Recursive diagonalization methods

As indicated by the Kronecker deltas in the above equation, the resulting Hamiltonian matrix is extremely sparse and its action onto a vector can be readily computed one term at a time.12,13 This property becomes very important for recursive diagonalization methods, which rely on matrix-vector multiplication ... 

In recent years, state-of-the-art recursive diagonalization methods have been applied to bound-states problems for LiCN,152 H20,12,117,239-241 CH2,242 HCN,i3,80,105-107,241 40,67,164,243-245 246 H0C1,247,248 N02,76,249-253... 

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... 

To understand the recursive diagonalization idea, it is instructive to examine the power method.18,20 Assuming that the eigenvector of H corresponding to its largest eigenvalue (bi, 1 = smax) is contained in an initial vector (q) ... 

The aforementioned applications of recursive methods in reaction dynamics do not involve diagonalization explicitly. In some quantum mechanical formulations of reactive scattering problems, however, diagonalization of sub-Hamiltonian matrices is needed. Recursive diagonalizers for Hermitian and real-symmetric matrices described earlier in this chapter have been used by several authors.73,81... 

SC) CAS-SDCI may be viewed as a set of d equations to be solved simultaneously (actually, the implementation of the method is a recursive diagonalization process). In contrast, ec-CCSD implies a unique diagonalization and a further resolution of a rather small set of non-linear equations. 

Diagonalization Homogeneous Recursive Filtering and a Low-Storage Method for the Calculation of Matrix Elements. 

We have generated a chain of variables with diagonal matrix elements a and nearest-neighbor interaction b . The explicit calculation of and b can be performed with standard programs for any operator expressed in a local basis. In Fig. 2 we give a schematic picture of the chain-model variables generated by the recursion method in a simple two-dimensional square lattice. 

As opposed to the Lanczos method, in which the diagonali2 tion of (3.24) is performed, the recursion method focuses on the construction of the diagonal Green s-function matrix element... 

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... 

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