Iterative diagonalization

For the calculation of the LDA ground-state one can proceed either via the direct" methods, i.e. via the glocal minimization of the total free energy with respect to the electronic degrees of freedom, or via the the diagonalization (for large PW basis-sets necessarily iterative diagonalization) of the KS Hamiltonian in combination with an iterative update of chai ge-density and potential. 

Figure 1 Convergence of the total energy and of the Hellmann-Feynman forces for ensembles of paramagnetic Fe atoms with 4 to 32 atoms. Part (a) shows the results of non-selfconsistent calculations performed with a fixed potential, part (b) the results of selfconsistent calculations. Full lines represent the RMM-DIIS (iterative diagonal-ization) results, broken lines the CGa (total-energy minimization) calculations. (4. text.

Section II discusses the real wave packet propagation method we have found useful for the description of several three- and four-atom problems. As with many other wave packet or time-dependent quantum mechanical methods, as well as iterative diagonalization procedures for time-independent problems, repeated actions of a Hamiltonian matrix on a vector represent the major computational bottleneck of the method. Section III discusses relevant issues concerning the efficient numerical representation of the wave packet and the action of the Hamiltonian matrix on a vector in four-atom dynamics problems. Similar considerations apply to problems with fewer or more atoms. Problems involving four or more atoms can be computationally very taxing. Modern (parallel) computer architectures can be exploited to reduce the physical time to solution and Section IV discusses some parallel algorithms we have developed. Section V presents our concluding remarks. 

We conclude that the problem of finding optimal NOs turns into the iterative diagonalization of Eq. (60) with a Fockian matrix, Eq. (66). The corresponding eigenfunctions are certainly orthonormal and optimize the total energy functional, Eq. (41). 

The one-electron equations (60) offer a new possibility for finding the optimal NSOs by iterative diagonalization of the Fockian, Eq. (66). The main advantage of this method is that the resulting orbitals are automatically orthogonal. The first calculations based on this diagonalization technique has confirmed its practical value [81]. 

The direct Cl equations are obtained by combining the normal Cl equations (3.3) with an iterative diagonalization procedure. (The same direct Cl equations can also be obtained within a perturbation theory approach). Since diagonalization procedures have been described in another set of lectures we will here only repeat the most essential results. The simplest iterative procedure is obtained by moving everything but the diagonal terms in the Cl equations over to the right hand side and assume that this side of the equations can be obtained from the Cl vector of the previous iteration C. ... 

Equation 5.82, a slight modification of Eq. 5.78, is the key equation in calculating the ab initio Fock matrix (you need memorize this equation only to the extent that the Fock matrix element consists of //corc, P, and the two-electron integrals). Each density matrix element Ptu represents the coefficients c for a particular pair of basis functions (f>, and (f> , summed over all the occupied MO s > /, (i 1,2,., n). We use the density matrix here just as a convenient way to express the Fock matrix elements, and to formulate the calculation of properties arising from electron distribution (Section 5.5.4), although there is far more to the density matrix concept [27]. Equation 5.82 enables the MO wavefunctions ij/ (which are linear combinations of the c s and s) and their energy levels e to be calculated by iterative diagonalization of the Fock matrix. 

Trouiller N, Martins JL, Efficient pseudopotentials for plane-wave calculations II Operators for fast iterative diagonalization, Phys Rev B, 43, 8861 (1991)... 

There is a range of iterative diagonalization routines to choose between, including classical orthogonal polynomial expansion methods [48], Davidson iteration[58] and Krylov subspace iteration methods. Here the popular Lanezos method[59] will be discussed in the context of finding the eigenstates of the surface Hamiltonian appearing in the hyperspherical coordinate method. 

We have shown how accurate boundary conditions can be implemented into the RBU model and illustrated the effect of using approximate boundary conditions. We have also described the guided spectral transform, GST, method for iterative diagonalization of large sparse matrices. 

One further observation must be made about the loss of spin symmetry in the Cl vector in the iterative diagonalization of the Hamiltonian. Even very slight deviations from (117), such as might occur from roundoff errors, become magnified in subsequent iterations and cause the iteration procedure to become numerically unstable because precise adherence to (117) is assumed if any of the Ms — 0 simplifications just described. If necessary, these difficulties can be avoided by explicitly enforcing the spin symmetry of any new vector in the subspace expansion. In this respect it is important to modify the diagonal elements of the Hamiltonian in the preconditioner for the subspace iteration method, as already discussed in section 3.2. 

Owing to the presence of the terms D" and e , which contain the unknown C , Eq. (39) has to be solved iteratively. Diagonal dominance is guaranteed if this is done in canonical MO basis, i.e. by writing C as... 

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