Potential energy surfaces contour plot

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.

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 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.

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 1. (Left) Potential energy surface for the LiNC/LiCN isomerizing system drawn as a contours plot. The minimum energy path connecting the two stable linear isomers, LiNC (0 = 180°) and LiCN (6 = 0), is shown as a dotted line. 

Utilizing Eqs. (34) to (39) in Eq. (33), the potential energy surface for the iodide ion-iodine system as a function of distance x from the electrode and the normalized solvent coordinate qig was determined as given in Fig. 15 as a contour plot. It is observed that far from the electrode surface, the ionic and the atomic states are separated by an energy barrier... 

Lowest 5 kcal mol 1 of the calculated overall potential energy surface for a model peptide of Ser-DHype(trans propyl)-Oic-Arg. The contour interval is 0.5 Kcalmol"1 and the highest (outermost) and lowest contour energy values are labeled. Superimposed on the contour plots are values for j/i+1 and fi+2 from each of the thirty structures generated from the NMR data corresponding to the tetrapeptide... 

Figure 4. Contour plots of the potential-energy surfaces for HCO, HNO, and HO2. The left-hand side shows the (R, r) dependence, with the angle 7 being fixed at the equilibrium in the well. The right-hand side highlights the (R, 7) dependence, with r fixed at the equilibrium. The spacing of the contours is 0.25 eV and the lowest contour is 0.25 eV above the minimum. Energy normalization is so that E = 0 corresponds to H + XO(re). The Jacobi coordinates R, r, and 7 are as described in the text, with 7 = 180° corresponding to H-X-O. (Reprinted, with permission of the Royal Society of Chemistry, from Ref. 34.)...

Figure 61. Contour plot of potential-energy surface for linear (He-H-H)+ system. Indicated internuclear separations are in atomic units. Energies of equipoten-tial lines are given in units of kcal/mol relative to +He asymptotic limit.453...

Figure 64. Contour plot of potential-energy surface for ground state of linear H3+. Values for / , and R2 are in atomic units, and energies are in electron volts. Indicated energies are with respect to dissociated particles. Cut through surface at large / , coincides with potential curve of H2 at small R2 and with curve of at large R2 (see Fig. 63).2...

Fig. 1. The path and the contour plot of the lowest potential energy surface in the T t2 system. By symmetry considerations, the 3D problem is reduced to a calculation in 2D (X, Y) coordinate space. The five paths have been computed with different initial conditions and the one (in the middle) that reaches the zero-point energy (the innermost ring) is the least action path.

Fig. 1.12. Two-dimensional polar plots of the potential energy surfaces (denoted below by Vx, Va, and Ve) of the three lowest electronic states of H2O. One of the O-H bonds is frozen at its equilibrium in the electronic ground state. The contours represent the potential energy as the other H atom swings around the O atom. Energies and distances are given in eV and A, respectively. The energy is normalized such that H + OH(2II, re) and H + OH(2E, re), respectively, correspond to E — 0. Vx is the empirical fit of Sorbie and Murrell (1975) whereas Va and Vb have been calculated by Staemmler and Palma (1985) and by Theodorakopoulos, Petsalakis, and Buenker (1985), respectively. The heavy arrows illustrate the main dissociation paths in the excited states.

Fig. 8.9. Contour plot of the potential energy surface of H2O in the BlA state as a function of the H-OH dissociation bond Rh-oh and the HOH bending angle a the other O-H bond is frozen at the equilibrium value in the ground electronic state. The energy normalization is such that E = 0 corresponds to H(2S ) + OH(2E, re). This potential is based on the ab initio calculations of Theodorakopulos, Petsalakis, and Buenker (1985). The structures at short H-OH distances are artifacts of the fitting procedure. The cross marks the equilibrium in the ground state and the ellipse indicates the breadth of the ground-state wavefunction. The heavy arrow illustrates the main dissociation path and the dashed line represents an unstable periodic orbit with a total energy of 0.5 eV above the dissociation threshold.

Fig. 9.9. Contour plot of the potential energy surface of H2O in the AlB state the bending angle is fixed at 104°. Superimposed are the total stationary wavefunctions I tot( ) defined in (2.70). The total energies are —2.6 eV and -2.0 eV corresponding to wavelengths of A = 180 nm and 165 nm, respectively. Energy normalization is such that E = 0 corresponds to three ground-state atoms.

Fig. 3.1.4 Contour plot of a potential energy surface for the reaction A + BC —> AB + C. The surface is shown as a function of the two internuclear distances Rab and Rbc at a fixed approach angle. The barrier (marked with an arrow) occurs in the entrance channel, i.e., an early barrier.

Fig. 3.1.6 Contour plot of the potential energy surface of Fig. 3.1.5. H2O is at its equilibrium bond angle of 104.5° and the inner contours correspond to the lowest energies. The minimum is at an OH distance of 1.81 a0, where 1 ao = 0.529 A.

Fig. 7.1.2 Contour plot (at a fixed bending angle) of the electronic ground-state potential energy surface for a (asymmetric) triatomic molecule. The last contour exceeds the dissociation limit of one of the bonds.

Figure 2 Contour plots of the potential energy surface along two-dimensional cuts through hie six-dimensional coordinate space of H2 in front of (1 00) metal surfaces determined by DFT-GGA calculations in the h-b-h geometry. The contour spacing is 0.1 eV...

It is important to note that DFT total-energy calculations do not provide a continuous potential energy surface, as one might naively assume from the inspection of Fig. 2. In fact, the elbow plots shown are based on a series of 50-100 DFT calculations with varying center of mass and H-H distance. The continuous representation is just a result of a contour plot routine that interpolates between the actually calculated energies. 

Figure 10 Potential energy surface of the dissociation of 02/Pt(l 11) determined by the ab initio derived tight-binding Hamiltonian. The coordinates in the figure are the O2 center-of-mass distance from the surface Z and the 0-0 interatomic distance d. The configurations of the remaining O2 degrees of freedom are illustrated in the insets. The contour spacing is 0.2 eV per O2 molecule. In (a) a trajectory of an O2 molecule with an initial kinetic energy of 0.6 eV scattered at Pt(l 11) is also plotted.

Figure 3 A contour plot of the potential energy surface described in the text, as a function of z and Zu the distance of the incident and target atoms, respectively, above the surface plane. The distance between the atoms parallel to the surface is held fixed at the gas-phase H-H bond length. The marked contours are in eV, with a contour spacing of 0.2 eV. Taken from Ref. [83].

Figure 34. Contour plot of the potential energy surface for the isomerization of 3-phospholene. X is the ring pucking coordinate and y is the PH inversion coordinate. [From C. C. Marston and N. De Leon, J. Chem. Phys. 91, 3392 (1989).]...

The potential energy surface for the isomerization reaction HCN —> CNH used below was proposed by Murrell, Carter, and Halonen [65]. A contour plot of this potential energy surface is presented in Fig. 36. The two local minima correspond clearly to HCN and CNH. Although there are quantitative differences between this potential energy surface and others, all of the surfaces are sufficiently similar that the qualitative character of the classical dynamics that each surface supports is the same. 

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