N-Body system

After this computer experiment, a great number of papers followed. Some of them attempted to simulate with the ab-initio data the properties of the ion in solution at room temperature [76,77], others [78] attempted to determine, via Monte Carlo simulations, the free energy, enthalpy and entropy for the reaction (24). The discrepancy between experimental and simulated data was rationalized in terms of the inadequacy of a two-body potential to represent correctly the n-body system. In addition, the radial distribution function for the Li+(H20)6 cluster showed [78] only one maximum, pointing out that the six water molecules are in the first hydration shell of the ion. The Monte Carlo simulation [77] for the system Li+(H20)2oo predicted five water molecules in the first hydration shell. A subsequent MD simulation [79] of a system composed of one Li+ ion and 343 water molecules at T=298 K, with periodic boundary conditions, yielded... 

MSN.60. I. Prigogine, C. George, and F. Henin, Dynamical and statistical descriptions of N-body systems, Physica 45, 418—434 (1969). 

To summarize, we have considered a quantum mechanical N-body system with dilation analytic potentials, Ey, and its dependence on the scaling parameter i] = t] (for some 0 < < 0, depending on V). To be more detailed, we need to restrict Hilbert space to a dense subspace [54], the so-called Nelson class N, which provides the domain over which the unbounded complex scaling is well-defined. Closing the subset < - D T) in ft, see [9] and references therein for a more detailed expose, one obtains the scaled version of the original partial differential equation... 

J. P. Leroy, R. Wallace, and H. Rabitz, Chem. Phys., 165, 89 (1992). A Method for Inverting Curvilinear Transformations of Relevance in the Quantum-Mechanical Hamiltonian Describing n-Body Systems. 

II. Basic Scheme of the Gauge-Theoretical Formalism for Internal Motions of n-Body Systems... 

The Newtonian gravitational force is the dominant force in the N-Body systems in the universe, as for example in a planetary system, a planet with its satellites, or a multiple stellar system. The long term evolution of the system depends on the topology of its phase space and on the existence of ordered or chaotic regions. The topology of the phase space is determined by the position and the stability character of the periodic orbits of the system (the fixed points of the Poincare map on a surface of section). Islands of stable motion exist around the stable periodic orbits, chaotic motion appears at unstable periodic orbits. This makes clear the importance of the periodic orbits in the study of the dynamics of such systems. 

Marchal C. (1971). Qualitative study of a n-body system a new condition of complete scattering. Astronomy and Astrophysics, 10, 2, p. 278-289. 

The procedure can clearly be extended to treat more than three particles, and this is done, e.g. in ref. [24]. It has also to be pointed out the fact that the hyperradius is a measure of the total inertia of the n-body system, and this can be a physical motivation for its candidacy as a proper nearly separable variable, invariant with respect to the choice of the set of Jacobi vectors. 

As equation (3) caimot be solved analytically for the n-body problem ( >2) numerical methods must be utilized to solve n-body systems. In practice, MD calculations are carried out in time steps of the order of 10" s and are repeated several thousands or even millions of times. 

D = I, for which / = 0 and / = 1 correspond to eigenstates of even and odd parity. A key theorem for S states of any N-body system is demonstrated for the N = Z case the D-dimensional Hamiltonian can be cast in the same form as D = Z, with the addition of a scalar centrifugal potential that contains the sole dependence on D as a quadratic polynomial. For two-electron atoms, the D —y oo limit and the first-order correction in 1/D are discussed for both the complete Hamiltonian and the Hartree-Fock approximation. 

The number of local directions in which the configurational (position) state can be varied is called the number of degrees of freedom Nj. For the N-body system in R without additional constraints there are Na = bt = 3N degrees of freedom. If r independent constraints are present the number of degrees of freedom is Na = b —r. 

The total energy of the N-body system is a function of positions and velocities. 

In addition to the energy, there may be other constants of motion which arise from the structure of the equations of motion. For example, if the full set of particles of an N-body system is included in the model, then the instantaneous force acting on particle i due to the presence of particle j will be the exact negative of the force acting on particle j due to particle t this is a direct consequence of Newton s third law (every action has an equal and opposite reaction). This means that the sum of all the forces will vanish. Since... 

For a few very special N-body systems, analytical solution is possible. 

As mentioned previously, for the N-body system, energy and total momentum are constants of motion—they do not change even as the bodies of the system move along their natural paths (defined by the equations of motion). Another term for constant of motion is first integral. In general, if we have a dynamical system z = /(z), first integral is a smooth function /(z) which is constant along solutions, for all values of the initial condition. Letz(f) f M be a solution, then... 

Initialization of an fee System. The density q of an N-body system in a cubic box of side L is given by... 

Hamilton s principle of least action provides a mechanism for deriving equations of motion from a Lagrangian. Recall from Chap. 1 that the Lagrangian for the N-body system is defined by... 

There are several other N-body systems that exhibit an exponential growth of M N) with i 0.07 and 0.16. In these statistical models the... 

M. Neumann and M. Zoppi, Phys. Rev. A, 40, 4572 (1989). Asymptotic Expansions and Effective Potentials for Almost Classical N-Body Systems. 

MD simulations provide the means to solve the equations of motion of the particles and output the desired physical quantities in the term of some microscopic information. In a MD simulation, one often wishes to explore the macroscopic properties of a system through the microscopic information. These conversions are performed on the basis of the statistical mechanics, which provide the rigorous mathematical expressions that relate macroscopic properties to the distribution and motion of the atoms and molecules of the N-body system. With MD simulations, one can study both thermodynamic properties and the time-dependent properties. Some quantities that are routinely calculated from a MD simulation include temperature, pressure, energy, the radial distribution function, the mean square displacement, the time correlation function, and so on (Allen and Tildesley 1989 Rapaport 2004). 

Excitons are quasiparticle-like excitations to the N-body system whereas most first principle calculations of crystals, namely electron energy band calculations, obtain results by taken an electron away or adding an electron to the system, the (N + l)-body system. The (N + l)-body approximation for the low-lying excitations of the N-body systems is in general not a very good one for organic solids. Thus, we will focus attention on the exciton in this set of lectures. 

In order to calculate the excitations to the N-body system, one must consider the second-order Green s function. In particular the derivation of the polarization propagator of the particle-hole (PH) excitation is the term that needs to be outlined. This term describes the response of the system to a perturbation of the form... 

It is interesting to note that Zao< involve ionizations of the (N+2)-body system as vi/ell as excitations of the N-body system. 

As can be seen by this paper the quasiparticle called an "exciton" is a very useful construct in describing the physics and chemistry of organic solids. It is not only a good definition to some excitations of the N-body system but can be used in describing the polarization field as was the case for the superconductivity model. 

---