Xa scattered-wave method

The SCF-Xa Scattered-wave Method.—An alternative to the LCAO-MO method for molecules is the Xa scattered wave or multiple-scattering Xa method, which was suggested less than 10 years ago114 and now has an extensive literative. The basic theory has been refined in several directions115 and there are several excellent reviews.116-180 The method is based on the scattered-wave method in solids181 and aims to solve the Fock equation,... 

Table I summarizes the results of our calculations, along with previous work on two- and three-iron clusters. As we have noted before, the general tendency of the Xa scattered wave method is to overestimate J for example, the experimental estimate of J for the oxidized linear three-iron cluster (18) is 300 cm, about 45% lower than the value we estimate. Estimates for J in oxidized two-iron clusters are smaller than our estimates by similar amounts (6). We expect the behavior to be about the same for the reduced species, but that the qualitative trends should be correct. We have discussed the results of the oxidized three-iron clusters in our previous paper (9), showing that the Xa calculations predict a ground state for the...

The ease with which the geometry of the metal-carbene complexes can adjust to accommodate the incoming olefin may be an important factor in determining the rate and stereoselectivity in a given metathesis reaction (Lee, J.B. 1981). Extended Hiickel MO calculations on Ti(=CH2)L2, where L = H, Cl, Cp, have shown that the completely planar molecule is easily distorted into a flattish pyramid with Ti at the apex, ready to receive the incoming donor olefin (Gregory 1985). Similarly, calculations employing the self-consistent-field-Xa-scattered-wave method on Mo(=CH2)(=NH)(OMe)2 show that the reaction with ethene at the COO faee to... 

Fig. 24. Contours of the (a) 5ai, and (b) lia molecular orbitals calculated by SCF-Xa scattered wave methods. The amplitude of the wave functions increase by a factor of three and five [for (a) and (b) respectively] with each incremental increase in the value of the contour label (507). 

Faucher, J.-R, Labarre, J.-F., and Shaw, R. A., Bonding in cyclophosphazenes. Quantum and experimental support for Dewar s island model, Z. Naturforsch., 31b, 677 (1976). Mitchell, K. A. R., Noodleman, L., and Paddock, N. L., Xa scattered wave calculations of the ionization energies of N3P3F6 and N4P4F8, Chem. Phys. Lett., 47, 265 (1977). Noodleman, L., Westwood, N. P. C., and Mitchell, K. A. R., Ionization energies and electronic structure of the cyclic phosphazene as determined by UV photoelectron spectroscopy and Xa scattered wave methods. Chem. Phys. Lett., 58, 252 (1978). 

YN Kucherenko, LM Sheludchenko, VZ Khiinovsky, VV Nemoshkalenko. Study of the electron states in nonstoichiometric titanium nitrides by the Xa-scattered-wave method. J Phys Chem Sohds 45 319, 1984. 

Let us briefly comment on the Xa scattered-wave (SW) method [174,175] which does not involve MuUiken s population analysis. 

The procedure for calculating energy level diagrams by the self-consistent field Xa scattered wave (SCF-Xa-SW) method is as follows (Sherman, 1984, 1991). An octahedral cluster such as [FeO6]10 is partitioned into a set of (overlapping) spheres centred about divalent iron and each oxygen atom, and these are surrounded by an outer sphere. Within each atomic sphere the one-electron Schrodinger equation... 

Results obtained by this method have been compared with the Xa scattered-wave results for FeCU anions.144 Results for ferrocene compare well with ab initio results and experiment.145 They appear to give better ionization potentials than the ab initio results of Veillard.83 This method has several clear advantages. The use of the LCAO form avoids the muffin-tin potentials and may make the method more easily visualizable for chemists. The numerical integration imposes no restriction on the form of the basic functions. STOs can be used as readily as CGTOs. The Ni problem is avoided since only N2 matrix elements need be stored. 

The self-consistent statistical exchange (Xa) multiple scattered wave method, which is well described in Ref. (58), has also been used by Bair a and Robaux (59) to assign the peaks in the UP-spectrum of CO/Ni. 

The cluster approach is, like the extended-surface approach, characterized by many different calculational schemes. A recent review stresses electronic aspects of bonding. In this review we have chosen to concentrate on geometric aspects. We shall discuss a number of major techniques in order of increasing computational complexity the extended H uckel theory, self-consistent Xa scattered wave calculations, and self-consistent ab initio Hartree-Fock and valence bond methods. In that order these techniques allow increasing accuracy. However, the cluster size must decrease simultaneously due to calculational complexity, ultimately reducing the degree of analogy with surfaces. 

I Extended Hiickel (9), Xa Scattered Wave (15), semiempirical SCF methods 16)) gave way to more sophisticated treatments, based both on ab initio Hartree-Fock SCF (17), Cl (18-20)) and on local density methods (DVM-Xa, 22), LCGTO-Xa (23), LCGTO-LSD-MP (24))- These more elaborate studies furnish structural and energetic information on chemisorption bonds -within the restriction of a local cluster model (9), of course. 

TTie three methods that have been most widely adopted to calculate DOS curves for clusters are (/) the semi-empirical extended-Huckel (EH) method, (n) the complete-neglect-of-differential-overlap (CNDO) method, and iii) the self-consistent-field Xa scattered-wave (Xa-SW) method. Calculations of electronic structures of many transition- and noble-metal clusters have been reported over the last decade. No attempt is made to summarize them all, because there appears to be a disparity of view among the expert practitioners as to the validities of the various methods of calculation. However, the story to date is as follows. 

The SCF-Xa scattered wave technique has been used by Basiri and Pan for calculation of the MO energies of the octahedral boranes BeHe z = 0, 2, and 4) this technique requires only a small fraction of the computer time of ab initio Hartree—Fock LCAO methods. These calculations confirm the stability of BeHe " by 110.8 kj/mol with respect to the monoanion and by 176.8 kJ/mol with respect to the neutral molecule. 

Fig. 2. Single-particle electronic structures of (a) H4Si4Hi2, (b) Si5H12 (host), (c) Si4H12 (vacancy), and (d) Li4Si4H12, using the scattered-wave Xa duster method (Reprinted with permission from the American Physical Society, DeLeo, G.G., Fowler, W.B., Watkins, G.D. (1984). Phys. Rev. B 29, 1819.)...

A basis set may be employed that is of the same form throughout the space of the system, or one in which the orbitals are expanded in different types of basis functions in different parts of space. Such partitioned bases are often used in solid-state calculations in which one must describe an overall wave function that is rapidly varying near the nuclei and slowly varying and free-electron-like when far from the nuclei. Such partitioned bases will be considered further in our discussion of band-theoretical calculations and the multiple-scattering Xa molecular-orbital method. 

Slater summarized the power of the Xa method at the end of the article from 1972 by stating that The Xa method provides a practical starting point for an a priori theory of such problems, as magnetism, a starting point which has not been practical with earlier techniques. The power of the Xa method was further accentuated in an article by Johnson on Scattering-Wave Theory of the Chemical Bond in 1973 [6], where he gives an overview of the traditional Hartree-Fock, HF, SCF method and also semi-empirical ones. He points out that the new Xa technique based on Scattering Wave Theory has... 

Experimental studies on the K0/Ka x-ray intensity ratio for 3d elements have shown [18-23] that this ratio changes under influence of the chemical environment of the 3d atom. Brunner et al. [22] explained their experimental results due to the change in screening of 3p electrons by 3d valence electrons as well as the polarization effect. Band et al. [24] used the scattered-wave (SW) Xa MO method [25] and calculated the chemical effect on the K0/Ka ratios for 3d elements. They performed the SW-Xa MO calculations for different chemical compounds of Cr and Mn. The spherically averaged self-consistent-field (SCF) potential and the total charge of valence electrons in the central atom, obtained by the MO calculations, were used to solve the Dirac equation for the central atom and the x-ray transition probabilities were calculated. 

Similar simplifications hold for the Xot. scattered-wave (SW) method (Johnson, 1973). Diamond (1973) symmetrized the 2fa-SW free-eleetron Green s function. Case and Yang (1980) used the invariance of the Green s function to eliminate the summation over the set atomic centers in the bra (or ket) of the equivalent of Eq. (22) for Xa-SW. 

Cotton, F. A., J. B. Harmon, and R. M. Hedges (1976), Calculation of the Ground State Electronic Structures and Electronic Spectra of Di- and Trisulfide Radical Anions by the Scattered Wave-SCF-Xa Method, J. Am. Chem. Soc. 98, 1417-1424. 

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