Bohmian approach

Although trajectories are not computed in QFD-DFT, it is clear that there is a strong connection between this approach and the trajectory or hydrodynamical picture of quantum mechanics [20], independently developed by Madelung [21], de Broglie [22], and Bohm [23], which is also known as Bohmian mechanics. From the same hydrodynamical equations, information not only about the system... 

Here we focus on yet another implementation, the single-particle hydrodynamic approach or QFD-DFT, which provides a natural link between DFT and Bohmian trajectories. The corresponding derivation is based on the realization that the density, p(r, t), and the current density, j(r, t) satisfy a coupled set of classical fluid, Navier-Stokes equations ... 

In Bohmian mechanics, the way the full problem is tackled in order to obtain operational formulas can determine dramatically the final solution due to the context-dependence of this theory. More specifically, developing a Bohmian description within the many-body framework and then focusing on a particle is not equivalent to directly starting from the reduced density matrix or from the one-particle TD-DFT equation. Being well aware of the severe computational problems coming from the first and second approaches, we are still tempted to claim that those are the most natural ways to deal with a many-body problem in a Bohmian context. 

The question then arises if a convenient mixed quantum-classical description exists, which allows to treat as quantum objects only the (small number of) degrees of freedom whose dynamics cannot be described by classical equations of motion. Apart in the limit of adiabatic dynamics, the question is open and a coherent derivation of a consistent mixed quantum-classical dynamics is still lacking. All the methods proposed so far to derive a quantum-classical dynamics, such as the linearized path integral approach [2,6,7], the coupled Bohmian phase space variables dynamics [3,4,9] or the quantum-classical Li-ouville representation [11,17—19], are based on approximations and typically fail to satisfy some properties that are expected to hold for a consistent mechanics [5,19]. 

Burghardt, I. Dynamics of coupled Bohmian and phase-space variables a moment approach to mixed quantum-classical dynamics. J. Chem. Phys. 122 94103 (2005). 

Virtually no aspect of chemistry is left untouched by the new insight gained through consideration of Bohmian mechanics, number theory and symmetry. A preliminary analysis of some of these aspects are presented here in an effort to stimulate further research, that may happen once familiarity with the new approach overcomes traditional scepticism. Most of the material in this book has been published in somewhat different form before. Chapters 1, 2, 4, 5, and 6 are elaborations based on recent review articles [8, 9, 10, 11] by the author and included here with permission by the relevant publishers. 

Recently [8-11] an alternative treatment to mix quantum mechanics with classical mechanics, based on Bohmian quantum trajectories was proposed. Briefly, the quantum subsystem is described by a time-dependent Schrodinger equation that depends parametrically on classical variables. This is similar to other approaches discussed above. The difference comes from the way the classical trajectories are calculated. In our approach, which was called mixed quantum-classical Bohmian (MQCB) trajectories, the wave packet is used to define de Broglie-Bohm quantum trajectories [12] which in turn are used to calculate the force acting on the classical variables. 

Putz, M. V. (2012). Valence atom with Bohmian quantum potential the golden ratio approach. Chemistry Central Journal, 6, 135/16 pp. (DOI 10.1186/1752-153X-6-135). 

The consequences of the joint consideration of Bohmian mechanics and the golden ratio for the main atomic systems will be explored, and the quantum chemical valence state will be accordingly described alongside the so-called universal electronegativity and chemical hardness, refining the work of (Parr Bartolotti, 1982) as well as generalizing the previous Bohmian-Boeyens approach (Boeyens, 2005, 2011). 

Characterizes electronegativity and chemical hardness by both quantum observability perspective as well as by Bohmian subquantum approach, under the most disputed and discussed parabolic eneigetic shape of valence electrons occupancy ... 

The following three sections describe the Bohmian quantum-classical approach [22,23] that uniquely solves the quantum back-reaction branching problem, the stochastic mean-field approximation [20] (SMF) that both resolves the back-reaction problem and incorporates the quantum decoherence and Franck-Condon overlap effects into NA-MD, and the quantized mean-field method [21] (QMF) that takes into account ZPE. The Bohmian and QMF approaches are illustrated by a model designed to capture some features of the O2 dissociation on a Pt surface. The concluding section summarizes the features of the methods and discusses further avenues for development and consideration. 

Numerical illustration of the Bohmian quantum-classical and quantized mean-field approaches... 

The Bohmian quantum-classical and QMF approaches [21,22] have been applied to a model intended as a simplified representation of gaseous oxygen interacting with a platinum surface, Ref. [13,91]. The model consists of a light particle q with mass m colliding with a heavier particle Q with mass M. The heavy particle is bound to an immobile surface. The total Hamiltonian for the system is given by... 

The Bohmian quantum-classical and QMF approaches have been tested with a model application designed to simulate the interaction of an oxygen molecule with a platinum surface [13,21,22,91]. With trajectory branching the Bohmian quantum-classical method recovers the correct asymptotic behavior of the scattering probability of the quantum subsystem. The QMF approach shows improvement in both the short and long time scattering probabilities. The improvement is achieved due to the proper treatment of ZPE. 

To recapitulate, the Bohmian quantum-classical, stochastic mean-field and quantized mean-field approaches described above are capable of reproducing quantum solvent effects that are crucial in simulation of NA chemical processes. The approaches are computationally simple and are particularly suitable for studies of large chemical systems. 

In this respect, the more deep approaches of quantum description of the chemical bonding advocates in making the required steps toward assessing the quantum particle of the chemical bond as based on the derived chemical field released at its turn by the fundamental electronic equations of motion either within Bohmian non-relativistic (Schrodinger) and to explore the first consequences. If successful, the present endeavor will contribute to celebrate the dream in unveiling the true particle-wave nature of the chemical bond. 

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