Single-particle move

Each atom of a molecule that rotates about an axis through its centre of mass, describes a circular orbit. The total rotational energy must therefore be a function of the molecular moment of inertia about the rotation axis and the angular momentum. The energy calculation for a complex molecule is of the same type as the calculation for a single particle moving at constant (zero) potential on a ring. 

The simplest form of scattering theory is for a single particle moving in a local external potential. If we ignore relativistic effects and return to the Schrodinger equation... 

The polarizable point dipole model has also been used in Monte Carlo simulations with single particle moves.When using the iterative method, a whole new set of dipoles must be computed after each molecule is moved. These updates can be made more efficient by storing the distances between all the particles, since most of them are unchanged, but this requires a lot of memory. The many-body nature of polarization makes it more amenable to molecular dynamics techniques, in which all particles move at once, compared to Monte Carlo methods where typically only one particle moves at a time. For nonpolarizable, pairwise-additive models, MC methods can be efficient because only the interactions involving the moved particle need to be recalculated [while the other (N - 1) x (]V - 1) interactions are unchanged]. For polarizable models, all N x N interactions are, in principle, altered when one particle moves. Consequently, exact polarizable MC calculations can be... 

During the 1970s and 1980s, density-functional theory became an important tool for calculating static electronic and structural properties of solids. The theory represents in principle an exact formulation of the many-electron problem in terms of a single particle moving in the mean field of the other electrons. All the difficulties associated with the solution of the many-electron problem are enclosed in this mean field, for which some approximation must be adopted. In practice, most calculations have been carried out using the local-density approximation (LDA), which has... 

An operator with the property exhibited in eqn (5.5) is said to be Hermitian if it satisfies this equation for all functions P defined in the function space in which the operator is defined. The mathematical requirement for Hermiticity of H expressed in eqn (5.5) places a corresponding physical requirement on the system—that there be a zero flux in the vector current through the surface S bounding the system To illustrate this and other properties of the total system we shall assume, without loss of generality, a form for H corresponding to a single particle moving under the influence of a scalar potential F(r)... 

In addition to variation of the maximum single-particle displacement, one may also attempt to increase the rate at which configurations evolve by moving several particles at a time. However, if the moves of these particles are independent, this is less efficient than a sequence of single-particle moves [2]. 

The above analysis was done for a single particle moving in one dimension, but can be extended to higher dimensions using the same procedure. The starting point is the multidimensional analog of Eq. (8.141), given by... 

For a single particle moving in three dimensions under the influence of a spherically symmetric potential K(R), the classical Hamiltonian is... 

In rarefied systems of particles, drops, or bubbles, the particle-particle interaction can be neglected in the first approximation then one deals with the behavior of a single particle moving in fluid. In this case, the streamline pattern depends on the particle shape, the flow type (translational or shear), and a number of other geometric factors. 

For the purpose of determining the stability of a solution, we need only determine the non-zero characteristic exponents. To do this we may not need to solve the whole system of linear equations of variation. This is also true for the purpose of iterating initial conditions to improve the closeness with which a solution closes, so as to approach more closely an exactly periodic solution. For example, consider the case of a system of two degrees of freedom, as for example a single particle moving in... 

Before studying the hydrogen atom, we shall consider the more general problem of a single particle moving under a central force. The results of this section will apply to any central-force problem, for example, the hydrogen atom (Section 6.5), the isotropic three-dimensional harmonic oscillator (Problem 6.1). 

The assumptions on which quantum mechanics is based may be given in the form of postulates I-VI, which are described next. For simplicity, we will restrict ourselves to a single particle moving along a single coordinate axis x (the mathematical foundations of quantum mechanics are given in Appendix B available at booksite.elsevier.com/978-0-444-59436-5 on p. el). 

Illustration of the Heisenberg uncertainty principle in the case of a single particle moving on the x axis, (al)... 

Fig. 18 a Macroion acceptance ratio and b macroion root-mean-square displacement per MC pass as a function of the cluster displacement parameter for System 11 from simulations with cluster trial moves using pd = 1.0 and indicated cluster radii R i in Ru units. At = Rm fhe single-particle move is recovered. In a, the acceptance ratio equal to 0.5, and in b, the relation = A /i/2 are given (dotted lines). Nm = 20 and Npass = 10 ... 

Some important characteristics of SCF-phonons can be illustrated very simply in terms of a single particle moving in a one-dimensional potential (Koehler, 1968). The Hamiltonian of this simple system is... 

For a particle moving in a fluid, the force acting on the surface of a particle depends only on the flow of the fluid in its immediate vicinity. For the simplest case, let us consider a single particle moving at a velocity relative to its immediate fluid around the particle. It is also assumed that the fluid is newtonian and that the Uf is constant. The fluid dynamic parameters can then be evaluated as follows. 

The subscripted i terms are reduced masses. The use of mass scaled coordinates casts the 3 body coUinear problem into that of a single particle moving on a potential eneigy surface skewed by the angle p,... 

---