Conditional Born–Oppenheimer Dynamics: Quantum Dynamics Simulations for the Model Porphine

Abstract

We report a new theoretical approach to solve adiabatic quantum molecular dynamics halfway between wave function and trajectory-based methods. The evolution of a N-body nuclear wave function moving on a 3N-dimensional Born− Oppenheimer potential-energy hyper-surface is rewritten in terms of single-nuclei wave functions evolving nonunitarily on a 3-dimensional potential-energy surface that depends parametrically on the configuration of an ensemble of generally defined trajectories. The scheme is exact and, together with the use of trajectory-based statistical techniques, can be exploited to circumvent the calculation and storage of many-body quantities (e.g., wave function and potential-energy surface) whose size scales exponentially with the number of nuclear degrees of freedom. As a proof of concept, we present numerical simulations of a 2-dimensional model porphine where switching from concerted to sequential double proton transfer (and back) is induced quantum mechanically. O n the basis of the Born−Huang expansion of the molecular wave function, 1 an exact description of adiabatic molecular dynamics requires the propagation of a nuclear wavepacket on the ground-state Born−Oppenheimer potential-energy surface (gs-BOPES). This propagation scheme is, somehow, computationally doubly prohibitive. Besides the computational burden associated with the propagation of the (many-body) nuclear wave function, the calculation of the gs-BOPES constitutes, per se, a time-independent problem that grows exponentially with the number of electrons and nuclei. In this respect, two main classes of computational methods have emerged depending on whether the knowledge of the gs-BOPES is required in the full configuration space, that is, fullquantum methods, 2,3 or only at certain reduced number of points, namely trajectory-based or direct methods.4,5 While methods for computing the energy of any configuration of nuclei have become quicker and more accurate, full-quantum dynamics calculations still become rapidly unfeasible for large molecules. Alternatively, direct dynamics notably reduce the computational cost of the simulations by avoiding partially, sometimes completely, the calculation of the full gs-BOPES (this can be done, for instance, by the use of reaction-path Hamiltonians 6,7 ). Nuclear quantum effects, however, can be hardly included systematically in this second class of methods. Up to date, only quantum-trajectory methods have the particularity of being able to describe all nuclear quantum effects (just as full-quantum methods) and being on-the-fly simultaneously.8−11 Unfortunately, these methods have serious problems in dealing with the so-called quantum potential, which gathers, by definition, all quantum information on the system. The mathematical structure of the quantum potential depends on the inverse of the quantum probability density, and thus, its manipulation entails serious instability problems. 12−14We report here an exact theoretical approach to solve adiabatic quantum molecular dynamics based o...

Cavity Born–Oppenheimer Approximation for Correlated Electron–Nuclear-Photon Systems

“…25,26 The discussed methods can be still improved, e.g. along the lines of a more accurate factorization method such as the exact factorization 54−56 known for electron–nuclear problems, or trajectory based methods 50,57 can be applied to simulate such systems dynamically. This work has direct implications on more complex correlated matter-photon problems that can be approximately solved employing the cavity Born–Oppenheimer approximation to better understand complex correlated light-matter coupled systems.…”

Entangled Quantum Dynamics of Many-Body Systems using Bohmian Trajectories

“…[25] which does not take entanglement into consideration. It was also employed recently in [35,36] in order to devise an approximate solution for electron-nuclear dynamics in molecular systems. In Fig.…”

“…• We use the split operator technique [36] in order to propagate the exact two-particle wavefunction in imaginary time (to generate the ground state) and in real time (as in Fig. 1) or to generate the exact Bohmian trajectories (as in Fig.…”

Different flavors of nonadiabatic molecular dynamics

“…Quantum (or Bohmian) trajectories were also proposed as a mean to go beyond the independent trajectory approximation of TSH . Algorithms have also been proposed exploiting the idea of conditional electronic wavefunctions …”