Contour potential-energy

Figure Al.6.27. Equipotential contour plots of (a) the excited- and (b), (c) ground-state potential energy surfaces. (Here a hamionic excited state is used because that is the way the first calculations were perfomied.) (a) The classical trajectory that originates from rest on the ground-state surface makes a vertical transition to the excited state, and subsequently undergoes Lissajous motion, which is shown superimposed, (b) Assuming a vertical transition down at time (position and momentum conserved) the trajectory continues to evolve on the ground-state surface and exits from chaimel 1. (c) If the transition down is at time 2 the classical trajectory exits from chaimel 2 (reprinted from [52]).

Figure A3.7.7. Two-dimensional contour plot of the Stark-Wemer potential energy surface for the F + H2 reaction near the transition state. 0 is the F-H-H bend angle.

Figure 3. Relaxed triangular plot [68] of the U3 ground-state potential energy surface using hyperspherical coordinates. Contours, are given by the expression (eV) — —0.56 -t- 0.045(n — 1) with n = 1,2,..,, where the dashed line indicates the level —0.565 eV. The dissociation limit indicated by the dense contouring implies Li2 X Sg ) -t- Li.

Fig. 5. The left hand side figure shows a contour plot of the potential energy landscape due to V4 with equipotential lines of the energies E = 1.5, 2, 3 (solid lines) and E = 7,8,12 (dashed lines). There are minima at the four points ( 1, 1) (named A to D), a local maximum at (0, 0), and saddle-points in between the minima. The right hand figure illustrates a solution of the corresponding Hamiltonian system with total energy E = 4.5 (positions qi and qs versus time t).

If we use a contour map to represent a three-dimensional surface, with each contour line representing constant potential energy, two vibrational coordinates can be illustrated. Figure 6.35 shows such a map for the linear molecule CO2. The coordinates used here are not normal coordinates but the two CO bond lengths rj and r2 shown in Figure 6.36(a). It is assumed that the molecule does not bend. 

Figure 6.35 Contour of potential energy as a function of two C-0 bond lengths, r- and r2, in CO2...

Fig. 12-7. Potential energy contour diagram showing the course of an aromatic substitution X+ + ArH - ArX + H+ (after Zollinger, 1956 a).

Figures 7-11 show potential energy contours for two-dimensional cuts through these three surfaces. The contour sets are labelled by 0. and where the z axis points from molecule 1 to...

Figure 14. Classical trajectories for the H + H2(v = l,j = 0) reaction representing a 1-TS (a-d) and a 2-TS reaction path (e-h). Both trajectories lead to H2(v = 2,/ = 5,k = 0) products and the same scattering angle, 0 = 50°. (a-c) 1-TS trajectory in Cartesian coordinates. The positions of the atoms (Ha, solid circles Hb, open circles He, dotted circles) are plotted at constant time intervals of 4.1 fs on top of snapshots of the potential energy surface in a space-fixed frame centered at the reactant HbHc molecule. The location of the conical intersection is indicated by crosses (x). (d) 1-TS trajectory in hyperspherical coordinates (cf. Fig. 1) showing the different H - - H2 arrangements (open diamonds) at the same time intervals as panels (a-c) the potential energy contours are for a fixed hyperradius of p = 4.0 a.u. (e-h) As above for the 2-TS trajectory. Note that the 1-TS trajectory is deflected to the nearside (deflection angle 0 = +50°), whereas the 2-TS trajectory proceeds via an insertion mechanism and is deflected to the farside (0 = —50°).

Figure 11. Contour plot of the adiabatic ground potential energy surface of the 2D model. The dashed line shows the seam surface. Taken from Ref. [27].

Figure 42. The potential energy contour felt by a hydrogen atom when the two atoms N (left side) and C (right side) are fixed. (See color insert.)...

Figure 10. Three-dimensional potential-energy surface for the H + C2H3 C2H4 addition reaction. The lower left plot is taken in the symmetry plane of the vinyl radical. The other plots are taken in parallel planes at distances of O.S. O a.u. from the symmetry plane (1 a.u. = 0.52918 A). Solid contours are positive, dashed contours are negative, and the zero-energy contour (defined to be the energy of the reactant asymptote) is shown with a heavy sohd fine. The contour increment is 1 kcalmoU. Reproduced from [57] by pentrission of the PCCP Owner Societies.

Figure 18. Contour plots of the potential energy surfaces of the first three electronic states of H2O. The polar plots depict the movement of one H atom around OH with an OH bond length fixed at 1.07 A. Energies are in electron volts relative to the ground electronic state. The X and B states are degenerate at the conical intersection (denoted by (g)) in the (a) H—OH geometry and (b) H—HO geometry. Reprinted fix)m [75] with permission from the American Association for the Advancement of Science.

Figure 12. Potential energy contour plots for He + I Cl(B,v = 3) and the corresponding probability densities for the n = 0, 2, and 4 intermolecular vibrational levels, (a), (c), and (e) plotted as a function of intermolecular angle, 0 and distance, R. Modified with permission from Ref. 40. The I Cl(B,v = 2/) rotational product state distributions measured following excitation to n = 0, 2, and 4 within the He + I Cl(B,v = 3) potential are plotted as black squares in (b), (d), and (f), respectively. The populations are normalized so that their sum is unity. The l Cl(B,v = 2/) rotational product state distributions calculated by Gray and Wozny [101] for the vibrational predissociation of He I Cl(B,v = 3,n = 0,/ = 0) complexes are shown as open circles in panel (b). Modified with permission from Ref. [51].

The VBSCF and EH-MOVB potential energy surfaces for the nucleophilic substitution reaction of HS and CH3CI are depicted in Figure 4-2. The energy contours determined using the EH-MOVB method (Figure 4-2A) are found to be in good accord... 

Fig. 16. Potential energy contours for the H + D2O system as a function of the OH and one OD bond length. In each panel, the energy has been minimized with respect to the remaining degrees-of-freedom in the vicinity of the minimum energy paths. In (a) the saddle point for the abstraction reaction, and in (b) the shallow < >, minimum for the exchange reaction are marked with X.

Fig. 1. Conformational energy diagram for the alanine dipeptide (adapted from Ramachandran et al., 1963). Energy contours are drawn at intervals of 1 kcal mol-1. The potential energy minima for p, ofR, and aL are labeled. The dependence of the sequential d (i, i + 1) distance (in A) on the 0 and 0 dihedral angles (Billeter etal., 1982) is shown as a set of contours labeled according to interproton distance at the right of the figure. The da (i, i + 1) distance depends only on 0 for trans peptide bonds (Wright et al., 1988) and is represented as a series of contours parallel to the 0 axis. Reproduced from Dyson and Wright (1991). Ann. Rev. Biophys. Chem. 20, 519-538, with permission from Annual Reviews.

Figure 9. Comparison of ab initio (full line) and ab m/rfo/interpolated (dashed line) potential energy surfaces for the first electronically excited state of Li + H2 system restricted to C2v geometry. Contours are labeled in eV. (Figure adapted from Ref. 125.)...

Fig. 5. Contour plot of the adiabatic potential-energy surface of an H atom in the (110) plane for the neutral H—B pair from a local-density pseudopotential calculation. The boron atom is at the center. For every hydrogen position, the B and Si atoms are allowed to relax, but only unrelaxed positions are indicated in the figure (Reprinted with permission from the American Physical Society, Denteneer, P.J.H., Van de Walle, C.G., and Pantelides, S.T. (1989). Phys. Rev. B 39, 10809.)...

Figure 8 Top Potential energy as a function of position for a one-dimensional system. Bottom Potential-energy contours for an atom moving in two dimensions.

The mechanism whereby an atom, adsorbed on a solid surface at X, moves towards a site of lower energy at Y, is also shown in figure 8. The contours of potential energy show that the route of lowest energy from X to Y passes through the saddle point at A. 

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