Configuration collinear

Figure 3. Contour piot of the eiectronic density of a (tripiet) eigenstate strongly scarred by the antisymmetric stretch orbit (left), in 2D configuration space (spanned by the electrons distances ri and r-2 from the nucleus, in the collinear configurations considered here). This eigenstate belong to the N = 9 series. The solid lines depict the associated classical periodic orbit. Autoionization rates of antisymmetric stretch singlet states (right) of the Nth autoionizing series of the helium spectrum, in ID (squares), 2D (circles), and 3D (diamonds) configuration space.

Hashimoto, M., Araki T., and Kawata, S. 2000. Molecular vibration imaging in the fingerprint region by use of coherent anti-Stokes Raman scattering microscopy with a collinear configuration. Opt. Lett. 25 1768-70. 

We focus on the nonlinear dynamics for the collinear configuration which we treat as part of the full multidimensional system. This is to be contrasted with two-degree-of-freedom models where the molecule is assumed to be frozen in some angular configuration, such that the bending degree of freedom is excluded from the dynamics. In our analysis, bending is taken into account in terms of linearized dynamics, which allows us to extend the results for the collinear situation to the full three-dimensional system. The restriction we must be aware of is that the three-dimensional system may have periodic orbits that are not of collinear type. 

For symmetric molecules XYX in the collinear configuration, the Hamiltonian is of the form... 

Here we consider an optical transition between Aj and E electronic states of a center of a trigonal symmetry. To describe the vibrations of the center we use the collinear-configurational approximation [27] in which only the central forces are taken into account in the optical center (taking account of deviations from this approximation, see later). If one restricts oneself to the linear vibronic coupling in the e state, then in this approximation the potential energy operators in the Ai and E electronic states can be presented in the form ... 

Above the collinear-configurational approximation (1) was used. If one wants to take into account the deviations from this approximation then in equation (3) the term V1 = Y.mwnQni

[Pg.139]

Here Hint is the anharmonic interaction in the collinear-configurational approximation (see Refs. [5,7]), Vmk = v/n,7i/2displacement operators of the host atoms with respect to the atom(s)of the mode (Y.i eimeim = M- We take into account that the strongly excited mode can be considered classically, and replace its coordinate operator by Q(t) = A cos( )/f), where A is the initial amplitude of the mode. Then... 

Figure 8.2 depicts a typical potential energy surface (PES) for a symmetric molecule ABA with intramolecular bond distances R and R2] the ABA bond angle is assumed to be 180° (collinear configuration). The PES is symmetric with respect to the line defined by Ri = R2 it has a saddle point at short distances and decreases monotonically from the saddle point out into the two identical product channels A + BA and AB + A (see also Figure 7.18). The shaded area indicates the Franck-Condon (FC) region accessed via photon absorption and the two arrows illustrate the main dissociation paths for the quantum mechanical wavepacket or, equivalently, a swarm of classical trajectories. Because no barrier obstructs dissociation, the majority of trajectories immediately evanesce in either one of the two product channels without ever returning to the vicinity of the FC point.

The system is based on the ability of a classical single-shot autocorrelator, widely used to measure the duration of picosecond and femtosecond laser pulses, to operate a time-to-space conversion [14], After being reflected by a biological sample, a probe pulse supports a temporal stretching due to the different internal reflections occurring at different depths. In the nonlinear crystal, if the backscattered probe pulse interacts with the pump pulse in a non-collinear configuration, the temporal stretching can be converted into a spatial stretch-... 

Fig. 3.1.1 A potential energy surface for a direct bimolecular reaction. The surface corresponds to a reaction like D + H — H—>D — H + Hata fixed approach angle, say in a collinear configuration specified by the D-H and H-H distances. These distances are measured along the two perpendicular axes. (Note that in this figure all energies above a fixed cut-off value Emax have been replaced by max.)...

Figure 15 Potential energy curves for two H atoms over a graphite surface, for the case where the target atom is initially chemisorbed, and the carbon lattice is allowed to relax for each configuration of the H atoms. Cases (a) and (b) correspond to the collinear and quasi-collinear configurations. The potential energy is plotted as a function of the distance of the incident H above the surface, for three positions of the target H 1.49 A (solid line, filled circles), 1.69 A (dashed line, open diamonds), and 1.89 A (dotted line, filled squares). The symbols correspond to the DFT calculations, and the lines correspond to the model PES. Taken from Ref. [90],...

Ab initio [515, 516] and semi-empirical calculations [517] of the reaction potential-energy surface show that the potential-energy barrier for reaction depends on the angle of the H—H—F transition state and is lowest for the collinear configuration, having a value 4 kJ mole-1. Thus, collisions involving a nearly collinear approach of F to H2 make the major contribution to reaction and give backward-scattered products. All the surfaces are of a repulsive type. 

Figure 25 Scheme of antiferromagnetic collinear configuration of the magnetic moments originating from the Mn(II) ions, which constitute the chiral 3-D network compound [104]. 

The semiempirical potential functions which have been used in trajectory calculations fall into two categories those based on pairwise interactions between the atoms, and more general interaction potentials which deal more directly with the three-atom reaction surface. In discussing their characteristics it is customary to concentrate on the surface for the collinear configuration since it is possible to represent this function by a contour diagram of the kind shown in Figure 1.13. 

Figure 5.4(a), strong

We define the order of the singular values as a > a2 > 31. The planar and collinear configurations give a3 0 and a2 a3 = 0, respectively. Furthermore, we let the sign of a3 specify the permutational isomers of the cluster [14]. That is, if (det Ws) = psl (ps2 x ps3) > 0, which is the case for isomer (A) in Fig. 12, fl3 >0. Otherwise, a3 < 0. Eigenvectors ea(a = 1,2,3) coincide with the principal axes of instantaneous moment of inertia tensor of the four-body system. We thereby refer to the principal-axis frame as a body frame. On the other hand, the triplet of axes (u1,u2,u3) or an SO(3) matrix U constitutes an internal frame. Rotation of the internal frame in a three-dimensional space, which is the democratic rotation in the four-body system, is parameterized by three... 

The approximately collinear configuration of M—C=N has been confirmed in a number of molecular cyanides (Table 21.8). In some of these cyanides there are intermolecular M—N distances less than the sums of the van der Waals radii, and in S(CN)2, for example, the two next nearest neighbours of S are coplanar with the intramolecular C atoms, suggesting that intermolecular interactions may be responsible for the small deviations of the M-C=N systems from exact collinearity. 

The limit value (p/o)l2 can easily be obtained in the following way it is the minimum of the ratio g2jv2 for collinear configurations with m2 between mi and m3 ... 

The multiple scattering pathways where the angle between the outgoing wave and the reflected wave is 90° (for example the n = 3 contribution for the M—O—O—M pathway) has zero amplitude in the plane wave approximation as can be seen from formula (12) in Sect. 3.1. Therefore the pathways for atoms in collinear configurations provide the main contribution to multiple scattering. 

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