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Extra resources for Advanced chemical oxidation treatment of dye wastewater and remediation of chlorinated aromatic

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41) −1 where mi = ti . In muffin-tin formalism, it is customary to introduce τ ij as the inverse of M ij . τ ij is called the scattering path operator [40], though it is not really an operator, but a matrix. 43) LL where G is the Fourier transformation of G0 and V is the volume of the Brillouin zone. It is also interesting to investigate how the introduction of an impurity on a site influences the scattering path operator. This means that t goes to t on one site. The result [39] is that τ obeys a Dyson-like equation: old,ij −1 new,ij τLnew,ij = τLold,ij .

One usually also introduces ul (k, r) = rRl (k, r) for notational reasons. 47) where the integration constants C may still depend on k. There is a well known expansion theorem to expand the plane wave solution (which are the solutions to the free particle equation) in terms of Bessel functions as well [39]: ∞ l ∗ ˆ il jl (kr)Ylm (ki )Ylm (ˆr). 49) and the rest vanish. 3 The connection between the t-matrices and the potential 33 just from trigonometric identities. Here δl (k) is called the phase shift for the l partial wave, and this parameter determines how the incoming wave is affected by the scattering.

The way to get around this problem is to use one basis set inside the potential spheres, and another basis set in the interstitial, that is, the space between the potential spheres. While it is possible to use partial waves in the potential spheres, the use of plane waves in the interstitial is still too badly convergent to be used with maximal computational efficiency, but there are other sets available. In the EMTO method, the basis set used is called screened spherical waves [58] and will be discussed and defined below.

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Advanced chemical oxidation treatment of dye wastewater and remediation of chlorinated aromatic


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