Unitary transformation integrable systems

Classical dynamics and orthodox quantum mechanics are constructed along the model of integrable systems in the sense of Poincare. Our aim is to construct dynamics for nonintegrable systems. As far as we know, this is a new attempt, which has its roots in the early work of the Brussels School [1-9]. The main result is that we have to replace the unitary transformation f/ by a nonunitary... 

The unitary transformation from the basis of the CMOs to the basis of the LMOs of the /-system does not change the covalent contribution to the effective crystal field. According to numerical estimates the resonance integrals (,/. between d-AO and LPs of the donor atoms by I0-H00 times overcomes the resonance integrals between d- AO and any other LMOs and thus dominates the resonance interaction of the d- and /-systems. So, as it has been shown in [71], restricting the summation in eq. (4.82) by the sum of diagonal elements (L = L) over only the LPs results in error in the estimated splitting of the (/-levels of 0.1 eV. This precision is comparable to that of the EHCF method itself. This estimate is described by the formula ... 

Assuming that the parameters of the system are not altered, the variation of the action integral in eqn (8.78) arises only from infinitesimal changes of the sets of commuting observables at the two times t and tj. We have previously seen (eqn (8.41)) that such transformations may be characterized in terms of the generators of infinitesimal unitary transformations jF(t,) and F(t2), which act on the eigenvectors and q,2> 2>- Comparing eqns (8.41) and... 

There is no need to use Lagrange multipliers between the solutions — even and odd — of each system because orthogonality is automatically ensured through spin functions. In addition, each of the two subspaces of spin orbitals can be subjected to a unitary transformation which eliminates the off-diagonal multipliers within each system. After the integration over the spin coordinates has been done, we obtain two effective eigenvalue problems connected only by the common Coulomb potential for the entire molecule ... 

A molecular orbital located on a certain fragment of a molecular system and spatially separated from other orbitals as much as possible. LMOs are derived from the occupied canonical molecular orbitals by a unitary transformation determined with respect to a physical criterion, e.g., by maximizing the sum of squares of the centroids of occupied molecular orbitals (Foster-Boys procedure) or by minimizing the sum of the exchange (or Coulomb) repulsion integrals between the occupied orbitals (Edmiston-Ruedenberg procedure). 

The implicit-midpoint (IM) scheme differs from IE above in that it is symmetric and symplectic. It is also special in the sense that the transformation matrix for the model linear problem is unitary, partitioning kinetic and potential-energy components identically. Like IE, IM is also A-stable. IM is (herefore a more reasonable candidate for integration of conservative systems, and several researchers have explored such applications [58, 59, 60, 61]. 

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