Hypersurface, energy

The energy hypersurface describes the energy of chemical systems with a given set of atoms as a function of the spatial distribution of the atomic nuclei. If one could compute the complete energy hypersurface for any set of atoms and analyse the corresponding data in a suitable m.anner, one could predict most of the relevant properties of molecular systems with the methods of theoretical physics and mathematics. 

however, is not feasible in the foreseeable future, because the computational effort would be extremely large, even in the case of rather small molecular systems. 

For the solution of many chemical problems the theoretical treatment of a few selected points and their vicinity on the energy hypersurface would suffice. 

in many cases, one does not know which points are the relevant ones. For chemistry it would be rather useful to have a method for providing a survey of those points and pathways on an energy hypersurface which are essential for the solution of a given problen without the necessity of a quantum mechanical treatment of wide areas of an energy hypersurface. 

In the present paper a simple mathematical model of the chemistry of a given set of atoms is presented which afford s precisely the latter, and may serve as a basis of computer programs for the deductive solution of a great variety of chemical problems. 

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Consider, at t = 0, some non-equilibrium ensemble density P g(P. q°) on the constant energy hypersurface S, such that it is nonnalized to one. By Liouville s theorem, at a later time t the ensemble density becomes ((t) t(p. q)), where q) is die function that takes die current phase coordinates (p, q) to their initial values time (0 ago the fimctioii ( ) is uniquely detemiined by the equations of motion. The expectation value of any dynamical variable ilat time t is therefore... 

Also we must bear in mind that the advancement of the coordinates fidfds two fiinctions (i) accurate calculation of dynamical properties, especially over times as long as typical correlation times x (ii) accurately staying on the constant-energy hypersurface, for much longer times Exact time reversibility is highly desirable (since the original equations... 

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. 

Polypeptide chains exist in an equilibrium between different conformations as a function of environment (solvent, other solutes, pH) and thermodynamic (temperature, pressure) conditions. If a polypeptide adopts a structurally ordered, stable conformation, one speaks of an equilibrium between a folded state, represented by the structured, densely populated conformer, and an unfolded state, represented by diverse, sparsely populated conformers. Although this equilibrium exists for polypeptide chains of any size, its thermodynamics and kinetics are typically different for oligopeptides and proteins. This can be broadly explained with reference to the different dimensionalities of the free-energy hypersurfaces of these two types of molecules. 

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. 

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