Davidsons Algorithm

Description Coupled to a Block-Davidson Algorithm An Efficient Scheme to Calculate Highly Excited Vibrational Levels. 

Relativistic generalization for G-spinor basis functions of the well-known McMurchie-Davidson algorithm [3] for direct evaluation of interaction integrals. 

Due to its dimensionality, a complete diagonalization of the Cl matrix is not feasible. One usually looks only for a few roots by means of an iterative procedure. For quantum chemical purposes, the Davidson algorithm or modifications thereof have proven to be well suited.135-137 In its original... 

Correction vectors lead to inhomogeneous linear equations of the form Ax = b that involve sparse matrices in the VB representation. An iterative small mar trix method gives rapid convergence using coordinate relaxation that resembles the Davidson algorithm for eigenvalues. We summarize the procedure here [48, 40]. We begin with a set of m orthonormal A-vectors Q,- i = 1... m and construct x the zeroth approximation to the true solution x as,... 

The McMurchie-Davidson algorithm [111] is used extensively for the calculation of two electron interaction integrals in nonrelativistic electronic structure calculations. The relativistic generalization [112] can be obtained immediately... 

Electron repulsion integrals may be evaluated by a straightforward generalization of the McMurchie-Davidson algorithm [34], using the definition of the two-spinor charge operator. The Coulomb interaction involves only the Eg-coefficients for q = 0, and results in G-spinor integrals of the form... 

It is here that the simplifications made possible by defining the , -coefficients is most apparent, since integrals over 612 may be be obtained by a simple extension of the McMurchie-Davidson algorithm for electron repulsion integrals. This approach conveys the advantages of simplicity, generality and efficiency when compared with those which exist in the literature [49, 50]. 

In the low-temperature (large-)8) limit, the partition functions are dominated by contributions from low-lying states, which can be determined selectively and inexpensively by the Davidson algorithm [11]. Hence, we propose approximating them by... 

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