Configuration representative point

From these considerations it is clear that complexes in spin equilibrium do not exist at the crossover point between high-spin and low-spin configurations represented on a Tanabe-Sugano diagram. The two states are electronic isomers with geometric and electronic structures well separated on either side of the crossover point. The energy required to reach the crossover point represents at least part of the activation energy for the spin state interconversion. 

The curvature of a wavefront appears transformed into the curvature of a mirror surface shaped so that it would focus the total wavefront into the point of ohservation.The reason is that a focusing mirror reflects light in such a way that the total wavefront arrives to the focal point at one point of time. Thus, a small flat wavefront that passes by will appear tilted at 45°. A larger flat wavefront will not only appear tilted but will also be transformed into a paraboloid whose focal point is the point of observation. A spherical wavefront appears transformed into an ellipsoid, where one focal point is the point source of light (A) and the other is the point of observation (B). This configuration represents one of the ellipsoids of the holodiagram. 

Figure 4.2. Projection in the xy plane of the unit sphere in configuration space, showing the initial orientation of the unit vectors ei, e2 before applying the symmetry operator T. Note that e3 is normal to the plane of the paper and points upwards towards the reader. Also shown are the positions of the representative point E after applying to configuration space the symmetry operators in rows 2 to 6 of Table 4.6. The unit vector b lies along the direction [I 1 0].

The PES of the water dimer system is characterised by three main stationary points the "quasi-linear" structure, representing the global minimum (Fig. 11), and the cyclic and bifurcated configurations (saddle points). 

It does not represent a stable state and corresponds to the atomic configuration at the highest point of the potential energy curve. The energy expended in bringing the system from the state / to this configuration represents the energy of activation for the substitution reaction. 

As Manassen et al. (1981) pointed out, the configurations represented in Fig. 10.5 have another disadvantage, namely the disparity between the small surface area desired to minimise the dark current in the storage electrode or half-cell, since this opposes and reduces the photocurrent, and the large surface area necessary to minimise storage polarisation losses and maximise storage capacity. 

Fi is the force on particle i caused by the other particles, the dots indicate the second time derivative and m is the molecular mass. The forces on particle i in a conservative system can be written as the gradient of the potential energy, V, C/, with respect to the coordinates of particle /. In most simulation studies, U is written as a sum of pairwise additive interactions, occasionally also three-particle and four-particle interactions are employed. The integration of Eq. (1) has to be done numerically. The simulation proceeds by repeated numerical integration for tens or hundreds of thousands of small time steps. The sequence of these time steps is a set of configurations, all of which have equal probability. The completely deterministic MD simulation scheme is usually performed for a fixed number of particles, iV in a fixed volume V. As the total energy of a conservative system is a constant of motion, the set of configurations are representative points in the microcanonical ensemble. Many variants of these two basic schemes, particularly of the Monte Carlo approach exist (see, e.g.. Ref. 19-23). 

For a classical mechanical system of / degrees of freedom we may define a y phase space. This is a Euclidian space of 2/ dimensions, one for each configuration coordinate (configuration space). . . /, and one for each momentum coordinate (momentum space) Px - pf The state of each system in the ensemble would be given by a "representative point in the y phase space. The state of the ensemble as a whole would then be a "doud of points in the y phase space. We may also define a phase space as that of one molecule in the system. If we have N molecules in the system, then the state of the system is determined by one point in y space or a doud of iV points in space. For identical molecules, a cloud of JV points in fi space represents the same physical situation with interchange of the N points. And since there are iV different but equivalent arrangements, there are N points in y space that correspond to equivalent clouds in fi space. 

These molecules, represented schematically in Fig. 21, exhibit host lattices based on the use of hydrogen bonding to build a hexameric unit. In the case of Dianin s compound, the hexamer consists of three molecules of one configuration, R, pointing upward, and three of the other configuration, R, pointing downward. When these units pack in the solid state, the result is the formation of a cavity, as shown in Fig. 22. 

The possibility of describing an elementary process in the adiabatic approximation means that the representative point in the configuration space of the nuclei always remains on one certain potential energy surface. It follows that the potential energy surfaces of the reactant and product molecules are parts of a common potential energy surface of the system. 

The totality of coordinates forms the configuration space of the system and the totality of coordinates and momenta forms the phase space. At any moment, the state of the system is determined by the functions Pk(t), Qk(t) which specify in the configuration or phase space the position of the point known as the representative point of the system. The time behaviour of the system is described by the motion of the representative point along the trajectory in the configuration or phase space. The coordinates and momenta are obtained from the equations of classical mechanics... 

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