90;
5 = 5.92.
There are a t present no well-characterized complexes of this type.
ESR spectra give information about the distribution of unpaired electrons in the ground state. This was discussed earlier as the first direct evidence for molecular orbitals. The values of the g splitting factors give information about the orbital degeneracy in a metal complex. Particularly helpful in many cases is a measurement of the anisotropy of the g value; that is, obtaining g,, g,, and g,. For octahedral and tetrag,; hut for a square hedral complexes, g, = g, planar complex, g, g, # g,. Also, since the values of g,, g,, and g. depend on which orbitals are occupied in the ground state and lowest excited states, ESR data aid in ordering the orbitals in low symmetry complexes. In summary, ESR information can be of considerable value in indicating electronic and molecular structure.
-
-
Acknowledgments
This paper was presented a t the Conference on Advances in Coordination Chemistry, held a t the Ohio State University during July and August, 1962. The author gratefully acknowledges useful discussions with the conference participants. The author thanks the National Science Foundation and the donors of the Petroleum Research Fund for support of his research in the transition metal complex area.
BALLHAUSEN, C. J., "Advances in the Chemistry of the Coordination Compounds," S. KIRSHNER, Editor, Macmillan, New York, 1961, p. 3. LIEHR,A. D., J . CHEM.EDUC.,39, 135 (1962). SUGANO, S., "Chemical Physics of Nonmetdlic Crystals," Benjamin, New York, 1962, p. 303. SUTTON, I.. E., J. CHEM.EDUC.,37, 498 (1960). Recent references which should prove helpful: (a) CILRLIN, R. I,., J. CnEa.E~uc.,40,135 (1963); ( b ) COTTON,F.A., "Chemical Applications of Group Theory," Interscience, New Yark, 1963. J$RGENSEN, C. K., Acla C h m . Scand., 9 , 1362 (1955). MOORE,C. E., "Atomic Energy Levels," NBS Circular No. 467, Vol. 11, 1952. KLEINER,W. H., J . Chem. Phys., 20, 1784 (1952). OWEN, J., AND STEVENS,K. W. H., Nature.. 171.836 . (1953). . . GRIFFITHS, J . H. E., OWEN,J., AND WARD,I. M.,Pmc. Roy. Soe. (London),A219, 526 (1953). GRIFFITHS,J . H. E., AND OWEN,J., Proc. Roy. Soc. (London), A226, 96 (1954).
BERNAL,I., EBSWORTH, E. A. V., AND WEIL, J. A,, P W C . Chem. Soe., 1959, 57. EBSWORTH, E. A. V., AND WEIL,J . A,, J . Phy8. Chem., 63, 1890 (19591. . .
TINKHAM, M., Proe. Roy. Soe. (London), A236, 535, 549 (1956).
BERNAL,I., AND HARRISON, S. E., J. Chen. Phys., 34, 102
.
(1961). ,
GRAY,H. B., BERNAL,I.,
AND BILLIG,E., J . A m . Chem. Soc., 8 4 , 3404 (1962). GIBSON,J. F., Nature, 196, 64 (1962). BERNAL, I., AND HOCKINGS, E. F., PTOC.C h m . Soe., 1962, 361. SHULMAN, R. G., AND JACCARINO, V., P h y ~Rev., . 108,1219 (1957). ~ ~, STOUT,J. W., AND SHULMAN, R. G., Phys. Rev., 118, 1136 (1960). SHULMAN, R. G., AND KNOX,K., P h y s Rev., 119, 94
figfin) \----,.
( 3 7 ) SHULMAN, R. G., AND KNOX,K., P h w . Rev. Letkrs, 4, 603 (1960). . . ( 3 8 ) TOWNES, C . H., AND DAILEY,B. P., J. C h a . Phys., 17,782 (1949). D., K U ~ T AY., , ITO,K., AND KUBO,M., ( 3 9 ) NAIKAMURA, J. Am. Chem. Soc., 8 2 , 5783 (1960).
Volume 41, Number 1, lanuary 1964
/
11
ITO,K., NAKAMURA, D., KURITA,Y., ITO, K., A N D KUBO, M., J. Am. Chem. Soe., 83, 4526 (1961). See ref. (81, p. 265. OWEN,J., P ~ cRoy. . Soe. (London),A227, 183 (1955). DUNN,T. M., J . Chem. Soe., 1959, 623. C. J., AND GRAY,H. B., Imrg. Chenz., 1, 111 BALLHAUSEN, (1962). HARE,C. R., BERNAL, I., A N D GRAY,H. B., In07g. Chem., 1, 831 (1962). ZAHNER, J. C., A N D DRICKAMER, H. G., J . Chem. Phys., 35, 1483 (1961); and references therein. BALLHAUSEN, C. J., AND LIEHR,A. U., J. Mo1. Sped?., 2, 342 (1958); ibid., 4, 190 (1960). BALLHAUSEN, C. J., Prog. Inorg. Chem., 2, 251 (1960). COTTON, F. A,, AND SODERBERG, R . H., J. Am. Chem. Soe., 84, 872 (1962). C. J., Acta Chem. Scand., GRAY,H. B., AND BALLHAUSEN, 15, 1327 (1961). OWEN,J., Discussions Fwadav Soe., 19, 127 (1955). GRAY,H. B., AND BALLHAUSEN, C. J., J . Am. Chem. Soe., 85, 260 (1963). W. C., J. CHEM.EDUC.,38, MANCH,W., AND FERNELIUS, 192 (1961). ROOTHAAN. C. C. J.. Rev. Mod. Phus.. 23.69 11951). , WOLFSBERG, M., AND HELMHOLZ, i . , 'hem. ~ .Phys., 20. 837 (1952). COTTON, F. A,, J . Chem. Soe., 1960, 5269. CARRINGTON, A., AND J~RGENSEN, C. K., Mol. Phys., 4,395 (1961).
.
12
/
Journal o f Chemical Education
A., AND SYMONS, M. C. R., J . Chem. Soc., (58) CARRINGTON, 1960, 889. A,, A N D SCHONLAND, D. S., Md. Phys., 3, (59) CARRINGTON, 331 (1960). J. R., LIEHR,A. D., A N D ADAMSON, A. W.: (60) PERUMAREDDI, J . Am. Chem. Soe., 85, 249 (1963). A. D., Can. J. Chem., (61) BADER,R. F. W., AND WESTLAND, 39, 2306 (1961). (62) GLIEIANN,G., Theorel. ehim. Ac2a (Bed.), 1, 14 (1962). (63) KONIG,E., Theoret. chim.Acta (Berl.), 1, 23 (1962). H., AND S C H ~ F E H. R , L., Zeit. phys. Chem., (64) HARTMANN, 197, 116 (1951). (65) GRAY,H. B., AND BEACH,N. A., J . Am. Chem. Soc., 85. 2922 (1963). (66) See ref. (7), pp. 110-11. F. A,, GOODGAME, D. M. L., AND GOODGAME, M., (67) COTTON, J . Am. Chem. Soe., 83, 4690 (1961). (68) See ref. (8),pp. 233-5. (69) BILLIG,E., unpublished results. (70) MCCLURE,D. S., "Advances in the Chemistry of the Coordination Compounds," S. KIRSHNER, Editor, MacrnilIan, New York, 1961, p. 498. R. G., AND SUGANO, S., Phys. Reo. Letlers, 7, (71) SHULMAN, 157 (1961). ~, (72) CHATT,J., AND HATTER,R. G., J . Chem. Soe., 1961, 772. . I , 337 (1962). (73) ROBIN,M. B., I w ~ g Chem., (74) NNMAN,C. S., J. Chem. Phys., 35, 323 (1961). (75) GRIPFITR,W. H., Qmvl. Rev., 1962, 188. (76) VISTE, A,. ANT) GRAY,H. B., to be published ~
