Crystal field potentials

The crystal field theory is based on the assumption that the chromophore ML of a coordination compound can be described by the model according to which the central atom with its valence d-orbitals is under the influence of point charges generated at the ligand positions. The electrostatic potential acting as a sum of the ligand contributions is termed the crystal field potential and it is 

The action of the crystal field potential should be considered in relationship to other perturbations such as the interelectron repulsion F 6 and eventually the spin-orbit coupling operator IT0. According to these relationships, several types of crystal field are distinguished (Table 8.8). The principal difference between them is which basis set is generated prior to the application of the actual perturbation operator. 

The matrix elements of the crystal field potential between the atomic orbitals can be written in the form 

Type Operator relationship Basis set Actual operator Method of calculation 

We shall restrict ourselves to the d-atomic orbitals only, so that the orbitals differ only in the magnetic quantum numbers m. The inverse of the electron-ligand separation can be expanded into the basis of the Legendre polynomials such an expansion follows an analogous philosophy to that for in a free atom. Hence 

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The origin of zero-field terms in the spin Hamiltonian is rooted in crystal-field theory, in which coordination complexes are represented as geometric structures of point charges (Stevens 1997). The crystal-field potential of these point charges is... 

Transition metal ions have an incompletely filled d-shell, i.e. their electron configuration is d". The optically active electrons are thus bound to central potential as well as experiencing crystal field potential, and are not shielded by outer electrons. Most transition metal ions are multi-valent. Mainly the number of 3d electrons and the crystal field determine their optical properties. Thus the groups below have similar optical behavior ... 

Crystal-Field Potential. Before we can discuss the magnetic interactions, we must consider the effect of the lattice on the ion, since this is by far the largest perturbation on the wave function of the ion. In crystal-field theory the electrons and nuclei of the surrounding atoms are replaced by a potential field of the same symmetry, and the effects of this crystal field on the metal ion are evaluated. Thus we take for V in Eq. (5) the expression... 

The crystal field potential obtained in this way will be of the general form... 

The derivation of the crystal field potential VC at a transition metal ion site in a crystal is seen to be conceptually straightforward, and to obtain the relative effects such a potential should have on the energies of the (/-orbital is, similarly, formally simple. We note that the (/-orbitals are mathematical functions of the form... 

JThe usual contribution of the crystal field potential to the Hamiltonian is much smaller than the Ft s and 4/. It should be recognized that all parameters, A , Ft and 4/ cannot be treated as free variables. The parameters Ft and 4/ are first adjusted to fit the free ion levels and then the crystal field parameters are considered to get a best fit with the crystal levels. The available crystal field parameters of the rare earth ions in LaCk and ethylsulphate hosts are compared in Table 43. 

The geometric model adopted here is one in which the O2 molecule provides a C2v component to the crystal field potential a-round the Cu11 ion, and together with the three framework oxygens of the SII site makes an additional tetrahedral (T ) component. This model has been theoretically analyzed earlier (30) and the... 

While the crystal field potential causes the symmetry lowering (on passing from Rs to its subgroup G), the spin-orbit interaction causes a passage from the group G to its double group G. ... 

Lanthanide complexes with axial symmetry (i.e., possessing at least a threefold axis, see sect. 2.4.2) are exclusively considered because the principal magnetic z axis coincides with the molecular symmetry axis (Forsberg et al., 1995) and the c 2 spherical tensor operators do not contribute to the crystal-field potentials (Gorller-Walrand and Binne-mans, 1996). The rhombic term of Bleaney s approach V6B Hi (eqs. (42), (46)) thus vanishes and the crystal-field independent methods (eqs. (51), (53)) can be used without complications. 

In the second step, the crystal-field potential is introduced to the Hamiltonian. The potential in the one-electron approximation can be written as follows ... 

The C are tensor operators, whose matrix elements again can be calculated exactly, whereas the crystal-field parameters Bk are regarded as adjustable parameters. The number of parameters for this potential is greatly reduced by the parity and triangular selection rules and finally by the point symmetry for the f-element ion in the crystal. Detailed information about the crystal-field potential has been given for example by Gorller-Walrand and Binnemans (1996). 

The quantities B (Rq) and are treated as adjustable parameters and Rq is a reference distance that can be chosen arbitrarily. In principle, the number of ligand shells considered for the calculation of intrinsic parameters is not limited, however, it is usually assumed that only the nearest neighbors of the rare-earth ion contribute significantly to the crystal-field potential. Thus, especially long-range interactions like electrostatic interactions are not accounted for explicitly. Because these interactions are most important for k = 2 parameters, in many cases only the k = 4, 6 intrinsic parameters have been considered. 

The crystal field interaction can be treated approximately as a point charge perturbation on the free-ion energy states, which have eigenfunctions constructed with the spherical harmonic functions, therefore, the effective operators of crystal field interaction may be defined with the tensor operators of the spherical harmonics Ck). Following Wyboume s formalism (Wyboume, 1965), the crystal field potential may be defined by ... 

M. Jacek and Z. Gajek, The Effective Crystal Field Potential, Elsevier, New York,... 

Hush, N. S. Pryce, M. H. L. (1958) Influence of the crystal field potential on interatomic separation in salts of divalent iron-group ions. J. Chem. Phys., 28, 424-9. 

Hush, N. S. and M. II. L. Pryce Influence of the Crystal-Field Potential on Interionic Separation in Salts of Divalent Iron-Group Ions. J. Chem. Phys. 28, 244 (1958). 

The value of k is limited to k 21, where / is the orbital angular momentum of the electrons. For f electrons k 6 k must also be even based on parity of the matrix elements involved in the crystal field potential. Thus k = 2,4 or 6 for f electrons. Allowed values of q have to follow the rule q k. Any further restrictions on q are dependent on the symmetry... 

According to Jorgensen and Judd, hypersensitivity may occur due to pseudoquadrupole transitions [66]. Consequently an ion embedded in an inhomogeneous dielectric would exhibit hypersensitive behavior. These normally weak electric quadrupole transitions are probably intensified and become hypersensitive transitions. Hypersensitivity can also occur in symmetries of spherical harmonics (Ymk, with k = 1) which form totally symmetrical representations. Thus this permits their inclusion in the crystal field potential. 

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