## INTENSITIES OF THE OPTICAL ABSORPTION SPECTRA OF OCTAHEDRAL COMPLEXES OF THE TRANSITION METAL IONS

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Ohio State University##### Abstract:

The origin of the optical absorption spectra of the octahedral complexes of the transition metal ions is not yet completely understood. It is well known that in the absence of symmetry-destroying vibronic (vibrational-electronic) perturbations, electric dipole transitions between the various $d^{n}$ electronic states of octahedrally coordinated inorganic complexes are rigorously forbidden $(g-| \rightarrow g).^{1}$ Thus to compute the intensities of these electric dipole transitions, it is necessary to consider the mixing of $d^{n}$ configurations, $\Psi^{0}_{g}(d)$, with $d^{n-1}p$ configurations, $\Psi^{0}_{u}(p), (n=1,2,3,4,5,6,7,8,9)$, under vibrational $perturbations.^{1}$ Previous calculations of this mixing have confined themselves to $Ti(III), V(III), Ni(II)$, and Cu(II) complexes and to the use of hydrogenic wave $functions.^{1}$ We have now extended these results in several directions. The algebraic structure of the oscillator strengths of the electric dipole transitions of $kd^{n}, (k=3,4,5), (n=1,2,3,4,5,6,7,8,9)$, complexes has been derived, and formal correspondences amongst these strengths deduced. The vibronic coupling constants which occur in the algebra are ultimately related with the integrals $G^{k}_{l}$ and $B^{k}_{l}$ defined $by^{2}$ $\begin{array}{l} G^{k}_{l} = \int^{\infty}_{0} R_{kd}(r) \frac{r^{l}_{<}}{r^{l+1}_{>}} R_{(k+1)p} (r) r^{2} dr\\ B^{k}_{l} = \frac{d}{dr_{0}}G^{k}_{l}\end{array}$ where $r_{0}$ is the metal-ligand bond distance. We have computed these integrals using (a) hydrogenic type wave functions with a variable effective nuclear charge, $Z_{eff, r}$ (b) Slater type wave functions with a Slater effective nuclear charge, $Z_{eff, r}$, and (c) Richardson et. al. type wave $functions.^{2}$ The oscillator strengths obtained by approximations (a) and (b) have been plotted as a function of the effective nuclear charge, $Z_{eff}$, and compared with those of approximation (c), and intensities have been reckoned at both $300^{\circ}K$ and $0^{\circ}K$. In addition we have also calculated the magnetic dipole and electric quadrupole transitions for the same complexes. For the electric dipole and quadrupole transitions we have evaluated the requisite matrix elements $<d\bigg|\overrightarrow{r}\bigg|p>, <d\bigg|\overrightarrow{\nabla}\bigg|p>, <d|r^{2}|d>$, and $<d\bigg|\overrightarrow{r \cdot} \overrightarrow{\nabla}\bigg|d>$, in the same three approximations, (a), (b), and (c) above. The relative and absolute goodness of the approximations is discussed and a plea for future caution entered.

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$^{1}$ A. D. Liehr and C. J. Ballhausen, Phys. Rev. 106, 1161 (1957); C. J. Ballhausen and A. D. Liehr, Mol. Phys. 2, 123 (1959). $^{2}$ A. D. Liehr, Adv. Chem. Phys. 5, 241 (1963). $^{3}$ J. W. Richardson, W. C. Nieuwpoort, R. R. Powell, and W. F. Edgell, J. Chem. Phys. 36, 1057 (1962); J. W. Richardson, R. R. Powell, and W. C. Nieuwpoort, ibid., 38, 796 (1963).

Author Institution: Mellon Institute

Author Institution: Mellon Institute

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