Energy contour

With the above in mind, it is sensible to modify the atomic orbital by treating the orbital exponent as a variational parameter. What we could do is vary for each value of the intemuclear separation 7 ab, and for each value of 7 ab calculate the energy with that particular orbital exponent. Just for illustration, I have calculated the energies for a range of orbital exponent and intemuclear distance pairs, and my results are shown as energy contours in Figure 3.3. 

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

Fig. 2. The energy contours for two GB(4.4, 20.0, 1, 1) molecules confined to a plane with their symmetry axes parallel. The contours are separated by 0.25 and range from 0 to —2.25...

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 15. Left-. Geometry of the surface 6 = 0 in Eq. (46) with fixed total angular momenta S and N. Properties of the special points A, B, C, and D are listed in Table I. All other permissible classical phase points lie on or inside the surface of the rounded tetrahedron. Right. Critical section at J. Continuous lines are energy contours for y = 0.5 and N/S = 4. Dashed lines are tangents to the section at D. Axes correspond to normalized coordinates, /NS and K JiN + S). Taken from Ref. [2] with permission of Elsevier.

In order to establish conditions for the isolation of the image of point D under the EM map, the projection is performed by first taking a section through the surface ct = 0 at fixed J, an example of which is shown in the right-hand panel of Fig. 15, for the critical value = S — N. The shaded area of the (K, plane defines the classically allowed range for the specified value. The lines indicate energy contours for y = 0.5. Those that touch the section correspond to relative equilibria of the Hamiltonian, whose values... 

The dashed lines show energy contours for the case c > b, which touch constant sections at = 0. The dotted line is an energy contour for > a, which also touches the 7 = N/2 section at... 

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 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 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 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.

Fig. 2.6 Potential energy contour diagram for a thermoneutral reaction.

FIGURE 2. An energy contour map for diethylmethylamine recomputed using the MM2-91 force field. The separation between contour lines is 1.0 kcal mol-1... 

Figure 7.1 Potential energy contour plot for H20 in U1(2) U2(2) (Benjamin and Levine, 1985), plotted as a function of rlre. The energy contours are 0.5 eV apart. N =...

Figure 7.2 Potential energy contour plot for S02 in Uj(2) TJ2(2) versus r/re...

Figure 2. Initial ( (/a) and final ( J/b) state potential-energy contours for the complete (two-mode) active space the abscissa refers to the inner-sphere mode and the ordinate governs the low-frequency active solvent mode. The difference in frequencies leads to a curved reaction path. Equilibrium coordinate values for the reactant ( j/A) and product ( J/b) states are labeled qA and qB, respectively. For the case of qin, qB° - qA° = Aqin°, as given by Eq. 16.

Figure 4. Conformational map for dihydropyran. Because of the double bond, 4 atoms are always almost coplanar and a limited number of conformations is probable. The energy contours are at 2 kcal/mol intervals, starting 1 kcal/mol above the minima. The favored conformations are half-chairs, and the easiest paths of transition between the two are through the boat forms. The symmetry of this energy map applies only to dihydropyran, and not to derivatives which cause increases and decreases in the sizes of the allowed (low-energy) areas. This map was calculated with MMP2(85) at increments of 0.1 A shift of the two non-planar atoms. Three of the carbon atoms were held in a plane while C6 and 01 were held at specific distances above and below the plane. Otherwise, the structure was fully relaxed at each increment. The reader may enjoy plotting the indicated path of conformational interchange (pseudorotation) on a copy of Figure 3.

Figure 2. A contour diagram of the conformational energy of p-cellobiose computed from eqn. (6) holfing constant all variables except < ), v see ref. 5 for details. The rigid glucose residue geometry was taken from ref. 23, and the valence angle p at 04 was chosen as 116 in accordance with the results of pertinent crystal structure determinations. Contours are drawn at 2,4, 6, 8,10,25, and 50 kcal/mol above the absolute minimum located near ( ), v = -20 , -30 higher energy contours are omitted.

Figure 4. Selected iso-n and and iso-h contours superimposed on the potential energy surface for maltose. The iso-energy contours are drawn by interpolation of 1 kcal/mol with respect to the relative energy minimum ( ). The iso-h = 0 contour divides the map into two regions, corresponding to right-handed (h>0) and left-handed (h<0) chirality.

Figure 9. Potential energy surface for cellobiose at 400 K. The trajectory of conformational changes during a portion of the simulation are shown on the left. Energy contours in the vicinity of minima 1-3 are shown on the right. Barrier heights 5.3 Kcal/mol between minima 1 and 2,1.3 Kcal/mol between 2 and 3. (MM2(85) functions).

Figure 6.6 Schematic illustration of a two dimensional energy surface with two local minima separated by a transition state. The dark curves are energy contours with energy equal to the transition state energy. The transition state is the intersection point of the two dark curves. Dashed (solid) curves indicate contours with energies lower (higher) than the transition state energy. The MEP is indicated with a dark line. Filled circles show the location of images used in an elastic band calculation.

MAECIS also contains a molecular conformation analysis system (4). This system allows the user to generate all possible conformations of the current molecule over a series of single bond rotations. Energy contour maps can be obtained for the various conformations and this allows for the selection of low energy conformations for further manipulation or calculations. 

Figure 13.4 Torsional potential energy surfaces about the two C-O bonds linking the anomeric centers of sucrose at the MM3 level (a), 2-tetrahydrofuranyl-2-tetrahydropyranyl ether at the MM3 level (b), the same ether at the HF/6-31G(d) level (c), and the sum of the difference between the last two with the first (d). Thus, the last surface may be viewed either as the effect of the sucrose hydroxyl groups on the energy surface, evaluated at the MM3 level, added to the framework surface calculated at the ab initio level, or as an MM3 surface that has been partially conected quantum mechanically. Solid triangles represent anomeric torsions in sucrose units found in various X-ray crystal structures. Note that the hybrid surface is the only one that clusters the large majority of these triangles within low-energy contours...

---