Potential energy surfaces symmetry considerations

Similar to the case without consideration of the GP effect, the nuclear probability densities of Ai and A2 symmetries have threefold symmetry, while each component of E symmetry has twofold symmetry with respect to the line defined by (3 = 0. However, the nuclear probability density for the lowest E state has a higher symmetry, being cylindrical with an empty core. This is easyly understand since there is no potential barrier for pseudorotation in the upper sheet. Thus, the nuclear wave function can move freely all the way around the conical intersection. Note that the nuclear probability density vanishes at the conical intersection in the single-surface calculations as first noted by Mead [76] and generally proved by Varandas and Xu [77]. The nuclear probability density of the lowest state of Aj (A2) locates at regions where the lower sheet of the potential energy surface has A2 (Ai) symmetry in 5s. Note also that the Ai levels are raised up, and the A2 levels lowered down, while the order of the E levels has been altered by consideration of the GP effect. Such behavior is similar to that encountered for the trough states [11]. 

The origin of a torsional barrier can be studied best in simple cases like ethane. Here, rotation about the central carbon-carbon bond results in three staggered and three eclipsed stationary points on the potential energy surface, at least when symmetry considerations are not taken into account. Quantum mechanically, the barrier of rotation is explained by anti-bonding interactions between the hydrogens attached to different carbon atoms. These interactions are small when the conformation of ethane is staggered, and reach a maximum value when the molecule approaches an eclipsed geometry. 

It should be stressed that although these symmetry considerations may allow one to anticipate barriers on reaction potential energy surfaces, they have nothing to do with the thermodynamic energy differences of such reactions. Symmetry says whether there will be symmetry-imposed barriers above and beyond any thermodynamic energy differences. The enthalpies of formation of reactants and products contain the information about the reaction s overall energy balance. 

Even if full potential energy surfaces are not calculated, simple EHT calculations, skilfully coupled with orbital symmetry considerations, can provide insight into complex reactivity problems. This is well exemplified by Hoffmann and Stohrer s analysis of substituent effects on the Cope rearrangement (28). 

Figure 1. Cartoon depicting symmetry considerations of the asymptotic wavefunction that influences which potential energy surface is accessed. Representative angular momentum quantum numbers relevant to the process are the total molecular angular momentum exclusive of nuclear spin (f) and the relative orbital angular momentum (/).

Figure 9.5 Five configurations of a Pt13 cluster that are local minima on this cluster s potential energy surface as characterized by GGA DFT calculations. The three clusters in the upper part of the figure were generated by hand based on symmetry considerations. The two clusters in the lower part of the figure were obtained using ab initio MD as described in the text. The energy of each cluster is defined relative to the lowest energy cluster.

Fig. 1. The path and the contour plot of the lowest potential energy surface in the T t2 system. By symmetry considerations, the 3D problem is reduced to a calculation in 2D (X, Y) coordinate space. The five paths have been computed with different initial conditions and the one (in the middle) that reaches the zero-point energy (the innermost ring) is the least action path.

Abstract When considering the work of Carl Ballhausen on vibrational spectra, it is suggested that his use of the Born-Oppenheimer approximation is capable of some refinement and extension in the light of later developments. A consideration of the potential energy surface in the context of a full Coulomb Schrodinger Hamiltonian in which translational and rotational motions are explicitly considered would seem to require a reformulation of the Born-Oppenheimer approach. The resulting potential surface for vibrational motion should be treated, allowing for the rotational motion and the nuclear permutational symmetry of the molecule. 

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