Periodic orbit condition

In the parabolic model the equations for caustics are simply Q+ = Q, and Q- = <2-- The periodic orbits inside the well are not described by (4.46), but they run along the borders of the rectangle formed by caustics. It is these trajectories that correspond to topologically irreducible contours on a two-dimensional torus [Arnold 1978] and lead to the quantization condition (4.47). 

The point q = p = 0 (or P = Q = 0) is a fixed point of the dynamics in the reactive mode. In the full-dimensional dynamics, it corresponds to all trajectories in which only the motion in the bath modes is excited. These trajectories are characterized by the property that they remain confined to the neighborhood of the saddle point for all time. They correspond to a bound state in the continuum, and thus to the transition state in the sense of Ref. 20. Because it is described by the two independent conditions q = 0 and p = 0, the set of all initial conditions that give rise to trajectories in the transition state forms a manifold of dimension 2/V — 2 in the full 2/V-dimensional phase space. It is called the central manifold of the saddle point. The central manifold is subdivided into level sets of the Hamiltonian in Eq. (5), each of which has dimension 2N — 1. These energy shells are normally hyperbolic invariant manifolds (NHIM) of the dynamical system [88]. Following Ref. 34, we use the term NHIM to refer to these objects. In the special case of the two-dimensional system, every NHIM has dimension one. It reduces to a periodic orbit and reproduces the well-known PODS [20-22]. 

Periodic boundary conditions, Monte Carlo heat flow simulation, nonequilibrium molecular dynamics, 79—81 Periodic-orbit dividing surface (PODS) geometric transition state theory, 196-201 transition state trajectory, 202-213 Perturbation theory, transition state trajectory, deterministically moving manifolds, 224-228... 

Most of the current implementations employ the original Car-Parrinello scheme based on DFT. The system is treated within periodic boundary conditions (PBC) and the Kohn-Sham (KS) one-electron orbitals are expanded in a basis set of plane waves (with wave vectors Gm) [48-50] ... 

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]. 

Consider the space state model R deflned by Eq.(52), showing an equilibrium point such that the matrix of the linearized system at this point has a real negative eigenvalue A and a pair of complex eigenvalues a j/3, j = /—1) with positive real parts 0.. In this situation, the equilibrium point has onedimensional stable manifold and two-dimensional unstable manifold. If the condition A < a is verified, it is possible that an homoclinic orbit appears, which tends to the equilibrium point. This orbit is very singular, and then the Shilnikov theorem asserts that every neighborhood of the homoclinic orbit contains infinite number of unstable periodic orbits. 

In this section, we arrive at the quantization condition expressed in terms of periodic orbits. The periodic-orbit contribution to the trace formula can be written as the logarithmic derivative of a so-called zeta function,... 

Because the periodic orbits are highly unstable in the present example (i.e., the slightest distortion destroys the periodicity and leads to dissociation) the trapped trajectories recur only once after returning to their origin they start again, with a new set of initial conditions, and most probably will dissociate in the second round. 

Gutzwiller, M.C. (1971). Periodic orbits and classical quantization conditions, J. Math. Phys. 12, 343-358. 

Electronic structure methods for studies of nanostructures can be divided broadly into supercell methods and real-space methods. Supercell methods use standard k-space electronic structure techniques separating periodically repeated nanostructures by distances large enough to neglect their interactions. Direct space methods do not need to use periodic boundary conditions. Various electronic structure methods are developed and applied using both approaches. In this section we will shortly discuss few popular but powerful electronic structure methods the pseudopotential method, linear muffin-tin orbital and related methods, and tight-binding methods. 

Recently, Wan and Tong (48) have performed an EH-type calculation for the d band of Ni using periodic boundary conditions. Using two Slater orbitals for each d orbital, they found good agreement with previous ab initio calculations for bulk Ni. The d band was calculated to be 2.48 eV wide, and the ionization potential was 6.38 eV. Despite the fact that no s orbitals were included in the calculations, the results suggest that EH is as reliable a procedure as more expensive ones for the calculation of bulk Ni properties. 

We evaluated adsorbed SO3 configuration on Pt (111) surface by using the first-principles calculations with a slab model in a periodic boundary condition. On the basis of the result of the calculations with a slab model, we evaluated the electronic states of SO3 in detail using the relativistic DV-Xa molecular orbital method. 

Most chemists are well acquainted with LCAO-MO theory. The numbers of atomic orbitals, even in large molecules, however, are miniscule compared to a nonmolecular solid, where the entire crystal can be considered one giant molecule. In a crystal there are in the order of 10 atomic orbitals, which is, for all practical purposes, an infinite number. The principle difference between applying the LCAO approach to solids, versus molecules, is the number of orbitals involved. Fortunately, periodic boundary conditions allow us to smdy solids by evaluating the bonding between atoms associated with a single lattice point. Thus, the lattice point is to the solid-state scientist, what the molecule is to the chemist. 

In this equation, N is equal to the number of unit cells in the crystal. Note how the function in Eq. 5.27 is the same as that of Eq. 5.19 for cyclic tt molecules, if a new index is defined ask = liij/Na. Bloch sums are simply symmetry-adapted linear combinations of atomic orbitals. However, whereas the exponential term in Eq. 5.19 is the character of the yth irreducible representation of the cychc group to which the molecule belongs, in Eq. 5.27 the exponential term is related to the character of the Mi irreducible representation of the cychc group of infinite order (Albright, 1985). This, in turn, may be replaced with the infinite linear translation group because of the periodic boundary conditions. It turns out that SALCs for any system with translational symmetry are con-stmcted in this same manner. Thus, as with cychc tt systems, there should never be a need to use the projection operators referred to earher to generate a Bloch sum. 

As an example, consider a two-dimensional dynamical mapping with the simplest periodic orbit—that is, a fixed point at (qo,po)- Suppose that the initial condition (qo,Po) is only infinitesimally shifted from (qo,Po) with = dq p q— po= dp. With one iteration of the map, (qo,Po) evolves to (q, p ), and the initial errors dq and dp are propagated to bq and 8p, which is given by... 

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