Koch cluster

The structure of crystalline FeO belongs to the NaCl type. When iron(II) oxide is prepared under normal conditions, the composition of the product (wustite) is always Fei 50. In order to retain overall electric neutrality, part of the Fe2+ is oxidized to Fe3+, and the chemical formula becomes Fc Fe. O. Since the radius of Fe3+ is small, the Fe3+ cations tend to occupy the tetrahedral holes to form a short-range ordered Fe40io cluster, which is called the Koch cluster of Fei 0, as shown in Fig. 10.1.3. The Koch clusters are distributed randomly in the crystal structure. To satisfy charge neutrality, the formation of a Koch cluster must be accompanied by the presence of six Fe2+ vacancies, one of which is located at the center of the cluster, and the remaining five are distributed randomly at the centers of the edges of the cubic unit cell. 

Structure of Koch cluster of Fei jO (black circle, Fe3+ large white circle, O2- small white circle represents an octahedral vacancy). 

FIGURE 3.37. Koch cluster in wustite. (From West, A.R., Solid State Chemistry and Its Applications, John Wiley Sons, New York, 1984. With permission.)... 

Koch H, Jensen HJA, Jorgensen P, Helgaker T (1990) Excitation-energies from the coupled cluster singles and doubles linear response function (CCSDLR) - applications to be, CH+, CO, and H2O. J Chem Phys 93 3345... 

Christiansen O, Koch H, Jorgensen P (1995) The second-order approximate coupled cluster singles and doubles model CC2. Chem Phys Lett 243 409 t 18... 

Koch MA, Wittenberg LO, Basu S, Jeyaraj DA, Gourzoididou E, Reinecke K, Odermatt A, Waldmann H, Compound library development guided by protein structure similarity clustering and natural product structure, Proc Natl Acad Sci USA 101 16721-16726, 2004. 

Single crystal X-ray diffraction and neutron diffraction studies of a quenched sample indicated that the vacancies were not spread statistically over the structure, but were arranged in clusters (Koch/Cohen clusters) consisting of 13 vacancies and four... 

Figure 1.11. Schematic diagrams of (a) and (b) Fei j O point defect clusters (after Koch and Cohen 1969) (c) clustering in fluorite-type structures and (d) the Cap2 structure.

FIGURE 5.26 The Koch-Cohen cluster illustrated with the back and front planes cut away for clarity. The central section with four tetrahedrally coordinated Fe ions is picked out in bold. 

Figure 5.44 depicts the central section of a possible defect cluster for FeO. (a) Determine the vacancy interstitial ratio for this cluster, (b) Assuming that this section is surrounded by Fe ions and oxide ions in octahedral sites as in the Koch-Cohen cluster, determine the formula of a sample made totally of such clusters, (c) Determine the numbers of Fe and Fe ions in octahedral sites. 

Figure 5.14 Koch-Cohen cluster of defects in (wiistite.) Hatched circles...

The defect structure of Fei O with the NaCl-type structure had been estimated to be a random distribution of iron vacancies. In 1960, Roth confirmed, by powder X-ray diffraction, that the defect structure of wiistite quenched from high temperatures consists of iron vacancies (Vp ) and interstitial iron (Fcj) (there are about half as many FCj as Vpe). This was a remarkable discovery in the sense that it showed that different types of crystal defects with comparable concentrations are able to exist simultaneously in a substance, Roth also proposed a structure model, named a Roth cluster, shown in Fig. 1.84. Later this model (defect complex = vacancy -F interstitial) was verified by X-ray diffraction on a single crystal and also by in-situ neutron diffraction experiments. Moreover, it has been shown that the defect complex arranges regularly and results in a kind of super-structure, the model structure of which (called a Koch-Cohen model) is shown in Fig. 1.85 together with the basic structures (a) and (b). 

Fig. 1.85 Defect complexes of wustite. (a) Basic structure (Roth cluster) (b) edge-sharing tetrahedra (6 2 complex) (c) cornersharing tetrahedra (Koch-Cohen complex).

Koch H, Jdrgen H, Jensen A, Jorgensen P, Helgaker T, Scuseria GE, Schaefer III HF (1990) Coupled cluster energy derivatives. Analytic Hessian for the closed-shell coupled cluster singles and doubles wave function Theory and applications. J Chem Phys 92 4924-4940... 

Koch H, Jprgensen P (1990) Coupled cluster response functions. J Chem Phys 93 3333—3344. 

Koch H, Christiansen O, Jorgensen P, Sanchez de Meras AM, Helgaker T (1997) The CC3 model An iterative coupled cluster approach including connected triples. J Chem Phys 106 1808—1818. 

Christiansen O, Koch H, Halkier A, Jprgensen P, Helgaker T, Sanchez de Meras A (1996) Large-scale calculations of excitation energies in coupled-cluster theory The singlet excited states of benzene. J Chem Phys 105 6921-6939. 

Christiansen O, Koch H, J0rgensen P (1996) Perturbative triple excitation corrections to coupled cluster singles and doubles excitation energies. J Chem Phys 105 1451-1459. 

Christiansen O, Koch H, Jprgensen P, Olsen J (1996) Excitation energies of H20, N2, and C2 in full configuration and coupled cluster theory. Chem Phys Lett 256 185-194. 

Kutzelnigg and Koch/77/ also introduced a unitary Fock space cluster operator O = exp(o) with an antihermitian Fock space cr, as in eq. (7.2.3). In this case, the transformed Fock space hamiltonian L should be brought into a form such that... 

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