Exponential convergence of the local diabatic representation for nonadiabatic eigenvalue problems

Discrete variable local diabatic representation (LDR) provides a divergence-free framework for exact conical intersection dynamics simulation. In this work, we investigate the convergence with respect to the number of "nuclear" grid points and "electronic" states of LDR for the eigenvalue problems using coupled oscillator models and a conical intersection model. The performance of LDR is compared with traditional approaches based on the Born-Huang ansatz and with the crude adiabatic representation. Our results demonstrate that for weak vibronic couplings, LDR shows a similar convergence rate to the exact Born-Huang representation including not only the first-order derivative couplings but also the diagonal Born-Oppenheimer corrections and second-order derivative couplings. For strong but non-diverging vibronic couplings, LDR shows a significantly faster convergence rate with respect to the number of grid points, hence the number of electronic structure computations, than the exact Born-Huang representation. The diagonal Born-Oppenheimer corrections and second-order derivative couplings are found to be important in the Born-Huang framework. For the conical intersection model, while the Born-Huang representation simply does not converge due to diverging derivative couplings, the LDR remains highly accurate and converges exponentially fast. The crude adiabatic representation in general shows a much slower convergence rate in all cases.