Theoretical studies of vibrationally excited polyatomic molecules using canonical Van Vleck perturbation theory

Abstract

The highly excited vibrational states of polyatomic molecules are investigated using canonical Van Vleck perturbation theory, implemented in a superoperator framework. This approach is used to transform a vibrational Hamiltonian to a new representation which has a form ideally suited to the study of the dynamics of interest. The key advantage is that the solution to the full problem is obtained in the new representation using significantly smaller basis sets than are needed to obtain the solutions in the original representation. The transformations are applied to the Hamiltonian operator itself, not the Hamiltonian matrix; this superoperator approach obviates the need for large basis sets. The tedious and complex algebra, that is required to perform these transformations, is readily implemented with FORTRAN codes. Combining these two features has enabled the investigations of vibrational dynamics in energy regimes and densities of states, unattainable by standard methods. These methods are applied to two model problems and to the study of the highly excited overtones of CHD3 with up to five quanta of excitation in the CH bond.

Quantum integrability and action operators in spin dynamics

“…The associated problems of convergence are known to occur in various perturbational approaches, including transformations to the quantum mechanical Birkhoff-Gustaffson normal form [29], algebraic quantization, [30] and quasidegenerate perturbation theory. [26,27] From the vantage point of a quantum-mechanical notion of integrability, the lack of convergence of the perturbational schemes now appears in a new light. It may open an avenue to test the criterion QJ(N ) of quantum integrability for a given system on a rigorous basis.…”

“…The KAM theorem provides information about which tori survive a specific perturbation. [25] Van Vleck's formulation of quantum-mechanical perturbation theory, [26,27] which is akin to the Lie transform method, [28] seems to be suitable for the implementation of the diagonalizing unitary transformation U for a one-spin system. Here the strategy is to systematically remove the (off-diagonal) creation and annihilation operators (or spin ladder operators) from the Hamiltonian in increasing orders.…”

“…for easy insertion of the solutions (27). The time average leaves the first term intact and wipes out the second one.…”

“…The time evolution of an arbitrary dynamical variable F = F ( S) = F(Ĵ) (here with F = F) is then obtained by insertion of (27). As in classical mechanics (Sec.…”

First-principles anharmonic quantum calculations for peptide spectroscopy: VSCF calculations and comparison with experiments

“…[14][15][16][17][18][19][20][21] It has been successfully used for small to medium sized biomolecules. [22][23][24][25][26] Other alternatives such as perturbation theory 27,28 and classical MD simulations 29 are also available. The former has been applied with good accuracy for small biomolecules.…”