Bohmian

The theories that should feature prominently in the understanding of chemical effects have been summarized in this volume, without demonstrating their application. The way forward has been indicated by Primas [67] and in the second volume of this work the practical use of modern concepts such as spontaneous symmetry breaking, non-local interaction, bohmian mechanics, number theories and space-time topology, to elucidate chemical effects will be explored. The aim is to stimulate renewed theoretical interest in chemistry. 

Q Time-Dependent Density Functional Theory from a Bohmian Perspective... 

Bohmian Mechanics A Trajectory Picture of Quantum Mechanics. 112... 

Bohmian Trajectories Describing Many-Body Systems. 114... 

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... 

The purpose of this chapter is to show and discuss the connection between TD-DFT and Bohmian mechanics, as well as the sources of lack of accuracy in DFT, in general, regarding the problem of correlations within the Bohmian framework or, in other words, of entanglement. In order to be self-contained, a brief account of how DFT tackles the many-body problem with spin is given in Section 8.2. A short and simple introduction to TD-DFT and its quantum hydrodynamical version (QFD-DFT) is presented in Section 8.3. The problem of the many-body wave function in Bohmian mechanics, as well as the fundamental grounds of this theory, are described and discussed in Section 8.4. This chapter is concluded with a short final discussion in Section 8.5. 

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 ... 

BOHMIAN MECHANICS A TRAJECTORY PICTURE OF QUANTUM MECHANICS... 

Apart from the operational, wave or action-based pictures of quantum mechanics provided by Heisenberg, Schrodinger, or Feynman, respectively, there is an additional, fully trajectory-based picture Bohmian mechanics [20,23]. Within this picture, the standard quantum formalism is understood in terms of trajectories defined... 

The Bohmian formalism follows from the Schrodinger one in the position representation after considering a change of variables, from the complex wave function field OP, P ) to the real fields (p, S) according to the transformation relation ... 

An alternative way to obtain the quantum trajectories is by formulating the Bohmian mechanics as a Newtonian-like theory. Then, Equation 8.29 gives rise to a generalized Newton s second law ... 

In the case of a many-body problem, the Bohmian mechanics for an A-body dynamics follows from the one for a single system, but replacing Equation 8.25 by... 

These new trajectories are the so-called reduced quantum trajectories [30], which are only explicitly related to the system reduced density matrix. The dynamics described by Equation 8.42 leads to the correct intensity (time evolution of which is described by Equation 8.40) when the statistics of a large number of particles are considered. Moreover, Equation 8.42 reduces to the well-known expression for the velocity held in Bohmian mechanics, when there is no interaction with the environment. 

Nowadays the success of DFT and TD-DFT is out of question in both the physics and chemistry communities. The numerical results obtained are most of cases in good agreement to those from experimental and other theoretical methods with a relative small computational effort. However, in this chapter, our goal is to present the TD-DFT from a Bohmian perspective and to analyze, from a conceptual level, some of the aspects which are deeply rooted in DFT. 

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). 

As in the hydrodynamic model the Bohmian interpretation assumes a wave function in polar form,... 

The interacting waves from myriads of charge centres constitute the electromagnetic radiation field. In particle physics the field connection between balanced charge centres is called a virtual photon. This equilibrium is equivalent to the postulated balance between classical and quantum potentials in Bohmian mechanics, which extends holistically over all space. 

In Bohmian formalism it may be argued that the reaction system, considered closed, is described at all times by an equation H Sf = IA > in the time-dependent wave function T(/l, />, 6, I)). The product states 4,a- 4,b and as well as the valence state are special solutions of this equa-... 

Bohmian mechanics refers to the ontological interpretation of quantum theory pioneered by Bohm [4]. The mathematical structure of the theory is not affected by the different interpretation and the same formalism adopted before [7] will be used here. 

Some of the chemical concepts with little or no quantum-mechanical meaning outside the Bohmian formulation but, well explained in terms of the new interpretation, include electronegativity, the valence state, chemical potential, metallization, chemical bonding, isomerism, chemical equilibrium, orbital angular momentum, bond strength, molecular shape, phase transformation, chirality and barriers to rotation. In addition, atomic stability is explained in terms of a simple physical model. The central new concepts in Bohmian mechanics are quantum potential and quantum torque. 

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