Local quadratic approximation

The simplest smooth fiuictioii which has a local miiiimum is a quadratic. Such a function has only one, easily detemiinable stationary point. It is thus not surprising that most optimization methods try to model the unknown fiuictioii with a local quadratic approximation, in the fomi of equation (B3.5.1). 

The propagation of the wavepacket is thereby reduced to the solution of coupled first-order differential equations for the parameters representing the Gaussian wavepacket, with the true potential being expanded about the instantaneous center of the wavepacket [i2(<),f(<)]. This propagation scheme is very appealing and efficient provided the basic assumptions are fulfilled. The essential prerequisite is that the locally quadratic approximation of the PES is valid over the spread of the wavepacket. This rules out bifurcation of the wavepacket, resonance effects, or strong an-harmonicities. 

This constrained nonlinear optimisation problem can be solved using a Successive Quadratic Programming (SQP) algorithm. In the SQP, at each iteration of optimisation a quadratic program (QP) is formed by using a local quadratic approximation to the objective function and a linear approximation to the nonlinear constraints. The resulting QP problem is solved to determine the search direction and with this direction, the next step length of the decision variable is specified. See Chen (1988) for further details. 

As mentioned earlier, it is possible to calculate the interference signal directly from (3.5) without resorting to adiabatic or locally quadratic approximations, by diagonalizing the Longuet-Higgins model Hamiltonian with the standard procedure [10]. Figure 1 shows the exact interfer-... 

Figure 2. The same signal as in Fig. 1. calculated in the adiabatic and locally quadratic approximations.

Fig. 10.9. Reaction path following on the Muller-Brown surface using Euler, Ishida, Morokuma, and Komomicki (IMK), local quadratic approximation (LQA), Hratchian-Schlegel (HS), and second-order Gonzalez-Schlegel (GS2) methods.

Nichols J A, Taylor H, Schmidt P P and Simons J 1990 Walking on potential energy surfaces J. Chem. Phys. 92 340-6 Simons J and Nichols J 1990 Strategies for walking on potential energy surfaces using local quadratic approximations... 

This equation can be solved in terms of the eigenvectors of F(rQ) to advance one step forward from 5( o) on the MEP. Much larger steps can be taken using this local quadratic approximation method [41,44]. Even more substantial increases in the step size have been achieved by accounting approximately for third derivatives of the energy along the path (although only ab initio second derivatives are actually computed [45]). 

Reaction Coordinate Comparison of Gradient and Local Quadratic Approximation Methods. 

Our objective in this chapter is to survey the theory and practice of the computation of transition state structure. We take the word structure in this context to encompass the geometry of the saddle point on the surface, a local quadratic approximation to the surface at that point (the second derivative matrix), and the nature of the reaction path in the region of the... 

A step size 5s of 0.01 ao was used in the local quadratic approximation for following the reaction path for all of the systems. The accuracy of the calculated reaction path was tested by comparing computed rate constants (including tunneling effects) with at least one other reaction path determined with another step size. In all cases the rate constants for temperatures above 300 K are numerically converged to better than 50% and those at 200 K are numerically accurate to about a factor of 2. Tests were also made... 

Direct dynamics calculations of the type just described, with all degrees of freedom included, are very expensive if the local quadratic approximations to the potential energy surface are obtained from an ab initio computation. In applications we have used a hybrid parameterized quantum-mechanical/force-field method, designed to simulate the CASSCF potential for ground and covalent excited states. A force field is used to describe the inert molecular a-framework, and a parameterized Heisenberg Hamiltonian is used to represent the CASSCF active orbitals in a valence bond space. Applications include azulene and benzene excited state decay dynamics. 

CLQA = corrected local quadratic approximation DDRP = dynamically defined reaction path DRP = dynamic reaction path ES = Euler stabilization method GS = Gonzalez and Schlegel method IMK = Ishida-Morokuma-Kormomicld method LQA = local quadratic approximation MB = Miillar-Brown method MEP = minimum energy path ODE = ordinary differential equations SDRP = steepest descent reaction path VRl = valley-ridge inflection. 

Standard numerical methods such as second-order Runge-Kutta could be used, but a more effective approach is to expand the potential energy surface in equation (2) to second order and integrate the resulting expression from Xi to 3c,+i. This yields the local quadratic approximation (LQA) of Page and Mclver " which is an explicit second-order method. 

Page and Mclver proposed an algorithm which takes the curvature term of equation (16) into account and, accordingly, leads to a local quadratic approximation of the RP at s, -In this way, a RP can be determined which has the correct curvature at the point of expansion. Special efforts have to be taken to determine the direction and curvature of the RP at the saddlepoint. Since the gradient vanishes at this point,. second and third energy derivatives (He.ssian and first derivatives of the Hessian), have to be calculated according to the rule of L Hospital. Hence, the local quadratic approximation of Page... 

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