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Paramagnetism of Caesium Titanium Alum Lucjan Dubicki and Mark J Paramagnetism of caesium titanium alum Lucjan Dubicki and Mark J. Riley Citation: The Journal of Chemical Physics 106, 1669 (1997); doi: 10.1063/1.473320 View online: http://dx.doi.org/10.1063/1.473320 View Table of Contents: http://scitation.aip.org/content/aip/journal/jcp/106/5?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Ab initio study of dynamical E × e Jahn-Teller and spin-orbit coupling effects in the transition-metal trifluorides TiF3, CrF3, and NiF3 J. Chem. Phys. 136, 084308 (2012); 10.1063/1.3687001 Multiple lines of conical intersections and nondegenerate ground state in T⊗t 2 Jahn–Teller systems J. Chem. Phys. 112, 8470 (2000); 10.1063/1.481449 Crystal-field splitting, magnetic interaction, and vibronic excitations of 244 Cm 3+ in YPO 4 and LuPO 4 J. Chem. Phys. 109, 6800 (1998); 10.1063/1.477326 Giant second-order Zeeman effect in titanium (III) doped caesium gallium alum J. Chem. Phys. 109, 2967 (1998); 10.1063/1.476885 Paramagnetism of caesium titanium alum and the Jahn–Teller interaction J. Chem. Phys. 107, 8275 (1997); 10.1063/1.475029 Reuse of AIP Publishing content is subject to the terms: https://publishing.aip.org/authors/rights-and-permissions. Downloaded to IP: 130.102.82.2 On: Tue, 18 Oct 2016 06:21:12 Paramagnetism of caesium titanium alum Lucjan Dubicki Research School of Chemistry, Australian National University, Canberra, 0200, Australia Mark J. Riley Department of Chemistry, University of Queensland, St Lucia, 4072, Australia ~Received 31 May 1996; accepted 16 October 1996! • Caesium titanium alum, CsTi~SO4!2 12H2O, is a b alum and exhibits a large trigonal field and a 2 dynamic Jahn–Teller effect. Exact calculations of the linear T2 ^ e Jahn–Teller coupling show that in the strict S6 site symmetry the ground multiplet consists of a Kramers doublet 2G6 with magnetic 21 splitting factors gi51.1 and g'50, a G4G5 doublet at ;60 cm with gi52.51 and g'50.06 and 21 another G4G5 doublet at ;270 cm with gi51.67 and g'51.83. The controversial g values observed below 4.2 K, gi51.25 and g'51.14, are shown to arise from low symmetry distortions. These distortions couple the vibronic levels and induce into the ground state the off-diagonal axial Zeeman interaction that exists between the first excited and the ground vibronic levels. © 1997 American Institute of Physics. @S0021-9606~97!01804-7# I. INTRODUCTION Since for positive y the ground state of the Ti31 ion will always remain 2G with g 50, the early attempts to account Crystals of caesium titanium alum, CsTi~SO ! •12H O 6 ' 4 2 2 for g 51.14 in CsTiS employed a negative value of y. The ~abbreviated as CsTiS!, have the cubic space group Pa3 with ' 2G ground state has g 50 by symmetry, regardless of four equivalent Ti31 ions in the unit cell. The nominally 6 ' whether there is Jahn–Teller coupling operating or not. The octahedral complex ion, Ti~H O!31 , is slightly compressed 2 6 earlier work culminated in the paper by Shing and Walsh7 along the molecular trigonal axis that lies along one of the who proposed a G ^e Jahn–Teller coupling with a very body diagonals of the unit cell. The site symmetry at the Ti31 8 small trigonal field. This model was based on the assumption ion is S and the fractional atomic coordinates of the sulphur 6 that the spin–orbit coupling of the Ti31 ion breaks the atoms identify the CsTiS alum to be of the b modification.1 2T ( 1 ) ^ e coupling into ^e and quenches the The paramagnetism and, in particular, the magnetic g 2 G8 G7 G8 pseudo Jahn–Teller coupling between G8 and the higher ly- values of CsTiS have been an unresolved problem for more 2 than forty years.2–7 Unlike the electron paramagnetic reso- ing G7 , where G8 and G7 are spin–orbit components of T2 in 31 8 the cubic limit. This analysis is not valid because the spin– nance ~EPR! of Ti doped into an a alum such as RbAlS 21 where twelve magnetically inequivalent sites were observed orbit constant, z;120 cm in bound Ti~III!, is comparable or even less than the Jahn–Teller stabilization energy that and correspond to one chemical species with rhombic sym- 31 11,13,14 metry, the single crystal EPR of CsTiS indicated one chemi- has been observed in several Ti complexes. 5 A new insight into the electronic structure of alums has cal species of trigonal symmetry with gi 1.25 and 15 g 51.14.3 The large linewidth of the EPR lines, about 250 been revealed by the theoretical work of Daul and Goursot, ' 16 gauss, and the variation of the magnetic susceptibility at low the electronic Raman measurements of vanadium alums temperatures indicated the presence of low lying paramag- and by more recent x-ray and neutron diffraction structure 17–20 netic states.3 determinations. The work of Best, Forsyth, and Tregenna-Piggott18–20 have been particularly definitive. All If the trigonal field that lifts the degeneracy of the t2 d-orbitals of the Ti31 ions has a positive sign, CsMS alums with metal ions having the electronic configu- ration, (tn) n,6, are b alums that are characterized by y5E(t2x0)2E(t2x6) is greater than zero, then the ground 2 M~III! ions coordinated by planar water molecules. The OH Kramers doublet in S6 symmetry has the symmetry classifi- 2 1 2 planes are all rotated about the M–O bonds by 20° with cation (t2x6) T2(2G6) where we use the complex trigonal ; 9 basis for t2 orbitals and the double group notation of Koster the sense that the oxygen pp lone pairs normal to the OH2 10 31 et al. plane are tilted towards the trigonal z axis of the M~OH2!6 3 1 The 2G6 doublet transforms as the M J56 2 components ion. The structure of CsTiS at 100 K is shown in Fig. 1~a! 3 31 of angular momentum J5 2 and has g'50. Indeed, for Ti displaying the S6 symmetry of the Ti~III! site. The ;20° 21 doped in Al2O3 where y5700 cm the observed g values twist of the planar water molecules is midway between the 11 19 are gi51.07 and g';0.0. The gi value differs appreciably Th and the ‘‘all horizontal’’ D3d symmetries. The angular from the prediction of the static ligand field model that gives overlap model shows that such a rotation generates a large 15 to first order, gi(2G6)5222Kz'0.5, where Kz is the effec- positive trigonal field, y. 1 2 tive orbital reduction factor for the (t2) T2 multiplet. The Consequently, the previous work on CsTiS has focused explanation of this discrepancy as well as accounting for the on the wrong part of the ligand field energy diagram. Since 21 energies of the low-lying vibronic states at 38 cm and 108 the trigonal splitting of the t2 orbitals ~see Sec. IV! is much cm21 was a major success for Ham’s effective Hamiltonian greater than the spin–orbit coupling of Ti31 or the Jahn– 2 11,12 treatment of the T2 ^ e Jahn–Teller problem. Teller coupling, it is the trigonal field which tends to break J. Chem. Phys. 106 (5), 1 February 1997 0021-9606/97/106(5)/1669/7/$10.00 © 1997 American Institute of Physics 1669 Reuse of AIP Publishing content is subject to the terms: https://publishing.aip.org/authors/rights-and-permissions. Downloaded to IP: 130.102.82.2 On: Tue, 18 Oct 2016 06:21:12 1670 L. Dubicki and M. J. Riley: Paramagnetism of caesium titanium alum 2 EJT5A1/\v for T2 ^ e coupling, where \v is the wave num- ber of the Jahn–Teller active e vibration. In the complex trigonal basis the coupling matrix becomes 1 1 1 u6 2 X1& u6 2 X2& u6 2 X0& 1 ^6 2 X1u 0 Q1 2Q2 1 1 ^6 2 X2u 2Q2 0 2Q1 3A 2 A1 ~2! 1 u6 2 X u S Q1 Q2 0 D 0 where Q65(7A1/2)(Qx 6 iQy) and Qx ,Qy are the two components of the degenerate e vibration. This Jahn–Teller coupling matrix was included in the 10310 electronic matrix containing the ligand field and spin–orbit coupling as given by Macfarlane, Wong, and Sturge.11 Each electronic basis function was then expanded in terms of n50,1,2,...,nv lev- els of a two-dimensional harmonic oscillator vibrational ba- sis. Matrix elements of the Jahn–Teller coupling were then evaluated and the resulting large complex matrix diagonal- ized. The total basis size without exploiting the vibronic 1 symmetries was N5103 23(nv11)(nv12). A fast Lanc- zos diagonalization routine for real symmetric matrices was used the find the lowest vibronic energies and wave func- tions. This meant a further doubling of the size of the origi- nal complex matrix. A typical calculation with nv521 re- quired the diagonalization of a 506035060 sparse matrix FIG. 1. ~a! The S6 symmetry of the b alum structure viewed down the with 16 148 nonzero off-diagonal matrix elements. trigonal axis. ~b! A schematic representation of the perturbations acting on In addition to these calculations, we also used an effec- the d electronic levels of a Ti~III! ion. 1 2 1 2 tive (t2) T2 basis corrected for mixing of (e ) E to the sec- ond order.
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