The von Neumann basis in non-Cartesian coordinates: Application to floppy triatomic molecules

Short-periodic-orbit method for excited chaotic eigenfunctions

An alternative method for the calculation of excited chaotic eigenfunctions in arbitrary energy windows is presented. We demonstrate the feasibility of using wave functions localized on unstable periodic orbits as efficient basis sets for this task in classically chaotic systems. The number of required localized wave functions is only of the order of the ratio t_{H}/t_{E}, with t_{H} the Heisenberg time and t_{E} the Ehrenfest time. As an illustration, we present convincing results for a coupled two-dimensional quartic oscillator with chaotic dynamics.

A-VCI: A flexible method to efficiently compute vibrational spectra

The adaptive vibrational configuration interaction algorithm has been introduced as a new method to efficiently reduce the dimension of the set of basis functions used in a vibrational configuration interaction process. It is based on the construction of nested bases for the discretization of the Hamiltonian operator according to a theoretical criterion that ensures the convergence of the method. In the present work, the Hamiltonian is written as a sum of products of operators. The purpose of this paper is to study the properties and outline the performance details of the main steps of the algorithm. New parameters have been incorporated to increase flexibility, and their influence has been thoroughly investigated. The robustness and reliability of the method are demonstrated for the computation of the vibrational spectrum up to 3000 cm^-1 of a widely studied 6-atom molecule (acetonitrile). Our results are compared to the most accurate up to date computation; we also give a new reference calculation for future work on this system. The algorithm has also been applied to a more challenging 7-atom molecule (ethylene oxide). The computed spectrum up to 3200 cm^-1 is the most accurate computation that exists today on such systems.

Systematically expanding nondirect product bases within the pruned multi-configuration time-dependent Hartree (MCTDH) method: A comparison with multi-layer MCTDH

We propose a pruned multi-configuration time-dependent Hartree (MCTDH) method with systematically expanding nondirect product bases and use it to solve the time-independent Schrödinger equation. No pre-determined pruning condition is required to select the basis functions. Using about 65 000 basis functions, we calculate the first 69 vibrational eigenpairs of acetonitrile, CH₃CN, to an accuracy better than that achieved in a previous pruned MCTDH calculation which required more than 100 000 basis functions. In addition, we compare the new pruned MCTDH method with the established multi-layer MCTDH (ML-MCTDH) scheme and determine that although ML-MCTDH is somewhat more efficient when low or intermediate accuracy is desired, pruned MCTDH is more efficient when high accuracy is required. In our largest calculation, the vast majority of the energies have errors smaller than 0.01 cm^-1.

Quantum Dynamics in Phase Space using Projected von Neumann Bases

We describe the mathematical underpinnings of the biorthogonal von Neumann method for quantum mechanical simulations (PvB). In particular, we present a detailed discussion of the important issue of nonorthogonal projection onto subspaces of biorthogonal bases, and how this differs from orthogonal projection. We present various representations of the Schrödinger equation in the reduced basis and discuss their relative merits. We conclude with illustrative examples and a discussion of the outlook and challenges ahead for the PvB representation.

Neural network iterative diagonalization method to solve eigenvalue problems in quantum mechanics

The neural network iterative diagonalization structure for computing the eigenstates of complex symmetric or Hermitian matrices.