Complex function saddle point

Let us take a simple example, namely a generic Sn2 reaction mechanism and construct the state functions for the active precursor and successor complexes. To accomplish this task, it is useful to introduce a coordinate set where an interconversion coordinate (%-) can again be defined. This is sketched in Figure 2. The reactant and product channels are labelled as Hc(i) and Hc(j), and the chemical interconversion step can usually be related to a stationary Hamiltonian Hc(ij) whose characterization, at the adiabatic level, corresponds to a saddle point of index one [89, 175]. The stationarity required for the interconversion Hamiltonian Hc(ij) defines a point (geometry) on the configurational space. We assume that the quantum states of the active precursor and successor complexes that have non zero transition matrix elements, if they exist, will be found in the neighborhood of this point. 

The expression for the thermal rate constant k(T) is given as a product of two functions an exponential function and a prefactor. The prefactor contains the partition function for the reaction complex, the supermolecule , at the saddle point (with the reaction coordinate omitted) and partition functions for the reactants. The second factor is an exponential with an argument that contains the energy difference between the zero-point energy level of the supermolecule at the saddle point and of the reactants. 

Fig. 4.2. Complex saddle points t s (left panel), ts (middle panel), and kxs (s = i,j) (right panel) for the pair of solutions having the shortest travel times as discussed in the text. The figure is for ATI, for a Keldysh parameter of 7 = 0.975, and emission parallel to the laser held. The panels present the paths in the complex plane that are followed by the saddle points as a function of the final energy of the electron at the detector, which is indicated by the numbers (in multiples of Up). The figure shows how the saddle points of a pair approach each other very closely near the classical cutoff at 10 Up, which is the classical cutoff of the ATI energy spectrum. The contribution of the orbit that is drawn dashed has to be dropped after the cutoff. From [30]...

Starting at a saddle point, a path of steepest descent can be defined on the potential energy surface by using the gradient function 8W/8Qj the path of steepest descent is uniquely determined by extremal values of the gradient unless a stationary point is reached (55). Besides the minima corresponding to the reactant and product asymptotes, a potential energy surface may exhibit some additional minima due to, e.g., van der Waals (59) complexes or intermediates (see later). In such cases, the reactant and product asymptote can be interconnected by several steepest descent paths and the construction... 

C. Fr jacques, in a private communication, has calculated the sum when it is assumed that the hypersurface in the vicinity of the saddle point is parabolic in shape, so that v" " is a function of the energy of the complex. The correction amounts to a factor of Vs for the simple system H + H2 (exchange). 

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. 

Calculations by this method have shown remarkable Insensitivity to the nonlinear parameters of the complex part of the trial function. For the 2s autolonlzlng state, for which Equation 14 Is the trial function, Chung and Davis (33) obtained Ej. 57.8483 eV and r 0.12468 eV which compare well with the experimental (34) values of Ej. 57.8210.04 eV and T 0.13810.15 eV. The saddle point complex-rotation method Is strictly speaking another variant of CCI, but It demonstrates the premium In accuracy and efficiency to be gained from a well chosen trial function. In this case one In which the Feshbach Q-space (resonance) part of the trial wave-function Is optimized. 

The integral is to be calculated over a path in the complex plane, cut along the negative half-axis (see Fig. 13.5). We must study I(n, S, u) for large values of vS and for this purpose we shall use the steepest descent method. The relevant saddle point corresponds to the minimum of the function... 

In our most straightforward implementation of VTST for gas-phase reactions, rather than allow arbitrary orientations of the dividing surface, we consider a one-parameter sequence of dividing surfaces that are defined in terms of a reaction path [12,13]. This procedure is applicable to complex problems, and it immediately provides a practical improvement over the conventional choice of placing the dividing surface at the saddle point. A robust choice for the reaction path is the minimum energy path (MEP), that is, the path of steepest descent in the mass-scaled coordinates [14]. The coordinates on this path are denoted q (j ) as a function of a progress variable s, and the path is defined by... 

The electronic partition function is usually equal to unity. In the transition state theory developed by Polanyi and Eyring, the transition complex is located at the top of the energy barrier (Figure 3.2) and the reaction can be presented a movement along a potential energy surface where the transition state is located at the saddle point. 

In the previous sections we have generated a differential equation which generates (at least in principle) a set of orbitals which ensure a stationary point in the single-determinant energy functional eqn ( 2.1). We have proceeded from an algebraic expression to a differential equation a considerable increase in complexity. There is an alternative approach which does not involve differential equations and attacks the problem of finding the turning points of eqn ( 2.1) directly. In fact, this method is only practicable for local minima in the functional because of the technical difficulties associated with the location of maxima, saddle points etc. 

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