High order finite difference algorithms for solving the Schrödinger equation in molecular dynamics

Hamiltonian Computational Chemistry: Geometrical Structures in Chemical Dynamics and Kinetics

The common geometrical (symplectic) structures of classical mechanics, quantum mechanics, and classical thermodynamics are unveiled with three pictures. These cardinal theories, mainly at the non-relativistic approximation, are the cornerstones for studying chemical dynamics and chemical kinetics. Working in extended phase spaces, we show that the physical states of integrable dynamical systems are depicted by Lagrangian submanifolds embedded in phase space. Observable quantities are calculated by properly transforming the extended phase space onto a reduced space, and trajectories are integrated by solving Hamilton's equations of motion. After defining a Riemannian metric, we can also estimate the length between two states. Local constants of motion are investigated by integrating Jacobi fields and solving the variational linear equations. Diagonalizing the symplectic fundamental matrix, eigenvalues equal to one reveal the number of constants of motion. For conservative systems, geometrical quantum mechanics has proved that solving the Schrödinger equation in extended Hilbert space, which incorporates the quantum phase, is equivalent to solving Hamilton's equations in the projective Hilbert space. In classical thermodynamics, we take entropy and energy as canonical variables to construct the extended phase space and to represent the Lagrangian submanifold. Hamilton's and variational equations are written and solved in the same fashion as in classical mechanics. Solvers based on high-order finite differences for numerically solving Hamilton's, variational, and Schrödinger equations are described. Employing the Hénon-Heiles two-dimensional nonlinear model, representative results for time-dependent, quantum, and dissipative macroscopic systems are shown to illustrate concepts and methods. High-order finite-difference algorithms, despite their accuracy in low-dimensional systems, require substantial computer resources when they are applied to systems with many degrees of freedom, such as polyatomic molecules. We discuss recent research progress in employing Hamiltonian neural networks for solving Hamilton's equations. It turns out that Hamiltonian geometry, shared with all physical theories, yields the necessary and sufficient conditions for the mutual assistance of humans and machines in deep-learning processes.

Resonant Fragmentation of the Water Cation by Electron Impact: a Wave-Packet Study

We have investigated the dissociation of a resonant state that can be formed in low energy electron scattering from H2 O+ . We have chosen the second triplet resonance above theB ˜ 2 A ' ${{{\tilde{\rm {B}}}}\;^2 {\rm{A{^\prime}}}}$ ( B ˜ 2 B 2 ) ${{\rm{(\tilde{B}}}\;^2 {\rm{B}}_2 )}$ state of H2 O+ whose autoionization mainly produces H2 O+ (X ˜ 2 A ' ' ${{{\tilde{\rm {X}}}}\;^2 {\rm{A{^\prime}{^\prime}}}}$ ). We have considered both dissociation of the resonant state itself, dissociative recombination (DR), or the dissociation of the H2 O+ cation after autodetachment, dissociative excitation (DE). The time-evolution of a wave packet on the potential energy surfaces of the resonance and cationic states shows, for the initial conditions studied, that the probability for DR is about 38 % while the probability for DE is negligible.

Nonadiabatic fragmentation of H2O+ and isotopomers. Wave packet propagation using ab initio wavefunctions

The nonadiabatic fragmentation of excited water cations (and isotopomers) is studied by propagating wave packets on ab initio potential energy surfaces.

Nonadiabatic Quantum Dynamics Predissociation of H2O(+)(B̃ (2)B2)

A quantum-mechanical study of the predissociation of H2O(+) (B̃ (2)B2) is carried out by using wave packet propagations on ab initio potential energy surfaces connected by nonadiabatic couplings. The simulations show that within the first 30 fs 80% of the initial wave packet is transferred from the B̃ (2)B2 to the à (2)A1 electronic state through a conical intersection. A much slower transfer (in the ps time scale) from the à(2)A1 to the X̃ (2)B1 state due to a Renner-Teller coupling determines the fragmentation branching ratios, which are in accordance with the experimental measurements.

Application of Coulomb wave function discrete variable representation to atomic systems in strong laser fields

We present an efficient and accurate grid method for solving the time-dependent Schrodinger equation for an atomic system interacting with an intense laser pulse. Instead of the usual finite difference (FD) method, the radial coordinate is discretized using the discrete variable representation (DVR) constructed from Coulomb wave functions. For an accurate description of the ionization dynamics of atomic systems, the Coulomb wave function discrete variable representation (CWDVR) method needs three to ten times fewer grid points than the FD method. The resultant grid points of the CWDVR are distributed unevenly so that one has a finer grid near the origin and a coarser one at larger distances. The other important advantage of the CWDVR method is that it treats the Coulomb singularity accurately and gives a good representation of continuum wave functions. The time propagation of the wave function is implemented using the well-known Arnoldi method. As examples, the present method is applied to multiphoton ionization of both the H atom and the H(-) ion in intense laser fields. The short-time excitation and ionization dynamics of H by an abruptly introduced static electric field is also investigated. For a wide range of field parameters, ionization rates calculated using the present method are in excellent agreement with those from other accurate theoretical calculations.