Distribution of eigenvalues

Figure 1. Nearest-neighbor spacing distributions of eigenvalues for a circle (left) and the Bunimovich stadium (right). Taken from Ref. (McDonald and Kaufman, 1979).

Odelius and co-workers reported some time ago an important study involving a combined quantum chemistry and molecular dynamics (MD) simulation of the ZFS fluctuations in aqueous Ni(II) (128). The ab initio calculations for hexa-aquo Ni(II) complex were used to generate an expression for the ZFS as a function of the distortions of the idealized 7), symmetry of the complex along the normal modes of Eg and T2s symmetries. An MD simulation provided a 200 ps trajectory of motion of a system consisting of a Ni(II) ion and 255 water molecules, which was analyzed in terms of the structure and dynamics of the first solvation shell of the ion. The fluctuations of the structure could be converted in the time variation of the ZFS. The distribution of eigenvalues of ZFS tensor was found to be consistent with the rhombic, rather than axial, symmetry of the tensor, which prompted the development of the analytical theory mentioned above (89). The time-correlation... 

A famous and only partly solved problem of this type is the linear chain of harmonically bound particles, in which the masses and spring constants are random.5 0 A related problem is the determination of the distribution of eigenvalues of a random matrix. )... 

The shape of the ellipsoid was on average non-cylindrical. In fact since the distribution of eigenvalues was fairly symmetric, it is possible to describe with simply a rhombic component. The TCF of the form of the EFG shows a rapid initial decay to a plateau with a order parameter of 0.8 of the time zero value. The reorientational motion is multi exponential and is the main cause of the decay of the EFG-TCF. 

The first point is straight forward, but the second has some important qualifications. Although row and column sums arc one, entries are not limited to the range zero to one. This interpretation of the UPSR as a measure of the distribution of eigenvalue-to-state associations, while complicated by entries outside the range zero to one, does nevertheless remain the single most important interpretation of the UPSR. 

The distribution of eigenvalues and hence of relaxation times thus depends on the value of k[N ]°. In the limit of n going to infinity the maximum and minimum values of the eigenvalues are given by... 

Dean developed a simple method (based on his so-called negative eigenvalue theorem) for determining the distribution of eigenvalues (density of states) in order to calculate the vibrational spectra of disordered systems. This method cannot be used — as we shall see — for simple topological reasons in the case of the electronic states of two- and three-dimensional solids (and therefore was generally not applied in... 

The author and cowoikers have introduced a number of theorems from the analytic theory of polynomial equations and perturbation theory for the purpose of gaming irrsight irrto the distribution of eigenvalues by simply knowing the Lanczos parameters. These theorerrrs, which include the Gershgorin circle theorems, enable one to constract spectral domains in the complex plane to which the eigenvalues of L) and are confined. It has been... 

By giving X different values throughout the spectrum and taking the difference of the number of negative i( v) s belonging to consecutive values, one can obtain a histogram for the distribution of eigenvalues (density of states) of for any desired accuracy. 

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