John R. Wasson' and Henry J. Stoklosa University o f Kentucky Lexington, 40506
Electronic Spectra of LOW-Symmetry
d' and 6 Ion Complexes
With recent years several articles (1-18) concerned with ligand field theory have appeared in this Journal. Theoretical aspects 1-7.9-1116) calculations (5,8,12,13,16,18) and experiments (I2,13,15,17) instrumental to the incorporation of ligand field theory into the undergraduate curriculum have been described. Perhaps the most straightforward ligand field calculations which can he profitably introduced into junior-senior and beginning graduate level courses are those for 3d1 systems so lucidly presented by Companion and Komarynsky (8). To he sure, their approach is not the only one (19-23/, but i t is particularly well-suited to the majority of students. In this paper we show how the formalism of Companion and Komarynsky (8) can he employed for the calculation of the electronic spectra of d9 complexes and how elementary group theory can he used to simplify the calculations and label electronic energy levels. Parameters involved in the calculations and various applications to low symmetry complexes, i.e., those which are not octahedral, tetrahedral or cubic eight-coordinate, are discussed. Finally, a relatively simple computer program to facilitate calculations is described. Theory
A detailed discussion of the theory of ligand field calculations for d l systems is to be found in the paper by Companion and Komarynsky (8). Using real d orbitals (familiar to students) as a hasis set, first-order perturbation theory for degenerate systems is applied once the perturhing electrostatic potential due to the ligands has been defined. The ligand field problem for any number of ligands in any geometrical arrangement then reduces to evaluation of integrals of the type
Table 1. Spectrochemical Functions of Ligands and Metal Ions (24)
Metal ion
g
Ligands
f
-
aEstirnated from the spectrum of VF& A value of 27.4 is obtained from the spectrum of VCln. The difficulty in obtaining a good value stems from the Jahn-Teller effectin cubic d' complexes. Table 2. Character Table for Point Grour, C d
scattan, F. A,, (1971);Ref. (271. which contributes to the shifting of all of the degenerate d levels by the same amount has been omitted) enables numerical evaluation of the energies of the individual d orbitals. The lowest energy d orhital(s) can be taken as the electronic ground state and the electronic absorption spectrum of a complex can be calculated by taking the differences in energies of other orhitals with the ground state. Parameters to Calculate Electronic Spectra
and solution of the determinental equation
for the five roots, E K , i.e., the energies of the five d orbitals. The latter is readilv accom~lishedusing even relatively small electronic digital computers. Evaluation of the H , , intemals is accomplished (8) by giving the B (angle the metal-iigand bond axis makes with the z-axis of the coordinate system chosen for the complex) and 6 (angle the projection of the metal-ligand bond axis into the xy plane makes with the x-axis of the coordinate system chosen for the complex) coordinates of each of the ligands and using Tables 1 and 2 in the paper by Companion and Komarynsky (8). The specification of values (in cm-' units) of the ligand field parameters ruz and ua for each of the ligands (note that the parameter ruo
This work was supported in part by Project Themis and the University of Kentucky Research Foundation. The authors are of the U.K. Computer Center for their grateful & the assistance. 'Author to whom correspondence should be addressed. 186
/ Journal of ~hem%alEducation
For an octahedral complex the energies of the one electron energy levels are given by
and
-
The separation between the ground state and the higher e , promotion energy, energy orbitals (the familiar tz, lODq is 5/3 aq, i.e., lODq = 5h a* or a* = 6Dq. Thus, it is possible to relate the theoretical parameter nq to the lieand field snlittineu for an octahedral c o m ~ l e xwhich is qualitatively discussed in freshman chemistry courses. The auantitv. rua- can be evaluated theoretically but, in practice, parameters necessary to fit calculated electronic spectra to observation are employed. Jngensen (24) has shown that lODq for cubic complexes may be estimated to a good approximation using the expression
-~
10Dq
" f(ligands).g(centralion) x 10' em-'
(1)
The function f is dependent on the number and type of ligauds surrounding the metal ion. Equation (1) provides
for the ready estimation of a 4 parameters. Some typical s listed inTahle 1. values off and P ~ a r a m e t e r are For d"com&xes which are not octahedral, tetrahedral or cubic eight-coordinate the solution of the ligand field problem necessitates (81 evaluation of both wz and ar. Generally, a 2 is obtained by assuming (8,16,25) an a z l a r ratio of about 1.0 rather than using theoretical ratios of two or more. This is due to the failure of the simple crystal field approach which has been discussed in the cited references. Since a d9 ion is the "hole-equivalent" (5.9.19-23) of a dl ion, the s i m of a n is reversed when calculations are performed for @ ion c&nplexes. Alternately, the energy level diagram for a d l complex can be inverted in orderto obtain tbe energy level scheme for a d9 complex. Application of Group Theory
Several articles in this Journal (26) and texts (27) have appeared which describe the application of group theory to chemical problems and they should be consulted for details. Here we only wish to show how readily elementary group theory can be used to simplify the calculations described by Companion and Komarynsky (81 and provide symmetry labels for one-electron energy levels. Once the d orbital states have been pro~erlylabeled, information regarding electronic absorption hand polarizations (5,271 can he readily obtained. When calculations are not done using the computer program described below, inspection of the character table for the noint mouD - . to which a comnlex helones meatlv reduces ;he number of matrix elements to ge i v a l u k e d usine the tables siven bv Comoanion and Komarvnskv I S the ~ &guti) (8). k o r example,'the complex ~ ~ O C (see belongs to the point group C4,. Inspection of the character table2 (Table 2) for the point group shows that the five d-orbitals dzz, d,z-,z, d,, and d,, and d,, belong to the Al, B1, Bz and E irreducihle representations of the point group C4,,. These provide the symmetry labels for the one-electron levels and show that the d,, and d,, orbitals belong to the same irreducible representation, E. States labeled according to the same irreducihle representation may "mix," i.e., share a common off-diagonal matrix element. Thus, the only off-diagonal matrix element which must he evaluated using the tables of Companion and Komarynsky 18) is Hzr, i.e.
8 Coordinate system used for calcuiatians of eftects at distortion electronic energy levels of the VOC1s3 ion.
on the
Instead of evaluating fifteen matrix elements, use of group theorv reduces the Drohlem for a d1 or d9 c o m ~ l e xhelonaing to point group CllUto the evaluation of six matrix elements. When calculations are being- done by hand this resuits in a great saving of time. Generally, it is good practice to state the coordinate system used in a particular calculation and it is strongly recommended that conventions (Jaffe, H. H., Orchin, M., Ref. (271, pp. 9-11) for the orientation of molecules in the Cartesian coordinate systems be followed. This tends to minimize confusion regarding notation and permits transferability of general results. Applications to Low Symmetry Complexes
Oxouanadium(1V) Complexes. Compounds containing the V = 02+ion have been extensively studied (281. In view of the popularity of these 3d1 complexes i t is worthwhile to consider the VOCl& ion and how effects of distortion of the complex on the electronic energy levels can he calculated usine the method described bv Com~anion and Komarynsky 6).The coordinate system employed in the calculations is eiven in the fieure. For a c o m ~ l e xbeC4" the energies of the d l&els are longing to point given by
ZThe right-hand side of the character table for a particular point group lists the coordinates x , y , and n as well as the squares and binary products of coordinates according to the transformation properties. Identification of the real d orbitals with the latter enables proper assignment of symmetry labels for one-electron levels. Texts listed in reference (26)should he consulted for additional details as well as discussion of polarized absorption bands in electronic spectra. Volume 50, Number 3, March 7973 / 187
In the preceding expressions no2 or 4 , 01% or 4 , and a 5 or 1 are the parameters for oxygen, the trans-axial chloride ion and equatorial chloride ions, respectively. The angle 8 is defined by the 0 = V - equatorial ligand bond. The structure of VOCla3- is not known hut that of VOC14(Hz0)2- (29) is. The angle 8 can thus he assumed to he 96". The octahedral complex VC1e2- exhibits an ahsorption hand a t 15,400 cm-1 (30) which can he attrih2T2, transition, i.e., Dq = 1540 cm-'. nted to the ZE, This is slightly less than the Dq = 1790 cm-' estimated using the parameters in Tahle 1. For chloride ion u 4 is thus taken to be 9240 cm-1. The axial and equatorial chloride ions are not expected to he bound to the vanadium equally well. The trans-axial chloride is expected to be found a t a greater distance from the metal and consequently the 014" parameter will he reduced (about 5-20%) relative to those for the equatorial chloride ions. The effect of reducing the a4* parameter from 9240 cm-' is given in Tahle 3. Reduction of asA to 50% of this initial value does not change the relative ordering of the energy levels. In practice, such parameter variation can he employed to obtain close agreement between observed and calculated optical spectra, u h O for the doubly bonded oxygen in VOC1s3- was taken to be 94,500 cm-'. This value is apparently high compared to the value which can he estimated from data on transition metal ions in oxide lattices. However, in oxide lattices, e.g., MgO, the oxygen hound to the transition metal ion is also hound to the counterion, e.g., Mg2+, and the crystal field exerted on the transition metal ion is attenuated relative to that expected for the monomeric hypothetical oxide ion complex, e.g., V0e8-. The value of mao used here has been found useful for the calculation of the electronic spectra of a numher of oxometal ion complexes. The m / a 4 ratios employed in these calculations were all set equal to 0.9. This ratio is consistent with previous work (25) as well as several other calculations (3i)we have performed. The calculated order of one-electron 3d levels for < d,, < d , ~ - ~ z< d22 which is the VOCLs3 is d, same ordering scheme calculated by Ballhausen and Gray (32) for VO(Hz0)52+ using molecular orbital theory. With en"= 0.85 a 4 " ( a 4 " = 9240 cm-I) the electronic transid X Y ) , ~2 (dl~-y2 dXy) and (drz tions UI (d,,, dyz dry) are predicted to appear at 12,803, 15,064 and 78,292 cm-1, respectively. The results for the first two ahsorption hands are in good agreement with the values of 12,100 cm-I and 14,300 cm-l reported by Kon and Sharpless (33) who also mention a third band, presumably an ol bz transition, a t 21,500 cm-1. A more extensive analysis (31) of the spectra of oxovanadium(1V) compounds employing both ligand field and molecular orbital theory shows that in addition to the third hand in the region cited by Kon and Sharpless (33) still another a l bz transition is expected a t about 79,000 cm-'. This shows that additional low energy transitions other than those calculated on the basis of the present model necessitate utilization of molecular orbital theory. The possibility of moderately low intensity, low energy charge-transfer transitions can also not he overlooked. The effects of distortion on the energy levels and electronic spectra of complexes are readily evaluated using the method of Companion and Komarynsky (8). For the VOC153- ion the effect of changing the 0 = V - equatorial chloride bond angle, 8, on the calculated transition energies is given in the figure. Such calculations are of importance in accounting for the distortions of complexes due to matrix effects.
-
-
-
-
-
-
Summary
In the preceding we have indicated how: (1) parameters can be chosen in order to perform somewhat empirical, yet chemically meaningful ligand field calculations of the 188
1 Journalof Chemical Education
Table 3. Effect of Trans-Axial Parameter Variation on the Calculated Optical Spectrum of the VOCIs3- Ion
0 13,007 15,064 15,064 5 12,940 15,064 10 12,871 15,064 15 12,803 20 12,734 15,064 15,064 25 12,666 15,064 30 12,599 35 12,530 15,064 40 12,462 15,064 15,064 45 12.39'5 15,064 50 12,325 en (equatorial chloride) = 9240 cm-I
79,324 78,987 78,640 78,292 77,944 77,596 77,259 76,902 76,564 76,206 75,858
electronic spectra for d l and d9 complexes, using the procedure described by Companion and Komarynsky (81; (2) elementary group t h e o l y c a u he employed to properly label the energy levels and electronic transitions and (3) provided an example of a calculation of the spectrum of a modestly low-symmetry complex ion containing different ligands and the effects of distortion on the spectrum. This procedure is readily extended to complexes possessing even lower symmetry and more different ligands. Computer Program
Calculations performed by hand are tedious and subject to a numher of simple mathematical errors. It is recommended that students work out the general formulas for the matrix elements for some complex or complexes possessing fairly high symmetry. However, a Fortran computer program has been written which can he used to check hand calculations as well as perform a large number of calculations in a short amount of time. The input for the program requires only 04, 0, 4 and the a z / w ratio for the ligands. Up to eight ligands are accomodated and the program can he readily expanded to handle any number of ligands. The program diagonalizes the secular determinant and prints out a label for the problem, the input data, the D and G parameters, matrix elements, secular determinant, the eigenvalues (one-electron energies) and the eigenvectors ("mixing" coefficients of the d-orbitals). The program is readily handled by relatively small computers such as IBM models 1130 and 1620 and can be expanded to perform a number of related calculations such as to calculate transition energies and take ratios of transition energies, etc. Literature Cited
(5) Carlin. R.L.,J. C H E M EOUC..40,135(19631. (6) Gray, H . B . J . CHEM.EDUC..dl,2(19641. (7) Bu~ch,D.H.,J.CHEM.EDUC.,41,7ll1964L rRi Comnanion. A. L.. and Kornarvnskv. M . A,. J. C H E M EDUC. 44.257 I19641 (9) cotton, F. A :, J.CHEM. ~ ~ ~ E . , 4 i ; 4 6 6 ( 1 9 6 4 1 . (10) Zuckerrnan, J. J., J. CHEM.EDUC.. 42,315(1965). (111 Kettle,S.F.A.,J.CHEM.EDUC..4?,21119961:46.339~19691. (121 Dunne,T.G., J . C H E M EDUC.,44,101 (1967). (13) n a p p , C . . s n d Johnmn,R.. J. CHEM.EDUC.,44.527 119611. (I41 Lpver,A.B.P.. J. CHEM. EDUC.. 45.71 11968). (151 Kilner, M.,endSrnith. J:M., J.CHEM. E D U C , 45,94(1968). 1161 Krirhnarnunhy, R., and Schaap. W. B.. J. CHEM. EOUC., 46. 799 11969): 47. 433 11970). (171 Brurnfitf,G., J.CHEM. EOUC., 46,250(19691. (131 Wasson. J. R..J. CHEM.EDUC.,47,371119701. (191 Grifflth. J. S... ,'Theow of Trsnsifion~MetaIIons." Carnhridee Univenitv Press. London, 1961. (MI . . Ballhausm. C. J.. "Intmduefion U, Lieand Field Theory," MeGraw-Hill Book Co., h e . , NeuYork. 1962. (21) watanabe, H., "O~eraforMethods in Ligand Field Theory." Prentice-Hall, Inc., Englewood Cliffs; Now Jeney. 1966. ,221 Sehlafer, H. L., and Gliemsn", G.. "Basic F'rinciplea of Ligsnd Field Theory." Wiley-lnfeneience. New York. 1969.
.~.
~~~~~p~~
.
I231 Sugano S., Tanabe, Y., snd Kamimura. H., "Multiplets of Transition-Metal Ions in Crystals," AesdemicPnss. New Yark. 1910. I241 Jogenson, C. K., Struefure and Bonding, 1, 3 (1966): aec also: Figgis, B. N., "Introduction ta Ligand Fields." Intencienc. Publishen. N- York. 1%6, p p 2425. (25) Spes, S. T., Perumanddi. J. R.. and Adsmson. A. W.. J Phya Chem.. 72, 1822 (1968); J. Amer Chrm Soc.. 90, 6626 119661: Companion. A. L., J. Phyr. Chrm., 73, 739 (1969): Donini, J. C., Hollebone, B. R., and Lever, A. B. P., J. Amm Chem Sac. 93. 6455 119711: Bellito, C.. Tomlinson, A. A. G., and Furlad, C., J, Chem. Soe.. lA1,3267(1971). I261 Orchin, 0.. and Jsffe, H. H., J. CHEM. EDUC., 47. 246, 372 and 510 (1970) snd references therein. I271 Jaffe. H. H., and Orchin. M., "Symmetry in Chemistxy." John Wlley B sons, he., New York, 1965. Sehonland, D.. "Moleculsr Symmetry." D. Van Nostrand Co., Ltd.. New York. 1965. Hall, L. H.. "GmupTheoryand Symmetry in Chem~
istry," MeGrsw Hill Book Co.. New York, 1969. h e c h . J. W., and N e m s n . D. Use Groups." Methuen and Co., Ltd.. Iandar, 1169. Cotton, F. A., "Chemical Applicatians of Gmup Theory." Wiley-Inteneienee. New York. 1911. (2nd d l . Nusshaum. A.. "Applied Gmup Theory for Chomisfs, Phyaicisfs and Engmeen," Prentice-Hall, Inc.. Engleruwd Cliffs. New Jeney, 1971. Faekler, J. P.. Jr.. "Symmetry in Coordination Chemistry," Academic h a s . New York, 1911. Orehin. M., and Jaffe. H. H.. "Symmetry. Orbi~llaand Spectra." WileyIntencience. Now Yark. 1971. I281 Selhin, J., Chem. Revs., 65.153l1965); Caard. Chem. Rsus.. 1,293(1966). (281 Atovmyan, L. O.,andAliev, Z.G..Zh. Srrukr. Khim., ll,782119701. I301 Kilty, P.A.,andNicholls,D..J Chem. Soc., 4915(1965). 1311 Wasson, J . R..and Sroklasa, H. J., vnpublishedxasults. I321 Ballhausen. C.J., andGray, H.B.,lnorg. Chem.. 1.111 (1962). (331 Kon. H., andsharpless, N. E., J Phys Chem., 70,105 11966). J.. "How to
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