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The Nephelauxetic Effect — Calculation and Accuracy of the Interelectronic Repulsion Parameters II

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The Nephelauxetic Effect — Calculation and Accuracy of the Interelectronic Repulsion Parameters II BAND 27 b ZEITSCHRIFT FÜR NATURFORSCHUNG HEFT 1 The Nephelauxetic Effect — Calculation and Accuracy of the Interelectronic Repulsion Parameters II. Application to d3 and d7 Single Crystal Spectra at Cryogenic Temperatures * E. KÖNIG Institut für Physikalische Chemie II, Universität Erlangen-Nürnberg, 8520 Erlangen, Germany (Z. Naturforsch. 27 b, 1—5 [1972] ; received September 28, 1971) Expressions are reviewed which may be used to determine 10 Dq and B from the spin-allowed bands in the optical spectra of d3 and d7 electron systems within octahedral and tetrahedral sym metry. Application to low-temperature single crystal spectra demonstrates that (i) the semi-empiri- cal ligand field theory reproduces transition energies with sufficient accuracy; (ii) differences in the values of 10 Dq and B observed with different fitting methods may be attributed to the in- accuracy of experimental data; (iii) there are generally valid values of B35 and /?33 for each com- plex ion. The semi-empirical ligand field theory provides those complex ions are considered where all three means to completely determine the electronic d — d spin-allowed d — d bands are observed. Room tem- spectra of transition metal ions of octahedral sym- perature solution and single crystal spectra of al- metry in terms of three parameters: the octahedral most fifty complexes and impurity ions of the splitting parameter 10 Dq ( = A) and the inter- transition metals were subject of the analysis. The electronic repulsion parameters (= Racah para- results 4 may be summarized as follows: meters) B and C which are linear combinations of (i) The accuracy of B and 10 Dq depends on the the Condon-Short ley parameters F2 and 1. method adopted to their calculation. Conse- An analogous statement applies to tetrahedral sym- quently, certain methods may be selected which metry and one or two additional parameters (e. g. provide the "best" possible fit to the experi- Ds and Dt in tetragonal environment) are required mental data; if the symmetry is lower than cubic. In general, the (ii) Given a specific method of calculation, an un- numerical values of the parameters B and C, as systematic variation in the deviations between determined from d — d spectra, are lower than the calculated and observed transition energies is values in a free transition metal ion. This observa- often encountered. It was suggested that this tion is well known as the nephelauxetic effect 2' 3. is due to insufficient accuracy of the experi- In the first part of this study 4, the author has re- mental room temperature (solution) data. cently reviewed and tested methods which may be In the present contribution, the same methods as used to determine 10 Dq, B, and C from electronic used previously4 will be applied to single crystal spectra. To focus the attention on the value of B, spectra measured at cryogenic temperatures. It will these methods were applied to the spin-allowed d — d be demonstrated that results somewhat different bands in high-spin d2, d3, d7, and d8 complexes of from those of room temperature spectra are ob- octahedral and tetrahedral microsymmetry. In the tained. expressions of the corresponding transition energies, the parameter C does not occur5. In addition, I. Ligand Field Theory of d3 and d7 Ions 10 Dq may always be fixed by a suitable choice of in Cubic Fields the calculation method. A convenient check on the accuracy of the employed numerical procedure is The general treatment of ligand field theory is provided by calculating the extra band energy, if adequately covered in several textbooks 7-9 to which Requests for reprints should be sent to Doz. Dr. E. KÖNIG, * For the first part of this study refer to E. KÖNIG, Struct. Institut für Phys. Chem. II der Universität Erlangen-Nürn- Bonding 9, 175 [1971]. berg, D-8520 Erlangen, Fahrstr. 17. G ZT-I'H)^ Dieses Werk wurde im Jahr 2013 vom Verlag Zeitschrift für Naturforschung This work has been digitalized and published in 2013 by Verlag Zeitschrift in Zusammenarbeit mit der Max-Planck-Gesellschaft zur Förderung der für Naturforschung in cooperation with the Max Planck Society for the Wissenschaften e.V. digitalisiert und unter folgender Lizenz veröffentlicht: Advancement of Science under a Creative Commons Attribution Creative Commons Namensnennung 4.0 Lizenz. 4.0 International License. E. KÖNIG reference is made here. With respect to the single (b) fitting the first and third band, crystal data available at present, we will concentrate 10 Dq = 2 ?>! — j>3 + 15 B, (8) primarily on the theory of d3 and d7 ions in cubic fields. The relevant energy expressions have been B = jo[-{2Vl~ ± ^ ~ + *»* + "1 >'3>,/!] , derived previously4. For convenience, we will (c) fitting the second and third band, briefly introduce those quantities and list explicitly those expressions which will be needed in the sub- 10 Dq= I (2 v2 — v3) +55, sequent numerical calculations. 3 ß= 2 Thus in the octahedral d configuration three 510 2) ±3{81rs -16^(r2-ra)}''•]. 4 spin-allowed transitions from the A2g ground state (9) to the excited states 4T2g, a 4Tlg, and b 4Tig are (d) fitting the difference between the first and expected. Within the approximation considered second band, here, the energy of the lowest transition is always 10 Z)g = — J'i, determined as vx (4A2g —> 4T2g) = 10 Dq. The ener- gies of the two higher transitions follow from B=(v2 + V3-3V1)/15. (10) >'2,3= 2 (15B + 3O0<7)+-2 t(15ß-10^)2 II. Application to Single Crystal Spectra + 12 B • 10 Dq]1'1. (1) In order to asses the accuracy of the parameter The parameter B may then be obtained according values of 10 Dq and B, the equations listed in sec- to four different methods: tion I will be applied below to some recent low tem- (a) fitting the second band, perature single crystal spectra. Following the first 2 2 5 = (2v1 + v2 -3vlv2)/ (15 v2 — 21 vt), (2) part of this study 4 it will be assumed, for the sake (b) fitting the third band, of argument, that the three-parameter (10 Dq, B, C) B = (2 Vi2 + f32 - 3 n r3) / (15 v3 - 27 vt), (3) theory is valid exactly. The question then arises about the significance of the calculated transition (c) fitting the sum of the second and third band, energies. In ligand field theory, all energy dif- B=(v2 + r3-3v1)/15, (4) ferences are calculated at a constant value of 10 Dq (d) fitting the difference between the second and (viz. "vertical" transitions in the Tanabe-Su- third band, g a no diagram). Since the relation 10 Dq ~ R~5 holds to a reasonable approximation 10, this is equi- B = ^ [3 f! ± (25 0>3 - v2)2 -16 V)I/2] • (5) valent to a fixed metal-ligand distance, R. In the spin-allowed d — d transitions considered here, the The expressions (1) to (5) apply to tetrahedral d7 states involved originate in different strong field ions as well. configurations g eg and, consequently, the potential In the octahedral d7 configuration, the ground minima of the excited state and the ground state do state is a4Tlg and the excited quartest states are, in not coincide. The calculated transition energy cor- the order of increasing energy, 4T2g, 4A2g, and responds, therefore, to the energy of a transition b 4Tlg . The energy of the three spin-allowed transi- from the zero-point vibrational level of the elec- tions is determined according to tronic ground state to an excited vibrational level of the excited state (cf. "vertical" transition ac- "i (a4Tig 4T2g) = I (10 Dq -15 B) + , cording to the Franck-Condon principle). v2 (a4Tig -> 4A2g) = vx +10 Dq , (6) As far as the comparison between theoretical and r3(a4Tlg-^b4Tig) = [(10Z)q + 15 5)2 experimental energies is concerned, two limiting — 12 5-10 Dq]1/!. conditions may be distinguished: There are again four different methods which may (i) In centrosymmetric (e.g. octahedral) com- be employed to obtain the parameters 10 Dq and B: plexes, all d — d transitions are rigorously forbidden (a) fitting the first and second band, on the basis of parity. The forbidden electronic 10 Dq — v2 — vl, transitions may gain intensity through coupling n B= (2v12-v1v2)(Uv2-27 vt), (7) to odd vibrations (vibronic mechanism ). At low THE NEPHELAUXETIC EFFECT 3 temperatures, each band thus consists of a progres- III. Results and Discussion sion in one or more even vibrations superimposed upon one quantum of the odd ("permitting") vibra- Results of the present analysis are compiled in tion. The no-phonon (0" —> 0') band is absent or Tables 1 to 3. For each compound, experimental of very weak intensity 12. Therefore, within reason- transition energies determined according to sec- able approximation, the calculated vertical transition tion II are listed in line 1. Subsequent lines contain energy should be associated with the maximum of the calculated transition energies, their deviation the vibronic band determined, in principle, at 0 °K. from the corresponding experimental value, dv — (ii) In non-centrosymmetric (e. g. tetrahedral) ''calc (in cm 1 and in percent), and the values complexes, the d — d transitions become partly al- of the parameters B3- and ßS5 . In Table 2, values lowed on account of mixing with odd-parity states of 10 Dq and of the deviation, (5(10Z)g), from the of the central ion (e.g. p states11). One observes, value of v2 — vt are listed in addition. Each line ap- at low temperatures, a progression in the totally plies to a different method marked with reference symmetric vibrational mode originating in the no- to section I. phonon (0"—>-0') band.
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