Simple leading eigenvalues

In order to get more experience with the newly proposed index ( ) we will consider the leading eigenvalue X of D/D matrices for several well-defined mathematical curves. We should emphasize that this approach is neither restricted to curves (chains) embedded on regular lattices, nor restricted to lattices on a plane. However, the examples that we will consider correspond to mathematical curves embedded on the simple square lattice associated with the Cartesian coordinates system in the plane, or a trigonal lattice. The selected curves show visibly distinct spatial properties. Some of the curves considered apparently are more and more folded as they grow. They illustrate the self-similarity that characterizes fractals. " A small portion of such curve has the appearance of the same curve in an earlier stage of the evolution. For illustration, we selected the Koch curve, the Hubert curve, the Sierpinski curve and a portion of another Sierpinski curve, and the Dragon curve. These are compared to a spiral, a double spiral, and a worm-curve. 

This scheme requires the exponential only of matrices that are diagonal or transformed to diagonal form by fast Fourier transforms. Unfortunately, this matrix splitting leads to time step restrictions of the order of the inverse of the largest eigenvalue of T/fi. A simple, Verlet-like scheme that uses no matrix splitting, is the following ... 

In the unrestricted treatment, the eigenvalue problem formulated by Pople and Nesbet (25) resembles closely that of closed-shell treatments.-On the other hand, the variation method in restricted open-shell treatments leads to two systems of SCF equations which have to be connected in one eigenvalue problem (26). This task is not a simple one the solution was done in different ways by Longuet-Higgins and Pople (27), Lefebvre (28), Roothaan (29), McWeeny (30), Huzinaga (31,32), Birss and Fraga (33), and Dewar with co-workers (34). 

Of the three eigenvalue equations, the one of interest to us is the vibrational equation. It has a particularly simple form when normal coordinates are employed because the classical kinetic and potential energies then have no cross terms (see eqns (9-2.17) and (9-2.18)) and this fact leads to a simple form for their quantum mechanical analogues (the kinetic energy and potential energy operators). The vibrational equation is thus... 

Here we have an exponential of the matrix D. The matrix is zero except on the diagonal (containing the eigenvalues Aj, A2,..., A/y), and, as Smith proves, this and the definition of a matrix exponential lead to the simple result that... 

This leads to a simple relationship between the two eigenvalues of the M matrix. Indeed, directly solving... 

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