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Path search by anharmonic downward distortion following (ADDF) method


Introduction of ADDF

Along reactive potential curves, the anharmonic downward distortion (ADD) arises and increases toward dissociation channel (DC) or transition state (TS). Therefore, the anharmonic downward distortion following (ADDF) method detects and follows the ADDs starting from local minima to find reaction paths. The path of ADDF is not identical to the IRC path. Nevertheless, typical ADDF paths pass through near actual TSs. In other words, the ADDF gives approximate TSs as the highest energy points of the ADDF paths. Thus, actual TSs can be obtained starting from such approximate TSs.

A direction in which ADD is maximal can be found as a minimum energy point on the isoenergy hypersurface of harmonic potential. This hypersurface is a hyperellipsoid in the multidimensional potential energy surface (PES). For simplicity, the ellipsoidal harmonic-iso-energy hypersurface is converted to a hypersphere by adopting the scaled normal coordinate qi = λi1/2Qi, where Qi and λi are the i-th normal coordinate and the corresponding eigenvalue, respectively. On the scaled hypersphere, the harmonic energy is constant. Thus, local minima of the real (anharmonic) potential correspond to maximal ADD points on the scaled hypersphere. ADDs can thus be found by energy minimizations on the scaled hypersphere. This technique for searching ADDs on the scaled hypersphere is called scaled hypersphere search (SHS).

All ADDs are not found as explicit local minima on the scaled hypersphere. ADDs overlap with each other and small ones may be hidden by the larger ones. The ADD is eliminated by adding a cos3θ function centered at the corresponding maximal ADD point, and small ADDs hidden by larger ones are found as explicit local minima on the scaled hypersphere on which large ADDs are eliminated by cos3θ functions, where at θ = π/2, the ADD function, i.e., cos3θ, is truncated. The iterative optimization and elimination (IOE) technique repeats optimization of ADD maximal points and elimination of ADDs by cos3θ functions until no ADD is found. At first ADD maximal points on the scaled hypersphere are located by the SHS technique starting from both positive and negative directions of mode 1, and then obtained ADDs are eliminated. The same procedure is applied to mode 2, mode 3, and so on. Once the procedure is completed for all modes, the same procedure is repeated from mode 1. This IOE procedure is continued until no new ADD is found. In default, the initial IOE is performed on the scaled hypersphere with a harmonic energy ε0 which is determined so that the geometrical displacement from the sphere center to the scaled hypersphere along the hardest mode becomes 0.03 Å, where the size can be changed to 0.01×n by the InitSize = n option.

Once all ADDs are located on the initial sphere with ε = ε0 (0th step), these ADDs are followed with increasing the sphere radius. In the pth steps, ADDs are searched on the scaled hypersphere with harmonic energy ε = p2εr, where εr is determined so that the geometrical displacement from the sphere center to the scaled hypersphere with ε = εr along the hardest mode becomes 0.1 Å, where the size can be changed to 0.01×n by the UpSize = n option. Although the smaller step size will give the better approximate TSs, this value was chosen to save computational costs by compromise. A simplified predictor-corrector IOE (PC-IOE) procedure is performed. At first, maximal ADD points on the last sphere are linearly projected onto the present sphere. Then, the optimization-elimination procedure is applied to all these points in descending order according to magnitude of ADD on the last sphere. These optimizations converge quickly because such starting points are close to ADD maxima. In our implementation, even when an ADDF path bifurcated, only one of the two is followed. Among the two, a closer one to the initial point is chosen in PC-IOE. In other words, the number of ADDF paths will not increase after the full IOE at the 0th step.

When an energy maximum is detected along an ADDF path, the maximum energy point, that is, approximate TS, is optimized to the corresponding true TS. TS optimization may fail due to the large step size (εr). If failed, the maximum energy region is further searched by the 2PSHS method in the reverse direction with decreasing the sphere radius. Then, the meta-IRC, mass-weighted steepest descent path starting from non-TS point, is computed from the final ADDF path point to reach corresponding product, i.e., the other MIN or a DC. When a molecule decomposed into two or more fragments, the path is considered to reach a DC. Discrimination of whether the obtained MIN is new or not and whether the molecule has decomposed are both made using the distance matrix. All ADDF paths, even those already reached MINs or DCs, are followed together until all of them reach MINs or DCs. This is because ADDs overlap with each other and some hidden small ADDs may have to be found with elimination of the larger ADDs. Once all ADDF paths reached MINs or DCs, an application to the starting MIN is completed.

For further details, see S. Maeda, et al., Bull. Chem. Soc. Jpn. 2014, 87, 1315–1334 (this paper is Open Access).