References
- Schäffer , C. E. 1982 . Physica , 114A : 28 – 49 .
- Schäffer , C. E. 1968 . Struct. Bond. , 5 : 68 – 95 .
- Schäffer , C. E. 1973 . Struct. Bond. , 14 : 69 – 110 .
- Griffith , J. S. 1961 . The Theory of Transition-Metal Ions , Cambridge : Cambridge University Press .
- Schäffer , C. E. 1974 . Theor. Chim. Acta , 34 : 237 – 243 .
- Schäffer , C. E. 1973 . “ Chap. 12 ” . In Wave Mechanics-The First Fifty Years , Edited by: Price , W. C. , Chissick , S. S. and Ravensdale , T. London : Butterworths .
- The angular overlap model, when applied to linearly ligating ligands, associates with a given central-ion-to-ligand system, Cr-F say, a set of empirical parameters eσ eφ, and e δ, and is then symmetry-based in the sense that the d orbitals associated with these energies are the dσ dφ, and dδ orbitals labeled by irreducible representations of the group C∞ν of Cr-F. Often eδ is put equal to zero. This is equivalent to introducing the new parameters e′σ = e σ - eδ and e′φ = e′φ - e δ. However, the subscripts of e′σ and e′φ have thereby strictly speaking lost their symmetry meaning
- A ligand field model may also be based upon a reducible space. If, for example, an s set and a d set of functions are combined, it is based upon a reducible space Vs(R3)
- What in this Comment has been called a “molecular scheme” was in Ref. 6 called an “orbital energy parametrization scheme” and in Ref. 5, a “ligand field parame-trization scheme”. The “atomic scheme” in this Comment was called the “spherical harmonic scheme” and the “crystal field parametrization scheme”, respectively
- Schäffer , C. E. 1966 . Theor. Chim. Acta , 4 : 166 – 173 .
- Harnung , S. E. and Schäffer , C. E. 1972 . Struct. Bond. , 12 : 201 – 255 .
- Löwdin , P.O. 1977 . Int. J. Quant. Chem. , 12 ( Suppl. 1 ) : 197 – 266 .
- Roothaan , C. C. J. and Detrich , J. H. 1983 . Phys. Rev. A , 27 : 29 – 56 .
- 1979 . “ Unitary Group Approach ” . In Spin Eigenfunctions -Construction and Use , New York : Plenum . See any of the now numerous sources dealing with the, e.g., Chapter 13 of R. Pauncz, or references therein
- Löwdin , P.O. Set Theory and Linear Algebra, Part I and II , Sweden : Department of Quantum Chemistry, Uppsala University . and Quantum Theory Project, University of Florida, Gainesville, Florida, U.S.A. Undated notes
- Schäffer , C. E. 1971 . Int. J. Quant. Chem. Symp. , 5 : 379 – 390 .
- The ligand field operator may further be looked upon as a tensor in the following way: as indicated by Eqs.(21)–(24), there is a natural isomorphic relationship between the four-dimensional operator space just mentioned and the direct or tensor product space where is the space spanned by the bras (θ and ϵ, just as Ve(o) is spanned by the kets θ and ϵ) is sometimes said to be complex conjugate to Ve(o) ). Thus, W is a mixed tensor of rank (or degree) 2 on a two-dimensional space (“mixed” because of the complex conjugation on just one factor in the above tensor product and “of rank 2” because there are two factors in the tensor product). It therefore has 2 × 2 = 4 components. The tensor components are the parameters S αβ
- The reality restriction is in fact a somewhat delicate question from the point of view of what we have here called molecular ligand field theory (see Introduction and Section III. 1). Atomic ligand field theory may be based on explicitly defined real functions within a Hilbert space where the Hamiltonian is composed of terms which are either just multiplication operators (potentials) or second-order differential operators (kinetic energy). In such a situation the Hamiltonian can only have real matrix elements in the basis chosen. The real basis functions generate real matrix representations when acted upon by the usual, explicitly defined rotation operators; thus, for example, the set {deθ, deϵ} mentioned in connection with Eqs.(10)–(13) generates the real form of e(O) defined by Eqs.(14) and (15). The property of the Hamiltonian having real matrix elements is then taken over in the molecular theory when using basis sets generating real matrix representations. Not all consequences of this have been satisfactorily investigated
- Operators transforming according to different irreducible representations can easily be seen to be mutually orthogonal with respect to the operator scalar product defined in Eq.(30). Thus, if A is an operator containing no totally symmetric terms, we have, since the identity operator I is totally symmetric, 0 = I = Tr(A), i.e., A is traceless
- Glerup , J. , Monsted , O. and Schäffer , C. E. 1976 . Inorg. Chem. , 15 : 1399 – 1407 .
- Determination of Ne θ and Ne ϵ using a purely symmetry-based nonadditive field model is a complicated problem in practice, but can in principle be carried out by performing a sufficient number of suitable experiments (for example, spectra of oriented crystals)
- The factor of normalization depends on the dimension of the space, i.e., on the value of the effective spin S. As the matrix of S2 z - (1/3)S2 is diagonal in the |S Ms ) basis, the sum of its elements squared is given by the expression , where Ms is the eigenvalue of the operator S. The factor of normalization for is the same as for owing to the relation between these two operators by an orthogonal transformation [cf. the relation of
- Hall , P. L. , Angel , B. R. and Jones , J. P. E. 1974 . J. Mag. Res. , 15 : 64 – 68 .
- Poole , C. P. and Farach , H. A. 1974 . J. Chem. Phys. , 61 : 2220 – 2221 .
- For trans, trans, trans-[CrBr2Cl2F2]3- the two preferred choices of tetragonal axis are the same, namely the F-Cr-F axis. The complex trans, trans, trans-[CrBr2F2(NH3)2]− may exemplify a different situation. In this complex the preferred tetragonal axis in the de subspace is Br-Cr-Br while it is F-Cr-F in the dt2 subspace. This emerges from a calculation of the normalized parameters based on the AOM parameters of Ref. 20, Table IV
- The two operators appearing on the right-hand side of Eq.(82) are those of a1(O) symmetry in the molecular scheme. They can be transformed into those of a1(O) symmetry in the atomic scheme by using the orthogonal Racah lemma matrix {dda1 }, described in Ref. 1. Thereby the operator Na ddg , mentioned in the main text, arises together with the spherically symmetric operator Na dds which is unobservable within the d space. Similar transformations from the molecular scheme to the atomic one can be made for the operators of e(O) symmetry using the Racah lemma matrix {dde} for the θ component as well as the ϵ component
- Schäffer , C. E. 1967 . Proc. R. Soc. London Ser. A , 297 : 96 – 133 .
- There are other ways of normalizing operators than the one used here. For example, Racah's unit operators and are within an l space, for 0 ≤k ≤2l, normalized to l/(2k + 1) (see Ref. 11, p. 243). An alternative way of stating this is to note that our normalized operators have reduced matrix elements of . Using this fact and Eq. 61 of Ref. 29, one obtains the relationship Eq k = Nq k between the reduced ligand field parameters and the parameters of the orthonormal operators scheme. The relationship6 between the parameters Eq k and the parameters Iq k which correspond to the operators being multiplicative and expressed by spherical harmonics3 normalized over the unit sphere to , where is given by Eqs.(41) and (21) of Ref. 29
- Harnung , S. E. and Schäffer , C. E. 1972 . Struc. Bond. , 12 : 257 – 295 .