Energy hypersurface calculations

Figure 5. CNDO total energy hypersurface calculations for XSSX X2SS systems. Counterclockwise are displayed the hyper-...

Thermolysis of dimethylphosphine oxide at >770K leads to the elimination of water to give 2-phosphapropene (46).27 Cleavage of the normally stable P=0 bond is explained by reference to an energy hypersurface calculated by MNDO methods. This suggests that there is an entropy-favoured dissipation of the activation energy stored in the... 

The failure of the Bom-Oppenheimer approximation for degenerate vibrational modes which can lift electronic degeneracies can thus account for the poor results obtained in frequency calculations which assume the validity of the Bom-Oppenheimer approximation in the energy hypersurface calculation. A case in point is Grein s comparison (749) of the results on BH4, CH4 and NHj (see also above, as well as (44—48, 50, 52—57)) which shows the discrepancies between the experimental and calculated values of co2(E), W8(Fa) and U4(p2) of the XY4 molecules the agreement with wi(Ai) is far better. 

Let us imagine the molecular dynamics on energy hypersurface calculated using a quantum-mechanical method (classical force helds are not appropriate since they offer non-hreakahle chemical bonds). The system is represented hy a point that slides downhill (with an increasing velocity) and climhs uphill (with deceasing velocity). The system has a certain kinetic energy because chemists usually heat their flasks. 

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... 

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...

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... 

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). 

Scheme 2. Relevant cutouts for the calculated C2H2O energy hypersurface (B3LYP/6-311++G ).

The first option involves the obtainement of an analytic function that reproduces the interaction energy between couples of molecules which has been calculated by solving the Schrodinger equation usually by means of an ab initio method. The advantge of this possibility is that information about any potential energy hypersurface point can be obtained from the calculation whereas experimentally this is not always possible. The practical procedure in order to build up an ab initio pair potential for the interaction between two molecules a and P can be divided in four steps. 

Spectroscopic applications usually require us to go beyond single-point electronic energy calculations or structure optimizations. Scans of the potential energy hypersurface or at least Taylor expansions around stationary points are needed to extract nuclear dynamics information. If spectral intensity information is required, dipole moment or polarizability hypersurfaces [202] have to be developed as well. If multiple relevant minima exist on the potential energy hyper surface, efficient methods to explore them are needed [203, 204],... 

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