Derivation of Molecular Spectra from the Polarized Spectra of Monoclinic Crystals and its Application to the Electronic Spectrum of Bis(methoxyacetato)diaquacopper(n) BY MICHAELA. HITCHMAN Chemistry Department, University of Tasmania, G.P.O. Box 252C, Hobart, Tasmania 7001, Australia Received 5th May, 1975 A method of determining the molecular absorption tensor of molecules having a single symmetry axis (pointgroups Cs,Cz and Czh)and crystallizing in a monoclinic space group is described. The method is applied to the electronic spectrum of bis(methoxyacetato)diaquacopper(u) and shows that the " d-d " electronic transitions are polarized approximately midway between the bond direc- A tions and the bisectors of OCuO bond angles, probably closer to the bond bisectors, with the most intense absorption occurring when the electric vector lies within theche late ring.A possible explanation for this is considered. The method is also used to derive the g-tensor of the molecule and the non-coincidence of the principal in-plane g-axes and absorption directions is discussed. There has been considerable interest 1-5 in the electronic structure of metal com- plexes having only a single axis of symmetry (point group C,,C2or C2h). In favour- able cases the ground states of such molecules may be studied by e.p.r. spectroscopy and considerable progress has been made in the theoretical interpretation of the g-tensors of molecules of this type.1-3 The intensity of absorption of light is also a tensor quantity, and the principles involved in the deduction of the molecular absorp- tion ellipsoid from single crystal measurements are similar to those encountered in the evaluation of the molecular g-tensor.Despite the fact that an increasing number of complexes of low symmetry are being studied by polarized crystal spectr~scopy,~-~ no attempt has as yet been made to derive the absorption ellipsoid in its complete form, though this should greatly help the elucidation both of the nature of the excited states, and of the intensity inducing mechanisms operating in complexes of this kind. This paper describes a method for deducing this information. DISCUSSION The complete specification of a tensor of a molecule having one symmetry axis z requires the measurement of 4 tensor elements, these being used to derive the 3 principal values and the orientation of the in-plane axes with respect to arbitrarily chosen x and y axes.The conditions limiting the experimental determination of these tensor elements have been discussed in detail for the'derivation of molecular g-tensors from crystals having exchange-narrowed e.p.r. signal^,^ and exactly similar arguments apply to the interpretation of the optical spectra of crystals. First, the crystal sym- metry must be monoclinic or triclinic; the subsequent discussion will be limited to electric dipole transitions in monoclinic crystals. Secondly, two measurements must be made on each of two crystal planes, and at least one of these planes must 54 M.A. HITCHMAN not contain the unique crystal axis. In optical spectroscopy the measurements must be made with the electric vector parallel to the two extinction directions of each crystal face. If this condition is not satisfied, rotation of the vector occurs as the light passes through the crystal and the knowledge of the precise orientation of the vector in the crystal coordinates is lost. Normally, the absorption is measured as a function of wavelength, thus yielding a spectrum. The measurement of the wave- length dependence of the absorption is complicated by the fact that except when it lies along the unique crystal axis, the extinction direction of a crystal face can in principle vary as a function of wavelength. However, while the possibility of such changes should always be investigated experimentally, in practice, for metal com- plexes they are usually found to be too small to be detected.To define the molecular absorption ellipsoid, a molecular coordinate system must be defined. For molecules of C,,C2 or C,, symmetry, z coincides with the unique axis of the point group, while x and y can be chosen as any orthogonal directions perpendicular to z. The first step in the derivation of the molecular spectra is the calculation, for each crystal spectrum, of the projections made by the electric vector upon the molecular axes of each of the two independently oriented molecules in a monoclinic space group, and qualitative discussions of molecular spectra have often been made by correlating changes in the relative intensities of absorption peaks with changes in these projec- tions.If it is assumed that the principal absorption directions coincide with the chosen molecular axes (i.e., if the off-diagonal elements of the absorption tensor are set equal to 0) then the calculation of the molecular spectra is straightforward and this process has been carried out for several c~mplexes.~-~~ It has been suggested lo that this calculation is best performed by comparing the changes in the absorbance with polarization direction for each crystal face [see eqn (4) and (5) of the Appendix], as this then excludes errors due to the measurements of crystal thickness and mini- mizes those due to variations in optical quality between crystal faces.While this method is adequate for molecules in which all of the molecular axes are defined by symmetry, e.g., point groups DZhor C,, (and, incidentally, is also applicable to orthorhombic crystal systems) it requires extension for the lower symmetry molecules under consideration here. The additional determination of the principal directions of the absorption tensor in the xy molecular plane is accomplished by the measurement of the appropriate off-diagonal tensor element, followed by diagonalization of the tensor. In order to deduce the four molecular tensor elements, use must be made of the four parameters defining the crystal absorption ellipsoid. The method described here therefore makes use not only of the absorption magnitudes, but also of the direction of maximum and minimum absorption in the crystal face not containing the unique axis.The mathe- matical procedure by which this is done is given in the Appendix. Note that the method assumes that the extinction directions coincide with the directions of maxi- mum and minimum absorption of light. In practice, this will almost always be the case in studies of spin-allowed "d-d" spectra, as the refractive index of the crystal (upon which the extinction directions depend) will be dominated by its imaginary component, which is proportional to the intensity of light absorption. However, the extinction directions and directions of maximum and minimum absorption are not formally required to be coincident,13 and this feature should always be checked experimentally.Particular care should be exercised in this respect when the ligands contain aromatic ring systems, as the neighbouring intense internal ligand transitions could affect the extinction directions. When experimental measurements can be made on three crystal faces, all four tensor elements can be obtained without making use of this directional information (see Appendix). SPECTRA OF Cu (RIIeOAc),(H,B), APPLICATION OF THE METHOD TO THE OPTICAL SPECTRUM OF B I s(MET HOXY A cE T AT 0)DI AQUAcoPPER (I I) As a typical application of the method we consider the optical spectrum of the complex bis(methoxyacetato)diaquacopper(II), [Cu(MeOAc),(H,O),]. This centro- symmetric complex has approximate C,, symmetry (fig.1) and crystallizes in the monoclinic space group P21/nwith unit cell parameters a = 692, b = 726, c = 1010 pm, y = 96.7".14 However, in the subsequent discussion the b and c axes will be interchanged to comply with the more usual convention defining b as the unique axis and p as the monoclinic angle. To deduce the molecular spectrum we define a molecu- lar coordinate system as follows : z lies along the Cu-O(H,O) direction, y is ortho- gonal to z and the Cu--O(acetate) direction, and x is orthogonal to y and z (i.e., x is almost exactly along the Cu-O(acetate) direction (fig. 1). The spectra of the (OlO), (001) and (100) crystal faces have been reported by Bew et a1.l' The extinction I 6 FIG.1.-The inolecular geometry of bis(inethoxyacetato)diaquacopper(rr) ;the hydrogen atoms have been omitted for the sake of clarity.The molecular coordinate system x,y and z, and directions of the in-plane g axes and electronic absorption axes are illustrated. At the band maximum EX makes an angle of 27+ 7" with the x axis, while the angle made by gx is -8f4". directions in the (010) face happen to lie along the directions of the a and c* axes and were found to be coincident with the directions of maximum and minimum absorption.16 The spectra measured with the electric vector along the a, b and c axes are shown in fig. 2 [the a spectrum is conimon to the (010) and (001) faces]. The principal molecular absorbances E~,cy and cZ (given in arbitrary units) calculated by the method given in the Appendix are also shown in fig.2, while the orientation of the principal absorbance axes x and y (principal tensor parameters will be indicated by bold type) at the band maximum is illustrated in fig. 1. DISCUSSION OF THE ERRORS INHERENT IN THE RESOLUTION OF MOLECULAR SPECTRA The errors occurring in the derivation of a molecular absorption spectrum are derived basically frorn two sources. First, there is the experimental error in the M. A. HITCHMAN measurements (absorbance, angle etc.) ; this is generally easily estimated. Secondly, there is an additional uncertainty due to the transformation from crystal to molecular coordinates, followed by the averaging over the two independently oriented molecules in the unit cell and solution of the set of simultaneous equations to obtain the absorp- tion tensor elements. It is this latter source of error, which has usually been ignored in the past, which can sometimes cause large uncertainties in the molecular spectra. l7 A major contributing factor here is the relative orientation of the two independent molecules in the unit cell, as this decides the degree of independence of the set of equations which must be solved to obtain the absorption tensor elements.The true accuracy of the molecular spectra can be estimated by calculating these for the widest possible extremes of experimental data, and this was done in the case of Cu(MeOAc),-(H20)2assuming a possible error of & 0.03 in all absorbance values and & 3" for the extinction direction in the ac crystal plane.The resulting uncertainty was k0.05, I L I I 1 15 10 energy/103 cm-l FIG,2.-The crystal and molecular spectra of bis(methoxyacetato)diaquacopper(Ir). The crystal spectra are given for the electric vector parallel to the a,b and c crystal axes. See text for the defini- tion of b and c, and the method of calculation of the x, y and z spectra. 0.04 and 0.06 absorbance units in cx, E~ and cZ, respectively, at the absorbance maxi- mum (these values were somewhat lower in the less intense regions of the spectrum). The uncertainty in the angle a between the principal absorption directions and the molecular axes was & 7" at the absorbance maximum, increasing to & 10" on either side of this (9 and 16 x lo3 cm-I). As expected, the uncertainty in the molecular spectra is somewhat greater than that in the crystal spectra, though the effect is not very pronounced, indicating that in this particular case the relative orientation of the molecules does not seriously jeopardize the resolution of the molecular spectra.It is noteworthy that the z spectrum shows a slight negative absorbance in the region 15 to 18 x lo3 cin-l suggesting the existence of some experimental error not taken into account in the present treatment. SPECTRA OF Cu (M~OAC)~(H~O)~ APPLICATION OF THE METHOD TO THE DETERMINATION OF MOLECULAR g-TENSORS The method described here for the determination of the electronic absorption ellipsoid is equally applicable to the deduction of the g-tensors of molecules of C2,or similar symmetry (or indeed to other tensors, such as molecular magnetic susceptibility).The appropriate values of g2 and q need only be substituted for the absorbance values in eqn (4) and (6) of the Appendix. The principal crystal g-values g1 = 2.088 5, g2 = 2.166 2, g3 = 2.344 5 have been reported l5 for Cu(Me0Ac) with g2 lying along the b crystal axis and g1 making an angle of y = -38" with the a crystal axis. Allowing an uncertainty of f0.03 in the values of g2, and 53" in y gives principal molecular g-values of g, = 2.036 k0.015, g, = 2.348 +0.008 and g, = 2.212+0.011, with g, making an angle of a = -8k4" with the molecular x axis. These compare favourably with the values of g, = 2.028 k0.004, g, = 2.368 +0.001, g, = 2.223&0.001, a = 0.6" found by Dawson et aZ.,3 and g, = 2.026 6, g, = 2.344 7, g, = 2.224 1 obtained by Bew et a2.l with g, defined along the Cu-0 (carboxylate) direction.INTERPRETATION OF RESULTS The molecular spectrum of Cu(MeOAc),(H,O), shows three peaks at -9.5, 12.25 and -16.25 x lo3cm-l (fig. 2). Each of these is considerably more intense in x, than y or z polarization, and the peak at -16.25x lo3cm-1 is absent from the z spectrum. The angle a defining the direction of in-plane polarization of the transi- tions is 26+ 10" at 9.5 x lo3cm-l, 27+7" at 12.25 x lo3cm-1 and 16+ 10" at 16.25 x lo3 cm-l, a positive sign indicating rotation from the x axis into the chelate ring. An angle of 40" would occur if the transitions were polarized along the bond angle bisectors, while one of -5" would be expected if the transitions were polarized most nearly along the bond directions.It would seem that the electronic transitions are polarized approximately midway between these extreme situations, though, particu- larly for the more accurately defined intense transition at 12.25 x lo3cm-l, the A polarization directions more nearly correspond to the bisectors of the OCuO bond angles.For a centrosymmetric complex such as Cu(MeOAc),(H,O), the "d-d" transi-tions occur by a vibronic intensity mechanism. In the C2,point-group vibrational modes are available to allow each "d-d" transition in every polarization, so that formal selection rules cannot be used to assign the observed spectrum.Note, how- ever, that one transition is absent in one polarization, that at -16.25 x lo3cm-l in y. If the eflectiue vibronic pointgroup were D2, with x, y and z defined as pre- viously (i,e., approximately along the bond directions), then the transition dxy-+ ~Z ~Z-would be forbidden in y polarization. This lends some support to the hypothe- sis that the peak at -16.25x lo3cm-l is due to a transition between the orbitals containing these functions (both dx2-y2 and dxybelong to A, in the C,, pointgroup). What seems fairly certain is that as quite strong bonds occur along each of the direc- tions x, y and z, the lowest energy peak at -9.25 x lo3cm-l is due to a transition between the level derived from d3z2-rz, and the ground state orbital.This would suggest that the most intense peak at 12.25 x lo3cm-l is due to one or both of the transitions from the two levels derived from the dxyand dyzorbitals. It should be stressed, however, that the assignment of the two higher energy peaks is tentative. In the earlier study of the optical spectrum of Cu(MeOAc),(H,O), it was assumed that the directions of polarization of the optical transitions were coincident with the M. A. HITCHMAN principal g-axes. However, the full analysis of the electronic spectrum presented here shows that the in-plane polarization directions are in fact quite different from the directions of the g-axes (fig. 1) as the latter are quite close to the bond directions. This clearly demonstrates the value of determining the direction of in-plane polariza- tion in the analysis of the spectra of complexes of this kind, and also shows the fallibility of the assumption which has sometimes been made 6* that the polariza- 59 tion directions of the optical spectra must coincide with the principal g-axes.As has been stressed el~ewhere,~. 19* 2o the g-tensor and electronic absorption tensor are derived from quite different sources, and in fact in a low symmetry complex of this type, the difference in the orientation of these tensors can provide useful information on the electronic structure of the compound. In a comparatively ionic complex such as Cu(MeOAc),(H,O), the g-tensor is dominated by the metal parts of the molecular orbital containing the unpaired electron.The orientation of the g-tensor therefore essentially reflects the effect of the ligand field upon the d-orbitals, and the very large difference between g, and g, and the close coincidence of the g axes with the Cu-0 (carboxylate) and Cu-0 (methoxy) bond directions results from the fact that the oxygen atom of the carboxylate group produces a much greater o-perturbation than that of the methoxy group.3 As shown previously,lg 3* l9 a large difference in 6-bonding power produces significant admixture of the dX2-,,z and d3z2-rZorbitals, and this tends to dominate the in-plane g anisotropy. The polarization directions of the '' d-d " electronic transitions, however, are dominated by the comparatively small ligand components of these molecular orbitals, so that it is the electronic structure of the ligands rather than simply that of the coordination polyhedron around the copper ion which is important in deciding the intensities and polarization properties of the electronic transitions.While the complicated electronic structure of the molecule precludes a detailed interpretation of the absorption intensities in Cu(MeOAc)(H,O), with the data presently available, it is noteworthy that many planar complexes of DZhsymmetry of the general form Cu(acetylacetonate), exhibit strong in-plane polarization of the electronic transitions, the absorption being most intense when the electric vector bisects the chelate ring.g* lo*21 There is also some evidence 22 that this is the case in complexes of the type trans-bis(N-alkylsalicyla1diminato)copper(II), which have a rather similar microsymmetry to Cu(MeOAc),(H,O),.In all of these complexes, as in Cu(MeOAc),(H,O),, the most intense spectrum occurs when the electric vector is directed approximately towards the centre of the chelate ring. The polarization properties of the "d-d " transitions in these complexes have been related to the symmetry of neighbouring charge transfer states in which an electron is trans- ferred from an essentially non-bonding set of" lone pair '' ligand orbitals to the half- filled orbital centred largely on the copper lop 23 The carboxylate and methoxy- oxygen atoms in Cu(MeOAc),(H,O), also contain "lone pair " orbitals lying approximately in the plane of the chelate ring and the fact that the polarization direc- tions of the electronic transitions lie more nearly along the bisectors of the bonds than along the bonds themselves, and the relative intensities of the x and y spectra, suggest that possibly a similar intensity mechanism may be making a significant contribution in this complex also.This is interesting, as Cu(MeOAc),(H,O), differs from the acetylacetonato complexes in the fact that the low symmetry of the ligand field produces a ground state wavefunction with lobes which differ in extent along x and y, the former being much larger than the latter.3 The intensity mechanisms operating in transition metal compounds are by no means well understood and it would seem that the investigation of the polarization directions of the electronic transitions in low symmetry complexes should provide useful information in this area.For instance, it would be interesting to know the in-plane polarization direc- SPECTRA OF Cu (McOAc),(H20), tions of a planar complex of C2h symmetry in which only one of the donor atoms of each chelating ligand has a ‘‘ lone pair ” of electrons available to take part in a charge transfer transition (e.g., a trans amino acid complex). APPENDIX We consider the absorption occurring when polarized light passes through a crystal. Let the orientation of the electric vector in the crystal be dehed by three angles y, 8 and 4 (fig. 3). Here 8 and 4 define the crystal face perpendicular to the light path, and y defines b a ii c FIG.3.-Definition of the angles 7,8and 4 used to define the position of the eIectric vector E in the crystal coordinates a, b, and c.the position of the electric vector E in the face. The projections of E on the crystal coor- dinates a, b and c are given by :24 Eu = Eccos /?* (1 -E,” sin2 p* -E,”)* (14 (the sign is decided by which of the quadrants defined by the (100) and (001) crystal planes E occupies; the positive sign is taken when E is in a quadrant containing the positive a axis) : Eb = sin y sin 8 (W E, = (sin tj cos 8-cos y tan 4) cos 4/sin p*. (14 These projections are converted to a set of orthogonal crystal coordinates a’,b’, c’ by the relationships : E,. L= E, sin /?*; Ebr = Eb; E,.= Ec-Ea COS /?* (2) The projections Exl,Ex, etc. made by the electric vector upon the two independent molecules (labelled 1 and 2) in the monoclinic cell are : Eyl = .yu’ Yb’ Yc’ ; E~2 -xb’xb’ 42 = Ya’ -Yb‘Yc’ -&] (3) 2,‘Ez1 [”‘zb’ 2,’ [&I EZ, -2,’ -2b’ Z,J [E~J Here the orthogonal molecular coordinate system is defined in accordance with the point- group C,,C2or C2h(Le., z is defined by symmetry, x and y are not) and xa*, etc., are the M. A. HITCHMAN projections niade by the molecular coordinate vectors upon the orthogonal crystal co-ordinates. The absorbance of light A by the crystal is given by : where gij is the molar extinction coefficient tensor defining the light absorbance of the mole- cule (note &,,,.zyx), c is the molar concentration of the complex and t is the crystal thickness.To measure ~~ithe absorbance must be measured with E along each of the extinction direc- tions of two crystal faces. Let the measured absorbances at a fixed wavelength be A(1) and 42) for the first face and 43) and A(4) for the second face. Substitution into eqn (4), followed by division of the equation containing A(1) by that containing A(2),and that con- taining A(3) by that containing A(4), and some straightforward algebra yields two equations of the form : (41)[Exl(2)2+Ex2(2)21/42) -ado2 -Ex2(1)21&xx+ -* * +2(A(l)[EX1(2)E,l(2>+ Ex2(2)EYZ (2)1/&2) -Ex1(1PY 1(1)-Ex2 (1142(1>IEx, = 0 (54 (A(3)[ExI (4124-Ex2(4)’I/A (4) -Ex1(3)2 -E,2@)2) Ex, -I---* +2 (A(3)[Ex1(4)Ey 1(4) + Ex2 (4)Ey2 (91/A(4) -Ex1(3)EyL(3) -Ex2 (3)Ey 2 (3)} = 0 (5b) Note that by taking the ratios A(l)/A(2)and A(3)/A(4),the thicknesses of the two crystal faces are removed from the equations.Note also, that when E,, = 0 (i.e., when the complex belongs to a pointgroup such as D2hwhere the axes are defined by symmetry elements) these two equations are sufficient to allow the ratios E,, : E~~ etc. to be determined. Let the second crystal face be one which does not contain the unique crystal axis. For this face (defined by 8 = 6’ and 4 = 4) the maximum absorbance A(4) occurs when E is defined by the angle y= ij, i.e., dA/dq = 0 when = #.* Differentiation of eqn (4) yields : From eqn (3) : From eqn (2) : From eqn (1) : d(E,)/dy = cos y sin 8 (9b) d(E,)/dV = [cos + (cos ycos 6 +sin ytan +)]/sinz /?*.(94 * As noted in the text, although there is no formal requirement that the extinction directions should coincide with the directions of maximum and minimum absorption, this should almost always be the case in practice where spin-allowed “d-d” transitions are being studied. If this condition is not met, the four tensor elements can be determined if the spectra of a third crystal face can be measured. The three equations of the form given in eqn (5) can then be solved for the ratios E~~ : Eyy :EZZ :Exy. = SPECTRA OF Cu (MeOAc),(H,O), Substitution of the values ij, 8 and $ into eqn (6), via the relationships given in eqn (7)-(9), plus the two expressions of the form given in eqn (5) yields three simultaneous equa- tions which may be solved to give the ratios of the tensor elements E,, :E,, :gZz :E,,,.Dia-gonalization yields the ratios of the principal tensor elements zx:E, : E= and the angle between the principal x and y axes and the molecular x and y axes. If the thickness of one of the crystal faces is known, substitution into eqn (4) readily gives the absolute values of the prin- cipal tensor elements. To generate the molecular spectrum the process is performed at successive wavelengths, the whole process being conveniently performed by a computer program.t M. A. Hitchman, C. D. Olsen and R. L. Belford, J. Chem. Phys., 1967, 50, 1195. M. A. Hitchman, B. W. Moores and R. L. Belford, Inorg. Chem., 1969, 8,1817.K. Dawson, M. A. Hitchman, C. K. Prout and F. J. C. Rossotti, J.C.S. Dalton, 1972,1509. B. W. Moores and R. L. Belford, Electron Spin Resonance of Metal Complexes, ed. T. F. Yen (Plenum, New York, 1969), p. 17. B. J. Hathaway and P. G. Hodgson, Spectr. Acta, 1974, 30A, 1465. C. W. Reimann, G. F. Kokosyka and H. C. Allen, J. Res. Nut. Bur. Stand. A, 1966,70, 1.’R. J. Dudley, B. J. Hathaway and P. G. Hodgson, J.C.S. Dalton, 1972, 882. J. Ferguson, J. Chem. Phys., 1961,35, 1612. R. L. Belford and J. W. Carmichael, Jr., J. Chem. Phys., 1967, 46,4515. lo M. A. Hitchman and R. L. Belford, Inorg. Chem., 1971, 10,984. l1 P. L. Meredith and R. A. Palmer, Inorg. Chem., 1971,10, 1049. l2 S. G. Lipson and H. Lipson, Optical Physics (Cambridge University Press, 1969), pp.308-311. l3 J. A. Mandarino, Amer. Miner., 1959, 44, 65. l4 C. K. Prout, R. A. Armstrong, J. R. Carruthers, J. G. Forrest, P. Murray-Rust and F. J. C. Rossotti, J. Chem. Soc. A, 1968, 2791. l5 M. J. Bew, D. E. Billing, R. J. Dudley and B. J. Hathaway, J. Chem. SOC.A, 1970, 2640. l6 B. J. Hathaway, personal communication. l7 T. S. Piper and R. L. Belford, Mol. Phys., 1962, 5, 1969. l8 D. E. Billing, R. Dudley, B. J. Hathaway, P. Nicholls and I. M. Procter, J. Chem. SOC.A, 1969, 265 ; B. J. Hathaway, M. J. Bew and D. E. Billing, J. Chem. SOC.A, 1970,1090. l9 M. A. Hitchman, J. Chem. SOC.A, 1970,4. 2o R. J. Dudley, B. J. Hathaway and P. G. Hodgson, J.C.S. Dalton, 1972, 882; R. J. Dudley,B. J. Hathaway, P. G. Hodgson, P. C. Power and D. J. Loose, J.C.S. Dalton, 1974, 1005; B. J. Hathaway and P. G. Hodgson, Spectrochirn. Acta A, 1973, 1465. 21 F. A. Cotton and J. J. Wise, Inorg. Chem., 1967, 6, 917; B. J. Hathaway, D. E. Billing and R. J. Dudley, J. Chem. Soc. A, 1970, 1420. 22 J. Ferguson, J. Chem. Phys., 1961, 35, 1612. 23 M. A. Hitchman, Znorg. Chem., 1974, 13, 2218. 24 M. A. Hitchman and R. L. Belford, Electron Spin Resonance of Metal Complexes, ed. T. F. Yen (Plenum, New York, 1969), p. 97. (PAPER 5/826) j. A copy of the program, written in ALGOL 3 is available on request from the author.