Conical Intersection Optimization Based on a Double Newton–Raphson Algorithm Using Composed Steps

An algorithm for conical intersection optimization based on a double Newton–Raphson step (DNR) has been implemented and tested in 11 cases using CASSCF as the electronic structure method. The optimization is carried out in redundant coordinates, and the steps are the sum of two independent Newton–Raphson steps. The first step is carried out to reach the energy degeneracy and uses the gradient of the energy difference between the crossing states and the so-called branching space Hessian. The second step minimizes the energy in the intersection space and uses the projected excited state gradient and the intersection space Hessian. The branching and intersection space Hessians are obtained with a Broyden–Fletcher–Goldfarb–Shanno update from the gradient difference and projected excited state gradients, respectively. In some cases, mixing of the quasi-degenerate states near the seam causes changes in the direction of the gradient difference vector and induces a loss of the degeneracy. This behavior is avoided switching to a composed step (CS) algorithm [Sicilia et al.<i> J. Chem. Theory Comput.</i> <b>2008</b>, <i>4</i>, 27], i.e., a hybrid DNR-CS implementation. Compared to the composed gradient (CG) [Bearpark et al. <i>Chem. Phys. Lett.</i> <b>1994</b>, <i>223</i>, 269] and hybrid CG-CS algorithms, the DNR-CS algorithm reaches the MECI in 30% and 15% less steps, respectively. The improvement occurs mostly because the approach to the seam is more efficient, and a degeneracy threshold of 0.001 hartree is reached at lower energies than in the CG and CG-CS cases