Energy Crossing-Point Location

g R) is the gradient evaluated at nuclear coordinate R, and E R) is the fth state energy. Given the fact that the gradient itself can also be expanded in a Taylor series, one can utilize the recursion to obtain the more familiar quadratic Newton-Raphson equation. 

Koga and Morokuma treated the MEXP search as a constrained optimization. In their Lagrange multiplier method, the energy of the fth state, E (R), is minimized with a constraint that the energy difference between two states, / and /, becomes zero. The Lagrangian is written as 

The Lagrangian is expanded in a Taylor series, analogous to Eq. [90] above, and one obtains the Newton-Raphson equation for the Lagrangian, 

The last two terms are the orthogonality and geometry constraints. The Newton-Raphson equations derived from the Lagrangian (Eq. [96]) are given 

Lagrange multipliers corresponding to energy, orthogonality, and geometry constraints, respectively. In the case of spin-forbidden reactions, the terms derived from in Eq. [96] should be neglected, and one should use the 

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The minimum energy crossing point on Ai and Bi surfaces of the phenyl cation (9 X=H) was located by Harvey et al. This point with C2V symmetry lies only 0.12 kcal/mol (CCSD[T]) above the triplet state minimum. At this point, spin-orbit coupling, evaluated at the CASSCF(6,7) level by using a one-electron operator with effective nuclear charges. 

Singlet and Triplet States of Phenyl Cation. A Hybrid Approach for Locating Minimum Energy Crossing Points between Noninteracting Potential Energy Surfaces. 

Anglada, J. M., 8c Bofill, J. M. (1997). A reduced-restricted-quasi-Newton-Raphson method for locating and optimizing energy crossing points between two potential energy surfaces. Journal of Computational Chemistry, 18(S), 992-1003. 

Quasi-Newton-Raphson Method for Locating and Optimizing Energy Crossing Points between Two Potential Energy Surfaces. 

Shown in Fig. 21c is the structure of another hypothetical diamond structure obtained by locating cubes at the nodes of a bcc lattice. Each atom is four coordinate here too. Figure 22 shows the energy difference curve for supercubane and cubic diamond, this time dominated by the fourth moment difference as expected from the squares present in the former. Notice that although the crossing points and the shapes of these curves are a little different from those of Fig, 6, the general features are quite similar. 

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