By I Prigogine
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Extra info for Advances in Chemical Physics, Vol.2 (Interscience 1958)
3) 0 We can see what happens: Either the wave propagates directly to r without impinging on the scatterer (the first term), or propagates freely to the scatterer, scatters, and then propagates freely to r. 1. A depiction of some possible two-center scattering events. scattering off of one scatterer (one each of tˆ1 and tˆ2 ), two terms where it scatters first off of one scatterer and then the next, and then an infinite number of more terms where it scatters back and forth between the two scatterers any number of times before coming to r.
45)). 49) we get: ∞ Ψki lim r→∞ = l [4π(2l + 1)]1/2 il A(k) l=0 m=−l i(kr− lπ 2 ) lπ − e−i(kr− 2 ) Ylm (ˆr)δm,0 2ikr eikr +f (k, θ, φ) . 56) lπ ×[ei(kr− 2 +δl ) − e−i(kr− 2 +δl ) ]Ylm (ˆr). 57) If we now compare the coefficients between these two expressions, we find: clm (k) = A(k) [4π(2l + 1)]1/2 il eiδl δm,0 . 46)) as: ∞ 4π(2l + 1) l iδl i e Rl (k, r)Yl,0 (θ). 60) l=0 where Pl (cos θ) are Legendre polynomials . 61) 34 Multiple scattering theory we can write ∞ f (k, θ) = (2l + 1)al (k)Pl (cos θ).
2 Hamiltonians and formal multiple scattering theory We are going to formulate this in terms of Hamiltonians, which turn up naturally in discussions about Green’s functions. In fact, the Green’s function is defined in terms of the Hamiltonian, as we saw in the beginning of this chapter. Let us define H0 to be the non-interacting Hamiltonian and H = H0 + V to be the Hamiltonian for the interacting system. We will assume that both systems have the same continuous (free electron) energy eigenstates, and that the interacting system also has a discrete spectrum of bound states below this continuum.
Advances in Chemical Physics, Vol.2 (Interscience 1958) by I Prigogine