Energy surfaces, projection

The potential energy surface projected on to the (j>o E plane is shown in Figure 7. There are now three identical transformations starting from the central trigonal bipyramid T0 at 4>d = 30.0°, 0E = 150.0°, depending on which of the three equivalent atoms forming the trigonal plane, B,... 

The potential energy surface projected onto the 0A—0B plane is shown in Figures 78a and 78b. The centre of the surface, at 0A = 0B = 45°, corresponds to the dodecahedron, labelled D[. Distortion of this dodecahedron by decreasing 0A to 30.9° and increasing 0B to 55.3° produces a square antiprism (At) with AjB j A4B4 and A2B2A3B3 comprising the two square faces. The potential... 

Typical potential energy surfaces projected on to the plane are shown in Figures 29-31... 

The potential energy surface projected onto the 0a b plane is shown in Figures 78a and 78b. The centre of the surface, at a = 45°, corresponds to the dodecahedron, labelled Dp... 

THE REPRESENTATION AND USE OF POTENTIAL ENERGY SURFACES projected matrix F (s) of eq. 

In the remainder of this section, we focus on the two lowest doublet states of Li3. Figures 3 and 4 show relaxed triangular plots [68] of the lower and upper sheets of the 03 DMBE III [69,70] potential energy surface using hyper-spherical coordinates. Each plot corresponds to a stereographic projection of the... 

Fig. 3. Projections on the (<1>, maps of the CICADA conformational search of the pentasaccharide. The dots indicate the values of all the optimized conformations determined by CICADA at each glycosidic linkange in 8 kcal/mol energy window For comparison, the isocontours, drawn in 1 Kcal/mol steps with an outer limit of 8 kcal/mol, represent the energy level of each disaccharide and calculated with the relaxed grid search approach. Dashed regions represent the locations of the low energy conformation of the pentasaccharide plotted on the potential energy surfaces of the constituting disaccharide segments...

Fig. 16 (a) R (D + RX) and P (D,+ + R + X ) zero-order potential energy surfaces. Rc and Pc are the caged systems, (b) Projection of the steepest descent paths on the X-Y plane J, transition state of the photoinduced reaction j, transition state of the ground state reaction W, point where the photoinduced reaction path crosses the intersection between the R and P zero-order surfaces R ., caged reactant system, (c) Oscillatory descent from W to J on the upper first-order potential energy surface obtained from the R and P zero-order surfaces. 

Fig. 4.5 Schematic projection of the energetics of a reaction. The diagram shows the Born-Oppenheimer energy surface mapped onto the reaction coordinate. The barrier height AE has its zero at the bottom of the reactant well. One of the 3n — 6 vibrational modes orthogonal to the reaction coordinate is shown in the transition state. H and D zero point vibrational levels are shown schematically in the reactant, product, and transition states. The reaction as diagrammed is slightly endothermic, AE > 0. The semiclassical reaction path follows the dash-dot arrows. Alternatively part of the reaction may proceed by tunneling through the barrier from reactants to products with a certain probability as shown with the gray arrow...

Fig. 5.1 A schematic projection of the 3n dimensional (per molecule) potential energy surface for intermolecular interaction. Lennard-Jones potential energy is plotted against molecule-molecule separation in one plane, the shifts in the position of the minimum and the curvature of an internal molecular vibration in the other. The heavy upper curve, a, represents the gas-gas pair interaction, the lower heavy curve, p, measures condensation. The lighter parabolic curves show the internal vibration in the dilute gas, the gas dimer, and the condensed phase. For the CH symmetric stretch of methane (3143.7 cm-1) at 300 K, RT corresponds to 8% of the oscillator zpe, and 210% of the LJ well depth for the gas-gas dimer (Van Hook, W. A., Rebelo, L. P. N. and Wolfsberg, M. /. Phys. Chem. A 105, 9284 (2001))...

Figure 2. Franck-Condon windows lVpc(Gi, r, v5) for the Na3(X) - N83(B) and for the Na3(B) Na3+ (X) + e transitions, X = 621 nm. The FC windows are evaluated as rather small areas of the lobes of vibrational wavefunctions that are transferred from one electronic state to the other. The vertical arrows indicate these regions in statu nascendi subsequently, the nascent lobes of the wavepackets move coherently to other domains of the potential-energy surfaces, yielding, e.g., the situation at t = 653 fs, which is illustrated in the figure. The snapshots of three-dimensional (3d) ab initio densities are superimposed on equicontours of the ab initio potential-energy surfaces of Na3(X), Na3(B), and Na3+ (X), adapted from Ref. 5 and projected in the pseudorotational coordinate space Qx r cos

A normal-mode representation of the Hamiltonian for the reduced system involves the diagonalization of the projected force constant matrix, which in turn generates a reduced-dimension potential-energy surface in terms of the mass-weighted coordinates of the reaction path [64] ... 

Figure 67 (a) The adiabatic potential energy surfaces (Mexican hat) (b) the normal coordinates and (c) th projection of the potential energy surface (a) warped by the inclusion of higher order terms viewed down the prinrip axes of (a), with R]T = radius of the minimum potential432... 

Figure 5 (a) Projection of the potential energy surface for [M(unidentate)s] on to the plane (in degrees). The five... 

Figure 10 Projection of the potential energy surface for [M(unidentate A)(unidentate B)4] on to the 4>t> 4>E plane (in degrees). The five faint contour lines are for successive 0.01 increments in X above the minima, and the five heavy contour lines are for successive 0.1 increments above the minima, at T0 and T2. R = 0.8. The positions of the trigonal bipyramids (T) and square pyramid (S) are shown...

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