Quantum Hydrodynamics with Trajectories: The Nonlinear Conservation Form Mixed/Discontinuous Galerkin Method with Applications in Chemistry

The basic idea emerging from quantum chemistry in the context of QHD is to employ the time-dependent Schrödinger equation (TDSE) to solve for the dynamical properties (probability densities, “particle” velocities, etc.) of chemical systems. In the same spirit in which the de Broglie-Bohm interpretation (see [4,5,16]) of quantum mechanics may be used to recover “trajectories” of individual fluid elements along the characteristics of motion of the solution, the QHD equations of Madelung and Bohm are derived as formally equivalent to the TDSE and thus comprise an alternative route to solutions which generate quantum trajectories that follow particles along their respective paths (see [38] and [18] for a comprehensive overview).

Figure 1: Here we have the intramolecular rearrangement of the aryl radical 2, 4, 6-tri-tert-butylephenyl to 3, 5-di-tert-butylneophyl (see [6] for details).

These solutions hold particular significance, where, in the context of the QHD formulation, it is possible to resolve the chemical dynamics of a vast number of reaction mechanisms known to have pathways dominated by quantum Figure 2: Here we show an enzymatic catalysis -an aromatic amine dehydrogenase (AADH) with a tryptophan tryptophyl quinone (TTQ) prosthetic group catalyzing the oxidative deamination of tryptamine with an electron transfer to an arsenate reductase enzyme (see [6] and [26] for details, PDB codes: 1nwp (azurin), 2agy (AADH)). tunneling regimes. Some of these systems include proton transfer reactions (for example see figure 1), conformational inversions, biologically important redox reactions in enzymatic catalysis reactions (see figure 2), and proton-coupled electron transfer reactions (refer to [27] and [26]). It is not yet clear if these types of methods may also have application at higher energies, for example in the halo nuclei tunneling occurring in fusion reactions (as seen, for example, in [17]).

Substantial research has been done in quantum hydrodynamics to find the best and fastest computational methodology for solving this system of equations. In the standard methodology presented using the quantum trajectory method (QTM), for example, solutions to the QHD equations are found by transforming the system of equations, which is generally posited in the Eulerian fixed coor-dinate framework (see [13,14,18,24]), into the same set of equations in the Lagrangian coordinate framework, which effectively follows solutions along particle trajectories; or along so-called “Bohmian trajectories.” The transformation from the Eulerian to the Lagrangian frame leads to a set of coupled equations which solve for two unknowns: the quantum action S(t, r) and the probability density or quantum amplitude (t, r) = R(t, r) along the trajectories r(t, x) (e.g. see [38] box 1.2). The obvious advantage of the Lagrangian framework is reduced computational times, since solutions are only computed along a set of chosen trajectories; while clearly the disadvantage is the possibility of obscuring structure hidden within the continuum of the full solution, which may only emerge properly in convergent numerical schemes, and also the increased complications of transposing into more complicated settings: such as with functional or time dependencies on the potential term V , or including dissipative or rotational vector fields.

In addition, the numerical solutions to the above mentioned Lagrangian formulations have demonstrated characteristic behaviors which introduce certain technical difficulties at the level of formal analysis. First, the system of equations are stiff, which is to say, solutions to the system may locally or globally vary rapidly enough to become numerical unstable without reducing numerically to extremely small timesteps. Furthermore, there exists the so-called “node problem,” which is characterized by singularity formation (see [38] for characterization of node types) along particle trajectories. Another issue which arises is obtaining unique solutions, since there is not a unique choice of trajectories in the Lagrangian formulation (see for example §6 and appendix A). And finally, boundary data is often treated without regard to the (often substantial) numerical residuals introduced in the weak entropy case, or taking into account consistency between the TDSE and the QHD system of equations (see for example [29] and §3).

We introduce an alternative formulation to the standard solutions described above in and S and tracked with respect to the Lagrangian coordinate frame which is motivated by work of Gardner, Cockburn, et al. (see [7,14,15]). Instead, we keep the system in its conservation form (instead of in a primitive variable form) in the Eulerian coordi

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