Saddle optimization method

As explained above, the QM/MM-FE method requires the calculation of the MEP. The MEP for a potential energy surface is the steepest descent path that connects a first order saddle point (transition state) with two minima (reactant and product). Several methods have been recently adapted by our lab to calculate MEPs in enzymes. These methods include coordinate driving (CD) [13,19], nudged elastic band (NEB) [20-25], a second order parallel path optimizer method [25, 26], a procedure that combines these last two methods in order to improve computational efficiency [27],... 

The optimization methods described in Sections 12.2-12.4 concentrate on locating stationary points on an energy surface. The important points for discussing chemical reactions are minima, corresponding to reactant(s) and product(s), and saddle points. 

If at all possible, stationary points found by any optimization method should be characterized by computing the Hessian and examining its eigenvalues. This is especially important for high-symmetry structures (where lower-energy, lower-symmetry structures may exist nearby) and for saddle points (which can be higher-order saddle points rather than first-order). 

Moreover, the second-generation MCSCF parametrizes the wave function in a way that enables the simultaneous optimization of spinors and Cl coefficients, in this context then called orbital or spinor rotation parameters and state transfer parameters, respectively. Then, a Newton-Raphson optimization method is employed which also requires the second derivatives of the MCSCF electronic energy with respect to the molecular spinor coefficients (more precisely, to the orbital rotation parameters) and to the Cl coefficients. As we have seen, in Hartree-Fock theory the second derivatives are usually not calculated to confirm that a solution of the SCF procedure has indeed reached a minimum with respect to the large component and not a saddle point. Now, these general MCSCF methods could, in principle, provide such information, although it is often not needed in practice. 

We now turn to methods for first-order saddle points. As already noted, saddle points present no problems in the local region provided the exact Hessian is calculated at each step. The problem with saddle point optimizations is that in the global region of the search, there are no simple criteria that allow us to select the step unambiguously. Thus, whereas for minimization methods it is often possible to give a proof of convergence with no significant restrictions on the function to be minimized, no such proofs are known for saddle-point methods, except, of course, for quadratic surfaces. Nevertheless, over the years several useful techniques have been developed for the determination of saddle points. We here discuss some of these techniques with no pretence at completeness. 

Saddle optimization, 329 Scalar relativistic corrections, 209 Scaled External Correlation (SEC), Scaled All Correlation (SAC) models, 169 Scaling of different methods with basis set, 145 Schrodinger equation, 2, 53 Schwarz inequality, for integral screening, 78 Second quantization, 411 Second-order corrections, in perturbation methods, 126... 

Techniques have been developed within the CASSCF method to characterize the critical points on the excited-state PES. Analytic first and second derivatives mean that minima and saddle points can be located using traditional energy optimization procedures. More importantly, intersections can also be located using constrained minimization [42,43]. Of particular interest for the mechanism of a reaction is the minimum energy path (MEP), defined as the line followed by a classical particle with zero kinetic energy [44-46]. Such paths can be calculated using intrinsic reaction coordinate (IRC) techniques... 

A more balanced description requires MCSCF based methods where the orbitals are optimized for each particular state, or optimized for a suitable average of the desired states (state averaged MCSCF). It should be noted that such excited state MCSCF solutions correspond to saddle points in the parameter space for the wave function, and second-order optimization techniques are therefore almost mandatory. In order to obtain accurate excitation energies it is normally necessarily to also include dynamical Correlation, for example by using the CASPT2 method. 

The reaction path formed by a sequence of points generated by constrained optimizations may be discontinuous. For methods where two points are gradually moved from the reactant and product sides (e.g. saddle and LTP), tliis means that tlie distance between end-points does not converge towards zero. 

---