Potential energy hypersurface

For a problem with many degrees of freedom, Eq. [16] is replaced by 

The conformational spaces of clusters of Lennard-Jones atoms were searched by application of the diffusion equation method for clusters of various sizes m from m = 5 to 55. For example, for m = 55, with 3m - 6 = 159 degrees of freedom, there are about 1045 local minima, and the global minimum (a MacKay icosahedron) was found in about 400 s on an IBM 3090 computer.205 

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Piela L, Kostrowicki J and Scheraga H A 1989 The multiple-minima problem in the conformational analysis of molecules. Deformation of the potential energy hypersurface by the diffusion equation method J. Phys. Chem. 93 3339... 

P. G. Mezey, Potential Energy Hypersurfaces Elsevier, Amsterdam (1987). 

Potential energy hypersurfaces form the basis for the complete description of a reacting chemical system, if they are throughly researched (see also part 2.2). Due to the fact that when the potential energy surface is known and therefore the geometrical and electronical structure of the educts, activated complexes, reactive intermediates, if available, as well as the products, are also known, the characterizations described in parts 3.1 and 3.2 can be carried out in theory. 

Table 23 contains the formation enthalpies for individual points of the potential energy hypersurface of the C4H9BF4 supermolecule, that is, a molecule which can be considered to be made up of the following components C2H, C2H4 and BF4. The same table provides further possibilities to divide the supermolecule C4HgBF4 into logical constituents. 

Table 23. Enthalpies of formation AHj (kj mol-1) in the gas phase (g) and in dichloro methane solution (s) calculated from separate molecules at selected points of the C4H9BF4 potential energy hypersurface...

Polyurethane foam 8, 27, 46, 72 Potential energy hypersurfaces (see Potential energy surfaces)... 

Ab initio MO and DFT calculations have revealed that S4 can exist as six isomers on the potential energy hypersurface. Their connectivities and relative energies (in kj mol ) [9] are shown in Scheme 1. 

Only the structures of di- and trisulfane have been determined experimentally. For a number of other sulfanes structural information is available from theoretical calculations using either density functional theory or ab initio molecular orbital theory. In all cases the unbranched chain has been confirmed as the most stable structure but these chains can exist as different ro-tamers and, in some cases, as enantiomers. However, by theoretical methods information about the structures and stabilities of additional isomeric sul-fane molecules with branched sulfur chains and cluster-like structures was obtained which were identified as local minima on the potential energy hypersurface (see later). 

By ab initio MO and density functional theoretical (DPT) calculations it has been shown that the branched isomers of the sulfanes are local minima on the particular potential energy hypersurface. In the case of disulfane the thiosulfoxide isomer H2S=S of Cg symmetry is by 138 kj mol less stable than the chain-like molecule of C2 symmetry at the QCISD(T)/6-31+G // MP2/6-31G level of theory at 0 K [49]. At the MP2/6-311G //MP2/6-3110 level the energy difference is 143 kJ mol" and the activation energy for the isomerization is 210 kJ mol at 0 K [50]. Somewhat smaller values (117/195 kJ mor ) have been calculated with the more elaborate CCSD(T)/ ANO-L method [50]. The high barrier of ca. 80 kJ mol" for the isomerization of the pyramidal H2S=S back to the screw-like disulfane structure means that the thiosulfoxide, once it has been formed, will not decompose in an unimolecular reaction at low temperature, e.g., in a matrix-isolation experiment. The transition state structure is characterized by a hydrogen atom bridging the two sulfur atoms. 

In the literature tetrathiosulfuranes have been discussed as possible intermediates in the thermal decomposition of sulfanes and other polysulfur compounds. High-level ab initio MO calculations have in fact revealed that such species are local minima on the potential energy hypersurface [34]. However, recent results show that both the Gibbs reaction energies as well as the activation enthalpies for reactions of the type... 

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... 

Mezey, P.G. (1987) Potential Energy Hypersurfaces, Elsevier, Amsterdam. 

Viviani, W., J.-L. Rivail, A. Perczel, and I. G. Csizmadia. 1993b. Peptide Models. 3. Conformational Potential Energy Hypersurface of Formyl-L-valinamide. J. Am. Chem. Soc. 115, 8321-8329. 

Fig. 5 Potential energy hypersurfaces as a function of the reaction coordinate for adiabatic (A, single-minimum potential B, double-minimum potential) and non-adiabatic (C) electron-transfer reactions.

Hoffmann s review.2) The number of specific examples mentioned in the text is severely limited in order to save space they can be easily found elsewhere.2) Instead, space is devoted to detailed discussion of topics likely to be less familiar to the organic chemist, such as some of the properties of potential energy hypersurfaces in multidimensional nuclear configuration space, etc. When in doubt, the author erred on the side of sounding too explicit and trivial at the risk of offending the reader with good physical background. 

Very high solvent viscosity also effectively alters shapes of potential energy hypersurfaces by making large changes in shape difficult or... 

A higher level of understanding would require a knowledge of molecular dynamics and presently represents a rather distant goal. In addition to reliable knowledge of the shapes of potential energy hypersurfaces, it would also require information such as vibronic coupling elements, densities of vibrational states, detailed mechanism of the action of the heat bath, etc. 

For a recent list of references to calculations of parts of ground state potential energy hypersurface see Salem, L. Accounts Chem. Res. 4, 322 (1971). 

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