Accelerating molecular property calculations with semiempirical preconditioning

Computing ab initio molecular linear response properties, e.g., electronic excitation energies and transition dipole moments, requires the solution of large eigenvalue problems or large systems of equations. These large eigenvalue problems or large systems of equations are commonly solved iteratively using Krylov space algorithms, such as the Davidson algorithm for eigenvalue problems. A critical ingredient in Krylov space algorithms is the preconditioner, which is used to generate optimal update vectors in each iteration. We propose to use semiempirical approximations as preconditioners to accelerate the calculation of ab initio properties. The crucial advantage to improving the preconditioner is that the converged result is unchanged, so there is no trade-off between accuracy and speedup. We demonstrate our approach by accelerating the calculation of electronic excitation energies and electric polarizabilities from linear response time-dependent density functional theory using the simplified time-dependent density functional theory semiempirical model. For excitation energies, the semiempirical preconditioner reduces the number of iterations on average by 37% and up to 70%. The semiempirical preconditioner reduces the number of iterations for computing the polarizability by 15% on average and up to 33%. Moreover, we show that the preconditioner can be further improved by tuning the empirical parameters that define the semiempirical model, leading to an additional reduction in the number of iterations by about 20%. Our approach bridges the gap between semiempirical models and ab initio methods and charts a path toward combining the speed of semiempirical models with the accuracy of ab initio methods.