Classical mechanics mixing system

In molecular dynamics applications there is a growing interest in mixed quantum-classical models various kinds of which have been proposed in the current literature. We will concentrate on two of these models the adiabatic or time-dependent Born-Oppenheimer (BO) model, [8, 13], and the so-called QCMD model. Both models describe most atoms of the molecular system by the means of classical mechanics but an important, small portion of the system by the means of a wavefunction. In the BO model this wavefunction is adiabatically coupled to the classical motion while the QCMD model consists of a singularly perturbed Schrddinger equation nonlinearly coupled to classical Newtonian equations, 2.2. 

In the mixed quantum-classical molecular dynamics (QCMD) model (see [11, 9, 2, 3, 5] and references therein), most atoms are described by classical mechanics, but an important small portion of the system by quantum mechanics. The full quantum system is first separated via a tensor product ansatz. The evolution of each part is then modeled either classically or quan-tally. This leads to a coupled system of Newtonian and Schrbdinger equations. 

The topic that is commonly referred to as statistical quantum mechanics deals with mixed ensembles only, although pure ensembles may be represented in the same formalism. There is an interesting difference with classical statistics arising here In classical mechanics maximum information about all subsystems is obtained as soon as maximum information about the total system is available. This statement is no longer valid in quantum mechanics. It may happen that the total system is represented by a pure ensemble and a subsystem thereof by a mixed ensemble. 

Finally, it must be remembered that DFT and AIMD can be incorporated into the so-called mixed quantum mechanical/molec-ular mechanical (QM/MM) hybrid schemes [12, 13]. In such methods, only the immediate reactive region of the system under investigation is treated by the quantum mechanical approach -the effects of the surroundings are taken into account by means of a classical mechanical force field description. These DFT/MM calculations enable realistic description of atomic processes (e.g. chemical reactions) that occur in complex heterogeneous envir-... 

The isotope independent potential energy surface was evaluated using a mixed quantum mechanics/molecular mechanics (QM/MM) method. The system (N atoms) was partitioned into Nqm quantum mechanical atoms and Nmm classical mechanical atoms. Nqm consisted of the 15 atom substrate (phospho-D-glycerate)... 

By definition, a mixed quantum-classical method treats the various degrees of freedom (DoF) of a system on a different dynamical footing—for example, quantum mechanics for the electronic DoF and classical mechanics for the... 

We presented the formal elements that are necessary to discuss the mixed quantum-classical mechanics, but we still need to define the dynamical quantities characterizing the two coupled systems to which we associate h and h2 as the values of the Planck constant. 

The effects of parity conservation are not relevant for classical mechanics. Classical systems are in fact arrangements of mixed parity, so that no new information is obtained by taking their mirror images. 

In this chapter, we discussed the principle quantum mechanical effects inherent to the dynamics of unimolecular dissociation. The starting point of our analysis is the concept of discrete metastable states (resonances) in the dissociation continuum, introduced in Sect. 2 and then amply illustrated in Sects. 5 and 6. Resonances allow one to treat the spectroscopic and kinetic aspects of unimolecular dissociation on equal grounds — they are spectroscopically measurable states and, at the same time, the states in which a molecule can be temporally trapped so that it can be stabilized in collisions with bath particles. The main property of quantum state-resolved unimolecular dissociation is that the lifetimes and hence the dissociation rates strongly fluctuate from state to state — they are intimately related to the shape of the resonance wave functions in the potential well. These fluctuations are universal in that they are observed in mode-specific, statistical state-specific and mixed systems. Thus, the classical notion of an energy dependent reaction rate is not strictly valid in quantum mechanics Molecules activated with equal amounts of energy but in different resonance states can decay with drastically different rates. 

For larger systems mixed QM-QCT or pure QCT codes are used. QCT codes integrate the equations of the classical mechanics to directly j)roduce the probability matrix (P) or even more averaged quantities. One has also the option. 

However, one should keep in mind that even if the ARET for this system is well described by classical mechanics, the diffraction process, being of purely quantum mechanical nature, cannot be treated by classical mechanics. Thus, one clearly sees how the MQCB method, as a mixed method, really combines classical mechanics with quantum mechanics in some degrees of freedom, when one is interested in effects that are of purely quantum mechanical nature. Comparing the MQCB method with the full quantum results presented in Figs 1 and 2, one clearly sees that both the diffraction probabilities as well as the ARET are extremely well described by the MQCB method. 

Beyond the purely classical treatments of these interfaces, where structural and dynamical events are probed, lie the most interesting systems to study systems in which reactions occur. Chemical reactions by their very nature usually require quantum mechanical treatments. Such studies may be purely quantum mechanical or mixed quantum/classical mechanical in nature. Because reactions occur in localized environments, classical treatments of much of the system can dramatically reduce simulation times. Some exploratory studies in this direction have been conducted, and the field is full of possibilities. " > > " ... 

Mixing Classical mechanical system that is ergodic and possesses additional properties associated with relaxation. 

Considerations of the quantum dynamics of bound molecules shows that, in the absence of the emission of radiation from energized molecules, all dynamics is quasiperiodic and regular. That is, quantum mechanics does not admit the possibility of long-time relaxation to a time-independent stationary state, a property that characterizes a classical mixing system. This property creates... 

The description of quantum-mechanical processes through a mixed quantum-classical (MQC) formulation has attracted considerable interest for more than seventy years. The reason for this is easy to understand As the numerical effort of a quantum-mechanical basis-set calculation increases exponentially with the number of nonseparable degrees of freedom (DoF), a straightforward quantum computation is restricted to only a few vibrational DoF of a polyatomic system. Classical mechanics, on the other hand. 

By definition, a mixed quantum-classical method treats the various degrees of freedom (DoF) of a system on a different djmamical footing, e.g. quantum mechanics for the electronic DoF and classical mechanics for the nuclear DoF. As was discussed above, some of the problems with these methods are related to inconsistencies inherent in this mixed quantum-classical ansatz. To avoid these problems, recently a conceptually different way to incorporate quantum mechanical DoF into a semiclassical or quasiclassical theory has been proposed, the so-called mapping approach. " In this formulation, the problem of a classical treatment of discrete DoF such as electronic states is bypassed by transforming the discrete quantum variables to continuous variables. In this section we briefly introduce the general concept of the mapping approach and discuss the quasiclassical implementation of this method as well as applications to the three models introduced above. The semiclassical version of the mapping approach is discussed in Sec. 7. 

We resume with a general theory of mixed quantum-classical representation of dynamics of a system composed of fast and slow subsystems, in which the former and latter are, respectively, treated quantum and classical mechanically. In the present case, the fast and slow subsystems happen to be electronic and nuclear parts, respectively. 

Nonadiabatic dynamics is a quantum phenomenon which occurs in systems that interact sufficiently strongly with their environments to cause a breakdown of the Born-Oppenheimer approximation. Nonadiabatic transitions play significant roles in many chemical processes such as proton and electron transfer events in solution and biological systems, and in the response of molecules to radiation fields and their subsequent relaxation. Since the bath in which the quantum dynamics of interest occurs often consists of relatively heavy molecules, its evolution can be modeled by classical mechanics to a high degree of accuracy. This observation has led to the development of mixed quantum-classical methods for nonadiabatic processes. 

Carbocation Stabilities Comparison of Theory and Experiment Force Fields A General Discussion Mixed Quantum-Classical Methods Molecular Mechanics Conjugated Systems. 

In order to illustrate the approximations involved when trying to mix quantum and classical mechanics we consider a simple system with just two degrees of freedom r and R. The R coordinate is the candidate for a classical treatment - it is, e.g., the translational motion of an atom relative to the center of mass of a diatomic or polyatomic molecule. This motion is slow compared to the vibrational motion of the diatom - here described by the r coordinate. Thus if we treat the latter quantum mechanically we could introduce a wavefunction t) and expand this in eigenstates of the molecule, i.e., eigenstates to the hamiltonian operator Hq for the isolated molecule. Thus we have... 

The basic strategy for the QM/MM method lies in the hybrid potential in which a classical MM potential is combined with a QM one (Field et al. 1990). The energy of the system, , is calculated by solving the Schrbdinger equation with an effective Hamiltonian, for the mixed quantum mechanical and classical mechanical system... 

---