Influence of Covalency upon Rare-Earth Ligand Field Splittings

J. D. Axe; Gerald Burns

doi:10.1103/PhysRev.152.331

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Influence of Covalency upon Rare-Earth Ligand Field Splittings

J. D. Axe and Gerald Burns

IBM Watson Research Center, Yorktown Heights, New York

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Phys. Rev. 152, 331 – Published 2 December, 1966

DOI: https://doi.org/10.1103/PhysRev.152.331

Abstract

Experimental results for the shift with uniaxial stress of the $({}^{2}F_{\frac{5}{2}}, Γ_{7}) \to ({}^{2}F_{\frac{7}{2}}, Γ_{7})$ laser transition in ${Tm}^{2 +}$ :Ca $F_{2}$ and ${Tm}^{2 +}$ :Sr $F_{2}$ are presented. The results, 1.75 ${cm}^{- 1}$ ${(d y n / {cm}^{2})}^{- 1}$ and 1.78 ${cm}^{- 1}$ ${(d y n / {cm}^{2})}^{- 1}$ , are used to calculate the radial dependence of the cubic ligand field splitting. The resulting dependence is somewhat larger than that predicted by the familiar electrostatic model for the splitting. Partially to determine its influence on the above result, we have considered the effect of covalency by means of a semiempirical molecular-orbital model. The overlap of the $4 f$ orbitals with the neighboring fluoride ions was calculated using Hartree-Fock wave functions and known internuclear distances. The off-diagonal elements of the interaction Hamiltonian were obtained from the Wolfsberg-Helmholz approximation $H_{ij} = \frac{2 S_{ij} (H_{ii} + H_{jj})}{2}$ . A range of reasonable values for the diagonal elements were obtained by analogy with those necessary to explain iron-series splittings. The largest group overlap of the $4 f$ wave function with $F^{1 -}$ ligands was found to be 3.6% and leads to a sizable (our best estimate in Ca $F_{2}$ is 50%) covalent contribution to the ligand field splitting. We have also investigated some of the consequences of a covalent contribution of this magnitude. The radial dependence of the covalent part of the energy is greater than for the electrostatic part. The resulting radial dependence is thus in better agreement with experiment. Transferred hyperfine effects are calculated and compared to experiment, but the extent of the agreement is hard to ascertain because of uncertainty of the sign of the experimental quantity and polarization effects. The calculated orbital reduction factor for ${Tm}^{2 +}$ :Ca $F_{2}$ is found to be much smaller than is observed. We have also calculated the expected variation of the ${(rare earth)}^{3 +} - F^{1 -}$ overlap as a function of atomic number.

References (47)

C. A. Hutchison and E. Y. Wong, J. Chem. Phys. 29, 754 (1958) E. Y. Wong and I. Richman, ibid. 36, 1889 (1962)
G. Burns, Phys. Rev. 128, 2121 (1962)
M. T. Hutchings and D. K. Ray, Proc. Roy. Soc. (London) 81, 633 (1963)
C. J. Lenander E. Y. Wong, J. Chem. Phys. 38, 2750 (1963)
D. K. Ray, Proc. Phys. Soc. (London) 82, 47 (1963)
R. E. Watson and A. J. Freeman, Phys. Rev. 133, A1571 (1964)
G. Burns, J. Chem. Phys. 42, 377 (1965)
J. M. Baker and J. P. Hurrell, Proc. Phys. Soc. (London) 82, 742 (1963)
R. G. Bessent and W. Hayes, Proc. Roy. Soc. (London) A285, 430 (1964)
C. K. Jorgensen, R. Pappalardo, and H. H. Smidtke, J. Chem. Phys. 39, 1422 (1963)
A. A. Kaplianskii, Opt. Spectry. (USSR) 7, 406 (1959)
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W. Hayes and J. W. Twidell, J. Chem. Phys. 35, 152 (1961)
G. Burns and J. D. Axe, Phys. Letters 19, 98 (1965)
C. J. Ballhausen and H. Gray, Molecular Orbital Theory (W. A. Benjamin, Inc., New York, 1964) C. J. Ballhausen, Ligand Field Theory (McGraw-Hill Book Company, Inc., New York, 1962)
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T. P. Das, Phys. Rev. 140, A1957 (1965)
J. C. Eisenstein and M. H. L. Pryce, Proc. Roy. Soc. (London) A255, 181 (1960) J. D. Axe, thesis [Lawrence Radiation Laboratory Report No. UCRL-9293, 1960 (unpublished)] J. D. AxeDocument No. 8127 from the Photoduplication Service, ADI Auxiliary Publications Project, Library of Congress, Washington 25, D. C. (unpublished)
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([42])
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[c1] C. A. Hutchison and E. Y. Wong, J. Chem. Phys. 29, 754 (1958) E. Y. Wong and I. Richman, ibid. 36, 1889 (1962)

[c2] G. Burns, Phys. Rev. 128, 2121 (1962)

[c3] M. T. Hutchings and D. K. Ray, Proc. Roy. Soc. (London) 81, 633 (1963)

[c4] C. J. Lenander E. Y. Wong, J. Chem. Phys. 38, 2750 (1963)

[c5] D. K. Ray, Proc. Phys. Soc. (London) 82, 47 (1963)

[c6] R. E. Watson and A. J. Freeman, Phys. Rev. 133, A1571 (1964)

[c7] G. Burns, J. Chem. Phys. 42, 377 (1965)

[c8] J. M. Baker and J. P. Hurrell, Proc. Phys. Soc. (London) 82, 742 (1963)

[c9] R. G. Bessent and W. Hayes, Proc. Roy. Soc. (London) A285, 430 (1964)

[c10] C. K. Jorgensen, R. Pappalardo, and H. H. Smidtke, J. Chem. Phys. 39, 1422 (1963)

[c11] A. A. Kaplianskii, Opt. Spectry. (USSR) 7, 406 (1959)

[c12] W. A. Runcimann and C. V. Stager, J. Chem. Phys. 38, 279 (1963)

[c13] H. G. Drickamer, in Solid State Physics, edited by F. Seitz and D. Turnbull (Academic Press Inc., New York, 1965), Vol. 17

[c14] Z. J. Kiss, Phys. Rev. 127, 718 (1962)

[c15] W. Hayes and J. W. Twidell, J. Chem. Phys. 35, 152 (1961)

[c16] G. Burns and J. D. Axe, Phys. Letters 19, 98 (1965)

[c17] C. J. Ballhausen and H. Gray, Molecular Orbital Theory (W. A. Benjamin, Inc., New York, 1964) C. J. Ballhausen, Ligand Field Theory (McGraw-Hill Book Company, Inc., New York, 1962)

[c18] A. A. Kaplianskii, Opt. i Spektroskopiya 16, 1031 (1964) [English transl.: Opt. Spectry. (USSR) 16, 557 (1964)]

[c19] B. Bleaney, Proc. Roy. Soc. (London) A277, 289 (1964)

[c20] Z. J. Kiss, Phys. Rev. 137, A1749 (1965)

[c21] Z. J. Kiss and H. A. Weakliem, Technical Report AFAL-TR-64-334, 1965, p. 33 (unpublished)

[c22] J. D. Eshelby, in Solid State Physics, edited by F. Seitz and D. Turnbull (Academic Press Inc., New York, 1956), Vol. 3, p. 79

[c23] J. Friedel, Phil. Mag. 46, 514 (1955)

[c24] F. D. Murnaghan, Finite Deformation of an Elastic Solid (John Wiley & Sons, Inc., New York, 1951), p. 119

[c25] H. Brooks, Impurities and Imperfections (American Society for Metals, Cleveland, Ohio, 1955), pp. 22-23

[c26] D. R. Huffman and M. H. Norwood, Phys. Rev. 117, 709 (1960) D. Gerlich, ibid. 136, A1366 (1964)

[c27] T. P. Das, Phys. Rev. 140, A1957 (1965)

[c28] J. C. Eisenstein and M. H. L. Pryce, Proc. Roy. Soc. (London) A255, 181 (1960) J. D. Axe, thesis [Lawrence Radiation Laboratory Report No. UCRL-9293, 1960 (unpublished)] J. D. AxeDocument No. 8127 from the Photoduplication Service, ADI Auxiliary Publications Project, Library of Congress, Washington 25, D. C. (unpublished)

[c29] P. W. Anderson, in Solid State Physics, edited F. Seitz and D. Turnbull (Academic Press Inc., New York, 1963), Vol. 14, p. 190ff

[c30] R. S. Mulliken, C. A. Reike, D. Orloff, and H. Orloff, J. Chem. Phys. 17, 1248 (1949)

[c31] D. H. Templeton and C. H. Dauben, J. Am. Chem. Soc. 76, 5237 (1954)

[c32] A. J. Freeman and R. E. Watson, Phys. Rev. 127, 2058 (1962)

[c33] T. R. McGuire and M. W. Shafer, J. Appl. Phys. 35, 984 (1964)

[c34] C. K. Jorgensen, Absorption Spectra and Chemical Bonding in Complexes (Pergamon Press, Ltd., Oxford, 1962) ([46])

[c35] ([42])

[c36] G. Burns and J. D. Axe, J. Chem. Phys. (to be published)

[c37] K. W. H. Stevens, Proc. Roy. Soc. (London) A219, 542 (1953)

[c38] M. Inoue, Phys. Rev. Letters 11, 196 (1963) R. Orbach and P. Pincus, Phys. Rev. 143, 168 (1966)

[c39] R. E. Watson and A. J. Freeman, Phys. Rev. Letters 6, 277 (1961)

[c40] A. A. Kaplianskii and A. K. Przhevuskii, Opt. Spectroskopiya 13, 882 (1962) [English transl.: Opt. Spectry. (USSR) 13, 508 (1962)] Z. J. Kiss (unpublished)

[c41] M. H. Cohen, and V. Heine, Phys. Rev. 122, 1821 (1961)

[c42] R. G. Schulman and S. Sugano, Phys. Rev. 130, 506 (1962) K. Knox, R. G. Shulman, and S. Sugano, ibid. 130, 512 (1962) S. Sugano and R. G. Shulman, ibid. 130, 517 (1962)

[c43] R. E. Watson and A. J. Freeman, Phys. Rev. 134, A1526 (1964)

[c44] E. Simanek and Z. Sroubek, Phys. Status Solidi 4, 251 (1964)

[c45] J. Hubbard, D. E. Rimmer, and F. R. A. Hopgood, Proc. Phys. Soc. 88, 13 (1966)

[c46] S. Sugano and Y. Tanabe, J. Phys. Soc. Japan 20, 1155 (1965)

[c47] D. W. Hafemeister and W. H. Flygare, J. Chem. Phys. 43, 795 (1965)

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Influence of Covalency upon Rare-Earth Ligand Field Splittings

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