Crystalline field Theory

In crystalline field theory, the valence electrons belong to ion A and the effect of the lattice is considered throngh the electrostahc field created by the snrronnding B ions at the A position. This electrostatic field is called the crystalline field. It is then assnmed that the valence electrons are localized in ion A and that the charge of B ions does not penetrate into the region occnpied by these valence electrons. Thns the Hamiltonian can be written as 

In order to apply quantum mechanical pertnrbation theory, the free ion term is usually written as 

Depending upon the size of the crystal field term //cf in comparison to these three free ion terms, different approaches can be considered to the solution of Equation (5.1) by perturbation methods  

To illustrate how the perturbation problem must be solved, we now describe one of the simplest cases, corresponding to an octahedral crystalline field acting on a single d valence electron. 

One of the simplest descriptions of the crystalline field occurs for the d outer electronic configuration (i.e., for a single d valence electron). This means that //ee = 0 and, consequently, there is no distinction between intermediate and strong crystalline fields. 

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Thus, MO theory is generally applied to interpretation of the so-called charge transfer spectra. However, for a great variety of centers in solids, crystalline field theory suffiees to provide at least a qnalitative interpretation of spectra. 

Gray HB, Ballhausen CJ (1961) The mixing of 3dn and 3dn 14 s states in crystalline field theory. Acta Chem Scand 15 1327... 

In the simple crystalline field theory, the influence of the environments is represented by the crystalline electric field, which is dominantly cubic in many cases. If we disregard the lower symmetry parts, the effect of the cubic field is expressed by the parameter A or lODq. Several attempts at the nonempirical calculation of the parameter were not very successful because of... 

For the Co atom the valence state of the orbitals s, p, d, is considered which, by the virtue of the crystalline field theory from the previous Section, are associated with the terms s Aj, p... 

Crystal-field theory (CFT) was constructed as the first theoretical model to account for these spectral differences. Its central idea is simple in the extreme. In free atoms and ions, all electrons, but for our interests particularly the outer or non-core electrons, are subject to three main energetic constraints a) they possess kinetic energy, b) they are attracted to the nucleus and c) they repel one another. (We shall put that a little more exactly, and symbolically, later). Within the environment of other ions, as for example within the lattice of a crystal, those electrons are expected to be subject also to one further constraint. Namely, they will be affected by the non-spherical electric field established by the surrounding ions. That electric field was called the crystalline field , but we now simply call it the crystal field . Since we are almost exclusively concerned with the spectral and other properties of positively charged transition-metal ions surrounded by anions of the lattice, the effect of the crystal field is to repel the electrons. 

A number of theoretical works have been devoted to the study of the hydrogen-deuterium exchange reaction. Hauffe (25) examined this reaction from the standpoint of the boundary layer theory of chemisorption. Dowden and co-workers (26) undertook a theoretical investigation of the hydrogen-deuterium exchange reaction from the viewpoint of the theory of crystalline fields. 

The crystal field theory (CFT) was developed for crystalline solids by the physicist Hans Bethe in 1929. The model takes into account the distance separating the positively and... 

Weak crystalline field //cf //so, Hq. In this case, the energy levels of the free ion A are only slightly perturbed (shifted and split) by the crystalline field. The free ion wavefunctions are then used as basis functions to apply perturbation theory, //cf being the perturbation Hamiltonian over the / states (where S and L are the spin and orbital angular momenta and. 1 = L + S). This approach is generally applied to describe the energy levels of trivalent rare earth ions, since for these ions the 4f valence electrons are screened by the outer 5s 5p electrons. These electrons partially shield the crystalline field created by the B ions (see Section 6.2). 

Crystal field theory has its origins in Hans Bethe s famous 1929 paper Splitting of terms in crystals. In that paper Bethe demonstrated what happens to the various states of an ion when it is placed in a crystalline environment of definite symmetry. Later, John Van Vleck showed that the results of that investigation would apply equally well to a transition-metal compound if it could be approximated as a metal ion surrounded by ligands which only interact electrostatically with the... 

Two-proton transfer in crystals of carboxylic acids has been studied thoroughly by the 7 -NMR and IINS methods. The proton spin-lattice relaxation time, measured by T,-NMR, is associated with the potential asymmetry A, induced by the crystalline field. The rate constant of thermally activated hopping between the acid monomers can be found from Tj using the theory of spin exchange [Look and Lowe, 1966] ... 

The paramagnetism in transition metal salts is affected by the presence of symmetry-dependent crystalline electric fields, which can distort the electronic configuration of the bare ion significantly, by modifying the hybridization. This is treated by ligand-field theory. 

The birth of crystal field theory is due to Bethe who in 1929 showed that open-shell energy levels in a crystalline environment could be associated with the irreducible representation labels of the site point group. Little experimental work was done at that time because... 

Pedersen T G, Johansen P M. (1997) Mean-field theory of photoinduced molecular reorientation in azobenzene liquid crystalline side-chain polymers. Phys Rev Lett 79 2470-2473... 

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