Two-dimensional potential energy

Olsen R A, Philipsen P H T, Baerends E J, Kroes G J and Louvik O M 1997 Direct subsurface adsorption of hydrogen on Pd(111) quantum mechanical calculations on a new two-dimensional potential energy surfaced. Chem. Phys. 106 9286... 

The ionization mechanism for nucleophilic substitution proceeds by rate-determining heterolytic dissociation of the reactant to a tricoordinate carbocation (also sometimes referred to as a carbonium ion or carbenium ion f and the leaving group. This dissociation is followed by rapid combination of the highly electrophilic carbocation with a Lewis base (nucleophile) present in the medium. A two-dimensional potential energy diagram representing this process for a neutral reactant and anionic nucleophile is shown in Fig. 

The coordinates involved are depicted in Scheme 1-2. Two-dimensional Potential Energy Surface (PES) scans of the r /r2 and r /rn surfaces were performed where the r values were restrained to values in a grid, and the geometry of the remaining atoms optimized, were followed by 10 ns windows of umbrella sampling... 

Figure 14-5. Side by side comparison of the two-dimensional potential energy surface for the transesterification reaction computed with B3LYP (left) and M06-2X (right)...

Figure 4 Two-dimensional potential energy curve for C-H bond decomposition.

The representation as a two-dimensional potential energy diagram is simple for diatomic molecules. But for polyatomic molecules, vibrational motion is more complex. If the vibrations are assumed to be simple harmonic, the net vibrational motion of TV-atomic molecule can be resolved into 3TV-6 components termed normal modes of ibrations (3TV-5 for... 

Fig. 2a-c. Two-dimensional potential energy surfaces. The Qx coordinate corresponds to the low frequency mode, the Qy coordinate corresponds to the high frequency mode. The forms of the coupling are a kxy = 0 (no coupling), b k>y = xy2 (quadratic coupling), and c kxy = xy (linear/Duschinsky coupling... 

Fig. 3.2. Two-dimensional potential energy surface V(R, 7) (dashed contours) for the photodissociation of C1CN, calculated by Waite and Dunlap (1986) the energies are given in eV. The closed contours represent the total dissociation wavefunction tot R,l E) defined in analogy to (2.70) in Section 2.5 for the vibrational problem. The energy in the excited state is Ef = 2.133 eV. The heavy arrow illustrates a classical trajectory starting at the maximum of the wavefunction and having the same total energy as in the quantum mechanical calculation. The remarkable coincidence of the trajectory with the center of the wavefunction elucidates Ehrenfest s theorem (Cohen-Tannoudji, Diu, and Laloe 1977 ch.III). Reprinted from Schinke (1990).

The photodissociation of methyl nitrite in the first absorption band, CH30N0(Si) — CH3O + NO(n, j), exemplifies indirect photodissociation (Hennig et al. 1987). Figure 1.11 shows the two-dimensional potential energy surface (PES) of the S electronic state as a function of the two O-N bonds. All other coordinates are frozen at the equilibrium values in the electronic ground state. Although these two modes suffice to illustrate the overall dissociation dynamics, a more realistic picture... 

Fig. 9.8. Two-dimensional potential energy surface for the photodissociation of CH3I fitted to reproduce the latest experimental data. The coordinates are given in atomic units. Adapted from Guo and Schatz (1990b).

Figure 8. Two-dimensional potential energy surfaces (schematic) for (a) early and (b) late barrier (B) of dissociation of H2 on a transition metal surface.

Figure 29. Two-dimensional potential energy diagram for the convesion of /-N2 (vertically adsorbed) to a-N2 (side-on bonded) [246-248] on the Fe(l 11) surface [249],...

This represents a slice through the two-dimensional potential energy surface defined by the two constrained torsion angles. The energy values and the associated torsion angles are recorded in the . dat file. This file should be copied to a new file and kept. 

Figure 7-2. Potential energy surface by Williams in the region of the transition structure in different representations [21] (a) Three-dimensional representation of the saddle-shaped potential energy surface (b) Two-dimensional potential energy curve produced by a vertical cut through the surface in (a) along the reaction path (indicated by bold dashed line) from reactants (R) to products (P) (c) Energy contours produced by horizontal cuts through the potential energy sufrace in (a). Adapted with permission from Reference [21],...

In order to evaluate the kinetic stability of covalently bound 02N—02C—C02—N02 it was decided to compute a two dimensional potential energy hypersurface... 

Fig. 7. Two-dimensional potential energy surface and static reaction path for the synchronous conrotatory motion of the terminal methylene groups. 2 a represents the value of the carbon ring angle. The abcissa gives the common value of both rotational angles 6 = dj = dj. TS denotes the position of a transition state. The energies are in kcal/mol...

We first treat separately the trajectories conesponding to the particular values 6° = 45° and —45°. Indeed, the total symmetry of the problem is such that, whenever the motion of both rotors at the starting point is either purely conrotatory or purely disrotatory, it keeps this particularity throughout the trajectory. Then the trajectory can be drawn on a two-dimensional potential energy surface such as that pictured in Fig. 7 and 8. 

The thermodynanucs of gas-phase species adsorbing on a surface can be separated into the classes of physical adsorption (physisorption) and chemical adsorption (chemisorption). The difference between the two classes of adsorption is related to the heat of adsorption, A Hads. In physisorption, weak attractive forces (van der Waals) such as dipole-dipole and induced dipole interactions drive the adsorption of species from the gas phase. For chemisorbed systems, chemical bonds associated with electron transfer between adsorbate and substrate are formed upon adsorption. Each of these interactions can be described by a two-dimensional potential-energy diagram as in Figures 10 and 11. 

For an octahedral geometry there are three equivalent elongated distortions, where the elongated axis is directed along either x,y, or z axes. These equivalent directions result in the threefold symmetry of the two-dimensional potential energy surface as a function of Qg, for the ground state of copper(II) as shown in Fig. 1 b. 

This two-dimensional potential energy surface is the lower energy solution obtained from the diagonalization of the potential energy operators in the E 0 e vibronic Hamiltonian acting within a ( 0>, >) electronic basis ... 

Later work evaluated the two-dimensional potential energy surface using various correlation treatments including many-body perturbation theory and coupled cluster techniques Evaluation of the vibrational spectrum was explicitly anharmonic in nature, mak-... 

We should mention here that even for the diatomic molecule Hgj, which is domi-nantely bound via van der Waals binding, our calculation in terms of the Xa-ap-proximation is valid to some extent. At least it results in a binding of the two atoms, which is non-trivial. The largest system so far is Hgj, which we calculated in terms of a total energy calculation . The two dimensional potential energy surface with its relatively flat minimum is presented in Fig. 4. 

Part of the (two-dimensional) potential energy diagram for the isomerization and decomposition reactions of 33 is shown in Figure 2. Obviously, the multistep pathway 33 - 34 - 35 - 28 (Scheme 5) represents the energetically most favoured reaction channel. Noteworthy is the fact that direct Cl elimination from the ring-opened... 

Figure 1. An illustration of a two-dimensional potential energy surface where two local minima and two saddle points exist.

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