The Ligand Field Hamiltonian

We now turn to the ligand-field potential which consists of the non-spherical parts only of the following expression (Sect. 5), 

VeQ is the electron-nuclear Coulomb potential at ligand Q, while / and K are modified Coulomb and exchange operators that act on spin-orbitals (x) as follows (c.f. 4-8), 

We shall not specify further the screened Coulomb interaction Uirn) which is taken to be some average of the true configuration dependent operator, as discussed in Sect. 5 instead it will be more illuminating to develop a one-electron theory of Vl p that parallels 

The partitioning operators P and (5 will be specified below. Now write 

we note that since the q are bond-orbitals, they are localized functions, and if we write V in terms of a cellular decomposition 

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When treating CF parameters in any of the two formalisms, non-specialists often overlook that the coefficients of the expansion of the CF potential (i.e. the values of CF parameters) depend on the choice of the coordinate system, so that conventions for assigning the correct reference framework are required. The conventional choice in which parameters are expressed requires the z-direction to be the principal symmetry axis, while the y-axis is chosen to coincide with a twofold symmetry axis (if present). Finally, the x-axis is perpendicular to both y- and z-axes, in such a way that the three axes form a right-handed coordinate system [31]. For symmetry in which no binary axis perpendicular to principal symmetry axis exists (e.g. C3h, Ctt), y is usually chosen so as to set one of the B kq (in Wybourne s approach) or Aq with q < 0 (in Stevens approach) to zero, thereby reducing the number of terms providing a non-zero imaginary contribution to the matrix elements of the ligand field Hamiltonian. Finally, for even lower symmetry (orthorhombic or monoclinic), the correct choice is such that the ratio of the Stevens parameter is restrained to X = /A (0, 1) and equivalently k =... 

Equation (5-9) can be regarded as a canonical (Lowdin) orthonormalisation of the set of vectors 0 , or equivalently as a polar decomposition of the operator 3l (J0rgensen 9) Thus the Schrodinger equation for the n-electron Hamiltonian, H, Eq. (2-2), can always be formally transformed to the eigenvalue problem (5-10 a) for the effective Hamiltonian,, acting in the subspace S sparmed by a finite set of orthonormal vectors 0 the ligand field Hamiltonian (1-5) must therefore be an approximation to this object. 

In the previous sections we have discussed the ligand-field theory from the point of view of quantum chemistry, and have presented an ab initio derivation of the ligand-field Hamiltonian (1-5). In principle this Hamiltonian can be constructed explicitly using the standard techniques of computational quantum chemistry, although in practice it is evident that this would be subject to the usual difficulties encountered with large molecules. Our concern in this section is with the use of Eq. (1-5) as the basis for a parameterisation scheme that permits the interpretation of the spectroscopic and magnetic properties of transition metal complexes in terms that are chemically intelligible. 

The AOM is concerned purely with the representation of the matrix elements of the ligand-field potential part of the ligand-field Hamiltonian in a basis of d-orbitals. The ligand-field potential which we discussed in Sect. 5, is divided into a superposition of nonoverlapping contributions by dividing the molecular space into a set of nonoverlapping cells. 

Thus, this solution defines the exact AI parameterization of the ligand field Hamiltonian. To the best of our knowledge, this is a new result. In practice, of course, the FCI equations cannot be solved for even the smallest transition metal complex. Hence one has to resort to approximation that still define an effective Hamiltonian or at least its energies. We next turn our attention to such approximations. 

So, ligand-field theory is the name given to crystal-field theory that is freely parameterized. The centrally important point is that ligand-field calculations, whether numerical or merely qualitative, explicitly or implicitly employ a ligand-field Hamiltonian, very much like the crystal-field Hamiltonian, operating upon a basis set of pure d orbitals. Instead of the crystal-field Hamiltonian (Eq. 6.15),... 

These values are in reasonable agreement with the values reported on the basis of far IR work (67a, 67b). In general the anisotropy measurements establish that the ligand field in the XFe(R2Dtc)2 complexes is rhombic and provide a method for estimating spin-Hamiltonian parameters. 

When S > 1/2 and the metal site symmetry is lower than Oh or Td, there is a term in the spin Hamiltonian, in addition to the Zeeman term (gfill), that will split the (2S + 1 )MS spin degeneracy even in the absence of a magnetic field.32 This is shown in Equation 1.8, where D in the first term describes the effect of an axial distortion of the ligand field (z / x y) and E in the second term accounts for the presence of a rhombic ligand field y). 

Equation 1.10 describes this non-Kramers doublet behavior and its fit to the VTVH MCD data in Figure 1.12a (with orientation averaging for a frozen solution) allows the spin Hamiltonian parameters to be obtained.29,30 These, in turn, can be related to the ligand field splittings of the t2g set of d-orbitals, as described in Ref. 7, which probe the re-interactions of the Fe(II)... 

It is conventional that the ligand field problem for systems with Na> d electrons requires the diagonalization of an effective Hamiltonian operator composed for the electronic kinetic energy T, and both one-electron ligand field terms, and two-electron Coulomb interactions ... 

The outline of the review is as follows in the next section (Sect. 2) we introduce the basic ideas of effective Hamiltonian theory based on the use of projection operators. The effective Hamiltonian (1-5) for the ligand field problem is constructed in several steps first by analogy with r-electron theory we use the group product function method of Lykos and Parr to define a set of n-electron wavefimctions which define a subspace of the full -particle Hilbert space in which we can give a detailed analysis of the Schrodinger equation for the full molecular Hamiltonian H (Sect. 3 and 4). This subspace consists of fully antisymmetrized product wavefimctions composed of a fixed ground state wavefunction, for the electrons in the molecule other than the electrons which are placed in states, constructed out of pure d-orbitals on the... 

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