Narrow Results

We introduce a new two-dimensional model with diagonal four spin exchange and an exactly known ground-state. Using variational ansätze and exact diagonalisation we calculate upper and lower bounds for the critical coupling of the model. Both for this model and for the Shastry–Sutherland model we study periodic systems up to system size 6×6.

Properties of shell model Hamiltonians, and in particular the tri-diagonal matrices obtained with the Lanczos procedure are exploited to determine the density of states and to estimate the ground-state energies. It will be shown that with a few parameters that are adjusted to the moments up to fourth order of the Hamiltonian, one can model the resultant tri-diagonal matrices. An estimate of the ground state energy can quickly be obtained from the modeled matrices, and the density of states for the full matrix can quickly be obtained using the WKB approximation. The accuracy of the ground state is dependent on high moments, six and greater.

Cross correlation (CC) functions C_if(E) play an important role in chemical physics. They appear in the description of reactive scattering, photo-dissociation, photo-electron spectroscopy and electron transfer to mention a few. In this paper, we discuss two methods based on the Lanczos algorithm to compute the CC function for several initial and final states at the same time, without diagonalization. The property of the coupled two-term recursions variant of the Lanczos algorithm that yields a decomposition of the tridiagonal Lanczos matrix is crucial. The first method, the quasi minimal-recursive residue generation method (QM-RRGM) is based on solving a set of linear equations whereas the second method is based on a band-Lanczos method. The computational cost is of the same order of magnitude for both methods and is given by the number of matrix-vector multiplications in the underlying Lanczos method. Only a small set of scalars needs to be updated each recursion. The methods are compared for a model problem, the continuum resonance Raman cross section for a collinear model of CH₂IBr. Both methods shows similar convergence properties. By adding a pre-conditioner, the rate of convergence can be increased dramatically.