Classical Newtonian

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. 

Notice that the solution is not identical to J but an approximation of it. The evolution of a and S in time may conveniently be described via the following classical Newtonian equations of motion Given the initial values... 

Various kinds of mixed quantum-classical models have been introduced in the literature. We will concentrate on the so-called quantum-classical molecular dynamics (QCMD) model, which consists of a Schrodinger equation coupled to classical Newtonian equations (cf. Sec. 2). 

During the late nineteenth century evidence began to accumulate that classical newtonian mechanics, which was completely successful on a macroscopic scale, was unsuccessful when applied fo problems on an atomic scale. 

How is physics, as it is currently practiced, deficient in its description of nature Certainly, as popularizations of physics frequently reniiiid us, theories such as Quantum Electrodynamics are successful to a reinarkiible degree in predicting the results of experiments. However, any reasonable measure of success requires that wc add the caveat, ...in the domain (or domains) for which the theory was developed. For example, classical Newtonian physics is perfectly correct in its description of slow-moving, macroscopic objects, but is fundamentally incorrect in its description of quantum and/or relativistic systems. 

Complex liquids seldom behave as classical Newtonian fluids thus, analysis of their behavior requires a thorough understanding of non-Newtonian rheology. The importance of this knowledge is illustrated by the following two examples ... 

Multiparticle collision dynamics describes the interactions in a many-body system in terms of effective collisions that occur at discrete time intervals. Although the dynamics is a simplified representation of real dynamics, it conserves mass, momentum, and energy and preserves phase space volumes. Consequently, it retains many of the basic characteristics of classical Newtonian dynamics. The statistical mechanical basis of multiparticle collision dynamics is well established. Starting with the specification of the dynamics and the collision model, one may verify its dynamical properties, derive macroscopic laws, and, perhaps most importantly, obtain expressions for the transport coefficients. These features distinguish MPC dynamics from a number of other mesoscopic schemes. In order to describe solute motion in solution, MPC dynamics may be combined with molecular dynamics to construct hybrid schemes that can be used to explore a variety of phenomena. The fact that hydrodynamic interactions are properly accounted for in hybrid MPC-MD dynamics makes it a useful tool for the investigation of polymer and colloid dynamics. Since it is a particle-based scheme it incorporates fluctuations so that the reactive and nonreactive dynamics in small systems where such effects are important can be studied. 

Theoretical models include those based on classical (Newtonian) mechanical methods—force field methods known as molecular mechanical methods. These include MM2, MM3, Amber, Sybyl, UFF, and others described in the following paragraphs. These methods are based on Hook s law describing the parabolic potential for the stretching of a chemical bond, van der Waal s interactions, electrostatics, and other forces described more fully below. The combination assembled into the force field is parameterized based on fitting to experimental data. One can treat 1500-2500 atom systems by molecular mechanical methods. Only this method is treated in detail in this text. Other theoretical models are based on quantum mechanical methods. These include ... 

Atomic. From spectroscopic studies, it is known lliat when an electron is bound to a positively charged nucleus only certain fixed energy levels are accessible to the electron. Before 1926, the old quantum theory considered that the motion of the electrons could be described by classical Newtonian mechanics in which the electrons move in well defined circular or elliptical orbits around the nucleus. However, the theory encountered numerous difficulties and in many instances there arose serious discrepancies between its predictions and experimental fact. 

Classical Newtonian mechanics assumes that a physical system can be kept under continuous observation without thereby disturbing it. This is reasonable when the system is a planet or even a spinning top, but is unacceptable for microscopic systems, such as an atom. To observe the motion of an election, it is necessary to ilium mate it with light of ultrashort wavelength (gamma rays) momentum is transferred from the radiation to the electron and the particle s velocity is. therefore, continuously disturbed. The effect upon a system of observing it can not be determined exactly, and this means that the state of a system at any time cannot be known with complete precision. As a consequence, predictions regarding the behavior of microscopic systems have to be made on a probability basis and complete certainty can rarely be achieved. This limitation is accepted and is made one of the foundation stones upon which the theory of quantum mechanics is constructed. 

An overview of the approaches that have been taken to linking different theoretical and computational modeling descriptions is also provided in Fig. 2 The first principles (QC) descriptions are based on the Schrodinger equation and the Bom Oppenheimer approximation as realized in most chemical applications by density functional [14] or Hartree-Fock [15] methods. Molecular dynamics (MD) methods [16], based on classical Newtonian me-... 

The geometric description of the light propagation and the kinetics description of motion were closely correlated in the history of science. Among the main evidence of classical Newtonian mechanics is Euclidean geometry based on optical effects. In Newtonian physics, space has an affine structure but time is absolute. The basic idea is the inertial system, and the relations are the linear force laws. The affine structure allows linear transformations in space between the inertial coordinate systems, but not in time. This is the Galilean transformation ... 

Numerous unsuccessful measurements were made to determine the motion of earth in the ether. These measurements were not able to give results compatible within the framework of classical Newtonian mechanics, even though that the earth has an orbital velocity v0 30,000 m/s (where v0 is velocity of the earth to the ether). In 1887 Michelson and Morley also determined the earth s orbital velocity by their precision interferometer [11], The updated arrangement of Michelson-Morley experiment (M-M experiment) can be seen in Fig. 1. 

In a strict sense, the classical Newtonian mechanics and the Maxwell s theory of electromagnetism are not compatible. The M-M-type experiments refuted the geometric optics completed by classical mechanics. In classical mechanics the inertial system was a basic concept, and the equation of motion must be invariant to the Galilean transformation Eq. (1). After the M-M experiments, Eq. (1) and so any equations of motion became invalid. Einstein realized that only the Maxwell equations are invariant for the Lorentz transformation. Therefore he believed that they are the authentic equations of motion, and so he created new concepts for the space, time, inertia, and so on. Within... 

The method of defining complexions depends on whether we are treating our systems by classical, Newtonian mechanics or by quantum theory. First we shall take up classical mechanics, for that is more familiar. But later, when we describe the methods of quantum theory, we shall observe that that theory is more correct and more fundamental for statistical purposes. In classical mechanics, a system is, df scrihpd by tV... 

Molecular mechanical calculations which are based on classical Newtonian mechanics [260—262], but use quantum mechanical concepts to formulate empirical equations. The parameters used are based entirely on experimental data. 

Let us attempt a rough trajectory through molecular dynamics. We have a system of N atoms obeying classical Newtonian mechan-... 

The potential energy surface for a reaction is a function of the 2>N — 6 coordinates that determine the relative positions of the atoms of the system (3A1 — 5 for a linear system). A particular configuration of the atoms is represented by a point on the potential energy surface, and as the atoms move during the reaction, so the point follows a trajectory over the surface, obeying the laws of classical (Newtonian) mechanics. A simple view of the classical rate of reaction is the net rate at which these trajectories pass from reactants to products. This idea is made more rigorous in the approach by Nordholm [24], discussed in the book by Gilbert and Smith [25]. 

Remark Every elliptic solution of the classical Newtonian equation is unstable (Stiefel and Scheifele, 1971). In order to make this statement more precise, let us provide the following... 

The Schrodinger time equation is closely related to the equation of classical Newtonian mechanics... 

The basic assumption in MM methods is that a molecule can be considered as a collection of classical Newtonian masses held together by hypothetical springs holding the atoms at an equilibrium position in configuration space [10]. The generic form of an MM force field is given by... 

For reactions in the gas phase, answers to a number of these questions have been developed over the past 70 years through a combination of experiment, analytic theory, simple models, and computational dynamics. In that field, the computational methods now range from classical Newtonian dynamics to fully accurate (and time-consuming) quantum mechanical techniques. Comparison between numerical results and experiment is commonplace, and the field is rich in theories and models that explain a large number of gas phase processes on the microscopic level. ... 

Molecular dynamics (MD) integrates the classical Newtonian differential equations of motion for surface and substrate atoms. These equations for an adatom trajectory Y(t) can be written in the form ... 

Let us use the Cartesian coordinate system once more, with 3N coordinates for the N nuclei Xi,i = 1,... 3N, where Xi, X2, X3 denote the x-, y-, and z-coordinates of atom 1 of mass Ml, etc. The ith coordinate is therefore associated with mass M, of the corresponding atom. The classical Newtonian equation of motion for an atom of mass M, and coordinate X/ is ... 

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