Direct diagonalization

In cubic closest-packing, consideration of the face-centered unit cell is a convenient way to get an impression of the arrangement of the interstices. The octahedral interstices are situated in the center of the unit cell and in the middle of each of its edges [Fig. 17.3(a)], The octahedra share vertices in the three directions parallel to the unit cell edges. They share edges in the directions diagonal to the unit cell faces. There are no face-sharing octahedra. 

As the modification of the original Hamiltonian matrix is involved in the diagonalization methods discussed above, we denote such approaches as direct diagonalization to distinguish them from the recursive ones discussed below. The direct diagonalization process is illustrated in Figure 1. 

As discussed, an alternative to direct diagonalization is by recursion. The recursive diagonalization approach has several attractive features, including more favorable scaling laws, which make it ideally suited for large eigenproblems. For example, some applications of linear-scaling recursive methods in... 

Because many physical systems possess certain types of symmetry, its adaptation has become an important issue in theoretical studies of molecules. For example, symmetry facilitates the assignment of energy levels and determines selection rules in optical transitions. In direct diagonalization, symmetry adaptation, often performed on a symmetrized basis, significantly reduces the numerical costs in diagonalizing the Hamiltonian matrix because the resulting block-diagonal structure of the Hamiltonian matrix allows for the separate... 

Then the (w -f 1) x (m -f 1) eigenvalue matrix can be constructed from (9) and can be solved exactly by direct diagonalization or with the aid of perturbation methods for the eigenvalues with the external magnetic field along any principal axis, (m -f 1) Energy values and therefore m Am, = 1... 

While direct diagonalization of the Hamiltonian matrix works well for situations in which there is a finite number of states, as in Fig. 9.1, it is clearly hopeless to try it in this case. A useful WKB approach was proposed by Edmonds12 and refined by Starace.13 Using the fact that azimuthal symmetry exists, Starace writes the wavefunction of the spinless Rydberg electron in cylindrical coordinates as13... 

We now study the disordered effective hamiltonian (4.4). Since a direct diagonalization of (4.4) is too hard, we shall have to use approximations which are conveniently expressed in the resolvent (or Green s function) formalism. The translation-invariant K sum in HeU is restricted to the optical wave vectors only (for K oj/c, RK / K 0I)- Therefore, it is possible to restrict the problem to this small part of the Brillouin zone using the projector operator... 

Summing up, we see that the traditional approach to impurity problems within the Green s-function formalism exploits the basic idea of splitting the problem into a perfect crystal described by the operator and a perturbation described by the operator U. The matrix elements of < are then calculated, usually by direct diagonalization of or by means of the recursion method. Following this traditional line of attack, one does not fully exploit the power of the memory function methods. They appear at most as an auxiliary (but not really essential) tool used to calculate the matrix elements of... 

The CPU scaling of the GST method with the size of the basis set exjjansion has been investigated in an application to CI-1-CH4. A scaling better than N, close to A"lnA in fact, was observed as illustrated in Fig. 6. This is cilso the scaling normally found in wavepacket Ccilculations and decidedly better than the api)roximately scaling observed in direct diagonalization methods. Note also that the GST method has only smcill requirements on primary memory. 

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