Angular momentum representation

From the two-center expansion of the Green function, this equation implies X(r) = -J2 Jl(r) J2 J2 

This derivation implies that these equations are valid under the condition that no adjacent nucleus should lie within the enclosing sphere of a given local cell. 

If the matrix C is not singular, which requires the number of basis functions to match the number of solid harmonics used to expand the Green function, a local r-matrix is defined by t = —SC l. The consistency condition expressed above in terms of C and S matrices then reduces to the simple matrix expression 

This is the fundamental equation of multiple scattering theory. It has the remarkable property of concentrating effects of the local potential function into the 

The Korringa-Kohn-Rostoker (KKR) method [207,204] is implemented by searching for the zeroes of the secular determinant det(r 1 — g), or det(7 — tg) if t is singular. 

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Figure 8.1 Schematic structure of the Hamiltonian matrix for a collision system in an external field expressed in the total angular momentum representation.

To take advantage from the pseudo-angular momentum representation we shall employ the technique of the irreducible tensor operators as suggested in Ref. [10]. One can easily establish the following interrelations between the matrices Orr and the orbital angular momentum operators ... 

The projection integrals (Yjo y>7) can be interpreted as the (discrete) angular momentum representation of the initial bending wavefunction in the electronic ground state. Employing the semiclassical limit for the spherical harmonics,... 

After about 50 fs (t) has reached the asymptotic region where the torque is essentially zero and the distribution does not alter any further, i.e., the dissociation is over. Figure 10.6 illustrates rotational excitation in the angular momentum representation, whereas Figure 10.2(b) manifests rotational excitation in the coordinate picture. 

Using the standard total angular momentum representation JMjft) eorrt sponding to the basis set of Eq.(96) (see for instance the book [1]) we obtain the following form of Hamiltonian for the radial nuclear motion (atomic units) ... 

The first method for solving the MST problem in angular momentum representation was made by Korringa [43] and Kohn and Rostocker [44] separately. The method came to be called the KKR method for electronic structure calculations and used the Green s function technique from Chapter 3 to solve the electronic structure problem. The separation into potential- and structure dependent parts made the method conceptually clean and also speeded up calculations, since the structural dependent part could be calculated once and for all for each structure. Furthermore, the Green s function technique made the method very suitable for the treatment of disordered alloys, since the Coherent Potential Approximation [45] could easily be implemented. 

Our current direction is to study dynamical processes in atomic and molecular two- and three-body systems. We use a technique which formally is based on the mathematical theory of dilation analytic functions. Numerically these results axe realized though a fully three-dimensional finite element method applied to a total angular-momentum representation. We here show how generalizations of our previously published two-body methods to three-body systems are possible without formal approximations. 

This method has some advantages compared to the total angular momentum representation [791. It is widely used in the theory of direct nuclear reactions (Glendenning [431. Satchler [721. see also Levine [541),... 

As described in Section 1.3.2 above, field-fi-ee scattering calculations are usually carried out in total angular momentum representations. The equations arising from each value of the total angular momentum are independent and may be solved separately. In the presence of an applied field, this symmetry is lost and there is much less advantage in working in a total angular momentum representation. 

For the angular part, the transformation matrix between the coupled angular momentum representation and the angular grid representation is given by ... 

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