The Canonical Structure Constants

In the k = 0 limit used in the atomic-sphere approximation, the wave equation (5.7) used to construct the tail of the partial wave (5.10) turns into the Laplace equation. Hence, in the definition of the muffin-tin orbitals (5.13,25) the spherical Bessel and Neumann functions should be substituted by the harmonic functions (r/S)z and (r/S) 1, respectively. By means of the small kr limits (5.8) of the spherical Bessel and Neumann functions, the expansion theorem (5.14) becomes 

Inspection of expansion (6.5) shows that it is invariant under a uniform change in the scale of the lattice. This is a direct consequence of the fact that the solutions of the Laplace equation are harmonic functions, and hence the structure constants given below are also invariant under uniform lattice sealings. 

In the limit of vanishing k the Bloch sum (5.31) of MTO tails may be written as the one-centre expansion 

The structure constants S, L introduced by Andersen [6.2,5] have a number 

As shown in [6.6] the LMTO-ASA Hamiltonian matrix may be transformed into the two-centre form [6.7] where the hopping integrals are products of potential parameters and the canonical structure constants. This result was already stated in Sect.2.5. A less accurate two-centre approximation based upon the KKR-ASA equations will be presented in Sect.8.1.2. The canonical structure constants which, after multipiication by the appropriate potential parameters, form the two-centre hopping integrals are 1 isted in Table 6.1. The 

---

Fig.2.2. Lay-out for the canonical structure constant matrix in the m representation (a), and in the i representation (b). In the latter the diagonal exhibits the canonical s, p, and d bands...

The STR may be used to calculate the canonical structure constants defined by (6.7-9) or (8.23,24). In a typical application the programme is executed once for a given crystal structure. It produces and stores on disk or tape a set of structure-constant matrices distributed on a suitable grid in the irreducible wedge of the Brillouin zone. Whenever that particular crystal structure is encountered, the structure constant matrices may be retrieved and used to set up the LMTO eigenvalue problem which, in turn, leads to the energy bands of the material considered. 

The basic output from STR is the canonical structure constants used in CANON and LMTO to calculate band structures. In addition, STR produces a file with real and reciprocal space vectors which is used by the combined correction term programme COR. This file may also be read by STR next time the same crystal structure is encountered, thus saving the time used to generate these vectors. 

Let us assume that the programme STR has been successfully compiled, and that the user wants to calculate and store the canonical structure constants for the bcc structure. Let us further assume that the data files needed have been created according to the attributes given in the listing, Sect.9.2.2. The user is now faced with the problem of generating the input data necessary to make the programme run. To help him choose the correct value of the various variables, Table 9.2 lists input data for four different cases explained in detail below. 

Rate constants and Arrhenius parameters for the reaction of Et3Si radicals with various carbonyl compounds are available. Some data are collected in Table 5.2 [49]. The ease of addition of EtsSi radicals was found to decrease in the order 1,4-benzoquinone > cyclic diaryl ketones, benzaldehyde, benzil, perfluoro propionic anhydride > benzophenone alkyl aryl ketone, alkyl aldehyde > oxalate > benzoate, trifluoroacetate, anhydride > cyclic dialkyl ketone > acyclic dialkyl ketone > formate > acetate [49,50]. This order of reactivity was rationalized in terms of bond energy differences, stabilization of the radical formed, polar effects, and steric factors. Thus, a phenyl or acyl group adjacent to the carbonyl will stabilize the radical adduct whereas a perfluoroalkyl or acyloxy group next to the carbonyl moiety will enhance the contribution given by the canonical structure with a charge separation to the transition state (Equation 5.24). 

In 1984, it was also realized that it was possible to transform the original, so called bare (or canonical), structure constants into other types of structure constants using so-called screening transformations [53]. This allowed one to trans-... 

In Chap.6 the atomic-sphere approximation is introduced and discussed, canonical structure constants are presented, and it is shown that the LMTO-ASA and KKR-ASA equations are mathematically equivalent in the sense that the KKR-ASA matrix is a factor of the LMTO-ASA secular matrix. In addition, we treat muffin-tin orbitals in the ASA, project out the i character of the eigenvectors, derive expressions for the spherically averaged electron density, and develop a correction to the ASA. 

Table 6.1. Canonical structure constants [6.8,9],in the two-centre notation of Stater and Roster [6.7]. The present, real structure constants are equal to those defined in (6.7,8.8) times 1 and S(a m, dm) = (-)t + S(dm,d m ), where m refers to the angular momentum. The vector from the first to the second orbital has a length R, and direction cosines i, m, and n. The distance S, which also enters the definition of the potential functions, is arbitrary. The entries not given in the table may be found by cyclically permuting the coordinates and direction cosines... 

The basic input to STR is the translational vectors spanning the unit cell of the crystal, and the basis vectors giving the positions of the individual atoms in the cell. With this information STR may in principle be used to calculate canonical structure constants of any crystal structure, the only limitation being that central processor time grows rapidly as the number of atoms per cell is increased. 

Early ESR studies demonstrated that the hyperfine coupling constant (ac 13) for 13C(car-bonyl)-substituted fluorenone radical anion is counterion-dependent. For the free ion, ac 13 = 2.75 Gauss. In contrast, when the counterion is Li+, ac 13 = 6.2 Gauss23. Consider Scheme 4 For the free ion, canonical structure 1 and 2 are contributors to the resonance hybrid. For the >C=0 / Li+ ion pair, association of Li+ with oxygen increases the relative contribution of canonical structure 1 to the resonance hybrid, resulting in greater spin density at carbon. The fact that spin (and charge density) varies as a function of counterion (and presumably solvent) will certainly affect the reactivity of the radical ion. However, very few quantitative studies exist which directly address this point. 

---