Potential-energy surfaces dimensionality

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.

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... 

Wiesenekker G, Kroes G J and Baerends E J 1996 An analytical six-dimensional potential energy surface for dissociation of molecular hydrogen on Cu(IOO) J. Chem. Phys. 104 7344... 

Ifnited atom force fields (see Ifnited versiisAll Atom Forcehiclds" on page 28 ) arc sometimes used for bioraoleciiles to decrease the number of nonbonded in teraction s and the computation time. Another reason for using a simplified poten tial is to reduce the dimensionality of the potential energy surface. This, in turn, allows for more samples of the surface. 

We envision a potential energy surface with minima near the equilibrium positions of the atoms comprising the molecule. The MM model is intended to mimic the many-dimensional potential energy surface of real polyatomic molecules. (MM is little used for very small molecules like diatomies.) Once the potential energy surface iias been established for an MM model by specifying the force constants for all forces operative within the molecule, the calculation can proceed. 

The dimensionality of a potential energy surface depends on the number of degrees of freedom in a molecule. If Vp s is a function of two variables, then a plot of the potential energy surface represents a 3D space. 

For multi-dimensional potential energy surfaces a convenient measure of the gradient vector is the root-mean-square (RMS) gradient described by... 

Potential Energy Surface. A many-dimensional function of the energy of a molecule in terms of the geometrical coordinates of the atoms. 

A potential energy surface can help us visualize the energy changes in the course of a reaction as a function of the locations of the atoms. In this three-dimensional... 

Figure 7. Two-dimensional cuts through the potential energy surface for planar HF-HF collisions including vibration. The quantity plotted in the figure is the total potential (in hartrees), which is defined as the sum of the interaction potential and the two diatomic potentials, with the zero of energy corresponding to two infinitely separated HF molecules, each at its classical equilibrium separation. This figure shows cuts through the r. plane (in bohrs) for 0 = 0 = = 0 and...

Figure 55. Two-dimensional coupled potential energy surfaces and the wavepacket motion, (a) Si — S2 surfaces and (b) Si — So surfaces. The black, gray, and white circles and dotted lines indicate the locations of the FC region. Si - S2 conical intersection minimum, 5MR Si — So conical intersection minimum, and seam lines, respectively. The solid arrows indicate the schematic wavepacket pathway in the case of natural photoisomerization starting from the vibrational ground state. Taken from Ref. [49].

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.

As briefly stated in the introduction, we may consider one-dimensional cross sections through the zero-order potential energy surfaces for the two spin states, cf. Fig. 9, in order to illustrate the spin interconversion process and the accompanying modification of molecular structure. The potential energy of the complex in the particular spin state is thus plotted as a function of the vibrational coordinate that is most active in the process, i.e., the metal-ligand bond distance, R. These potential curves may be taken to represent a suitable cross section of the metal 3N-6 dimensional potential energy hypersurface of the molecule. Each potential curve has a minimum corresponding to the stable... 

The total Hamiltonian is the sum of the two terms H = H + //osc- The way in which the rate constant is obtained from this Hamiltonian depends on whether the reaction is adiabatic or nonadiabatic, concepts that are explained in Fig. 2.2, which shows a simplified, one-dimensional potential energy surface for the reaction. In the absence of an electronic interaction between the reactant and the metal (i.e., all Vk = 0), there are two parabolic surfaces one for the initial state labeled A, and one for the final state B. In the presence of an electronic interaction, the two surfaces split at their intersection point. When a thermal fluctuation takes the system to the intersection, electron transfer can occur in this case, the system follows the path... 

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