Second-order saddle point

Orbital-based methods can be used to compute transition structures. When a negative frequency is computed, it indicates that the geometry of the molecule corresponds to a maximum of potential energy with respect to the positions of the nuclei. The transition state of a reaction is characterized by having one negative frequency. Structures with two negative frequencies are called second-order saddle points. These structures have little relevance to chemistry since it is extremely unlikely that the molecule will be found with that structure. 

Or, more precisely, a firsl-order saddle paint, where the order indicates the number of dimensions in which the saddle point is a maximum. A second-order saddle point would be a maximum in two dimensions and a minimum in all others. Transition structures are first-order saddle points. 

G and Huzinaga basis sets used for K, Rb and Cs, 6-31+G and 6-31+G on C used for C3H5. 6-31+G. Number of imaginary frequencies is given in parentheses (1) a transition state (2) a second-order saddle point. 

An equivalent linear structure of CoP-CO was found to be a second-order saddle point (N-x = 2) when the 6-31G(d) basis set... 

The MEP for inversion corresponds to 6 = 0 and is characterized by the barrier height VWhen C/2V, > 1, apart from this MEP, there is a path that includes two segments described by Eq. (8.42) and a second-order saddle point. The barrier along this path is greater than V, and equal to U,(l + 2V0/C). The transverse frequency along the straight-line MEP for inversion has a minimum at the saddle point q = 0, 0 = 0 consequently, the vibrationally adiabatic barrier is lower than the static one. 

What kind(s) of stationary points do you think a second-order saddle point connects ... 

We recently investigated the potential energy surface for dissociation of 3 at the CAS(12,12) level [19]. The dissociation process was found to proceed via a transition state of C2V symmetry (5TS). Earlier studies had proposed a D2h transition state [24, 29], however, it was shown that the D2h structure is a second order saddle point rather than a transition state. Single point calculations at... 

Min, global energy minimum Min, energy minimum TS, transition state SP 2, second-order saddle point... 

Fig. 3. Methane inversion at QCISD(T)/6-311+0(3df,2p)//CISD/6-311G, rel. energies in kcal mol" number of imaginary frequencies (NIMAG) in parentheses NIMAG = 0 (minimum), 1 (transition state), 2 (second order saddle point)...

Quantum chemical calculations of the Ne potential-energy hypersurface have shown that the qualitative shape shongly depends on the choice of the theoretical method and basis set. All the geometries represented in Scheme 6 have been shown to be minima on the potential smface, but most of them do not possess minima at all the levels of theories applied. Hexaazabenzene (13), for example, has a minimum for a stmctme at the HF level of theory. However, this geometry is a second-order saddle point with the density functional theory (DFT) and also at the MP2 level of theory. D2 hexaazabenzene has a minimum structure at DFT, but at the CCSD(T)/aug-cc-pVDZ level, the D2 geometry resembles a van der Waals complex of two N3 units, whereas it is a minimum structure at the CCSD(T)/cc-pVTZ level of theory. Similar behavior has been observed for most of the other isomers. 

Smith ct al. also reinvestigated the question of the trifurcated structure and found it to be a second-order saddle point, nearly 2 kcal/mol higher in energy than the minimum. 

Figure 29.2 Illustration of stepwise and concerted double proton transfer in naphthazarin, showing the equilibrium configurations (MIN), the (unstable) intermediates (INT), the second-order saddle point (SP2), and the location of the first-order saddle points (SPl).

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