Topology of potential energy surfaces

In this chapter, we will first discuss the general topology of potential energy surfaces that have been calculated for elementary chemical reactions. Second, we will consider the solution of the electronic Schrodinger equation with a focus on analytical results and elementary concepts rather than detailed computational procedures. 

On the theoretical side the H20-He systems has a sufficiently small number of electrons to be tackled by the most sophisticated quantum-chemical techniques, and in the last two decades several calculations by various methods of electronic structure theory have been attempted [77-80]. More recently, new sophisticated calculations appeared in the literature they exploited combined symmetry - adapted perturbation theory SAPT and CCSD(T), purely ab initio SAPT [81,82], and valence bond methods [83]. A thorough comparison of the topology, the properties of the stationary points, and the anisotropy of potential energy surfaces obtained with coupled cluster, Moller-Plesset, and valence bond methods has been recently presented [83]. 

OM Becker, M Karplus. The topology of multidimensional potential energy surfaces Theory and application to peptide stiaicture and kinetics. I Chem Phys 106 1495-1517, 1997. 

In such a case the last choice is to take the direction of the eigenvector of the only one nonzero eigenvalue of the rank one Hessian matrix of the difference between the two adiabatic potential energies [51]. In the vicinity of conical intersection, the topology of the potential energy surface can be described by the diadiabatic Hamiltonian in the form... 

Perczel, A., J. G. Angyan, M. Kajtar, W. Viviani, J.-L. Rivail, J.-F. Marcoccia, and I. G. Csizmadia. 1991a. Peptides Models. 1. Topology of Selected Peptide Conformational Potential Energy Surfaces (Glycine and Alanine Derivatives), J. Am. Chem. Soc. 113, 6256-6265. 

Figure 11 Simplified two-dimensional schematic of a multidimensional potential energy surface as a function of its configurational degrees of freedom. The landscape topology is specified by the density, whereas the system s elevation on the landscape is dictated by temperature. Reprinted with permission from Ref. 6.

In Figure 6, one-dimensional cuts through the PES in the direction of the 1,3-interaction distance R in 42 are shown. Either structures 42a, 42b, 42c or all three of them can occupy stationary points on the potential energy surface (PES) and, according to the topology of the PES, different chemical situations can be distinguished. 

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