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Quantum-classical equations of motion

Abstract. A rigorous derivation of quantum-classical equations of motion is still lacking. The framework proposed so far to describe in a consistent way the dynamics of a mixed quantum-classical system using systematic approximations have failed. A recent attempt to solve the inconsistencies of quantum-classical approximated methods by introducing a group-theoretical approach is discussed in detail. The new formulation which should restore the consistency of the proposed quantum-classical dynamics and statistical mechanics will be shown to produce, instead, a purely classical description. In spite of that, the discussed approach remains interesting since it could produce non-trivial formulations. [Pg.437]

The formulation of a consistent scheme to derive mixed quantum-classical equations of motion is a relevant goal both from a purely theoretical point of view and for practical (e.g., computational) applications. [Pg.437]

This result concludes the presentation of the Heisenberg group approach as the powerful tool that allows to derive classical mechanics as a formal limit of quantum mechanics, for h —> 0. The most important ingredients that have been introduced to obtain this result are the Fourier-like representation of observables and equations of motion and the definition of the antiderivative operator. These elements will be used in section 5 to derive a similiar procedure for a mixed quantum-classical mechanics. An ansatz on the quantum-classical equations of motion will be necessary, but the subsequent application of Heisenberg group formalism will be a straightforward generalization of what has been done so far. [Pg.451]

This section represents the conclusive part of our work on the quantum-classical equations of motion derived in section 5, following the prescriptions of Ref. [15]. We will show an alternative derivation of the quantum-classical equation of motion (60), obtained by taking the limit hi —> h, h2 — 0 in eq.(44), which is an ansatz on the mixed dynamical generator, after making some remarks on the equation of motion itself and on the operators used as generators in the representation of the group D", i.e., the position Xj>ol and momentum haD a operators. [Pg.457]

An attempt to solve the difficulties and inconsistencies arising from an approximated derivation of quantum-classical equations of motion was made some time ago [15] to restore the properties that are expected to hold within a consistent formulation of dynamics and statistical mechanics, and are instead missed by the existing approximate methods. We refer not only to the properties that the Lie brackets, which generate the dynamics, satisfy in a full quantum and full classical formulation, e.g., the bi-linearity and anti-symmetry properties, the Jacobi identity and the Leibniz rule12, but also to statistical mechanical properties, like the time translational invariance of equilibrium correlation functions [see eq.(8)]. [Pg.462]

The mixed quantum-classical equations of motion can now be formulated in the form of Hamilton s equations... [Pg.138]

While one can imagine various routes to obtain the quantum-classical equations of motion, one way to disentangle these different contributions is to suppose that the particles comprising have mass M while those of scale variables so that distances are measured on the scale characteristic of the light particles, Ato = with eg some characteristic energy of the system,... [Pg.528]

Rather than carrying out a linear response derivation to obtain correlation function expressions for transport coefficients based on the quantum-classical equations of motion, in this section we show how transport coefficients can be obtained by a different route. We take as a starting point the quantum mechanical expression for a transport coefficient and consider a limit where the dynamics is approximated by quantum-classical dynamics [17,18]. The advantage of this approach is that the full quantum equilibrium structure can... [Pg.532]

Both the BO dynamics and Gaussian wavepacket methods described above in Section n separate the nuclear and electronic motion at the outset, and use the concept of potential energy surfaces. In what is generally known as the Ehrenfest dynamics method, the picture is still of semiclassical nuclei and quantum mechanical electrons, but in a fundamentally different approach the electronic wave function is propagated at the same time as the pseudoparticles. These are driven by standard classical equations of motion, with the force provided by an instantaneous potential energy function... [Pg.290]

Depending on the desired level of accuracy, the equation of motion to be numerically solved may be the classical equation of motion (Newton s), a stochastic equation of motion (Langevin s), a Brownian equation of motion, or even a combination of quantum and classical mechanics (QM/MM, see Chapter 11). [Pg.39]

The temporal behavior of molecules, which are quantum mechanical entities, is best described by the quantum mechanical equation of motion, i.e., the time-dependent Schrdd-inger equation. However, because this equation is extremely difficult to solve for large systems, a simpler classical mechanical description is often used to approximate the motion executed by the molecule s heavy atoms. Thus, in most computational studies of biomolecules, it is the classical mechanics Newtonian equation of motion that is being solved rather than the quantum mechanical equation. [Pg.42]

Because the mapping approach treats electronic and nuclear dynamics on the same dynamical footing, its classical limit can be employed to study the phase-space properties of a nonadiabatic system. With this end in mind, we adopt a onemode two-state spin-boson system (Model IVa), which is mapped on a classical system with two degrees of freedom (DoF). Studying various Poincare surfaces of section, a detailed phase-space analysis of the problem is given, showing that the model exhibits mixed classical dynamics [123]. Furthermore, a number of periodic orbits (i.e., solutions of the classical equation of motion that return to their initial conditions) of the nonadiabatic system are identified and discussed [125]. It is shown that these vibronic periodic orbits can be used to analyze the nonadiabatic quantum dynamics [126]. Finally, a three-mode model of nonadiabatic photoisomerization (Model III) is employed to demonstrate the applicability of the concept of vibronic periodic orbits to multidimensional dynamics [127]. [Pg.326]

Molecular mechanics lies conceptually between quantum mechanics and classical mechanics, in that data obtained from quantum mechanical calculations are incorporated into a theoretical framework established by the classical equations of motion. The Bom-Oppenheimer approximation, used in quantum mechanics, states that Schrddinger s equation can be separated into a part that describes the motion of electrons and a part that describes the motion of nuclei, and that these can be treated independently. Quantum mechanics is concerned with the properties of electrons molecular mechanics is concerned with the nuclei, while electrons are treated in a classical electrostatic manner. [Pg.47]

Using a cumulant expansion, we have shown how to obtain quantum corrections to purely classical equations of motion. Quantum correction reduces chaos in... [Pg.421]

In the dynamical approach, one attempts to solve directly the quantum-mechanical or classical equations of motion for a system, Such a direct approach is practicable, for example, for treating the binary collisions between molecules in a gas, by either classical or quantum-mechanical methods.3 However, in a dense system such as a liquid, only the classical equations are tractable,4 even with high-speed computers,... [Pg.80]

These expressions for the moments can be evaluated as equilibrium averages, without actually solving for all the quantum states of the system, or without solving the classical equations of motion for the classical trajectories, In the quantum-mechanical case, these equilibrium averages, Eq. (10), can be rewritten as traces, which can then be evaluated in any convenient basis. Thus the difficult step of solving for all the quantum states can be avoided in evaluating moments. [Pg.83]

The fragmentation of a molecule in its ground electronic state is commonly known as unimolecular dissociation [26-28]. [For a recent review see Ref. 29 and the Faraday Discussion of the Chemical Society, vol. 102 (1995).] Because of its importance in several areas of physical chemistry, such as combustion or atmospheric kinetics, there is a high demand of accurate unimolecular dissociation rates. On the other hand, however, the calculation of reliable dissociation rates by dynamical methods (i.e., the solution of the classical or quantum mechanical equations of motion) is, for obvious technical problems, prohibited for all but a few simple molecules. For... [Pg.750]

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]. [Pg.438]

This expression will be used in the next section to obtain the classical equation of motion as a straighforward formal limit of the quantum equation of motion. [Pg.448]

If the interaction potential V/ depends only on R, energy cannot flow between the translational and the vibrational modes the full or half collision is elastic (no final state interaction). The resulting quantum mechanical or classical equations of motion separate in two uncoupled blocks and the motions in R and r evolve independently of each other. [Pg.203]

The Hamilton-Jacobi form of the classical equations of motion has been shown to have provided the basis for the quantum-mechanical formulations according to Sommerfeld, Heisenberg, Schrodinger and Bohm. Each of these formulations inspired its own peculiar interpretation of quantum effects, despite their common basis. Each of the different points of view still has its adherents and the debates about their relative merits continue. Closer scrutiny shows that the Sommerfeld and Heisenberg systems assume quanta to be particles in the classical sense, although Heisenberg considered electronic positions to be fundamentally unobservable. [Pg.85]


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See also in sourсe #XX -- [ Pg.163 , Pg.437 , Pg.440 , Pg.451 , Pg.452 , Pg.457 , Pg.462 ]




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